{"schema_version":"ocme-site-catalogue-v1","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"domains":[{"id":"foundations_logic","title":{"zh-Hant":"基礎與邏輯","en":"Foundations and logic"}},{"id":"arithmetic_number_systems","title":{"zh-Hant":"算術與數系","en":"Arithmetic and number systems"}},{"id":"algebra","title":{"zh-Hant":"代數","en":"Algebra"}},{"id":"geometry","title":{"zh-Hant":"幾何","en":"Geometry"}},{"id":"topology","title":{"zh-Hant":"拓撲","en":"Topology"}},{"id":"analysis","title":{"zh-Hant":"分析","en":"Analysis"}},{"id":"number_theory","title":{"zh-Hant":"數論","en":"Number theory"}},{"id":"combinatorics_discrete","title":{"zh-Hant":"組合與離散數學","en":"Combinatorics and discrete mathematics"}},{"id":"probability_statistics","title":{"zh-Hant":"機率與統計","en":"Probability and statistics"}},{"id":"dynamical_systems","title":{"zh-Hant":"動力系統","en":"Dynamical systems"}},{"id":"differential_equations","title":{"zh-Hant":"微分方程","en":"Differential equations"}},{"id":"numerical_computational","title":{"zh-Hant":"數值與計算數學","en":"Numerical and computational mathematics"}},{"id":"optimization_operations_research","title":{"zh-Hant":"最佳化與作業研究","en":"Optimization and operations research"}},{"id":"mathematical_physics","title":{"zh-Hant":"數理物理","en":"Mathematical physics"}},{"id":"mathematical_biology","title":{"zh-Hant":"數學生物","en":"Mathematical biology"}},{"id":"economics_finance","title":{"zh-Hant":"經濟與金融數學","en":"Mathematics of economics and finance"}},{"id":"computer_science_mathematics","title":{"zh-Hant":"計算機科學數學","en":"Mathematics of computer science"}},{"id":"mathematics_education","title":{"zh-Hant":"數學教育","en":"Mathematics education"}},{"id":"history_philosophy_methodology","title":{"zh-Hant":"歷史、哲學與方法論","en":"History, philosophy and methodology"}},{"id":"formalized_mathematics","title":{"zh-Hant":"形式化數學","en":"Formalized mathematics"}}],"knowledge":[{"id":"mko-point-line-plane","title":{"zh-Hant":"點、線、面","en":"Points, lines, and planes"},"summary":{"zh-Hant":"點、線與面提供描述位置、方向、共線、共面與入射關係的基礎幾何語言。","en":"Points, lines, and planes provide foundational geometric language for position, direction, collinearity, coplanarity, and incidence."},"type":"concept","domain":"geometry","formalization":"not_formalized","tags":["幾何","點","線","面","AI候選","待形式化"]},{"id":"mko-angle","title":{"zh-Hant":"角","en":"Angle"},"summary":{"zh-Hant":"在初等歐幾里得幾何中，角由具有共同端點的兩條射線形成，並可用度數描述其張開量。","en":"In elementary Euclidean geometry, an angle is formed by two rays with a common endpoint, and its opening can be described in degrees."},"type":"concept","domain":"geometry","formalization":"not_formalized","tags":["幾何","角","度數","AI候選","待形式化"]},{"id":"mko-triangle","title":{"zh-Hant":"三角形","en":"Triangle"},"summary":{"zh-Hant":"在平面中，三個相異且不共線的點與連接它們的三條線段形成非退化三角形。","en":"In the plane, three distinct noncollinear points together with the three segments joining them form a nondegenerate triangle."},"type":"concept","domain":"geometry","formalization":"not_formalized","tags":["幾何","三角形","頂點","邊","AI候選","待形式化"]},{"id":"mko-right-triangle","title":{"zh-Hant":"直角三角形","en":"Right triangle"},"summary":{"zh-Hant":"具有一個內角為直角的三角形。","en":"A triangle having one interior angle that is a right angle."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["幾何","三角形","定義"]},{"id":"mko-euclidean-length","title":{"zh-Hant":"歐幾里得長度","en":"Euclidean length"},"summary":{"zh-Hant":"平面上兩點之間由座標差平方和的正平方根所定義的距離。","en":"The distance between two points in the plane is defined as the nonnegative square root of the sum of squared coordinate differences."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["幾何","距離","定義"]},{"id":"mko-euclid-pythagorean-theorem","title":{"zh-Hant":"畢達哥拉斯定理","en":"Pythagorean theorem"},"summary":{"zh-Hant":"直角三角形兩股長度的平方和，等於斜邊長度的平方。","en":"In a right triangle, the sum of the squares of the two leg lengths equals the square of the hypotenuse length."},"type":"theorem","domain":"geometry","formalization":"formalized_declared_side_model","tags":["幾何","三角形","定理","FELRA","Lean","Mathlib","語義橋"]},{"id":"mko-natural-number","title":{"zh-Hant":"自然數","en":"Natural number"},"summary":{"zh-Hant":"自然數是用於計數與離散次序的基本數；本條目採用包含 0 的慣例。","en":"Natural numbers are basic numbers for counting and discrete order; this entry adopts the convention that they include 0."},"type":"concept","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["自然數","數系","基礎","AI候選","待形式化"]},{"id":"mko-integer","title":{"zh-Hant":"整數","en":"Integer"},"summary":{"zh-Hant":"整數由零、正整數與負整數組成，擴充自然數以表示相反方向與差額。","en":"The integers consist of zero, the positive integers, and the negative integers, extending natural numbers to represent opposite directions and differences."},"type":"concept","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["整數","數系","負數","AI候選","待形式化"]},{"id":"mko-rational-number","title":{"zh-Hant":"有理數","en":"Rational number"},"summary":{"zh-Hant":"有理數可表示為兩個整數之比，且分母不可為零。","en":"A rational number can be represented as a ratio of two integers with a nonzero denominator."},"type":"concept","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["有理數","數系","分數","比值","AI候選","待形式化"]},{"id":"mko-real-number","title":{"zh-Hant":"實數","en":"Real number"},"summary":{"zh-Hant":"實數包含有理數，並以完備化補入有理數系統中缺少的極限點。","en":"The real numbers contain the rational numbers and complete the rational system by adding its missing limit points."},"type":"concept","domain":"analysis","formalization":"not_formalized","tags":["實數","數系","完備化","分析","AI候選","待形式化"]},{"id":"mko-arithmetic-operations","title":{"zh-Hant":"四則運算","en":"Arithmetic operations"},"summary":{"zh-Hant":"加、減、乘、除是依指定數系與定義域操作兩個數量的基本運算家族。","en":"Addition, subtraction, multiplication, and division are a family of basic operations on two quantities within a specified number system and domain."},"type":"concept","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["算術","運算","數系","AI候選","待形式化"]},{"id":"mko-variable-expression","title":{"zh-Hant":"變數與代數式","en":"Variables and algebraic expressions"},"summary":{"zh-Hant":"代數式以變數、常數與已定義運算表示依輸入而定的數學量。","en":"An algebraic expression uses variables, constants, and defined operations to represent a mathematical quantity depending on its inputs."},"type":"concept","domain":"algebra","formalization":"not_formalized","tags":["代數","變數","表達式","AI候選","待形式化"]},{"id":"mko-equation","title":{"zh-Hant":"方程式","en":"Equation"},"summary":{"zh-Hant":"方程式要求兩個表達式在指定變數範圍內取相等值；解是使等式成立的允許值。","en":"An equation requires two expressions to take equal values over a specified variable domain; a solution is an allowed value that makes the equality true."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["代數","方程式","解集","等式","AI候選","待形式化"]},{"id":"mko-linear-equation","title":{"zh-Hant":"一元一次方程式","en":"Linear equation in one variable"},"summary":{"zh-Hant":"一元一次方程式可寫成 ax+b=0，且一次係數 a 必須非零。","en":"A linear equation in one variable can be written as ax+b=0 with a nonzero linear coefficient a."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["代數","一次方程式","唯一解","非零係數","AI候選","待形式化"]},{"id":"mko-set","title":{"zh-Hant":"集合","en":"Set"},"summary":{"zh-Hant":"集合把指定脈絡中的若干對象視為一個整體，並以元素隸屬來描述其內容。","en":"A set treats selected objects within a specified context as a whole and describes its contents through element membership."},"type":"concept","domain":"foundations_logic","formalization":"not_formalized","tags":["集合","基礎","外延","AI候選","待形式化"]},{"id":"mko-proposition","title":{"zh-Hant":"命題","en":"Proposition"},"summary":{"zh-Hant":"命題是在指定邏輯脈絡中可被斷言或否定，並可談論真假的有意義陳述。","en":"A proposition is a meaningful statement that can be asserted or denied and discussed as true or false within a specified logical context."},"type":"concept","domain":"foundations_logic","formalization":"not_formalized","tags":["邏輯","命題","基礎","AI候選","待形式化"]},{"id":"mko-logical-connectives","title":{"zh-Hant":"邏輯連接詞","en":"Logical connectives"},"summary":{"zh-Hant":"否定、合取、析取、蘊涵與雙條件用來由既有命題形成複合命題。","en":"Negation, conjunction, disjunction, implication, and biconditional form compound propositions from existing propositions."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["邏輯","連接詞","合取","蘊涵","AI候選","待形式化"]},{"id":"mko-proof","title":{"zh-Hant":"數學證明","en":"Mathematical proof"},"summary":{"zh-Hant":"數學證明是在明示的前提與推理規則下，建立指定結論的可檢查論證。","en":"A mathematical proof is a checkable argument establishing a specified conclusion from explicit premises and rules of inference."},"type":"concept","domain":"history_philosophy_methodology","formalization":"not_formalized","tags":["證明","邏輯","方法論","AI候選","待形式化"]},{"id":"mko-set-membership","title":{"zh-Hant":"集合隸屬","en":"Set membership"},"summary":{"zh-Hant":"符號 x∈A 表示元素 x 屬於集合 A。","en":"The notation x∈A states that the element x belongs to the set A."},"type":"definition","domain":"foundations_logic","formalization":"formalized_lean_set_semantics","tags":["集合","基礎","關係","Lean","Mathlib"]},{"id":"mko-subset","title":{"zh-Hant":"子集合","en":"Subset"},"summary":{"zh-Hant":"A⊆B 表示 A 的每個元素也都是 B 的元素。","en":"A⊆B means that every element of A is also an element of B."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["集合","子集合","包含","關係","AI候選","待形式化"]},{"id":"mko-set-operations","title":{"zh-Hant":"集合運算","en":"Set operations"},"summary":{"zh-Hant":"聯集、交集、差集與相對補集以元素隸屬條件從既有集合構造新集合。","en":"Union, intersection, difference, and relative complement construct new sets from existing sets through element-membership conditions."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["集合","聯集","交集","差集","補集","AI候選","待形式化"]},{"id":"mko-sample-space","title":{"zh-Hant":"樣本空間","en":"Sample space"},"summary":{"zh-Hant":"樣本空間是針對一個已指定隨機試驗所列出的所有可能基本結果。","en":"A sample space lists all possible elementary outcomes for one specified random experiment."},"type":"concept","domain":"probability_statistics","formalization":"not_formalized","tags":["機率","樣本空間","結果","AI候選","待形式化"]},{"id":"mko-event","title":{"zh-Hant":"事件","en":"Event"},"summary":{"zh-Hant":"事件描述隨機試驗中關心的一組可能結果。","en":"An event describes a set of outcomes of interest in a random experiment."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["機率","事件","樣本空間","子集合","AI候選","待形式化"]},{"id":"mko-function-mapping","title":{"zh-Hant":"函數映射","en":"Function mapping"},"summary":{"zh-Hant":"符號 f:X→Y 表示函數 f 從定義域 X 映射到陪域 Y。","en":"The notation f:X→Y states that the function f maps the domain X into the codomain Y."},"type":"definition","domain":"foundations_logic","formalization":"formalized_total_function_core","tags":["函數","映射","基礎","Lean","Mathlib"]},{"id":"mko-tends-to-relation","title":{"zh-Hant":"趨近關係","en":"Tends-to relation"},"summary":{"zh-Hant":"符號 x→a 表示變數或索引所描述的狀態朝目標 a 趨近。","en":"The notation x→a says that the state described by a variable or index approaches the target a."},"type":"concept","domain":"analysis","formalization":"formalized_filter_tendsto_core","tags":["極限","分析","趨近","Filter","Lean","Mathlib"]},{"id":"mko-mathematical-induction","title":{"zh-Hant":"數學歸納法","en":"Mathematical induction"},"summary":{"zh-Hant":"數學歸納法以基礎情況與歸納步驟證明自然數索引的整個命題族。","en":"Mathematical induction proves an entire natural-number-indexed family of propositions through a base case and an induction step."},"type":"concept","domain":"history_philosophy_methodology","formalization":"not_formalized","tags":["證明","歸納法","自然數","AI候選","待形式化"]},{"id":"mko-counting-principle","title":{"zh-Hant":"基本計數原理","en":"Basic counting principles"},"summary":{"zh-Hant":"加法與乘法原理把互斥分支或連續選擇的有限數量組合成總數。","en":"The addition and multiplication principles combine finite counts across disjoint branches or successive choices."},"type":"concept","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["組合","計數","加法原理","乘法原理","AI候選"]},{"id":"mko-factorial","title":{"zh-Hant":"階乘","en":"Factorial"},"summary":{"zh-Hant":"自然數 n 的階乘由 0!=1 與後繼遞迴式定義，等於 1 到 n 的正整數乘積。","en":"The factorial of a natural number n is defined by 0!=1 and a successor recurrence, and equals the product of the positive integers from 1 through n."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["組合","階乘","自然數","遞迴","AI候選","待形式化"]},{"id":"mko-absolute-value","title":{"zh-Hant":"絕對值","en":"Absolute value"},"summary":{"zh-Hant":"實數 x 的絕對值是 x 到 0 的非負距離，並依 x 的正負號分段取 x 或 -x。","en":"The absolute value of a real number x is its nonnegative distance from 0, taking x or -x according to the sign of x."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["分析","實數","絕對值","距離","AI候選","待形式化"]},{"id":"mko-fraction","title":{"zh-Hant":"分數","en":"Fraction"},"summary":{"zh-Hant":"分數 p/q 是有理數的一種表示；分子與分母為整數，分母必須非零。","en":"A fraction p/q is a representation of a rational number whose numerator and denominator are integers and whose denominator is nonzero."},"type":"definition","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["分數","AI候選","待形式化"]},{"id":"mko-ratio-proportion","title":{"zh-Hant":"比與比例","en":"Ratio and proportion"},"summary":{"zh-Hant":"本條目把數值比 a:b 解讀為 a/b，把比例定義為兩個合法比值相等。","en":"This entry interprets the numerical ratio a:b as a/b and defines a proportion as the equality of two valid ratios."},"type":"definition","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["比與比例","AI候選","待形式化"]},{"id":"mko-domain-codomain-range","title":{"zh-Hant":"定義域、陪域與值域","en":"Domain, codomain, and range"},"summary":{"zh-Hant":"總函數 f:A→B 的定義域是 A，陪域是 B，值域是實際取得的輸出集合 f[A]。","en":"For a total function f:A→B, A is the domain, B is the codomain, and f[A] is the set of outputs actually attained."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["定義域、陪域與值域","AI候選","待形式化"]},{"id":"mko-composition","title":{"zh-Hant":"函數合成","en":"Function composition"},"summary":{"zh-Hant":"合成 g∘f 先求 f 的輸出，再把它送入 g；合成前須確認中間值可作為 g 的輸入。","en":"The composition g∘f first evaluates f and then supplies its output to g; the intermediate value must be an allowed input of g."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["函數合成","AI候選","待形式化"]},{"id":"mko-sequence","title":{"zh-Hant":"數列","en":"Sequence"},"summary":{"zh-Hant":"本條目採從 0 起始的實數數列 a:N→R；數列項依索引排列，不能任意重排成集合。","en":"This entry uses real sequences a:N→R indexed from 0. Sequence terms are ordered by their indices and cannot be rearranged into an unordered set."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["數列","AI候選","待形式化"]},{"id":"mko-permutation-combination","title":{"zh-Hant":"排列與組合","en":"Permutations and combinations"},"summary":{"zh-Hant":"由 n 個相異物件不重複取 k 個，有序選取是排列，無序選取是組合。","en":"When k distinct objects are selected without replacement from n distinct objects, an ordered selection is a permutation and an unordered selection is a combination."},"type":"concept","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["排列與組合","AI候選","待形式化"]},{"id":"mko-coordinate-plane","title":{"zh-Hant":"座標平面","en":"Coordinate plane"},"summary":{"zh-Hant":"固定原點、互相垂直的有向實數軸與单位後，平面點用有序實數對 (x,y) 表示。","en":"After fixing an origin, perpendicular oriented real axes, and units, a point in the plane is represented by an ordered pair of real numbers (x,y)."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["座標平面","AI候選","待形式化"]},{"id":"mko-power-root","title":{"zh-Hant":"自然數冪與主平方根","en":"Natural-number powers and principal square roots"},"summary":{"zh-Hant":"本輪限定實數底數的自然數冪，以及非負實數的主平方根，分開冪運算、根值與解集。","en":"This entry covers natural-number powers of real bases and principal square roots of nonnegative real numbers, keeping powers, root values, and solution sets distinct."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["自然數冪與主平方根","AI候選","待形式化"]},{"id":"mko-polynomial","title":{"zh-Hant":"一元有理係數多項式","en":"Univariate polynomials over the rationals"},"summary":{"zh-Hant":"把多項式視為有限支撐的有理係數資料；合併同次項後再讀取係數與次數。","en":"A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["一元有理係數多項式","AI候選","待形式化"]},{"id":"mko-factorization","title":{"zh-Hant":"有理係數多項式因式分解","en":"Polynomial factorization over the rationals"},"summary":{"zh-Hant":"因式分解是在指定係數域中把多項式寫成因子乘積；驗收比較全部係數而不是若干代入值。","en":"Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有理係數多項式因式分解","AI候選","待形式化"]},{"id":"mko-quadratic-equation","title":{"zh-Hant":"一元實二次方程式","en":"Quadratic equations over the reals"},"summary":{"zh-Hant":"對非零二次係數的實方程，用判別式區分實根數，將方程式、根集合與重根分開。","en":"For a real equation with a nonzero quadratic coefficient, the discriminant classifies its distinct real roots. The equation, root set, and repeated roots are kept distinct."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["一元實二次方程式","AI候選","待形式化"]},{"id":"mko-linear-function","title":{"zh-Hant":"仿射函數與一次函數","en":"Affine functions and degree-one functions"},"summary":{"zh-Hant":"實函數 f(x)=mx+b 稱仿射函數；m≠0 時為一次函數，m=0 時為常數函數。","en":"A real function f(x)=mx+b is affine. If m≠0 it has degree one; if m=0 it is constant."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["仿射函數與一次函數","AI候選","待形式化"]},{"id":"mko-function-graph","title":{"zh-Hant":"實值函數圖形","en":"Graphs of real-valued functions"},"summary":{"zh-Hant":"固定直角座標系後，實值函數的圖形是全部合法輸入與其唯一輸出組成的有序對集合。","en":"In a fixed Cartesian coordinate system, the graph of a real-valued function is the set of ordered pairs of every permitted input with its unique output."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["實值函數圖形","AI候選","待形式化"]},{"id":"mko-probability","title":{"zh-Hant":"有限樣本空間上的機率","en":"Probability on finite sample spaces"},"summary":{"zh-Hant":"在非空有限樣本空間上指定非負且總和為一的權重，事件機率為其成員權重之和。","en":"Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限樣本空間上的機率","AI候選","待形式化"]},{"id":"mko-quantifier","title":{"zh-Hant":"量詞與明示論域","en":"Quantifiers and explicit domains"},"summary":{"zh-Hant":"全稱與存在量詞必須綁定論域；古典真值語義下，空論域的全稱為真、存在為假。","en":"Universal and existential quantifiers must bind a domain. Under classical truth semantics, a universal statement over the empty domain is true and an existential statement is false."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["量詞與明示論域","AI候選","待形式化"]},{"id":"mko-counterexample","title":{"zh-Hant":"反例與論域資格","en":"Counterexamples and domain eligibility"},"summary":{"zh-Hant":"反例必須屬於被聲稱的論域且使全稱述詞失敗；域外值和找不到反例都不能自動證明原命題。","en":"A counterexample must belong to the claimed domain and falsify the universal predicate. An outside value or failure to find a counterexample does not by itself prove the original statement."},"type":"definition","domain":"history_philosophy_methodology","formalization":"not_formalized","tags":["反例與論域資格","AI候選","待形式化"]},{"id":"mko-relation","title":{"zh-Hant":"二元關係與集合投影","en":"Binary relations and set projections"},"summary":{"zh-Hant":"二元關係是指定 A×B 中有序對的集合，不要求每個輸入唯一或必有輸出。","en":"A binary relation is a set of ordered pairs in a specified A×B. An input need not have a unique output or any output."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["二元關係與集合投影","AI候選","待形式化"]},{"id":"mko-graph","title":{"zh-Hant":"有限簡單無向圖","en":"Finite simple undirected graphs"},"summary":{"zh-Hant":"本輪限定有限簡單無向圖，邊是兩個相異頂點的無序集合，度數只計相鄰頂點。","en":"This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["有限簡單無向圖","AI候選","待形式化"]},{"id":"mko-vector","title":{"zh-Hant":"固定基底下的平面向量","en":"Plane vectors in a fixed basis"},"summary":{"zh-Hant":"本輪把平面向量限制為固定基底下的 R² 有序對，逐分量定義加法與純量乘法。","en":"This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["固定基底下的平面向量","AI候選","待形式化"]},{"id":"mko-system-linear-equations","title":{"zh-Hant":"二元聯立一次方程組與退化邊界","en":"Linear systems in two variables and degenerate constraints"},"summary":{"zh-Hant":"同時滿足兩個實數線性限制才是解；非零行列式給唯一解，奇異與零列須另判一致性。","en":"A solution must satisfy two real linear constraints simultaneously. A nonzero determinant gives a unique solution; singular systems and zero rows require separate consistency checks."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["二元聯立一次方程組與退化邊界","AI候選","待形式化"]},{"id":"mko-conditional-probability","title":{"zh-Hant":"有限模型的條件機率","en":"Conditional probability in finite models"},"summary":{"zh-Hant":"條件事件必須有正機率；有限模型中 P(A|B)=P(A∩B)/P(B)，方向不能任意交換。","en":"The conditioning event must have positive probability. In a finite model, P(A|B)=P(A∩B)/P(B), and the direction cannot be freely exchanged."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型的條件機率","AI候選","待形式化"]},{"id":"mko-injective","title":{"zh-Hant":"單射與不同輸入的辨識","en":"Injectivity and distinct inputs"},"summary":{"zh-Hant":"單射要求不同合法輸入不能有同一輸出，不要求覆蓋整個陪域。","en":"Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["單射與不同輸入的辨識","AI候選","待形式化"]},{"id":"mko-surjective","title":{"zh-Hant":"滿射與陪域覆蓋","en":"Surjectivity and codomain coverage"},"summary":{"zh-Hant":"滿射要求陪域每個元素都有合法原像；相同輸出可以有多個原像。","en":"Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["滿射與陪域覆蓋","AI候選","待形式化"]},{"id":"mko-bijective","title":{"zh-Hant":"雙射與唯一原像","en":"Bijectivity and unique preimages"},"summary":{"zh-Hant":"雙射同時單射與滿射，等價於陪域每個元素恰有一個合法原像。","en":"A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["雙射與唯一原像","AI候選","待形式化"]},{"id":"mko-inverse-function","title":{"zh-Hant":"雙射的反函數","en":"Inverse functions of bijections"},"summary":{"zh-Hant":"雙射 f:A→B 的反函數 g:B→A 須驗證兩個方向的還原，不能把交換資料當成無條件反函數。","en":"For a bijection f:A→B, an inverse g:B→A restores values in both directions; exchanging a table is not an unconditional inverse construction."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["雙射的反函數","AI候選","待形式化"]},{"id":"mko-random-variable","title":{"zh-Hant":"有限樣本空間的實值隨機變數","en":"Real random variables on finite sample spaces"},"summary":{"zh-Hant":"有限模型中的隨機變數是基本結果到實數的總函數；它不是基本結果，也不是權重本身。","en":"A random variable in a finite model is a total function from elementary outcomes to real numbers, distinct from an outcome or its probability weight."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限樣本空間的實值隨機變數","AI候選","待形式化"]},{"id":"mko-expectation","title":{"zh-Hant":"有限模型的期望值","en":"Expectation in finite models"},"summary":{"zh-Hant":"有限期望是合法機率權重與隨機變數值的加權總和，線性性不要求獨立。","en":"Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型的期望值","AI候選","待形式化"]},{"id":"mko-variance","title":{"zh-Hant":"有限機率模型的變異數","en":"Variance in finite probability models"},"summary":{"zh-Hant":"本輪變異數為 E[(X−E[X])²]，不是 n−1 分母的樣本變異數；零權重離群值不影響它。","en":"This entry defines variance as E[(X−E[X])²], distinct from sample variance with an n−1 denominator. Zero-weight outliers do not affect it."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限機率模型的變異數","AI候選","待形式化"]},{"id":"mko-order-relation","title":{"zh-Hant":"實數大小與順序","en":"Usual order on the real numbers"},"summary":{"zh-Hant":"在實數論域比較大小；精確有理數交叉乘積、有限極值及含端點區間是本輪切片。","en":"The usual real order compares real values and supports finite nonempty extrema, closed intervals, and translation by a common addend."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["實數大小與順序","AI候選","待形式化"]},{"id":"mko-quadratic-function","title":{"zh-Hant":"頂點式二次函數","en":"Quadratic functions in vertex form"},"summary":{"zh-Hant":"實數全論域的 q(x)=a(x−h)²+k、a≠0；頂點、全域極值與對稱點的明示切片。","en":"A nonzero-coefficient quadratic on all of R has a vertex, an axis of symmetry, and a global minimum or maximum determined by the coefficient sign."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["頂點式二次函數","AI候選","待形式化"]},{"id":"mko-rate-of-change","title":{"zh-Hant":"兩點間的平均變化率","en":"Average rate of change between distinct inputs"},"summary":{"zh-Hant":"平均變化率是輸出差除以非零輸入差，不是瞬時導數。","en":"Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["兩點間的平均變化率","AI候選","待形式化"]},{"id":"mko-distance-formula","title":{"zh-Hant":"等單位正交座標的歐氏距離","en":"Euclidean distance in an equal-unit orthonormal frame"},"summary":{"zh-Hant":"在同一等尺度正交座標系，用非負主平方根讀取二維歐氏距離。","en":"The nonnegative L2 distance in one equal-unit orthonormal coordinate frame is the principal square root of the sum of squared coordinate differences."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["等單位正交座標的歐氏距離","AI候選","待形式化"]},{"id":"mko-circle","title":{"zh-Hant":"圓的邊界與座標判定","en":"Circle boundaries and zero-radius degeneracy"},"summary":{"zh-Hant":"圓是到固定中心距離恰等於半徑的點集，不是含內部的圓盤。","en":"A circle here is the boundary point set at a nonnegative radius from its center, not the filled disk; radius zero yields a singleton."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["圓的邊界與座標判定","AI候選","待形式化"]},{"id":"mko-area","title":{"zh-Hant":"座標矩形與單位格的面積","en":"Exact area of half-open rectangles and unit-cell unions"},"summary":{"zh-Hant":"先精確處理半開矩形及有限單位格聯集，含重疊、觸邊與空矩形；不冒充一般面積理論。","en":"This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["座標矩形與單位格的面積","AI候選","待形式化"]},{"id":"mko-descriptive-statistics","title":{"zh-Hant":"有限觀測列的描述統計","en":"Descriptive summaries of nonempty observation lists"},"summary":{"zh-Hant":"保留重數的非空觀測列，精確計算均值、中位數、全距與所有最大頻率值。","en":"Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限觀測列的描述統計","AI候選","待形式化"]},{"id":"mko-matrix","title":{"zh-Hant":"有限矩陣的尺寸、索引與轉置","en":"Finite matrices: dimensions, indexing, and transpose"},"summary":{"zh-Hant":"矩陣以行、列組織元素；本輪使用非空有限實矩陣及精確整數陣列切片。","en":"Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限矩陣的尺寸、索引與轉置","AI候選","待形式化"]},{"id":"mko-linear-transformation","title":{"zh-Hant":"矩陣表示的實平面線性映射","en":"Matrix-represented linear maps on the real plane"},"summary":{"zh-Hant":"先在 R²→R² 的矩陣切片驗證加法與純量相容，不把帶非零平移的仿射式叫線性映射。","en":"Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["矩陣表示的實平面線性映射","AI候選","待形式化"]},{"id":"mko-determinant","title":{"zh-Hant":"實二階方陣的行列式","en":"Determinants of real 2×2 matrices"},"summary":{"zh-Hant":"2×2實矩陣的 ad−bc、非零行列式可逆性、乘積與純量縮放切片。","en":"This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實二階方陣的行列式","AI候選","待形式化"]},{"id":"mko-eigenvalue","title":{"zh-Hant":"實二階方陣的特徵值與非零特徵向量","en":"Real eigenvalues and nonzero eigenvectors of 2×2 matrices"},"summary":{"zh-Hant":"特徵向量須非零；候選特徵空間核含零向量，實二階矩陣可沒有實特徵值。","en":"An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實二階方陣的特徵值與非零特徵向量","AI候選","待形式化"]},{"id":"mko-congruence","title":{"zh-Hant":"明示對應的非退化三角形全等","en":"Congruence of nondegenerate triangles with explicit correspondence"},"summary":{"zh-Hant":"限定正交等單位平面中的標記非退化三角形，以三組對應邊長判全等，允許鏡射。","en":"In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["明示對應的非退化三角形全等","AI候選","待形式化"]},{"id":"mko-similarity","title":{"zh-Hant":"正倍率的標記三角形相似切片","en":"Labeled triangle similarity with a specified positive scale"},"summary":{"zh-Hant":"指定由A到B的正長度倍率 k，平方長度與指定面積以 k² 縮放，不把負座標係數當負長度倍率。","en":"Specify the positive length scale k from A to B; squared lengths and a given area scale by k². A negative coordinate coefficient is not a negative length scale."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["正倍率的標記三角形相似切片","AI候選","待形式化"]},{"id":"mko-geometric-transformations","title":{"zh-Hant":"平面座標的基本等距與均勻縮放","en":"Plane-coordinate isometries and uniform scaling"},"summary":{"zh-Hant":"平移、原點逆時針四分之一圈的整數倍旋轉、指定軸鏡射是全平面等距；均勻縮放先分清零、±1與其他係數。","en":"Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["平面座標的基本等距與均勻縮放","AI候選","待形式化"]},{"id":"mko-sequence-limit","title":{"zh-Hant":"數列極限與精確尾界","en":"Sequence limits and exact tail cutoffs"},"summary":{"zh-Hant":"從0起算的實數列以任意精度、某一自然尾界及整個尾段定義極限；本輪精確處理倒數尾項模型。","en":"A zero-based real sequence converges when every precision admits a natural cutoff controlling the entire tail; this pulse treats an exact reciprocal-tail model."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["數列極限與精確尾界","AI候選","待形式化"]},{"id":"mko-function-limit","title":{"zh-Hant":"去心實數論域的函數極限","en":"Function limits on punctured real domains"},"summary":{"zh-Hant":"在a附近排除a本身，以ε-δ控制函數值；點值不屬於去心極限條件。","en":"An ε–δ limit controls values near a while excluding a itself; the center value is not part of the punctured-limit condition."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["去心實數論域的函數極限","AI候選","待形式化"]},{"id":"mko-continuity","title":{"zh-Hant":"實函數的點值與連續","en":"Point values and continuity of real functions"},"summary":{"zh-Hant":"在全實數論域連續需要點值存在且與兩側極限一致；補點與原函數連續不可混用。","en":"Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實函數的點值與連續","AI候選","待形式化"]},{"id":"mko-derivative","title":{"zh-Hant":"實函數在一點的導數與差商","en":"Real derivatives at a point and finite difference quotients"},"summary":{"zh-Hant":"導數是非零步差商的兩側極限，不是任何一個有限步差商；絕對值尖點另保留不存在。","en":"A derivative is the two-sided limit of nonzero-step quotients, not any one finite-step quotient; an absolute-value corner retains nonexistence."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實函數在一點的導數與差商","AI候選","待形式化"]},{"id":"mko-derivative-rules","title":{"zh-Hant":"多項式的和、乘積與合成微分","en":"Differentiating polynomial sums, products, and compositions"},"summary":{"zh-Hant":"四條微分規則先保留可微前提；本輪只在有限整數係數多項式上精確計算。","en":"Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["多項式的和、乘積與合成微分","AI候選","待形式化"]},{"id":"mko-antiderivative","title":{"zh-Hant":"多項式原函數與常數選擇","en":"Polynomial antiderivatives and the choice of constant"},"summary":{"zh-Hant":"原函數須在宣告區間導回f；本輪全R多項式以常數項固定0的有理係數原函數作可重播代表。","en":"An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["多項式原函數與常數選擇","AI候選","待形式化"]},{"id":"mko-definite-integral","title":{"zh-Hant":"有限區間上的多項式定積分","en":"Signed polynomial definite integrals on finite intervals"},"summary":{"zh-Hant":"連續多項式的有向定積分與精確矩公式；有限左端黎曼和只是有限和，不冒充一般積分或微積分基本定理完成。","en":"Continuous polynomials have oriented definite integrals with exact moment formulas; a finite left Riemann sum is not a general integral or a completed general FTC bridge."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["有限區間上的多項式定積分","AI候選","待形式化"]},{"id":"mko-fundamental-theorem-calculus","title":{"zh-Hant":"微積分基本定理：連續前提與有向端點","en":"Fundamental theorem of calculus: continuity and oriented endpoints"},"summary":{"zh-Hant":"在全實數連續前提下，累積定積分的導數還原被積函數；合法原函數給有向積分端點差。","en":"With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["微積分基本定理：連續前提與有向端點","AI候選","待形式化"]},{"id":"mko-tree","title":{"zh-Hant":"非空有限樹：連通、無環與唯一道路","en":"Nonempty finite trees: connectivity, cycles, and unique paths"},"summary":{"zh-Hant":"非空有限簡單無向圖同時連通且無環才是樹；保留單點、度1葉及加一條新邊的唯一環。","en":"A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions."},"type":"concept","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["非空有限樹：連通、無環與唯一道路","AI候選","待形式化"]},{"id":"mko-topological-space","title":{"zh-Hant":"拓撲空間","en":"Topological space"},"summary":{"zh-Hant":"拓撲空間由底集合與一族開集組成；開集族包含空集與全空間，並對任意聯集與有限交封閉。","en":"A topological space consists of an underlying set and a family of open sets containing the empty set and whole space and closed under arbitrary unions and finite intersections."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲空間","AI候選","待形式化"]},{"id":"mko-natural-divisibility","title":{"zh-Hant":"自然數整除","en":"Natural-number divisibility"},"summary":{"zh-Hant":"自然數 a 整除 b，表示存在自然數乘數 k 使 b=a·k；零的兩側行為必須分開。","en":"A natural number a divides b when a natural multiplier k satisfies b=a·k; the two roles of zero must be distinguished."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["自然數整除","AI候選","待形式化"]},{"id":"mko-gcd-coprime","title":{"zh-Hant":"最大公因數與互質","en":"Greatest common divisor and coprimality"},"summary":{"zh-Hant":"gcd(a,b) 是同時整除 a、b 且被每個公因數整除的自然數；gcd=1 時稱互質。","en":"The natural number gcd(a,b) divides both a and b and is divisible by every common divisor; the inputs are coprime when their gcd is 1."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["最大公因數與互質","AI候選","待形式化"]},{"id":"mko-prime-number","title":{"zh-Hant":"自然數質數","en":"Prime natural number"},"summary":{"zh-Hant":"自然數質數是大於或等於 2，且正因數只有 1 與自身的自然數。","en":"A prime natural number is at least 2 and has no positive divisors other than 1 and itself."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["自然數質數","AI候選","待形式化"]},{"id":"mko-modular-congruence","title":{"zh-Hant":"自然數模同餘","en":"Natural-number congruence modulo a positive modulus"},"summary":{"zh-Hant":"固定正模數 n 時，a 與 b 模 n 同餘表示兩者除以 n 的餘數相同。","en":"For a fixed positive modulus n, two natural numbers a and b are congruent modulo n when they have the same remainder on division by n."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["自然數模同餘","AI候選","待形式化"]},{"id":"mko-decision-problem","title":{"zh-Hant":"決定問題","en":"Decision problem"},"summary":{"zh-Hant":"決定問題由輸入集合 X 與 yes-instance 子集合 Y⊆X 指定；decider 必須對每個合法輸入終止並回傳正確布林答案。","en":"A decision problem specifies an input set X and a yes-instance subset Y⊆X; a decider must terminate on every legal input and return the correct Boolean answer."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["決定問題","AI候選","待形式化"]},{"id":"mko-lean-formal-statement","title":{"zh-Hant":"證明助理中的形式陳述","en":"Formal statements in a proof assistant"},"summary":{"zh-Hant":"形式陳述是在固定版本、imports、環境與 local context 中成功 elaboration，且型別為 Prop 的核心項。","en":"A formal statement is a core term of type Prop obtained by successful elaboration in a fixed version, import environment and local context."},"type":"definition","domain":"formalized_mathematics","formalization":"not_formalized","tags":["證明助理中的形式陳述","AI候選","待形式化"]},{"id":"mko-open-closed-set","title":{"zh-Hant":"開集與閉集","en":"Open and closed sets"},"summary":{"zh-Hant":"在固定拓撲空間中，開集屬於拓撲；閉集的補集為開集，同一集合可以同時開且閉。","en":"In a fixed topological space, open sets belong to the topology and closed sets have open complements; one set can be both open and closed."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["開集與閉集","AI候選","待形式化"]},{"id":"mko-neighborhood","title":{"zh-Hant":"鄰域","en":"Neighborhood"},"summary":{"zh-Hant":"集合 N 是點 x 的鄰域，當且僅當 N 包含某個含 x 的開集；N 本身不必開。","en":"A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["鄰域","AI候選","待形式化"]},{"id":"mko-topological-continuity","title":{"zh-Hant":"拓撲連續映射","en":"Continuous maps between topological spaces"},"summary":{"zh-Hant":"拓撲映射連續，表示目標空間每個開集的逆像在來源空間仍是開集。","en":"A map is topologically continuous when every open set in its target has an open preimage in its source."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲連續映射","AI候選","待形式化"]},{"id":"mko-computational-model","title":{"zh-Hant":"抽象計算模型","en":"Abstract computational model"},"summary":{"zh-Hant":"有限狀態計算模型以狀態集、初始狀態集、步驟關係與停止狀態集描述可能執行。","en":"A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["抽象計算模型","AI候選","待形式化"]},{"id":"mko-algorithm-correctness","title":{"zh-Hant":"算法正確性","en":"Algorithm correctness"},"summary":{"zh-Hant":"算法正確性分開部分正確性與終止性；總正確性需要兩者同時成立。","en":"Algorithm correctness separates partial correctness from termination; total correctness requires both."},"type":"concept","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["算法正確性","AI候選","待形式化"]},{"id":"mko-asymptotic-resource-bound","title":{"zh-Hant":"漸近資源界","en":"Asymptotic resource bounds"},"summary":{"zh-Hant":"漸近上界、下界與緊界必須相對輸入大小、成本模型及全尾段量化定義。","en":"Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["漸近資源界","AI候選","待形式化"]},{"id":"mko-lean-proof-term","title":{"zh-Hant":"證明項","en":"Proof term"},"summary":{"zh-Hant":"證明項是在固定形式環境與 context 中具有命題 P 型別，並可由 kernel 檢查的項 t。","en":"A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel."},"type":"concept","domain":"formalized_mathematics","formalization":"not_formalized","tags":["證明項","AI候選","待形式化"]},{"id":"mko-discrete-dynamical-system","title":{"zh-Hant":"離散時間動力系統","en":"Discrete-time dynamical system"},"summary":{"zh-Hant":"離散時間動力系統以狀態空間、自映射與自然數時間描述逐步演化；有限軌道核對不自動給出一般穩定性。","en":"A discrete-time dynamical system uses a state space, a self-map and natural-number time to describe stepwise evolution; a finite orbit check does not establish general stability."},"type":"definition","domain":"dynamical_systems","formalization":"not_formalized","tags":["離散時間動力系統","AI候選","待形式化"]},{"id":"mko-ordinary-differential-equation","title":{"zh-Hant":"常微分方程","en":"Ordinary differential equation"},"summary":{"zh-Hant":"常微分方程關聯未知單變數函數及其有限階導數；候選函數的殘差核對不等於存在唯一性定理。","en":"An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems."},"type":"definition","domain":"differential_equations","formalization":"not_formalized","tags":["常微分方程","AI候選","待形式化"]},{"id":"mko-numerical-approximation-error","title":{"zh-Hant":"數值近似與誤差","en":"Numerical approximation and error"},"summary":{"zh-Hant":"絕對誤差是近似值與參照值之差的絕對值；相對誤差另要求參照值非零並除以其絕對值。","en":"Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value."},"type":"definition","domain":"numerical_computational","formalization":"not_formalized","tags":["數值近似與誤差","AI候選","待形式化"]},{"id":"mko-optimization-problem","title":{"zh-Hant":"最佳化問題","en":"Optimization problem"},"summary":{"zh-Hant":"最佳化問題在明示可行集合上尋找使目標函數最小或最大的點；最優值與最優解集合需分開。","en":"An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["最佳化問題","AI候選","待形式化"]},{"id":"mko-dimensional-analysis","title":{"zh-Hant":"量綱分析","en":"Dimensional analysis"},"summary":{"zh-Hant":"量綱分析用基本量綱指數向量追蹤乘、除與冪，並核對方程兩側量綱是否一致。","en":"Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides."},"type":"concept","domain":"mathematical_physics","formalization":"not_formalized","tags":["量綱分析","AI候選","待形式化"]},{"id":"mko-discrete-exponential-growth","title":{"zh-Hant":"離散指數成長模型","en":"Discrete exponential growth model"},"summary":{"zh-Hant":"固定時間步與固定非負因子 r 的離散模型滿足 x_(n+1)=r x_n，因此 x_n=x_0 r^n。","en":"With a fixed time step and fixed nonnegative factor r, the discrete recurrence x_(n+1)=r*x_n gives x_n=x_0*r^n."},"type":"definition","domain":"mathematical_biology","formalization":"not_formalized","tags":["離散指數成長模型","AI候選","待形式化"]},{"id":"mko-compound-interest","title":{"zh-Hant":"複利模型","en":"Compound-interest model"},"summary":{"zh-Hant":"固定每期利率 i 且無額外現金流時，餘額滿足 B_(n+1)=(1+i)B_n，故 B_n=B_0(1+i)^n。","en":"With a fixed rate i per period and no additional cash flows, balances satisfy B_(n+1)=(1+i)B_n and hence B_n=B_0(1+i)^n."},"type":"definition","domain":"economics_finance","formalization":"not_formalized","tags":["複利模型","AI候選","待形式化"]},{"id":"mko-mathematical-representation","title":{"zh-Hant":"數學表徵","en":"Mathematical representation"},"summary":{"zh-Hant":"數學表徵把同一對象的文字規則、符號式、有限表格、序對與座標資料分開記錄，並以明示語義橋核對一致性。","en":"Mathematical representation separately records verbal rules, symbolic expressions, finite tables, ordered pairs and coordinate data for the same object, using explicit semantic bridges to check agreement."},"type":"concept","domain":"mathematics_education","formalization":"not_formalized","tags":["數學表徵","AI候選","待形式化"]},{"id":"mko-diagnostic-mathematical-assessment","title":{"zh-Hant":"診斷性數學評量記錄","en":"Diagnostic mathematical assessment records"},"summary":{"zh-Hant":"診斷性數學評量記錄把題目、期望、作答與明示判定規則綁在一起，只產生可追溯的規則符合狀態。","en":"A diagnostic mathematical assessment record binds an item, its expected result, a response and an explicit judgment rule, producing only a traceable rule-agreement status."},"type":"concept","domain":"mathematics_education","formalization":"not_formalized","tags":["診斷性數學評量記錄","AI候選","待形式化"]},{"id":"mko-orbit-fixed-periodic","title":{"zh-Hant":"軌道、固定點與週期點","en":"Orbits, fixed points and periodic points"},"summary":{"zh-Hant":"反覆迭代自映射形成軌道；固定點週期為1，週期點的最小正週期須和任意返回時間分開。","en":"Repeated iteration of a self-map forms an orbit. Fixed points have period 1; the least positive period of a periodic point must be distinguished from an arbitrary return time."},"type":"concept","domain":"dynamical_systems","formalization":"not_formalized","tags":["軌道、固定點與週期點","AI候選","待形式化"]},{"id":"mko-initial-value-problem","title":{"zh-Hant":"初值問題","en":"Initial-value problem"},"summary":{"zh-Hant":"初值問題在微分方程外指定某點的函數值或導數值；候選核對與存在唯一性是不同責任。","en":"An initial-value problem adds function or derivative values at a specified point to a differential equation; candidate checks are distinct from existence and uniqueness."},"type":"definition","domain":"differential_equations","formalization":"not_formalized","tags":["初值問題","AI候選","待形式化"]},{"id":"mko-bisection-method","title":{"zh-Hant":"二分法的有限 bracket 更新","en":"Finite sign-bracket updates in the bisection method"},"summary":{"zh-Hant":"二分法在已提供的端點與中點符號資料上取中點並保留符號相反或零值的半區，有限步寬度減半。","en":"Using supplied endpoint and midpoint signs, bisection takes a midpoint and retains the opposite-sign half or a detected zero; finite nonterminal bisections halve the interval width."},"type":"algorithm","domain":"numerical_computational","formalization":"not_formalized","tags":["二分法的有限 bracket 更新","AI候選","待形式化"]},{"id":"mko-linear-programming","title":{"zh-Hant":"線性規劃","en":"Linear programming"},"summary":{"zh-Hant":"線性規劃在線性等式或不等式限制的可行域上最佳化線性目標函數；本輪題庫只驗完整有限候選網格。","en":"Linear programming optimizes a linear objective over a feasible domain defined by linear equalities or inequalities; this exercise slice checks only a complete explicit finite candidate grid."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["線性規劃","AI候選","待形式化"]},{"id":"mko-discrete-logistic-growth","title":{"zh-Hant":"離散 logistic 增量模型","en":"Discrete logistic increment model"},"summary":{"zh-Hant":"固定容量K與0≤r≤1時，本條目以x_(n+1)=x_n+r x_n(1-x_n/K)描述每步受限增量。","en":"For fixed capacity K and 0≤r≤1, this entry uses x_(n+1)=x_n+r*x_n*(1−x_n/K) to describe each capacity-limited discrete increment."},"type":"definition","domain":"mathematical_biology","formalization":"not_formalized","tags":["離散 logistic 增量模型","AI候選","待形式化"]},{"id":"mko-harmonic-oscillator-model","title":{"zh-Hant":"線性簡諧振子模型","en":"Linear harmonic-oscillator model"},"summary":{"zh-Hant":"無阻尼線性簡諧振子以m x''+k x=0及m,k>0定義；本輪不宣稱三角函數閉式解。","en":"The undamped linear harmonic oscillator is defined by m*x″+k*x=0 with m,k>0; this slice asserts no trigonometric closed-form solution."},"type":"definition","domain":"mathematical_physics","formalization":"not_formalized","tags":["線性簡諧振子模型","AI候選","待形式化"]},{"id":"mko-present-value-discounting","title":{"zh-Hant":"現值與折現","en":"Present value and discounting"},"summary":{"zh-Hant":"固定正累積因子與期數時，未來金額除以累積因子冪得到現值。","en":"For a fixed positive accumulation factor and period count, divide a future amount by the factor power to obtain present value."},"type":"definition","domain":"economics_finance","formalization":"not_formalized","tags":["現值與折現","AI候選","待形式化"]},{"id":"mko-complex-number","title":{"zh-Hant":"複數","en":"Complex number"},"summary":{"zh-Hant":"複數以實數有序對(a,b)表示a+bi，並以i²=-1定義加減乘法。","en":"A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1."},"type":"definition","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["複數","AI候選","待形式化"]},{"id":"mko-real-vector-space","title":{"zh-Hant":"實向量空間","en":"Real vector space"},"summary":{"zh-Hant":"實向量空間配備向量加法與實數純量乘法，滿足加法交換群及相容的純量作用公理。","en":"A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實向量空間","AI候選","待形式化"]},{"id":"mko-finite-group","title":{"zh-Hant":"有限群","en":"Finite group"},"summary":{"zh-Hant":"有限群由有限集合與完整封閉Cayley表構成，並滿足結合律、唯一單位元及每元素逆元。","en":"A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群","AI候選","待形式化"]},{"id":"mko-metric-space","title":{"zh-Hant":"度量空間","en":"Metric space"},"summary":{"zh-Hant":"度量空間由集合與距離函數組成，距離非負、只在相同點為零、對稱且滿足三角不等式。","en":"A metric space has a carrier set and a distance function that is nonnegative, zero exactly at equal points, symmetric and satisfies the triangle inequality."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["度量空間","AI候選","待形式化"]},{"id":"mko-finite-discrete-probability-distribution","title":{"zh-Hant":"有限離散機率分布","en":"Finite discrete probability distribution"},"summary":{"zh-Hant":"有限離散分布是在有限結果值上指定非負機率質量，全部質量精確和為一。","en":"A finite discrete distribution assigns nonnegative probability masses to finitely many outcome values, with exact total mass one."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限離散機率分布","AI候選","待形式化"]},{"id":"mko-complex-modulus-conjugate","title":{"zh-Hant":"複數模與共軛","en":"Complex modulus and conjugation"},"summary":{"zh-Hant":"複數共軛保留實部並反轉虛部；模平方為實部與虛部平方和。","en":"Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares."},"type":"concept","domain":"analysis","formalization":"not_formalized","tags":["複數模與共軛","AI候選","待形式化"]},{"id":"mko-linear-combination-span","title":{"zh-Hant":"線性組合與生成","en":"Linear combinations and span"},"summary":{"zh-Hant":"線性組合是有限個向量的純量加權和；生成空間收集全部有限線性組合。","en":"A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["線性組合與生成","AI候選","待形式化"]},{"id":"mko-subgroup-homomorphism","title":{"zh-Hant":"子群與群同態","en":"Subgroups and group homomorphisms"},"summary":{"zh-Hant":"子群在繼承運算下自身成群；群同態保持乘法，kernel與image由完整映射表導出。","en":"A subgroup is a group under the inherited operation; a group homomorphism preserves multiplication, and its kernel and image follow from the complete map table."},"type":"concept","domain":"algebra","formalization":"not_formalized","tags":["子群與群同態","AI候選","待形式化"]},{"id":"mko-metric-convergence","title":{"zh-Hant":"度量空間中的收斂","en":"Convergence in a metric space"},"summary":{"zh-Hant":"度量空間序列收斂到x，表示每個正epsilon終有所有n≥N落入x的epsilon球。","en":"A sequence in a metric space converges to x when, for every positive epsilon, eventually all n≥N lie in the epsilon-ball around x."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["度量空間中的收斂","AI候選","待形式化"]},{"id":"mko-sigma-algebra-measure-space","title":{"zh-Hant":"σ-代數與測度空間","en":"Sigma-algebras and measure spaces"},"summary":{"zh-Hant":"σ-代數含空集並對補集與可數聯集封閉；測度非負、空集為零且可數可加。","en":"A sigma-algebra contains the empty set and is closed under complements and countable unions; a measure is nonnegative, zero on the empty set and countably additive."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["σ-代數與測度空間","AI候選","待形式化"]},{"id":"mko-discrete-stochastic-process","title":{"zh-Hant":"離散隨機過程","en":"Discrete-time stochastic process"},"summary":{"zh-Hant":"離散隨機過程是共同機率空間上的離散時間隨機變數族；邊際分布不決定聯合律。","en":"A discrete-time stochastic process is a time-indexed family of random variables on one probability space; marginal distributions do not determine its joint law."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["離散隨機過程","AI候選","待形式化"]},{"id":"mko-multivariable-real-function","title":{"zh-Hant":"多變數實函數","en":"Multivariable real-valued function"},"summary":{"zh-Hant":"多變數實函數把明示實向量域中的輸入映到實數；座標角色與定義域是函數資料。","en":"A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["多變數實函數","AI候選","待形式化"]},{"id":"mko-linear-independence-basis","title":{"zh-Hant":"線性獨立與基底","en":"Linear independence and bases"},"summary":{"zh-Hant":"線性獨立排除非平凡零和關係；基底同時線性獨立並生成已宣告空間。","en":"Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space."},"type":"concept","domain":"algebra","formalization":"not_formalized","tags":["線性獨立與基底","AI候選","待形式化"]},{"id":"mko-measurable-function","title":{"zh-Hant":"可測函數","en":"Measurable function"},"summary":{"zh-Hant":"來源與目標各固定sigma代數時，函數可測表示每個目標可測集的逆像皆在來源sigma代數。","en":"With fixed source and target sigma-algebras, a function is measurable when the preimage of every target measurable set belongs to the source sigma-algebra."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["可測函數","AI候選","待形式化"]},{"id":"mko-finite-markov-chain","title":{"zh-Hant":"有限狀態 Markov 鏈","en":"Finite-state Markov chain"},"summary":{"zh-Hant":"有限時間齊次Markov鏈以row-stochastic轉移核與需要時的初始律指定有限路徑機率。","en":"A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限狀態 Markov 鏈","AI候選","待形式化"]},{"id":"mko-partial-derivative","title":{"zh-Hant":"偏導數","en":"Partial derivative"},"summary":{"zh-Hant":"偏導數固定其他座標，只沿一個座標切片取h趨近零的單變數導數。","en":"A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["偏導數","AI候選","待形式化"]},{"id":"mko-finite-convex-combination","title":{"zh-Hant":"有限凸組合","en":"Finite convex combination"},"summary":{"zh-Hant":"有限凸組合用非負精確權重加權有限非空向量列，且權重總和恰為一。","en":"A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["有限凸組合","AI候選","待形式化"]},{"id":"mko-partial-differential-equation","title":{"zh-Hant":"偏微分方程","en":"Partial differential equation"},"summary":{"zh-Hant":"偏微分方程關聯多變數未知函數與一個或多個偏導數；殘差與主部分類需固定算子convention。","en":"A partial differential equation relates a multivariable unknown function and one or more partial derivatives; residuals and principal-part classification require a fixed operator convention."},"type":"definition","domain":"differential_equations","formalization":"not_formalized","tags":["偏微分方程","AI候選","待形式化"]},{"id":"mko-convex-set","title":{"zh-Hant":"凸集合","en":"Convex set"},"summary":{"zh-Hant":"集合凸表示任兩點及每個lambda∈[0,1]形成的整條線段仍在集合內。","en":"A set is convex when the entire segment formed by every pair of its points and every lambda in [0,1] remains in the set."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["凸集合","AI候選","待形式化"]},{"id":"mko-boundary-value-problem","title":{"zh-Hant":"邊界值問題","en":"Boundary-value problem"},"summary":{"zh-Hant":"邊界值問題把定義域內的微分方程責任與邊界上的值或導數資料同時指定；候選通過不等於適定性。","en":"A boundary-value problem specifies both an interior differential equation and boundary values or derivative data; successful candidate checks do not establish general well-posedness."},"type":"definition","domain":"differential_equations","formalization":"not_formalized","tags":["邊界值問題","AI候選","待形式化"]},{"id":"mko-convex-function","title":{"zh-Hant":"凸函數","en":"Convex function"},"summary":{"zh-Hant":"凸域上的函數為凸，若每個兩點凸組合處的函數值不超過相同權重的端點函數值。","en":"A function on a convex domain is convex when its value at every two-point convex combination is at most the same weighted combination of endpoint values."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["凸函數","AI候選","待形式化"]},{"id":"mko-convex-optimization-problem","title":{"zh-Hant":"凸最佳化問題","en":"Convex optimization problem"},"summary":{"zh-Hant":"凸最佳化問題在凸可行域上最小化凸目標；問題資格、可行性、求值與全域最優是不同責任。","en":"Convex optimization minimizes a convex objective on a convex feasible domain; problem qualification, feasibility, evaluation and global optimality are distinct obligations."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["凸最佳化問題","AI候選","待形式化"]},{"id":"mko-finite-abelian-group","title":{"zh-Hant":"有限阿貝爾群","en":"Finite abelian group"},"summary":{"zh-Hant":"有限群若完整Cayley表對所有元素對都滿足交換律，則為有限阿貝爾群。","en":"A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限阿貝爾群","AI候選","待形式化"]},{"id":"mko-real-normed-vector-space","title":{"zh-Hant":"實賦範向量空間","en":"Real normed vector space"},"summary":{"zh-Hant":"實向量空間配備正定、絕對齊次且滿足三角不等式的實值範數。","en":"A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實賦範向量空間","AI候選","待形式化"]},{"id":"mko-cauchy-sequence","title":{"zh-Hant":"Cauchy序列","en":"Cauchy sequence"},"summary":{"zh-Hant":"度量空間序列為Cauchy，若每個正epsilon最終所有兩項彼此距離皆小於epsilon。","en":"A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["Cauchy序列","AI候選","待形式化"]},{"id":"mko-finite-joint-distribution","title":{"zh-Hant":"有限聯合分布","en":"Finite joint distribution"},"summary":{"zh-Hant":"共同有限機率空間上多個隨機變數的值組與精確質量共同指定有限聯合分布。","en":"Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限聯合分布","AI候選","待形式化"]},{"id":"mko-covariance","title":{"zh-Hant":"協方差","en":"Covariance"},"summary":{"zh-Hant":"同一有限聯合分布下協方差為E[XY]-E[X]E[Y]；零協方差通常不推出獨立。","en":"Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["協方差","AI候選","待形式化"]},{"id":"mko-graph-isomorphism","title":{"zh-Hant":"圖同構","en":"Graph isomorphism"},"summary":{"zh-Hant":"有限簡單圖同構是頂點雙射，對所有頂點對完整保持鄰接與非鄰接。","en":"An isomorphism of finite simple graphs is a vertex bijection preserving both adjacency and nonadjacency for every pair."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["圖同構","AI候選","待形式化"]},{"id":"mko-equivalence-relation","title":{"zh-Hant":"等價關係","en":"Equivalence relation"},"summary":{"zh-Hant":"固定載體上的關係若完整滿足反身、對稱與傳遞，則為等價關係。","en":"A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["等價關係","AI候選","待形式化"]},{"id":"mko-finite-unital-ring","title":{"zh-Hant":"有限含一環","en":"Finite unital ring"},"summary":{"zh-Hant":"有限含一環在同一完整有限載體上具加法阿貝爾群、乘法結合律、左右分配律及與零不同的乘法單位元。","en":"On one complete finite carrier, this unital-ring protocol requires an additive abelian group, associative multiplication, both distributive laws and a multiplicative identity different from zero."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限含一環","AI候選","待形式化"]},{"id":"mko-norm-induced-metric","title":{"zh-Hant":"範數誘導度量","en":"Norm-induced metric"},"summary":{"zh-Hant":"同一實賦範向量空間上的範數以向量差的範數定義度量，度量公理由範數公理全稱推出。","en":"On the same real normed vector space, taking the norm of a vector difference defines a metric whose axioms follow universally from the norm axioms."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["範數誘導度量","AI候選","待形式化"]},{"id":"mko-complete-metric-space","title":{"zh-Hant":"完備度量空間","en":"Complete metric space"},"summary":{"zh-Hant":"每個Cauchy序列都收斂到空間內某點的度量空間稱為完備。","en":"A metric space is complete when every Cauchy sequence converges to a point still belonging to that space."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["完備度量空間","AI候選","待形式化"]},{"id":"mko-real-bilinear-form","title":{"zh-Hant":"實雙線性形式","en":"Real bilinear form"},"summary":{"zh-Hant":"實向量空間上的雙線性形式分別對兩個輸入保持向量加法與實純量乘法。","en":"A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實雙線性形式","AI候選","待形式化"]},{"id":"mko-general-linear-map","title":{"zh-Hant":"實向量空間間線性映射","en":"Linear map between real vector spaces"},"summary":{"zh-Hant":"一般實向量空間間映射若對所有向量與實純量保持線性組合，則為線性映射。","en":"A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實向量空間間線性映射","AI候選","待形式化"]},{"id":"mko-event-independence","title":{"zh-Hant":"事件獨立","en":"Event independence"},"summary":{"zh-Hant":"兩事件獨立以交集機率等於邊際機率乘積定義；此定義不依賴零機率條件式。","en":"Two events are independent when their intersection probability equals the product of their marginal probabilities; this definition does not rely on conditioning on a zero-probability event."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["事件獨立","AI候選","待形式化"]},{"id":"mko-graph-walk-simple-path","title":{"zh-Hant":"圖walk與simple path","en":"Graph walks and simple paths"},"summary":{"zh-Hant":"有限無向簡單圖的walk要求每對連續頂點相鄰；simple path另要求所有頂點互異。","en":"In a finite undirected simple graph, a walk requires each consecutive pair of vertices to be adjacent; a simple path additionally requires all vertices to be distinct."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["圖walk與simple path","AI候選","待形式化"]},{"id":"mko-finite-commutative-ring","title":{"zh-Hant":"有限交換環","en":"Finite commutative ring"},"summary":{"zh-Hant":"有限含一環若乘法對全部元素對交換，則為有限交換環；零因子仍可能存在。","en":"A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限交換環","AI候選","待形式化"]},{"id":"mko-finite-two-sided-ideal","title":{"zh-Hant":"有限雙邊理想","en":"Finite two-sided ideal"},"summary":{"zh-Hant":"有限環子集若為加法子群且吸收父環的左右乘法，則為雙邊理想。","en":"A subset of a finite ring is a two-sided ideal when it is an additive subgroup and absorbs multiplication by parent-ring elements on both sides."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限雙邊理想","AI候選","待形式化"]},{"id":"mko-real-banach-space","title":{"zh-Hant":"實Banach空間","en":"Real Banach space"},"summary":{"zh-Hant":"在自身範數誘導度量下完備的實賦範向量空間稱為實Banach空間。","en":"A real normed vector space is a Banach space when it is complete in the metric induced by its own norm."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實Banach空間","AI候選","待形式化"]},{"id":"mko-real-inner-product-space","title":{"zh-Hant":"實內積空間","en":"Real inner-product space"},"summary":{"zh-Hant":"對稱且正定的實雙線性形式構成實內積；半正定或抽樣為正不足。","en":"A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實內積空間","AI候選","待形式化"]},{"id":"mko-bounded-linear-operator","title":{"zh-Hant":"有界線性算子","en":"Bounded linear operator"},"summary":{"zh-Hant":"賦範空間間線性映射若有單一非負常數統一控制所有輸出範數，則為有界線性算子。","en":"A linear map between normed spaces is bounded when one nonnegative constant uniformly controls all output norms relative to input norms."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["有界線性算子","AI候選","待形式化"]},{"id":"mko-random-variable-independence","title":{"zh-Hant":"隨機變數獨立","en":"Independence of random variables"},"summary":{"zh-Hant":"同一有限joint law下，隨機變數獨立要求全部值對的joint mass分解為邊際質量乘積。","en":"Under one finite joint law, independence requires every value-pair mass to factor as the product of its marginal masses."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["隨機變數獨立","AI候選","待形式化"]},{"id":"mko-connected-component","title":{"zh-Hant":"連通分量","en":"Connected component"},"summary":{"zh-Hant":"有限無向簡單圖中，由存在路徑所定義的頂點等價類恰為連通分量。","en":"In a finite undirected simple graph, connected components are exactly the vertex equivalence classes defined by the existence of a path."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["連通分量","AI候選","待形式化"]},{"id":"mko-finite-integral-domain","title":{"zh-Hant":"有限整環","en":"Finite integral domain"},"summary":{"zh-Hant":"有限交換含一環若1不等於0且沒有非零零因子，則為有限整環。","en":"A finite commutative unital ring with 1 distinct from 0 is an integral domain when it has no nonzero zero divisors."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限整環","AI候選","待形式化"]},{"id":"mko-polynomial-ring-over-finite-commutative-ring","title":{"zh-Hant":"有限交換環上的多項式環","en":"Polynomial ring over a finite commutative ring"},"summary":{"zh-Hant":"係數取自有限交換環的普通多項式以有限支撐係數序列表示，但所有任意有限次多項式合成的載體仍是無限。","en":"Ordinary polynomials over a finite commutative coefficient ring have finite-support coefficient sequences, but the collection over all finite degrees is still infinite."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限交換環上的多項式環","AI候選","待形式化"]},{"id":"mko-inner-product-induced-norm","title":{"zh-Hant":"內積誘導範數","en":"Inner-product-induced norm"},"summary":{"zh-Hant":"同一實內積以向量自內積的非負平方根定義範數；三角不等式依Cauchy–Schwarz全稱證書。","en":"On the same real inner-product space, the norm is the nonnegative square root of the self-inner-product; the triangle inequality uses the universal Cauchy–Schwarz certificate."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["內積誘導範數","AI候選","待形式化"]},{"id":"mko-conditional-expectation","title":{"zh-Hant":"有限條件期望","en":"Finite conditional expectation"},"summary":{"zh-Hant":"有限機率空間中，相對分割生成sigma代數的條件期望在每個正質量區塊取加權平均並保持區塊積分。","en":"Relative to a partition-generated sigma-algebra on a finite probability space, conditional expectation takes the weighted mean on each positive-mass block and preserves block integrals."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限條件期望","AI候選","待形式化"]},{"id":"mko-finite-filtration","title":{"zh-Hant":"有限filtration","en":"Finite filtration"},"summary":{"zh-Hant":"有限時間filtration是在同一機率空間上隨時間資訊遞增的sigma代數序列。","en":"A finite-time filtration is a sequence of sigma-algebras on the same probability space whose information increases with time."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限filtration","AI候選","待形式化"]},{"id":"mko-iid-sequence","title":{"zh-Hant":"獨立同分布序列","en":"Independent identically distributed sequence"},"summary":{"zh-Hant":"IID序列同時要求每個變數有相同分布，且每個有限子族滿足mutual independence。","en":"An IID sequence requires every variable to have the same distribution and every finite subfamily to be mutually independent."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["獨立同分布序列","AI候選","待形式化"]},{"id":"mko-equivalence-class","title":{"zh-Hant":"等價類","en":"Equivalence class"},"summary":{"zh-Hant":"固定完整等價關係下，元素a的等價類收集全部與a相關的載體元素。","en":"For a fixed complete equivalence relation, the class of an element a consists of every carrier element related to a."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["等價類","AI候選","待形式化"]},{"id":"mko-finite-field","title":{"zh-Hant":"有限域","en":"Finite field"},"summary":{"zh-Hant":"有限交換含一環中每個非零元素皆有乘法逆元時構成有限域。","en":"A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限域","AI候選","待形式化"]},{"id":"mko-real-hilbert-space","title":{"zh-Hant":"實Hilbert空間","en":"Real Hilbert space"},"summary":{"zh-Hant":"在同一內積誘導範數與度量下完備的實內積空間為實Hilbert空間。","en":"A real inner-product space is Hilbert when it is complete in the norm and metric induced by that same inner product."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實Hilbert空間","AI候選","待形式化"]},{"id":"mko-martingale","title":{"zh-Hant":"有限Martingale","en":"Finite martingale"},"summary":{"zh-Hant":"相對同一filtration適應且可積的有限過程，若每一步下一值條件期望等於現在值a.s.，則為Martingale。","en":"An integrable finite process adapted to the same filtration is a martingale when the next value's conditional expectation equals the current value almost surely at every step."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限Martingale","AI候選","待形式化"]},{"id":"mko-quotient-set","title":{"zh-Hant":"商集","en":"Quotient set"},"summary":{"zh-Hant":"完整等價關係的全部互異等價類組成商集，並分割原載體。","en":"The distinct equivalence classes of a complete equivalence relation form its quotient set and partition the original carrier."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["商集","AI候選","待形式化"]},{"id":"mko-cardinality-comparison","title":{"zh-Hant":"集合基數比較","en":"Cardinality comparison of sets"},"summary":{"zh-Hant":"明示source與target間的單射、滿射、雙射分別提供有限集合基數比較證書。","en":"An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate."},"type":"concept","domain":"foundations_logic","formalization":"not_formalized","tags":["集合基數比較","AI候選","待形式化"]},{"id":"mko-cumulative-distribution-function","title":{"zh-Hant":"累積分布函數","en":"Cumulative distribution function"},"summary":{"zh-Hant":"實值隨機變數的CDF以F(t)=P(X小於等於t)定義，為單調右連續且有限離散跳躍等於點質量。","en":"For a real-valued random variable, F(t)=P(X≤t) is nondecreasing and right-continuous; in a finite discrete law, its jump is the point mass."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["累積分布函數","AI候選","待形式化"]},{"id":"mko-convergence-in-probability","title":{"zh-Hant":"依機率收斂","en":"Convergence in probability"},"summary":{"zh-Hant":"X_n依機率收斂到X，若對每個epsilon>0，偏差超過epsilon的機率趨近0。","en":"X_n converges in probability to X when, for every epsilon>0, the probability of deviation greater than epsilon tends to zero."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["依機率收斂","AI候選","待形式化"]},{"id":"mko-at-most-countable-set","title":{"zh-Hant":"至多可數集","en":"At-most-countable set"},"summary":{"zh-Hant":"集合若有限或存在到自然數的明示單射，則為至多可數；不等同可數無限。","en":"A set is at most countable if it is finite or has an explicit injection into the natural numbers; this is not the same as countably infinite."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["至多可數集","AI候選","待形式化"]},{"id":"mko-bipartite-graph","title":{"zh-Hant":"二分圖","en":"Bipartite graph"},"summary":{"zh-Hant":"有限無向簡單圖若全體頂點可分為兩個互斥部分且每條邊跨部，則為二分圖。","en":"A finite undirected simple graph is bipartite when all vertices split into two disjoint parts and every edge crosses between the parts."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["二分圖","AI候選","待形式化"]},{"id":"mko-convergence-in-distribution","title":{"zh-Hant":"依分布收斂","en":"Convergence in distribution"},"summary":{"zh-Hant":"極限CDF每個連續點上F_n逐點收斂到F時，稱依分布收斂。","en":"Convergence in distribution means that F_n tends pointwise to F at every continuity point of the limiting CDF."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["依分布收斂","AI候選","待形式化"]},{"id":"mko-finite-quotient-ring","title":{"zh-Hant":"有限環的商環","en":"Quotient ring of a finite ring"},"summary":{"zh-Hant":"有限環依proper雙邊理想的加法cosets形成商環，運算須與代表元選擇無關。","en":"Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限環的商環","AI候選","待形式化"]},{"id":"mko-graph-coloring","title":{"zh-Hant":"圖著色","en":"Graph coloring"},"summary":{"zh-Hant":"proper k-coloring使用至多k種顏色並使每條邊兩端顏色不同。","en":"A proper k-coloring uses at most k colors and gives different colors to the endpoints of every edge."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["圖著色","AI候選","待形式化"]},{"id":"mko-vector-subspace","title":{"zh-Hant":"實向量子空間","en":"Real vector subspace"},"summary":{"zh-Hant":"實向量空間的非空子集若含零向量並對向量加法與每個實純量乘法封閉，則為向量子空間。","en":"A nonempty subset of a real vector space is a subspace when it contains zero and is closed under vector addition and multiplication by every real scalar."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["實向量子空間","AI候選","待形式化"]},{"id":"mko-closure-and-interior","title":{"zh-Hant":"拓撲閉包與內部","en":"Topological closure and interior"},"summary":{"zh-Hant":"集合閉包是所有閉超集的交，內部是所有開子集的聯；兩者相對於指定拓撲。","en":"The closure of a set is the intersection of all closed supersets, and its interior is the union of all open subsets, relative to the specified topology."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲閉包與內部","AI候選","待形式化"]},{"id":"mko-open-cover-and-subcover","title":{"zh-Hant":"開覆蓋與子覆蓋","en":"Open cover and subcover"},"summary":{"zh-Hant":"開覆蓋由指定拓撲的開集組成且聯集包含目標集合；子覆蓋須取自原覆蓋並仍覆蓋同一目標。","en":"An open cover consists of open sets in the specified topology whose union contains the target; a subcover selects from that original cover and still covers the same target."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["開覆蓋與子覆蓋","AI候選","待形式化"]},{"id":"mko-almost-sure-convergence","title":{"zh-Hant":"幾乎必然收斂","en":"Almost-sure convergence"},"summary":{"zh-Hant":"同一機率空間上的隨機變數序列若在零機率例外集外逐點收斂，則幾乎必然收斂。","en":"Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["幾乎必然收斂","AI候選","待形式化"]},{"id":"mko-graph-matching","title":{"zh-Hant":"圖匹配","en":"Graph matching"},"summary":{"zh-Hant":"有限圖的matching是彼此不共享端點的邊子集；空matching合法。","en":"A matching in a finite graph is an edge subset whose edges share no endpoints; the empty matching is allowed."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["圖匹配","AI候選","待形式化"]},{"id":"mko-euler-totient-function","title":{"zh-Hant":"Euler φ函數","en":"Euler totient function"},"summary":{"zh-Hant":"Euler φ(n)計數一個完整剩餘代表系中與n互質的代表；約定φ(1)=1。","en":"Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["Euler φ函數","AI候選","待形式化"]},{"id":"mko-finite-automaton","title":{"zh-Hant":"確定有限自動機","en":"Deterministic finite automaton"},"summary":{"zh-Hant":"DFA由有限狀態集、有限字母表、總轉移函數、唯一初始狀態與接受狀態子集組成。","en":"A DFA consists of finite states, a finite alphabet, a total transition function, one initial state and a subset of accepting states."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["確定有限自動機","AI候選","待形式化"]},{"id":"mko-vector-space-dimension","title":{"zh-Hant":"有限維向量空間的維度","en":"Dimension of a finite-dimensional vector space"},"summary":{"zh-Hant":"有限維向量空間的維度是任一有限基底的元素數；基底不同但基數不變。","en":"The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限維向量空間的維度","AI候選","待形式化"]},{"id":"mko-linear-map-kernel","title":{"zh-Hant":"線性映射的核","en":"Kernel of a linear map"},"summary":{"zh-Hant":"線性映射T的核是全部滿足T(x)=0的來源向量，並形成來源子空間。","en":"The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["線性映射的核","AI候選","待形式化"]},{"id":"mko-linear-map-image","title":{"zh-Hant":"線性映射的像","en":"Image of a linear map"},"summary":{"zh-Hant":"線性映射的像是所有輸出T(x)的集合；矩陣映射的像等於column span。","en":"The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["線性映射的像","AI候選","待形式化"]},{"id":"mko-topological-boundary","title":{"zh-Hant":"拓撲邊界","en":"Topological boundary"},"summary":{"zh-Hant":"集合A的拓撲邊界為closure(A)減去interior(A)，亦等於closure(A)交closure(complement A)。","en":"The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲邊界","AI候選","待形式化"]},{"id":"mko-compact-topological-space","title":{"zh-Hant":"緊緻拓撲空間","en":"Compact topological space"},"summary":{"zh-Hant":"拓撲空間緊緻，若每個開覆蓋都有有限子覆蓋；每個有限拓撲空間皆緊緻。","en":"A space is compact when every open cover has a finite subcover; every finite topological space is compact."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["緊緻拓撲空間","AI候選","待形式化"]},{"id":"mko-bayes-theorem","title":{"zh-Hant":"貝氏定理","en":"Bayes theorem"},"summary":{"zh-Hant":"正分母條件下，Bayes定理以共同intersection與priors/likelihoods精確反轉條件機率。","en":"With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["貝氏定理","AI候選","待形式化"]},{"id":"mko-regular-language","title":{"zh-Hant":"正規語言","en":"Regular language"},"summary":{"zh-Hant":"正規語言是存在一個完整DFA可辨識其全部字串的語言。","en":"A language is regular when a complete DFA recognizes all and exactly its words."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["正規語言","AI候選","待形式化"]},{"id":"mko-matrix-rank","title":{"zh-Hant":"矩陣秩","en":"Matrix rank"},"summary":{"zh-Hant":"矩陣秩是row/column space維度，可由精確rank-preserving消去的pivot數計算。","en":"Matrix rank is the dimension of the row or column space and can be computed as the number of pivots after exact rank-preserving elimination."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["矩陣秩","AI候選","待形式化"]},{"id":"mko-pointwise-convergence-of-functions","title":{"zh-Hant":"函數列逐點收斂","en":"Pointwise convergence of functions"},"summary":{"zh-Hant":"同一domain上的函數列逐點收斂，若每個固定x的值序列收斂至f(x)，且N可依x與epsilon改變。","en":"A function sequence on one domain converges pointwise when its value sequence converges at each fixed x; the witness N may depend on x and epsilon."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["函數列逐點收斂","AI候選","待形式化"]},{"id":"mko-infinite-series","title":{"zh-Hant":"無窮級數","en":"Infinite series"},"summary":{"zh-Hant":"無窮級數以partial-sum sequence的極限定義；項趨零必要但不充分。","en":"An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["無窮級數","AI候選","待形式化"]},{"id":"mko-parametric-statistical-model","title":{"zh-Hant":"參數統計模型","en":"Parametric statistical model"},"summary":{"zh-Hant":"參數統計模型為每個theta指定同一有限可測sample space上的完整機率law P_theta。","en":"A parametric statistical model assigns each parameter theta a complete probability law P_theta on the same finite measurable sample space."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["參數統計模型","AI候選","待形式化"]},{"id":"mko-connected-topological-space","title":{"zh-Hant":"連通拓撲空間","en":"Connected topological space"},"summary":{"zh-Hant":"拓撲空間連通，若不存在由兩個不交非空開集組成的separation；等價於無非平凡clopen subset。","en":"A space is connected when it has no separation into two disjoint nonempty open sets, equivalently no nontrivial clopen subset."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["連通拓撲空間","AI候選","待形式化"]},{"id":"mko-maximum-matching","title":{"zh-Hant":"最大匹配","en":"Maximum matching"},"summary":{"zh-Hant":"最大匹配是在完整有限圖全部合法matching中邊數最大者；可有多個且不必perfect。","en":"A maximum matching has the greatest edge count among all valid matchings of the complete finite graph; it may be nonunique and need not be perfect."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["最大匹配","AI候選","待形式化"]},{"id":"mko-modular-inverse","title":{"zh-Hant":"模反元素","en":"Modular inverse"},"summary":{"zh-Hant":"n≥2時，a模n有乘法反元素恰當gcd(a,n)=1；反元素在模n意義唯一。","en":"For n≥2, a has a multiplicative inverse modulo n exactly when gcd(a,n)=1; the inverse is unique as a residue class modulo n."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["模反元素","AI候選","待形式化"]},{"id":"mko-rank-nullity-theorem","title":{"zh-Hant":"秩－零度定理","en":"Rank-nullity theorem"},"summary":{"zh-Hant":"有限維線性映射T:V→W滿足dim V=dim ker T+rank T。","en":"A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T)."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["秩－零度定理","AI候選","待形式化"]},{"id":"mko-uniform-convergence-of-functions","title":{"zh-Hant":"函數列一致收斂","en":"Uniform convergence of functions"},"summary":{"zh-Hant":"一致收斂要求同一N(epsilon)同時控制domain全部x。","en":"Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["函數列一致收斂","AI候選","待形式化"]},{"id":"mko-absolute-convergence-of-series","title":{"zh-Hant":"級數絕對收斂","en":"Absolute convergence of series"},"summary":{"zh-Hant":"sum |a_n|收斂時sum a_n絕對收斂；絕對收斂推出收斂，反向不一般成立。","en":"A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["級數絕對收斂","AI候選","待形式化"]},{"id":"mko-statistical-estimator","title":{"zh-Hant":"統計估計量","en":"Statistical estimator"},"summary":{"zh-Hant":"估計量是固定且不依未知theta的可測total statistic；觀察後數值是estimate。","en":"An estimator is a fixed measurable total statistic independent of the unknown parameter; its observed numerical value is an estimate."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["統計估計量","AI候選","待形式化"]},{"id":"mko-chinese-remainder-theorem","title":{"zh-Hant":"中國剩餘定理","en":"Chinese remainder theorem"},"summary":{"zh-Hant":"正兩兩互質模數的同餘系統在模乘積意義下存在唯一解。","en":"For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["中國剩餘定理","AI候選","待形式化"]},{"id":"mko-spanning-tree","title":{"zh-Hant":"生成樹","en":"Spanning tree"},"summary":{"zh-Hant":"生成樹是包含父圖全部頂點的連通無圈子圖。","en":"A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["生成樹","AI候選","待形式化"]},{"id":"mko-lp-convergence","title":{"zh-Hant":"Lp收斂","en":"Lp convergence"},"summary":{"zh-Hant":"共同機率空間上固定整數p≥1時，E|X_n−X|^p→0稱Lp收斂。","en":"On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["Lp收斂","AI候選","待形式化"]},{"id":"mko-direct-sum","title":{"zh-Hant":"子空間直和","en":"Direct sum of subspaces"},"summary":{"zh-Hant":"二個子空間之和若每個向量有唯一分解便是內直和；要宣稱覆蓋母空間還須另證生成。","en":"The sum of two subspaces is an internal direct sum when every vector in that sum has a unique decomposition; covering an ambient space is a separate spanning obligation."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["子空間直和","AI候選","待形式化"]},{"id":"mko-dual-vector-space","title":{"zh-Hant":"對偶向量空間","en":"Dual vector space"},"summary":{"zh-Hant":"實向量空間的對偶空間由所有實值線性泛函組成；座標covector表示依賴基底。","en":"The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["對偶向量空間","AI候選","待形式化"]},{"id":"mko-euclidean-algorithm","title":{"zh-Hant":"Euclidean演算法","en":"Euclidean algorithm"},"summary":{"zh-Hant":"Euclidean演算法以精確帶餘除法、gcd invariant與嚴格餘數下降求最大公因數。","en":"The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders."},"type":"algorithm","domain":"number_theory","formalization":"not_formalized","tags":["Euclidean演算法","AI候選","待形式化"]},{"id":"mko-eulerian-trail","title":{"zh-Hant":"Euler trail與circuit","en":"Euler trails and circuits"},"summary":{"zh-Hant":"Euler trail恰用每條邊一次；非零邊支撐連通且奇度數為0或2時分別有circuit或open trail。","en":"An Euler trail uses each edge once; on connected nonzero-edge support, zero odd-degree vertices give a circuit and two give an open Euler trail."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["Euler trail與circuit","AI候選","待形式化"]},{"id":"mko-homeomorphism","title":{"zh-Hant":"同胚","en":"Homeomorphism"},"summary":{"zh-Hant":"同胚是雙射、連續且逆映射亦連續的映射；一般continuous bijection仍可能不是同胚。","en":"A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["同胚","AI候選","待形式化"]},{"id":"mko-sampling-distribution","title":{"zh-Hant":"抽樣分布","en":"Sampling distribution"},"summary":{"zh-Hant":"固定theta時，估計量把樣本分布pushforward成統計量的抽樣分布；全部theta形成一族laws。","en":"At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["抽樣分布","AI候選","待形式化"]},{"id":"mko-weak-law-large-numbers","title":{"zh-Hant":"弱大數律","en":"Weak law of large numbers"},"summary":{"zh-Hant":"IID且具有有限variance時，Chebyshev界給樣本平均依機率收斂到共同mean。","en":"For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["弱大數律","AI候選","待形式化"]},{"id":"mko-chromatic-number","title":{"zh-Hant":"色數","en":"Chromatic number"},"summary":{"zh-Hant":"有限簡單圖的色數是proper頂點著色所需顏色數的最小值，需同時具備上界與下界證書。","en":"The chromatic number of a finite simple graph is the minimum number of colors in a proper vertex coloring; equality needs both upper and lower certificates."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["色數","AI候選","待形式化"]},{"id":"mko-clique-and-independent-set","title":{"zh-Hant":"Clique與獨立集","en":"Cliques and independent sets"},"summary":{"zh-Hant":"Clique內每對不同頂點相鄰，獨立集內每對不同頂點不相鄰；最大與極大不可混同。","en":"Every pair of distinct vertices of a clique is adjacent; every pair of an independent set is nonadjacent. Inclusion-maximal and maximum-cardinality are different."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["Clique與獨立集","AI候選","待形式化"]},{"id":"mko-estimator-bias","title":{"zh-Hant":"估計量偏誤","en":"Estimator bias"},"summary":{"zh-Hant":"估計量在theta的偏誤是其theta抽樣分布期望減去目標參數；無偏要求對所有theta成立。","en":"Bias at theta is the estimator expectation under its theta sampling law minus the target parameter; unbiasedness requires zero bias at every theta."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["估計量偏誤","AI候選","待形式化"]},{"id":"mko-estimator-variance","title":{"zh-Hant":"估計量變異數","en":"Estimator variance"},"summary":{"zh-Hant":"估計量變異數逐theta衡量sampling law相對自身期望的二次離差，不等同偏誤或MSE。","en":"At each theta, estimator variance is the second central moment of its sampling law around its own mean; it is not bias or MSE by itself."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["估計量變異數","AI候選","待形式化"]},{"id":"mko-fermat-little-theorem","title":{"zh-Hant":"Fermat小定理","en":"Fermat's little theorem"},"summary":{"zh-Hant":"p為質數時a^p同餘a；若p不整除a則可化為a^(p-1)同餘1，逆命題不成立。","en":"For a prime p, a^p is congruent to a modulo p; if p does not divide a, a^(p−1) is congruent to1 modulo p. The converse fails."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["Fermat小定理","AI候選","待形式化"]},{"id":"mko-linear-diophantine-equation","title":{"zh-Hant":"線性Diophantine方程","en":"Linear Diophantine equation"},"summary":{"zh-Hant":"非退化整數方程ax+by=c有解恰當gcd(a,b)整除c，全部解由一個整數參數給出。","en":"A nondegenerate integer equation a*x+b*y=c has a solution exactly when gcd(a,b) divides c; all solutions have one integer parameter."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["線性Diophantine方程","AI候選","待形式化"]},{"id":"mko-quotient-vector-space","title":{"zh-Hant":"商向量空間","en":"Quotient vector space"},"summary":{"zh-Hant":"V依子空間W的cosets形成商向量空間，運算須與代表元選擇無關。","en":"Cosets of a subspace W form the quotient vector space V/W, whose operations must be independent of representative choices."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["商向量空間","AI候選","待形式化"]},{"id":"mko-finite-group-action","title":{"zh-Hant":"有限群作用","en":"Finite group action"},"summary":{"zh-Hant":"有限群在有限集合上的作用是滿足單位與乘法相容律的total action table。","en":"An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群作用","AI候選","待形式化"]},{"id":"mko-supremum-and-infimum","title":{"zh-Hant":"上確界與下確界","en":"Supremum and infimum"},"summary":{"zh-Hant":"非空且有界實數集合的supremum是最小上界，infimum是最大下界，兩者未必屬於集合。","en":"For a nonempty appropriately bounded real set, the supremum is its least upper bound and the infimum its greatest lower bound; either need not belong to the set."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["上確界與下確界","AI候選","待形式化"]},{"id":"mko-topological-basis","title":{"zh-Hant":"拓撲基底","en":"Topological basis"},"summary":{"zh-Hant":"拓撲基底以覆蓋carrier與逐點局部交細化兩條公理生成拓撲。","en":"A topological basis generates a topology through carrier coverage and pointwise local intersection refinement."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲基底","AI候選","待形式化"]},{"id":"mko-markov-inequality","title":{"zh-Hant":"Markov不等式","en":"Markov inequality"},"summary":{"zh-Hant":"非負且有限期望的隨機變數在正閾值上的尾機率不超過E[X]/a。","en":"For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["Markov不等式","AI候選","待形式化"]},{"id":"mko-pigeonhole-principle","title":{"zh-Hant":"鴿籠原理","en":"Pigeonhole principle"},"summary":{"zh-Hant":"有限total map的domain比codomain大時必有兩個不同元素共享像。","en":"A total map from a larger finite domain into a smaller finite codomain sends two distinct source elements to the same value."},"type":"theorem","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["鴿籠原理","AI候選","待形式化"]},{"id":"mko-euler-theorem-number-theory","title":{"zh-Hant":"Euler定理","en":"Euler's theorem in number theory"},"summary":{"zh-Hant":"n>=2且gcd(a,n)=1時，a的phi(n)次方模n同餘於1。","en":"For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["Euler定理","AI候選","待形式化"]},{"id":"mko-subsequence","title":{"zh-Hant":"子數列","en":"Subsequence"},"summary":{"zh-Hant":"子數列由嚴格遞增的自然數索引映射選取原數列項並保留次序。","en":"A subsequence selects terms by a strictly increasing natural-index map and preserves their original order."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["子數列","AI候選","待形式化"]},{"id":"mko-finite-group-orbit","title":{"zh-Hant":"有限群作用的軌道","en":"Orbit of a finite group action"},"summary":{"zh-Hant":"有限群作用下，一點被全部群元素送到的像集合形成軌道，全部軌道分割carrier。","en":"The images of one point under all group elements form its orbit; all orbits partition the point carrier."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群作用的軌道","AI候選","待形式化"]},{"id":"mko-monotone-sequence-convergence-theorem","title":{"zh-Hant":"單調有界數列收斂定理","en":"Monotone bounded sequence convergence theorem"},"summary":{"zh-Hant":"非遞減且有上界的實數列收斂至其項集合supremum；非遞增版本對偶收斂至infimum。","en":"A nondecreasing real sequence bounded above converges to the supremum of its terms; the nonincreasing bounded-below version converges to the infimum."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["單調有界數列收斂定理","AI候選","待形式化"]},{"id":"mko-subspace-topology","title":{"zh-Hant":"子空間拓撲","en":"Subspace topology"},"summary":{"zh-Hant":"Y⊆X的相對拓撲由全部母空間open sets與Y的交集組成。","en":"For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["子空間拓撲","AI候選","待形式化"]},{"id":"mko-chebyshev-inequality","title":{"zh-Hant":"Chebyshev不等式","en":"Chebyshev inequality"},"summary":{"zh-Hant":"有限均值與變異數的隨機變數偏離均值至少epsilon的機率不超過variance/epsilon²。","en":"For a variable with finite mean and variance, the probability of absolute deviation at least epsilon is at most variance/epsilon^2."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["Chebyshev不等式","AI候選","待形式化"]},{"id":"mko-inclusion-exclusion-principle","title":{"zh-Hant":"容斥原理","en":"Inclusion-exclusion principle"},"summary":{"zh-Hant":"有限indexed set family的union cardinality是全部非空index subsets交集大小的交替和。","en":"The union cardinality of a finite indexed set family is the alternating sum of intersection sizes over every nonempty index subset."},"type":"theorem","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["容斥原理","AI候選","待形式化"]},{"id":"mko-unique-prime-factorization","title":{"zh-Hant":"整數唯一質因數分解","en":"Unique prime factorization of integers"},"summary":{"zh-Hant":"每個n>1有prime factorization且prime multiset唯一，只允許因數次序差異。","en":"Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["整數唯一質因數分解","AI候選","待形式化"]},{"id":"mko-finite-directed-graph","title":{"zh-Hant":"有限有向圖","en":"Finite directed graph"},"summary":{"zh-Hant":"有限loop-free simple digraph由有限vertices與不含loop、無平行同向重邊的ordered-pair edge set組成。","en":"Under the loop-free simple convention, a finite digraph has a finite vertex carrier and an ordered-pair edge set without loops or duplicate same-direction edges."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["有限有向圖","AI候選","待形式化"]},{"id":"mko-finite-group-stabilizer","title":{"zh-Hant":"有限群作用的穩定子群","en":"Stabilizer of a finite group action"},"summary":{"zh-Hant":"指定點的stabilizer是有限群作用中固定該點的全部群元素，並形成子群。","en":"The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群作用的穩定子群","AI候選","待形式化"]},{"id":"mko-bolzano-weierstrass-theorem-real","title":{"zh-Hant":"實數列Bolzano－Weierstrass定理","en":"Bolzano-Weierstrass theorem for real sequences"},"summary":{"zh-Hant":"每個整體有界的實數列都有一個以嚴格遞增無限索引取得的收斂子數列。","en":"Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["實數列Bolzano－Weierstrass定理","AI候選","待形式化"]},{"id":"mko-compact-subset","title":{"zh-Hant":"緊緻子集","en":"Compact subset"},"summary":{"zh-Hant":"K⊆X在subspace topology下compact，等價地每個relative-open cover有有限subcover。","en":"A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["緊緻子集","AI候選","待形式化"]},{"id":"mko-law-of-total-expectation","title":{"zh-Hant":"全期望公式","en":"Law of total expectation"},"summary":{"zh-Hant":"同一有限機率空間上，完整partition各cell的conditional mean按cell probability加權後等於E[X]。","en":"On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X]."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["全期望公式","AI候選","待形式化"]},{"id":"mko-binomial-theorem","title":{"zh-Hant":"二項式定理","en":"Binomial theorem"},"summary":{"zh-Hant":"交換乘法環境中，natural exponent n的二項式冪以binomial coefficients完整展開。","en":"With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients."},"type":"theorem","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["二項式定理","AI候選","待形式化"]},{"id":"mko-mobius-function","title":{"zh-Hant":"Möbius函數","en":"Mobius function"},"summary":{"zh-Hant":"Möbius函數在positive integers上由完整prime multiplicities決定：mu(1)=1、含平方質因數為0、squarefree取parity sign。","en":"On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["Möbius函數","AI候選","待形式化"]},{"id":"mko-context-free-grammar","title":{"zh-Hant":"上下文無關文法","en":"Context-free grammar"},"summary":{"zh-Hant":"CFG由有限且不交的nonterminals、terminals、start與單一nonterminal左側的finite productions生成語言。","en":"A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["上下文無關文法","AI候選","待形式化"]},{"id":"mko-orbit-stabilizer-theorem","title":{"zh-Hant":"軌道－穩定子定理","en":"Orbit-stabilizer theorem"},"summary":{"zh-Hant":"完整有限群作用中，同一點的orbit大小乘以stabilizer大小等於群階。","en":"For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["軌道－穩定子定理","AI候選","待形式化"]},{"id":"mko-mean-value-theorem","title":{"zh-Hant":"微分中值定理","en":"Mean value theorem"},"summary":{"zh-Hant":"若a<b，f在[a,b]連續且在(a,b)可微，則某個內點c的導數等於端點割線斜率。","en":"If a<b, f is continuous on [a,b] and differentiable on (a,b), some interior c has derivative equal to the endpoint secant slope."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["微分中值定理","AI候選","待形式化"]},{"id":"mko-continuous-image-of-compact","title":{"zh-Hant":"緊緻集的連續像","en":"Continuous image of a compact set"},"summary":{"zh-Hant":"若K在X中compact且f:X→Y連續，則image f(K)在Y中compact。","en":"If K is compact in X and f:X→Y is continuous, then f(K) is compact in Y."},"type":"theorem","domain":"topology","formalization":"not_formalized","tags":["緊緻集的連續像","AI候選","待形式化"]},{"id":"mko-law-of-total-variance","title":{"zh-Hant":"全變異數公式","en":"Law of total variance"},"summary":{"zh-Hant":"同一有限law與完整partition生成的G下，總變異數等於cell內變異數期望加cell均值變異數。","en":"For the same finite law and partition-generated G, total variance is the expected within-cell variance plus the variance of the cell means."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["全變異數公式","AI候選","待形式化"]},{"id":"mko-recurrence-relation","title":{"zh-Hant":"遞迴關係","en":"Recurrence relation"},"summary":{"zh-Hant":"order-r recurrence以明示initial values與total single-valued state update逐項決定sequence。","en":"An order-r recurrence uses explicit initial values and a total single-valued state update to determine successive sequence terms."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["遞迴關係","AI候選","待形式化"]},{"id":"mko-mobius-inversion-finite-divisors","title":{"zh-Hant":"有限因數格上的Möbius反演","en":"Mobius inversion on finite divisor sets"},"summary":{"zh-Hant":"若F(n)是所有正因數d上的f(d)總和，則以Möbius權重可精確反演f(n)。","en":"If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n)."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["有限因數格上的Möbius反演","AI候選","待形式化"]},{"id":"mko-pushdown-automaton","title":{"zh-Hant":"下推自動機","en":"Pushdown automaton"},"summary":{"zh-Hant":"nondeterministic PDA以有限控制、input與stack transitions運算，本輪固定讀完input且到final state接受。","en":"A nondeterministic PDA uses finite control, input and stack transitions; this slice accepts only after reading the whole input in a final state."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["下推自動機","AI候選","待形式化"]},{"id":"mko-lagrange-theorem-finite-groups","title":{"zh-Hant":"有限群的Lagrange定理","en":"Lagrange theorem for finite groups"},"summary":{"zh-Hant":"有限群G的子群H以等大左陪集分割G，因此|G|=[G:H]|H|且|H|整除|G|。","en":"Equal-size left cosets of a subgroup H partition the finite group G, so |G|=[G:H]*|H| and |H| divides |G|."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["有限群的Lagrange定理","AI候選","待形式化"]},{"id":"mko-higher-order-derivative","title":{"zh-Hant":"高階導數","en":"Higher-order derivative"},"summary":{"zh-Hant":"以f^(0)=f及f^(n+1)=(f^(n))'遞迴定義每個存在的有限階導數。","en":"Define each existing finite derivative recursively by f^(0)=f and f^(n+1)=(f^(n))'."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["高階導數","AI候選","待形式化"]},{"id":"mko-hausdorff-space","title":{"zh-Hant":"Hausdorff空間","en":"Hausdorff space"},"summary":{"zh-Hant":"拓撲空間X若每對不同點都有不交open neighborhoods即為Hausdorff。","en":"A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["Hausdorff空間","AI候選","待形式化"]},{"id":"mko-continuous-real-distribution-function","title":{"zh-Hant":"連續實值分布函數","en":"Continuous real distribution function"},"summary":{"zh-Hant":"本節點以CDF本身連續定義continuous distribution；CDF仍須nondecreasing、right-continuous且兩端極限0與1。","en":"Here a continuous distribution means that its CDF is continuous; the CDF must still be nondecreasing, right-continuous and have endpoint limits0 and1."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["連續實值分布函數","AI候選","待形式化"]},{"id":"mko-catalan-numbers","title":{"zh-Hant":"Catalan數","en":"Catalan numbers"},"summary":{"zh-Hant":"Catalan數採C_0=1，並以convolution recurrence或兩個exact closed forms定義同一zero-based sequence。","en":"With C_0=1, the convolution recurrence and two exact closed forms define the same zero-based Catalan sequence."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["Catalan數","AI候選","待形式化"]},{"id":"mko-quadratic-residue","title":{"zh-Hant":"二次剩餘","en":"Quadratic residue"},"summary":{"zh-Hant":"本節點採odd prime p上的nonzero convention：a mod p非零且等於某個nonzero square時為quadratic residue。","en":"Under this odd-prime nonzero convention, a class is a quadratic residue when its normalized nonzero value is a square of a nonzero class modulo p."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["二次剩餘","AI候選","待形式化"]},{"id":"mko-nondeterministic-finite-automaton","title":{"zh-Hant":"非確定有限自動機","en":"Nondeterministic finite automaton"},"summary":{"zh-Hant":"epsilon-NFA以狀態、input symbol或epsilon對應可能後繼集合，存在一條full-input final path即接受。","en":"An epsilon-NFA relates a state and input symbol or epsilon to possible successor states; one full-input final path suffices for acceptance."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["非確定有限自動機","AI候選","待形式化"]},{"id":"mko-cauchy-theorem-finite-groups","title":{"zh-Hant":"有限群的Cauchy定理","en":"Cauchy theorem for finite groups"},"summary":{"zh-Hant":"若質數p整除有限群G的階，則G含有一個exact order為p的元素。","en":"If a prime p divides the order of a finite group G, G contains an element of exact order p."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["有限群的Cauchy定理","AI候選","待形式化"]},{"id":"mko-taylor-polynomial","title":{"zh-Hant":"Taylor多項式","en":"Taylor polynomial"},"summary":{"zh-Hant":"在中心a由f的零階到n階導數值依factorial縮放組成n階Taylor多項式。","en":"At center a, derivative values of orders0 through n, scaled by their factorials, form the order-n Taylor polynomial."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["Taylor多項式","AI候選","待形式化"]},{"id":"mko-compact-subset-closed-in-hausdorff","title":{"zh-Hant":"Hausdorff空間中的緊緻集為閉集","en":"A compact subset of a Hausdorff space is closed"},"summary":{"zh-Hant":"若K是Hausdorff空間X的緊緻子集，則K在X中為閉集。","en":"If K is compact in the Hausdorff space X, then K is closed in X."},"type":"theorem","domain":"topology","formalization":"not_formalized","tags":["Hausdorff空間中的緊緻集為閉集","AI候選","待形式化"]},{"id":"mko-improper-integral","title":{"zh-Hant":"瑕積分","en":"Improper integral"},"summary":{"zh-Hant":"無界區間或含奇點的積分以適當有限區間積分的one-sided limits定義，且每個必要分段須獨立收斂。","en":"Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["瑕積分","AI候選","待形式化"]},{"id":"mko-power-series-radius-of-convergence","title":{"zh-Hant":"冪級數收斂半徑","en":"Radius of convergence of a power series"},"summary":{"zh-Hant":"冪級數在中心c周圍具有內部絕對收斂、外部發散的半徑R，邊界點須逐案判定。","en":"Around its center c, a power series has absolute convergence inside radius R and divergence outside; endpoints require separate decisions."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["冪級數收斂半徑","AI候選","待形式化"]},{"id":"mko-legendre-symbol","title":{"zh-Hant":"Legendre符號","en":"Legendre symbol"},"summary":{"zh-Hant":"對奇質數p，Legendre符號以0、1、-1分別編碼zero class、nonzero quadratic residue與nonresidue。","en":"For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["Legendre符號","AI候選","待形式化"]},{"id":"mko-subset-construction-nfa-to-dfa","title":{"zh-Hant":"NFA到DFA的子集合構造","en":"Subset construction from an NFA to a DFA"},"summary":{"zh-Hant":"以epsilon-NFA狀態子集合為DFA狀態，start、transition與acceptance按epsilon closure定義即可保持語言。","en":"Use epsilon-NFA state subsets as DFA states, defining start, transitions and acceptance with epsilon closure to preserve the language."},"type":"theorem","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["NFA到DFA的子集合構造","AI候選","待形式化"]},{"id":"mko-finite-normal-subgroup","title":{"zh-Hant":"有限正規子群","en":"Finite normal subgroup"},"summary":{"zh-Hant":"有限群G的子群H若對每個g皆滿足gHg^{-1}=H，則H為正規子群；等價地每個左陪集等於相應右陪集。","en":"A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限正規子群","AI候選","待形式化"]},{"id":"mko-real-exponential-function","title":{"zh-Hant":"實指數函數","en":"Real exponential function"},"summary":{"zh-Hant":"實指數函數以對每個實數絕對收斂的冪級數E(x)=sum_{n≥0}x^n/n!定義，並滿足E(x+y)=E(x)E(y)與E(x)>0。","en":"Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實指數函數","AI候選","待形式化"]},{"id":"mko-analytic-ordinary-generating-function","title":{"zh-Hant":"解析普通生成函數","en":"Analytic ordinary generating function"},"summary":{"zh-Hant":"數列(a_n)在明示收斂域內以ordinary power series A(x)=sum_{n≥0}a_nx^n解析編碼。","en":"An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["解析普通生成函數","AI候選","待形式化"]},{"id":"mko-euler-criterion-quadratic-residue","title":{"zh-Hant":"Euler二次剩餘判準","en":"Euler criterion for quadratic residues"},"summary":{"zh-Hant":"若p為奇質數且p不整除a，則a^((p-1)/2)模p等於Legendre符號(a/p)的模p代表。","en":"For odd prime p and a not divisible by p, a^((p−1)/2) modulo p is the modulo-p representative of the Legendre symbol (a/p)."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["Euler二次剩餘判準","AI候選","待形式化"]},{"id":"mko-turing-machine-decision-model","title":{"zh-Hant":"Turing機判定模型","en":"Turing-machine decision model"},"summary":{"zh-Hant":"固定deterministic single-tape Turing機語義；decider必須對每個合法輸入停機並給正確accept/reject答案。","en":"Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["Turing機判定模型","AI候選","待形式化"]},{"id":"mko-gradient-vector","title":{"zh-Hant":"梯度向量","en":"Gradient vector"},"summary":{"zh-Hant":"在固定standard Euclidean coordinates中，可微實值函數的一階偏導依座標排列成梯度向量。","en":"In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["梯度向量","AI候選","待形式化"]},{"id":"mko-finite-flow-network","title":{"zh-Hant":"有限流網路","en":"Finite flow network"},"summary":{"zh-Hant":"有限有向網路中的非負流須逐邊不超過capacity，並在source與sink以外每個頂點滿足流量守恆。","en":"A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["有限流網路","AI候選","待形式化"]},{"id":"mko-finite-group-homomorphism-kernel","title":{"zh-Hant":"有限群同態的核","en":"Kernel of a finite group homomorphism"},"summary":{"zh-Hant":"有限群同態f:G→Q映到目標identity的完整原像是G的正規子群，且同態fibers恰為kernel的陪集。","en":"For a finite group homomorphism f:G→Q, the full preimage of the target identity is a normal subgroup of G, and nonempty fibers are its cosets."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群同態的核","AI候選","待形式化"]},{"id":"mko-finite-quotient-group","title":{"zh-Hant":"有限商群","en":"Finite quotient group"},"summary":{"zh-Hant":"有限群G依正規子群N的陪集形成商群，元素是cosets且乘法(gN)(hN)=(gh)N必須與代表元無關。","en":"Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限商群","AI候選","待形式化"]},{"id":"mko-finite-group-conjugacy-class","title":{"zh-Hant":"有限群共軛類","en":"Conjugacy class in a finite group"},"summary":{"zh-Hant":"有限群G以conjugation g·x=gxg^{-1}作用於自身，其orbits形成共軛類並分割G。","en":"A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群共軛類","AI候選","待形式化"]},{"id":"mko-standard-normal-distribution","title":{"zh-Hant":"標準常態分布","en":"Standard normal distribution"},"summary":{"zh-Hant":"標準常態Z以Gaussian density phi(x)=(2pi)^(-1/2)exp(-x^2/2)與其連續CDF Phi定義，並具有mean 0、variance 1。","en":"The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["標準常態分布","AI候選","待形式化"]},{"id":"mko-estimator-mean-squared-error","title":{"zh-Hant":"估計量均方誤差","en":"Mean squared error of an estimator"},"summary":{"zh-Hant":"scalar parameter theta下，估計量T的MSE_theta(T)=E_theta[(T-theta)^2]由完整sampling distribution逐theta定義。","en":"For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["估計量均方誤差","AI候選","待形式化"]},{"id":"mko-linear-program-standard-form","title":{"zh-Hant":"線性規劃標準形","en":"Standard form of a linear program"},"summary":{"zh-Hant":"本節點固定唯一primal convention：maximize c^T x，subject to Ax≤b與x≥0。","en":"This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0."},"type":"definition","domain":"optimization_operations_research","formalization":"not_formalized","tags":["線性規劃標準形","AI候選","待形式化"]},{"id":"mko-vertex-cover","title":{"zh-Hant":"頂點覆蓋","en":"Vertex cover"},"summary":{"zh-Hant":"有限simple undirected graph的頂點子集C若與每條edge至少一端相交，則C為vertex cover。","en":"A vertex subset C covers a finite simple undirected graph if every edge has at least one endpoint in C."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["頂點覆蓋","AI候選","待形式化"]},{"id":"mko-first-isomorphism-theorem-finite-groups","title":{"zh-Hant":"有限群第一同構定理","en":"First isomorphism theorem for finite groups"},"summary":{"zh-Hant":"有限群同態f:G→Q由kernel cosets誘導出G/ker f到im f的群同構，而非自動同構到整個codomain。","en":"A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain."},"type":"theorem","domain":"algebra","formalization":"not_formalized","tags":["有限群第一同構定理","AI候選","待形式化"]},{"id":"mko-finite-group-center","title":{"zh-Hant":"有限群中心","en":"Center of a finite group"},"summary":{"zh-Hant":"有限群G的中心Z(G)由與每個群元素交換的全部元素構成，且是G的正規子群。","en":"The center consists of all elements commuting with every group element and is a normal subgroup."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有限群中心","AI候選","待形式化"]},{"id":"mko-bias-variance-decomposition","title":{"zh-Hant":"偏誤－變異數分解","en":"Bias-variance decomposition"},"summary":{"zh-Hant":"逐參數theta，scalar estimator的MSE等於其variance加上bias平方。","en":"For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["偏誤－變異數分解","AI候選","待形式化"]},{"id":"mko-linear-program-duality","title":{"zh-Hant":"線性規劃對偶","en":"Linear-programming duality"},"summary":{"zh-Hant":"對固定primal max c^Tx、Ax≤b、x≥0，其dual為min b^Ty、A^Ty≥c、y≥0；feasible pair滿足弱對偶。","en":"For primal maximize c^T*x with A*x≤b,x≥0, the dual minimizes b^T*y with A^T*y≥c,y≥0; feasible pairs satisfy weak duality."},"type":"theorem","domain":"optimization_operations_research","formalization":"not_formalized","tags":["線性規劃對偶","AI候選","待形式化"]},{"id":"mko-max-flow-min-cut-theorem","title":{"zh-Hant":"最大流最小割定理","en":"Max-flow min-cut theorem"},"summary":{"zh-Hant":"有限容量網路的最大feasible flow value等於全部s–t cuts中的最小capacity。","en":"In a finite capacitated network the largest feasible flow value equals the least capacity among all source–sink cuts."},"type":"theorem","domain":"optimization_operations_research","formalization":"not_formalized","tags":["最大流最小割定理","AI候選","待形式化"]},{"id":"mko-minimum-vertex-cover","title":{"zh-Hant":"最小頂點覆蓋","en":"Minimum vertex cover"},"summary":{"zh-Hant":"有限simple graph中，cardinality在全部vertex covers裡最小的候選是minimum vertex cover。","en":"A minimum vertex cover has least cardinality among all vertex covers of the finite simple graph."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["最小頂點覆蓋","AI候選","待形式化"]},{"id":"mko-many-one-reduction","title":{"zh-Hant":"多對一歸約","en":"Many-one reduction"},"summary":{"zh-Hant":"A≤mB表示存在total computable f，使每個x皆滿足x∈A恰當f(x)∈B。","en":"A≤mB means a total computable map f satisfies x∈A exactly when f(x)∈B for every x."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["多對一歸約","AI候選","待形式化"]},{"id":"mko-contraction-mapping-theorem","title":{"zh-Hant":"壓縮映射定理","en":"Contraction mapping theorem"},"summary":{"zh-Hant":"非空完備度量空間上的self-map若以固定q<1壓縮所有距離，則有唯一fixed point且Picard orbit收斂。","en":"A self-map on a nonempty complete metric space with one global contraction constant q<1 has a unique fixed point and convergent Picard iterates."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["壓縮映射定理","AI候選","待形式化"]},{"id":"mko-intermediate-value-theorem","title":{"zh-Hant":"介值定理","en":"Intermediate value theorem"},"summary":{"zh-Hant":"連續實函數在閉區間端點值之間取到每個中間值。","en":"A continuous real function on a closed interval attains each value between its endpoint values."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["介值定理","AI候選","待形式化"]},{"id":"mko-taylor-remainder-theorem","title":{"zh-Hant":"Taylor餘項定理","en":"Taylor remainder theorem"},"summary":{"zh-Hant":"足夠高階可微函數的Taylor approximation error由Lagrange remainder與derivative bound控制。","en":"Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["Taylor餘項定理","AI候選","待形式化"]},{"id":"mko-finite-maximum-likelihood-estimator","title":{"zh-Hant":"有限模型最大似然估計量","en":"Maximum-likelihood estimator in a finite model"},"summary":{"zh-Hant":"有限參數模型中，MLE選取使觀察likelihood最大的參數，tie需固定且與theta無關。","en":"A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型最大似然估計量","AI候選","待形式化"]},{"id":"mko-finite-sufficient-statistic","title":{"zh-Hant":"有限模型充分統計量","en":"Sufficient statistic in a finite model"},"summary":{"zh-Hant":"finite model中，給定statistic value後sample conditional law不依theta時，T對theta充分。","en":"A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型充分統計量","AI候選","待形式化"]},{"id":"mko-topological-ordering-dag","title":{"zh-Hant":"有向無環圖的拓撲排序","en":"Topological ordering of a directed acyclic graph"},"summary":{"zh-Hant":"有限有向圖存在topological ordering恰當其無directed cycle。","en":"A finite directed graph has a topological ordering exactly when it has no directed cycle."},"type":"theorem","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["有向無環圖的拓撲排序","AI候選","待形式化"]},{"id":"mko-multiplicative-order-mod-n","title":{"zh-Hant":"模n乘法階","en":"Multiplicative order modulo n"},"summary":{"zh-Hant":"gcd(a,n)=1時，ord_n(a)是a^k≡1 mod n的最小正整數k。","en":"For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["模n乘法階","AI候選","待形式化"]},{"id":"mko-series-comparison-test","title":{"zh-Hant":"級數比較判別法","en":"Comparison tests for series"},"summary":{"zh-Hant":"非負項級數可由逐項或最終逐項上界與下界傳遞收斂或發散。","en":"Eventual termwise bounds transfer convergence from a nonnegative upper series or divergence from a nonnegative lower series."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["級數比較判別法","AI候選","待形式化"]},{"id":"mko-ratio-test-series","title":{"zh-Hant":"級數比值判別法","en":"Ratio test for series"},"summary":{"zh-Hant":"非零項絕對值相鄰比極限小於一時絕對收斂，大於一時發散，等於一時不判定。","en":"For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["級數比值判別法","AI候選","待形式化"]},{"id":"mko-uniform-limit-continuity-theorem","title":{"zh-Hant":"一致極限保持連續","en":"A uniform limit of continuous functions is continuous"},"summary":{"zh-Hant":"同一domain上的連續函數列若一致收斂至f，則極限函數f連續。","en":"Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["一致極限保持連續","AI候選","待形式化"]},{"id":"mko-finite-product-topology","title":{"zh-Hant":"有限乘積拓撲","en":"Finite product topology"},"summary":{"zh-Hant":"有限多個拓撲空間的Cartesian product由open rectangles生成乘積拓撲。","en":"Open rectangles generate the topology of a finite Cartesian product of topological spaces."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["有限乘積拓撲","AI候選","待形式化"]},{"id":"mko-quotient-topology","title":{"zh-Hant":"商拓撲","en":"Quotient topology"},"summary":{"zh-Hant":"滿射q:X→Y使U⊆Y恰在q逆像開時為開，從而賦予Y商拓撲。","en":"For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["商拓撲","AI候選","待形式化"]},{"id":"mko-path-connected-space","title":{"zh-Hant":"道路連通空間","en":"Path-connected space"},"summary":{"zh-Hant":"空間中任意兩點若可由單位區間的連續映射連接，則空間道路連通。","en":"A space is path connected when every pair of its points is joined by a continuous map from the unit interval."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["道路連通空間","AI候選","待形式化"]},{"id":"mko-topological-connected-component","title":{"zh-Hant":"拓撲連通分支","en":"Connected component of a topological space"},"summary":{"zh-Hant":"指定點的連通分支是包含該點的極大連通子集；所有分支分割空間且各自閉。","en":"The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲連通分支","AI候選","待形式化"]},{"id":"mko-central-limit-theorem-finite-variance-iid","title":{"zh-Hant":"有限變異數IID中央極限定理","en":"Central limit theorem for finite-variance IID sequences"},"summary":{"zh-Hant":"共同law且0<σ²<∞的無限IID序列，其正確中心化與sqrt(n)標準化和依分布收斂至標準常態。","en":"An infinite IID sequence with common finite positive variance has centered sums scaled by sigma*sqrt(n) converging in distribution to the standard normal law."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["有限變異數IID中央極限定理","AI候選","待形式化"]},{"id":"mko-strong-law-large-numbers-finite-valued-iid","title":{"zh-Hant":"有限值IID強大數律","en":"Strong law of large numbers for finite-valued IID sequences"},"summary":{"zh-Hant":"有限值無限IID序列的樣本平均幾乎必然收斂至共同期望。","en":"The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation."},"type":"theorem","domain":"probability_statistics","formalization":"not_formalized","tags":["有限值IID強大數律","AI候選","待形式化"]},{"id":"mko-finite-confidence-interval-coverage","title":{"zh-Hant":"有限模型信賴區間覆蓋率","en":"Confidence-interval coverage in a finite model"},"summary":{"zh-Hant":"有限模型中預先固定的interval-valued procedure逐參數覆蓋真值的機率，定義其frequentist coverage。","en":"Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限模型信賴區間覆蓋率","AI候選","待形式化"]},{"id":"mko-wilson-theorem","title":{"zh-Hant":"Wilson定理","en":"Wilson theorem"},"summary":{"zh-Hant":"整數n>1為質數，恰當(n-1)!模n同餘於-1。","en":"An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n."},"type":"theorem","domain":"number_theory","formalization":"not_formalized","tags":["Wilson定理","AI候選","待形式化"]},{"id":"mko-p-adic-valuation-integers","title":{"zh-Hant":"整數p進賦值","en":"p-adic valuation of integers"},"summary":{"zh-Hant":"對質數p與非零整數a，v_p(a)是p整除|a|的最高次數；本節固定v_p(0)=infinity。","en":"For prime p and nonzero integer a, v_p(a) is the largest exponent of p dividing |a|; this node fixes v_p(0)=infinity."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["整數p進賦值","AI候選","待形式化"]},{"id":"mko-regular-expression","title":{"zh-Hant":"正規表示式","en":"Regular expression"},"summary":{"zh-Hant":"正規表示式由empty language、epsilon、alphabet symbols、union、concatenation與Kleene star遞迴建構語言。","en":"Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["正規表示式","AI候選","待形式化"]},{"id":"mko-pumping-lemma-regular-languages","title":{"zh-Hant":"正規語言Pumping引理","en":"Pumping lemma for regular languages"},"summary":{"zh-Hant":"每個正規語言都有p，使每個長度至少p的成員word存在可反覆pumping且仍留在語言的分解。","en":"Every regular language has a positive pumping length such that each sufficiently long member word has a split that remains in the language for every pump count."},"type":"theorem","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["正規語言Pumping引理","AI候選","待形式化"]},{"id":"mko-newton-method","title":{"zh-Hant":"Newton根求解方法","en":"Newton method for root finding"},"summary":{"zh-Hant":"在導數非零的目前點，以切線零點更新x至x-f(x)/f'(x)；收斂是另需局部假設的責任。","en":"At a qualified point with nonzero derivative, Newton method moves to the tangent-line zero x−f(x)/f_prime(x); convergence requires separate local hypotheses."},"type":"algorithm","domain":"numerical_computational","formalization":"not_formalized","tags":["Newton根求解方法","AI候選","待形式化"]},{"id":"mko-trapezoidal-quadrature","title":{"zh-Hant":"複合梯形數值積分","en":"Composite trapezoidal quadrature"},"summary":{"zh-Hant":"在嚴格遞增partition上逐段以端點平均乘各自寬度，得到定積分的複合梯形近似。","en":"On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral."},"type":"algorithm","domain":"numerical_computational","formalization":"not_formalized","tags":["複合梯形數值積分","AI候選","待形式化"]},{"id":"mko-gradient-descent","title":{"zh-Hant":"梯度下降法","en":"Gradient descent"},"summary":{"zh-Hant":"在固定Euclidean coordinates中以正步長沿負梯度更新；下降與收斂仍需目標及步長假設。","en":"In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses."},"type":"algorithm","domain":"optimization_operations_research","formalization":"not_formalized","tags":["梯度下降法","AI候選","待形式化"]},{"id":"mko-lagrange-multiplier-condition","title":{"zh-Hant":"等式約束Lagrange乘數必要條件","en":"Necessary Lagrange multiplier condition for equality constraints"},"summary":{"zh-Hant":"正則等式約束局部極值點的目標梯度，必為constraint gradients的線性組合。","en":"At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients."},"type":"theorem","domain":"optimization_operations_research","formalization":"not_formalized","tags":["等式約束Lagrange乘數必要條件","AI候選","待形式化"]}],"guides":[{"id":"derivation-mko-point-line-plane-01","title":{"zh-Hant":"點、線、面：在座標模型中判定點是否在直線上","en":"Points, lines and planes: deciding incidence in the coordinate model"},"summary":{"zh-Hant":"在座標平面模型 ℝ² 中，設 P=(1,2)、Q=(3,6)。判定 R=(4,8) 與 S=(2,5) 是否在通過 P、Q 的直線 L 上，並據此判定 P、Q、R 是否共線。","en":"In the coordinate-plane model ℝ², let P=(1,2) and Q=(3,6). Decide whether R=(4,8) and S=(2,5) lie on the line L through P and Q, and hence whether P, Q, R are collinear."},"mko_ids":["mko-point-line-plane"]},{"id":"derivation-mko-angle-01","title":{"zh-Hant":"角：對頂角相等的逐步證明","en":"Angle: a step-by-step proof that vertical angles are equal"},"summary":{"zh-Hant":"直線 AB 與直線 CD 是兩條相異直線，相交於 O，且 O 在 A、B 之間，也在 C、D 之間。證明對頂角相等：m∠AOC=m∠BOD 且 m∠COB=m∠DOA；再以 m∠AOC=35° 為例算出四個角。","en":"Distinct lines AB and CD meet at O, with O between A and B and between C and D. Prove that vertical angles are equal: m∠AOC=m∠BOD and m∠COB=m∠DOA; then, for m∠AOC=35°, compute all four angles."},"mko_ids":["mko-angle"]},{"id":"derivation-mko-triangle-01","title":{"zh-Hant":"三角形：在座標模型中確認三點界定非退化三角形","en":"Triangle: confirming in the coordinate model that three points determine a nondegenerate triangle"},"summary":{"zh-Hant":"在座標平面模型 ℝ² 中，設 A=(0,0)、B=(4,0)、C=(1,3)。確認 A、B、C 兩兩相異且不共線，因而依 MKO 界定非退化三角形 ABC；寫出三條邊與各頂點的對邊，並說明三頂點的平均點 G=(5/3,1) 不在三角形 ABC 上。另設 D=(2,1)、E=(6,3)，說明相異三點 A、D、E 共線，不界定非退化三角形。","en":"In the coordinate-plane model ℝ², let A=(0,0), B=(4,0), C=(1,3). Confirm that A, B, C are pairwise distinct and noncollinear, so that by the MKO they determine the nondegenerate triangle ABC; write out its three sides and the side opposite each vertex, and show that the average of the vertices, G=(5/3,1), is not on triangle ABC. Then, with D=(2,1) and E=(6,3), show that the distinct points A, D, E are collinear and so do not determine a nondegenerate triangle."},"mko_ids":["mko-triangle"]},{"id":"derivation-mko-right-triangle-01","title":{"zh-Hant":"直角三角形：用座標判定哪一個內角是直角","en":"Right triangle: deciding in coordinates which interior angle is right"},"summary":{"zh-Hant":"在座標平面模型 ℝ² 中，設 A=(1,1)、B=(5,3)、C=(0,3)。證明 A、B、C 構成非退化三角形，且其在 A 的內角等於 90°（即 π/2），所以 ABC 是直角三角形；說明另外兩角不是直角、任何非退化三角形至多一個直角，並指出斜邊與兩股。再以 D=(1,4) 說明三角形 ABD 不是直角三角形。","en":"In the coordinate-plane model ℝ², let A=(1,1), B=(5,3), C=(0,3). Prove that A, B, C form a nondegenerate triangle whose interior angle at A equals 90° (that is, π/2), so ABC is a right triangle; show that the other two angles are not right and that any nondegenerate triangle has at most one right angle, and identify the hypotenuse and the legs. Then, with D=(1,4), show that triangle ABD is not a right triangle."},"mko_ids":["mko-right-triangle"]},{"id":"derivation-mko-euclidean-length-01","title":{"zh-Hant":"歐幾里得長度：由座標精確計算兩點距離","en":"Euclidean length: computing the distance between two points exactly from coordinates"},"summary":{"zh-Hant":"在以 ℝ² 表示的歐幾里得平面中，設 P=(1,−2)、Q=(4,2)、R=(2,3)。精確算出 d(P,Q)=5；算出 d(P,R)=√26，並以平方比較證明 5.09<√26<5.1，說明 5.1 只是近似值。另對一般的實數坐標證明：距離有定義且非負、d(Q,P)=d(P,Q)、d(P,P)=0，以及 d(P,Q)=0 ⇔ P=Q。","en":"In the Euclidean plane represented by ℝ², let P=(1,−2), Q=(4,2), R=(2,3). Compute d(P,Q)=5 exactly; compute d(P,R)=√26 and prove by comparing squares that 5.09<√26<5.1, showing that 5.1 is only an approximation. For general real coordinates, also prove that the distance is defined and nonnegative, that d(Q,P)=d(P,Q), that d(P,P)=0, and that d(P,Q)=0 ⇔ P=Q."},"mko_ids":["mko-euclidean-length"]},{"id":"derivation-mko-euclid-pythagorean-theorem-01","title":{"zh-Hant":"畢達哥拉斯定理：在座標模型中以內積證明","en":"Pythagorean theorem: a proof by dot products in the coordinate model"},"summary":{"zh-Hant":"在以 ℝ² 表示的歐幾里得平面中（距離為歐幾里得長度），證明：對任何直角三角形 ABC，若直角在 C，兩股長 a=d(C,B)、b=d(C,A)，斜邊長 c=d(B,A)，則 a²+b²=c²。再以 C=(1,2)、B=(4,6)、A=(−7,8) 驗算（25+100=125），並以非直角三角形 E=(0,0)、F=(4,0)、H=(1,2) 說明沒有直角時關係一般不成立。","en":"In the Euclidean plane represented by ℝ² (with Euclidean length as distance), prove: for every right triangle ABC with the right angle at C, legs of lengths a=d(C,B) and b=d(C,A), and hypotenuse of length c=d(B,A), a²+b²=c². Then check the example C=(1,2), B=(4,6), A=(−7,8) (25+100=125), and use the non-right triangle E=(0,0), F=(4,0), H=(1,2) to show that without a right angle the relation fails in general."},"mko_ids":["mko-euclid-pythagorean-theorem"]},{"id":"derivation-mko-natural-number-01","title":{"zh-Hant":"自然數：以最小歸納集判定成員，並分辨數與數字符號","en":"Natural number: deciding membership through the smallest inductive set, and telling numbers from numerals"},"summary":{"zh-Hant":"依 OCME 慣例（0∈N），把 MKO 的「0 以及由 0 經有限次後繼所得的數」讀成「實數中包含 0 且對 x↦x+1 封閉的最小集合」。據此證明 0∈N、3∈N；說明二進位符號「11」與十進位符號「3」表示同一個自然數；證明 −1∉N 與 1/2∉N；並與「自然數從 1 開始」的慣例 N⁺=N∖{0} 比較最小元素。","en":"Under the OCME convention (0∈N), read the MKO's '0 and the numbers obtained from 0 by finitely many successor steps' as 'the smallest set of reals that contains 0 and is closed under x↦x+1'. On this basis prove 0∈N and 3∈N; show that the binary numeral '11' and the decimal numeral '3' denote the same natural number; prove −1∉N and 1/2∉N; and compare least elements with the convention N⁺=N∖{0}, in which natural numbers start at 1."},"mko_ids":["mko-natural-number"]},{"id":"derivation-mko-integer-01","title":{"zh-Hant":"整數：判定成員、負號的意義與對負號的封閉性","en":"Integer: deciding membership, the meaning of the minus sign, and closure under negation"},"summary":{"zh-Hant":"在實數中，取 Z={0}∪N⁺∪{−n ∣ n∈N⁺}（N 依 OCME 慣例含 0，N⁺=N∖{0}，−n 是 n 的加法反元素）。證明 −7∈Z；說明十進位符號「3.0」表示整數 3；證明 −2.5∉Z；由加法反元素的唯一性證明 −(−a)=a，特別是 −(−4)=4；對所有 z∈Z 證明 −z∈Z；並以 5−8=5+(−8)=−3 分辨二元減法與一元負號。","en":"Inside the reals, take Z={0}∪N⁺∪{−n ∣ n∈N⁺} (N contains 0 by the OCME convention, N⁺=N∖{0}, and −n is the additive inverse of n). Prove −7∈Z; show that the decimal numeral '3.0' denotes the integer 3; prove −2.5∉Z; prove −(−a)=a from uniqueness of additive inverses, in particular −(−4)=4; prove −z∈Z for every z∈Z; and use 5−8=5+(−8)=−3 to tell binary subtraction from the unary minus sign."},"mko_ids":["mko-integer"]},{"id":"derivation-mko-rational-number-01","title":{"zh-Hant":"有理數：同一個有理數的不同分數表示、交叉相乘與約分","en":"Rational number: different fraction representations of one rational, cross-multiplication and reduction"},"summary":{"zh-Hant":"依 MKO 的定義（存在整數 p、q，q≠0，使 x=p/q），證明 6/(−8)、−3/4、9/(−12) 表示同一個有理數，並把 6/(−8) 精確約成正規形式 −3/4；以交叉相乘證明 2/3≠3/5；說明「5/0」不表示任何數；說明有限小數 −1.25=−5/4 與整數 7=7/1 都是有理數；並在分母為正時用交叉相乘證明 −3/4<−2/3。","en":"Using the MKO's definition (there are integers p, q with q≠0 and x=p/q), prove that 6/(−8), −3/4 and 9/(−12) represent the same rational number, and reduce 6/(−8) exactly to the canonical form −3/4; prove 2/3≠3/5 by cross-multiplication; show that '5/0' denotes no number; show that the finite decimal −1.25=−5/4 and the integer 7=7/1 are rational; and, with positive denominators, prove −3/4<−2/3 by cross-multiplication."},"mko_ids":["mko-rational-number"]},{"id":"derivation-mko-real-number-01","title":{"zh-Hant":"實數：在 Cauchy 完備化中證明「循環的 9」與有理數 1 是同一個實數","en":"Real number: proving in the Cauchy completion that 'repeating nines' and the rational 1 are the same real"},"summary":{"zh-Hant":"在以有理數 Cauchy 序列的等價類建構的實數中（有理數 q 嵌入為常數列的類 ι(q)），令 a_n=1−10^{−n}（n≥1），即有限小數 0.9、0.99、0.999 等。證明 (a_n) 是 Cauchy 序列，且與常數列 1 等價，所以 [(a_n)]=ι(1)；證明 ι 是單射，因此有限小數 0.999=999/1000 所表示的實數不等於 1；並說明有限小數 0.375=3/8 是有理數，在 ℝ 中就是 ι(3/8)。","en":"In the reals constructed as equivalence classes of Cauchy sequences of rationals (a rational q embedded as the class ι(q) of its constant sequence), let a_n=1−10^{−n} (n≥1), the finite decimals 0.9, 0.99, 0.999 and so on. Prove that (a_n) is a Cauchy sequence equivalent to the constant sequence 1, so [(a_n)]=ι(1); prove that ι is injective, so the real denoted by the finite decimal 0.999=999/1000 is not 1; and show that the finite decimal 0.375=3/8 is rational and is ι(3/8) in ℝ."},"mko_ids":["mko-real-number"]},{"id":"derivation-mko-arithmetic-operations-01","title":{"zh-Hant":"四則運算：同一組輸入在 ℕ、ℤ、ℚ 中是否有定義","en":"Arithmetic operations: whether the same inputs give a defined result in ℕ, ℤ and ℚ"},"summary":{"zh-Hant":"取 a=3、b=5。在 ℕ（含 0）、ℤ、ℚ 三個數系中，判定 a+b、a·b、a−b、a÷b 是否有定義；有定義時求出結果，並判定結果屬於哪些數系。再判定 3÷0 與 0÷0 在這三個數系中是否有定義。","en":"Take a=3 and b=5. In each of the number systems ℕ (including 0), ℤ and ℚ, decide whether a+b, a·b, a−b and a÷b are defined; when they are, compute the result and decide which systems it belongs to. Then decide whether 3÷0 and 0÷0 are defined in these three systems."},"mko_ids":["mko-arithmetic-operations"]},{"id":"derivation-mko-variable-expression-01","title":{"zh-Hant":"變數與代數式：式子的值由取值、作用域與數系共同決定","en":"Variables and expressions: a value depends on the assignment, the scope and the number system"},"summary":{"zh-Hant":"設 E=x+1、F=(x+1)÷2、H=Σ_{x=1}^{3}(x+1)。(1) 在 ℤ 中求 E 在 x=2 與 x=5 的值，並證明 E 沒有與 x 無關的固定值；(2) 求 F 在 x=4 時於 ℚ 與 ℤ 中的值；(3) 在外部指定 x=5 時求 H；(4) 判定 (x+1)+x 與 2x+1、x·x 與 2x 在 ℤ 上是否等價；(5) 區分表達式 x+1 與方程 x+1=3，並求方程在 ℤ 中的解集。","en":"Let E=x+1, F=(x+1)÷2 and H=Σ_{x=1}^{3}(x+1). (1) In ℤ, evaluate E at x=2 and at x=5 and prove that E has no fixed value independent of x; (2) evaluate F at x=4 in ℚ and in ℤ; (3) evaluate H when x=5 is assigned from outside; (4) decide whether (x+1)+x and 2x+1, and whether x·x and 2x, are equivalent on ℤ; (5) distinguish the expression x+1 from the equation x+1=3 and find the solution set of the equation in ℤ."},"mko_ids":["mko-variable-expression"]},{"id":"derivation-mko-equation-01","title":{"zh-Hant":"方程式：在定義域內解 x²/(x−1)=1/(x−1)","en":"Equation: solving x²/(x−1)=1/(x−1) inside its domain"},"summary":{"zh-Hant":"在允許範圍 ℝ 中，求方程 x²/(x−1)=1/(x−1) 的解集；並說明為什麼兩邊乘以 x−1 後得到的方程 x²=1 有兩個解，而原方程只有一個。","en":"With allowed domain ℝ, find the solution set of the equation x²/(x−1)=1/(x−1), and explain why the equation x²=1 obtained by multiplying both sides by x−1 has two solutions while the original has only one."},"mko_ids":["mko-equation"]},{"id":"derivation-mko-set-01","title":{"zh-Hant":"集合：由元素刻畫、外延相等，以及不受限制的建構為何失敗","en":"Sets: characterized by elements, equal by extensionality, and why unrestricted formation fails"},"summary":{"zh-Hant":"在論域 ℤ 中，設 A={x∈ℤ ∣ x²=4}、B={−2,2}、C={2,−2,2}、D={2}。證明 A=B 與 C=B；證明 D≠B，並區分元素 2 與集合 {2}。最後證明：在任何帶有隸屬關係的論域 U 中，都不存在 R∈U 使「y∈R ⇔ y∉y」對所有 y∈U 成立，因此不是每個條件都能形成集合。","en":"In the universe ℤ, let A={x∈ℤ ∣ x²=4}, B={−2,2}, C={2,−2,2} and D={2}. Prove A=B and C=B; prove D≠B and distinguish the element 2 from the set {2}. Finally prove that in any domain U with a membership relation there is no R∈U with 'y∈R ⇔ y∉y' for all y∈U, so not every condition forms a set."},"mko_ids":["mko-set"]},{"id":"derivation-mko-proposition-01","title":{"zh-Hant":"命題：判定候選語句是否為命題，並區分「是命題」與「已證明為真」","en":"Proposition: deciding which candidate sentences are propositions, and separating 'is a proposition' from 'is proved true'"},"summary":{"zh-Hant":"在入門古典語境中，逐一判定下列候選是否為命題；是命題者再判定真假並給出理由：s1「7 是質數」；s2「4 是質數」；s3「7 是質數嗎？」；s4「計算 3+4。」；s5「x>0」（x 是未指定的整數變數），以及 s5 的封閉版本：代入 x=2 與 x=−1、加上 ∀x∈ℤ、加上 ∃x∈ℤ。","en":"In an introductory classical context, decide for each candidate whether it is a proposition, and for each proposition decide whether it is true, with reasons: s1 '7 is prime'; s2 '4 is prime'; s3 'Is 7 prime?'; s4 'Compute 3+4.'; s5 'x>0' (x an unassigned integer variable), together with closed versions of s5: substituting x=2 and x=−1, prefixing ∀x∈ℤ, and prefixing ∃x∈ℤ."},"mko_ids":["mko-proposition"]},{"id":"derivation-mko-logical-connectives-01","title":{"zh-Hant":"邏輯連接詞：以整除命題檢驗 ∨、→、↔ 與括號","en":"Logical connectives: testing ∨, →, ↔ and brackets on divisibility statements"},"summary":{"zh-Hant":"對整數 n，令 P(n) 為「4∣n」，Q(n) 為「2∣n」。(1) 判定「4∣12 ∨ 6∣12」的真假，說明析取是含括或；(2) 證明 ∀n∈ℤ (P(n)→Q(n))；(3) 證明 ∀n∈ℤ (Q(n)→P(n)) 為假，因而 ∀n∈ℤ (P(n)↔Q(n)) 為假；(4) 判定 P(3)→Q(3) 與「2+2=4 → 7 是奇數」的真假，說明蘊涵不是因果；(5) 比較 ¬Q(2)∧P(2) 與 ¬(Q(2)∧P(2)) 的真假。","en":"For an integer n let P(n) be '4∣n' and Q(n) be '2∣n'. (1) Decide the truth of '4∣12 ∨ 6∣12' and explain that disjunction is inclusive; (2) prove ∀n∈ℤ (P(n)→Q(n)); (3) prove that ∀n∈ℤ (Q(n)→P(n)) is false, and hence that ∀n∈ℤ (P(n)↔Q(n)) is false; (4) decide the truth of P(3)→Q(3) and of '2+2=4 → 7 is odd', explaining that implication is not causation; (5) compare the truth of ¬Q(2)∧P(2) and ¬(Q(2)∧P(2))."},"mko_ids":["mko-logical-connectives"]},{"id":"derivation-mko-proof-01","title":{"zh-Hant":"數學證明：證明 n²+n 恆為偶數，並以反例否定「n²+n+41 恆為質數」","en":"Mathematical proof: proving n²+n is always even, and refuting 'n²+n+41 is always prime' by a counterexample"},"summary":{"zh-Hant":"(1) 在整數的指定語境中，由明示的前提與規則證明定理 T：對每個整數 n，2∣(n²+n)；再由 T 以反證法推出沒有整數 n 使 n²+n 為奇數。(2) 檢驗主張 C：「對每個自然數 n，n²+n+41 是質數」；說明 n=0 到 39 的 40 個正例為何不構成證明，並以 n=40 的反例否定 C。","en":"(1) In a stated context for the integers, prove from explicit premises and rules the theorem T: for every integer n, 2∣(n²+n); then derive from T by contradiction that no integer n makes n²+n odd. (2) Test the claim C: 'for every natural number n, n²+n+41 is prime'; explain why the 40 positive cases n=0 to 39 do not constitute a proof, and refute C by the counterexample n=40."},"mko_ids":["mko-proof"]},{"id":"derivation-mko-set-membership-01","title":{"zh-Hant":"集合隸屬：x∈A 是命題，並與 ⊆、= 及程式的成員測試區分","en":"Set membership: x∈A is a proposition, kept apart from ⊆, = and a program's membership test"},"summary":{"zh-Hant":"在型別化語境中，設 A={1,2,3}、B={1,2}、𝓕={{1,2},{3}}（元素是整數集合的集合）、E={n∈ℤ ∣ 2∣n}。判定下列命題的真假並給出理由：2∈A；4∈A；B⊆A；A⊆B；B∈𝓕；{1}∈𝓕；1000000∈E；7∈E。並說明 1∈B 與 B∈𝓕 為何不給出 1∈𝓕，以及程式語言的成員測試為何不能取代集合隸屬的語義。","en":"In a typed context let A={1,2,3}, B={1,2}, 𝓕={{1,2},{3}} (a set whose elements are sets of integers) and E={n∈ℤ ∣ 2∣n}. Decide with reasons whether each holds: 2∈A; 4∈A; B⊆A; A⊆B; B∈𝓕; {1}∈𝓕; 1000000∈E; 7∈E. Also explain why 1∈B and B∈𝓕 do not give 1∈𝓕, and why a programming language's membership test cannot replace the semantics of set membership."},"mko_ids":["mko-set-membership"]},{"id":"derivation-mko-subset-01","title":{"zh-Hant":"子集合：以任意元素證明包含，並以反例否定包含","en":"Subset: proving inclusion from an arbitrary element and refuting it by a counterexample"},"summary":{"zh-Hant":"在論域 ℤ 中，令 A 為 6 的倍數之集、B 為偶數之集、C 為 4 的倍數之集。證明 A⊆B，且這個包含是嚴格的（A≠B）；證明 A⊄C，雖然 A 與 C 有共同元素 12；說明 B⊆B 成立（⊆ 允許相等）；並證明 {6}⊆A ⇔ 6∈A，同時分辨兩個語句的型別角色。","en":"In the universe ℤ, let A be the set of multiples of 6, B the set of even integers and C the set of multiples of 4. Prove A⊆B and that this inclusion is strict (A≠B); prove A⊄C even though A and C share the element 12; note that B⊆B holds (⊆ allows equality); and prove {6}⊆A ⇔ 6∈A while separating the type roles of the two statements."},"mko_ids":["mko-subset"]},{"id":"derivation-mko-set-operations-01","title":{"zh-Hant":"集合運算：在明示全集中計算聯集、交集、差集與補集","en":"Set operations: union, intersection, difference and complement in an explicit universe"},"summary":{"zh-Hant":"全集 U={1,2,3,4,5,6,7,8}，A={2,4,6,8}，B={3,4,5,6}。逐元素算出 A∪B、A∩B、A∖B、B∖A、U∖A、U∖B；說明 A∖B≠B∖A；說明全集換成 U′={1,2,3,4,5,6,7,8,9,10} 時 A 的補集改變；對任意集合證明 U∖(A∪B)=(U∖A)∩(U∖B) 並以本例核對；再以 A∪B=A∪(B∖A) 說明聯集為何有 6 個元素而不是 8 個。","en":"Universe U={1,2,3,4,5,6,7,8}, A={2,4,6,8}, B={3,4,5,6}. Compute A∪B, A∩B, A∖B, B∖A, U∖A and U∖B element by element; show A∖B≠B∖A; show that the complement of A changes when the universe is U′={1,2,3,4,5,6,7,8,9,10}; prove U∖(A∪B)=(U∖A)∩(U∖B) for arbitrary sets and check it on this example; and use A∪B=A∪(B∖A) to explain why the union has 6 elements, not 8."},"mko_ids":["mko-set-operations"]},{"id":"derivation-mko-sample-space-01","title":{"zh-Hant":"樣本空間：擲兩次硬幣的兩種觀察尺度與不等可能的結果","en":"Sample space: two observation scales for two coin tosses, and outcomes that are not equally likely"},"summary":{"zh-Hant":"隨機試驗：把一枚硬幣擲兩次。(1) 若依次記錄結果，寫出樣本空間 S，證明它包含所有可能基本結果且恰有 4 個元素；(2) 若只記錄正面次數，寫出樣本空間 S′={0,1,2}；(3) 在另外給定的機率模型 P（S 的每個有序結果機率 1/4）下，算出 S′ 各結果的機率 1/4、1/2、1/4，說明同一樣本空間中的結果不必等可能；(4) 以 S 上另一個合法的非均勻模型 Q 說明樣本空間本身不決定機率；(5) 分辨結果 (H,T) 與事件 {(H,T),(T,H)}。","en":"Random experiment: toss a coin twice. (1) Recording the results in order, write the sample space S, and prove that it contains every possible elementary outcome and has exactly 4 elements; (2) recording only the number of heads, write the sample space S′={0,1,2}; (3) under a separately given probability model P (each ordered outcome in S has probability 1/4), compute the probabilities 1/4, 1/2, 1/4 of the outcomes in S′, showing that outcomes in one sample space need not be equally likely; (4) use another valid, non-uniform model Q on S to show that the sample space by itself does not fix probabilities; (5) distinguish the outcome (H,T) from the event {(H,T),(T,H)}."},"mko_ids":["mko-sample-space"]},{"id":"derivation-mko-event-01","title":{"zh-Hant":"事件：在完整事件族與較粗事件族中判定事件","en":"Event: deciding what counts as an event under the full event family and under a coarser one"},"summary":{"zh-Hant":"擲一顆骰子一次，S={1,2,3,4,5,6}。(1) 在有限初等模型 F=𝒫(S) 中，說明 E={2,4,6} 是事件，判定觀察結果為 4 與 3 時 E 是否發生；分辨結果 4 與事件 {4}；說明空事件與必然事件。(2) 在一般模型 (S,F₀) 中，F₀={∅,O,E,S}，O={1,3,5}：證明 F₀ 是 S 上的事件族（σ-代數），並證明 {6}∉F₀，所以雖然 S 有限且 {6}⊆S，{6} 在這個模型中不是事件。","en":"Roll a die once, S={1,2,3,4,5,6}. (1) In the finite elementary model F=𝒫(S), show that E={2,4,6} is an event, and decide whether E occurs when the observed outcome is 4 and when it is 3; distinguish the outcome 4 from the event {4}; describe the empty and the sure event. (2) In the general model (S,F₀) with F₀={∅,O,E,S} and O={1,3,5}: prove that F₀ is an event family (σ-algebra) on S, and prove {6}∉F₀, so although S is finite and {6}⊆S, {6} is not an event in this model."},"mko_ids":["mko-event"]},{"id":"derivation-mko-function-mapping-01","title":{"zh-Hant":"函數映射：判定對應是否為函數，並分辨陪域、值域與圖形","en":"Function mapping: deciding whether a rule is a function, and separating codomain, range and graph"},"summary":{"zh-Hant":"設 X={−2,−1,0,1,2}、Y={0,1,2,3,4,5}，f:X→Y，f(x)=x²。證明 f 是函數；算出值域 f(X)={0,1,4} 並說明 f(X)≠Y；寫出圖形 Γ_f；說明把陪域改成 {0,1,4} 會得到不同的函數。再判定兩個非例：反向規則 s(y)=「X 中滿足 x²=y 的 x」不是 Y→X 的函數；g(x)=x+4 不是 X→Y 的函數。","en":"Let X={−2,−1,0,1,2}, Y={0,1,2,3,4,5} and f:X→Y, f(x)=x². Prove that f is a function; compute the range f(X)={0,1,4} and show f(X)≠Y; write down the graph Γ_f; and show that changing the codomain to {0,1,4} gives a different function. Then decide two non-examples: the reverse rule s(y)='the x in X with x²=y' is not a function Y→X, and g(x)=x+4 is not a function X→Y."},"mko_ids":["mko-function-mapping"]},{"id":"derivation-mko-tends-to-relation-01","title":{"zh-Hant":"趨近關係：x→a 不是 x=a，也不保證函數值收斂","en":"Tends-to relation: x→a is not x=a, and does not guarantee that function values converge"},"summary":{"zh-Hant":"在 (ℝ, d(x,y)=|x−y|) 中：(1) 證明數列 x_n=1+1/n 趨近 1，而每一項都有 x_n≠1；(2) 對 g(x)=1（x>1）、g(x)=0（x≤1），證明沿 x_n 有 g(x_n)→1，但 lim_{x→1} g(x) 不存在。由此說明 x→1 本身既不要求 x 等於 1，也不決定函數值是否收斂。","en":"In (ℝ, d(x,y)=|x−y|): (1) prove that the sequence x_n=1+1/n tends to 1 while every term satisfies x_n≠1; (2) for g(x)=1 (x>1) and g(x)=0 (x≤1), prove that g(x_n)→1 along x_n, but lim_{x→1} g(x) does not exist. This shows that x→1 by itself neither requires x to equal 1 nor decides whether function values converge."},"mko_ids":["mko-tends-to-relation"]},{"id":"derivation-mko-counting-principle-01","title":{"zh-Hant":"基本計數原理：數字互異的三位偶數有幾個","en":"Basic counting principles: how many three-digit even numbers have distinct digits"},"summary":{"zh-Hant":"計算 100 到 999 之間、三個數字兩兩相異的偶數共有幾個。要求：只把互斥的類別相加；每次相乘之前，先確認每個階段的可選數對所有先前選擇都相同，不同時先分支。","en":"Count the even integers from 100 to 999 whose three digits are pairwise distinct. Requirements: add only disjoint classes, and before each multiplication confirm that the number of options at every stage is the same for all earlier choices; when it is not, branch first."},"mko_ids":["mko-counting-principle"]},{"id":"derivation-mko-factorial-01","title":{"zh-Hant":"階乘：遞迴定義的唯一性、乘積形式與 5! 的計算","en":"Factorial: uniqueness of the recursive definition, the product form and computing 5!"},"summary":{"zh-Hant":"只用 F(0)=1 與 F(n+1)=(n+1)F(n)（n∈ℕ，ℕ 含 0）：(1) 證明滿足這兩條的函數 ℕ→ℕ 只有一個；(2) 證明對所有 n 有 F(n)=∏_{k=1}^{n} k（空乘積為 1），所以 n>0 時 n! 是從 1 到 n 所有整數的乘積；(3) 證明每個 F(n)≥1，特別是 0!=1≠0；(4) 由遞迴逐步算出 5!=120。","en":"Using only F(0)=1 and F(n+1)=(n+1)F(n) (n∈ℕ, with 0∈ℕ): (1) prove that at most one function ℕ→ℕ satisfies these two equations; (2) prove F(n)=∏_{k=1}^{n} k for every n (the empty product being 1), so that for n>0, n! is the product of all integers from 1 to n; (3) prove F(n)≥1 for every n, in particular 0!=1≠0; (4) compute 5!=120 step by step from the recursion."},"mko_ids":["mko-factorial"]},{"id":"derivation-mko-absolute-value-01","title":{"zh-Hant":"絕對值：由分段定義證明非負性、零點與 |a−b|=|b−a|","en":"Absolute value: nonnegativity, the zero case and |a−b|=|b−a| from the piecewise definition"},"summary":{"zh-Hant":"只用分段定義（x≥0 時 |x|=x，x<0 時 |x|=−x）與實數的序和加法，證明對所有實數 x、a、b：(1) |x|≥0；(2) |x|=0 ⇔ x=0；(3) |−x|=|x|；(4) |a−b|=|b−a|。並計算 |−3|、|2−7|、|7−2|，說明「|x|=x 對所有 x 成立」是錯的。","en":"Using only the piecewise definition (|x|=x when x≥0, |x|=−x when x<0) together with the order and addition of the reals, prove for all reals x, a, b: (1) |x|≥0; (2) |x|=0 ⇔ x=0; (3) |−x|=|x|; (4) |a−b|=|b−a|. Then compute |−3|, |2−7| and |7−2|, and show that '|x|=x for every x' is false."},"mko_ids":["mko-absolute-value"]},{"id":"derivation-mko-fraction-01","title":{"zh-Hant":"分數：判定等值、化為最簡正分母形式與分數加法","en":"Fractions: deciding equal value, normalizing to lowest terms with a positive denominator, and adding fractions"},"summary":{"zh-Hant":"(1) 判定 (−6)/(−8) 與 9/12 是否表示同一個有理數；(2) 把 (−6)/(−8) 化成最簡、正分母的表示，並說明數值不變；(3) 計算 3/4+5/6，並說明「分子加分子、分母加分母」得到的 8/10 不是和；(4) 說明 3/0 不是合法的分數表示。","en":"(1) Decide whether (−6)/(−8) and 9/12 represent the same rational number; (2) bring (−6)/(−8) to lowest terms with a positive denominator and show that the value is unchanged; (3) compute 3/4+5/6 and show that 'numerator plus numerator over denominator plus denominator', giving 8/10, is not the sum; (4) explain why 3/0 is not a valid fraction representation."},"mko_ids":["mko-fraction"]},{"id":"derivation-mko-ratio-proportion-01","title":{"zh-Hant":"比與比例：解 3:4=x:10、代回核對與檢查正比例資料","en":"Ratio and proportion: solving 3:4=x:10, checking back, and testing data for direct proportion"},"summary":{"zh-Hant":"在有理數中：(1) 證明對 b≠0、d≠0，比例 a:b=x:d 恰有一個有理解 x=ad/b；(2) 用它解 3:4=x:10 並代回核對；(3) 說明 4:3=x:10 的解不同，比的順序有意義；(4) 判定資料 (2,5)、(4,10)、(6,15)、(8,21) 是否符合同一個正比例 y=kx。","en":"Over the rationals: (1) prove that for b≠0 and d≠0 the proportion a:b=x:d has exactly one rational solution, x=ad/b; (2) use it to solve 3:4=x:10 and check back; (3) show that 4:3=x:10 has a different solution, so the order of a ratio matters; (4) decide whether the data (2,5), (4,10), (6,15), (8,21) fit a single direct proportion y=kx."},"mko_ids":["mko-ratio-proportion"]},{"id":"derivation-mko-domain-codomain-range-01","title":{"zh-Hant":"定義域、陪域與值域：有限表格與 h(x)=x²+1 的值域","en":"Domain, codomain and range: a finite table and the range of h(x)=x²+1"},"summary":{"zh-Hant":"(1) 設 A={−1,0,1,2}、B={0,1,2,3,4,5}、f(x)=x²+1：驗證 f:A→B 是總函數，求值域 f[A]，並說明 f[A]⊆B 但 f[A]≠B；(2) 說明輸出落在宣告陪域外或漏掉輸入的資料不是本條目的總函數；(3) 對 h:ℝ→ℝ、h(x)=x²+1，證明值域是 [1,∞)，不等於陪域 ℝ。","en":"(1) With A={−1,0,1,2}, B={0,1,2,3,4,5} and f(x)=x²+1: verify that f:A→B is a total function, find the range f[A], and show f[A]⊆B but f[A]≠B; (2) explain why data with an output outside the declared codomain, or with a missing input, is not a total function in the sense of this entry; (3) for h:ℝ→ℝ, h(x)=x²+1, prove that the range is [1,∞), which differs from the codomain ℝ."},"mko_ids":["mko-domain-codomain-range"]},{"id":"derivation-mko-composition-01","title":{"zh-Hant":"函數合成：受限合成的允許輸入、求值順序與不交換","en":"Function composition: the permitted inputs of a restricted composition, evaluation order and non-commutativity"},"summary":{"zh-Hant":"設 f:ℝ→ℝ，f(x)=x−3；g:[0,∞)→ℝ，g(u)=√u。(1) 求 g∘f 的允許輸入集合 D={x∈dom(f) ∣ f(x)∈dom(g)}，並證明 D=[3,∞)；(2) 依「先 f 後 g」計算 (g∘f)(7) 與 (g∘f)(4)，並說明 (g∘f)(1) 沒有定義；(3) 說明 f∘g 是 [0,∞) 上的總函數合成，並由 (f∘g)(4)=−1≠1=(g∘f)(4) 得到 g∘f≠f∘g；(4) 說明合成不是函數值的乘法。","en":"Let f:ℝ→ℝ, f(x)=x−3, and g:[0,∞)→ℝ, g(u)=√u. (1) Find the permitted inputs D={x∈dom(f) ∣ f(x)∈dom(g)} of g∘f and prove D=[3,∞); (2) evaluate (g∘f)(7) and (g∘f)(4) by 'f first, then g', and explain why (g∘f)(1) is undefined; (3) show that f∘g is a total composition on [0,∞), and conclude g∘f≠f∘g from (f∘g)(4)=−1≠1=(g∘f)(4); (4) show that composition is not multiplication of function values."},"mko_ids":["mko-composition"]},{"id":"derivation-mko-sequence-01","title":{"zh-Hant":"數列：由初始條件與遞迴確定通項，以及索引與有限前綴的陷阱","en":"Sequences: fixing the general term from an initial condition and a recurrence, and the pitfalls of indexing and finite prefixes"},"summary":{"zh-Hant":"設實數數列 a:ℕ→ℝ（ℕ 含 0）滿足 a₀=5、a_{n+1}=a_n+3。(1) 證明恰有一個這樣的數列，且 a_n=5+3n；(2) 計算 a₀ 到 a₄ 與 a₁₀，說明 a₀ 是起始項、a₁₀ 是從 1 數起的第 11 個值；(3) 寫出長度 L=4 的有限數列及其索引集合；(4) 說明前綴 5、8、11 不能唯一決定數列。","en":"Let the real sequence a:ℕ→ℝ (with 0∈ℕ) satisfy a₀=5 and a_{n+1}=a_n+3. (1) Prove that exactly one such sequence exists and that a_n=5+3n; (2) compute a₀ to a₄ and a₁₀, explaining that a₀ is the initial term and a₁₀ is the 11th value when counting from 1; (3) write down the finite sequence of length L=4 and its index set; (4) show that the prefix 5, 8, 11 does not determine the sequence."},"mko_ids":["mko-sequence"]},{"id":"derivation-mko-permutation-combination-01","title":{"zh-Hant":"排列與組合：以計數證明 P(n,k)=n!/(n−k)! 與 C(n,k)=n!/(k!(n−k)!)","en":"Permutations and combinations: proving P(n,k)=n!/(n−k)! and C(n,k)=n!/(k!(n−k)!) by counting"},"summary":{"zh-Hant":"設 X 是恰有 n 個相異物件的集合，n、k 為自然數且 0≤k≤n，選取無放回、不重複。證明：從 X 有序選取 k 個的方法數 P(n,k)=n!/(n−k)!，無序選取 k 個的方法數 C(n,k)=n!/(k!(n−k)!)，且 k=0 時兩者都等於 1。最後以 X={a,b,c,d,e}、k=3 算出 P(5,3)=60 與 C(5,3)=10。","en":"Let X be a set of exactly n distinct objects, with natural numbers n, k and 0≤k≤n, and selection without replacement and without repetition. Prove that the number of ordered selections of k objects from X is P(n,k)=n!/(n−k)!, that the number of unordered selections is C(n,k)=n!/(k!(n−k)!), and that both equal 1 when k=0. Finally compute P(5,3)=60 and C(5,3)=10 for X={a,b,c,d,e} and k=3."},"mko_ids":["mko-permutation-combination"]},{"id":"derivation-mko-coordinate-plane-01","title":{"zh-Hant":"座標平面：判定點在哪條軸或哪個象限，並比較有序對","en":"Coordinate plane: deciding axis or quadrant membership and comparing ordered pairs"},"summary":{"zh-Hant":"在固定的平面直角座標系中，給定 A=(−3,2)、B=(2,−3)、B′=(6/3,−3)、C=(0,−4)、O=(0,0)、E=(−1/2,−5)。先證明兩軸只交於原點、每個點要嘛在軸上、要嘛恰在一個象限中；再逐點判定它所在的軸或象限，並判定 A 與 B、B 與 B′ 是否為同一點。","en":"In a fixed Cartesian coordinate system, let A=(−3,2), B=(2,−3), B′=(6/3,−3), C=(0,−4), O=(0,0) and E=(−1/2,−5). First prove that the axes meet only at the origin and that every point either lies on an axis or lies in exactly one quadrant; then decide for each point its axis or quadrant, and decide whether A and B, and B and B′, are the same point."},"mko_ids":["mko-coordinate-plane"]},{"id":"derivation-mko-power-root-01","title":{"zh-Hant":"自然數冪與主平方根：由遞迴算冪，並證明 √(a²)=|a|","en":"Natural-number powers and principal square roots: computing powers by recursion and proving √(a²)=|a|"},"summary":{"zh-Hant":"只用遞迴定義 a^0=1、a^(n+1)=a^n·a 算出 (−2)^n（n=0,1,2,3,4）與 0^0、0^3；證明非負平方根的唯一性，並證明對每個實數 a，√(a²)=|a|；再求 √49、√((−5)²)、方程 y²=9 的實數解集，並說明 √(−4) 在實數中沒有定義。","en":"Using only the recursive definition a^0=1, a^(n+1)=a^n·a, compute (−2)^n for n=0,1,2,3,4 and 0^0, 0^3; prove that a nonnegative square root is unique and that √(a²)=|a| for every real a; then find √49, √((−5)²) and the real solution set of y²=9, and explain why √(−4) is not defined in the reals."},"mko_ids":["mko-power-root"]},{"id":"derivation-mko-polynomial-01","title":{"zh-Hant":"一元有理係數多項式：合併同次項、係數向量、次數與零多項式","en":"Univariate rational polynomials: combining like terms, coefficient vectors, degree and the zero polynomial"},"summary":{"zh-Hant":"在 ℚ[x] 中，設 p(x)=4x³+3x²−x+2−4x³+x，q 的係數向量為 [−1,1,0,0]，r(x)=x²−x²+3，z(x)=5x−5x，t(x)=5x。求 p、q、r 的係數與次數，判定 z 是零多項式並區分 z 與 r；以係數計算 p+q 與 p·q 並求次數；最後說明 p(1)=t(1) 不能推出 p=t。","en":"In ℚ[x], let p(x)=4x³+3x²−x+2−4x³+x, let q have coefficient vector [−1,1,0,0], and let r(x)=x²−x²+3, z(x)=5x−5x and t(x)=5x. Find the coefficients and degrees of p, q and r; show that z is the zero polynomial and distinguish it from r; compute p+q and p·q from coefficients and find their degrees; finally explain why p(1)=t(1) does not give p=t."},"mko_ids":["mko-polynomial"]},{"id":"derivation-mko-factorization-01","title":{"zh-Hant":"有理係數多項式因式分解：以係數比對驗證提公因子、二項式乘積與平方差","en":"Polynomial factorization over the rationals: verifying common factors, binomial products and differences of squares by comparing coefficients"},"summary":{"zh-Hant":"在 ℚ[x] 中：(一) 把 p(x)=3x²+3x−18 分解為 3(x+3)(x−2)；(二) 把 q(x)=2x²−50 分解為 2(x−5)(x+5)；每個分解都以展開並比對每個係數驗證。(三) 證明 x²+25≠(x−5)(x+5)，且 x²+25 不能寫成兩個一次因式的乘積。(四) 說明 x³ 與 x 在 x=−1,0,1 取值相同卻不是同一個多項式。","en":"In ℚ[x]: (1) factor p(x)=3x²+3x−18 as 3(x+3)(x−2); (2) factor q(x)=2x²−50 as 2(x−5)(x+5); verify each factorization by expanding and comparing every coefficient. (3) Prove that x²+25≠(x−5)(x+5) and that x²+25 is not a product of two degree-one factors. (4) Explain why x³ and x take equal values at x=−1, 0, 1 and still are different polynomials."},"mko_ids":["mko-factorization"]},{"id":"derivation-mko-linear-function-01","title":{"zh-Hant":"仿射函數與一次函數：由兩點求 m、b，判定次數、零點與是否為線性映射","en":"Affine and degree-one functions: finding m and b from two points, and deciding degree, zero and linearity"},"summary":{"zh-Hant":"仿射函數 f:ℝ→ℝ，f(x)=mx+b 的圖形通過 (1,4) 與 (3,0)。求 m、b；判定 f 是否為次數一的函數、求它的零點，並判定 f 是否為線性代數意義下的 ℝ 線性映射。再證明兩個仿射函數在 ℝ 上相等當且僅當斜率與截距都相同，據此說明 f(2)=k(2)（k(x)=x）不能推出 f=k；最後處理 m=0 的兩個退化例子 g(x)=5 與 h(x)=0。","en":"The graph of the affine function f:ℝ→ℝ, f(x)=mx+b, passes through (1,4) and (3,0). Find m and b; decide whether f has degree one, find its zero, and decide whether f is an ℝ-linear map in the sense of linear algebra. Then prove that two affine functions are equal on ℝ exactly when their slopes and intercepts agree, and use it to explain why f(2)=k(2) (with k(x)=x) does not give f=k; finally treat the two degenerate examples with m=0, g(x)=5 and h(x)=0."},"mko_ids":["mko-linear-function"]},{"id":"derivation-mko-function-graph-01","title":{"zh-Hant":"實值函數圖形：成員判定與有限表格的總函數驗證","en":"Graphs of real-valued functions: membership tests and verifying a finite table as a total function"},"summary":{"zh-Hant":"(一) 設 D=[0,2]、f:D→ℝ、f(x)=x²−1。先證明 (u,v)∈G_f ⇔ u∈D 且 v=f(u)，再判定 (1,0)、(−1,0)、(2,2)、(2,3) 是否在 G_f 上。(二) 設 D_0={0,1,2}、B={0,1,4}。證明一個集合 Γ 是某個總函數 D_0→B 的圖形，當且僅當 Γ⊆D_0×B 且每個輸入恰有一個輸出；據此判定 T={(0,0),(1,1),(2,4)} 是、而 T_1={(0,0),(2,4)}、T_2={(0,0),(1,1),(1,4),(2,4)}、T_3={(0,0),(1,1),(2,4),(3,9)} 都不是。","en":"(1) Let D=[0,2], f:D→ℝ, f(x)=x²−1. First prove (u,v)∈G_f ⇔ u∈D and v=f(u), then decide whether (1,0), (−1,0), (2,2) and (2,3) lie on G_f. (2) Let D_0={0,1,2} and B={0,1,4}. Prove that a set Γ is the graph of some total function D_0→B exactly when Γ⊆D_0×B and every input has exactly one output; use it to decide that T={(0,0),(1,1),(2,4)} is such a graph, while T_1={(0,0),(2,4)}, T_2={(0,0),(1,1),(1,4),(2,4)} and T_3={(0,0),(1,1),(2,4),(3,9)} are not."},"mko_ids":["mko-function-graph"]},{"id":"derivation-mko-probability-01","title":{"zh-Hant":"有限樣本空間上的機率：驗證權重模型、計算事件機率與補事件","en":"Probability on finite sample spaces: validating a weight model, computing event probabilities and complements"},"summary":{"zh-Hant":"S={1,2,3,4}，事件族為全冪集，權重 w_1=1/2、w_2=1/3、w_3=1/6、w_4=0。驗證這是合法的有限機率模型；由定義證明 P(∅)=0、P(S)=1、互斥可加與 P(S∖E)=1−P(E)；計算 P({1,3})，說明它不等於 |E|/|S|；說明 {4} 非空但機率為 0；並判定兩組候選權重 (1/2,1/3,1/3,0) 與 (1/2,2/3,−1/6,0) 都不合法。","en":"Let S={1,2,3,4} with the full powerset as event family and weights w_1=1/2, w_2=1/3, w_3=1/6, w_4=0. Verify that this is a legal finite probability model; prove from the definition the identities P(∅)=0 and P(S)=1, additivity for disjoint events, and the complement identity P(S∖E)=1−P(E); compute P({1,3}) and explain why it differs from |E|/|S|; show that {4} is nonempty yet has probability 0; and decide that the candidate weights (1/2,1/3,1/3,0) and (1/2,2/3,−1/6,0) are both illegal."},"mko_ids":["mko-probability"]},{"id":"derivation-mko-quantifier-01","title":{"zh-Hant":"量詞與明示論域：在有限整數論域上判定全稱、存在及其否定","en":"Quantifiers and explicit domains: deciding universal and existential statements and their negations on a finite integer domain"},"summary":{"zh-Hant":"在明示有限論域 U={−2,−1,0,1,2,3} 上，判定 φ₁：∀x∈U, x²<10；φ₂：∀x∈U, x²>x；φ₃：∃x∈U, x²=2 的真假；正確寫出 φ₂ 與 φ₃ 的否定；並說明論域為空集合時全稱與存在陳述的真假。","en":"On the explicit finite domain U={−2,−1,0,1,2,3}, decide the truth of φ₁: ∀x∈U, x²<10; φ₂: ∀x∈U, x²>x; φ₃: ∃x∈U, x²=2; write the negations of φ₂ and φ₃ correctly; and settle the truth of universal and existential statements when the domain is the empty set."},"mko_ids":["mko-quantifier"]},{"id":"derivation-mko-counterexample-01","title":{"zh-Hant":"反例與論域資格：n²+n+11 的質數性、失敗集合與最小反例","en":"Counterexamples and domain eligibility: primality of n²+n+11, the failure set and the least counterexample"},"summary":{"zh-Hant":"設 U={n∈ℤ ∣ 0≤n≤12}、V={n∈ℤ ∣ 0≤n≤9}、m(n)=n²+n+11、P(n)：「m(n) 是質數」。判定 φ_U：∀n∈U P(n)；求失敗集合 F_U 與最小反例；判定候選值 −11 的資格；並判定 φ_V：∀n∈V P(n)。","en":"Let U={n∈ℤ ∣ 0≤n≤12}, V={n∈ℤ ∣ 0≤n≤9}, m(n)=n²+n+11 and P(n): 'm(n) is prime'. Decide φ_U: ∀n∈U P(n); find the failure set F_U and the least counterexample; decide whether the candidate −11 is eligible; and decide φ_V: ∀n∈V P(n)."},"mko_ids":["mko-counterexample"]},{"id":"derivation-mko-relation-01","title":{"zh-Hant":"二元關係：定義域、關係像與逆關係的逐步計算","en":"Binary relations: computing the domain, images and the converse step by step"},"summary":{"zh-Hant":"設 A={1,2,3,4}、B={2,3,4,5,6}、R={(1,2),(1,3),(3,3),(4,6)}、C={1,3}。核對 R⊆A×B；計算 Dom(R)、R[C]、R[∅]、R⁻¹ 與 Dom(R⁻¹)；判定 R 是否為函數 A→B、R⁻¹ 是否為函數 B→A。","en":"Let A={1,2,3,4}, B={2,3,4,5,6}, R={(1,2),(1,3),(3,3),(4,6)} and C={1,3}. Check R⊆A×B; compute Dom(R), R[C], R[∅], R⁻¹ and Dom(R⁻¹); decide whether R is a function A→B and whether R⁻¹ is a function B→A."},"mko_ids":["mko-relation"]},{"id":"derivation-mko-graph-01","title":{"zh-Hant":"有限簡單無向圖：鄰居、度數與握手公式","en":"Finite simple undirected graphs: neighbours, degrees and the handshake formula"},"summary":{"zh-Hant":"設 V={1,2,3,4,5}，邊以端點對清單輸入：(1,2)、(2,1)、(2,3)、(3,4)、(2,4)、(4,2)，每一項表示一條無向邊。把輸入正規化為邊集合 E，求每個頂點的鄰居集合 N(v) 與度數 deg(v)，核對度數和等於 2|E|，並對每個有限簡單無向圖證明握手公式 Σ_{v∈V} deg(v)=2|E|。","en":"Let V={1,2,3,4,5}, with edges entered as a list of endpoint pairs: (1,2), (2,1), (2,3), (3,4), (2,4), (4,2), each entry denoting an undirected edge. Normalize the input to the edge set E, find the neighbour set N(v) and degree deg(v) of every vertex, check that the degree sum equals 2|E|, and prove the handshake formula Σ_{v∈V} deg(v)=2|E| for every finite simple undirected graph."},"mko_ids":["mko-graph"]},{"id":"derivation-mko-vector-01","title":{"zh-Hant":"固定基底下的平面向量：逐分量運算、相等判定與減法方向","en":"Plane vectors in a fixed basis: componentwise operations, equality and the direction of subtraction"},"summary":{"zh-Hant":"在固定標準基底下，設 u=(3,−1)、v=(−2,4)、w=(1,3)、k=−2。逐分量計算 u+v、u−v、v−u、ku、0u、2u+v；求滿足 u+x=v 的向量 x；核對 k(u+v)=ku+kv；並判定 u=w 與 u=(−1,3) 是否成立。","en":"In the fixed standard basis let u=(3,−1), v=(−2,4), w=(1,3) and k=−2. Compute u+v, u−v, v−u, ku, 0u and 2u+v componentwise; find the vector x with u+x=v; check k(u+v)=ku+kv; and decide whether u=w and whether u=(−1,3)."},"mko_ids":["mko-vector"]},{"id":"derivation-mko-system-linear-equations-01","title":{"zh-Hant":"二元聯立一次方程組：唯一解公式、兩列核對與 D=0 的一致性判定","en":"Linear systems in two variables: the unique-solution formula, checking both rows, and consistency when D=0"},"summary":{"zh-Hant":"(I) 解 2x+3y=7、x−y=1：證明 D=ae−bd≠0 時的唯一解公式，求解並代入兩列核對；(II) 判定 x+2y=3、2x+4y=7；(III) 判定 x+2y=3、2x+4y=6；並以 (IV) 0x+0y=1、0x+0y=2 說明 D 與兩個 Cramer 分子都為零時不能斷定有無限多解。","en":"(I) Solve 2x+3y=7, x−y=1: prove the unique-solution formula when D=ae−bd≠0, solve, and check by substituting into both rows; (II) decide x+2y=3, 2x+4y=7; (III) decide x+2y=3, 2x+4y=6; and use (IV) 0x+0y=1, 0x+0y=2 to show that D and both Cramer numerators being zero does not imply infinitely many solutions."},"mko_ids":["mko-system-linear-equations"]},{"id":"derivation-mko-conditional-probability-01","title":{"zh-Hant":"有限模型的條件機率：正分母、兩個方向與零機率條件","en":"Conditional probability in finite models: positive denominators, the two directions and null conditioning events"},"summary":{"zh-Hant":"在 S={1,2,3,4,5,6}、權重 w=(1,2,3,2,4,0)/12 的有限模型中，設 A={2,3}、B={3,4,5,6}、Z={6}。驗證模型合法，計算 P(A|B) 與 P(B|A)，與等可能計數比 |A∩B|/|B| 及乘積 P(A)P(B) 比較，並判定 P(A|Z) 是否有定義。","en":"In the finite model on S={1,2,3,4,5,6} with weights w=(1,2,3,2,4,0)/12, let A={2,3}, B={3,4,5,6} and Z={6}. Validate the model, compute P(A|B) and P(B|A), compare them with the equiprobable count ratio |A∩B|/|B| and with the product P(A)P(B), and decide whether P(A|Z) is defined."},"mko_ids":["mko-conditional-probability"]},{"id":"derivation-mko-injective-01","title":{"zh-Hant":"單射與不同輸入的辨識：函數表的資格、碰撞見證與纖維","en":"Injectivity and distinct inputs: table qualification, collision witnesses and fibers"},"summary":{"zh-Hant":"設 A={1,2,3,4}、B={1,2,3,4,5,6}，函數表 f：(1,4),(2,6),(3,1),(4,5)；g：(1,4),(2,6),(2,6),(3,4),(4,6)；h：(1,4),(2,6),(2,5),(3,1),(4,5)。確認各表是否為總函數 A→B；判定 f、g 是否單射並列出纖維；找出 g 按字典序的首碰撞；判定 g 限制到 C={1,2} 是否單射；並說明空定義域的情形。","en":"Let A={1,2,3,4}, B={1,2,3,4,5,6}, with function tables f: (1,4),(2,6),(3,1),(4,5); g: (1,4),(2,6),(2,6),(3,4),(4,6); h: (1,4),(2,6),(2,5),(3,1),(4,5). Decide which tables are total functions A→B; decide whether f and g are injective and list their fibers; find g's first collision in lexicographic order; decide whether g restricted to C={1,2} is injective; and treat the empty domain."},"mko_ids":["mko-injective"]},{"id":"derivation-mko-surjective-01","title":{"zh-Hant":"滿射與陪域覆蓋：值域、原像與陪域變更","en":"Surjectivity and codomain coverage: range, preimages and changing the codomain"},"summary":{"zh-Hant":"設 A={1,2,3,4,5}、B={1,2,3}，函數表 f：(1,2),(2,1),(3,3),(4,2),(5,1)。確認 f 是總函數 A→B，求值域 f[A] 與各原像，判定 f 是否滿射並找出 t=2 的首個原像；再把陪域換成 B2={1,2,3,4} 與 B3={1,2} 判定合法性與滿射性；最後處理 ∅→∅ 與 ∅→{7}。","en":"Let A={1,2,3,4,5}, B={1,2,3} and the function table f: (1,2),(2,1),(3,3),(4,2),(5,1). Confirm f is a total function A→B, find the range f[A] and the preimages, decide whether f is surjective and find the first preimage of t=2; then replace the codomain by B2={1,2,3,4} and by B3={1,2}, deciding legality and surjectivity; finally treat ∅→∅ and ∅→{7}."},"mko_ids":["mko-surjective"]},{"id":"derivation-mko-bijective-01","title":{"zh-Hant":"雙射：以纖維概況判定給定函數，並與「是否存在某個雙射」分開","en":"Bijectivity: testing given functions by fiber profiles, kept apart from whether some bijection exists"},"summary":{"zh-Hant":"設 A={1,2,3}、B={4,5,6}，函數 f、g:A→B 由表給出：f(1)=5、f(2)=6、f(3)=4；g(1)=4、g(2)=4、g(3)=6。先對一般總函數證明「雙射 ⇔ 每個 b∈B 的纖維恰有一個元素」，再按陪域 4、5、6 的順序列出 f 與 g 的完整纖維概況，判定 f 是雙射、g 既非單射也非滿射。最後回答另一類問題「是否存在某個雙射」：A→B 存在（f 即是）、{1,2,3}→{4,5} 不存在、∅→∅ 恰有一個空雙射。","en":"Let A={1,2,3}, B={4,5,6}, and let f, g:A→B be given by tables: f(1)=5, f(2)=6, f(3)=4; g(1)=4, g(2)=4, g(3)=6. First prove, for any total function, the equivalence: bijective ⇔ the fiber of every b∈B has exactly one element. Then list the complete fiber profiles of f and g in the codomain order 4, 5, 6, and decide that f is bijective while g is neither injective nor surjective. Finally answer the different question of whether some bijection exists: one exists A→B (f is one), none exists {1,2,3}→{4,5}, and ∅→∅ has exactly one, the empty bijection."},"mko_ids":["mko-bijective"]},{"id":"derivation-mko-inverse-function-01","title":{"zh-Hant":"反函數：交換有限雙射表並求非零斜率仿射函數的反函數","en":"Inverse functions: exchanging a finite bijection table and inverting an affine map with nonzero slope"},"summary":{"zh-Hant":"（一）A={1,2,3}、B={4,5,6}，p(1)=5、p(2)=6、p(3)=4。先確認 p 是雙射，再交換有序對得到候選 q，驗證 q 是總函數 B→A，並逐側檢查 q(p(x))=x 與 p(q(y))=y。（二）f:ℝ→ℝ，f(x)=−4x+6（m=−4≠0）：求反函數 g(y)=uy+v，u、v 以最簡、正分母的有理數表示，並對所有實數驗證兩側恆等式。（三）說明 h(1)=4、h(2)=4、h(3)=6 的交換表不是 B→A 的函數，以及 f^{-1} 不是 1/f。","en":"(1) A={1,2,3}, B={4,5,6}, p(1)=5, p(2)=6, p(3)=4. First confirm that p is a bijection, then exchange its ordered pairs to get a candidate q, validate q as a total function B→A, and check q(p(x))=x and p(q(y))=y separately. (2) f:ℝ→ℝ, f(x)=−4x+6 (m=−4≠0): find the inverse g(y)=uy+v with u, v reduced rationals with positive denominators, and verify both identities for all reals. (3) Show that the exchanged table of h(1)=4, h(2)=4, h(3)=6 is not a function B→A, and that f^{-1} is not 1/f."},"mko_ids":["mko-inverse-function"]},{"id":"derivation-mko-random-variable-01","title":{"zh-Hant":"有限樣本空間的隨機變數：值域、原像事件與點機率","en":"Random variables on a finite sample space: range, preimage events and point probabilities"},"summary":{"zh-Hant":"S={1,2,3,4}，事件族 F=𝒫(S)，權重 w₁=1/2、w₂=1/4、w₃=1/8、w₄=1/8。隨機變數 X:S→ℝ 由表給出：X(1)=1、X(2)=0、X(3)=1、X(4)=4（即 X(s)=(s−2)²）。求 X 的值域、原像事件 {X=1}、{X≥1} 與 {X∈{2,3}}，以及每個值的點機率 P(X=t)，並證明這些點機率之和為 1。","en":"S={1,2,3,4}, event family F=𝒫(S), weights w₁=1/2, w₂=1/4, w₃=1/8, w₄=1/8. The random variable X:S→ℝ is given by the table X(1)=1, X(2)=0, X(3)=1, X(4)=4 (that is, X(s)=(s−2)²). Find the range of X, the preimage events {X=1}, {X≥1} and {X∈{2,3}}, and the point probability P(X=t) of each value, and prove that these point probabilities sum to 1."},"mko_ids":["mko-random-variable"]},{"id":"derivation-mko-expectation-01","title":{"zh-Hant":"有限模型的期望值：線性性的證明與相依變數的實例","en":"Expectation in finite models: proving linearity, with an instance of dependent variables"},"summary":{"zh-Hant":"在非空有限樣本空間 S、非負且總和為 1 的權重 w_s 下，證明 E[c]=c、E[aX+b]=aE[X]+b、E[X+Y]=E[X]+E[Y]（X、Y 定義在同一個 S 上，不假設獨立）。再在 S={1,2,3}、w₁=1/2、w₂=1/4、w₃=1/4，X=(2,−2,0)、Y=(1,3,−1) 上計算 E[X]、E[Y]、E[X+Y]、E[−3X+5]，說明 X、Y 不獨立而線性性仍成立，並以 E[XY]≠E[X]E[Y] 說明線性性不推出乘積公式。","en":"For a nonempty finite sample space S with nonnegative weights w_s summing to 1, prove E[c]=c, E[aX+b]=aE[X]+b and E[X+Y]=E[X]+E[Y] (X, Y on the same S, independence not assumed). Then on S={1,2,3} with w₁=1/2, w₂=1/4, w₃=1/4, X=(2,−2,0), Y=(1,3,−1), compute E[X], E[Y], E[X+Y] and E[−3X+5], show that X and Y are not independent while linearity still holds, and use E[XY]≠E[X]E[Y] to show that linearity does not give a product formula."},"mko_ids":["mko-expectation"]},{"id":"derivation-mko-variance-01","title":{"zh-Hant":"有限機率模型的變異數：平移、縮放與零變異數的刻畫","en":"Variance in finite probability models: translation, scaling and the zero-variance characterization"},"summary":{"zh-Hant":"在有限合法機率模型中證明 Var(X)≥0、Var(X+c)=Var(X)、Var(kX)=k²Var(X)，以及 Var(X)=0 ⇔ X 在所有正權重結果上取同一個值。再以 S={1,2,3,4}、w₁=1/2、w₂=1/4、w₃=1/4、w₄=0、X=(1,3,−1,10) 計算 μ=1、Var(X)=2、Var(X+5)=2、Var(−3X)=18，並以 Z=(2,2,2,7) 說明零權重結果上的不同值不影響零變異數。","en":"In a valid finite probability model, prove Var(X)≥0, Var(X+c)=Var(X), Var(kX)=k²Var(X), and Var(X)=0 ⇔ X takes one common value at every positive-weight outcome. Then with S={1,2,3,4}, w₁=1/2, w₂=1/4, w₃=1/4, w₄=0 and X=(1,3,−1,10), compute μ=1, Var(X)=2, Var(X+5)=2 and Var(−3X)=18, and use Z=(2,2,2,7) to show that a different value at a zero-weight outcome does not prevent zero variance."},"mko_ids":["mko-variance"]},{"id":"derivation-mko-order-relation-01","title":{"zh-Hant":"實數的通常順序：加法保序、正分母交叉比較、有限極值與閉區間","en":"The usual order on ℝ: translation, cross-multiplication with positive denominators, finite extrema and closed intervals"},"summary":{"zh-Hant":"證明對所有實數 a、b、c：a<b ⇔ a+c<b+c；q、s>0 時 p/q<r/s ⇔ ps<rq；非空有限實數集合有最小元素，且最小元素唯一。據此比較 M={−5/9, 1/3, −7/12, −1/2}（照此次序列出）的元素，求 M 的最小與最大元素，並判定兩端點與 1/2 是否屬於閉區間 [min M, max M]。","en":"Prove for all reals a, b, c that a<b ⇔ a+c<b+c; that p/q<r/s ⇔ ps<rq when q, s>0; and that a finite nonempty set of reals has a minimum, which is unique. Use these to compare the elements of M={−5/9, 1/3, −7/12, −1/2} (listed in this order), find the minimum and maximum of M, and decide whether the two endpoints and 1/2 belong to the closed interval [min M, max M]."},"mko_ids":["mko-order-relation"]},{"id":"derivation-mko-quadratic-function-01","title":{"zh-Hant":"頂點式二次函數：求值、對稱、全域最大值與限制區間","en":"Quadratic functions in vertex form: evaluation, symmetry, global maximum and a restricted domain"},"summary":{"zh-Hant":"q:ℝ→ℝ，q(x)=−2(x−3)²+5（a=−2、h=3、k=5）。依序求 q(1) 及其對稱點；證明對所有實數 x 有 q(6−x)=q(x)；證明 q 的全域最大值為 5、只在 x=3 取得，頂點為 (3,5)，且 q 沒有全域最小值；再說明限制在 [4,6] 時最大值是 3（在 x=4），以及 a=0 時結論不能沿用。","en":"q:ℝ→ℝ, q(x)=−2(x−3)²+5 (a=−2, h=3, k=5). Evaluate q(1) step by step and find its mirror point; prove q(6−x)=q(x) for every real x; prove that the global maximum of q is 5, attained only at x=3, that the vertex is (3,5), and that q has no global minimum; then show that on the restricted domain [4,6] the maximum is 3 (at x=4), and that the conclusions do not carry over when a=0."},"mko_ids":["mko-quadratic-function"]},{"id":"derivation-mko-rate-of-change-01","title":{"zh-Hant":"兩點間的平均變化率：資格、端點對調、仿射與二次公式","en":"Average rate of change between distinct inputs: eligibility, swapping endpoints, the affine and quadratic formulas"},"summary":{"zh-Hant":"（一）已知某指定函數 f 的 f(2)=7、f(−4)=3：判定資格並求平均變化率，化為最簡正分母分數，並說明同時對調兩端點不改變結果。（二）證明 f(x)=mx+b 的平均變化率為 m，並以 ℓ(x)=−3x+7 在 −2 與 4 之間核對。（三）證明 q(x)=a(x−h)²+k（a≠0）的平均變化率為 a(x₀+x₁−2h)，以 q(x)=−2(x−3)²+5 在 1 與 4 之間核對；並以對稱端點、改變第二端點與相同輸入三個情形說明它不是導數，也不能對相同輸入填 0。","en":"(1) For a specified function f with f(2)=7 and f(−4)=3: check eligibility, compute the average rate of change as a reduced fraction with positive denominator, and show that swapping both endpoints does not change it. (2) Prove that f(x)=mx+b has average rate m, and check it with ℓ(x)=−3x+7 between −2 and 4. (3) Prove that q(x)=a(x−h)²+k (a≠0) has average rate a(x₀+x₁−2h), check it with q(x)=−2(x−3)²+5 between 1 and 4, and use symmetric endpoints, a changed second endpoint and equal inputs to show that it is not a derivative and that 0 must not be assigned when the inputs coincide."},"mko_ids":["mko-rate-of-change"]},{"id":"derivation-mko-distance-formula-01","title":{"zh-Hant":"歐氏距離公式：計算距離、拒絕負根並以平方比較","en":"Euclidean distance formula: computing, rejecting the negative root and comparing by squares"},"summary":{"zh-Hant":"在同一個等單位正交座標系中，設 P=(1,2)、Q=(4,6)、R=(6,−1)。求 d(P,Q)，並說明候選值 −5 為何不是距離；求 d(P,R) 的精確形式並判定它不是整數；以平方比較 d(P,Q) 與 d(P,R)；最後核對重合點的距離為零，以及距離與兩點的書寫順序無關。","en":"In one equal-unit orthonormal Cartesian frame let P=(1,2), Q=(4,6) and R=(6,−1). Find d(P,Q) and explain why the candidate −5 is not a distance; find the exact form of d(P,R) and decide that it is not an integer; compare d(P,Q) with d(P,R) through their squares; finally check that coincident points have distance zero and that the distance does not depend on the order in which the two points are written."},"mko_ids":["mko-distance-formula"]},{"id":"derivation-mko-circle-01","title":{"zh-Hant":"圓：以距離平方判定點在圓內、圓上或圓外，並處理零半徑","en":"Circle: classifying points as inside, on or outside by squared distance, including radius zero"},"summary":{"zh-Hant":"在等單位正交座標系中，取中心 O=(2,−1)、半徑 r=5。判定 A=(5,3)、B=(4,1)、E=(−2,3) 各在圓內、圓上或圓外；驗證四個軸向點 O±(5,0)、O±(0,5) 都在 C(O,5) 上，並由兩個相對的軸向點算出直徑長 2r=10。再把半徑改為 r=0，證明 C(O,0)={O}、沒有圓內點，並判定 B 的位置。","en":"In an equal-unit orthonormal frame take center O=(2,−1) and radius r=5. Decide whether each of A=(5,3), B=(4,1), E=(−2,3) is strictly inside, on, or outside the circle; verify that the four axial points O±(5,0), O±(0,5) lie on C(O,5), and compute from two opposite axial points the diameter length 2r=10. Then change the radius to r=0, prove that C(O,0)={O} and that no point is strictly inside, and classify B."},"mko_ids":["mko-circle"]},{"id":"derivation-mko-area-01","title":{"zh-Hant":"面積：半開矩形的交集、聯集與單位格重算","en":"Area: intersection and union of half-open rectangles, recounted by unit cells"},"summary":{"zh-Hant":"在等單位正交座標系中，設 A=[0,4)×[0,3)、B=[2,6)×[1,5)。求 A 與 B 的面積，求出交集 A∩B 及其面積，用包含排除求 A∪B 的面積，並以「不同單位格的個數」獨立重算。再設 C=[4,7)×[0,3)（與 A 只在 x=4 處相觸）與 Z=[3,3)×[0,5)（寬為零），求 A∩C、A∪C 的面積與 Z 的面積。","en":"In an equal-unit orthonormal frame let A=[0,4)×[0,3) and B=[2,6)×[1,5). Find the areas of A and B, find the intersection A∩B and its area, find the area of A∪B by inclusion and exclusion, and recount it independently as the number of distinct unit cells. Then let C=[4,7)×[0,3) (touching A only at x=4) and Z=[3,3)×[0,5) (zero width), and find the areas of A∩C, A∪C and Z."},"mko_ids":["mko-area"]},{"id":"derivation-mko-descriptive-statistics-01","title":{"zh-Hant":"描述統計：保留重數計算平均數、中位數、全距與眾數集合","en":"Descriptive statistics: mean, median, range and mode set with repetitions kept"},"summary":{"zh-Hant":"對觀測列 D=(4,10,1,3,8,1,6,3)，保留重複值，求觀測數 n、算術平均、中位數、全距與本輪約定的眾數集合；證明一般性質「最小值 ≤ 平均 ≤ 最大值」並用它核對；最後說明若先去重會得到不同（錯誤）的平均與中位數。","en":"For the observation list D=(4,10,1,3,8,1,6,3), keeping repeated values, find the number of observations n, the arithmetic mean, the median, the range and the mode set under this entry's convention; prove the general fact 'minimum ≤ mean ≤ maximum' and use it as a check; finally show that deduplicating first would give a different (wrong) mean and median."},"mko_ids":["mko-descriptive-statistics"]},{"id":"derivation-mko-matrix-01","title":{"zh-Hant":"矩陣：尺寸、一起算索引、逐項加法與轉置","en":"Matrices: dimensions, one-based indexing, entrywise addition and transpose"},"summary":{"zh-Hant":"設 M=[[2,−1,0],[5,3,7]]、N=[[1,4,−2],[0,−3,6]]（每個內層括號是一行，row）。求兩者尺寸，讀出 M(1,2)、M(2,3)，判定 M(3,1) 是否存在；求 M+N、Mᵀ、Nᵀ；對任意同尺寸實矩陣證明 (A+B)ᵀ=Aᵀ+Bᵀ 與 (Aᵀ)ᵀ=A，並在 M、N 上核對；說明 M+Mᵀ 無定義；最後以反例區分轉置與「逐行倒序」及「反矩陣」，並把程式的零起算索引對應回一起算索引。","en":"Let M=[[2,−1,0],[5,3,7]] and N=[[1,4,−2],[0,−3,6]] (each inner bracket is one row). Find their dimensions, read off M(1,2) and M(2,3), and decide whether M(3,1) exists; compute M+N, Mᵀ and Nᵀ; prove (A+B)ᵀ=Aᵀ+Bᵀ and (Aᵀ)ᵀ=A for all real matrices of equal size and check them on M and N; explain why M+Mᵀ is undefined; finally use counterexamples to separate the transpose from reversing the rows and from the inverse matrix, and translate zero-based program indices back to one-based indices."},"mko_ids":["mko-matrix"]},{"id":"derivation-mko-linear-transformation-01","title":{"zh-Hant":"線性映射：證明 v↦Mv 線性，且仿射式 Mv+b 線性當且僅當 b=0","en":"Linear maps: proving v↦Mv is linear, and Mv+b is linear exactly when b=0"},"summary":{"zh-Hant":"對任意 2×2 實矩陣 M，證明 T(v)=Mv 是 ℝ²→ℝ² 的線性映射（對所有向量 u、v 與實數 c 滿足加法相容與純量相容）；證明仿射式 F(v)=Mv+b 線性當且僅當 b=0。以 M₀=[[2,−1],[3,4]]、b=(1,−2)ᵀ、u=(1,1)ᵀ、w=(2,−3)ᵀ 為例：算出 M₀u，說明 F 不線性並給出具體失效等式；以數字示範 T 的加法相容；最後以 g(x,y)=(x²,y) 說明有限測點不能證明任意映射線性。","en":"For every 2×2 real matrix M, prove that T(v)=Mv is a linear map ℝ²→ℝ² (additivity and homogeneity for all vectors u, v and real c); prove that the affine map F(v)=Mv+b is linear exactly when b=0. For the example M₀=[[2,−1],[3,4]], b=(1,−2)ᵀ, u=(1,1)ᵀ, w=(2,−3)ᵀ: compute M₀u, show that F is not linear with an explicit failing equation, and illustrate additivity of T numerically; finally use g(x,y)=(x²,y) to show that finitely many test points cannot prove that an arbitrary map is linear."},"mko_ids":["mko-linear-transformation"]},{"id":"derivation-mko-determinant-01","title":{"zh-Hant":"二階行列式：乘積公式、縮放公式與實數域可逆判準","en":"2×2 determinants: the product rule, the scaling rule and the invertibility criterion over ℝ"},"summary":{"zh-Hant":"對所有 2×2 實矩陣證明 det(AB)=det(A)det(B) 與 det(tM)=t²det(M)，並證明 M 在實數域可逆當且僅當 det(M)≠0。以 A=[[2,1],[−1,3]]、B=[[1,−2],[4,0]] 核對乘積公式並顯示 AB≠BA；以 t=−3 核對縮放公式；求 A⁻¹，並說明 A 雖是整數矩陣且 det(A)=7≠0，卻沒有整數反矩陣；以 C=[[1,2],[3,4]]（det=−2）說明非零行列式不必為正。","en":"For all 2×2 real matrices prove det(AB)=det(A)det(B) and det(tM)=t²det(M), and prove that M is invertible over ℝ exactly when det(M)≠0. With A=[[2,1],[−1,3]] and B=[[1,−2],[4,0]], check the product rule and show AB≠BA; check the scaling rule with t=−3; find A⁻¹ and explain why A, although an integer matrix with det(A)=7≠0, has no integer inverse; use C=[[1,2],[3,4]] (det=−2) to show that a nonzero determinant need not be positive."},"mko_ids":["mko-determinant"]},{"id":"derivation-mko-eigenvalue-01","title":{"zh-Hant":"二階實矩陣的特徵值：特徵多項式、判別式與非零特徵向量","en":"Eigenvalues of 2×2 real matrices: characteristic polynomial, discriminant and nonzero eigenvectors"},"summary":{"zh-Hant":"對 2×2 實矩陣證明：λ 是實特徵值 ⇔ det(M−λI)=0 ⇔ λ²−tr(M)λ+det(M)=0，且互異實特徵值的個數由 Δ=tr(M)²−4det(M) 的正負決定（負、零、正分別為 0、1、2）。以 M=[[4,1],[2,3]] 求出全部互異實特徵值與各自的非零特徵向量，說明候選 λ=3 的核 K₃={0} 而 3 不是特徵值，並拒絕候選特徵對 (3,0)；再以旋轉矩陣 R=[[0,−1],[1,0]]（Δ<0）與 J=[[3,1],[0,3]]（Δ=0）對照。","en":"For 2×2 real matrices prove: λ is a real eigenvalue ⇔ det(M−λI)=0 ⇔ λ²−tr(M)λ+det(M)=0, and the number of distinct real eigenvalues is decided by the sign of Δ=tr(M)²−4det(M) (negative, zero, positive give 0, 1, 2). For M=[[4,1],[2,3]] find all distinct real eigenvalues with nonzero eigenvectors, show that for the candidate λ=3 the kernel is K₃={0} and 3 is not an eigenvalue, and reject the candidate eigenpair (3,0); then contrast with the rotation R=[[0,−1],[1,0]] (Δ<0) and J=[[3,1],[0,3]] (Δ=0)."},"mko_ids":["mko-eigenvalue"]},{"id":"derivation-mko-congruence-01","title":{"zh-Hant":"全等：以 SSS 切片判定有明示對應的三角形全等","en":"Congruence: deciding congruence of labeled triangles by the SSS slice"},"summary":{"zh-Hant":"在同一個等單位正交平面中，A1=(0,0)、A2=(4,0)、A3=(0,3)，B1=(1,1)、B2=(1,5)、B3=(4,1)，對應明示為 Ai↔Bi。先以行列式驗證兩組點都是非退化三角形，再依 12、13、23 的順序精確比較平方邊長，判定 G(A,B)；說明 B 的方向與 A 相反卻仍全等；最後找出另一個對應 A1↔B1、A2↔B3、A3↔B2 的首個不等見證，並說明退化點組不合格。","en":"In one equal-unit orthonormal plane, let A1=(0,0), A2=(4,0), A3=(0,3) and B1=(1,1), B2=(1,5), B3=(4,1), with explicit correspondence Ai↔Bi. First verify by determinants that both triples are nondegenerate triangles, then compare squared side lengths exactly in the order 12, 13, 23 to decide G(A,B); explain why B has the opposite orientation and is still congruent; finally find the first unequal witness for the other correspondence A1↔B1, A2↔B3, A3↔B2, and show that a degenerate triple is not eligible."},"mko_ids":["mko-congruence"]},{"id":"derivation-mko-similarity-01","title":{"zh-Hant":"相似：以指定正倍率判定標記三角形相似並求面積","en":"Similarity: deciding labeled triangle similarity at a specified positive scale and finding the area"},"summary":{"zh-Hant":"在同一等單位正交平面中，A1=(0,0)、A2=(4,0)、A3=(0,2)，B1=(1,1)、B2=(1,7)、B3=(4,1)，對應 Ai↔Bi，倍率方向為 |BiBj|=k|AiAj|，指定 k=3/2。判定兩者在 k=3/2 下相似，並由已知面積 Area(A)=4 求 Area(B)；再說明 k=2 只被否定為見證、k=−3/2 不合資格，以及方向相反不改變正倍率。","en":"In one equal-unit orthonormal plane, let A1=(0,0), A2=(4,0), A3=(0,2) and B1=(1,1), B2=(1,7), B3=(4,1), with correspondence Ai↔Bi and scale direction |BiBj|=k|AiAj|, at the specified k=3/2. Decide that they are similar at k=3/2, and from the given area Area(A)=4 find Area(B); then explain why k=2 is rejected only as a witness, why k=−3/2 is not eligible, and why opposite orientation does not change the positive scale."},"mko_ids":["mko-similarity"]},{"id":"derivation-mko-geometric-transformations-01","title":{"zh-Hant":"幾何變換：證明基本等距並分類均勻縮放","en":"Geometric transformations: proving the basic isometries and classifying uniform scaling"},"summary":{"zh-Hant":"在等單位正交實平面中證明：平移、原點逆時針 90° 旋轉 R90(x,y)=(−y,x) 以及 MKO 列出的其他旋轉與軸鏡射都是等距映射；半轉 (−x,−y) 不是所列四條軸中任何一條的鏡射；均勻縮放 Hc(x,y)=(cx,cy) 使每個距離乘 |c|。據此分類 c=0、c=−1、c=−2、c=2，並以 P=(3,1)、Q=(−1,4) 做數值核對。","en":"On the equal-unit orthonormal real plane, prove that translations, the counterclockwise quarter-turn R90(x,y)=(−y,x) about the origin, and the other rotations and axis reflections listed in the MKO are isometries; that the half-turn (−x,−y) is not a reflection in any of the listed axes; and that uniform scaling Hc(x,y)=(cx,cy) multiplies every distance by |c|. Use this to classify c=0, c=−1, c=−2 and c=2, with a numerical check on P=(3,1), Q=(−1,4)."},"mko_ids":["mko-geometric-transformations"]},{"id":"derivation-mko-sequence-limit-01","title":{"zh-Hant":"數列極限：倒數模型的精確尾界與 ε-N 證明","en":"Sequence limits: the exact tail cutoff and an ε-N proof for the reciprocal model"},"summary":{"zh-Hant":"對 u(n)=2−3/(n+1)（n∈ℕ0，即 MKO 模型 L=2、c=−3、d=1）：(1) 證明候選尾界 N 對 ε 合法當且僅當 3/(N+1)<ε；(2) 求 ε=1/2 的最小合法 N，並說明 N=5 為何不合法；(3) 以 ε-N 定義證明 u(n)→2；(4) 說明 N 必須依 ε 選取。","en":"For u(n)=2−3/(n+1) with n∈ℕ0 (the MKO model with L=2, c=−3, d=1): (1) prove that a candidate cutoff N is valid for ε exactly when 3/(N+1)<ε; (2) find the least valid N for ε=1/2 and explain why N=5 fails; (3) prove u(n)→2 from the ε-N definition; (4) show that N must be chosen after ε."},"mko_ids":["mko-sequence-limit"]},{"id":"derivation-mko-function-limit-01","title":{"zh-Hant":"函數極限：可去因子商、仿射 δ 界與不等的單側極限","en":"Function limits: a removable-factor quotient, affine δ bounds and unequal one-sided limits"},"summary":{"zh-Hant":"在去心實數論域上：(1) 以 ε-δ 證明 f(x)=(x²−9)/(x−3) 滿足 lim_(x→3) f(x)=6，而 f(3) 仍無定義；(2) 對仿射 g(x)=4x−1 在 a=2、ε=1/2，證明 δ=1/8 合法（等號 |m|δ=ε 允許），δ=1/7 不合法；(3) 對分支函數 h(x)=x+1（x<1）、h(x)=2x+3（x>1），求左右極限 2 與 5，並證明兩側極限不存在。","en":"On punctured real domains: (1) prove by ε-δ that f(x)=(x²−9)/(x−3) has lim_(x→3) f(x)=6 while f(3) stays undefined; (2) for the affine g(x)=4x−1 at a=2 with ε=1/2, prove that δ=1/8 is valid (the equality |m|δ=ε is allowed) and δ=1/7 is not; (3) for the branch function h(x)=x+1 (x<1), h(x)=2x+3 (x>1), find the one-sided limits 2 and 5 and prove that the two-sided limit does not exist."},"mko_ids":["mko-function-limit"]},{"id":"derivation-mko-continuity-01","title":{"zh-Hant":"連續：分支函數的中心值與可去商的唯一補點","en":"Continuity: centre values of branch functions and the unique continuous extension across a removable singularity"},"summary":{"zh-Hant":"(1) 對 f_V(x)=2x+1（x<1）、f_V(1)=V、f_V(x)=5x−2（x>1），證明 V=3 時 f_V 在 1 連續，V=4 時不連續；(2) 對 g(x)=2x+1（x<1）、g(1)=3、g(x)=5x−1（x>1），證明 g 在 1 不連續；(3) 對 q(x)=(x²−4)/(x−2)（論域 ℝ∖{2}），證明全實數連續延拓在 2 的值必為 4，且 4 確實可行。","en":"(1) For f_V(x)=2x+1 (x<1), f_V(1)=V, f_V(x)=5x−2 (x>1), prove that f_V is continuous at 1 when V=3 and not when V=4; (2) for g(x)=2x+1 (x<1), g(1)=3, g(x)=5x−1 (x>1), prove that g is not continuous at 1; (3) for q(x)=(x²−4)/(x−2) on ℝ∖{2}, prove that a continuous extension to all real numbers must take the value 4 at 2, and that 4 does work."},"mko_ids":["mko-continuity"]},{"id":"derivation-mko-derivative-01","title":{"zh-Hant":"導數：由差商極限求二次式導數並判定尖點無導數","en":"The derivative: a quadratic's derivative from the difference-quotient limit, and no derivative at a corner"},"summary":{"zh-Hant":"對 f(t)=3t²−2t+5：先對 h≠0 化簡差商 (f(1+h)−f(1))/h=4+3h，再以 ε-δ 證明 f'(1)=4；推導一般公式：差商為 2a2x+a1+a2h、導數為 2a2x+a1；說明有限步 h=1/10 的差商 4.3 不是導數、a2=0 的仿射邊界；最後證明 g(t)=|t−2| 在 t=2 沒有導數（不是 0），而在 t>2 處導數為 1。","en":"For f(t)=3t²−2t+5: first simplify the difference quotient for h≠0, (f(1+h)−f(1))/h=4+3h, then prove f'(1)=4 by ε-δ; derive the general formulas: the quotient is 2a2x+a1+a2h and the derivative 2a2x+a1; explain why the finite-step quotient 4.3 at h=1/10 is not the derivative, and the affine boundary a2=0; finally prove that g(t)=|t−2| has no derivative at t=2 (not 0), while its derivative is 1 for t>2."},"mko_ids":["mko-derivative"]},{"id":"derivation-mko-derivative-rules-01","title":{"zh-Hant":"微分法則：以展開求導核對乘積與鏈式規則","en":"Derivative rules: checking the product and chain rules against differentiating the expansion"},"summary":{"zh-Hant":"以升冪係數列表 f=[2,0,1]、g=[1,3]（即 f(x)=2+x²、g(x)=1+3x）：先由導數定義證明三次以下多項式的求導公式；再以兩條獨立路徑求 (fg)' 與 (f∘g)'——展開後求導，以及使用乘積、鏈式規則——並確認兩者對每個 x 相同；同時核對和與純量倍的規則，指出 f'g'、漏乘內導數、外導數取錯點、反讀係數等錯誤，最後求 x=1 的值。","en":"With the ascending coefficient lists f=[2,0,1] and g=[1,3] (that is f(x)=2+x² and g(x)=1+3x): first prove from the definition of the derivative the differentiation formula for polynomials of degree at most three; then find (fg)' and (f∘g)' along two independent routes, differentiating the expansion and applying the product and chain rules, and confirm they agree at every x; also check the sum and scalar rules, expose the errors f'g', a missing inner derivative, an outer derivative at the wrong point and reversed coefficient reading, and evaluate at x=1."},"mko_ids":["mko-derivative-rules"]},{"id":"derivation-mko-antiderivative-01","title":{"zh-Hant":"原函數：多項式代表原函數、常數族與候選驗收","en":"Antiderivatives: the representative polynomial antiderivative, the constant family and checking candidates"},"summary":{"zh-Hant":"對 f=[3,−4,6]（f(x)=3−4x+6x²）：由定義證明四次以下多項式的求導公式；構造代表原函數 F0=[0,3,−2,2] 並驗證 F0'=f；證明 F0+C 對每個實數 C 都是原函數，且在至多五個係數的多項式中每個原函數都是 F0+C；驗收候選 G=[7,3,−2,2]（接受）、H=[0,3,−4,2]（拒絕）、K=[0,3,−3,8/3]（兩個測點相符仍拒絕）；求 F0(2)+C（C=−5），並說明原函數不是反函數。","en":"For f=[3,−4,6] (f(x)=3−4x+6x²): prove from the definition the derivative formula for polynomials of degree at most four; build the representative antiderivative F0=[0,3,−2,2] and verify F0'=f; prove that F0+C is an antiderivative for every real C and that, among polynomials with at most five coefficients, every antiderivative is F0+C; check the candidates G=[7,3,−2,2] (accept), H=[0,3,−4,2] (reject) and K=[0,3,−3,8/3] (reject although it matches at two test points); evaluate F0(2)+C with C=−5; and explain why an antiderivative is not an inverse function."},"mko_ids":["mko-antiderivative"]},{"id":"derivation-mko-definite-integral-01","title":{"zh-Hant":"定積分：以左端和的極限與矩公式計算 ∫₀³(x²−4)dx","en":"Definite integral: computing ∫₀³(x²−4)dx as a limit of left sums and by the moment formula"},"summary":{"zh-Hant":"設 f(x)=x²−4、a=0、b=3。(1) 只用 Riemann 定義（所有合法標記分割和在網格趨 0 時的共同極限）與 f 可積的前提，由 N 等分左端和 L_N 的極限求出 I(f,0,3)；(2) 用本 MKO 明列的矩公式獨立再算一次並核對；(3) 求反向端點的 I(f,3,0)；(4) 以 N=3、N=50 說明有限左端和不等於積分，並說明有號積分不是非負面積。","en":"Let f(x)=x²−4, a=0, b=3. (1) Using only the Riemann definition (the common limit of all valid tagged-partition sums as the mesh tends to zero) and the premise that f is integrable, find I(f,0,3) as the limit of the N-piece left sums L_N; (2) recompute it independently with the moment formula listed in this MKO and compare; (3) find the reversed integral I(f,3,0); (4) use N=3 and N=50 to show that a finite left sum is not the integral, and explain why the signed integral is not a nonnegative area."},"mko_ids":["mko-definite-integral"]},{"id":"derivation-mko-fundamental-theorem-calculus-01","title":{"zh-Hant":"微積分基本定理：由連續性證明 G′=f，再推出 ∫_a^b f=F(b)−F(a)","en":"Fundamental theorem of calculus: proving G′=f from continuity, then ∫_a^b f=F(b)−F(a)"},"summary":{"zh-Hant":"設 f:ℝ→ℝ 在全 ℝ 連續，a∈ℝ，G(x)=∫_a^x f(t)dt 為有向積分。(1) 證明 G 在每一點可微且 G′(x)=f(x)；(2) 若 F:ℝ→ℝ 可微且 F′=f，證明對任意實數 a、b 有 ∫_a^b f(t)dt=F(b)−F(a)；(3) 以 f(t)=6t²−2t+1 與候選 F(x)=2x³−x²+x+1/2 算出 ∫_2^{−1} f(t)dt，並說明 G(2)=0 不推出 G′(2)=0、上下限在 x=0 重合也不推出導數為 0。","en":"Let f:ℝ→ℝ be continuous on all of ℝ, a∈ℝ, and G(x)=∫_a^x f(t)dt an oriented integral. (1) Prove that G is differentiable at every point with G′(x)=f(x); (2) if F:ℝ→ℝ is differentiable with F′=f, prove ∫_a^b f(t)dt=F(b)−F(a) for arbitrary real a, b; (3) with f(t)=6t²−2t+1 and the candidate F(x)=2x³−x²+x+1/2, compute ∫_2^{−1} f(t)dt, and explain why G(2)=0 does not give G′(2)=0 and why bounds coinciding at x=0 do not give a zero derivative."},"mko_ids":["mko-fundamental-theorem-calculus"]},{"id":"derivation-mko-tree-01","title":{"zh-Hant":"樹：判定一個 6 頂點圖是樹，求葉、唯一道路與加邊後的唯一環","en":"Trees: deciding that a 6-vertex graph is a tree, with its leaves, a unique path and the unique cycle after adding an edge"},"summary":{"zh-Hant":"設 G=(V,E)，V={1,2,3,4,5,6}，E={{1,2},{2,3},{2,4},{4,5},{4,6}}。(1) 依 MKO 定義證明 G 是樹（V 非空、連通、無簡單環）；(2) 列出葉；(3) 求從 3 到 6 的道路並證明它唯一；(4) 加入邊 {3,6}，寫出所形成的簡單環並證明它是唯一的簡單環；(5) 對照：三角形加一個孤立點雖有 |E|=|V|−1 卻不是樹；單點無邊圖是樹但沒有葉。","en":"Let G=(V,E) with V={1,2,3,4,5,6} and E={{1,2},{2,3},{2,4},{4,5},{4,6}}. (1) Prove from the MKO's definition that G is a tree (V nonempty, connected, no simple cycle); (2) list its leaves; (3) find the path from 3 to 6 and prove it is unique; (4) add the edge {3,6}, write the simple cycle it creates and prove it is the only simple cycle; (5) contrast: a triangle plus an isolated vertex has |E|=|V|−1 but is not a tree; a single vertex without edges is a tree with no leaf."},"mko_ids":["mko-tree"]},{"id":"derivation-mko-topological-space-01","title":{"zh-Hant":"拓撲空間：完整驗證三點集合上的一個鏈狀開集族，並辨認非拓撲","en":"Topological space: fully verifying a chain of open sets on a three-point set, and recognising non-topologies"},"summary":{"zh-Hant":"設 X={a,b,c}（a、b、c 相異）。(1) 證明 τ₁={∅,{a},{a,b},X} 是 X 上的拓撲，涵蓋全部 16 個子族（含空族）；(2) 證明 τ₂={∅,{a},{b},X} 不是拓撲，並指出它通過了哪些檢查；(3) 比較 τ₁ 與不可分拓撲、離散拓撲的粗細；(4) 在 ℕ 上，證明 τ_fin={ℕ 的有限子集}∪{ℕ} 對兩兩聯集與有限交集封閉，卻不是拓撲。","en":"Let X={a,b,c} with a, b, c distinct. (1) Prove that τ₁={∅,{a},{a,b},X} is a topology on X, covering all 16 subfamilies (including the empty one); (2) prove that τ₂={∅,{a},{b},X} is not a topology, and say which checks it does pass; (3) compare τ₁ with the indiscrete and discrete topologies; (4) on ℕ, prove that τ_fin={finite subsets of ℕ}∪{ℕ} is closed under pairwise unions and finite intersections but is not a topology."},"mko_ids":["mko-topological-space"]},{"id":"derivation-mko-natural-divisibility-01","title":{"zh-Hant":"自然數整除：以乘數見證判定、處理 0 的情形並證明遞移性","en":"Natural-number divisibility: deciding by a multiplier witness, handling zero, and proving transitivity"},"summary":{"zh-Hant":"依 MKO 定義（a∣b ⇔ 存在自然數 k 使 b=a·k）：(1) 證明 6∣84；(2) 證明 6∤87；(3) 證明 4∤10，雖然 10/4=5/2 是一個數；(4) 證明每個 a 都整除 0；(5) 證明 0∣b ⇔ b=0，並說明 0∣0 的見證不唯一、0∤5；(6) 證明 a>0 時見證至多一個（若 a∣b，則既有見證唯一）；(7) 證明整除的自反性與遞移性，並由 3∣6、6∣84 求出 3∣84 的見證。","en":"From the MKO's definition (a∣b ⇔ there is a natural k with b=a·k): (1) prove 6∣84; (2) prove 6∤87; (3) prove 4∤10, although 10/4=5/2 is a number; (4) prove that every a divides 0; (5) prove 0∣b ⇔ b=0, and explain why the witness for 0∣0 is not unique and why 0∤5; (6) prove that for a>0 there is at most one witness (and when a∣b that existing witness is unique); (7) prove reflexivity and transitivity of divisibility, and from 3∣6 and 6∣84 find a witness for 3∣84."},"mko_ids":["mko-natural-divisibility"]},{"id":"derivation-mko-gcd-coprime-01","title":{"zh-Hant":"最大公因數與互質：以整除刻畫驗證歐幾里得算法","en":"Greatest common divisor and coprimality: checking Euclid's algorithm against the divisibility characterization"},"summary":{"zh-Hant":"依 MKO 的整除刻畫（d∣a、d∣b，且每個公因數 c 都整除 d）：(1) 證明滿足刻畫的 d 至多一個；(2) 證明歐幾里得算法每一步保持公因數集合且必會停止，並以它算出 gcd(84,36)=12，再直接驗證 12 滿足刻畫；(3) 說明公因數 6 不是 gcd；(4) 算出 gcd(8,15)=1，說明 8 與 15 互質但都不是質數；(5) 處理含 0 的情形 gcd(0,1)=1、gcd(0,6)=6、gcd(0,0)=0。","en":"From the MKO's divisibility characterization (d∣a, d∣b, and every common divisor c divides d): (1) prove that at most one d satisfies it; (2) prove that each step of Euclid's algorithm preserves the set of common divisors and that the algorithm stops, use it to compute gcd(84,36)=12, and verify directly that 12 satisfies the characterization; (3) explain why the common divisor 6 is not the gcd; (4) compute gcd(8,15)=1 and explain that 8 and 15 are coprime although neither is prime; (5) treat the zero cases gcd(0,1)=1, gcd(0,6)=6 and gcd(0,0)=0."},"mko_ids":["mko-gcd-coprime"]},{"id":"derivation-mko-prime-number-01","title":{"zh-Hant":"自然數質數：證明 97 是質數、91 是合數，並排除 0 與 1","en":"Prime natural numbers: proving 97 prime and 91 composite, and excluding 0 and 1"},"summary":{"zh-Hant":"依 MKO 定義（p 為質數 ⇔ 2≤p 且 p 的每個自然數因數 d 都滿足 d=1 或 d=p）：(1) 說明 0 與 1 不是質數；(2) 證明 2 是質數；(3) 以見證 7 證明 91 是合數，並說明只測 2、3、5 會誤判；(4) 證明「若 n=dk 且 d、k≥2，則 min(d,k)²≤n」，據此只需檢查 2 到 9，證明 97 是質數；(5) 證明 2 是唯一的偶質數。","en":"From the MKO's definition (p is prime ⇔ 2≤p and every natural divisor d of p satisfies d=1 or d=p): (1) explain why 0 and 1 are not prime; (2) prove that 2 is prime; (3) prove that 91 is composite with the witness 7, and show that testing only 2, 3, 5 misjudges it; (4) prove 'if n=dk with d, k≥2 then min(d,k)²≤n', and use it to prove that 97 is prime by checking only 2 to 9; (5) prove that 2 is the only even prime."},"mko_ids":["mko-prime-number"]},{"id":"derivation-mko-modular-congruence-01","title":{"zh-Hant":"自然數模同餘：餘數定義、見證形式、運算保持與取消的限制","en":"Congruence modulo a positive modulus: the remainder definition, the witness form, preservation by operations and the limits of cancellation"},"summary":{"zh-Hant":"依 MKO 定義（n>0，a≡b (mod n) ⇔ a mod n=b mod n）：(1) 判定 100≡23 (mod 7)；(2) 證明它等價於見證形式「存在自然數 k、l 使 a+nk=b+nl」，並給出 100 與 23 的見證；(3) 說明同餘是等價關係；(4) 證明加法與乘法保持同餘，並由 15≡1 (mod 7) 得 115≡24、1500≡23 (mod 7)；(5) 以 2·3≡2·0 (mod 6) 說明不能無條件取消公因子；(6) 說明不同模數不能串接，並說明模數 0 被排除。","en":"From the MKO's definition (n>0, a≡b (mod n) ⇔ a mod n=b mod n): (1) decide 100≡23 (mod 7); (2) prove that this is equivalent to the witness form 'there are naturals k, l with a+nk=b+nl', and give witnesses for 100 and 23; (3) explain why congruence is an equivalence relation; (4) prove that addition and multiplication preserve congruence, and from 15≡1 (mod 7) obtain 115≡24 and 1500≡23 (mod 7); (5) use 2·3≡2·0 (mod 6) to show that a common factor cannot be cancelled unconditionally; (6) explain why different moduli cannot be chained, and why the modulus 0 is excluded."},"mko_ids":["mko-modular-congruence"]},{"id":"derivation-mko-decision-problem-01","title":{"zh-Hant":"決定問題：在明示有限輸入集上證明試除程序是 decider","en":"Decision problem: proving that a trial-division procedure is a decider on an explicit finite input set"},"summary":{"zh-Hant":"取輸入集合 X={0,1,2,3,4,5,6,7,8,9}、規格 P(x)：「x 是質數」與 yes-instance 集合 Y={x∈X ∣ P(x)}。在明示的 while 程式計算模型 W 中，證明試除程序 D1 對每個 x∈X 都終止，且 D1(x)=true 當且僅當 x∈Y，因此 D1 是 (X,Y) 的 decider；再說明只在 yes-instance 上停止的 D2 與只看奇偶的 D3 都不是 decider。","en":"Take the input set X={0,1,2,3,4,5,6,7,8,9}, the specification P(x): 'x is prime', and the yes-instance set Y={x∈X ∣ P(x)}. In the explicit while-program model W, prove that the trial-division procedure D1 terminates on every x∈X and that D1(x)=true exactly when x∈Y, so D1 is a decider for (X,Y); then show that D2, which halts only on yes-instances, and D3, which tests only parity, are not deciders."},"mko_ids":["mko-decision-problem"]},{"id":"derivation-mko-lean-formal-statement-01","title":{"zh-Hant":"證明助理中的形式陳述：逐層讀懂一個 Lean 陳述並區分「是陳述」與「已證明」","en":"Formal statements in a proof assistant: reading a Lean statement layer by layer and separating 'is a statement' from 'is proved'"},"summary":{"zh-Hant":"以源碼字串 s：`∀ (a b : Nat), a ≤ b → a < b + 1` 為教學對象，在一個明示固定但未實際執行的環境 E 中逐層說明：它的約束變數與型別、假設與結論、若 elaboration 成功應得到的核心項 S、為何 S 的型別會是 Prop、一個證明項必須建立什麼；再給出其數學內容的非正式證明，並以 s′：`∀ (a : Nat), a - 1 < a`（是 Prop 但為假）與 s″：`∀ (a : Int), a - 1 < a`（為真）對照。全程未執行 Lean，不宣稱任何內容可解析、可 elaboration、已編譯或已被 Lean 證明。","en":"Take the source string s: `∀ (a b : Nat), a ≤ b → a < b + 1` as a teaching object and, in an environment E that is fixed explicitly but never executed, explain layer by layer: its bound variables and their types, its hypothesis and conclusion, the core term S that elaboration would have to produce if it succeeded, why S would have type Prop, and what a proof term would have to establish; then give an informal proof of its mathematical content and contrast it with s′: `∀ (a : Nat), a - 1 < a` (a Prop, but false) and s″: `∀ (a : Int), a - 1 < a` (true). Lean is not run at any point, and nothing here claims that any text parses, elaborates, compiles or is proved by Lean."},"mko_ids":["mko-lean-formal-statement"]},{"id":"derivation-mko-open-closed-set-01","title":{"zh-Hant":"開集與閉集：在四點拓撲中完整分類十六個子集","en":"Open and closed sets: classifying all sixteen subsets of a four-point topology"},"summary":{"zh-Hant":"設 X={1,2,3,4}，τ={∅,{1},{1,2},{3,4},{1,3,4},X}。先驗證 τ 是 X 上的拓撲，再列出全部閉集，並把 X 的十六個子集分成「既開又閉」「只開」「只閉」「既不開也不閉」四類；最後證明在任何拓撲空間中 ∅ 與 X 都既開又閉，以及兩個閉集的聯集仍是閉集。","en":"Let X={1,2,3,4} and τ={∅,{1},{1,2},{3,4},{1,3,4},X}. First verify that τ is a topology on X, then list all closed sets and sort the sixteen subsets of X into four classes: clopen, open only, closed only, and neither; finally prove that in every topological space ∅ and X are clopen, and that the union of two closed sets is closed."},"mko_ids":["mko-open-closed-set"]},{"id":"derivation-mko-neighborhood-01","title":{"zh-Hant":"鄰域：在四點拓撲中搜尋開集見證並列出全部鄰域","en":"Neighborhood: searching for open witnesses and listing all neighborhoods in a four-point topology"},"summary":{"zh-Hant":"設 X={1,2,3,4}，τ={∅,{1},{1,2},{3,4},{1,3,4},X}。判定 N1={1,2,3} 與 N2={2,3,4} 是否為點 2 的鄰域；證明在有限拓撲中，點 x 的鄰域恰是包含 O_x（含 x 的全部開集之交）的子集，據此列出點 1、2、3、4 的全部鄰域；證明鄰域向上封閉；並說明同一集合在離散拓撲下的判定可能不同。","en":"Let X={1,2,3,4} and τ={∅,{1},{1,2},{3,4},{1,3,4},X}. Decide whether N1={1,2,3} and N2={2,3,4} are neighborhoods of the point 2; prove that in a finite topology the neighborhoods of x are exactly the subsets containing O_x (the intersection of all open sets containing x), and use this to list all neighborhoods of the points 1, 2, 3 and 4; prove that neighborhoods are closed upwards; and show that the verdict for the same set can change under the discrete topology."},"mko_ids":["mko-neighborhood"]},{"id":"derivation-mko-topological-continuity-01","title":{"zh-Hant":"拓撲連續映射：在有限空間中用逆像判定連續，並與開映射區分","en":"Continuous maps between topological spaces: deciding continuity by preimages in finite spaces and separating it from openness"},"summary":{"zh-Hant":"設 X={1,2,3}，τX={∅,{1},{1,2},X}；Y={a,b}，τY={∅,{a},Y}。對三個以完整表格給出的映射 f（1↦a、2↦a、3↦b）、g（1↦a、2↦b、3↦a）、c（全部 ↦b），用逆像條件判定是否連續，並判定是否為開映射；證明連續映射的合成仍連續並以 q:Y→Z 驗證；最後數出 X→Y 的全部連續映射。","en":"Let X={1,2,3} with τX={∅,{1},{1,2},X}, and Y={a,b} with τY={∅,{a},Y}. For three maps given by complete tables, f (1↦a, 2↦a, 3↦b), g (1↦a, 2↦b, 3↦a) and c (everything ↦b), decide continuity by the preimage condition and decide whether each is an open map; prove that a composite of continuous maps is continuous and check it with q:Y→Z; finally count all continuous maps X→Y."},"mko_ids":["mko-topological-continuity"]},{"id":"derivation-mko-computational-model-01","title":{"zh-Hant":"抽象計算模型：在七狀態模型中截斷停止邊、計算可達集並區分停止、卡住與不終止","en":"Abstract computational model: truncating halt edges, computing reachability and separating halting, stuck and non-terminating runs in a seven-state model"},"summary":{"zh-Hant":"給定有限模型 (S,Init,Step,Halt)：S={s0,s1,s2,s3,s4,s5,s6}，Init={s0}，Halt={s3}，原始邊資料 {(s0,s1),(s0,s2),(s1,s3),(s1,s5),(s2,s4),(s4,s2),(s3,s6)}。依 MKO 規則截斷停止狀態的外出邊，以不動點迭代算出可達集，找出停止狀態、卡住狀態與可達循環，給出合法的停止、卡住與無窮執行，排除三條不合法的 trace，並說明為何尚未窮盡可達圖的有限深度探索，不能僅憑未見反例就斷言所有極大執行終止。","en":"Given the finite model (S,Init,Step,Halt) with S={s0,s1,s2,s3,s4,s5,s6}, Init={s0}, Halt={s3} and raw edge data {(s0,s1),(s0,s2),(s1,s3),(s1,s5),(s2,s4),(s4,s2),(s3,s6)}: truncate edges out of halt states as the MKO requires, compute the reachable set by fixpoint iteration, identify halt states, stuck states and reachable cycles, exhibit legal halting, stuck and infinite executions, rule out three illegal traces, and explain why depth-bounded exploration that has not exhausted the reachable graph cannot certify termination of all maximal executions merely from the absence of an observed counterexample."},"mko_ids":["mko-computational-model"]},{"id":"derivation-mko-algorithm-correctness-01","title":{"zh-Hant":"算法正確性：非確定性相鄰交換排序的部分正確、終止與總正確","en":"Algorithm correctness: partial correctness, termination and total correctness of nondeterministic adjacent-swap sorting"},"summary":{"zh-Hant":"在有限狀態模型 M 中，狀態是 (1,2,3) 的六個排列，初始狀態 (3,2,1)，每步可非確定性地交換任一個相鄰逆序對，Halt={(1,2,3)}。以後置條件「遞增排列且為輸入的重排」證明 M 的部分正確性、終止性與總正確性：用排名函數 inv（逆序數）證明每條執行至多 3 步，且每條極大執行停在 Halt。再以變體 M′（允許交換任何相鄰對）說明一條成功路徑與部分正確都不足以得到總正確，以變體 M″（錯誤的停止條件）說明終止不足以得到正確。","en":"In a finite-state model M whose states are the six permutations of (1,2,3), with initial state (3,2,1), each step may nondeterministically swap any adjacent inverted pair, and Halt={(1,2,3)}. With the postcondition 'increasing, and a rearrangement of the input', prove partial correctness, termination and total correctness of M, using the ranking function inv (the number of inversions) to show every execution takes at most 3 steps and every maximal execution stops in Halt. Then use the variant M′ (any adjacent pair may be swapped) to show that one successful path and partial correctness do not give total correctness, and the variant M″ (a wrong halting condition) to show that termination does not give correctness."},"mko_ids":["mko-algorithm-correctness"]},{"id":"derivation-mko-asymptotic-resource-bound-01","title":{"zh-Hant":"漸近資源界：證明 3n²+5n+2∈Θ(n²) 並排除 Θ(n)","en":"Asymptotic resource bounds: proving 3n²+5n+2∈Θ(n²) and ruling out Θ(n)"},"summary":{"zh-Hant":"設成本函數 f(n)=3n²+5n+2（n 從 0 起），比較函數 g(n)=n² 與 h(n)=n。以明示的全尾段 witness 證明 f∈O(g)（c=10、N=1）、f∈Ω(g)（c=3、N=0），故 f∈Θ(g)；說明 N=0 對 O 不可行、c=3 對 O 不可行、另一組 witness c=4、N=6 也可行；再證明 f∉O(h)，因此 f∈Ω(h) 但 f∉Θ(h)。","en":"Let the cost function be f(n)=3n²+5n+2 (n starting at 0), with comparison functions g(n)=n² and h(n)=n. With explicit full-tail witnesses prove f∈O(g) (c=10, N=1) and f∈Ω(g) (c=3, N=0), hence f∈Θ(g); show that N=0 cannot work for O, that c=3 cannot work for O, and that another witness c=4, N=6 also works; then prove f∉O(h), so f∈Ω(h) but f∉Θ(h)."},"mko_ids":["mko-asymptotic-resource-bound"]},{"id":"derivation-mko-lean-proof-term-01","title":{"zh-Hant":"證明項：逐部分解讀 λh. ⟨snd h, fst h⟩ 為何是 P∧Q→Q∧P 的證明項","en":"Proof term: reading λh. ⟨snd h, fst h⟩ part by part as a proof term for P∧Q→Q∧P"},"summary":{"zh-Hant":"在一個明示的合成小型命題型別系統 ST 中（對應 MKO 假設 a2），取相異命題原子 P、Q：逐部分說明項 t=λh:P∧Q. ⟨snd h, fst h⟩ 的每一部分建立了什麼、具有什麼型別，推導判斷 ⊢ t : P∧Q→Q∧P，並說明 t 的型別為什麼與命題 S=P∧Q→Q∧P 相符；再證明外觀相近的 t′=λh:P∧Q. ⟨fst h, snd h⟩ 不是 S 的證明項。最後區分「ST 中的型別推導」「Lean kernel 接受」與「已保存的 proof-term receipt」三者。","en":"In an explicit synthetic small propositional type system ST (matching MKO assumption a2), with distinct propositional atoms P and Q: explain part by part what each piece of the term t=λh:P∧Q. ⟨snd h, fst h⟩ establishes and what type it has, derive the judgment ⊢ t : P∧Q→Q∧P, and explain why the type of t matches the proposition S=P∧Q→Q∧P; then prove that the similar-looking t′=λh:P∧Q. ⟨fst h, snd h⟩ is not a proof term for S. Finally separate three things: a typing derivation in ST, acceptance by the Lean kernel, and a saved proof-term receipt."},"mko_ids":["mko-lean-proof-term"]},{"id":"derivation-mko-discrete-dynamical-system-01","title":{"zh-Hant":"離散時間動力系統：在七個狀態上迭代 T(x)=(3x+1) mod 7","en":"Discrete-time dynamical system: iterating T(x)=(3x+1) mod 7 on seven states"},"summary":{"zh-Hant":"狀態空間 X={0,1,2,3,4,5,6}，更新映射 T(x)=(3x+1) mod 7。由 x_0=0 依 x_(n+1)=T(x_n) 算出軌道，證明對所有自然數 n 有 x_(n+6)=x_n，據此求 x_2026；完整枚舉固定點；判定 A={0,1,2,4,5,6} 是正向不變子集而 B={0,1} 不是，並證明從 0 出發的軌道永遠不會到達固定點。","en":"State space X={0,1,2,3,4,5,6}, update map T(x)=(3x+1) mod 7. From x_0=0 compute the orbit given by x_(n+1)=T(x_n), prove x_(n+6)=x_n for every natural number n and use it to find x_2026; enumerate the fixed points completely; decide that A={0,1,2,4,5,6} is forward invariant while B={0,1} is not, and prove that the orbit from 0 never reaches the fixed point."},"mko_ids":["mko-discrete-dynamical-system"]},{"id":"derivation-mko-ordinary-differential-equation-01","title":{"zh-Hant":"常微分方程：判定階數並核對候選解 y=x² 與 y=x²+x","en":"Ordinary differential equation: finding the order and checking the candidates y=x² and y=x²+x"},"summary":{"zh-Hant":"在區間 I=ℝ 上考慮常微分方程 y″−2y′+y=x²−4x+2。(1) 判定它的階數；(2) 核對 y₁(x)=x² 在整個 ℝ 上是解（殘差恆為零）；(3) 證明 y₂(x)=x²+x 的殘差是 x−2，只在 x=2 為零，所以它不是 ℝ 上的解；(4) 核對初值 y(0)=0、y′(0)=0，並引用二階線性方程的存在唯一定理（含其假設）判定 y₁ 是此初值問題的唯一解；(5) 說明沒有初值時解不唯一。","en":"On the interval I=ℝ consider the ordinary differential equation y″−2y′+y=x²−4x+2. (1) Determine its order; (2) check that y₁(x)=x² is a solution on all of ℝ (its residual is identically zero); (3) prove that y₂(x)=x²+x has residual x−2, which vanishes only at x=2, so it is not a solution on ℝ; (4) check the initial values y(0)=0, y′(0)=0 and, citing the existence and uniqueness theorem for second-order linear equations with its hypotheses, conclude that y₁ is the unique solution of this initial value problem; (5) show that without initial values the solution is not unique."},"mko_ids":["mko-ordinary-differential-equation"]},{"id":"derivation-mko-numerical-approximation-error-01","title":{"zh-Hant":"數值近似與誤差：以 0.33 近似 1/3 的絕對誤差、相對誤差與容許範圍","en":"Numerical approximation and error: absolute error, relative error and tolerance for 0.33 approximating 1/3"},"summary":{"zh-Hant":"以精確有理數計算：近似值 a=33/100 相對參照值 x=1/3 的絕對誤差與相對誤差；判定 a 是否在容許誤差 τ₁=1/250、τ₂=1/400、τ₃=1/300 之內（τ₃ 為端點情形）；證明 |a−x|≤τ ⇔ x−τ≤a≤x+τ；比較放大 1000 倍後的誤差；說明參照值為 0 時相對誤差沒有定義，以及誤差數值本身不能說明誤差來源。","en":"Using exact rationals: compute the absolute and relative error of the approximation a=33/100 to the reference x=1/3; decide whether a is within the tolerances τ₁=1/250, τ₂=1/400 and τ₃=1/300 (τ₃ is the endpoint case); prove |a−x|≤τ ⇔ x−τ≤a≤x+τ; compare the errors after scaling by 1000; explain why relative error is undefined for a zero reference and why an error value alone does not reveal the source of the error."},"mko_ids":["mko-numerical-approximation-error"]},{"id":"derivation-mko-optimization-problem-01","title":{"zh-Hant":"最佳化問題：在九個整數候選中先判可行、再比較目標值","en":"Optimization problem: feasibility first, then objective values, over nine integer candidates"},"summary":{"zh-Hant":"候選集合 C={0,1,2}² 共九點，限制 x+2y≥3，目標函數 f(x,y)=x+y。以「先判定可行性、再逐一比較」的掃描演算法求最小值，並以不變量證明結果：最優值 m=2，argmin={(0,2),(1,1)}（兩個平手）；說明 (0,0) 與 (2,0) 雖然目標值不大於 2 卻不能入選；求最大值；並說明同一限制在實數非負象限上的最小值為 3/2，其達到點 (0,3/2) 不在宣告的有限候選集合中。","en":"The candidate set C={0,1,2}² has nine points, the constraint is x+2y≥3 and the objective is f(x,y)=x+y. Minimize with a scan algorithm that first decides feasibility and then compares values one by one, and prove the result with an invariant: optimal value m=2, argmin={(0,2),(1,1)} (a tie of two); explain why (0,0) and (2,0) are excluded although their objective values do not exceed 2; find the maximum; and show that under the same constraint over the real nonnegative quadrant, the minimum is 3/2, attained at (0,3/2), which is not in the declared finite candidate set."},"mko_ids":["mko-optimization-problem"]},{"id":"derivation-mko-dimensional-analysis-01","title":{"zh-Hant":"量綱分析：以指數向量核對 ½mv²=mgh 並排除 v=2gh","en":"Dimensional analysis: checking ½mv²=mgh with exponent vectors and ruling out v=2gh"},"summary":{"zh-Hant":"以基本量綱 L、M、T 的整數指數向量 d(Q)=(a,b,c)：算出速度、加速度與兩種能量式的量綱；判定 ½mv²=mgh 量綱一致、v=2gh 不一致且無法以純數常數修正、v²=4gh 量綱一致卻在等加速度模型中錯誤；判定 m+v 不能相加；由 E=C·m^α v^β 解出 α=1、β=2；並說明量綱相同不代表單位換算因子相同。","en":"With integer exponent vectors d(Q)=(a,b,c) over the base dimensions L, M, T: compute the dimensions of speed, acceleration and two energy expressions; decide that ½mv²=mgh is dimensionally consistent, that v=2gh is not and cannot be repaired by a pure-number constant, and that v²=4gh is consistent but wrong in the constant-acceleration model; decide that m+v cannot be added; solve E=C·m^α v^β for α=1 and β=2; and explain that equal dimensions do not mean equal unit conversion factors."},"mko_ids":["mko-dimensional-analysis"]},{"id":"derivation-mko-discrete-exponential-growth-01","title":{"zh-Hant":"離散指數成長模型：由 x_(n+1)=r·x_n 歸納證明 x_n=x_0·r^n","en":"Discrete exponential growth: proving x_n=x_0·r^n from x_(n+1)=r·x_n by induction"},"summary":{"zh-Hant":"在固定時間步、常數因子 r≥0 的模型 x_(n+1)=r·x_n 中，以數學歸納法證明對每個自然數 n 有 x_n=x_0·r^n，並說明遞迴式與閉式描述同一數列；以 x_0=8、r=3/2 計算 x_1 到 x_4 並與閉式比對；說明 r>1、r=1、0≤r<1 與 x_0=0 的行為；判定資料 8、12、18、26 不符合任何以 x_0=8 開始的固定因子模型；並說明每步因子 3/2 不是連續成長率 1/2。","en":"In the model x_(n+1)=r·x_n with a fixed time step and constant factor r≥0, prove by induction that x_n=x_0·r^n for every natural number n, and explain that the recurrence and the closed form describe the same sequence; with x_0=8 and r=3/2 compute x_1 to x_4 and compare with the closed form; describe the behaviour for r>1, r=1, 0≤r<1 and x_0=0; decide that the data 8, 12, 18, 26 fit no fixed-factor model starting at x_0=8; and explain why the per-step factor 3/2 is not the continuous growth rate 1/2."},"mko_ids":["mko-discrete-exponential-growth"]},{"id":"derivation-mko-compound-interest-01","title":{"zh-Hant":"複利模型：本金 1000、每年 2%、三年的餘額與單利比較","en":"Compound-interest model: principal 1000 at 2% per year for three years, compared with simple interest"},"summary":{"zh-Hant":"在期別為一年、每期利率 i=1/50（2%）、期間內無存提款與費用的模型中，以歸納證明 B_n=B_0(1+i)^n，算出 B_0=1000 時三年後的餘額 132651/125（即 1061.208）與利息；與單利 B_0(1+ni)=1060 比較，並以 Bernoulli 不等式證明複利餘額對每個 n 都不小於單利；說明期別混用與期間現金流如何使單一冪次公式失效。","en":"In the model with one-year periods, rate i=1/50 (2%) per period and no deposits, withdrawals or fees, prove B_n=B_0(1+i)^n by induction, compute the balance after three years from B_0=1000, namely 132651/125 (that is 1061.208), and the interest; compare with simple interest B_0(1+ni)=1060 and prove by Bernoulli's inequality that the compound balance is never below the simple one for any n; explain how mixing period units and cash flows during the period break the single-power formula."},"mko_ids":["mko-compound-interest"]},{"id":"derivation-mko-mathematical-representation-01","title":{"zh-Hant":"數學表徵：判定文字、符號、表格與序對是否表示同一個函數","en":"Mathematical representation: deciding whether words, a formula, tables and ordered pairs represent the same function"},"summary":{"zh-Hant":"固定對象 f:D→ℤ，D={0,1,2,3}，f(x)=2x−1。判定：文字「輸入加倍後減 1」對應 2x−1；表格 T1={(0,−1),(1,1),(2,3),(3,5)} 表示 f；序對集合是 f 的圖形；由兩列 (1,1)、(3,5) 唯一恢復仿射規則；表格 T2 把 (3,5) 換成 (3,6) 後不表示 f，也不表示任何仿射規則；含重複輸入的 T3 不是函數表；定義在 ℤ 上的 g(x)=2x−1 與 f 不是同一個函數；以及表格不提供未列點的值。","en":"Fix the object f:D→ℤ with D={0,1,2,3} and f(x)=2x−1. Decide: the words 'double the input, then subtract 1' correspond to 2x−1; the table T1={(0,−1),(1,1),(2,3),(3,5)} represents f; the set of ordered pairs is the graph of f; the affine rule is recovered uniquely from the two rows (1,1) and (3,5); the table T2, with (3,5) replaced by (3,6), represents neither f nor any affine rule; T3, with a repeated input, is not a function table; g(x)=2x−1 defined on ℤ is not the same function as f; and a table supplies no values at unlisted points."},"mko_ids":["mko-mathematical-representation"]},{"id":"derivation-mko-diagnostic-mathematical-assessment-01","title":{"zh-Hant":"診斷性數學評量記錄：一道分數加法題的合成記錄與判定","en":"Diagnostic assessment records: synthetic records and judgments for one fraction-addition item"},"summary":{"zh-Hant":"對題目 Q-FRAC-01「計算 3/4+1/6，答案寫成分數 a/b（a、b 為整數，b≠0）」，先精確算出期望結果 e，再寫出判定規則 R_val（作答的值等於 e 即符合）與較嚴格的 R_low（值相等且為最簡分數）。據此判定三份合成記錄：D1 作答 4/10、D2 作答 22/24、D3 作答 22/24 但規則欄空白；並證明 D1 的「不符合」不能推出任何關於真實學生能力或錯誤原因的結論。","en":"For the item Q-FRAC-01, 'compute 3/4+1/6 and write the answer as a fraction a/b with integers a, b and b≠0', first compute the expected result e exactly, then state the judgment rule R_val (a response matches when its value equals e) and the stricter rule R_low (equal value and lowest terms). Use them to judge three synthetic records: D1 with response 4/10, D2 with response 22/24, and D3 with response 22/24 but an empty rule field; and show that the label 'no match' on D1 supports no conclusion about a real student's ability or error cause."},"mko_ids":["mko-diagnostic-mathematical-assessment"]},{"id":"derivation-mko-orbit-fixed-periodic-01","title":{"zh-Hant":"軌道、固定點與週期點：完整分類 T(x)=(2x+1) mod 6 的六個狀態","en":"Orbits, fixed points and periodic points: classifying all six states of T(x)=(2x+1) mod 6"},"summary":{"zh-Hant":"在狀態空間 X={0,1,2,3,4,5} 上取自映射 T(x)=(2x+1) mod 6。求出全部固定點與最小週期 2 的週期點；證明對最小週期 p 的點，T^n(x)=x 當且僅當 p 整除 n，並用它說明 T^4(1)=1 不表示 1 的最小週期是 4；最後證明 0、2、4 不是週期點，雖然它們的軌道一步之後就進入循環。","en":"On the state space X={0,1,2,3,4,5} take the self-map T(x)=(2x+1) mod 6. Find all fixed points and all periodic points of least period 2; prove that for a point of least period p, T^n(x)=x exactly when p divides n, and use this to explain why T^4(1)=1 does not make 4 the least period of 1; finally prove that 0, 2 and 4 are not periodic, although their orbits enter a cycle after one step."},"mko_ids":["mko-orbit-fixed-periodic"]},{"id":"derivation-mko-initial-value-problem-01","title":{"zh-Hant":"初值問題：逐一審核四個多項式候選","en":"Initial-value problem: auditing four polynomial candidates"},"summary":{"zh-Hant":"初值問題：在 ℝ 上 y''(x)=6x，且 y(0)=1、y'(0)=2。逐一審核四個整係數多項式候選 y_A=x³+2x+1、y_B=x³+1、y_C=x²+2x+1、y_D=3x⁵−15x⁴+21x³+2x+1：分開核對方程殘差與兩個初值，判定哪一個是此初值問題的解，並說明只核初值、只核一個初值、或只在有限網格上核殘差為何都不夠。","en":"Initial-value problem: y''(x)=6x on ℝ with y(0)=1 and y'(0)=2. Audit the four integer-coefficient polynomial candidates y_A=x³+2x+1, y_B=x³+1, y_C=x²+2x+1 and y_D=3x⁵−15x⁴+21x³+2x+1 one by one: check the equation residual and the two initial values separately, decide which candidate solves this initial-value problem, and explain why checking only the initial data, only one initial value, or the residual only on a finite grid is not enough."},"mko_ids":["mko-initial-value-problem"]},{"id":"derivation-mko-bisection-method-01","title":{"zh-Hant":"二分法：四步 bracket 更新、不變量與根存在的獨立責任","en":"Bisection: four bracket updates, the invariant and the separate proof obligation for root existence"},"summary":{"zh-Hant":"對 f(x)=x²−2，從 [1,2] 與所提供的端點符號 f(1)<0、f(2)>0 出發，依所提供的中點符號執行四步二分，追蹤 bracket；證明「端點異號且 a<b」是不變量、n 步後寬度為 1/2^n、在每次所需符號可取得且更新有效的延伸執行中，寬度停止準則必在有限步內終止；再證明這些 bracket 資料本身不證明有根，並另外引用中間值定理，對連續的 f 得出最終 bracket 內有根。","en":"For f(x)=x²−2, start from [1,2] with the supplied endpoint signs f(1)<0 and f(2)>0, run four bisection steps with the supplied midpoint signs and trace the brackets; prove that 'opposite endpoint signs and a<b' is an invariant, that the width after n steps is 1/2^n, and that, in an extended run with every needed sign available and effective updates, a width-based stopping rule terminates after finitely many steps; then show that the bracket data alone do not prove a root exists, and separately cite the intermediate value theorem to conclude, for the continuous f, that the final bracket contains a root."},"mko_ids":["mko-bisection-method"]},{"id":"derivation-mko-linear-programming-01","title":{"zh-Hant":"線性規劃：在明示候選中求最佳，並說明它不是連續最優","en":"Linear programming: the best supplied candidate, and why it is not the continuous optimum"},"summary":{"zh-Hant":"限制 3x+2y≤12、x+2y≤6、−x≤0、−y≤0，目標為最大化 L(x,y)=x+y。對明示的七個候選 P1=(0,0)、P2=(4,0)、P3=(3,1)、P4=(2,2)、P5=(0,3)、P6=(4,1)、P7=(3,2)，逐限制核對可行性，只在可行候選間比較目標值，求出候選中的最大值與全部最佳點；再以可行點 Q=(3,3/2) 與一個不等式證書證明連續可行域上的最大值是 9/2，而不是候選中的 4。","en":"Constraints 3x+2y≤12, x+2y≤6, −x≤0, −y≤0; objective: maximize L(x,y)=x+y. For the seven supplied candidates P1=(0,0), P2=(4,0), P3=(3,1), P4=(2,2), P5=(0,3), P6=(4,1), P7=(3,2), check feasibility constraint by constraint, compare objective values only among feasible candidates, and find the best value among the candidates and all points attaining it; then use the feasible point Q=(3,3/2) and an inequality certificate to prove that the maximum over the continuous feasible region is 9/2, not the candidate value 4."},"mko_ids":["mko-linear-programming"]},{"id":"derivation-mko-discrete-logistic-growth-01","title":{"zh-Hant":"離散 logistic 增量模型：[0,K] 的不變性、固定點與一條短軌道","en":"Discrete logistic increment model: invariance of [0,K], fixed points and a short orbit"},"summary":{"zh-Hant":"對模型 G(x)=x+r x(1−x/K)（K>0、0≤r≤1），證明：若 0≤x≤K，則 0≤G(x)≤K，因此從 [0,K] 出發的整條軌道都留在 [0,K]；0 與 K 是固定點，且 r>0 時它們是僅有的固定點。再以 K=8、r=1、x0=2 精確算出 x1、x2、x3，並以 r=3 的例子說明條件 r≤1 不可省。","en":"For the model G(x)=x+r x(1−x/K) with K>0 and 0≤r≤1, prove: if 0≤x≤K then 0≤G(x)≤K, so the whole orbit starting in [0,K] stays in [0,K]; 0 and K are fixed points, and for r>0 they are the only ones. Then compute x1, x2, x3 exactly for K=8, r=1, x0=2, and use an example with r=3 to show that the condition r≤1 cannot be dropped."},"mko_ids":["mko-discrete-logistic-growth"]},{"id":"derivation-mko-harmonic-oscillator-model-01","title":{"zh-Hant":"線性簡諧振子模型：資格、k/m、多項式候選的殘差與單點能量","en":"Linear harmonic oscillator: eligibility, k/m, residuals of a polynomial candidate and single-state energy"},"summary":{"zh-Hant":"取 m=2、k=8。核對模型資格並算出 k/m；對多項式候選 x_c(t)=1−2t² 算出殘差 R(t)=m x_c''+k x_c，在指定點 t=0、1/2、1 求值，並證明 x_c 不是任何正長度區間上的解；證明方程 2x''+8x=0 沒有非零多項式解；最後算出單一狀態 (x,v)=(1,0) 的能量 E=4，並以能量導數恆等式說明單點能量值為何不證明守恆。","en":"Take m=2 and k=8. Check that the parameters satisfy the model's requirements and compute k/m; for the polynomial candidate x_c(t)=1−2t², compute the residual R(t)=m x_c''+k x_c, evaluate it at the specified points t=0, 1/2, 1, and prove that x_c is not a solution on any interval of positive length; prove that the equation 2x''+8x=0 has no nonzero polynomial solution; finally compute the energy E=4 of the single state (x,v)=(1,0), and use the energy-derivative identity to explain why a single energy value proves no conservation."},"mko_ids":["mko-harmonic-oscillator-model"]},{"id":"derivation-mko-present-value-discounting-01","title":{"zh-Hant":"現值與折現：證明折現與複利互逆，並算出 1331 的三年現值","en":"Present value and discounting: proving that discounting inverts compounding, and the three-year present value of 1331"},"summary":{"zh-Hant":"在固定每期利率 i（1+i>0）與期數 n 下，證明 (1+i)^n>0，因此現值 PV_n(F)=F/(1+i)^n 有定義；證明 PV_n 與複利 A_n(P)=P(1+i)^n 互為反函數。再以 i=1/10（每年）、n=3、F=1331 算出現值 1000，用逐年折現與 roundtrip 交叉核對，並說明把折現寫成乘法或混用年率與月率為何錯誤。","en":"For a fixed rate i per period (1+i>0) and a period count n, prove that (1+i)^n>0, so the present value PV_n(F)=F/(1+i)^n is defined; prove that PV_n and compounding A_n(P)=P(1+i)^n are inverse functions of each other. Then, with i=1/10 per year, n=3 and F=1331, compute the present value 1000, cross-check by year-by-year discounting and a round trip, and explain why writing discounting as a multiplication, or mixing annual and monthly rates, is wrong."},"mko_ids":["mko-present-value-discounting"]},{"id":"derivation-mko-complex-number-01","title":{"zh-Hant":"複數：以有序對計算 (2+3i)(4−5i) 並判定相等","en":"Complex numbers: computing (2+3i)(4−5i) as an ordered pair and deciding equality"},"summary":{"zh-Hant":"設 z=2+3i、w=4−5i。以有序對（實部, 虛部係數）求 z+w、z−w 與 z·w。乘法先由分配律與 i²=−1 推出一般公式 (a+bi)(c+di)=(ac−bd)+(ad+bc)i，再代入數值，並以直接展開交叉核對；最後判定 z·w 是否等於實數 23。","en":"Let z=2+3i and w=4−5i. Find z+w, z−w and z·w as ordered pairs (real part, imaginary coefficient). For the product, first derive the general formula (a+bi)(c+di)=(ac−bd)+(ad+bc)i from distributivity and i²=−1, then substitute the numbers and cross-check by direct expansion; finally decide whether z·w equals the real number 23."},"mko_ids":["mko-complex-number"]},{"id":"derivation-mko-real-vector-space-01","title":{"zh-Hant":"實向量空間：在 ℝ³ 核對公理實例並證明一條分配律","en":"Real vector space: auditing axiom instances in ℝ³ and proving one distributive law"},"summary":{"zh-Hant":"在 ℝ³ 中取 u=(1,−2,3)、v=(4,0,−1)、a=2、b=−3。（一）確認維度相同；（二）精確核對 a(u+v)=au+av、(a+b)u=au+bu、(ab)u=a(bu)、1u=u、u+(−u)=0 五個公理實例；（三）對所有實數 a 與所有 u、v∈ℝ³ 證明 a(u+v)=au+av；（四）以 ℝ² 上改造的純量乘法說明實例通過不等於公理成立。","en":"In ℝ³ take u=(1,−2,3), v=(4,0,−1), a=2, b=−3. (1) Confirm the dimensions agree; (2) check exactly the five axiom instances a(u+v)=au+av, (a+b)u=au+bu, (ab)u=a(bu), 1u=u and u+(−u)=0; (3) prove a(u+v)=au+av for every real a and all u, v∈ℝ³; (4) use a modified scalar multiplication on ℝ² to show that successful instances do not establish an axiom."},"mko_ids":["mko-real-vector-space"]},{"id":"derivation-mko-finite-group-01","title":{"zh-Hant":"有限群：以完整 Cayley 表核對群公理","en":"Finite group: auditing the group axioms from a complete Cayley table"},"summary":{"zh-Hant":"給兩個完整運算表。（甲）G={0,1,2,3} 配上表 T，證明 (G,*) 是有限群：封閉、全部 4³=64 個三元組的結合律、雙側單位元、雙側逆元。（乙）K={0,1,2} 配上表 ∘：它對稱、有雙側單位元、每個元素都有雙側逆元，卻在三元組 (1,1,2) 違反結合律，因此不是群；同時說明含 0 的 19 個三元組全部通過，抽樣會誤判。","en":"Two complete operation tables are given. (A) For G={0,1,2,3} with table T, prove that (G,*) is a finite group: closure, associativity for all 4³=64 triples, a two-sided identity and two-sided inverses. (B) K={0,1,2} with table ∘ is symmetric, has a two-sided identity and two-sided inverses for every element, yet violates associativity at the triple (1,1,2), so it is not a group; also show that all 19 triples containing 0 pass, so sampling would misjudge it."},"mko_ids":["mko-finite-group"]},{"id":"derivation-mko-metric-space-01","title":{"zh-Hant":"度量空間：逐格與逐三元組核對有限距離表","en":"Metric space: auditing a finite distance table cell by cell and triple by triple"},"summary":{"zh-Hant":"給三張完整整數距離表。（甲）X={0,1,2,3} 上的表 d（四點環上的步數距離）：逐格核非負、對角線為零、非對角線不為零、對稱，並以涵蓋全部 4³=64 個三元組的分情形論證核三角不等式，得出 (X,d) 是度量空間。（乙）Y={0,1,2} 上的表 e 通過前四項卻在 (0,1,2) 違反三角不等式。（丙）Z={0,1,2} 上的表 f 非負、對稱、對角線為零，卻有非對角零值，違反分離性。","en":"Three complete integer distance tables are given. (A) On X={0,1,2,3}, table d (step distance on a 4-cycle): check nonnegativity, zero diagonal, nonzero off-diagonal entries and symmetry cell by cell, and the triangle inequality by a case argument covering all 4³=64 triples, concluding that (X,d) is a metric space. (B) Table e on Y={0,1,2} passes the first four checks but violates the triangle inequality at (0,1,2). (C) Table f on Z={0,1,2} is nonnegative, symmetric and zero on the diagonal, yet has an off-diagonal zero and violates separation."},"mko_ids":["mko-metric-space"]},{"id":"derivation-mko-finite-discrete-probability-distribution-01","title":{"zh-Hant":"有限離散機率分布：核 PMF 資格、事件機率、期望與推前分布","en":"Finite discrete probability distribution: validating a PMF, event probability, expectation and pushforward"},"summary":{"zh-Hant":"隨機變數 X 取值 −1、0、1、2，質量 p(−1)=1/6、p(0)=1/3、p(1)=1/4、p(2)=1/4。（一）先核 p 是否為合格 PMF；（二）求 P(X≥1)；（三）求 E[X]；（四）求 Y=X² 的推前分布、證明它仍是 PMF，求 E[Y] 並以 Σx²p(x) 交叉核對；（五）判定兩個候選 q、r 不是 PMF：q 每項介於 0 與 1 但總和不為 1，r 總和為 1 但有負項。","en":"A random variable X takes the values −1, 0, 1, 2 with masses p(−1)=1/6, p(0)=1/3, p(1)=1/4, p(2)=1/4. (1) First check that p is a valid PMF; (2) find P(X≥1); (3) find E[X]; (4) find the pushforward distribution of Y=X², prove it is again a PMF, find E[Y] and cross-check it against Σx²p(x); (5) decide that two candidates q and r are not PMFs: every mass of q lies between 0 and 1 but the total is not 1, while r sums to 1 but has a negative mass."},"mko_ids":["mko-finite-discrete-probability-distribution"]},{"id":"derivation-mko-complex-modulus-conjugate-01","title":{"zh-Hant":"複數模與共軛：證明 z·z̄=|z|² 與共軛保持運算","en":"Complex modulus and conjugate: proving z·z̄=|z|² and that conjugation preserves the operations"},"summary":{"zh-Hant":"對 z=a+bi、w=c+di（a、b、c、d 為實數）證明：（一）z·z̄=(a²+b²)+0i，即模平方的實數嵌入；（二）共軛保持加法：(z+w)‾=z̄+w̄；（三）共軛保持乘法：(zw)‾=z̄·w̄；（四）由此得 |zw|²=|z|²|w|²。再以 z=3−4i、w=1+2i 精確核對，並區分模平方與模。","en":"For z=a+bi and w=c+di (a, b, c, d real) prove: (1) z·z̄=(a²+b²)+0i, the real embedding of the modulus squared; (2) conjugation preserves addition: (z+w)‾=z̄+w̄; (3) conjugation preserves multiplication: (zw)‾=z̄·w̄; (4) hence |zw|²=|z|²|w|². Then check everything exactly for z=3−4i and w=1+2i, distinguishing the modulus squared from the modulus."},"mko_ids":["mko-complex-modulus-conjugate"]},{"id":"derivation-mko-linear-combination-span-01","title":{"zh-Hant":"線性組合與生成：以有限係數 witness 判定 span 成員","en":"Linear combinations and span: deciding membership using finite coefficient lists as witnesses"},"summary":{"zh-Hant":"（一）在 ℝ² 中，v₁=(1,2)、v₂=(3,−1)，判定 b=(7,0) 是否屬於 span{v₁,v₂}：先證明兩個向量的生成集恰為 {c₁v₁+c₂v₂}，再解係數、證明唯一，並由 witness 重建 b。（二）在 ℝ³ 中，u₁=(1,0,1)、u₂=(0,1,1)，證明 t=(2,3,4) 不屬於 span{u₁,u₂}，而 t'=(2,3,5) 屬於：同為三維並不決定成員資格。","en":"(1) In ℝ², with v₁=(1,2) and v₂=(3,−1), decide whether b=(7,0) lies in span{v₁,v₂}: first prove that the span of two vectors is exactly {c₁v₁+c₂v₂}, then solve for the coefficients, prove they are unique, and rebuild b from the witness. (2) In ℝ³, with u₁=(1,0,1) and u₂=(0,1,1), prove that t=(2,3,4) is not in span{u₁,u₂} while t'=(2,3,5) is: having three coordinates does not decide membership."},"mko_ids":["mko-linear-combination-span"]},{"id":"derivation-mko-subgroup-homomorphism-01","title":{"zh-Hant":"子群與群同態：在 ℤ/4 與 ℤ/2 的完整表上核對","en":"Subgroups and homomorphisms: auditing on the complete tables of ℤ/4 and ℤ/2"},"summary":{"zh-Hant":"G={0,1,2,3} 以 (a+b) mod 4 為運算，K={0,1} 以 (a+b) mod 2 為運算，兩者都給出完整 Cayley 表。（一）證明 H={0,2} 是 G 的子群，H'={0,1} 不是；（二）證明映射表 f(0)=0、f(1)=1、f(2)=0、f(3)=1 是同態 G→K（涵蓋全部 16 個輸入對）；（三）求 kernel 與 image，說明 f 滿射但不單射，因此不是同構；（四）說明常數映射 h≡1 不是同態。","en":"G={0,1,2,3} with operation (a+b) mod 4 and K={0,1} with operation (a+b) mod 2 are given by complete Cayley tables. (1) Prove that H={0,2} is a subgroup of G and H'={0,1} is not; (2) prove that the mapping table f(0)=0, f(1)=1, f(2)=0, f(3)=1 is a homomorphism G→K, covering all 16 input pairs; (3) find the kernel and the image, and show that f is surjective but not injective, hence not an isomorphism; (4) show that the constant map h≡1 is not a homomorphism."},"mko_ids":["mko-subgroup-homomorphism"]},{"id":"derivation-mko-metric-convergence-01","title":{"zh-Hant":"度量空間中的收斂：有限度量表上最終常值序列的極限與唯一性","en":"Convergence in a metric space: limit and uniqueness for an ultimately constant sequence on a finite metric table"},"summary":{"zh-Hant":"在三點集合 X={a,b,c} 上給定完整度量表 d(a,b)=1、d(a,c)=2、d(b,c)=3/2。序列 x_0=a、x_1=c、x_2=a，並明示對所有 n≥3 有 x_n=b。(1) 核對 d 是度量；(2) 以 ε-N 定義證明 x_n→b，並找出各 ε 下最小的有效 N；(3) 證明度量空間中極限唯一，並直接給出 x_n 不收斂到 a、也不收斂到 c 的見證；(4) 用一個在有限尾段上與 x_n 完全一致、極限卻不同的序列，說明只核有限尾段不足以判定極限是否為指定的 b。","en":"On the three-point set X={a,b,c} a complete metric table is given: d(a,b)=1, d(a,c)=2, d(b,c)=3/2. The sequence has x_0=a, x_1=c, x_2=a, and it is declared that x_n=b for every n≥3. (1) Check that d is a metric; (2) prove x_n→b from the ε-N definition and find the least valid N for each ε; (3) prove that limits in a metric space are unique, and give direct witnesses that x_n converges neither to a nor to c; (4) using a sequence that agrees with x_n on a finite tail but has a different limit, show that checking a finite tail cannot decide whether the limit is the specified b."},"mko_ids":["mko-metric-convergence"]},{"id":"derivation-mko-sigma-algebra-measure-space-01","title":{"zh-Hant":"σ-代數與測度空間：四點載體上由原子生成的 σ-代數與一個機率測度","en":"Sigma-algebras and measure spaces: an atom-generated sigma-algebra and a probability measure on a four-point carrier"},"summary":{"zh-Hant":"在 X={0,1,2,3} 上，以原子 A_1={0}、A_2={1,2}、A_3={3} 生成的 8 個集合構成 Σ，並以 m_1=1/2、m_2=1/3、m_3=1/6 定義 μ(∪_{i∈I}A_i)=Σ_{i∈I}m_i。證明 Σ 是 σ-代數（含可數聯集的有限歸約），證明 μ 是測度（含可數可加性的有限歸約）且為機率測度；再說明 𝒢={∅,{0},{1},X} 不是 σ-代數，以及只改一個值的 ν 不是測度。","en":"On X={0,1,2,3}, the 8 sets generated by the atoms A_1={0}, A_2={1,2}, A_3={3} form Σ, and μ(∪_{i∈I}A_i)=Σ_{i∈I}m_i with m_1=1/2, m_2=1/3, m_3=1/6. Prove that Σ is a sigma-algebra (with the finite reduction of countable unions), that μ is a measure (with the finite reduction of countable additivity) and a probability measure; then show that 𝒢={∅,{0},{1},X} is not a sigma-algebra and that ν, which changes a single value, is not a measure."},"mko_ids":["mko-sigma-algebra-measure-space"]},{"id":"derivation-mko-discrete-stochastic-process-01","title":{"zh-Hant":"離散隨機過程：由完整路徑分布導出邊際與兩時刻聯合分布，並檢驗 Markov 性與平穩性","en":"Discrete-time stochastic process: marginals and two-time joint laws from a complete path distribution, with Markov and stationarity checks"},"summary":{"zh-Hant":"時域 {0,1,2}、狀態 {0,1}。完整路徑支撐與精確 PMF 為 (0,0,0):1/4、(0,1,1):1/4、(1,0,1):1/8、(1,1,0):1/8、(1,1,1):1/4。(1) 核對支撐與 PMF，建立機率空間並說明 X_0、X_1、X_2 是同一空間上的隨機變數；(2) 求三個時刻的邊際分布與 (X_1,X_2) 的聯合分布；(3) 證明這個過程不平穩、不具 Markov 性；(4) 構造一個邊際相同但聯合分布不同的路徑律，說明邊際表不等於完整過程。","en":"Horizon {0,1,2}, states {0,1}. The complete path support with exact PMF is (0,0,0):1/4, (0,1,1):1/4, (1,0,1):1/8, (1,1,0):1/8, (1,1,1):1/4. (1) Check the support and the PMF, build the probability space and explain why X_0, X_1, X_2 are random variables on the same space; (2) find the marginal law at each time and the joint law of (X_1,X_2); (3) prove that this process is neither stationary nor Markov; (4) construct a path law with the same marginals but a different joint law, showing that the marginal tables are not the whole process."},"mko_ids":["mko-discrete-stochastic-process"]},{"id":"derivation-mko-multivariable-real-function-01","title":{"zh-Hant":"多變數實函數：二變數二次多項式的係數次序、宣告定義域、切片與點值","en":"Multivariable real-valued function: coefficient order, declared domain, slices and point values of a bivariate quadratic"},"summary":{"zh-Hant":"係數向量依 [1,x,y,x²,xy,y²] 次序為 c=(1,−2,3,1,2,−1)，宣告定義域為閉矩形 R=[−1,2]×[0,3]。(1) 寫出 f 並確認 f:R→ℝ 是函數；(2) 分別核對 x、y 邊界，判定 (2,3)、(3,1)、(0,−1) 是否在 R 中；(3) 計算 f(2,1) 與 f(1,2)，並證明交換 x、y 係數得到的 g 是不同的函數；(4) 求 y=1 的 x-切片，並說明給定的 y=1 切片與指定四個點值各自都不足以決定 f。","en":"The coefficient vector in the order [1,x,y,x²,xy,y²] is c=(1,−2,3,1,2,−1), and the declared domain is the closed rectangle R=[−1,2]×[0,3]. (1) Write f and confirm that f:R→ℝ is a function; (2) check the x and y bounds separately to decide whether (2,3), (3,1), (0,−1) lie in R; (3) compute f(2,1) and f(1,2), and prove that g, obtained by swapping the x and y coefficients, is a different function; (4) find the x-slice at y=1 and show that neither the given y=1 slice nor the four specified point values determine f."},"mko_ids":["mko-multivariable-real-function"]},{"id":"derivation-mko-linear-independence-basis-01","title":{"zh-Hant":"線性獨立與基底：以精確消去證明 ℝ² 的一組有序基底，並以兩個反例分開兩條責任","en":"Linear independence and bases: proving an ordered basis of ℝ² by exact elimination, with two counterexamples that separate the two obligations"},"summary":{"zh-Hant":"在 ℝ² 中令 v_1=(1,2)、v_2=(3,5)、v_3=(1,1)、w=(4,7)。(1) 以精確消去證明 (v_1,v_2) 線性獨立；(2) 對任意 (p,q) 求出係數，證明 span(v_1,v_2)=ℝ²，因而 (v_1,v_2) 是有序基底；(3) 求 w 的唯一座標；(4) 證明 (v_1) 獨立但不生成 ℝ²，而 (v_1,v_2,v_3) 生成 ℝ² 但不獨立，且 w 在其中的表示不唯一。","en":"In ℝ² let v_1=(1,2), v_2=(3,5), v_3=(1,1), w=(4,7). (1) Prove by exact elimination that (v_1,v_2) is linearly independent; (2) find the coefficients for an arbitrary (p,q), proving span(v_1,v_2)=ℝ², so that (v_1,v_2) is an ordered basis; (3) find the unique coordinates of w; (4) prove that (v_1) is independent but does not span ℝ², while (v_1,v_2,v_3) spans ℝ² but is not independent, and w has more than one representation in it."},"mko_ids":["mko-linear-independence-basis"]},{"id":"derivation-mko-measurable-function-01","title":{"zh-Hant":"可測函數：以完整列舉逐一核逆像，並以合成、指示函數與像集陷阱對照","en":"Measurable function: checking every preimage by complete enumeration, with composition, indicators and the image-set trap"},"summary":{"zh-Hant":"X={0,1,2,3}、Σ_X={∅,{0,1},{2,3},X}；Y={0,1,2}、Σ_Y={∅,{0},{1,2},Y}；Z={0,1}、Σ_Z 為冪集。(1) 以完整列舉證明 f=(0↦0,1↦0,2↦1,3↦2) 可測，而 g=(0↦0,1↦1,2↦1,3↦2) 不可測；(2) 證明可測映射的合成可測，並把 φ∘f 讀回為指示函數；(3) 證明指示函數 1_E 可測若且唯若 E∈Σ_X；(4) 用 m 與 k 說明「檢查像集」既會誤拒也會誤收。","en":"X={0,1,2,3}, Σ_X={∅,{0,1},{2,3},X}; Y={0,1,2}, Σ_Y={∅,{0},{1,2},Y}; Z={0,1} with Σ_Z the power set. (1) Prove by complete enumeration that f=(0↦0,1↦0,2↦1,3↦2) is measurable and g=(0↦0,1↦1,2↦1,3↦2) is not; (2) prove that a composite of measurable maps is measurable, and read φ∘f back as an indicator function; (3) prove that an indicator 1_E is measurable exactly when E∈Σ_X; (4) use m and k to show that 'checking images' can both wrongly reject and wrongly accept."},"mko_ids":["mko-measurable-function"]},{"id":"derivation-mko-finite-markov-chain-01","title":{"zh-Hant":"有限狀態 Markov 鏈：核對行隨機矩陣，計算兩步時刻律與路徑機率","en":"Finite-state Markov chain: checking a row-stochastic matrix and computing two-step time laws and path probabilities"},"summary":{"zh-Hant":"狀態依序為 1、2、3，轉移矩陣 P 的行為 (1/2,1/2,0)、(1/4,1/2,1/4)、(0,1/3,2/3)，初始律 μ_0=(1/2,1/2,0)。(1) 核對 P 是行隨機矩陣，並說明列和不必為 1；(2) 證明一步公式 μ_{n+1}=μ_nP 且結果仍是機率向量，算出 μ_1 與 μ_2；(3) 證明路徑機率公式並計算 P(X_0=1,X_1=2,X_2=3)、P(X_0=2,X_1=3,X_2=3)；(4) 說明 P 單獨一般不能決定完整的無條件時刻律與路徑律。","en":"The states are ordered 1, 2, 3; the transition matrix P has rows (1/2,1/2,0), (1/4,1/2,1/4), (0,1/3,2/3), and the initial law is μ_0=(1/2,1/2,0). (1) Check that P is row-stochastic and explain why column sums need not be 1; (2) prove the one-step formula μ_{n+1}=μ_nP and that the result is again a probability vector, and compute μ_1 and μ_2; (3) prove the path-probability formula and compute P(X_0=1,X_1=2,X_2=3) and P(X_0=2,X_1=3,X_2=3); (4) explain why P alone generally does not determine complete unconditional marginal and path laws."},"mko_ids":["mko-finite-markov-chain"]},{"id":"derivation-mko-partial-derivative-01","title":{"zh-Hant":"偏導數：由差商極限推出二次多項式的偏導公式，並計算一點的梯度","en":"Partial derivative: deriving the partial-derivative formulas of a quadratic from the difference-quotient limit, and computing a gradient"},"summary":{"zh-Hant":"對係數依 [1,x,y,x²,xy,y²] 排列的二次多項式 f，從差商極限的定義證明 ∂f/∂x=c_1+2c_3x+c_4y、∂f/∂y=c_2+c_4x+2c_5y。再對 c=(1,−2,3,1,2,−1) 算出 (1,2) 的偏導與梯度 (4,1)，並說明單一非零 h 的差商（5 或 41/10）與同時移動兩個座標（極限 5）都不是 x 偏導。","en":"For a quadratic f with coefficients in the order [1,x,y,x²,xy,y²], prove from the difference-quotient limit that ∂f/∂x=c_1+2c_3x+c_4y and ∂f/∂y=c_2+c_4x+2c_5y. Then for c=(1,−2,3,1,2,−1) compute the partials and the gradient (4,1) at (1,2), and show that a single nonzero-h quotient (5 or 41/10) and moving both coordinates at once (limit 5) are not the x-partial."},"mko_ids":["mko-partial-derivative"]},{"id":"derivation-mko-finite-convex-combination-01","title":{"zh-Hant":"有限凸組合：以精確權重核對見證，並區分負權重、近似權重與唯一性","en":"Finite convex combination: checking a witness with exact weights, and separating negative weights, approximate weights and uniqueness"},"summary":{"zh-Hant":"在 ℝ² 中令 v_1=(0,0)、v_2=(6,0)、v_3=(0,6)。(1) 核對 λ=(1/6,1/3,1/2) 是凸權重並重建 w=(2,3)；(2) 說明零權重允許、兩點凸組合是線段點；(3) 證明 u=(4,4) 不是 v_1、v_2、v_3 的凸組合（唯一的仿射權重含 −1/3），並以線性泛函上界給第二個證明；(4) 說明 0.333 三個權重不合格；(5) 加入 v_4=(3,3) 後，同一點有兩組不同的凸權重。","en":"In ℝ² let v_1=(0,0), v_2=(6,0), v_3=(0,6). (1) Check that λ=(1/6,1/3,1/2) are convex weights and reconstruct w=(2,3); (2) show that zero weights are allowed and that two-point convex combinations are segment points; (3) prove that u=(4,4) is not a convex combination of v_1, v_2, v_3 (its only affine weights include −1/3), with a second proof by a linear-functional bound; (4) show that three weights of 0.333 do not qualify; (5) after adding v_4=(3,3), exhibit two different convex weight vectors for the same point."},"mko_ids":["mko-finite-convex-combination"]},{"id":"derivation-mko-partial-differential-equation-01","title":{"zh-Hant":"偏微分方程：常係數二階算子的主部分類與二次多項式候選的符號殘差","en":"Partial differential equation: principal-part classification and symbolic residuals of quadratic polynomial candidates"},"summary":{"zh-Hant":"在 ℝ² 上給定線性常係數算子 L(u)=u_xx+2u_xy−3u_yy+u_x 與右端 g(x,y)=2x+y−2。先依明示的主部約定 a u_xx+b u_xy+c u_yy 與判別式 b²−4ac 對 L 分類；再對三個二次多項式候選 u_A=x²+xy+y²、u_B=x²+2xy、u_C=x²+xy+y²+y 精確計算符號殘差 R=L(u)−g，判定哪些候選在全 ℝ² 上滿足 L(u)=g，並說明 u_B 的點殘差在整條直線 y=−8 上為零，為何仍不構成解。","en":"On ℝ², take the linear constant-coefficient operator L(u)=u_xx+2u_xy−3u_yy+u_x and the right-hand side g(x,y)=2x+y−2. First classify L using the explicit principal-part convention a u_xx+b u_xy+c u_yy with discriminant b²−4ac; then, for the three quadratic polynomial candidates u_A=x²+xy+y², u_B=x²+2xy and u_C=x²+xy+y²+y, compute the symbolic residual R=L(u)−g exactly, decide which candidates satisfy L(u)=g on all of ℝ², and explain why u_B is still not a solution although its point residual vanishes on the whole line y=−8."},"mko_ids":["mko-partial-differential-equation"]},{"id":"derivation-mko-convex-set-01","title":{"zh-Hant":"凸集合：半空間與軸對齊盒的完整凸性證書，及有限點集的中點反例","en":"Convex set: complete convexity certificates for a halfspace and an axis-aligned box, and a midpoint counterexample for a finite point set"},"summary":{"zh-Hant":"在實向量空間 ℝ² 中：（一）證明閉半空間 H={(x,y) ∣ 2x−3y≤6} 是凸集合；（二）證明非空軸對齊盒 B=[−1,2]×[0,3] 是凸集合；兩個證明都必須對所有 p、q 與所有 λ∈[0,1] 成立。（三）用一個精確的中點見證否證有限點集 F={(0,0),(2,0),(0,2)} 的凸性，並說明為何抽查有限個 λ、或找不到反例，都不能證明凸性。","en":"In the real vector space ℝ²: (1) prove that the closed halfspace H={(x,y) ∣ 2x−3y≤6} is convex; (2) prove that the nonempty axis-aligned box B=[−1,2]×[0,3] is convex; both proofs must hold for all p, q and all λ∈[0,1]. (3) Refute the convexity of the finite point set F={(0,0),(2,0),(0,2)} with one exact midpoint witness, and explain why checking finitely many λ, or failing to find a counterexample, cannot prove convexity."},"mko_ids":["mko-convex-set"]},{"id":"derivation-mko-boundary-value-problem-01","title":{"zh-Hant":"邊界值問題：矩形上 Poisson–Dirichlet 問題的內部與四邊分開核對","en":"Boundary-value problem: checking the interior and all four edges separately for a Poisson–Dirichlet problem on a rectangle"},"summary":{"zh-Hant":"在矩形 Ω=(0,2)×(0,1) 上宣告 BVP：內部 Δu=u_xx+u_yy=4；Dirichlet 資料為左邊 u(0,y)=y²、右邊 u(2,y)=y²+2y+4（0≤y≤1）、下邊 u(x,0)=x²、上邊 u(x,1)=x²+x+1（0≤x≤2）。對二次多項式候選 u=x²+xy+y² 與 v=2x²+xy−2x+y，分別記錄內部 Poisson 義務（符號 Laplacian 恆等式）與四條邊的 Dirichlet 義務（精確限制到整條邊後逐係數比較），判定誰滿足此宣告的 BVP，並說明 v 為何在內部方程成立、四個角點也都吻合的情況下仍不滿足。","en":"On the rectangle Ω=(0,2)×(0,1) declare the BVP: in the interior Δu=u_xx+u_yy=4; Dirichlet data u(0,y)=y² on the left, u(2,y)=y²+2y+4 on the right (0≤y≤1), u(x,0)=x² on the bottom and u(x,1)=x²+x+1 on the top (0≤x≤2). For the quadratic polynomial candidates u=x²+xy+y² and v=2x²+xy−2x+y, record separately the interior Poisson obligation (a symbolic Laplacian identity) and the Dirichlet obligation on all four edges (exact restriction to each whole edge, then coefficient comparison); decide which candidate satisfies this declared BVP, and explain why v fails although its interior equation holds and it matches the data at all four corners."},"mko_ids":["mko-boundary-value-problem"]},{"id":"derivation-mko-convex-function-01","title":{"zh-Hant":"凸函數：二次函數 Jensen gap 的完整證書與單一 Jensen 反例","en":"Convex function: a complete certificate via the Jensen gap of a quadratic, and a single Jensen counterexample"},"summary":{"zh-Hant":"（一）證明 f(x)=2x²−3x+1 在區間 I=[−1,4] 上是凸函數：先確認 I 是凸集合，再對所有 x,y∈I 與所有 λ∈[0,1] 證明 f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y)，方法是證明二次函數的 Jensen gap 等於 aλ(1−λ)(x−y)²。（二）由同一恆等式得仿射函數 ℓ(x)=3x−5 在 I 上取等號。（三）用一個精確的 Jensen 違反否證 g(x)=−x²+x 在 J=[0,2] 上的凸性，並說明幾個通過的樣本為何不能證明凸性。","en":"(1) Prove that f(x)=2x²−3x+1 is convex on the interval I=[−1,4]: first confirm that I is convex, then prove f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y) for all x,y∈I and all λ∈[0,1], by showing that the Jensen gap of a quadratic equals aλ(1−λ)(x−y)². (2) From the same identity, the affine function ℓ(x)=3x−5 gives equality on I. (3) Refute the convexity of g(x)=−x²+x on J=[0,2] with one exact Jensen violation, and explain why a few passing samples cannot prove convexity."},"mko_ids":["mko-convex-function"]},{"id":"derivation-mko-convex-optimization-problem-01","title":{"zh-Hant":"凸最佳化問題：資格核對、可行性、單點求值與兩候選比較，以及它們不能推出什麼","en":"Convex optimization problem: qualification, feasibility, single-point evaluation and a two-candidate comparison, and what they do not imply"},"summary":{"zh-Hant":"問題 P：在可行域 X=[0,5] 上最小化 f(x)=x²−6x+10。（一）分別核對可行域凸與目標凸，判定 P 具凸問題資格；（二）判定候選 x₁=1、x₂=4、x₃=−1 的可行性；（三）求 f(1)、f(4) 並比較兩個可行候選；（四）說明兩候選中較低者不因此是 P 的最優解，並以 P′：在 X′=(0,1] 上最小化 x² 說明凸問題不一定有最小值點。","en":"Problem P: minimize f(x)=x²−6x+10 over the feasible domain X=[0,5]. (1) Check separately that the domain and the objective are convex, so that P qualifies as a convex problem; (2) decide feasibility of the candidates x₁=1, x₂=4, x₃=−1; (3) evaluate f(1) and f(4) and compare the two feasible candidates; (4) explain why the candidate with the lower objective value is not thereby an optimal solution of P, and use P′: minimize x² over X′=(0,1], to show that a convex problem need not have a minimizer."},"mko_ids":["mko-convex-optimization-problem"]},{"id":"derivation-mko-finite-abelian-group-01","title":{"zh-Hant":"有限阿貝爾群：由完整 Cayley 表先核群資格、再核交換性","en":"Finite abelian group: from a complete Cayley table, group qualification first, then commutativity"},"summary":{"zh-Hant":"載體 G={0,1,2,3} 上給出完整的 4×4 Cayley 表（行 a、列 b 的格子是 a*b）：第 0 行 (1,2,3,0)，第 1 行 (2,3,0,1)，第 2 行 (3,0,1,2)，第 3 行 (0,1,2,3)。依序核對封閉、結合、唯一單位元與逆元，再對所有 a、b 核對 a*b=b*a，判定 (G,*) 是否為有限阿貝爾群；求元素 0 與 1 的階與它們生成的循環子群；並以 {0,1,2} 上的對稱表 a∘b=|a−b| 說明「表對稱」不能代替群資格。","en":"On the carrier G={0,1,2,3} a complete 4×4 Cayley table is given (the cell in row a, column b is a*b): row 0 is (1,2,3,0), row 1 is (2,3,0,1), row 2 is (3,0,1,2), row 3 is (0,1,2,3). Check closure, associativity, a unique identity and inverses in that order, then check a*b=b*a for all a, b, to decide whether (G,*) is a finite abelian group; find the orders of the elements 0 and 1 and the cyclic subgroups they generate; and use the symmetric table a∘b=|a−b| on {0,1,2} to show that a symmetric table cannot replace group qualification."},"mko_ids":["mko-finite-abelian-group"]},{"id":"derivation-mko-real-normed-vector-space-01","title":{"zh-Hant":"實賦範向量空間：ℓ¹ 與 ℓ^∞ 公式的全稱範數證書，以及兩個非範數公式","en":"Real normed vector space: universal norm certificates for the ℓ¹ and ℓ^∞ formulas, and two formulas that are not norms"},"summary":{"zh-Hant":"在 ℝⁿ（n≥1 固定）上，對所有向量 u、v 與所有實數 a，證明 ‖v‖₁=Σ|v_i| 與 ‖v‖_∞=max|v_i| 都滿足範數的三條公理：非負且 ‖v‖=0 ⇔ v=0、‖av‖=|a|‖v‖、‖u+v‖≤‖u‖+‖v‖。再在 ℝ³ 中對整數座標向量 u=(3,−4,1)、v=(−1,2,5) 計算兩種範數與三角不等式、齊次性的實例；最後說明 φ(v)=|v₁| 與 ψ(v)=(‖v‖₁)² 不是範數。","en":"On ℝⁿ (n≥1 fixed), for all vectors u, v and all reals a, prove that ‖v‖₁=Σ|v_i| and ‖v‖_∞=max|v_i| both satisfy the three norm axioms: nonnegativity with ‖v‖=0 ⇔ v=0, ‖av‖=|a|‖v‖, and ‖u+v‖≤‖u‖+‖v‖. Then, in ℝ³, compute both norms and instances of the triangle inequality and homogeneity for the integer-coordinate vectors u=(3,−4,1) and v=(−1,2,5); finally show that φ(v)=|v₁| and ψ(v)=(‖v‖₁)² are not norms."},"mko_ids":["mko-real-normed-vector-space"]},{"id":"derivation-mko-cauchy-sequence-01","title":{"zh-Hant":"Cauchy 序列：倒數序列的完整證書、序列在其中無極限的空間、交替反例與最終常值模型","en":"Cauchy sequence: a complete certificate for the reciprocal sequence, a space in which it has no limit, an alternating counterexample and an ultimately constant model"},"summary":{"zh-Hant":"（一）證明 x_n=1/n（n≥1）在 (ℝ, |s−t|) 中是 Cauchy 序列：對每個 ε>0 給出 N，使所有 m,n≥N 都有 |1/m−1/n|<ε。（二）證明同一序列在 X=(0,1]（限制度量）中仍是 Cauchy，但在 X 中沒有極限。（三）證明 y_n=(−1)^n 不是 Cauchy。（四）證明最終常值序列 z（z₁=0、z₂=7、n≥3 時 z_n=5）是 Cauchy。並說明有限尾段為何不能證明 Cauchy。","en":"(1) Prove that x_n=1/n (n≥1) is a Cauchy sequence in (ℝ, |s−t|): for every ε>0 give N such that |1/m−1/n|<ε for all m,n≥N. (2) Prove that the same sequence is still Cauchy in X=(0,1] with the restricted metric, but has no limit in X. (3) Prove that y_n=(−1)^n is not Cauchy. (4) Prove that the ultimately constant sequence z (z₁=0, z₂=7, z_n=5 for n≥3) is Cauchy. Also explain why a finite tail cannot prove the Cauchy property."},"mko_ids":["mko-cauchy-sequence"]},{"id":"derivation-mko-finite-joint-distribution-01","title":{"zh-Hant":"有限聯合分布：驗證聯合質量表並由同一 joint law 聚合邊際與事件","en":"Finite joint distribution: validating a joint mass table and aggregating marginals and events from one joint law"},"summary":{"zh-Hant":"給定支撐 S={(0,0),(0,1),(1,0),(1,1),(2,1)} 與質量 p(0,0)=1/6、p(0,1)=1/3、p(1,0)=1/4、p(1,1)=1/12、p(2,1)=1/6。驗證 (S,p) 是二維有限聯合分布；由同一個 joint law 算出 X 與 Y 的邊際、矩形事件 P(X∈{1,2}, Y=1) 與單點質量 P(X=2, Y=0)；最後構造一個邊際相同但不同的 joint law，說明兩個邊際一般不能反推聯合分布。","en":"Given the support S={(0,0),(0,1),(1,0),(1,1),(2,1)} with masses p(0,0)=1/6, p(0,1)=1/3, p(1,0)=1/4, p(1,1)=1/12, p(2,1)=1/6, verify that (S,p) is a two-dimensional finite joint distribution; from this one joint law compute the marginals of X and Y, the rectangle event P(X∈{1,2}, Y=1) and the single-pair mass P(X=2, Y=0); finally construct a different joint law with the same marginals, showing that two marginals do not in general determine the joint distribution."},"mko_ids":["mko-finite-joint-distribution"]},{"id":"derivation-mko-covariance-01","title":{"zh-Hant":"協方差：對稱性、仿射規則與「零協方差不蘊含獨立」","en":"Covariance: symmetry, the affine rule, and why zero covariance does not imply independence"},"summary":{"zh-Hant":"在同一個精確有限 joint law 上：(1) 證明 Cov(Y,X)=Cov(X,Y)；(2) 證明 Cov(aX+b,cY+d)=ac·Cov(X,Y)；(3) 證明 X、Y 獨立時 Cov(X,Y)=0，因此非零協方差可否證獨立；(4) 對 X 均勻取 −1、0、1 且 Y=X² 的 joint law，算出 E[X]、E[Y]、E[XY] 與 Cov(X,Y)=0，並證明 X、Y 不獨立。","en":"On one exact finite joint law: (1) prove Cov(Y,X)=Cov(X,Y); (2) prove Cov(aX+b,cY+d)=ac·Cov(X,Y); (3) prove that independence of X and Y gives Cov(X,Y)=0, so a nonzero covariance refutes independence; (4) for the joint law with X uniform on −1, 0, 1 and Y=X², compute E[X], E[Y], E[XY] and Cov(X,Y)=0, and prove that X and Y are not independent."},"mko_ids":["mko-covariance"]},{"id":"derivation-mko-graph-isomorphism-01","title":{"zh-Hant":"圖同構：以完整映射表逐項核對雙射與鄰接當且僅當","en":"Graph isomorphism: checking bijectivity and both directions of adjacency preservation from a complete mapping table"},"summary":{"zh-Hant":"G 的頂點為 1、2、3、4，邊為 {1,2}、{2,3}、{3,4}、{1,4}；H 的頂點為 a、b、c、d，邊為 {a,c}、{b,c}、{b,d}、{a,d}。證明映射 f：1↦a、2↦c、3↦b、4↦d 是圖同構 G→H；說明雙射 g：1↦a、2↦b、3↦c、4↦d 不是同構；並證明路徑 P（邊 {1,2}、{2,3}、{3,4}）與 H 不同構，即使 f 把 P 的每條邊都送到 H 的邊。","en":"G has vertices 1, 2, 3, 4 and edges {1,2}, {2,3}, {3,4}, {1,4}; H has vertices a, b, c, d and edges {a,c}, {b,c}, {b,d}, {a,d}. Prove that f: 1↦a, 2↦c, 3↦b, 4↦d is a graph isomorphism G→H; show that the bijection g: 1↦a, 2↦b, 3↦c, 4↦d is not an isomorphism; and prove that the path P (edges {1,2}, {2,3}, {3,4}) is not isomorphic to H, even though f sends every edge of P to an edge of H."},"mko_ids":["mko-graph-isomorphism"]},{"id":"derivation-mko-equivalence-relation-01","title":{"zh-Hant":"等價關係：在完整有限關係表上逐項核對三公理並輸出等價類","en":"Equivalence relation: checking the three axioms on a complete finite relation and emitting the classes"},"summary":{"zh-Hant":"在 X={0,1,2,3,4} 上，關係 R 完整列出為 13 個有序對：(0,0)、(1,1)、(2,2)、(3,3)、(4,4)、(0,2)、(2,0)、(1,3)、(3,1)、(1,4)、(4,1)、(3,4)、(4,3)。逐項核對反身、對稱、傳遞，證明 R 是等價關係並輸出等價類；再說明去掉 (3,4)、(4,3) 後的 R′ 雖反身且對稱，卻不是等價關係。","en":"On X={0,1,2,3,4} the relation R is listed completely as 13 ordered pairs: (0,0), (1,1), (2,2), (3,3), (4,4), (0,2), (2,0), (1,3), (3,1), (1,4), (4,1), (3,4), (4,3). Check reflexivity, symmetry and transitivity item by item, prove that R is an equivalence relation and emit its classes; then show that R′, obtained by removing (3,4) and (4,3), is reflexive and symmetric but not an equivalence relation."},"mko_ids":["mko-equivalence-relation"]},{"id":"derivation-mko-finite-unital-ring-01","title":{"zh-Hant":"有限含一環：證明模 4 的加法表與乘法表構成含一環","en":"Finite unital ring: proving that the mod-4 addition and multiplication tables form a unital ring"},"summary":{"zh-Hant":"在載體 X={0,1,2,3} 上給定完整的加法表 ⊕ 與乘法表 ⊙（即模 4 的加法與乘法）。對所有元素對與三元組證明：(X,⊕) 是阿貝爾群；⊙ 結合；⊙ 對 ⊕ 左右分配；存在唯一的雙側乘法單位元 1，且 1≠0。並指出 2⊙2=0 不妨礙它是含一環。","en":"On the carrier X={0,1,2,3}, complete addition and multiplication tables ⊕ and ⊙ (addition and multiplication mod 4) are given. Prove for all pairs and triples: (X,⊕) is an abelian group; ⊙ is associative; ⊙ distributes over ⊕ on both sides; there is a unique two-sided multiplicative identity 1, and 1≠0. Note also that 2⊙2=0 does not stop it from being a unital ring."},"mko_ids":["mko-finite-unital-ring"]},{"id":"derivation-mko-norm-induced-metric-01","title":{"zh-Hant":"範數誘導度量：由範數公理證明 d(x,y)=‖x−y‖ 是度量，並在 ℝ³ 上以 l1 與 l∞ 精確計算","en":"Norm-induced metric: proving from the norm axioms that d(x,y)=‖x−y‖ is a metric, with exact l1 and l∞ computations in ℝ³"},"summary":{"zh-Hant":"設 (V,‖·‖) 是實賦範向量空間。證明 d(x,y)=‖x−y‖ 滿足非負、分離、對稱與三角不等式；證明 ℝⁿ 上的 l1 與 l∞ 公式都是範數，所以都誘導度量；再在 ℝ³ 上對 x=(1,−2,3)、y=(4,0,−1)、z=(2,5,1) 精確計算 d1 與 d∞。","en":"Let (V,‖·‖) be a real normed vector space. Prove that d(x,y)=‖x−y‖ is nonnegative, separates points, is symmetric and satisfies the triangle inequality; prove that the l1 and l∞ formulas on ℝⁿ are norms, so both induce metrics; then compute d1 and d∞ exactly in ℝ³ for x=(1,−2,3), y=(4,0,−1), z=(2,5,1)."},"mko_ids":["mko-norm-induced-metric"]},{"id":"derivation-mko-complete-metric-space-01","title":{"zh-Hant":"完備度量空間：以最小正距離證明有限度量空間完備，並以 (0,1] 對照","en":"Complete metric space: proving every finite metric space complete by the least positive distance, contrasted with (0,1]"},"summary":{"zh-Hant":"證明：每個至少有一點的有限度量空間 (X,d) 都是完備的（單點情形直接成立；至少兩點時用最小正距離 δ 證明每個 Cauchy 序列最終常值）。在 X0={0,1,3}、d(x,y)=|x−y| 上算出 δ=1，給一個最終常值證書與一個明示的非 Cauchy 序列；最後證明子空間 Y=(0,1] 不完備：x_n=1/n 在 Y 中是 Cauchy，但在 Y 中沒有極限。","en":"Prove that every finite metric space (X,d) with at least one point is complete (immediate for a single point; with at least two points, the least positive distance δ forces every Cauchy sequence to be ultimately constant). On X0={0,1,3} with d(x,y)=|x−y|, compute δ=1, give an ultimately constant certificate and an explicit non-Cauchy sequence; finally prove that the subspace Y=(0,1] is not complete: x_n=1/n is Cauchy in Y but has no limit in Y."},"mko_ids":["mko-complete-metric-space"]},{"id":"derivation-mko-real-bilinear-form-01","title":{"zh-Hant":"實雙線性形式：以矩陣公式 x^T A y 分槽證明雙線性並精確求值","en":"Real bilinear form: proving bilinearity slot by slot from x^T A y, with exact evaluation"},"summary":{"zh-Hant":"設 A=[[1,2],[−1,3]]，B(x,y)=x^T A y（x、y∈ℝ²）。分別證明 B 對第一槽與第二槽都是線性的（雙線性）；對 x=(2,−1)、y=(1,4) 精確求 B(x,y)=7 與 B(y,x)=−20，說明雙線性不要求對稱；並證明黑箱函數 f(x,y)=x1y1+(x2³−x2)y2 雖通過若干加法樣本卻不是雙線性。","en":"Let A=[[1,2],[−1,3]] and B(x,y)=x^T A y for x, y∈ℝ². Prove separately that B is linear in the first slot and in the second slot (bilinear); evaluate exactly B(x,y)=7 and B(y,x)=−20 for x=(2,−1), y=(1,4), showing that bilinearity does not require symmetry; and prove that the black-box function f(x,y)=x1y1+(x2³−x2)y2 passes some additivity samples but is not bilinear."},"mko_ids":["mko-real-bilinear-form"]},{"id":"derivation-mko-general-linear-map-01","title":{"zh-Hant":"實向量空間間線性映射：矩陣映射的全稱證書與仿射、黑箱反例","en":"Linear maps between real vector spaces: a universal certificate for a matrix map, with affine and black-box counterexamples"},"summary":{"zh-Hant":"設 A 為第一行 (2,−1,0)、第二行 (1,3,4) 的 2×3 整數矩陣，T:ℝ³→ℝ²，T(x)=Ax。（一）證明對所有 u,v∈ℝ³ 與所有實數 α,β，T(αu+βv)=αT(u)+βT(v)。（二）證明 S(x)=Ax+b（b=(1,0)）不是線性映射，並證明對任意 b∈ℝ²，x↦Ax+b 線性當且僅當 b=0。（三）說明黑箱 F(x)=Ax+(x₁²−x₁,0) 通過一個樣本測試，卻不是線性映射。","en":"Let A be the 2-by-3 integer matrix with first row (2,−1,0) and second row (1,3,4), and let T:ℝ³→ℝ², T(x)=Ax. (1) Prove that T(αu+βv)=αT(u)+βT(v) for all u,v∈ℝ³ and all reals α,β. (2) Prove that S(x)=Ax+b with b=(1,0) is not linear, and that for every b∈ℝ² the map x↦Ax+b is linear if and only if b=0. (3) Show that the black box F(x)=Ax+(x₁²−x₁,0) passes one sample test and yet is not linear."},"mko_ids":["mko-general-linear-map"]},{"id":"derivation-mko-event-independence-01","title":{"zh-Hant":"事件獨立：以交集乘積判定，並核對條件式、補集與零機率事件","en":"Event independence: deciding by the product rule, with the conditional form, complements and a zero-probability event"},"summary":{"zh-Hant":"公正骰子模型：Ω={1,2,3,4,5,6}，每個結果的質量為 1/6。設 A={2,4,6}、B={1,2,3,4}、D={1,3,5}、Z=∅。（一）以乘積定義判定 A 與 B 是否獨立，並在 P(B)>0 下核對 P(A|B)=P(A)。（二）證明 A 與 B 獨立推出 A 與 Bᶜ 獨立，並以數字核對。（三）判定互斥事件 A、D 是否獨立。（四）判定 A 與零機率事件 Z 是否獨立，並說明條件式為何不能使用。","en":"Fair-die model: Ω={1,2,3,4,5,6}, each outcome with mass 1/6. Let A={2,4,6}, B={1,2,3,4}, D={1,3,5} and Z=∅. (1) Decide by the product definition whether A and B are independent, and with P(B)>0 check P(A|B)=P(A). (2) Prove that independence of A and B implies independence of A and Bᶜ, and check it numerically. (3) Decide whether the mutually exclusive events A and D are independent. (4) Decide whether A and the zero-probability event Z are independent, and explain why the conditional form cannot be used."},"mko_ids":["mko-event-independence"]},{"id":"derivation-mko-graph-walk-simple-path-01","title":{"zh-Hant":"圖的 walk 與 simple path：逐對核對、長度與 trail 的區別","en":"Graph walks and simple paths: pair-by-pair checks, length and the difference from trails"},"summary":{"zh-Hant":"有限無向無 loop 簡單圖 G：V={0,1,2,3,4}，E={{0,1},{1,2},{2,3},{0,3},{1,3},{3,4}}。對頂點序列 W₁=(0,1,2,3,4)、W₂=(0,1,3,1,2)、W₃=(0,2,3)、W₄=(2,3,0,1,3,4)、W₅=(4)、W₆=(1,1) 逐一判定是否為 walk、長度為何、是否為 simple path；並證明 simple path 不重複邊，以及本圖中 simple path 的長度至多為 4。","en":"Finite undirected loop-free simple graph G: V={0,1,2,3,4}, E={{0,1},{1,2},{2,3},{0,3},{1,3},{3,4}}. For the vertex sequences W₁=(0,1,2,3,4), W₂=(0,1,3,1,2), W₃=(0,2,3), W₄=(2,3,0,1,3,4), W₅=(4) and W₆=(1,1), decide whether each is a walk, its length, and whether it is a simple path; also prove that a simple path repeats no edge and that in this graph a simple path has length at most 4."},"mko_ids":["mko-graph-walk-simple-path"]},{"id":"derivation-mko-finite-commutative-ring-01","title":{"zh-Hant":"有限交換環：逐元素對核對 ℤ₆ 的交換律，並以上三角矩陣環為非交換反例","en":"Finite commutative ring: checking commutativity of ℤ₆ over all element pairs, with an upper-triangular matrix ring as a non-commutative counterexample"},"summary":{"zh-Hant":"（一）在已通過有限含一環資格的 ℤ₆={0,1,2,3,4,5}（a⊕b=(a+b) mod 6、a⊗b=(ab) mod 6）上，核對全部 36 個有序元素對的 a⊗b=b⊗a，判定 ℤ₆ 是否為有限交換環。（二）說明 ℤ₆ 有非零零因子且 2 不可逆，所以不是整環或域。（三）在已通過資格的 ℱ₂ 上 2×2 上三角矩陣環 U 中，說明若干乘積可交換，但 U 不是交換環。","en":"(1) On ℤ₆={0,1,2,3,4,5} (a⊕b=(a+b) mod 6, a⊗b=(ab) mod 6), already qualified as a finite unital ring, check a⊗b=b⊗a for all 36 ordered element pairs and decide whether ℤ₆ is a finite commutative ring. (2) Show that ℤ₆ has nonzero zero divisors and that 2 is not invertible, so it is neither an integral domain nor a field. (3) In the qualified ring U of 2-by-2 upper-triangular matrices over 𝔽₂, show that several products commute and yet U is not commutative."},"mko_ids":["mko-finite-commutative-ring"]},{"id":"derivation-mko-finite-two-sided-ideal-01","title":{"zh-Hant":"有限雙邊理想：在上三角矩陣環中分開核對左吸收與右吸收","en":"Finite two-sided ideal: checking left and right absorption separately in an upper-triangular matrix ring"},"summary":{"zh-Hant":"父環 U 是 𝔽₂ 上 2×2 上三角矩陣環（元素記為 (a,b,c)，代表第一行 (a,b)、第二行 (0,c)），已通過有限含一環資格。（一）證明 I={(0,0,0),(0,1,0)} 是 U 的雙邊理想。（二）證明 L={(0,0,0),(1,0,0)} 滿足左吸收，但不是雙邊理想。（三）證明對角子環 D={(a,0,c)} 含 1 卻不是理想。（四）說明 {0} 與 U 本身是雙邊理想。","en":"The parent ring U is the ring of 2-by-2 upper-triangular matrices over 𝔽₂ (elements written (a,b,c) for the matrix with first row (a,b) and second row (0,c)), already qualified as a finite unital ring. (1) Prove that I={(0,0,0),(0,1,0)} is a two-sided ideal of U. (2) Prove that L={(0,0,0),(1,0,0)} satisfies left absorption but is not a two-sided ideal. (3) Prove that the diagonal subring D={(a,0,c)} contains 1 but is not an ideal. (4) Show that {0} and U itself are two-sided ideals."},"mko_ids":["mko-finite-two-sided-ideal"]},{"id":"derivation-mko-real-banach-space-01","title":{"zh-Hant":"實 Banach 空間：(ℝ², ℓ¹) 的範數、誘導度量與完備性","en":"Real Banach space: norm, induced metric and completeness of (ℝ², ℓ¹)"},"summary":{"zh-Hant":"在 V=ℝ² 上取 ℓ¹ 範數 ‖x‖₁=|x₁|+|x₂| 與同一範數誘導的度量 d₁(x,y)=‖x−y‖₁。（一）證明 ‖·‖₁ 是範數、d₁ 是度量，並算出 d₁((3,−1),(1,2))。（二）由實數的 Cauchy 完備性完整證明 (ℝ²,d₁) 完備，所以 (ℝ²,‖·‖₁) 是實 Banach 空間；並引用有限維實賦範空間完備的命名定理。（三）證明 ‖·‖₁ 不滿足平行四邊形律，所以不是由任何內積誘導。","en":"On V=ℝ² take the ℓ¹ norm ‖x‖₁=|x₁|+|x₂| and the metric induced by that same norm, d₁(x,y)=‖x−y‖₁. (1) Prove that ‖·‖₁ is a norm and d₁ a metric, and compute d₁((3,−1),(1,2)). (2) Prove completely, from the Cauchy completeness of the reals, that (ℝ²,d₁) is complete, so (ℝ²,‖·‖₁) is a real Banach space; also cite the named theorem that finite-dimensional real normed spaces are complete. (3) Prove that ‖·‖₁ violates the parallelogram law and hence is induced by no inner product."},"mko_ids":["mko-real-banach-space"]},{"id":"derivation-mko-real-inner-product-space-01","title":{"zh-Hant":"實內積空間：以 Sylvester 條件為 ℝ² 上的矩陣形式作全稱正定證書","en":"Real inner-product space: Sylvester conditions as a universal positive-definiteness certificate for a matrix form on ℝ²"},"summary":{"zh-Hant":"設 M 為第一行 (2,1)、第二行 (1,3) 的整數對稱矩陣，⟨x,y⟩_M=xᵀMy（x,y∈ℝ²）。（一）分別證明 ⟨·,·⟩_M 雙線性、對稱、正定，所以是 ℝ² 上的內積；正定性以 2×2 Sylvester 條件（完整證明其充分與必要方向）作全稱證書。（二）判定 N（行 (1,2),(2,1)）、P（行 (1,1),(1,1)）、R（行 (2,1),(0,3)）給出的形式為何不是內積。","en":"Let M be the symmetric integer matrix with first row (2,1) and second row (1,3), and ⟨x,y⟩_M=xᵀMy for x,y∈ℝ². (1) Prove separately that ⟨·,·⟩_M is bilinear, symmetric and positive definite, hence an inner product on ℝ²; positive definiteness is certified universally by the 2-by-2 Sylvester conditions, with both sufficiency and necessity proved in full. (2) Decide why the forms given by N (rows (1,2),(2,1)), P (rows (1,1),(1,1)) and R (rows (2,1),(0,3)) are not inner products."},"mko_ids":["mko-real-inner-product-space"]},{"id":"derivation-mko-bounded-linear-operator-01","title":{"zh-Hant":"有界線性算子：矩陣在 ℓ^∞ 與 ℓ¹ 下的精確誘導算子範數","en":"Bounded linear operator: exact induced operator norms of a matrix under ℓ^∞ and ℓ¹"},"summary":{"zh-Hant":"設 A 為第一行 (1,−2)、第二行 (3,4) 的整數矩陣，T(x)=Ax:ℝ²→ℝ²。（一）在兩端都用 ℓ^∞ 範數時，證明對所有 x 有 ‖Ax‖_∞≤7‖x‖_∞，且沒有更小的常數可用，所以 ‖T‖_{∞→∞}=7（最大行絕對值和）。（二）兩端都用 ℓ¹ 範數時，證明 ‖T‖_{1→1}=6（最大列絕對值和）。（三）說明非最佳常數、有限輸入與值域無界這三個陷阱。","en":"Let A be the integer matrix with first row (1,−2) and second row (3,4), and T(x)=Ax:ℝ²→ℝ². (1) With the ℓ^∞ norm at both ends, prove that ‖Ax‖_∞≤7‖x‖_∞ for all x and that no smaller constant works, so ‖T‖_{∞→∞}=7 (the maximum absolute row sum). (2) With the ℓ¹ norm at both ends, prove ‖T‖_{1→1}=6 (the maximum absolute column sum). (3) Explain three traps: non-optimal constants, finitely many inputs, and unbounded range."},"mko_ids":["mko-bounded-linear-operator"]},{"id":"derivation-mko-random-variable-independence-01","title":{"zh-Hant":"隨機變數獨立：完整核對有限聯合表，零協方差只是後果","en":"Independence of random variables: checking the complete finite joint table, with zero covariance only a consequence"},"summary":{"zh-Hant":"（一）聯合表 J₁：X∈{0,1}、Y∈{0,1,2}，P(0,0)=1/12、P(0,1)=1/12、P(0,2)=1/6、P(1,0)=1/6、P(1,1)=1/6、P(1,2)=1/3。由聯合表聚合兩個邊際，核對全部 6 個值對的乘積式，判定 X、Y 是否獨立；並證明獨立推出任意座標事件 {X∈S} 與 {Y∈T} 獨立與 E[XY]=E[X]E[Y]。（二）聯合表 J₂：X 在 {−1,0,1} 上均勻、Y=X²。證明 Cov(X,Y)=0 但 X、Y 不獨立。（三）聯合表 J₃ 說明只核一個值對不夠。","en":"(1) Joint table J₁: X∈{0,1}, Y∈{0,1,2}, P(0,0)=1/12, P(0,1)=1/12, P(0,2)=1/6, P(1,0)=1/6, P(1,1)=1/6, P(1,2)=1/3. Aggregate both marginals from the joint table, check the product equation at all 6 value pairs, and decide whether X and Y are independent; also prove that independence implies independence of any coordinate events {X∈S} and {Y∈T} and E[XY]=E[X]E[Y]. (2) Joint table J₂: X uniform on {−1,0,1} and Y=X². Prove Cov(X,Y)=0 although X and Y are not independent. (3) Use joint table J₃ to show that checking a single value pair is not enough."},"mko_ids":["mko-random-variable-independence"]},{"id":"derivation-mko-connected-component-01","title":{"zh-Hant":"連通分量：以完整 BFS 求出有限簡單圖的全部分量","en":"Connected components: finding every component of a finite simple graph by complete BFS"},"summary":{"zh-Hant":"設 G=(V,E)，V={0,1,2,3,4,5,6}，E 恰為 {0,1}、{0,2}、{1,2}、{3,4}、{4,5} 五條邊。先證明「有 path 相連」是 V 上的等價關係，再以完整 BFS 求出每個等價類，證明 BFS 回傳的集合恰為該類，並得出 Q(G)={{0,1,2},{3,4,5},{6}}，其中孤立頂點 6 自成單點分量。","en":"Let G=(V,E) with V={0,1,2,3,4,5,6} and E consisting of exactly the five edges {0,1}, {0,2}, {1,2}, {3,4}, {4,5}. First prove that 'joined by a path' is an equivalence relation on V; then compute each equivalence class by complete BFS, prove that the set BFS returns is exactly that class, and conclude Q(G)={{0,1,2},{3,4,5},{6}}, where the isolated vertex 6 forms a singleton component."},"mko_ids":["mko-connected-component"]},{"id":"derivation-mko-finite-integral-domain-01","title":{"zh-Hant":"有限整環：以完整乘法表判定 ℤ/5ℤ 與 ℤ/6ℤ，再由有限性推出可逆","en":"Finite integral domain: deciding ℤ/5ℤ and ℤ/6ℤ from complete tables, then deriving invertibility from finiteness"},"summary":{"zh-Hant":"以完整乘法表判定 ℤ/5ℤ 是有限整環、ℤ/6ℤ 不是；再把「每個非零元可逆」作為定義之後的定理，對任何有限整環用有限性證明，並在 ℤ/5ℤ 中求出全部逆元，同時說明 ℤ/6ℤ 中 2 為何沒有逆元。","en":"Using complete multiplication tables, decide that ℤ/5ℤ is a finite integral domain and ℤ/6ℤ is not; then prove, as a theorem that comes after the definition, that every nonzero element of any finite integral domain is invertible, using finiteness; find all inverses in ℤ/5ℤ and explain why 2 has no inverse in ℤ/6ℤ."},"mko_ids":["mko-finite-integral-domain"]},{"id":"derivation-mko-polynomial-ring-over-finite-commutative-ring-01","title":{"zh-Hant":"有限交換環上的多項式環：在 (ℤ/4ℤ)[x] 中做加法、卷積與次數核對","en":"Polynomial ring over a finite commutative ring: addition, convolution and degree checks in (ℤ/4ℤ)[x]"},"summary":{"zh-Hant":"在 R=ℤ/4ℤ 上取 f=1+2x、g=2x。依係數序列定義計算 f+g、f·g、f·f、g·g 並正規化；證明首項係數公式 (fg)_{m+n}=f_m g_n，說明為何此處 deg(fg)=deg f+deg g 失敗；最後證明 R[x] 不是有限環。","en":"Over R=ℤ/4ℤ take f=1+2x and g=2x. Using the coefficient-sequence definition, compute and normalize f+g, f·g, f·f and g·g; prove the top-coefficient formula (fg)_{m+n}=f_m g_n and explain why deg(fg)=deg f+deg g fails here; finally prove that R[x] is not a finite ring."},"mko_ids":["mko-polynomial-ring-over-finite-commutative-ring"]},{"id":"derivation-mko-inner-product-induced-norm-01","title":{"zh-Hant":"內積誘導範數：證明 √⟨x,x⟩ 是範數並在 ℝ² 正定矩陣下精確計算","en":"Inner-product-induced norm: proving √⟨x,x⟩ is a norm and computing exactly for a positive-definite matrix on ℝ²"},"summary":{"zh-Hant":"在任一實內積空間中證明 N(v)=√⟨v,v⟩ 是範數：非負與分離、絕對齊次、三角不等式（經由完整證明的 Cauchy–Schwarz）。再在 ℝ² 上以 A=[[2,1],[1,2]] 定義的內積，對 x=(1,0)、y=(0,1) 精確算出平方範數、距離平方，並做 Cauchy–Schwarz、三角不等式與平行四邊形恆等式的實例核對。","en":"In any real inner-product space, prove that N(v)=√⟨v,v⟩ is a norm: nonnegativity and separation, absolute homogeneity, and the triangle inequality (via a fully proved Cauchy–Schwarz inequality). Then, on ℝ² with the inner product given by A=[[2,1],[1,2]], compute exactly the squared norms and the squared distance for x=(1,0) and y=(0,1), and check Cauchy–Schwarz, the triangle inequality and the parallelogram identity on this instance."},"mko_ids":["mko-inner-product-induced-norm"]},{"id":"derivation-mko-conditional-expectation-01","title":{"zh-Hant":"有限條件期望：由分割生成的 σ 代數上求 E[X|G]，含零質量區塊","en":"Finite conditional expectation: computing E[X|G] for a partition-generated σ-algebra, including a zero-mass block"},"summary":{"zh-Hant":"在 Ω={0,1,2,3,4} 上，質量依序為 1/8、3/8、1/4、1/4、0，X 的值依序為 4、0、2、6、10，G 由分割 {0,1}、{2,3}、{4} 生成。由條件期望的兩項定義（G 可測、在每個 G 事件上積分相同）求出 E[X|G] 的全部版本，證明它在兩個正質量區塊上必為 1 與 4，在零質量區塊 {4} 上可取任意值（只幾乎處處決定），並說明逐點條件化、只核總期望與不加權平均為何都錯。","en":"On Ω={0,1,2,3,4} with masses 1/8, 3/8, 1/4, 1/4, 0 and X taking the values 4, 0, 2, 6, 10 in order, let G be generated by the partition {0,1}, {2,3}, {4}. From the two defining conditions of conditional expectation (G-measurable, same integral on every event of G), find all versions of E[X|G]: prove they must equal 1 and 4 on the two positive-mass blocks and may take any value on the zero-mass block {4} (determined only almost surely), and explain why pointwise conditioning, checking only the total expectation, and the unweighted mean are all wrong."},"mko_ids":["mko-conditional-expectation"]},{"id":"derivation-mko-finite-filtration-01","title":{"zh-Hant":"有限 filtration：以分割細化核對資訊嵌套，並區分 adapted 與獨立","en":"Finite filtration: checking information nesting by partition refinement, and separating it from adaptedness and independence"},"summary":{"zh-Hant":"在 Ω={0,1,2,3}（兩次擲幣：0=正正、1=正反、2=反正、3=反反）上，以分割編碼 P_0、P_1、P_2 表示時刻 0、1、2 的 σ 代數。證明「較晚分割細化較早分割 ⇔ σ 代數包含」，逐相鄰時刻完整核對，得出 (F_t) 是 filtration；給出不是 filtration 的分割序列；並證明 filtration 既不使每個過程 adapted，也不推出獨立。","en":"On Ω={0,1,2,3} (two coin tosses: 0=HH, 1=HT, 2=TH, 3=TT), represent the σ-algebras at times 0, 1, 2 by partition rows P_0, P_1, P_2. Prove that 'the later partition refines the earlier one ⇔ the σ-algebras are nested', check every adjacent pair of times completely and conclude that (F_t) is a filtration; give a sequence of partitions that is not a filtration; and show that a filtration neither makes every process adapted nor implies independence."},"mko_ids":["mko-finite-filtration"]},{"id":"derivation-mko-iid-sequence-01","title":{"zh-Hant":"獨立同分布序列：由 Bernoulli(1/3) 柱集乘積公式證明 IID","en":"IID sequence: proving IID from the Bernoulli(1/3) cylinder product formula"},"summary":{"zh-Hant":"設 p=1/3，柱集證書規定每個有限 0-1 word w（長度 n、含 k 個 1）的機率為 p^k(1−p)^{n−k}。證明：此族一致且每層總和為 1；由此（配合引用的 Kolmogorov 存在定理）得到 {0,1}^ℕ 上的機率；每個 X_i 的完整邊際都是 Bernoulli(1/3)；每個有限子族都獨立。再以三個對照說明：檢查過的 prefix、pairwise 獨立與相同期望都不足以宣稱無限 IID。","en":"Let p=1/3, and let the cylinder certificate give each finite binary word w (length n, with k ones) the probability p^k(1−p)^{n−k}. Prove that this family is consistent with total mass 1 at every level; obtain from it (with the cited Kolmogorov existence theorem) a probability on {0,1}^ℕ; show that every X_i has the complete marginal Bernoulli(1/3) and that every finite subfamily is independent. Then use three contrasts to show that a checked prefix, pairwise independence and equal expectations are each insufficient for an infinite IID claim."},"mko_ids":["mko-iid-sequence"]},{"id":"derivation-mko-equivalence-class-01","title":{"zh-Hant":"等價類：在完整有限關係上先認證、再求類，並以非等價關係作對照","en":"Equivalence class: certify a complete finite relation first, then form classes, with a non-equivalence as contrast"},"summary":{"zh-Hant":"在 X={0,1,2,3,4,5} 上，R 由 12 個明列的有序對給出。先完整認證 R 是等價關係，再依定義 [a]={x∈X ∣ aRx} 求出全部類；證明 aRb ⇒ [a]=[b]、任兩類相等或不交、各類覆蓋 X，得 K(R)={{0,3},{1,4},{2,5}}。最後以 S=R∪{(0,1),(1,0)} 說明未通過傳遞性的關係不能形成等價類。","en":"On X={0,1,2,3,4,5}, R is given by 12 explicitly listed ordered pairs. First certify completely that R is an equivalence relation, then compute every class from the definition [a]={x∈X ∣ aRx}; prove that aRb ⇒ [a]=[b], that any two classes are equal or disjoint and that the classes cover X, giving K(R)={{0,3},{1,4},{2,5}}. Finally use S=R∪{(0,1),(1,0)} to show that a relation failing transitivity cannot form equivalence classes."},"mko_ids":["mko-equivalence-class"]},{"id":"derivation-mko-finite-field-01","title":{"zh-Hant":"完整表認證 F₃ 與有限整環引理","en":"Certifying F3 and the finite-domain lemma"},"summary":{"zh-Hant":"F₃ 是域，2/2=1；有限整環的非零乘法映射給逆元。","en":"F3 is a field and 2/2=1; nonzero multiplication maps supply inverses in a finite domain."},"mko_ids":["mko-finite-field"]},{"id":"derivation-mko-real-hilbert-space-01","title":{"zh-Hant":"同一矩陣誘導的有限維 Hilbert 證書","en":"A finite-dimensional Hilbert certificate from one matrix"},"summary":{"zh-Hant":"驗 R^2 配 A=[[2,1],[1,2]] 的內積、範數、度量與完備性一致。","en":"Verify the consistent inner product, norm, metric and completeness of R^2 with A=[[2,1],[1,2]]."},"mko_ids":["mko-real-hilbert-space"]},{"id":"derivation-mko-martingale-01","title":{"zh-Hant":"兩次公平擲幣的完整鞅檢查","en":"A complete two-coin martingale audit"},"summary":{"zh-Hant":"在同一空間逐時核鞅的全部責任。","en":"Audit every martingale obligation on one space."},"mko_ids":["mko-martingale"]},{"id":"derivation-mko-quotient-set-01","title":{"zh-Hant":"模2關係在四元素載體上的商集","en":"The quotient of a four-element carrier by parity"},"summary":{"zh-Hant":"X/R={ {0,2},{1,3} }，且投影值是整個類。","en":"X/R={{0,2},{1,3}}, and projection values are whole classes."},"mko_ids":["mko-quotient-set"]},{"id":"derivation-mko-cardinality-comparison-01","title":{"zh-Hant":"完整映射表的基數證書","en":"Cardinality certificates from full maps"},"summary":{"zh-Hant":"有單射A→B且|A|=2<3=|B|；反向滿射非單射。","en":"There is an injection A to B with |A|=2<3=|B|; a reverse surjection is not injective."},"mko_ids":["mko-cardinality-comparison"]},{"id":"derivation-mko-cumulative-distribution-function-01","title":{"zh-Hant":"三點分布的完整CDF","en":"The complete CDF of a three-point law"},"summary":{"zh-Hant":"算出全實軸CDF、右連續值與跳躍。","en":"Compute the CDF on the whole real line, including its right-continuous values and its jumps."},"mko_ids":["mko-cumulative-distribution-function"]},{"id":"derivation-mko-convergence-in-probability-01","title":{"zh-Hant":"eventual equality的全稱尾端證書","en":"A universal tail certificate from eventual equality"},"summary":{"zh-Hant":"以共同有限空間明證依機率收斂。","en":"Prove convergence in probability on a common finite space."},"mko_ids":["mko-convergence-in-probability"]},{"id":"derivation-mko-at-most-countable-set-01","title":{"zh-Hant":"三條自然數列的全稱編碼","en":"A universal code for three natural-number tracks"},"summary":{"zh-Hant":"N₀×{0,1,2}以c(n,j)=3n+j單射到N₀，故至多可數。","en":"N0 times {0,1,2} injects into N0 by c(n,j)=3n+j and is at most countable."},"mko_ids":["mko-at-most-countable-set"]},{"id":"derivation-mko-bipartite-graph-01","title":{"zh-Hant":"含孤立點的二分證書","en":"A bipartition certificate with an isolated vertex"},"summary":{"zh-Hant":"核兩部的全部責任，區分失敗partition與非二分。","en":"Audit every bipartition obligation and distinguish one failed partition from nonbipartiteness."},"mko_ids":["mko-bipartite-graph"]},{"id":"derivation-mko-convergence-in-distribution-01","title":{"zh-Hant":"Bernoulli參數尾端的CDF證書","en":"A Bernoulli parameter-tail CDF certificate"},"summary":{"zh-Hant":"在極限CDF連續點證依分布收斂。","en":"Prove distributional convergence at continuity points of the limiting CDF."},"mko_ids":["mko-convergence-in-distribution"]},{"id":"derivation-mko-finite-quotient-ring-01","title":{"zh-Hant":"Z/4Z 對偶數理想取商","en":"Quotienting Z/4Z by its even ideal"},"summary":{"zh-Hant":"I={0,2}為proper雙邊理想，R/I有兩個cosets及良定雙表。","en":"I={0,2} is a proper two-sided ideal; R/I has two cosets and well-defined operation tables."},"mko_ids":["mko-finite-quotient-ring"]},{"id":"derivation-mko-graph-coloring-01","title":{"zh-Hant":"完整著色與非最優greedy trace","en":"A complete coloring and a nonoptimal greedy trace"},"summary":{"zh-Hant":"逐邊驗proper coloring並展示上界非自動最優。","en":"Audit a proper coloring edgewise and show an upper bound need not be optimal."},"mko_ids":["mko-graph-coloring"]},{"id":"derivation-mko-vector-subspace-01","title":{"zh-Hant":"以全稱封閉認證過原點直線","en":"Universal closure of a line through zero"},"summary":{"zh-Hant":"U={(x,2x):x∈R}是R²子空間，仿射平移U+(0,1)不是。","en":"U={(x,2x):x in R} is a subspace of R2, whereas U+(0,1) is not."},"mko_ids":["mko-vector-subspace"]},{"id":"derivation-mko-closure-and-interior-01","title":{"zh-Hant":"在鏈狀三點拓撲求閉包與內部","en":"Closure and interior in a three-point chain topology"},"summary":{"zh-Hant":"求 A={b} 的閉包及內部。","en":"Find the closure and interior of A={b}."},"mko_ids":["mko-closure-and-interior"]},{"id":"derivation-mko-open-cover-and-subcover-01","title":{"zh-Hant":"從原覆蓋索引選取子覆蓋","en":"Selecting a subcover by original indices"},"summary":{"zh-Hant":"核對 K={a,b} 的指定開覆蓋與子覆蓋索引。","en":"Audit a supplied open cover of K={a,b} and its subcover indices."},"mko_ids":["mko-open-cover-and-subcover"]},{"id":"derivation-mko-almost-sure-convergence-01","title":{"zh-Hant":"逐原子尾端與零質量例外","en":"Atomwise tails and a null exception"},"summary":{"zh-Hant":"完整列出每個原子的收斂或失敗證書。","en":"Give a convergence or failure certificate for every atom."},"mko_ids":["mko-almost-sure-convergence"]},{"id":"derivation-mko-graph-matching-01","title":{"zh-Hant":"P4中maximal與maximum的分界","en":"Maximal versus maximum in P4"},"summary":{"zh-Hant":"完整核選邊與端點，再分清三種匹配性質。","en":"Audit selected edges and endpoints, then distinguish matching properties."},"mko_ids":["mko-graph-matching"]},{"id":"derivation-mko-euler-totient-function-01","title":{"zh-Hant":"逐代表計算φ(12)","en":"Counting phi(12) representative by representative"},"summary":{"zh-Hant":"φ(12)=4，φ(1)=1，且本例φ(3·4)=φ(3)φ(4)。","en":"phi(12)=4, phi(1)=1, and here phi(3*4)=phi(3)*phi(4)."},"mko_ids":["mko-euler-totient-function"]},{"id":"derivation-mko-finite-automaton-01","title":{"zh-Hant":"完整DFA的1011追蹤","en":"Tracing 1011 through a complete DFA"},"summary":{"zh-Hant":"指定DFA拒絕1011，並接受空字ε。","en":"The specified DFA rejects 1011 and accepts epsilon."},"mko_ids":["mko-finite-automaton"]},{"id":"derivation-mko-vector-space-dimension-01","title":{"zh-Hant":"先證基底再讀取維度","en":"Prove a basis before reading dimension"},"summary":{"zh-Hant":"B=((1,1),(1,−1))是R²有序基底，故dim R²=2。","en":"B=((1,1),(1,-1)) is an ordered basis of R2, so dim R2=2."},"mko_ids":["mko-vector-space-dimension"]},{"id":"derivation-mko-linear-map-kernel-01","title":{"zh-Hant":"完整求解二階矩陣的核","en":"The full kernel of a two-by-two map"},"summary":{"zh-Hant":"T(x,y)=(x+2y,2x+4y)的核為span{(−2,1)}。","en":"The kernel of T(x,y)=(x+2y,2x+4y) is span{(-2,1)}."},"mko_ids":["mko-linear-map-kernel"]},{"id":"derivation-mko-linear-map-image-01","title":{"zh-Hant":"用一致性求完整像空間","en":"The full image from consistency"},"summary":{"zh-Hant":"im T=span{(1,2)}；(3,6)在像內，(0,1)不在像內。","en":"im T=span{(1,2)}; (3,6) lies in the image and (0,1) does not."},"mko_ids":["mko-linear-map-image"]},{"id":"derivation-mko-topological-boundary-01","title":{"zh-Hant":"邊界與補集的精確對照","en":"Exact boundary comparison with the complement"},"summary":{"zh-Hant":"計算鏈狀拓撲中 A={b} 與其補集的邊界。","en":"Compute the boundaries of A={b} and its complement in a chain topology."},"mko_ids":["mko-topological-boundary"]},{"id":"derivation-mko-compact-topological-space-01","title":{"zh-Hant":"有限拓撲的全稱緊緻證書","en":"A universal compactness certificate for a finite topology"},"summary":{"zh-Hant":"證明三點鏈狀拓撲對每個開覆蓋都有有限子覆蓋。","en":"Prove every open cover of the three-point chain topology has a finite subcover."},"mko_ids":["mko-compact-topological-space"]},{"id":"derivation-mko-bayes-theorem-01","title":{"zh-Hant":"四原子檢測模型的Bayes反轉","en":"Bayes reversal in a four-atom test model"},"summary":{"zh-Hant":"由完整joint law算prior、likelihood與posterior。","en":"Compute priors, likelihoods and posteriors from a complete joint law."},"mko_ids":["mko-bayes-theorem"]},{"id":"derivation-mko-regular-language-01","title":{"zh-Hant":"任意字長的parity invariant","en":"A parity invariant for arbitrary word length"},"summary":{"zh-Hant":"Σ={0,1}上含偶數個1的全部word構成正規語言。","en":"All words over {0,1} with an even number of 1s form a regular language."},"mko_ids":["mko-regular-language"]},{"id":"derivation-mko-matrix-rank-01","title":{"zh-Hant":"三階矩陣的完整RREF與上下界","en":"Full RREF and rank bounds"},"summary":{"zh-Hant":"A=[[1,2,3],[2,4,6],[1,1,1]]的row與column rank均為2。","en":"A=[[1,2,3],[2,4,6],[1,1,1]] has row and column rank 2."},"mko_ids":["mko-matrix-rank"]},{"id":"derivation-mko-pointwise-convergence-of-functions-01","title":{"zh-Hant":"有限定義域的逐點尾項證書","en":"Pointwise tail certificates on a finite domain"},"summary":{"zh-Hant":"證 f_n(p)=1+2/(n+1), f_n(q)=-2-3/(n+1) 逐點趨於 f。","en":"Prove pointwise convergence of f_n(p)=1+2/(n+1), f_n(q)=-2-3/(n+1) to f."},"mko_ids":["mko-pointwise-convergence-of-functions"]},{"id":"derivation-mko-infinite-series-01","title":{"zh-Hant":"幾何與望遠鏡級數的完整尾項","en":"Complete tails for geometric and telescoping series"},"summary":{"zh-Hant":"用精確部分和與餘項證兩個級數收斂。","en":"Prove convergence of two series with exact partial sums and remainders."},"mko_ids":["mko-infinite-series"]},{"id":"derivation-mko-parametric-statistical-model-01","title":{"zh-Hant":"兩個參數值、四個樣本點的完整模型","en":"A complete model with two parameter values and four sample points"},"summary":{"zh-Hant":"逐參數驗law並區分likelihood與posterior。","en":"Validate every parameter law and distinguish likelihood from posterior."},"mko_ids":["mko-parametric-statistical-model"]},{"id":"derivation-mko-connected-topological-space-01","title":{"zh-Hant":"排除全部非平凡開閉集","en":"Excluding every nontrivial clopen set"},"summary":{"zh-Hant":"判斷三點鏈狀拓撲是否連通。","en":"Decide connectedness of a three-point chain topology."},"mko_ids":["mko-connected-topological-space"]},{"id":"derivation-mko-maximum-matching-01","title":{"zh-Hant":"三邊圖的全部八個matching候選","en":"All eight edge-subset candidates on a three-edge graph"},"summary":{"zh-Hant":"完整2³枚舉求最大matching與全部witness。","en":"Enumerate all2³ subsets for the maximum matching and all witnesses."},"mko_ids":["mko-maximum-matching"]},{"id":"derivation-mko-modular-inverse-01","title":{"zh-Hant":"17模43的正規化逆元","en":"The normalized inverse of 17 modulo 43"},"summary":{"zh-Hant":"17模43的唯一逆類為38；17·38=15·43+1。","en":"The unique inverse residue class of 17 modulo 43 is represented by 38; 17*38=15*43+1."},"mko_ids":["mko-modular-inverse"]},{"id":"derivation-mko-rank-nullity-theorem-01","title":{"zh-Hant":"三個來源座標的秩零度","en":"Rank and nullity for three source coordinates"},"summary":{"zh-Hant":"T:R³→R²，A=[[1,2,3],[2,4,6]]有rank1、nullity2，3=1+2。","en":"For T:R3 to R2, A=[[1,2,3],[2,4,6]] has rank 1 and nullity 2, with 3=1+2."},"mko_ids":["mko-rank-nullity-theorem"]},{"id":"derivation-mko-uniform-convergence-of-functions-01","title":{"zh-Hant":"三點域的精確上確界誤差","en":"Exact supremum error on a three-point domain"},"summary":{"zh-Hant":"證 g_n(a)=1/(n+2),g_n(b)=-4/(n+2),g_n(c)=2/(n+2) 一致趨零。","en":"Prove g_n(a)=1/(n+2),g_n(b)=-4/(n+2),g_n(c)=2/(n+2) converges uniformly to zero."},"mko_ids":["mko-uniform-convergence-of-functions"]},{"id":"derivation-mko-absolute-convergence-of-series-01","title":{"zh-Hant":"絕對收斂與交錯調和的雙證書","en":"Absolute convergence and the two certificates for alternating harmonic"},"summary":{"zh-Hant":"區分負比幾何級數的絕對收斂與交錯調和的條件收斂。","en":"Distinguish absolute convergence of a negative-ratio geometric series from conditional convergence of alternating harmonic."},"mko_ids":["mko-absolute-convergence-of-series"]},{"id":"derivation-mko-statistical-estimator-01","title":{"zh-Hant":"同一資料函數與兩個參數law","en":"One data function under two parameter laws"},"summary":{"zh-Hant":"核total estimator並計算逐theta期望。","en":"Validate a total estimator and its parameterwise expectations."},"mko_ids":["mko-statistical-estimator"]},{"id":"derivation-mko-chinese-remainder-theorem-01","title":{"zh-Hant":"三模數CRT的構造與唯一類","en":"Constructing a three-modulus CRT class"},"summary":{"zh-Hant":"x≡2 mod3、3 mod5、2 mod7的全部整數解為23+105t。","en":"All integer solutions of x=2 mod 3, x=3 mod 5, x=2 mod 7 are 23+105t."},"mko_ids":["mko-chinese-remainder-theorem"]},{"id":"derivation-mko-spanning-tree-01","title":{"zh-Hant":"方形父圖內的spanning tree","en":"A spanning tree inside a square parent graph"},"summary":{"zh-Hant":"核全頂點、選邊存在、connected及acyclic。","en":"Audit all vertices, edge membership, connectivity and acyclicity."},"mko_ids":["mko-spanning-tree"]},{"id":"derivation-mko-lp-convergence-01","title":{"zh-Hant":"有限law的L平方精確尾端","en":"An exact L-squared tail on a finite law"},"summary":{"zh-Hant":"計算二次誤差並提供任意容忍值的尾端。","en":"Compute the second error moment and give a tail for any tolerance."},"mko_ids":["mko-lp-convergence"]},{"id":"derivation-mko-direct-sum-01","title":{"zh-Hant":"R³中唯一分解與覆蓋","en":"Unique decomposition and coverage in R3"},"summary":{"zh-Hant":"R³=span{e₁}⊕span{e₂,e₃}，唯一性與覆蓋分別成立。","en":"R3=span{e1} direct-sum span{e2,e3}, with uniqueness and coverage separately verified."},"mko_ids":["mko-direct-sum"]},{"id":"derivation-mko-dual-vector-space-01","title":{"zh-Hant":"雙基底與covector換基","en":"Dual basis and covector change"},"summary":{"zh-Hant":"對b₁=(1,1)、b₂=(1,−1)，φ₁=(x+y)/2、φ₂=(x−y)/2為雙基底。","en":"For b1=(1,1),b2=(1,-1), phi1=(x+y)/2,phi2=(x-y)/2 form the dual basis."},"mko_ids":["mko-dual-vector-space"]},{"id":"derivation-mko-euclidean-algorithm-01","title":{"zh-Hant":"帶符號輸入的Euclid與回代","en":"Euclid and back-substitution for signed inputs"},"summary":{"zh-Hant":"gcd(−84,30)=6，且(−84)·1+30·3=6。","en":"gcd(-84,30)=6, with (-84)*1+30*3=6."},"mko_ids":["mko-euclidean-algorithm"]},{"id":"derivation-mko-eulerian-trail-01","title":{"zh-Hant":"含孤立頂點的Euler circuit","en":"An Euler circuit with an isolated vertex"},"summary":{"zh-Hant":"完整核每條邊恰一次與非孤立支撐條件。","en":"Audit every edge exactly once and the nonisolated support condition."},"mko_ids":["mko-eulerian-trail"]},{"id":"derivation-mko-homeomorphism-01","title":{"zh-Hant":"兩個有限鏈狀空間的同胚","en":"A homeomorphism between finite chain spaces"},"summary":{"zh-Hant":"用全部逆像核對重命名映射是同胚。","en":"Audit a relabeling homeomorphism using every open preimage."},"mko_ids":["mko-homeomorphism"]},{"id":"derivation-mko-sampling-distribution-01","title":{"zh-Hant":"依fiber推送的完整抽樣分布","en":"A complete fibre-pushforward sampling distribution"},"summary":{"zh-Hant":"對兩個theta完整合併U的相同值outcomes。","en":"Aggregate equal-valued outcomes of U under both theta laws."},"mko_ids":["mko-sampling-distribution"]},{"id":"derivation-mko-weak-law-large-numbers-01","title":{"zh-Hant":"公平Bernoulli平均的Chebyshev尾端","en":"A Chebyshev tail for fair Bernoulli averages"},"summary":{"zh-Hant":"由全序列IID前提導出WLLN。","en":"Derive the WLLN from a whole-sequence IID premise."},"mko_ids":["mko-weak-law-large-numbers"]},{"id":"derivation-mko-chromatic-number-01","title":{"zh-Hant":"三角形全部二色配置的排除","en":"Excluding every two-color assignment on a triangle"},"summary":{"zh-Hant":"以上界與完整低色數搜尋證chi=3。","en":"Prove chi=3 using an upper witness and complete lower-color search."},"mko_ids":["mko-chromatic-number"]},{"id":"derivation-mko-clique-and-independent-set-01","title":{"zh-Hant":"三頂點圖的全部clique與independent sets","en":"All cliques and independent sets on three vertices"},"summary":{"zh-Hant":"完整2³子集分類求omega與alpha。","en":"Classify all2³ subsets to obtain omega and alpha."},"mko_ids":["mko-clique-and-independent-set"]},{"id":"derivation-mko-estimator-bias-01","title":{"zh-Hant":"完整參數集合上的偏誤表","en":"A bias table over the whole parameter set"},"summary":{"zh-Hant":"逐theta求EthetaU-theta，區分觀察誤差。","en":"Compute EthetaU-theta at every theta, distinct from observed error."},"mko_ids":["mko-estimator-bias"]},{"id":"derivation-mko-estimator-variance-01","title":{"zh-Hant":"同law二次矩與中心矩交叉核對","en":"Same-law second and centered moment cross-checks"},"summary":{"zh-Hant":"對全部theta算variance並核對中心定義。","en":"Compute variance at all theta values and verify centered definitions."},"mko_ids":["mko-estimator-variance"]},{"id":"derivation-mko-fermat-little-theorem-01","title":{"zh-Hant":"模7的Fermat證書與合數邊界","en":"Fermat modulo 7 and a composite boundary"},"summary":{"zh-Hant":"(−2)^6≡1 mod7；倍數7只可用a^7≡a版本；通過測試不證質數。","en":"(-2)^6 is congruent to 1 modulo 7; for multiples of 7, only the congruence a^7≡a modulo 7 applies; passing a Fermat test does not prove primality."},"mko_ids":["mko-fermat-little-theorem"]},{"id":"derivation-mko-linear-diophantine-equation-01","title":{"zh-Hant":"負係數方程的全部整數解","en":"All integer solutions with a negative coefficient"},"summary":{"zh-Hant":"−6x+9y=15全部解為x=5+3t,y=5+2t，t∈Z。","en":"All integer solutions of -6x+9y=15 are x=5+3t,y=5+2t, t in Z."},"mko_ids":["mko-linear-diophantine-equation"]},{"id":"derivation-mko-quotient-vector-space-01","title":{"zh-Hant":"對角線的商空間","en":"Quotient by the diagonal line"},"summary":{"zh-Hant":"R²/span{(1,1)}的基底為[(0,1)]，維度1；零coset是整條對角線。","en":"R2/span{(1,1)} has basis [(0,1)] and dimension 1; its zero coset is the whole diagonal."},"mko_ids":["mko-quotient-vector-space"]},{"id":"derivation-mko-finite-group-action-01","title":{"zh-Hant":"六元素群對四點的完整作用","en":"A complete six-element action on four points"},"summary":{"zh-Hant":"明示24格表為有效群作用，但不是free或transitive。","en":"The explicit 24-entry table is a valid group action, but it is neither free nor transitive."},"mko_ids":["mko-finite-group-action"]},{"id":"derivation-mko-supremum-and-infimum-01","title":{"zh-Hant":"有限最大值與未取到的區間上確界","en":"Finite maxima and unattained interval suprema"},"summary":{"zh-Hant":"求 A={-2,1/3,5/2} 與 B=(0,1) 的 sup/inf，逐一核界與最小性。","en":"Find sup/inf of A={-2,1/3,5/2} and B=(0,1), checking bounds and extremality."},"mko_ids":["mko-supremum-and-infimum"]},{"id":"derivation-mko-topological-basis-01","title":{"zh-Hant":"有限鏈的基底與全部生成開集","en":"A finite-chain basis and every generated open"},"summary":{"zh-Hant":"驗 B={{a},{a,b},X} 是指定拓撲的基底。","en":"Verify B={{a},{a,b},X} is a basis for the specified topology."},"mko_ids":["mko-topological-basis"]},{"id":"derivation-mko-markov-inequality-01","title":{"zh-Hant":"由逐點不等式得到Markov界","en":"Deriving a Markov bound pointwise"},"summary":{"zh-Hant":"證有限law的界並比較exact tail。","en":"Prove the finite-law bound and compare its exact tail."},"mko_ids":["mko-markov-inequality"]},{"id":"derivation-mko-pigeonhole-principle-01","title":{"zh-Hant":"完整映射的碰撞與ceil負載","en":"Collision and ceiling load in a complete map"},"summary":{"zh-Hant":"從全部fibers證碰撞及廣義下界。","en":"Use all fibres to prove a collision and the generalized lower bound."},"mko_ids":["mko-pigeonhole-principle"]},{"id":"derivation-mko-euler-theorem-number-theory-01","title":{"zh-Hant":"模10的完整unit排列","en":"The full unit permutation modulo 10"},"summary":{"zh-Hant":"對a=−3,n=10，φ(10)=4且a⁴≡1 mod10。","en":"For a=-3,n=10, phi(10)=4 and a^4=1 modulo 10."},"mko_ids":["mko-euler-theorem-number-theory"]},{"id":"derivation-mko-subsequence-01","title":{"zh-Hant":"仿射索引的全稱子數列證書","en":"An all-index subsequence certificate for affine indices"},"summary":{"zh-Hant":"由 a_n=3n-2、k(j)=1+2j 構造並再取子數列。","en":"Construct a subsequence of a_n=3n-2 using k(j)=1+2j, then take a further subsequence."},"mko_ids":["mko-subsequence"]},{"id":"derivation-mko-finite-group-orbit-01","title":{"zh-Hant":"完整orbits與partition","en":"Complete orbits and their partition"},"summary":{"zh-Hant":"X的全部互異orbits為{0,1,2}與{3}，同orbit關係是等價關係。","en":"The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation."},"mko_ids":["mko-finite-group-orbit"]},{"id":"derivation-mko-monotone-sequence-convergence-theorem-01","title":{"zh-Hant":"有界單調尾與上確界論證","en":"A bounded monotone tail and the supremum argument"},"summary":{"zh-Hant":"對 a_n=3-2/(n+1) 證單調、有界、sup=3 及極限。","en":"For a_n=3-2/(n+1), prove monotonicity, boundedness, supremum 3 and convergence."},"mko_ids":["mko-monotone-sequence-convergence-theorem"]},{"id":"derivation-mko-subspace-topology-01","title":{"zh-Hant":"相對開集不等於母空間開集","en":"Relative openness differs from ambient openness"},"summary":{"zh-Hant":"計算 Y={b,c} 的子空間拓撲與全部相對閉集。","en":"Compute the subspace topology and all relative closed sets of Y={b,c}."},"mko_ids":["mko-subspace-topology"]},{"id":"derivation-mko-chebyshev-inequality-01","title":{"zh-Hant":"平方偏差的Chebyshev推導","en":"Chebyshev from squared deviations"},"summary":{"zh-Hant":"由Markov證Chebyshev並算exact例。","en":"Derive Chebyshev from Markov and compute an exact instance."},"mko_ids":["mko-chebyshev-inequality"]},{"id":"derivation-mko-inclusion-exclusion-principle-01","title":{"zh-Hant":"三集合全部七個交集項","en":"All seven intersection terms for three sets"},"summary":{"zh-Hant":"完整核交替和與每元素的重數。","en":"Audit the full alternating sum and each element's multiplicity."},"mko_ids":["mko-inclusion-exclusion-principle"]},{"id":"derivation-mko-unique-prime-factorization-01","title":{"zh-Hant":"存在、Euclid引理與唯一質因數","en":"Existence, Euclid's lemma and unique factors"},"summary":{"zh-Hant":"正整數n>1存在唯一質因數多重集合；360=2³·3²·5，負號與units另列。","en":"Every positive integer n>1 has a unique prime multiset; 360=2^3*3^2*5, with signs and units separate."},"mko_ids":["mko-unique-prime-factorization"]},{"id":"derivation-mko-finite-directed-graph-01","title":{"zh-Hant":"方向、全部degree與walk核對","en":"Orientation, all degrees and a walk audit"},"summary":{"zh-Hant":"按loop-free simple convention核完整digraph。","en":"Audit a fully specified digraph under the declared loop-free simple convention."},"mko_ids":["mko-finite-directed-graph"]},{"id":"derivation-mko-finite-group-stabilizer-01","title":{"zh-Hant":"point0的完整固定群元素","en":"All group elements fixing point zero"},"summary":{"zh-Hant":"Stab_G(0)={e,s}為子群；Stab_G(3)=G，與orbit及action kernel不同。","en":"Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions."},"mko_ids":["mko-finite-group-stabilizer"]},{"id":"derivation-mko-bolzano-weierstrass-theorem-real-01","title":{"zh-Hant":"振盪尾的兩個收斂子數列","en":"Two convergent subsequences of an oscillatory tail"},"summary":{"zh-Hant":"對 a_n=(-1)^n+1/(n+1) 給有界性與兩個完整收斂子數列。","en":"For a_n=(-1)^n+1/(n+1), provide boundedness and two complete convergent subsequences."},"mko_ids":["mko-bolzano-weierstrass-theorem-real"]},{"id":"derivation-mko-compact-subset-01","title":{"zh-Hant":"有限子空間的緊緻性","en":"Compactness of a finite subspace"},"summary":{"zh-Hant":"證 K={b,c} 在指定母空間中緊緻，並核相對開覆蓋。","en":"Prove K={b,c} compact in its ambient space and audit a relative-open cover."},"mko_ids":["mko-compact-subset"]},{"id":"derivation-mko-law-of-total-expectation-01","title":{"zh-Hant":"含非空零質量塊的全期望","en":"Total expectation with a nonempty null block"},"summary":{"zh-Hant":"完整核partition並以兩條路計算E[X]。","en":"Audit the partition and compute E[X] by two routes."},"mko_ids":["mko-law-of-total-expectation"]},{"id":"derivation-mko-binomial-theorem-01","title":{"zh-Hant":"選取y因子的完整二項展開","en":"A full binomial expansion by choosing y factors"},"summary":{"zh-Hant":"證一般係數責任並完整展開n=4。","en":"Establish coefficient counting and expand n=4 completely."},"mko_ids":["mko-binomial-theorem"]},{"id":"derivation-mko-mobius-function-01","title":{"zh-Hant":"完整因數分解與μ(30)","en":"Complete factorization and mu(30)"},"summary":{"zh-Hant":"μ(30)=−1，μ(12)=0，μ(1)=1；30的全部divisor值加總為0。","en":"mu(30)=-1, mu(12)=0, mu(1)=1; the sum over every divisor of 30 is zero."},"mko_ids":["mko-mobius-function"]},{"id":"derivation-mko-context-free-grammar-01","title":{"zh-Hant":"CFG的完整production與推導witness","en":"A CFG production list and derivation witness"},"summary":{"zh-Hant":"grammar S→aSb或ε具有aaabbb與ε的有限推導witness。","en":"The grammar S to aSb or epsilon has finite derivation witnesses for aaabbb and epsilon."},"mko_ids":["mko-context-free-grammar"]},{"id":"derivation-mko-orbit-stabilizer-theorem-01","title":{"zh-Hant":"fibers確為stabilizer左陪集","en":"Fibers as left cosets of the stabilizer"},"summary":{"zh-Hant":"point0的orbit大小3、stabilizer大小2；三個fiber均為左陪集，所以6=3·2。","en":"At point 0 the orbit has size 3 and stabilizer size 2; all three fibers are left cosets, giving 6=3*2."},"mko_ids":["mko-orbit-stabilizer-theorem"]},{"id":"derivation-mko-mean-value-theorem-01","title":{"zh-Hant":"二次多項式的均值點","en":"The mean-value point of a quadratic"},"summary":{"zh-Hant":"對 f(t)=A t^2+B t+C 在 a<b 上直接構造均值點。","en":"Directly construct a mean-value point for f(t)=A*t^2+B*t+C on a<b."},"mko_ids":["mko-mean-value-theorem"]},{"id":"derivation-mko-continuous-image-of-compact-01","title":{"zh-Hant":"拉回覆蓋再推回索引","en":"Pull back covers and return the same indices"},"summary":{"zh-Hant":"以明示有限映射呈現緊緻像的覆蓋論證。","en":"Demonstrate the compact-image cover proof with an explicit finite map."},"mko_ids":["mko-continuous-image-of-compact"]},{"id":"derivation-mko-law-of-total-variance-01","title":{"zh-Hant":"within與between的精確分解","en":"Exact within-between variance decomposition"},"summary":{"zh-Hant":"用含null塊的完整law分解variance。","en":"Decompose variance in a complete law with a null block."},"mko_ids":["mko-law-of-total-variance"]},{"id":"derivation-mko-recurrence-relation-01","title":{"zh-Hant":"oldest-to-newest二階遞迴trace","en":"An oldest-to-newest order-two recurrence trace"},"summary":{"zh-Hant":"對齊state、係數與全部初值，生成a5並證唯一性責任。","en":"Align states, coefficients and all initial values, generate a5 and explain uniqueness."},"mko_ids":["mko-recurrence-relation"]},{"id":"derivation-mko-mobius-inversion-finite-divisors-01","title":{"zh-Hant":"仿射函數在divisor poset的反演","en":"Divisor-poset inversion of an affine function"},"summary":{"zh-Hant":"f(k)=2k+1、n=6時，F(6)=28且反演得到f(6)=13；n=1亦roundtrip。","en":"For f(k)=2k+1 at n=6, F(6)=28 and inversion returns f(6)=13; n=1 roundtrips too."},"mko_ids":["mko-mobius-inversion-finite-divisors"]},{"id":"derivation-mko-pushdown-automaton-01","title":{"zh-Hant":"bottom-first PDA的aabb接受run","en":"A bottom-first PDA run accepting aabb"},"summary":{"zh-Hant":"指定PDA有讀完aabb到final f的有限run，末stack仍為[Z]。","en":"The specified PDA has a finite run consuming aabb and ending at final f with stack [Z]."},"mko_ids":["mko-pushdown-automaton"]},{"id":"derivation-mko-lagrange-theorem-finite-groups-01","title":{"zh-Hant":"全部左陪集的Lagrange計數","en":"Lagrange counting from all left cosets"},"summary":{"zh-Hant":"H={e,s}有三個互異左陪集，每個大小2，所以6=[G:H]|H|=3·2。","en":"H={e,s} has three distinct left cosets of size 2, giving 6=[G:H]|H|=3*2."},"mko_ids":["mko-lagrange-theorem-finite-groups"]},{"id":"derivation-mko-higher-order-derivative-01","title":{"zh-Hant":"四次多項式的全部導數階","en":"Every derivative order of a quartic"},"summary":{"zh-Hant":"由低到高係數 [2,-3,0,5,-2] 計算所有階導數。","en":"Compute all derivative orders for low-to-high coefficients [2,-3,0,5,-2]."},"mko_ids":["mko-higher-order-derivative"]},{"id":"derivation-mko-hausdorff-space-01","title":{"zh-Hant":"三點離散空間的全部點對分離","en":"Separating every pair in a discrete three-point space"},"summary":{"zh-Hant":"完整核三點離散空間的 Hausdorff 性，並對照一個失敗點對。","en":"Exhaustively audit Hausdorffness of a discrete three-point space and contrast a failed pair."},"mko_ids":["mko-hausdorff-space"]},{"id":"derivation-mko-continuous-real-distribution-function-01","title":{"zh-Hant":"有平坦段的連續CDF","en":"A continuous CDF with a flat interval"},"summary":{"zh-Hant":"從knots完整構造CDF並核atomless性。","en":"Construct a complete knot-based CDF and audit atomlessness."},"mko_ids":["mko-continuous-real-distribution-function"]},{"id":"derivation-mko-catalan-numbers-01","title":{"zh-Hant":"Catalan卷積與整除證書","en":"Catalan convolution and a divisibility certificate"},"summary":{"zh-Hant":"從C0完整生成C4並核兩個closed forms。","en":"Generate C4 from C0 and check both closed forms."},"mko_ids":["mko-catalan-numbers"]},{"id":"derivation-mko-quadratic-residue-01","title":{"zh-Hant":"完整模7平方表","en":"The complete nonzero square table modulo 7"},"summary":{"zh-Hant":"模7非零平方剩餘為{1,2,4}；−5≡2有根3、4，3無根。","en":"Modulo 7, the nonzero quadratic residues are {1,2,4}; the class of -5 equals the class of 2 and has root classes 3 and 4, while 3 has no square root modulo 7."},"mko_ids":["mko-quadratic-residue"]},{"id":"derivation-mko-nondeterministic-finite-automaton-01","title":{"zh-Hant":"NFA的existential有限路徑","en":"An existential finite accepting NFA path"},"summary":{"zh-Hant":"指定NFA接受aab，即使另一分支dead；epsilon loop本身不接受。","en":"The specified NFA accepts aab despite a dead alternative branch; an epsilon loop alone does not accept."},"mko_ids":["mko-nondeterministic-finite-automaton"]},{"id":"derivation-mko-cauchy-theorem-finite-groups-01","title":{"zh-Hant":"prime因子的exact-order witnesses","en":"Exact-order witnesses for the prime divisors"},"summary":{"zh-Hant":"此六元素群的prime divisors2、3分別有s與r的exact-order witnesses；無order6元素。","en":"In this six-element group, prime divisors 2,3 have exact-order witnesses s,r; there is no element of order 6."},"mko_ids":["mko-cauchy-theorem-finite-groups"]},{"id":"derivation-mko-taylor-polynomial-01","title":{"zh-Hant":"中心平移與 factorial 的 Taylor 多項式","en":"A Taylor polynomial retaining the center and factorials"},"summary":{"zh-Hant":"以中心 a=1、order 2 構造指定三次多項式的 T_2。","en":"Construct T_2 at center a=1 for the specified cubic polynomial."},"mko_ids":["mko-taylor-polynomial"]},{"id":"derivation-mko-compact-subset-closed-in-hausdorff-01","title":{"zh-Hant":"緊緻把分離鄰域縮為有限個","en":"Compactness reduces separating neighborhoods to finitely many"},"summary":{"zh-Hant":"證 Hausdorff 空間中的緊緻子集 K 閉。","en":"Prove a compact subset K of a Hausdorff space is closed."},"mko_ids":["mko-compact-subset-closed-in-hausdorff"]},{"id":"derivation-mko-improper-integral-01","title":{"zh-Hant":"每個無界端與奇點分別取極限","en":"Separate limits at every infinite endpoint and singularity"},"summary":{"zh-Hant":"計算三種明示冪函數瑕積分，並區分 principal value。","en":"Compute three explicit power-law improper integrals and distinguish principal value."},"mko_ids":["mko-improper-integral"]},{"id":"derivation-mko-power-series-radius-of-convergence-01","title":{"zh-Hant":"半徑、端點與完整係數律","en":"Radius, endpoints and complete coefficient laws"},"summary":{"zh-Hant":"對幾何係數 2^n 求 R，並展示端點及 0/infinity 邊界。","en":"Find R for geometric coefficients 2^n, and exhibit endpoint and zero/infinity boundaries."},"mko_ids":["mko-power-series-radius-of-convergence"]},{"id":"derivation-mko-legendre-symbol-01","title":{"zh-Hant":"獨立平方表定義三值Legendre符號","en":"The three-valued Legendre symbol from squares"},"summary":{"zh-Hant":"模7符號對0、1、2、3、4、5、6依序為0、1、1、−1、1、−1、−1。","en":"Modulo 7 the symbol on 0,1,2,3,4,5,6 is respectively 0,1,1,-1,1,-1,-1."},"mko_ids":["mko-legendre-symbol"]},{"id":"derivation-mko-subset-construction-nfa-to-dfa-01","title":{"zh-Hant":"epsilon-NFA的完整reachable subset DFA","en":"The complete reachable-subset DFA of an epsilon-NFA"},"summary":{"zh-Hant":"完整列轉移並以prefix induction證語言等價。","en":"List every transition and prove language equivalence by prefix induction."},"mko_ids":["mko-subset-construction-nfa-to-dfa"]},{"id":"derivation-mko-finite-normal-subgroup-01","title":{"zh-Hant":"全部共軛認證rotation子群正規","en":"Normality of rotations from every conjugate"},"summary":{"zh-Hant":"N={e,r,r²}為正規子群；H={e,s}雖為子群但不正規。","en":"N={e,r,r2} is normal; H={e,s} is a subgroup but is not normal."},"mko_ids":["mko-finite-normal-subgroup"]},{"id":"derivation-mko-real-exponential-function-01","title":{"zh-Hant":"全級數的加法律與嚴格正性","en":"Addition and strict positivity of the full exponential series"},"summary":{"zh-Hant":"從 E(x)=sum x^n/n! 證全實數絕對收斂、加法律、逆元與正性。","en":"From E(x)=sum x^n/n!, prove absolute convergence for all real x, addition, inverses and positivity."},"mko_ids":["mko-real-exponential-function"]},{"id":"derivation-mko-analytic-ordinary-generating-function-01","title":{"zh-Hant":"含初值邊界項的Fibonacci型OGF","en":"A Fibonacci-type OGF with initial boundary terms"},"summary":{"zh-Hant":"在證明收斂的域內從遞迴推出OGF。","en":"Derive the OGF from a recurrence on a justified convergence domain."},"mko_ids":["mko-analytic-ordinary-generating-function"]},{"id":"derivation-mko-euler-criterion-quadratic-residue-01","title":{"zh-Hant":"兩條獨立路徑核Euler判準","en":"Two independent routes for Euler's criterion"},"summary":{"zh-Hant":"p=7,a=−4時Legendre值−1且a³≡6≡−1 mod7；2則給+1。","en":"For p=7,a=-4 the Legendre value is -1 and a^3=6=-1 mod 7; 2 gives +1."},"mko_ids":["mko-euler-criterion-quadratic-residue"]},{"id":"derivation-mko-turing-machine-decision-model-01","title":{"zh-Hant":"二元字串偶數1的scan decider","en":"A scan decider for an even number of ones"},"summary":{"zh-Hant":"完整machine、trace、ranking與全input正確性。","en":"Give the complete machine, trace, ranking and all-input correctness."},"mko_ids":["mko-turing-machine-decision-model"]},{"id":"derivation-mko-gradient-vector-01","title":{"zh-Hant":"座標梯度、駐點與可微的區別","en":"Coordinate gradients, stationarity and differentiability"},"summary":{"zh-Hant":"求指定二元多項式梯度並核兩個常見錯誤推論。","en":"Compute the gradient of a specified bivariate polynomial and audit two false inferences."},"mko_ids":["mko-gradient-vector"]},{"id":"derivation-mko-finite-flow-network-01","title":{"zh-Hant":"逐邊與逐內點核可行流","en":"Checking every edge and internal vertex of a feasible flow"},"summary":{"zh-Hant":"核一個含反平行、零容量邊與零流量點的完整網路流值。","en":"Audit a complete flow network with antiparallel edges, a zero-capacity edge and a zero-flow vertex."},"mko_ids":["mko-finite-flow-network"]},{"id":"derivation-mko-finite-group-homomorphism-kernel-01","title":{"zh-Hant":"C₄到C₂的kernel、fibers與單射性","en":"Kernel, fibers and injectivity from C4 to C2"},"summary":{"zh-Hant":"f:C₄→C₂，f(a)=a mod2的kernel為{0,2}且正規；fibers是其cosets，f非單射。","en":"For f:C4 to C2, f(a)=a mod 2, the normal kernel is {0,2}; fibers are its cosets and f is not injective."},"mko_ids":["mko-finite-group-homomorphism-kernel"]},{"id":"derivation-mko-finite-quotient-group-01","title":{"zh-Hant":"全部代表pairs的商群表","en":"A quotient table checked on every representative pair"},"summary":{"zh-Hant":"G/N的兩cosets A=N、B={s,rs,r²s}具良定乘法AA=A,AB=BA=B,BB=A。","en":"G/N has cosets A=N,B={s,rs,r2s} and well-defined products AA=A,AB=BA=B,BB=A."},"mko_ids":["mko-finite-quotient-group"]},{"id":"derivation-mko-finite-group-conjugacy-class-01","title":{"zh-Hant":"完整共軛作用與三個classes","en":"Full conjugation action and three classes"},"summary":{"zh-Hant":"全部共軛類是{e}、{r,r²}、{s,rs,r²s}；r與s不共軛。","en":"The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate."},"mko_ids":["mko-finite-group-conjugacy-class"]},{"id":"derivation-mko-standard-normal-distribution-01","title":{"zh-Hant":"標準常態的對稱推導與named facts","en":"Standard-normal symmetry with explicit named facts"},"summary":{"zh-Hant":"清楚分開Gaussian前提與由它們推得的CDF對稱性。","en":"Separate Gaussian prerequisites from derived CDF symmetry."},"mko_ids":["mko-standard-normal-distribution"]},{"id":"derivation-mko-estimator-mean-squared-error-01","title":{"zh-Hant":"不用分解公式的直接MSE計算","en":"Direct MSE without the decomposition formula"},"summary":{"zh-Hant":"逐theta直接加權loss，保留bias-variance為後續。","en":"Weight losses directly at every theta, leaving decomposition for later."},"mko_ids":["mko-estimator-mean-squared-error"]},{"id":"derivation-mko-linear-program-standard-form-01","title":{"zh-Hant":"slack 的雙向點對應","en":"The two-way point correspondence for slack variables"},"summary":{"zh-Hant":"把固定 maximize/≤/nonnegative LP 轉為 equality slack form，並核雙向可行點與 objective。","en":"Convert a fixed maximize/≤/nonnegative LP to equality slack form and verify both feasible-point directions and objective."},"mko_ids":["mko-linear-program-standard-form"]},{"id":"derivation-mko-vertex-cover-01","title":{"zh-Hant":"星形圖的cover與刪點證書","en":"Vertex covers and deletion certificates on a star"},"summary":{"zh-Hant":"逐邊核cover並區分minimal與minimum。","en":"Audit covers edgewise and distinguish minimal from minimum."},"mko_ids":["mko-vertex-cover"]},{"id":"derivation-mko-first-isomorphism-theorem-finite-groups-01","title":{"zh-Hant":"非滿射同態的正確image同構","en":"The correct image isomorphism for a nonsurjection"},"summary":{"zh-Hant":"f:C₄→C₄,f(a)=2a mod4給C₄/{0,2}≅{0,2}，不是整個codomain C₄。","en":"For f:C4 to C4,f(a)=2a mod 4, C4/{0,2} is isomorphic to {0,2}, not the full codomain C4."},"mko_ids":["mko-first-isomorphism-theorem-finite-groups"]},{"id":"derivation-mko-finite-group-center-01","title":{"zh-Hant":"全部交換rows認證群中心","en":"Certifying the center from all commutation rows"},"summary":{"zh-Hant":"Z(G)={e}，直接驗subgroup及normal；G非abelian，r雖與自身交換卻不central。","en":"Z(G)={e}, directly a normal subgroup; G is nonabelian and r is not central despite commuting with itself."},"mko_ids":["mko-finite-group-center"]},{"id":"derivation-mko-bias-variance-decomposition-01","title":{"zh-Hant":"展開cross term的bias-variance推導","en":"Bias-variance derivation with the cross term visible"},"summary":{"zh-Hant":"在同一law逐符號證明分解，並核兩個theta。","en":"Prove the decomposition symbol by symbol in one law and check both theta values."},"mko_ids":["mko-bias-variance-decomposition"]},{"id":"derivation-mko-linear-program-duality-01","title":{"zh-Hant":"弱對偶鏈與零間隙最優證書","en":"The weak-duality chain and a zero-gap optimality certificate"},"summary":{"zh-Hant":"完整構造對偶，核 x=(1,3)、y=(1,1) 可行並由零間隙證雙最優。","en":"Construct the full dual, check feasibility of x=(1,3), y=(1,1), and certify both optimal by zero gap."},"mko_ids":["mko-linear-program-duality"]},{"id":"derivation-mko-max-flow-min-cut-theorem-01","title":{"zh-Hant":"完整四個 cut 與等值雙證書","en":"All four cuts and an equal-value two-sided certificate"},"summary":{"zh-Hant":"枚舉四點網路全部 s-t cuts，給值 5 的可行流與同值 cut。","en":"Enumerate every s-t cut of a four-vertex network and give a feasible flow of value 5 together with a cut of capacity 5."},"mko_ids":["mko-max-flow-min-cut-theorem"]},{"id":"derivation-mko-minimum-vertex-cover-01","title":{"zh-Hant":"P3全部八子集求minimum cover","en":"All eight subsets for minimum cover on P3"},"summary":{"zh-Hant":"完整分類所有covers與全部minimizers。","en":"Classify every cover and all minimizers."},"mko_ids":["mko-minimum-vertex-cover"]},{"id":"derivation-mko-many-one-reduction-01","title":{"zh-Hant":"完整有限reduction與方向保持","en":"A complete finite reduction with preserved direction"},"summary":{"zh-Hant":"逐x核iff、組合及decider transfer。","en":"Check every x equivalence, composition and decider transfer."},"mko_ids":["mko-many-one-reduction"]},{"id":"derivation-mko-contraction-mapping-theorem-01","title":{"zh-Hant":"有限度量的完整收縮證書","en":"A complete contraction certificate on a finite metric"},"summary":{"zh-Hant":"在三點離散度量與常值 self-map 上核完備、唯一不動點及誤差界。","en":"Audit completeness, a unique fixed point and the error bound for a constant self-map on a three-point discrete metric."},"mko_ids":["mko-contraction-mapping-theorem"]},{"id":"derivation-mko-intermediate-value-theorem-01","title":{"zh-Hant":"多項式 bracket 與精確中間值","en":"A polynomial bracket and an exact intermediate value"},"summary":{"zh-Hant":"對 f(x)=x^2-1 在 [0,2] 找值 0，並保留端點及非唯一邊界。","en":"Find value zero for f(x)=x^2-1 on [0,2], retaining endpoint and nonuniqueness boundaries."},"mko_ids":["mko-intermediate-value-theorem"]},{"id":"derivation-mko-taylor-remainder-theorem-01","title":{"zh-Hant":"三次多項式的精確 Lagrange 餘項","en":"An exact Lagrange remainder for a cubic"},"summary":{"zh-Hant":"以 f(x)=x^3、a=1、n=2、x=2 核精確餘項及有效導數界。","en":"For f(x)=x^3, a=1, n=2, x=2, audit the exact remainder and a qualified derivative bound."},"mko_ids":["mko-taylor-remainder-theorem"]},{"id":"derivation-mko-finite-maximum-likelihood-estimator-01","title":{"zh-Hant":"含正質量 tie 與全零似然欄的 MLE","en":"MLE with a positive-mass tie and an all-zero likelihood column"},"summary":{"zh-Hant":"完整算argmax及固定tie rule的total estimator。","en":"Compute every argmax and a total estimator using a fixed tie rule."},"mko_ids":["mko-finite-maximum-likelihood-estimator"]},{"id":"derivation-mko-finite-sufficient-statistic-01","title":{"zh-Hant":"逐fiber條件law的充分性證書","en":"A sufficiency certificate from every fibre conditional law"},"summary":{"zh-Hant":"完整核正質量fiber跨theta不變，保留null未定義。","en":"Audit invariance across theta on every positive-mass fibre, retaining undefined null conditions."},"mko_ids":["mko-finite-sufficient-statistic"]},{"id":"derivation-mko-topological-ordering-dag-01","title":{"zh-Hant":"三頂點DAG的全部排列與Kahn trace","en":"All permutations and a Kahn trace for a three-vertex DAG"},"summary":{"zh-Hant":"完整列valid orders並核source-removal invariant。","en":"List every valid order and audit source-removal invariants."},"mko_ids":["mko-topological-ordering-dag"]},{"id":"derivation-mko-multiplicative-order-mod-n-01","title":{"zh-Hant":"第一個回到1的冪","en":"The first power returning to one"},"summary":{"zh-Hant":"ord₈(−5)=ord₈(3)=2，φ(8)=4且2整除4，但order不等φ。","en":"ord_8(-5)=ord_8(3)=2, with phi(8)=4 and 2 dividing 4 but not equal to it."},"mko_ids":["mko-multiplicative-order-mod-n"]},{"id":"derivation-mko-series-comparison-test-01","title":{"zh-Hant":"最終比較的方向與有限前綴","en":"The direction of eventual comparison and finite prefixes"},"summary":{"zh-Hant":"證一個有任意有限前綴的級數由幾何尾上比較收斂，並示下比較發散。","en":"Prove convergence using geometric upper comparison despite a finite prefix, and show divergent lower comparison."},"mko_ids":["mko-series-comparison-test"]},{"id":"derivation-mko-ratio-test-series-01","title":{"zh-Hant":"比值極限、幾何支配與 L=1","en":"Ratio limits, geometric domination and L=1"},"summary":{"zh-Hant":"以完整有理比值證 L<1 收斂及 L>1 發散，並核 L=1 無結論。","en":"Use complete rational ratios for L<1 convergence and L>1 divergence, then audit indecision at L=1."},"mko_ids":["mko-ratio-test-series"]},{"id":"derivation-mko-uniform-limit-continuity-theorem-01","title":{"zh-Hant":"固定一個近似函數的 epsilon/3 傳遞","en":"An epsilon/3 transfer through one fixed approximant"},"summary":{"zh-Hant":"對 [0,1] 上的仿射擾動證一致收斂，並逐步傳遞連續性。","en":"Prove uniform convergence of affine perturbations on [0,1] and transfer continuity step by step."},"mko_ids":["mko-uniform-limit-continuity-theorem"]},{"id":"derivation-mko-finite-product-topology-01","title":{"zh-Hant":"四點乘積的全部開集","en":"Every open set in a four-point product"},"summary":{"zh-Hant":"由兩個 Sierpinski 因子生成全部乘積開集並核投影。","en":"Generate every product open from two Sierpinski factors and audit projections."},"mko_ids":["mko-finite-product-topology"]},{"id":"derivation-mko-quotient-topology-01","title":{"zh-Hant":"完整逆像條件決定商拓撲","en":"The full preimage condition determines the quotient topology"},"summary":{"zh-Hant":"由三點到兩點的滿射計算商拓撲與開像失敗。","en":"Compute a quotient topology for a three-to-two-point surjection and show an open-image failure."},"mko_ids":["mko-quotient-topology"]},{"id":"derivation-mko-path-connected-space-01","title":{"zh-Hant":"凸盒中的全稱路徑與拼接","en":"Universal paths and concatenation in a convex box"},"summary":{"zh-Hant":"證 [0,1]^2 路徑連通，並核兩段路徑的接點。","en":"Prove [0,1]^2 path connected and audit a two-piece concatenation."},"mko_ids":["mko-path-connected-space"]},{"id":"derivation-mko-topological-connected-component-01","title":{"zh-Hant":"完整子集表找包含極大的連通分量","en":"Finding inclusion-maximal components from the complete subset table"},"summary":{"zh-Hant":"求 X={a,b,c}、tau={空集,{a},{b,c},X} 的全部連通分量。","en":"Find every component of X={a,b,c}, tau={empty,{a},{b,c},X}."},"mko_ids":["mko-topological-connected-component"]},{"id":"derivation-mko-central-limit-theorem-finite-variance-iid-01","title":{"zh-Hant":"有限variance IID的CLT資格與格點audit","en":"CLT qualification and a finite lattice audit"},"summary":{"zh-Hant":"清楚區分finite-n moments與named CLT應用。","en":"Separate finite-n moments from the named CLT application."},"mko_ids":["mko-central-limit-theorem-finite-variance-iid"]},{"id":"derivation-mko-strong-law-large-numbers-finite-valued-iid-01","title":{"zh-Hant":"有限值IID的強律資格與固定n界線","en":"Finite-valued IID strong-law qualification and fixed-n boundaries"},"summary":{"zh-Hant":"對齊named SLLN前提，並明示prefix不是a.s.證明。","en":"Match the named SLLN hypotheses while separating prefix audits from a.s. proof."},"mko_ids":["mko-strong-law-large-numbers-finite-valued-iid"]},{"id":"derivation-mko-finite-confidence-interval-coverage-01","title":{"zh-Hant":"逐參數coverage、平均與selection","en":"Pointwise coverage, averaging and selection"},"summary":{"zh-Hant":"完整計算固定區間程序的pointwise及worst-case coverage。","en":"Compute pointwise and worst-case coverage of a fixed interval procedure."},"mko_ids":["mko-finite-confidence-interval-coverage"]},{"id":"derivation-mko-wilson-theorem-01","title":{"zh-Hant":"完整逆元配對與Wilson雙向證明","en":"Complete inverse pairing and both Wilson directions"},"summary":{"zh-Hant":"n>1時(n−1)!≡−1 modn恰當n prime；模7完整逆元表核6!≡6。","en":"For n>1, (n-1)!=-1 mod n exactly when n is prime; the full modulo-7 inverse table checks 6!=6 mod 7."},"mko_ids":["mko-wilson-theorem"]},{"id":"derivation-mko-p-adic-valuation-integers-01","title":{"zh-Hant":"帶符號整數的3進因子抽取","en":"Extracting 3-adic factors of signed integers"},"summary":{"zh-Hant":"v₃(−108)=3、v₃(18)=2、v₃(−1944)=5、v₃(−90)=2；零用infinity。","en":"v3(-108)=3,v3(18)=2,v3(-1944)=5,v3(-90)=2; zero uses infinity."},"mko_ids":["mko-p-adic-valuation-integers"]},{"id":"derivation-mko-regular-expression-01","title":{"zh-Hant":"含epsilon的star閉包與完整長度切片","en":"Star closure with epsilon and a complete length slice"},"summary":{"zh-Hant":"typed AST遞迴語義、完整bounded枚舉與membership責任。","en":"Give typed AST semantics, complete bounded enumeration and membership obligations."},"mko_ids":["mko-regular-expression"]},{"id":"derivation-mko-pumping-lemma-regular-languages-01","title":{"zh-Hant":"DFA迴圈量詞與a^n b^n反證","en":"DFA-loop quantifiers and the a^n b^n contradiction"},"summary":{"zh-Hant":"可見量詞次序、全pump count迴圈證明及全部合法splits。","en":"Expose quantifier order, the all-pump-count loop argument and every legal split."},"mko_ids":["mko-pumping-lemma-regular-languages"]},{"id":"derivation-mko-newton-method-01","title":{"zh-Hant":"合法 Newton 更新與有限 trace","en":"Qualified Newton updates and a finite trace"},"summary":{"zh-Hant":"對 x^2-2 從 1 做兩步精確 Newton，並核 root-first 與循環邊界。","en":"Take two exact Newton steps for x^2-2 from one, and audit root-first stopping and a cycle boundary."},"mko_ids":["mko-newton-method"]},{"id":"derivation-mko-trapezoidal-quadrature-01","title":{"zh-Hant":"非均勻梯形值與帶符號誤差","en":"Nonuniform trapezoids and signed error"},"summary":{"zh-Hant":"對 f(x)=x^2、nodes (0,1/2,2) 計算 T、I-T 與絕對誤差。","en":"For f(x)=x^2 and nodes (0,1/2,2), compute T, I-T and absolute error."},"mko_ids":["mko-trapezoidal-quadrature"]},{"id":"derivation-mko-gradient-descent-01","title":{"zh-Hant":"正對角二次函數的安全下降步","en":"Safe descent steps for a positive diagonal quadratic"},"summary":{"zh-Hant":"以 alpha=1/4 對指定二次函數做兩步並證每步下降條件。","en":"Take two steps with alpha=1/4 for a specified quadratic and prove the per-step descent condition."},"mko_ids":["mko-gradient-descent"]},{"id":"derivation-mko-lagrange-multiplier-condition-01","title":{"zh-Hant":"正則乘數與完整平方距離解","en":"Regular multipliers and a complete squared-distance solution"},"summary":{"zh-Hant":"minimize ‖x-(1,2)‖^2 subject to x_1+x_2=0，求點、乘數並獨立證最小。","en":"Minimize the squared distance to (1,2) subject to x_1+x_2=0; find the point and multiplier and independently prove the minimum."},"mko_ids":["mko-lagrange-multiplier-condition"]}],"batches":[{"id":"batch-20261015-advance-20261119-r07-optimization-problem-foundations","title":{"zh-Hant":"最佳化密度：有限local/global與證書邊界基礎 · 脈衝 R07","en":"Complete Finite Neighborhoods, Improvement Witnesses and Attained Bounds"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-optimization-problem"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","title":{"zh-Hant":"梯度密度：線性、負梯度方向與二次差分基礎 · 脈衝 R06","en":"Ordered Gradient Linearity, Negative Directions and Metric Identification"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-gradient-vector"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","title":{"zh-Hant":"數值誤差密度：residual、forward error與conditioning基礎 · 脈衝 R05","en":"Affine Residual Scaling, Forward Error and Tolerance Conventions"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-numerical-approximation-error"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","title":{"zh-Hant":"等式約束Lagrange乘數必要條件基礎 · 脈衝 R04","en":"Regular Equality Constraints, Multiplier Residuals and Necessity Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-lagrange-multiplier-condition"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","title":{"zh-Hant":"梯度下降法基礎 · 脈衝 R03","en":"Exact Diagonal-Quadratic Gradient Steps and Strict Safe-Step Traces"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-gradient-descent"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations","title":{"zh-Hant":"複合梯形數值積分基礎 · 脈衝 R02","en":"Nonuniform Composite Trapezoids, Affine Exactness and Signed Reference Error"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-trapezoidal-quadrature"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261119-r01-newton-method-foundations","title":{"zh-Hant":"Newton根求解方法基礎 · 脈衝 R01","en":"Root-First Newton Updates, Exact Finite Orbits and Convergence Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-newton-method"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","title":{"zh-Hant":"正規語言Pumping引理基礎 · 脈衝 R07","en":"DFA Loop Invariants, Pumping Quantifiers and Every Legal Adversarial Split"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-pumping-lemma-regular-languages"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261118-r06-regular-expression-foundations","title":{"zh-Hant":"正規表示式基礎 · 脈衝 R06","en":"Typed Regular Expressions, Complete Bounded Languages and Membership"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-regular-expression"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","title":{"zh-Hant":"整數p進賦值基礎 · 脈衝 R05","en":"Prime-Power Extraction, Product Additivity and Sum Cancellation"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-p-adic-valuation-integers"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261118-r04-wilson-theorem-foundations","title":{"zh-Hant":"Wilson定理基礎 · 脈衝 R04","en":"Factorial Congruence, Complete Inverse Pairing and the Modulus-Two Exception"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-wilson-theorem"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","title":{"zh-Hant":"有限模型信賴區間覆蓋率基礎 · 脈衝 R03","en":"Precommitted Finite Intervals, Pointwise Coverage and Selection Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-finite-confidence-interval-coverage"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations","title":{"zh-Hant":"有限值IID強大數律基礎 · 脈衝 R02","en":"Finite-Valued IID Strong-Law Scope and Fixed-n Audits"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-strong-law-large-numbers-finite-valued-iid"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","title":{"zh-Hant":"有限變異數IID中央極限定理基礎 · 脈衝 R01","en":"Declared IID CLT Premises, Exact Sum Moments and Finite-Lattice Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","title":{"zh-Hant":"拓撲連通分支基礎 · 脈衝 R07","en":"Inclusion-Maximal Connected Subspaces and Finite Clopen Partitions"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-topological-connected-component"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","title":{"zh-Hant":"道路連通空間基礎 · 脈衝 R06","en":"Explicit Interval and Box Paths with Full Continuity Obligations"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-path-connected-space"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","title":{"zh-Hant":"商拓撲基礎 · 脈衝 R05","en":"Surjective Fibers, Saturated Opens and Final-Topology Scope"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-quotient-topology"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","title":{"zh-Hant":"有限乘積拓撲基礎 · 脈衝 R04","en":"Complete Rectangle-Union Topologies and Coordinate Continuity"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-finite-product-topology"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","title":{"zh-Hant":"一致極限保持連續基礎 · 脈衝 R03","en":"Whole-Domain Error Bounds and a Fixed-Approximant Continuity Transfer"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-uniform-limit-continuity-theorem"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261117-r02-ratio-test-series-foundations","title":{"zh-Hant":"級數比值判別法基礎 · 脈衝 R02","en":"Absolute Ratio Limits, Domination and Limit-One Inconclusiveness"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-ratio-test-series"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261117-r01-series-comparison-test-foundations","title":{"zh-Hant":"級數比較判別法基礎 · 脈衝 R01","en":"Nonnegative Tail Comparisons and Correct Convergence Directions"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-series-comparison-test"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","title":{"zh-Hant":"模n乘法階基礎 · 脈衝 R07","en":"Qualified Units, First Positive Returns and Phi Divisibility"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-multiplicative-order-mod-n"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations","title":{"zh-Hant":"有向無環圖的拓撲排序基礎 · 脈衝 R06","en":"All Topological Orders, Source-Removal Traces and Explicit Cycles"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-topological-ordering-dag"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations","title":{"zh-Hant":"有限模型充分統計量基礎 · 脈衝 R05","en":"Complete Fibers, Positive-Mass Conditional Laws and Sufficiency"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-finite-sufficient-statistic"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","title":{"zh-Hant":"有限模型最大似然估計量基礎 · 脈衝 R04","en":"Complete Likelihood Argmax Sets and Fixed Total Tie Rules"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-finite-maximum-likelihood-estimator"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","title":{"zh-Hant":"Taylor餘項定理基礎 · 脈衝 R03","en":"Exact Polynomial Remainders, Lagrange Witnesses and Qualified Bounds"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-taylor-remainder-theorem"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","title":{"zh-Hant":"介值定理基礎 · 脈衝 R02","en":"Inclusive IVT Witnesses and Mode-Specific Continuity Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-intermediate-value-theorem"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations","title":{"zh-Hant":"壓縮映射定理基礎 · 脈衝 R01","en":"Global Contraction Qualifications, Picard Orbits and Exact Tail Bounds"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-contraction-mapping-theorem"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","title":{"zh-Hant":"多對一歸約基礎 · 脈衝 R07","en":"Total Iff Maps, Directed Composition and Forward Decider Transfer"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-many-one-reduction"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261115-r06-minimum-vertex-cover-foundations","title":{"zh-Hant":"最小頂點覆蓋基礎 · 脈衝 R06","en":"All Minimum Covers and Explicit Nonminimum Witnesses"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-minimum-vertex-cover"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations","title":{"zh-Hant":"最大流最小割定理基礎 · 脈衝 R05","en":"Complete Directed Cuts and Equal-Value Flow Optimality"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-max-flow-min-cut-theorem"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","title":{"zh-Hant":"線性規劃對偶基礎 · 脈衝 R04","en":"Fixed Dual Conventions, Feasible Weak Chains and Zero-Gap Certificates"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-linear-program-duality"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","title":{"zh-Hant":"偏誤－變異數分解基礎 · 脈衝 R03","en":"Theta-Specific Bias-Variance Identities and Centering"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-bias-variance-decomposition"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","title":{"zh-Hant":"有限群中心基礎 · 脈衝 R02","en":"Complete Centers, Normality and Singleton-Conjugacy Equivalences"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-15","count":100,"mko_ids":["mko-finite-group-center"],"state":"candidate_validated"},{"id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","title":{"zh-Hant":"有限群第一同構定理基礎 · 脈衝 R01","en":"Qualified Quotient-to-Image Isomorphisms and 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objects."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"先用熟悉的計數對象建立數學概念與符號的區分。","en":"Begin with familiar counting objects to distinguish mathematical concepts from their symbols."}},{"id":"mko-set","position":2,"requirement":"required","rationale":{"zh-Hant":"再以集合整理對象、元素與整體之間的語言。","en":"Use sets to organize objects and the relation between elements and wholes."}},{"id":"mko-proposition","position":3,"requirement":"required","rationale":{"zh-Hant":"引入可被斷言、判定與證明的數學陳述。","en":"Introduce mathematical statements that can be asserted, evaluated, and proved."}},{"id":"mko-logical-connectives","position":4,"requirement":"required","rationale":{"zh-Hant":"用否定、合取、析取、蘊涵與雙條件組合命題。","en":"Use logical connectives—negation, conjunction, disjunction, implication, and biconditional—to combine propositions."}},{"id":"mko-proof","position":5,"requirement":"required","rationale":{"zh-Hant":"以明示前提與合法推理建立結論。","en":"Use explicit premises and valid inference steps to establish a conclusion."}}]},{"id":"path-domain-coverage-wave-one","title":{"zh-Hant":"第一波領域覆蓋基礎","en":"Wave-One Domain Foundations"},"summary":{"zh-Hant":"由自然數、集合與命題分支進入算術、方程、機率、歸納、階乘與基本組合計數。","en":"Branch from natural numbers, sets, and propositions into arithmetic, equations, probability, induction, factorials, and elementary combinatorial counting."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"提供算術、歸納與有限計數的自然數基礎。","en":"Supply the natural-number foundation for arithmetic, induction, and finite counting."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"提供樣本空間作為結果集合的基礎。","en":"Supply the set foundation for treating a sample space as a collection of outcomes."}},{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"提供歸納法所處理命題族的邏輯基礎。","en":"Supply the logical foundation for proposition families handled by induction."}},{"id":"mko-arithmetic-operations","position":2,"requirement":"required","rationale":{"zh-Hant":"解鎖代數表達與基本計數中的數量組合。","en":"Unlock quantitative combination in algebraic expressions and elementary counting."}},{"id":"mko-sample-space","position":2,"requirement":"optional","rationale":{"zh-Hant":"從集合語言進入機率統計的結果論域。","en":"Move from set language into the outcome domain of probability and statistics."}},{"id":"mko-mathematical-induction","position":2,"requirement":"optional","rationale":{"zh-Hant":"從自然數與命題進入結構化證明方法。","en":"Move from natural numbers and propositions into a structured proof method."}},{"id":"mko-variable-expression","position":3,"requirement":"optional","rationale":{"zh-Hant":"從算術運算進入含變數的代數表達。","en":"Move from arithmetic operations into algebraic expressions containing variables."}},{"id":"mko-counting-principle","position":3,"requirement":"optional","rationale":{"zh-Hant":"以自然數與算術運算組合有限選擇數量。","en":"Combine counts of finite choices using natural numbers and arithmetic operations."}},{"id":"mko-factorial","position":3,"requirement":"optional","rationale":{"zh-Hant":"以自然數乘法與後繼遞迴建立有限乘積。","en":"Use natural-number multiplication and successor recursion to build a finite product."}},{"id":"mko-equation","position":4,"requirement":"optional","rationale":{"zh-Hant":"由表達式進入等式、解集與代入驗證。","en":"Move from expressions into equality, solution sets, and substitution checks."}},{"id":"mko-linear-equation","position":5,"requirement":"optional","rationale":{"zh-Hant":"以非零一次係數收斂到唯一解與退化分類。","en":"Use a nonzero linear coefficient to obtain a unique solution and distinguish the degenerate constant cases."}}]},{"id":"path-set-language-to-topology-candidate","title":{"zh-Hant":"集合語言到拓撲前置候選路徑","en":"Set Language toward Topology"},"summary":{"zh-Hant":"從集合、元素隸屬、子集合與基本集合運算建立閱讀拓撲空間定義所需的集合語言。","en":"Build the set-language prerequisites needed to read the definition of a topological space, progressing through sets, membership, subsets, and basic set operations."},"audience":["upper_secondary","undergraduate"],"review":"candidate","nodes":[{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"先確立集合是由隸屬條件刻畫的對象。","en":"Begin with sets as objects characterized by membership conditions."}},{"id":"mko-set-membership","position":2,"requirement":"required","rationale":{"zh-Hant":"再以元素是否屬於集合表達運算定義。","en":"Use element membership to express the definitions of later set operations."}},{"id":"mko-subset","position":3,"requirement":"required","rationale":{"zh-Hant":"由逐元素隸屬建立集合之間的包含關係。","en":"Develop the subset relation between sets from element-wise membership implication."}},{"id":"mko-set-operations","position":3,"requirement":"required","rationale":{"zh-Hant":"平行建立聯集、交集、差集與相對補集。","en":"In parallel, develop union, intersection, difference, and complement relative to an explicit universe."}}]},{"id":"path-probability-events-candidate","title":{"zh-Hant":"樣本空間到事件候選路徑","en":"From Sample Spaces to Events"},"summary":{"zh-Hant":"由集合語言與樣本空間建立事件作為結果子集合的精確邊界。","en":"Use set language and sample spaces to establish the precise boundary of an event as a set of outcomes."},"audience":["upper_secondary","undergraduate"],"review":"candidate","nodes":[{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"先建立集合語言。","en":"Begin with set language."}},{"id":"mko-set-membership","position":2,"requirement":"required","rationale":{"zh-Hant":"區分結果元素與結果集合。","en":"Distinguish an outcome element from a set of outcomes."}},{"id":"mko-subset","position":3,"requirement":"required","rationale":{"zh-Hant":"建立集合之間的包含關係。","en":"Establish inclusion between sets."}},{"id":"mko-sample-space","position":3,"requirement":"required","rationale":{"zh-Hant":"指定試驗的全部可能結果。","en":"Specify all possible outcomes of the experiment."}},{"id":"mko-event","position":4,"requirement":"required","rationale":{"zh-Hant":"把事件定位為樣本空間中的結果集合。","en":"Place an event as a set of outcomes inside the sample-space model."}}]},{"id":"path-number-system-extension-candidate","title":{"zh-Hant":"自然數、整數、有理數、實數到絕對值候選路徑","en":"From Natural Numbers to Absolute Value on the Real Line"},"summary":{"zh-Hant":"由計數用的自然數進入整數與有理數，以完備化語言建立實數入口，再用絕對值表達數線距離。","en":"Move from natural numbers used for counting to integers and rational numbers, enter the real-number system through completion language, and then express number-line distance with absolute value."},"audience":["general_learner","lower_secondary"],"review":"candidate","nodes":[{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"先建立自然數與零的慣例。","en":"Begin by fixing the convention for natural numbers and zero."}},{"id":"mko-integer","position":2,"requirement":"required","rationale":{"zh-Hant":"加入負方向形成整數。","en":"Add a negative direction to form the integers."}},{"id":"mko-rational-number","position":3,"requirement":"required","rationale":{"zh-Hant":"以兩個整數之比建立有理數，並排除零分母。","en":"Represent rational numbers as ratios of two integers while excluding a zero denominator."}},{"id":"mko-real-number","position":4,"requirement":"required","rationale":{"zh-Hant":"由有理數嵌入與完備化進入實數系。","en":"Enter the real-number system through the rational embedding and completion."}},{"id":"mko-absolute-value","position":5,"requirement":"optional","rationale":{"zh-Hant":"以分段函數表示實數到零的非負距離。","en":"Use a piecewise function to express the nonnegative distance from a real number to zero."}}]},{"id":"path-pulse-foundations-20260930","title":{"zh-Hant":"七輪基礎擴充：數值、函數、計數與座標","en":"Seven-Round Foundation Expansion: Numbers, Functions, Counting, and Coordinates"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-counting-principle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-factorial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-point-line-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-fraction","position":2,"requirement":"optional","rationale":{"zh-Hant":"分數 p/q 是有理數的一種表示；分子與分母為整數，分母必須非零。","en":"Treat p/q as a rational-number representation with integer numerator and denominator and a nonzero denominator."}},{"id":"mko-ratio-proportion","position":3,"requirement":"optional","rationale":{"zh-Hant":"本條目把數值比 a:b 解讀為 a/b，把比例定義為兩個合法比值相等。","en":"Interpret the numerical ratio a:b as a/b and a proportion as equality between two valid ratios."}},{"id":"mko-domain-codomain-range","position":4,"requirement":"optional","rationale":{"zh-Hant":"總函數 f:A→B 的定義域是 A，陪域是 B，值域是實際取得的輸出集合 f[A]。","en":"Distinguish the domain A, declared codomain B, and actually attained range f[A] of a total function f:A→B."}},{"id":"mko-composition","position":5,"requirement":"optional","rationale":{"zh-Hant":"合成 g∘f 先求 f 的輸出，再把它送入 g；合成前須確認中間值可作為 g 的輸入。","en":"Evaluate g∘f by applying f first and verify that the intermediate value is an allowed input of g."}},{"id":"mko-sequence","position":6,"requirement":"optional","rationale":{"zh-Hant":"本條目採從 0 起始的實數數列 a:N→R；數列項依索引排列，不能任意重排成集合。","en":"Use zero-based real sequences a:N→R whose terms retain their index order rather than becoming an unordered set."}},{"id":"mko-permutation-combination","position":7,"requirement":"optional","rationale":{"zh-Hant":"由 n 個相異物件不重複取 k 個，有序選取是排列，無序選取是組合。","en":"Distinguish ordered permutations from unordered combinations when selecting distinct objects without replacement."}},{"id":"mko-coordinate-plane","position":8,"requirement":"optional","rationale":{"zh-Hant":"固定原點、互相垂直的有向實數軸與单位後，平面點用有序實數對 (x,y) 表示。","en":"Represent plane points by ordered real pairs after fixing an origin, perpendicular oriented axes, and units."}}]},{"id":"path-pulse-foundations-20261003","title":{"zh-Hant":"七輪代數與機率基礎：冪、恆等式、函數與有限模型","en":"Seven-Round Algebra and Probability Foundations: Powers, Identities, Functions, and Finite Models"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-variable-expression","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-event","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-ratio-proportion","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":2,"requirement":"optional","rationale":{"zh-Hant":"本輪限定實數底數的自然數冪，以及非負實數的主平方根，分開冪運算、根值與解集。","en":"This entry covers natural-number powers of real bases and principal square roots of nonnegative real numbers, keeping powers, root values, and solution sets distinct."}},{"id":"mko-polynomial","position":3,"requirement":"optional","rationale":{"zh-Hant":"把多項式視為有限支撐的有理係數資料；合併同次項後再讀取係數與次數。","en":"A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree."}},{"id":"mko-factorization","position":4,"requirement":"optional","rationale":{"zh-Hant":"因式分解是在指定係數域中把多項式寫成因子乘積；驗收比較全部係數而不是若干代入值。","en":"Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values."}},{"id":"mko-quadratic-equation","position":5,"requirement":"optional","rationale":{"zh-Hant":"對非零二次係數的實方程，用判別式區分實根數，將方程式、根集合與重根分開。","en":"For a real equation with a nonzero quadratic coefficient, the discriminant classifies its distinct real roots. The equation, root set, and repeated roots are kept distinct."}},{"id":"mko-linear-function","position":6,"requirement":"optional","rationale":{"zh-Hant":"實函數 f(x)=mx+b 稱仿射函數；m≠0 時為一次函數，m=0 時為常數函數。","en":"A real function f(x)=mx+b is affine. If m≠0 it has degree one; if m=0 it is constant."}},{"id":"mko-function-graph","position":7,"requirement":"optional","rationale":{"zh-Hant":"固定直角座標系後，實值函數的圖形是全部合法輸入與其唯一輸出組成的有序對集合。","en":"In a fixed Cartesian coordinate system, the graph of a real-valued function is the set of ordered pairs of every permitted input with its unique output."}},{"id":"mko-probability","position":8,"requirement":"optional","rationale":{"zh-Hant":"在非空有限樣本空間上指定非負且總和為一的權重，事件機率為其成員權重之和。","en":"Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes."}}]},{"id":"path-pulse-foundations-20261004","title":{"zh-Hant":"每日七輪第一期：量詞、離散結構與線性／条件模型","en":"Daily Seven-Round Cycle I: Quantifiers, Discrete Structures, Linear Systems, and Conditional Models"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proof","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":2,"requirement":"optional","rationale":{"zh-Hant":"全稱與存在量詞必須綁定論域；古典真值語義下，空論域的全稱為真、存在為假。","en":"Universal and existential quantifiers must bind a domain. Under classical truth semantics, a universal statement over the empty domain is true and an existential statement is false."}},{"id":"mko-counterexample","position":3,"requirement":"optional","rationale":{"zh-Hant":"反例必須屬於被聲稱的論域且使全稱述詞失敗；域外值和找不到反例都不能自動證明原命題。","en":"A counterexample must belong to the claimed domain and falsify the universal predicate. An outside value or failure to find a counterexample does not by itself prove the original statement."}},{"id":"mko-relation","position":4,"requirement":"optional","rationale":{"zh-Hant":"二元關係是指定 A×B 中有序對的集合，不要求每個輸入唯一或必有輸出。","en":"A binary relation is a set of ordered pairs in a specified A×B. An input need not have a unique output or any output."}},{"id":"mko-graph","position":5,"requirement":"optional","rationale":{"zh-Hant":"本輪限定有限簡單無向圖，邊是兩個相異頂點的無序集合，度數只計相鄰頂點。","en":"This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors."}},{"id":"mko-vector","position":6,"requirement":"optional","rationale":{"zh-Hant":"本輪把平面向量限制為固定基底下的 R² 有序對，逐分量定義加法與純量乘法。","en":"This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise."}},{"id":"mko-system-linear-equations","position":7,"requirement":"optional","rationale":{"zh-Hant":"同時滿足兩個實數線性限制才是解；非零行列式給唯一解，奇異與零列須另判一致性。","en":"A solution must satisfy two real linear constraints simultaneously. A nonzero determinant gives a unique solution; singular systems and zero rows require separate consistency checks."}},{"id":"mko-conditional-probability","position":8,"requirement":"optional","rationale":{"zh-Hant":"條件事件必須有正機率；有限模型中 P(A|B)=P(A∩B)/P(B)，方向不能任意交換。","en":"The conditioning event must have positive probability. In a finite model, P(A|B)=P(A∩B)/P(B), and the direction cannot be freely exchanged."}}]},{"id":"path-pulse-foundations-20261005","title":{"zh-Hant":"函數可逆性與有限機率矩數：每日七輪","en":"Function Invertibility and Finite Probability Moments: Seven-Round Workflow"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-composition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sample-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-injective","position":2,"requirement":"optional","rationale":{"zh-Hant":"單射要求不同合法輸入不能有同一輸出，不要求覆蓋整個陪域。","en":"Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain."}},{"id":"mko-surjective","position":3,"requirement":"optional","rationale":{"zh-Hant":"滿射要求陪域每個元素都有合法原像；相同輸出可以有多個原像。","en":"Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages."}},{"id":"mko-bijective","position":4,"requirement":"optional","rationale":{"zh-Hant":"雙射同時單射與滿射，等價於陪域每個元素恰有一個合法原像。","en":"A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage."}},{"id":"mko-inverse-function","position":5,"requirement":"optional","rationale":{"zh-Hant":"雙射 f:A→B 的反函數 g:B→A 須驗證兩個方向的還原，不能把交換資料當成無條件反函數。","en":"For a bijection f:A→B, an inverse g:B→A restores values in both directions; exchanging a table is not an unconditional inverse construction."}},{"id":"mko-random-variable","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限模型中的隨機變數是基本結果到實數的總函數；它不是基本結果，也不是權重本身。","en":"A random variable in a finite model is a total function from elementary outcomes to real numbers, distinct from an outcome or its probability weight."}},{"id":"mko-expectation","position":7,"requirement":"optional","rationale":{"zh-Hant":"有限期望是合法機率權重與隨機變數值的加權總和，線性性不要求獨立。","en":"Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence."}},{"id":"mko-variance","position":8,"requirement":"optional","rationale":{"zh-Hant":"本輪變異數為 E[(X−E[X])²]，不是 n−1 分母的樣本變異數；零權重離群值不影響它。","en":"This entry defines variance as E[(X−E[X])²], distinct from sample variance with an n−1 denominator. Zero-weight outliers do not affect it."}}]},{"id":"path-pulse-foundations-20261006","title":{"zh-Hant":"實數順序、變化率與座標幾何、資料摘要：七輪脈衝","en":"Real Order, Average Rates, Coordinate Geometry, and Data Summaries: Seven-Round Workflow"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quadratic-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-ratio-proportion","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-euclidean-length","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-point-line-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":2,"requirement":"optional","rationale":{"zh-Hant":"在實數論域比較大小；精確有理數交叉乘積、有限極值及含端點區間是本輪切片。","en":"The usual real order compares real values and supports finite nonempty extrema, closed intervals, and translation by a common addend."}},{"id":"mko-quadratic-function","position":3,"requirement":"optional","rationale":{"zh-Hant":"實數全論域的 q(x)=a(x−h)²+k、a≠0；頂點、全域極值與對稱點的明示切片。","en":"A nonzero-coefficient quadratic on all of R has a vertex, an axis of symmetry, and a global minimum or maximum determined by the coefficient sign."}},{"id":"mko-rate-of-change","position":4,"requirement":"optional","rationale":{"zh-Hant":"平均變化率是輸出差除以非零輸入差，不是瞬時導數。","en":"Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative."}},{"id":"mko-distance-formula","position":5,"requirement":"optional","rationale":{"zh-Hant":"在同一等尺度正交座標系，用非負主平方根讀取二維歐氏距離。","en":"The nonnegative L2 distance in one equal-unit orthonormal coordinate frame is the principal square root of the sum of squared coordinate differences."}},{"id":"mko-circle","position":6,"requirement":"optional","rationale":{"zh-Hant":"圓是到固定中心距離恰等於半徑的點集，不是含內部的圓盤。","en":"A circle here is the boundary point set at a nonnegative radius from its center, not the filled disk; radius zero yields a singleton."}},{"id":"mko-area","position":7,"requirement":"optional","rationale":{"zh-Hant":"先精確處理半開矩形及有限單位格聯集，含重疊、觸邊與空矩形；不冒充一般面積理論。","en":"This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases."}},{"id":"mko-descriptive-statistics","position":8,"requirement":"optional","rationale":{"zh-Hant":"保留重數的非空觀測列，精確計算均值、中位數、全距與所有最大頻率值。","en":"Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities."}}]},{"id":"path-pulse-foundations-20261007","title":{"zh-Hant":"矩陣、實二階線性代數與標記三角形、座標變換：七輪脈衝","en":"Matrices, Real 2×2 Linear Algebra, Labeled Triangles, and Coordinate Transformations: Seven-Round Workflow"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quadratic-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-triangle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-distance-formula","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-ratio-proportion","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-matrix","position":2,"requirement":"optional","rationale":{"zh-Hant":"矩陣以行、列組織元素；本輪使用非空有限實矩陣及精確整數陣列切片。","en":"Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances."}},{"id":"mko-linear-transformation","position":3,"requirement":"optional","rationale":{"zh-Hant":"先在 R²→R² 的矩陣切片驗證加法與純量相容，不把帶非零平移的仿射式叫線性映射。","en":"Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear."}},{"id":"mko-determinant","position":4,"requirement":"optional","rationale":{"zh-Hant":"2×2實矩陣的 ad−bc、非零行列式可逆性、乘積與純量縮放切片。","en":"This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices."}},{"id":"mko-eigenvalue","position":5,"requirement":"optional","rationale":{"zh-Hant":"特徵向量須非零；候選特徵空間核含零向量，實二階矩陣可沒有實特徵值。","en":"An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues."}},{"id":"mko-congruence","position":6,"requirement":"optional","rationale":{"zh-Hant":"限定正交等單位平面中的標記非退化三角形，以三組對應邊長判全等，允許鏡射。","en":"In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection."}},{"id":"mko-similarity","position":7,"requirement":"optional","rationale":{"zh-Hant":"指定由A到B的正長度倍率 k，平方長度與指定面積以 k² 縮放，不把負座標係數當負長度倍率。","en":"Specify the positive length scale k from A to B; squared lengths and a given area scale by k². A negative coordinate coefficient is not a negative length scale."}},{"id":"mko-geometric-transformations","position":8,"requirement":"optional","rationale":{"zh-Hant":"平移、原點逆時針四分之一圈的整數倍旋轉、指定軸鏡射是全平面等距；均勻縮放先分清零、±1與其他係數。","en":"Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients."}}]},{"id":"path-pulse-foundations-20261008","title":{"zh-Hant":"實數極限、連續、導數與多項式積分：七輪脈衝","en":"Real Limits, Continuity, Derivatives, and Polynomial Integration: Seven-Round Workflow"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-tends-to-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-factorization","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-rate-of-change","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-polynomial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-composition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-area","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence-limit","position":2,"requirement":"optional","rationale":{"zh-Hant":"從0起算的實數列以任意精度、某一自然尾界及整個尾段定義極限；本輪精確處理倒數尾項模型。","en":"A zero-based real sequence converges when every precision admits a natural cutoff controlling the entire tail; this pulse treats an exact reciprocal-tail model."}},{"id":"mko-function-limit","position":3,"requirement":"optional","rationale":{"zh-Hant":"在a附近排除a本身，以ε-δ控制函數值；點值不屬於去心極限條件。","en":"An ε–δ limit controls values near a while excluding a itself; the center value is not part of the punctured-limit condition."}},{"id":"mko-continuity","position":4,"requirement":"optional","rationale":{"zh-Hant":"在全實數論域連續需要點值存在且與兩側極限一致；補點與原函數連續不可混用。","en":"Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct."}},{"id":"mko-derivative","position":5,"requirement":"optional","rationale":{"zh-Hant":"導數是非零步差商的兩側極限，不是任何一個有限步差商；絕對值尖點另保留不存在。","en":"A derivative is the two-sided limit of nonzero-step quotients, not any one finite-step quotient; an absolute-value corner retains nonexistence."}},{"id":"mko-derivative-rules","position":6,"requirement":"optional","rationale":{"zh-Hant":"四條微分規則先保留可微前提；本輪只在有限整數係數多項式上精確計算。","en":"Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials."}},{"id":"mko-antiderivative","position":7,"requirement":"optional","rationale":{"zh-Hant":"原函數須在宣告區間導回f；本輪全R多項式以常數項固定0的有理係數原函數作可重播代表。","en":"An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials."}},{"id":"mko-definite-integral","position":8,"requirement":"optional","rationale":{"zh-Hant":"連續多項式的有向定積分與精確矩公式；有限左端黎曼和只是有限和，不冒充一般積分或微積分基本定理完成。","en":"Continuous polynomials have oriented definite integrals with exact moment formulas; a finite left Riemann sum is not a general integral or a completed general FTC bridge."}}]},{"id":"path-pulse-foundations-20261009","title":{"zh-Hant":"Core80最後種子與五輪既有題族密度","en":"Final Core80 Seeds and Five Rounds of Existing-Family Depth"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-tends-to-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-factorization","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-polynomial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-composition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-area","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-linear-transformation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-determinant","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-quadratic-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-derivative-rules","position":2,"requirement":"optional","rationale":{"zh-Hant":"原有至多3次全R多項式切片：二階導數點值、乘積導數候選值的資格、兩個有序合成導數，以及和的導數完整升冪係數。合成可能升次，不截去高項。","en":"Deepen the full-real degree-at-most-three polynomial slice with second derivatives, product-value eligibility, ordered composition derivatives, and complete derivative coefficients without truncation."}},{"id":"mko-tree","position":2,"requirement":"optional","rationale":{"zh-Hant":"非空有限簡單無向圖同時連通且無環才是樹；保留單點、度1葉及加一條新邊的唯一環。","en":"A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions."}},{"id":"mko-function-limit","position":2,"requirement":"optional","rationale":{"zh-Hant":"在既有去心實數模型內加厚：m≠0且|m|δ>ε時取h=ε/|m|，得到0<h<δ但誤差=ε；點值與極限成對輸出；左右極限保持順序；商只在x≠a求值。","en":"Deepen the punctured model with a failing δ witness h=ε/|m|, center-versus-limit pairs, ordered one-sided limits, and quotient values only at x≠a."}},{"id":"mko-eigenvalue","position":2,"requirement":"optional","rationale":{"zh-Hant":"原有實二階方陣整數切片：降冪特徵多項式(1,−tr,det)、精確判別式、Mv−λv有序殘差、非零向量第一失配座標。沒有對角化、Jordan或複數譜擴充。","en":"Deepen real 2×2 integer matrices with descending characteristic coefficients, exact discriminants, ordered kernel residuals, and a nonzero vector’s first mismatch; no diagonalization or complex-spectrum expansion."}},{"id":"mko-antiderivative","position":3,"requirement":"optional","rationale":{"zh-Hant":"原有全R多項式與正分母有理係數切片：指定原函數係數、共享F0的兩常數差、由F(a)=V解C，以及候選F的導數係數。保持升冪、最簡分數與常數0規約。","en":"Deepen the rational-coefficient slice with a specified primitive coefficient, shared-primitive constant differences, anchor-selected constants, and candidate derivative coefficients."}},{"id":"mko-definite-integral","position":4,"requirement":"optional","rationale":{"zh-Hant":"原有多項式矩與有向有限區間切片：任意端點順序的三段積分；奇函數資格與對稱積分成對輸出；有序端點仿射左端有限和的精確誤差；非零常數函數積分的端點反解。","en":"Deepen oriented moments with three-segment additivity, paired oddness/symmetric-integral output, signed finite-left-sum error, and endpoint recovery for a nonzero constant integrand."}},{"id":"mko-fundamental-theorem-calculus","position":5,"requirement":"optional","rationale":{"zh-Hant":"在全實數連續前提下，累積定積分的導數還原被積函數；合法原函數給有向積分端點差。","en":"With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference."}}]},{"id":"path-pulse-foundations-20261010","title":{"zh-Hant":"七輪跨域密度：代數、集合、距離、有限概率與量詞","en":"Cross-Domain Depth: Algebra, Sets, Distances, Finite Probability, and Quantifiers"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-variable-expression","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-euclidean-length","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-polynomial","position":2,"requirement":"optional","rationale":{"zh-Hant":"固定整數升冪係數至多3次的Q[x]切片；和與乘積输出完整正規化係數，乘積可升至6次；同一多項式雙點有序求值；p(x+c)完整係數以精確二項式展開核對。不把單點相同當多項式恆等。","en":"Use the integer-coefficient slice of Q[x] through input degree 3: complete normalized sum/product coefficients (products through degree 6), ordered evaluation at two points, and complete coefficients of p(x+c) by exact expansion. Equality at one point is not polynomial identity."}},{"id":"mko-set-operations","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限明示整數集合正規化重複與排序；聯／交大小對、(A\\B)∪(B\\A)、U\\(A∪B)與(U\\A)∩(U\\B)的完整結果、三段有序分割。補集先要求A、B⊆明示U。只封閉此有限实例，不新增一般集合論形式證明。","en":"Normalize finite explicit integer sets; compute union/intersection cardinalities, symmetric differences, both relative De Morgan sets, and ordered three-part partitions. Require A,B⊆the declared U before complements; finite instances do not create general set-theory formal Evidence."}},{"id":"mko-distance-formula","position":2,"requirement":"optional","rationale":{"zh-Hant":"所有向量是同一等單位正交實平面從共同原點出發的整數座標差；精確平方對、候選非負資格與平方缺陷、有三個具索引距離的含同距分組排序，以及兩個完全平方距離的整數和。不是非正交／不等尺度或Prod預設metric；不拿兩段和當另一段距離。","en":"Vectors are integer coordinate differences from a common origin in one equal-unit orthogonal real plane. Check ordered squares, candidate sign and square defect, indexed three-distance tie groups, and sums of two integer principal distances. Exclude nonorthogonal/unequal-scale and default product-metric interpretations; a two-segment sum is not another segment’s length."}},{"id":"mko-expectation","position":2,"requirement":"optional","rationale":{"zh-Hant":"同一有限合法歸一化整數值模型：逐點wX向量、E[X−Y]、(E[XY],E[X]E[Y])有序對與給定均值下的正質量未知整數值反解。只有共同S與權重，不假設X/Y獨立；數值相等也不推出独立。未知slot明示null、正權重且解為整數；不做樣本估計或一般積分存在性。","en":"On a shared normalized finite integer-valued model, compute wX contributions, E[X−Y], ordered (E[XY],E[X]E[Y]), and an integer value at an explicit null slot of positive mass from a known mean. No independence, sample-estimation, or general integrability claim; equal numerical expressions do not establish independence."}},{"id":"mko-quantifier","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限明示整數U及可判定大小／相等／非零整除述詞。完整真假元素分割、(¬∀P,¬∃P)、三個有序候選的域內反例资格mask，及有限U/V上(滿足∃y內條件的x數,滿足∀x內條件的y數)。空域全稱真、存在假；候選域外無资格；嵌套枚舉不交换量詞、不外推全部整數或一般一階逻輯證明。","en":"Enumerate finite explicit integer domains with order, equality, or nonzero-divisibility predicates: true/false partitions, (¬∀P,¬∃P), typed counterexample masks, and ordered nested existence/universality counts on U/V. Empty-domain universal is True and existential False. Do not exchange quantifiers or extrapolate to all integers or general first-order proofs."}},{"id":"mko-conditional-probability","position":3,"requirement":"optional","rationale":{"zh-Hant":"S=0至n−1、全冪集、非負整數mass與明示正分母D，Σmass=D才為歸一化模型。B質量嚴格正，雙向另要求A正。枚舉交集／B概率對、雙向條件值、對全部singleton在B條件下的概率向量、候選值與真正條件值的有號缺陷。非空但mass0不能條件化；不宣稱Bayes或独立性。","en":"Use explicit finite S, its full powerset, nonnegative integer masses, and positive D with total mass D. Require positive B mass, and positive A mass for both directions. Compute probability pairs, directional conditionals, every conditional singleton weight, and signed candidate defects. Nonempty zero-mass events cannot condition; no Bayes or independence proof is claimed."}},{"id":"mko-variance","position":3,"requirement":"optional","rationale":{"zh-Hant":"合法有限概率模型：逐項w(X−μ)²、(E[X²],μ²)精確對、先構造共模型X−Y再求其變異數，以及相對第一個正權重值的第一個正支持失配索引。不是n−1樣本估計；保留零權重異值；不把Var(X−Y)拆成兩個Var，也不聲稱協方差理論完成。","en":"Compute w(X−μ)², exact (E[X²],μ²), variance after forming the shared-model difference, and the first positive-support mismatch relative to the first positive-weight value. Preserve zero-weight outliers; do not use n−1 sample variance, split Var(X−Y) into two variances, or claim complete covariance theory."}}]},{"id":"path-pulse-foundations-20261011","title":{"zh-Hant":"七輪基礎補缺：自然數、表達式、集合與有限趨近模型","en":"Foundation Gaps: Natural Numbers, Expressions, Sets, and Finite Limit Models"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-natural-number","position":1,"requirement":"optional","rationale":{"zh-Hant":"固定N包含0。列有限後繼窗、雙點的部分自然前驅、有限閉區間大小與三個有理候選的N資格；0無自然前驅，不混用截斷pred、負數或分數。","en":"Fix N including 0; list successor windows, two partial predecessors, closed-interval sizes, and rational candidates for N. Zero has no natural predecessor in this protocol; do not conflate it with truncated pred, negative values, or fractions."}},{"id":"mko-set","position":1,"requirement":"optional","rationale":{"zh-Hant":"明示整數型別的有限集合。去重排序、相等及雙向差、候選元素的內外分割、至多4元素的全部子集合列舉；空集合的冪集只有空集合，不宣稱無限制概括或新集合論公理。","en":"Use finite sets of an explicit integer type: deduplication/sorting, equality with both differences, candidate membership partitions, and complete powersets of at most four elements. The empty set’s powerset contains the empty set; no unrestricted comprehension or new axiom is asserted."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-arithmetic-operations","position":2,"requirement":"optional","rationale":{"zh-Hant":"固定有理數或N輸入的Q結果切片。四則有序結果、括號雙式、逆元候選有號缺陷、自然數差商的資格；除零未定義，負差不截成0，商不是整除。","en":"Use rational arithmetic, or natural inputs with Q results: ordered four operations, parenthesized pairs, inverse candidates with signed defects, and natural eligibility of differences/quotients. Division by zero is undefined, negative differences are not truncated to zero, and quotients are not integer division."}},{"id":"mko-subset","position":2,"requirement":"optional","rationale":{"zh-Hant":"同一整數型別的有限集合。全部A不在B的元素、非嚴格／嚴格包含及雙向見證、三段包含鏈的阻擋元素、三個候選反例的域內資格；空集包含與相等不是嚴格包含，有限檢查不當普遍集合證明。","en":"Compute all A-outside-B obstructions, non-strict/strict inclusion with both witnesses, three chain-obstruction sets, and typed candidate masks. Empty-set inclusion and equality are not strict inclusion, and finite checks are not universal set proofs."}},{"id":"mko-sample-space","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限且明示的可能結果協議。完整有序直積、記錄映射像及原結果纖維、候選清單的漏列／域外項、基本結果和事件隸屬的雙mask；不指定概率，不假設均勻、獨立或由未知試驗推得完整性。","en":"Use explicitly declared possible-outcome protocols: ordered products, recording images and original fibers, missing/outside candidate outcomes, and separate outcome/event masks. Assign no probabilities and infer neither uniformity, independence, nor completeness of an unknown experiment."}},{"id":"mko-tends-to-relation","position":2,"requirement":"optional","rationale":{"zh-Hant":"來源及目標均明示為有限主濾子，映射在完整有限型別上為總函數。核對像生成集、f(S)不在T的阻擋點、反例候選資格與兩條合成像計算；允許空生成集的bottom濾子。只關閉明示有限模型，不把裸箭頭當一般數列／點極限，不新增形式Evidence。","en":"Both filters are explicit finite principal filters, with a total map on complete finite types. Check image generators, f(S)-outside-T obstructions, typed candidates, and two composition-image routes; empty generators and the bottom filter are allowed. This finite model does not interpret a bare arrow as a general sequence/point limit or create new formal Evidence."}},{"id":"mko-variable-expression","position":3,"requirement":"optional","rationale":{"zh-Hant":"只指定Q中的四種小表達式，不建立通用AST。比較兩環境值、同時交換兩變數、內外層詞法遮蔽及a/(x-b)的原式定義域；零分子也不消除排除點，代數式不當方程式。","en":"Specify four small expressions over Q, not a universal AST: two environments, simultaneous variable exchange, lexical shadowing, and the original domain of a/(x−b). A zero numerator does not remove an excluded point; expressions are not equations."}}]},{"id":"path-pulse-foundations-20261012","title":{"zh-Hant":"七輪基礎推理與幾何：歸納、計數、命題、型別與長度","en":"Foundational Reasoning and Geometry: Induction, Counting, Typing, and Length"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-proposition","position":1,"requirement":"optional","rationale":{"zh-Hant":"固定古典整數比較陳述與有號算術見證。未代入的自由變數標未封閉；相等與否定分開；候選自報真假只作待核資料，與實際真假和正式證明項分離。不建立通用邏輯語言或新形式Evidence。","en":"Use classical integer comparison statements and signed arithmetic witnesses. Mark unassigned free variables as open; separate equality and negation, candidate reports, actual truths, and proof terms. Do not create a universal logic language or new formal Evidence."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-set-membership","position":1,"requirement":"optional","rationale":{"zh-Hant":"有限同型別集合與完整特徵表。Fin(n)外候選只標不合型別，不盲讀謂詞；隸屬判定與插入／刪除結果、有限明示論域上的謂詞集合外延。保留Set α→Prop既有Evidence，不把程式容器當全部集合論。","en":"Use finite typed sets and complete characteristic tables. Outside-Fin(n) candidates are ill-typed, not blindly evaluated predicate inputs; distinguish membership from insertion/deletion and finite predicate extension. Preserve existing Set α→Prop Evidence without treating containers as all of set theory."}},{"id":"mko-right-triangle","position":1,"requirement":"optional","rationale":{"zh-Hant":"候選整數角度先核正角與和180的協議，再檢90；兩個90加零角不能通過。合格直角資料標記頂點、對邊及銳角值；π有理倍數精確換算，不用容差、不取模、不把候選當已構造的非退化三角形。","en":"Qualify candidate positive integer angles and exact sum 180 before checking 90; two 90s and a zero fail. Preserve vertex/opposite-side labels and ordered acute angles; convert rational π multiples exactly, without tolerance or periodic wrapping. Candidate data are not a constructed nondegenerate triangle."}},{"id":"mko-euclidean-length","position":1,"requirement":"optional","rationale":{"zh-Hant":"同一等單位正交L2實平面的整數座標切片。保留有向座標差及長度平方；完全平方才給整數主距離；中點為精確有理數，兩段平方分別核對。軸上路徑與端點直線長分開，零距離不排除；不重播舊浮點參考Evidence或聲稱全部度量公理完成。","en":"Use integer coordinates in one equal-unit orthogonal Euclidean L2 plane. Preserve signed displacement and squared lengths; give integer principal distances only for perfect squares. Check exact rational midpoints and both half-segment squares; separate axis path sums from direct endpoint length, retain zero distances, and do not claim all metric axioms or replay old floating-point Evidence as new."}},{"id":"mko-mathematical-induction","position":2,"requirement":"optional","rationale":{"zh-Hant":"N從0起，分清歸納假設n與目標n+1。只核對明示整數遞推的有限前綴、仿射候選的基礎／步缺陷與有限檢查覆蓋；不把有限樣本當成任意n的歸納證明。","en":"N starts at 0; distinguish hypothesis index n from target n+1. Check explicit integer recurrence prefixes, affine-candidate base/step defects, and finite coverage only; finite samples are not induction for arbitrary n."}},{"id":"mko-counting-principle","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限選擇協議明示互斥標籤或完整有序直積，包含空分支與零選項；後續寬度可依前一步變化，候選等寬法核對資格和缺陷，不從總數巧合推得等寬或概率獨立。兩集合重疊由實際外延計數修正。","en":"Explicitly tagged disjoint choices or full ordered products include empty branches and zero options. Later widths may depend on earlier choices; check candidate equal-width eligibility separately from total defects. Matching totals imply neither equal widths nor probabilistic independence; correct overlaps by actual set extension."}},{"id":"mko-function-mapping","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整有限型別上的總映射，允許空定義域及唯一空表，非空源到空陪域不可宣稱總函數。圖形、實際值域與未用陪域分開；型別外查詢標不合型別；兩張同型別總表逐點核差異。保留既有總函數core Evidence，不混作部分、有副作用或會失敗的程式函數。","en":"Total maps on complete finite types allow the empty domain and its unique empty table, but no total map from a nonempty source into an empty target. Distinguish graph, range, and unused codomain; outside queries are ill-typed, and compare both complete tables pointwise. Preserve existing total-function core Evidence; exclude partial, effectful, or failing programs."}}]},{"id":"path-pulse-foundations-20261013","title":{"zh-Hant":"七輪有限結構：畢氏、數系、推理、事件與標記三角","en":"Finite Structures: Pythagoras, Number Systems, Inference, Events, and Labeled Triangles"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-right-triangle","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-euclidean-length","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-sample-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-point-line-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-euclid-pythagorean-theorem","position":2,"requirement":"optional","rationale":{"zh-Hant":"正整數兩股與明示歐氏直角邊長協議。平方欄位有序，完全平方才取正整數斜邊或缺股；候選負c即使平方殘差0也不合非退化長度。保留既有FELRA、向量核心及宣告side model證據，不宣稱任意仿射橋已完成。","en":"Use positive integer legs in an explicit Euclidean right-side protocol. Keep square fields ordered; only perfect squares yield positive integer hypotenuses or missing legs. A negative candidate c fails nondegenerate-length eligibility despite zero square residual. Preserve prior FELRA, vector-core, and declared side-model Evidence; no arbitrary affine bridge is completed."}},{"id":"mko-integer","position":2,"requirement":"optional","rationale":{"zh-Hant":"明示Z及有理候選。歐式整除要求正b及0≤r<b，負a不用截斷餘數；淨位移與步量分開；相反數及雙向差有序；負整數、零與非整數分數的資格和實值分組，有限切片不當環公理。","en":"Use explicit Z and rational candidates. Euclidean division requires b>0 and 0≤r<b, not truncated negative remainders. Separate net displacement from step magnitude, retain ordered opposites and directional differences, and classify zero/negative integers versus noninteger fractions; finite slices are not ring axioms."}},{"id":"mko-logical-connectives","position":2,"requirement":"optional","rationale":{"zh-Hant":"兩份完整八行0/1古典檔表逐行做固定合取／析取、材料蘊涵／雙條件、兩路德摩根與含括／排他或。材料蘊涵不作因果，逐行真假不當所有推理規則證明或通用邏輯語言。","en":"Evaluate two complete eight-row 0/1 classical tables for conjunction/disjunction, material implication/biconditional, both De Morgan routes, and inclusive/exclusive disjunction. Material implication is not causation; rowwise truths do not prove all inference rules or establish a general logic language."}},{"id":"mko-event","position":2,"requirement":"optional","rationale":{"zh-Hant":"明示非空有限S，區分F=P(S)與分塊聯集族。補事件／聯交外補、分塊的部分相交見證及最小閉包、候選觀測資格與首發生；有限S不自動讓全部子集可測，沒有概率或獨立假設。","en":"Declare a nonempty finite S and distinguish F=P(S) from partition-union families. Compute complement/union/intersection, partial-block witnesses and least containing events, typed observations, and first occurrence. Finite S does not automatically make every subset measurable, and probabilities/independence are not supplied."}},{"id":"mko-triangle","position":2,"requirement":"optional","rationale":{"zh-Hant":"整數標記A/B/C的歐氏頂點協議。相異／共線見證、合格三角形的對邊平方、循環及交換標記的座標行列式、對邊中點與三段平方；候選退化資料不冒稱三角形，有限座標核對不作一般幾何構造證明。","en":"Use integer-labeled Euclidean A/B/C vertex data: distinctness/noncollinearity witnesses, qualified opposite-side squares, cyclic/swapped coordinate determinants, opposite midpoints, and three squared segments. Degenerate candidates are not triangles and finite coordinates are not general construction proofs."}},{"id":"mko-rational-number","position":3,"requirement":"optional","rationale":{"zh-Hant":"分子與非零整數分母，包含負分母表示，答案正分母精確規約。和差、次序及精確差、除法資格與回乘、缺失加數回代，不用浮點或近似，不聲稱完整Rat構造或域公理。","en":"Represent rationals by integer numerators and nonzero denominators, including negative input denominators; outputs are exact reduced positive-denominator fractions. Check sum/difference, order and signed difference, guarded division/multiplication back, and missing-addend substitution. No floating-point approximation or complete Rat/field construction is claimed."}},{"id":"mko-proof","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限Pi標籤、給定前提與單前提MP邊。最短推導深度、型別資格、候選序列逐行審查與刪規則影響；無種子循環不自證，錯行不加入可用集。只是明示玩具推理協議，不當通用邏輯完備性、人審或形式Evidence。","en":"Use finite proposition labels, given facts, and single-premise MP edges. Check shortest depths, typing, candidate traces row by row, and rule-removal effects. Unseeded cycles cannot prove themselves and invalid rows never become premises. This toy protocol is neither general logical completeness nor human review or formal Evidence."}}]},{"id":"path-pulse-foundations-20261014","title":{"zh-Hant":"七輪基礎深化：角度、數線、方程、階乘與圓邊界","en":"Foundation Transforms: Angles, Number Lines, Equations, Factorials, and Circle Boundaries"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-point-line-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-variable-expression","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-euclidean-length","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-distance-formula","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-angle","position":2,"requirement":"optional","rationale":{"zh-Hant":"0到180度非定向角的精確度數協議。拆分先核資格，互餘與互補分开，雙門檻殘差和三段累加只作明示算術；超180累加量不冒充本條目角，不由示意圖、射線長或定向旋轉推測量值。","en":"Use exact nondirected degree values in [0,180]: guard splits, distinguish complements/supplements, and compute two threshold defects and three-part budgets. Sums above 180 are not angles in this entry. Infer no measurements from sketches, ray lengths, or directed rotations."}},{"id":"mko-real-number","position":2,"requirement":"optional","rationale":{"zh-Hant":"只核有理點在R的嵌入切片：有序中點、有限唯一點列及閉包區間、含端點的閉區間交集與分離距離、四個點的有向偏移及距離。接觸是點區間而非空；不是Cauchy完備性、無理數分類或一般連續性證明。","en":"Check the rational-point embedding slice only: ordered midpoints, finite distinct points and least containing closed intervals, inclusive interval intersections/separation, and four directed offsets/distances. Touching is a point interval, not empty; the containing interval is not the point set’s topological closure. No Cauchy completeness, irrational classification, or general continuity proof."}},{"id":"mko-equation","position":2,"requirement":"optional","rationale":{"zh-Hant":"給定Q上的原式雙側值、有限候選與可逆條件。乘零能改變解集；含x-h分母時先排除x=h，交乘殘差零不恢復未定義點。同量平移核兩側值及殘差，不建立通用AST或一般方程求解完備性。","en":"Check original Q equation sides, finite candidates, and reversibility premises. Multiplication by zero may change solutions; with denominator x−h first exclude x=h, and cross-multiplication cannot restore it. Check both shifted sides and defects without a universal AST or general solving-completeness claim."}},{"id":"mko-factorial","position":2,"requirement":"optional","rationale":{"zh-Hant":"N從0起且0!=1，用BigInt核相鄰階乘精確比、含空乘積的窗口、候選自報的真值與遞迴缺陷、省去一次已存在因子後回乘。報告用有界正規十進整數字串，不以浮點或遞迴自洽冒充正確階乘；不聲稱排列、整除性或漸近普遍證明。","en":"Use N from 0, 0!=1, and exact BigInt products: adjacent ratios, empty-product windows, independently checked reports/base and recurrence defects, and omission of one existing factor with multiplication back. Bound canonical decimal reports; recurrence self-consistency and floating-point values are not factorial correctness. No general permutation, divisibility, or asymptotic proof."}},{"id":"mko-circle","position":2,"requirement":"optional","rationale":{"zh-Hant":"等單位正交歐氏框架與非負整數半徑。三點平方殘差分類但圓只取邊界；軸向兩個對徑端點、半徑非負尺度乘算、水平截線平方餘量與整數候選。r=0只有中心且軸向表示折疊，負餘量不取根，不聲稱一般圓API或相似定理的形式橋。","en":"Use one equal-unit orthogonal Euclidean frame and a nonnegative integer radius. Check three squared residual/classification sets but only boundary membership, axis antipodes, radius-parameter scaling, and fixed-horizontal-offset square remainders. r=0 is only the center with collapsed endpoints; negative remainders have no real root. No general circle-API or similarity-theorem formal bridge."}},{"id":"mko-linear-equation","position":3,"requirement":"optional","rationale":{"zh-Hant":"a非零才屬Q上一元一次方程；規一列、參數切片的退化常數等式、根周有理偏移回代與兩個一次方程的根／共同解。a=0可能全Q或無解但不是一次，有限核對不作一般線性系統或唯一性形式Evidence。","en":"Only a≠0 gives a one-variable linear equation over Q. Check normalized rows, parameter-dependent constant degeneracies, exact offsets around roots, and two rows’ roots/common solutions. a=0 can yield all Q or no solution but is not linear; no general linear-system or uniqueness formal Evidence."}},{"id":"mko-absolute-value","position":3,"requirement":"optional","rationale":{"zh-Hant":"只核有理嵌入的分段絕對值：負分母表示先正規化，三點距離與路徑差分開，閉半徑四點代入與有理目標的根／三候選核對。負目標無根，零根去重；有限切片非全部實數性質、三角不等式或一般不等式求解證明。","en":"Normalize rational representations before branch decisions; distinguish three-point distances from path defects, inclusive four-point radius tests, and rational-target roots/candidate audits. Negative targets have no roots and zero roots are deduplicated. Finite slices are not all real absolute-value properties, a universal triangle inequality, or general inequality solving."}}]},{"id":"path-pulse-foundations-20261015","title":{"zh-Hant":"七輪型別資料與選取：分數、比例、映射、合成與座標","en":"Typed Data and Selection: Fractions, Ratios, Maps, Compositions, and Coordinates"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-counting-principle","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-factorial","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-point-line-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-fraction","position":2,"requirement":"optional","rationale":{"zh-Hant":"合法整數分數表示，原分母非零且負分母可用。真加法對照分子分母分別加的候選、四值相鄰差鏈、倒數先核零、兩次原始縮放表示與零縮放不合法；表示與數值分開，不把有限算術當商集或良定性形式證明。","en":"Keep legal original integer fractions with nonzero denominators, allowing negative denominators. Compare true addition with naive numerator/denominator addition, ordered difference chains, zero-guarded reciprocals, and two raw scaled representations. Zero scaling is illegal; representation/value separation is not a quotient well-definedness formal proof."}},{"id":"mko-domain-codomain-range","position":2,"requirement":"optional","rationale":{"zh-Hant":"Fin(n)到Fin(m)完整表的域／陪域／值域、兩個候選陪域值的原像、單格替換與值域增刪、一列協議候選表稽核。未使用格也須合型；重複同值資料列違反一列協議但不代表多值函數，不混同資料表示稽核與一般函數證明。","en":"Use complete Fin(n)→Fin(m) data for domain/codomain/range inventory, two typed-candidate preimages, one-cell replacement with range differences, and one-row protocol audits. Even unused cells must type-check; repeated equal-value rows violate the representation protocol without implying a multivalued function. Data audits are not general function proofs."}},{"id":"mko-sequence","position":2,"requirement":"optional","rationale":{"zh-Hant":"N從0起的明示有限索引、給定仿射規則的窗口、同四項前綴的兩個有限延伸、五項前綴和與有序差。次序與重複保留；空窗口不灌水、相同候選不算不同延伸，有限前綴不推定唯一無限規則、收斂或無窮級數。","en":"Use explicit finite zero-based indices, declared affine-recurrence windows, two finite extensions of a common four-term prefix, and five prefix sums/ordered differences. Keep order/repetition and genuinely empty windows; equal candidates are not different extensions. Finite prefixes imply neither a unique infinite rule nor convergence or an infinite series."}},{"id":"mko-permutation-combination","position":2,"requirement":"optional","rationale":{"zh-Hant":"相異標記Fin(n)物件無放回。有序／無序計數及k!與空選取、三個候選取樣資格、含／排物件0的組合分割、固定前r個物件順序後餘量與完成計數比。空選取只有一種，不套重複、不可區分、環形模型或擅稱概率。","en":"Select distinct labeled Fin(n) objects without replacement: ordered/unordered counts, k! and the unique empty selection, three candidate masks, inclusion/exclusion of label 0, and completion counts after a fixed ordered prefix. Do not use repetition, indistinguishable, or circular models, or silently call a counting ratio a probability."}},{"id":"mko-coordinate-plane","position":2,"requirement":"optional","rationale":{"zh-Hant":"同一正交有向xy框架中的整數有序對：四點欄位／軸／象限、逐分量差與相等、交換欄位候選、三對的首個不等。原點同屬兩軸，軸點不屬象限；交換欄位不作幾何反射形式橋，分量差不當距離或图形尺寸。","en":"Use ordered integer pairs in one orthogonal directed xy frame: four-point columns/axis/quadrant masks, component defects/equality, swapped-field candidates, and the first unequal pair. The origin lies on both axes and axis points are not in quadrants. A field swap is not a formal geometric reflection bridge; components are not distances or rendered dimensions."}},{"id":"mko-ratio-proportion","position":3,"requirement":"optional","rationale":{"zh-Hant":"Q數值比先核後項非零：正反向比、同k的完整三點律、同單位非負總量的正權分配、候選比例雙後項與交乘殘差。零總量的0:0仍非法，交乘零不恢復非法比；有限模型不外推因果、所有輸入或幾何相似。","en":"Numeric ratios in Q require nonzero second components: forward/reverse guards, a complete three-point common-k law, positive-weight same-unit nonnegative-total partitions, and candidate proportions with both denominator guards. Zero-total 0:0 remains illegal despite zero cross defects. Finite models imply no causality, universal law, or geometric similarity."}},{"id":"mko-composition","position":3,"requirement":"optional","rationale":{"zh-Hant":"先f後g且先核中間域；完整表路徑、g在明示子域的總表限制、兩序Q仿射合成與常數差、三總表的兩種括號路徑。未入g域不求值，未使用的g/h格仍須合型；合成不當乘法，不作全部函數等式形式證明。","en":"Evaluate f before g and guard the intermediate type/domain first. Check complete paths, g restricted to a declared subdomain while retaining original pairings, both Q affine orders and constant differences, and two parenthesizations of three maps. Affine output pairs are [slope,constant], not ascending coefficients. Outside g’s domain is not evaluated and unused g/h cells remain typed; no universal function-identity formal proof."}}]},{"id":"path-pulse-foundations-20261016","title":{"zh-Hant":"七輪線性與有限映射：向量、矩陣、方程及單滿射","en":"Seven Rounds of Linear and Finite-Mapping Depth"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-linear-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-vector","position":2,"requirement":"optional","rationale":{"zh-Hant":"固定標準基底下的整數向量：兩純量加權、相反差、純量和兩側與候選分量殘差。零純量与零向量合法，逐分量不當內積或長度。","en":"Use integer vectors in a fixed standard basis: two weighted terms, opposite directional differences, two scalar-addition routes, and candidate component residuals. Zero scalars/vectors are legal; component arithmetic is not an inner product or length."}},{"id":"mko-matrix","position":2,"requirement":"optional","rationale":{"zh-Hant":"非空1至5行列整數矩陣：轉置再轉置、同尺寸加法的兩側轉置、三個1起算索引資格與候選轉置逐項殘差。形狀不合先拒絕求值，不當求逆或反序。","en":"Use complete nonempty 1–5-row/column integer matrices: transpose round trips, both transpose-of-sum routes, three 1-based index guards, and candidate-transpose residuals. Reject mismatched shapes before subtraction; transpose is neither inversion nor element reversal."}},{"id":"mko-injective","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整Fin總表單射：列出全部碰撞、移去一個輸入後保留原標籤、附加輸出的陪域資格與限制到明示子集合。空源单射真，碰撞必有兩個不同輸入；重複表示不當多值函數。","en":"Validate complete finite total tables before enumerating every distinct-input collision, deleting an input with original labels retained, guarding appended outputs, or restricting to an explicit source subset. Empty sources are injective; a valid append may create nine labels and collisions still use the original numbering. Representation repetition is not a multivalued function."}},{"id":"mko-surjective","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整Fin總表滿射：全纖維與漏項、更換陪域先容納全部輸出、刪輸入保原標籤後漏項、兩個候選目標的資格與首原像。空到空滿射真，空到非空不滿；值域不自動當陪域。","en":"Enumerate all declared codomain fibers/missing targets; a changed codomain must contain every existing output before evaluating its surjectivity. Preserve original source labels on deletion and distinguish typed empty fibers from outside-codomain candidates. Empty-to-empty is surjective, empty-to-nonempty is not; actual range is not automatically the declared codomain."}},{"id":"mko-linear-transformation","position":3,"requirement":"optional","rationale":{"zh-Hant":"明示全R²到R²列向量模型Mv：基底列重建、加法兩側、純量兩側與Mv+b的零點障礙。零矩陣可以線性，有限測值不替任意函數證明線性。","en":"Declare the full R²→R² column-vector matrix model Mv: reconstruct basis-image columns, compare additivity/scalar routes, and test the zero-point obstruction for Mv+b. The zero matrix is linear; finite values do not prove arbitrary functions linear."}},{"id":"mko-determinant","position":3,"requirement":"optional","rationale":{"zh-Hant":"R上2×2有號行列式：交换兩行的負號、t²縮放、通常行乘列的兩序乘積、絕對值候選誤判。實數域非零可逆不搬至整數環，負行列式不取絕對值。","en":"Retain signed real 2×2 determinants: swapped-row sign, t² matrix scaling, ordinary row-by-column products in both orders, and absolute-value candidate audits. Nonzero real determinants permit real-field invertibility, not a transfer to integer-ring invertibility; matching product determinants do not make the products equal."}},{"id":"mko-system-linear-equations","position":3,"requirement":"optional","rationale":{"zh-Hant":"R上整數係數的兩元兩式：有理候選逐式殘差、係數與增廣秩區分無解／唯一／無限多解、加倍列消元與第一列縮放。t=0不是可逆列操作，不能只看行列式零便稱無解。","en":"Use integer-coefficient two-variable/two-equation systems over R: substitute rational candidates into both rows, distinguish solution classes with coefficient/augmented ranks, add a multiple of one complete row to the other, and scale the first row. A zero determinant does not decide inconsistency; zero scaling is not invertible even if a redundant system’s solution set happens to remain unchanged."}}]},{"id":"path-pulse-foundations-20261017","title":{"zh-Hant":"七輪離散結構：雙射、反函數、關係、圖樹與反例","en":"Seven Rounds of Discrete Structure: Bijections, Inverses, Relations, Graphs, Trees, and Counterexamples"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-injective","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-surjective","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-composition","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-linear-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-proof","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-bijective","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整Fin總表把單射、滿射與雙射分開；全纖維／碰撞／漏項、交換關係資格、交換兩個輸出標籤、基數存在性與給定表分開。相同基數不審定給定f；空到空雙射真。","en":"Validate complete finite total maps before separating injectivity, surjectivity and bijectivity. Enumerate fibers, collisions and missing targets, qualify the converse relation, transpose output labels and distinguish cardinality equality from the given map. Empty-to-empty is bijective."}},{"id":"mko-relation","position":2,"requirement":"optional","rationale":{"zh-Hant":"R是明示Fin來源×目標的有序對集合：完整投影／漏源、子集合像與全纖維、逆關係再逆、單值與總表分開。關係可多值或缺源；重複對與集合列項正規化，逆關係不自動給反函數。","en":"Use ordered-pair sets with explicit finite source and target: projections and missing sources, restricted images and fibers, double converse, single-valuedness and totality. Normalize repeated pairs; a relation may be multivalued or partial and its converse is not automatically an inverse function."}},{"id":"mko-order-relation","position":2,"requirement":"optional","rationale":{"zh-Hant":"R通常全序的正原分母有理切片：有序三筆穩定排序、兩閉區間交會與端點、三個會員與有號餘量、純量正向／反向／零合併。要求實際新數值，不只換平移參數重出相同比較；不外推偏序或複數。","en":"Use the usual real total order on rational inputs with positive original denominators. Compute stable three-value sorts, inclusive interval intersections and membership slacks, and actual scaled values with positive/negative/zero direction guards. Do not extrapolate to complex numbers or arbitrary partial orders."}},{"id":"mko-counterexample","position":2,"requirement":"optional","rationale":{"zh-Hant":"明示有限Fin論域與完整布林述詞表：全失敗集及最小反例、三個候選的域資格、限制子域後見證是否仍有效、明示加入一個新述詞值。域外不查P也不當反例；空域全稱量化真，有限枚舉不證無限域。","en":"Use a declared finite domain and complete Boolean predicate table. Enumerate failures and least witnesses; guard candidates by the domain, recheck witnesses under restriction, and explicitly provide any new predicate value on extension. Outside inputs are not counterexamples; empty universal quantification is true and finite enumeration proves no infinite-domain claim."}},{"id":"mko-inverse-function","position":3,"requirement":"optional","rationale":{"zh-Hant":"完整Fin總表先雙射再建反表；候選g逐格合型後才核兩側identity；查詢先陪域資格。不把反函數當函數值倒數。R到R仿射恢復限非零斜率，兩個合法有理輸入以最簡正分母回代。","en":"Construct inverse tables only after bijectivity; type every candidate inverse cell before checking both identities independently. Guard queries by declared codomain. An inverse is not a reciprocal; affine recovery on R requires nonzero slope and exact normalized rational substitution."}},{"id":"mko-graph","position":3,"requirement":"optional","rationale":{"zh-Hant":"簡單無向Fin圖：完整鄰接矩陣／度數與握手、全頂點鄰居／非鄰居分割、有效邊toggle的度數改變、明示子集合誘導圖且保原標籤。反向或重複邊不增加邊，無自環；本graph輪不作連通或樹判定。","en":"Use complete simple undirected finite graphs: adjacency matrices and degree sums, neighbor/nonneighbor partitions, valid edge toggles and induced subsets with original labels. Reversed or repeated edges count once; exclude loops and do not infer adjacency from drawings or claim tree qualification here."}},{"id":"mko-tree","position":4,"requirement":"optional","rationale":{"zh-Hant":"非空樹資格與完整簡單無向圖資料：components／有環與空域互達、原樹與端點資格後唯一道路、度1葉刪除保原標籤、原樹上新增不存在邊的唯一環。空圖不是樹；單點無邊是樹但度0非葉，邊數不單獨判樹。","en":"Require a nonempty connected acyclic simple undirected graph. Check components/cycles, original-tree and endpoint guards before paths, degree-one leaf deletion without relabeling, and a new nonedge’s unique cycle. Empty graphs are not trees; a singleton is a tree but its degree-zero vertex is not a leaf. Edge count alone does not qualify a tree."}}]},{"id":"path-pulse-foundations-20261018","title":{"zh-Hant":"擴展數學地圖第一批：拓撲、數論、決定問題與形式陳述","en":"First Extended Atlas Slice: Topology, Number Theory, Decision Problems, and Formal Statements"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":2,"requirement":"optional","rationale":{"zh-Hant":"拓撲空間由底集合與一族開集組成；開集族包含空集與全空間，並對任意聯集與有限交封閉。","en":"A topological space consists of an underlying set and a family of open sets containing the empty set and whole space and closed under arbitrary unions and finite intersections."}},{"id":"mko-natural-divisibility","position":3,"requirement":"optional","rationale":{"zh-Hant":"自然數 a 整除 b，表示存在自然數乘數 k 使 b=a·k；零的兩側行為必須分開。","en":"A natural number a divides b when a natural multiplier k satisfies b=a·k; the two roles of zero must be distinguished."}},{"id":"mko-gcd-coprime","position":4,"requirement":"optional","rationale":{"zh-Hant":"gcd(a,b) 是同時整除 a、b 且被每個公因數整除的自然數；gcd=1 時稱互質。","en":"The natural number gcd(a,b) divides both a and b and is divisible by every common divisor; the inputs are coprime when their gcd is 1."}},{"id":"mko-prime-number","position":5,"requirement":"optional","rationale":{"zh-Hant":"自然數質數是大於或等於 2，且正因數只有 1 與自身的自然數。","en":"A prime natural number is at least 2 and has no positive divisors other than 1 and itself."}},{"id":"mko-modular-congruence","position":6,"requirement":"optional","rationale":{"zh-Hant":"固定正模數 n 時，a 與 b 模 n 同餘表示兩者除以 n 的餘數相同。","en":"For a fixed positive modulus n, two natural numbers a and b are congruent modulo n when they have the same remainder on division by n."}},{"id":"mko-decision-problem","position":7,"requirement":"optional","rationale":{"zh-Hant":"決定問題由輸入集合 X 與 yes-instance 子集合 Y⊆X 指定；decider 必須對每個合法輸入終止並回傳正確布林答案。","en":"A decision problem specifies an input set X and a yes-instance subset Y⊆X; a decider must terminate on every legal input and return the correct Boolean answer."}},{"id":"mko-lean-formal-statement","position":8,"requirement":"optional","rationale":{"zh-Hant":"形式陳述是在固定版本、imports、環境與 local context 中成功 elaboration，且型別為 Prop 的核心項。","en":"A formal statement is a core term of type Prop obtained by successful elaboration in a fixed version, import environment and local context."}}]},{"id":"path-pulse-foundations-20261019","title":{"zh-Hant":"擴展數學地圖第二批：拓撲語言、計算正確性、漸近界與證明項","en":"Second Extended Atlas Slice: Topological Language, Computational Correctness, Asymptotic Bounds, and Proof Terms"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proof","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-lean-formal-statement","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-open-closed-set","position":2,"requirement":"optional","rationale":{"zh-Hant":"在固定拓撲空間中，開集屬於拓撲；閉集的補集為開集，同一集合可以同時開且閉。","en":"In a fixed topological space, open sets belong to the topology and closed sets have open complements; one set can be both open and closed."}},{"id":"mko-neighborhood","position":3,"requirement":"optional","rationale":{"zh-Hant":"集合 N 是點 x 的鄰域，當且僅當 N 包含某個含 x 的開集；N 本身不必開。","en":"A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open."}},{"id":"mko-topological-continuity","position":4,"requirement":"optional","rationale":{"zh-Hant":"拓撲映射連續，表示目標空間每個開集的逆像在來源空間仍是開集。","en":"A map is topologically continuous when every open set in its target has an open preimage in its source."}},{"id":"mko-computational-model","position":5,"requirement":"optional","rationale":{"zh-Hant":"有限狀態計算模型以狀態集、初始狀態集、步驟關係與停止狀態集描述可能執行。","en":"A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set."}},{"id":"mko-algorithm-correctness","position":6,"requirement":"optional","rationale":{"zh-Hant":"算法正確性分開部分正確性與終止性；總正確性需要兩者同時成立。","en":"Algorithm correctness separates partial correctness from termination; total correctness requires both."}},{"id":"mko-asymptotic-resource-bound","position":7,"requirement":"optional","rationale":{"zh-Hant":"漸近上界、下界與緊界必須相對輸入大小、成本模型及全尾段量化定義。","en":"Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail."}},{"id":"mko-lean-proof-term","position":8,"requirement":"optional","rationale":{"zh-Hant":"證明項是在固定形式環境與 context 中具有命題 P 型別，並可由 kernel 檢查的項 t。","en":"A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel."}}]},{"id":"path-pulse-foundations-20261020","title":{"zh-Hant":"應用數學基礎第一批：動力、微分方程、數值、最佳化、物理、生物與金融","en":"First Applied Foundations: Dynamics, ODEs, Numerical Error, Optimization, Physics, Biology, and Finance"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-ratio-proportion","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-discrete-dynamical-system","position":2,"requirement":"optional","rationale":{"zh-Hant":"離散時間動力系統以狀態空間、自映射與自然數時間描述逐步演化；有限軌道核對不自動給出一般穩定性。","en":"A discrete-time dynamical system uses a state space, a self-map and natural-number time to describe stepwise evolution; a finite orbit check does not establish general stability."}},{"id":"mko-ordinary-differential-equation","position":3,"requirement":"optional","rationale":{"zh-Hant":"常微分方程關聯未知單變數函數及其有限階導數；候選函數的殘差核對不等於存在唯一性定理。","en":"An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems."}},{"id":"mko-numerical-approximation-error","position":4,"requirement":"optional","rationale":{"zh-Hant":"絕對誤差是近似值與參照值之差的絕對值；相對誤差另要求參照值非零並除以其絕對值。","en":"Absolute error is the absolute difference between approximation and reference; relative error additionally requires a nonzero reference and divides by its absolute value."}},{"id":"mko-optimization-problem","position":5,"requirement":"optional","rationale":{"zh-Hant":"最佳化問題在明示可行集合上尋找使目標函數最小或最大的點；最優值與最優解集合需分開。","en":"An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects."}},{"id":"mko-dimensional-analysis","position":6,"requirement":"optional","rationale":{"zh-Hant":"量綱分析用基本量綱指數向量追蹤乘、除與冪，並核對方程兩側量綱是否一致。","en":"Dimensional analysis tracks products, quotients and powers through basic-dimension exponent vectors and checks agreement between equation sides."}},{"id":"mko-discrete-exponential-growth","position":7,"requirement":"optional","rationale":{"zh-Hant":"固定時間步與固定非負因子 r 的離散模型滿足 x_(n+1)=r x_n，因此 x_n=x_0 r^n。","en":"With a fixed time step and fixed nonnegative factor r, the discrete recurrence x_(n+1)=r*x_n gives x_n=x_0*r^n."}},{"id":"mko-compound-interest","position":8,"requirement":"optional","rationale":{"zh-Hant":"固定每期利率 i 且無額外現金流時，餘額滿足 B_(n+1)=(1+i)B_n，故 B_n=B_0(1+i)^n。","en":"With a fixed rate i per period and no additional cash flows, balances satisfy B_(n+1)=(1+i)B_n and hence B_n=B_0(1+i)^n."}}]},{"id":"path-pulse-foundations-20261021","title":{"zh-Hant":"應用數學基礎第二批：表徵、診斷、週期、初值、二分、線性規劃與受限成長","en":"Second Applied Foundations: Representation, Assessment Records, Periodicity, Initial Values, Bisection, Linear Programming and Capacity-Limited Growth"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-variable-expression","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-counterexample","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-discrete-dynamical-system","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-ordinary-differential-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-domain-codomain-range","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-numerical-approximation-error","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-optimization-problem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-system-linear-equations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-discrete-exponential-growth","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-mathematical-representation","position":2,"requirement":"optional","rationale":{"zh-Hant":"數學表徵把同一對象的文字規則、符號式、有限表格、序對與座標資料分開記錄，並以明示語義橋核對一致性。","en":"Mathematical representation separately records verbal rules, symbolic expressions, finite tables, ordered pairs and coordinate data for the same object, using explicit semantic bridges to check agreement."}},{"id":"mko-diagnostic-mathematical-assessment","position":3,"requirement":"optional","rationale":{"zh-Hant":"診斷性數學評量記錄把題目、期望、作答與明示判定規則綁在一起，只產生可追溯的規則符合狀態。","en":"A diagnostic mathematical assessment record binds an item, its expected result, a response and an explicit judgment rule, producing only a traceable rule-agreement status."}},{"id":"mko-orbit-fixed-periodic","position":4,"requirement":"optional","rationale":{"zh-Hant":"反覆迭代自映射形成軌道；固定點週期為1，週期點的最小正週期須和任意返回時間分開。","en":"Repeated iteration of a self-map forms an orbit. Fixed points have period 1; the least positive period of a periodic point must be distinguished from an arbitrary return time."}},{"id":"mko-initial-value-problem","position":5,"requirement":"optional","rationale":{"zh-Hant":"初值問題在微分方程外指定某點的函數值或導數值；候選核對與存在唯一性是不同責任。","en":"An initial-value problem adds function or derivative values at a specified point to a differential equation; candidate checks are distinct from existence and uniqueness."}},{"id":"mko-bisection-method","position":6,"requirement":"optional","rationale":{"zh-Hant":"二分法在已提供的端點與中點符號資料上取中點並保留符號相反或零值的半區，有限步寬度減半。","en":"Using supplied endpoint and midpoint signs, bisection takes a midpoint and retains the opposite-sign half or a detected zero; finite nonterminal bisections halve the interval width."}},{"id":"mko-linear-programming","position":7,"requirement":"optional","rationale":{"zh-Hant":"線性規劃在線性等式或不等式限制的可行域上最佳化線性目標函數；本輪題庫只驗完整有限候選網格。","en":"Linear programming optimizes a linear objective over a feasible domain defined by linear equalities or inequalities; this exercise slice checks only a complete explicit finite candidate grid."}},{"id":"mko-discrete-logistic-growth","position":8,"requirement":"optional","rationale":{"zh-Hant":"固定容量K與0≤r≤1時，本條目以x_(n+1)=x_n+r x_n(1-x_n/K)描述每步受限增量。","en":"For fixed capacity K and 0≤r≤1, this entry uses x_(n+1)=x_n+r*x_n*(1−x_n/K) to describe each capacity-limited discrete increment."}}]},{"id":"path-pulse-foundations-20261022","title":{"zh-Hant":"現代結構第一批：振子、折現、複數、向量空間、有限群、度量與分布","en":"First Modern Structures: Oscillators, Discounting, Complex Numbers, Vector Spaces, Finite Groups, Metrics and Distributions"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-ordinary-differential-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative-rules","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-compound-interest","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-fraction","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proof","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-harmonic-oscillator-model","position":2,"requirement":"optional","rationale":{"zh-Hant":"無阻尼線性簡諧振子以m x''+k x=0及m,k>0定義；本輪不宣稱三角函數閉式解。","en":"The undamped linear harmonic oscillator is defined by m*x″+k*x=0 with m,k>0; this slice asserts no trigonometric closed-form solution."}},{"id":"mko-present-value-discounting","position":3,"requirement":"optional","rationale":{"zh-Hant":"固定正累積因子與期數時，未來金額除以累積因子冪得到現值。","en":"For a fixed positive accumulation factor and period count, divide a future amount by the factor power to obtain present value."}},{"id":"mko-complex-number","position":4,"requirement":"optional","rationale":{"zh-Hant":"複數以實數有序對(a,b)表示a+bi，並以i²=-1定義加減乘法。","en":"A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1."}},{"id":"mko-real-vector-space","position":5,"requirement":"optional","rationale":{"zh-Hant":"實向量空間配備向量加法與實數純量乘法，滿足加法交換群及相容的純量作用公理。","en":"A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms."}},{"id":"mko-finite-group","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限群由有限集合與完整封閉Cayley表構成，並滿足結合律、唯一單位元及每元素逆元。","en":"A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element."}},{"id":"mko-metric-space","position":7,"requirement":"optional","rationale":{"zh-Hant":"度量空間由集合與距離函數組成，距離非負、只在相同點為零、對稱且滿足三角不等式。","en":"A metric space has a carrier set and a distance function that is nonnegative, zero exactly at equal points, symmetric and satisfies the triangle inequality."}},{"id":"mko-finite-discrete-probability-distribution","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限離散分布是在有限結果值上指定非負機率質量，全部質量精確和為一。","en":"A finite discrete distribution assigns nonnegative probability masses to finitely many outcome values, with exact total mass one."}}]},{"id":"path-pulse-foundations-20261023","title":{"zh-Hant":"現代結構第二批：複數不變量、生成、群同態、收斂、測度、過程與多變數函數","en":"Second Modern Structures: Complex Invariants, Spans, Group Homomorphisms, Convergence, Measure, Processes and Multivariable Functions"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-complex-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-system-linear-equations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-metric-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-discrete-probability-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-complex-modulus-conjugate","position":2,"requirement":"optional","rationale":{"zh-Hant":"複數共軛保留實部並反轉虛部；模平方為實部與虛部平方和。","en":"Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares."}},{"id":"mko-linear-combination-span","position":3,"requirement":"optional","rationale":{"zh-Hant":"線性組合是有限個向量的純量加權和；生成空間收集全部有限線性組合。","en":"A linear combination is a scalar-weighted sum of finitely many vectors; the span collects every finite linear combination."}},{"id":"mko-subgroup-homomorphism","position":4,"requirement":"optional","rationale":{"zh-Hant":"子群在繼承運算下自身成群；群同態保持乘法，kernel與image由完整映射表導出。","en":"A subgroup is a group under the inherited operation; a group homomorphism preserves multiplication, and its kernel and image follow from the complete map table."}},{"id":"mko-metric-convergence","position":5,"requirement":"optional","rationale":{"zh-Hant":"度量空間序列收斂到x，表示每個正epsilon終有所有n≥N落入x的epsilon球。","en":"A sequence in a metric space converges to x when, for every positive epsilon, eventually all n≥N lie in the epsilon-ball around x."}},{"id":"mko-sigma-algebra-measure-space","position":6,"requirement":"optional","rationale":{"zh-Hant":"σ-代數含空集並對補集與可數聯集封閉；測度非負、空集為零且可數可加。","en":"A sigma-algebra contains the empty set and is closed under complements and countable unions; a measure is nonnegative, zero on the empty set and countably additive."}},{"id":"mko-discrete-stochastic-process","position":7,"requirement":"optional","rationale":{"zh-Hant":"離散隨機過程是共同機率空間上的離散時間隨機變數族；邊際分布不決定聯合律。","en":"A discrete-time stochastic process is a time-indexed family of random variables on one probability space; marginal distributions do not determine its joint law."}},{"id":"mko-multivariable-real-function","position":8,"requirement":"optional","rationale":{"zh-Hant":"多變數實函數把明示實向量域中的輸入映到實數；座標角色與定義域是函數資料。","en":"A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data."}}]},{"id":"path-pulse-foundations-20261024","title":{"zh-Hant":"現代結構第三批：基底、可測性、Markov、偏導、PDE與凸幾何","en":"Third Modern Structures: Bases, Measurability, Markov Chains, Partials, PDEs and Convex Geometry"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-linear-combination-span","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-system-linear-equations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-determinant","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-matrix","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-multivariable-real-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-rate-of-change","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-ratio-proportion","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative-rules","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-independence-basis","position":2,"requirement":"optional","rationale":{"zh-Hant":"線性獨立排除非平凡零和關係；基底同時線性獨立並生成已宣告空間。","en":"Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space."}},{"id":"mko-measurable-function","position":3,"requirement":"optional","rationale":{"zh-Hant":"來源與目標各固定sigma代數時，函數可測表示每個目標可測集的逆像皆在來源sigma代數。","en":"With fixed source and target sigma-algebras, a function is measurable when the preimage of every target measurable set belongs to the source sigma-algebra."}},{"id":"mko-finite-markov-chain","position":4,"requirement":"optional","rationale":{"zh-Hant":"有限時間齊次Markov鏈以row-stochastic轉移核與需要時的初始律指定有限路徑機率。","en":"A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities."}},{"id":"mko-partial-derivative","position":5,"requirement":"optional","rationale":{"zh-Hant":"偏導數固定其他座標，只沿一個座標切片取h趨近零的單變數導數。","en":"A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero."}},{"id":"mko-finite-convex-combination","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限凸組合用非負精確權重加權有限非空向量列，且權重總和恰為一。","en":"A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one."}},{"id":"mko-partial-differential-equation","position":7,"requirement":"optional","rationale":{"zh-Hant":"偏微分方程關聯多變數未知函數與一個或多個偏導數；殘差與主部分類需固定算子convention。","en":"A partial differential equation relates a multivariable unknown function and one or more partial derivatives; residuals and principal-part classification require a fixed operator convention."}},{"id":"mko-convex-set","position":8,"requirement":"optional","rationale":{"zh-Hant":"集合凸表示任兩點及每個lambda∈[0,1]形成的整條線段仍在集合內。","en":"A set is convex when the entire segment formed by every pair of its points and every lambda in [0,1] remains in the set."}}]},{"id":"path-pulse-foundations-20261025","title":{"zh-Hant":"現代結構 Wave3 收束：邊界值、凸函數、凸最佳化與四項密度深化","en":"Modern Structures Wave3 Closure: Boundary Values, Convex Functions, Convex Optimization and Four Density Extensions"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-partial-differential-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-domain-codomain-range","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-convex-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-optimization-problem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-combination-span","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-system-linear-equations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-determinant","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-matrix","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-multivariable-real-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-rate-of-change","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-boundary-value-problem","position":2,"requirement":"optional","rationale":{"zh-Hant":"邊界值問題把定義域內的微分方程責任與邊界上的值或導數資料同時指定；候選通過不等於適定性。","en":"A boundary-value problem specifies both an interior differential equation and boundary values or derivative data; successful candidate checks do not establish general well-posedness."}},{"id":"mko-convex-function","position":2,"requirement":"optional","rationale":{"zh-Hant":"凸域上的函數為凸，若每個兩點凸組合處的函數值不超過相同權重的端點函數值。","en":"A function on a convex domain is convex when its value at every two-point convex combination is at most the same weighted combination of endpoint values."}},{"id":"mko-linear-independence-basis","position":2,"requirement":"optional","rationale":{"zh-Hant":"深化既有R²基底語義，以顯式重建與非平凡零和證書區分span、dependence與座標唯一性。","en":"Deepen the existing R² basis semantics through explicit reconstruction and nontrivial zero-sum certificates, separating span, dependence and coordinate uniqueness."}},{"id":"mko-measurable-function","position":2,"requirement":"optional","rationale":{"zh-Hant":"深化完整有限可測空間上的逆像補集／聯集律、常值映射及來源sigma代數細化。","en":"Deepen complement and union preimage laws, constant maps and source-sigma refinement on complete finite measurable spaces."}},{"id":"mko-finite-markov-chain","position":2,"requirement":"optional","rationale":{"zh-Hant":"深化精確有限時間齊次kernel的n步矩陣、端點joint與給定初態的條件路徑機率。","en":"Deepen exact n-step matrices, endpoint joint probabilities and paths conditional on a given initial state in a finite time-homogeneous kernel."}},{"id":"mko-partial-derivative","position":2,"requirement":"optional","rationale":{"zh-Hant":"深化二次多項式的x／y偏導多項式、固定x的y切片及符號差商消去後極限。","en":"Deepen the x/y partial polynomials of a quadratic, fixed-x y-slices and limits after symbolic difference-quotient cancellation."}},{"id":"mko-convex-optimization-problem","position":3,"requirement":"optional","rationale":{"zh-Hant":"凸最佳化問題在凸可行域上最小化凸目標；問題資格、可行性、求值與全域最優是不同責任。","en":"Convex optimization minimizes a convex objective on a convex feasible domain; problem qualification, feasibility, evaluation and global optimality are distinct obligations."}}]},{"id":"path-pulse-foundations-20261026","title":{"zh-Hant":"結構深度Wave4第一批：阿貝爾、範數、Cauchy、joint law、協方差、圖同構與等價關係","en":"Structural Depth Wave4, First Slice: Abelian Groups, Norms, Cauchy Sequences, Joint Laws, Covariance, Graph Isomorphisms and Equivalence Relations"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-metric-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-discrete-probability-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sample-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-bijective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-logical-connectives","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-abelian-group","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群若完整Cayley表對所有元素對都滿足交換律，則為有限阿貝爾群。","en":"A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements."}},{"id":"mko-real-normed-vector-space","position":3,"requirement":"optional","rationale":{"zh-Hant":"實向量空間配備正定、絕對齊次且滿足三角不等式的實值範數。","en":"A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality."}},{"id":"mko-cauchy-sequence","position":4,"requirement":"optional","rationale":{"zh-Hant":"度量空間序列為Cauchy，若每個正epsilon最終所有兩項彼此距離皆小於epsilon。","en":"A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart."}},{"id":"mko-finite-joint-distribution","position":5,"requirement":"optional","rationale":{"zh-Hant":"共同有限機率空間上多個隨機變數的值組與精確質量共同指定有限聯合分布。","en":"Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution."}},{"id":"mko-covariance","position":6,"requirement":"optional","rationale":{"zh-Hant":"同一有限聯合分布下協方差為E[XY]-E[X]E[Y]；零協方差通常不推出獨立。","en":"Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence."}},{"id":"mko-graph-isomorphism","position":7,"requirement":"optional","rationale":{"zh-Hant":"有限簡單圖同構是頂點雙射，對所有頂點對完整保持鄰接與非鄰接。","en":"An isomorphism of finite simple graphs is a vertex bijection preserving both adjacency and nonadjacency for every pair."}},{"id":"mko-equivalence-relation","position":8,"requirement":"optional","rationale":{"zh-Hant":"固定載體上的關係若完整滿足反身、對稱與傳遞，則為等價關係。","en":"A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity."}}]},{"id":"path-pulse-foundations-20261027","title":{"zh-Hant":"結構深度Wave4根節點：環、度量完備性、線性結構、獨立與圖路徑","en":"Structural Depth Wave4 Roots: Rings, Metric Completeness, Linear Structures, Independence and Graph Paths"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-abelian-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-normed-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-metric-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cauchy-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-metric-convergence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-combination-span","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-event","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-unital-ring","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限含一環在同一完整有限載體上具加法阿貝爾群、乘法結合律、左右分配律及與零不同的乘法單位元。","en":"On one complete finite carrier, this unital-ring protocol requires an additive abelian group, associative multiplication, both distributive laws and a multiplicative identity different from zero."}},{"id":"mko-norm-induced-metric","position":3,"requirement":"optional","rationale":{"zh-Hant":"同一實賦範向量空間上的範數以向量差的範數定義度量，度量公理由範數公理全稱推出。","en":"On the same real normed vector space, taking the norm of a vector difference defines a metric whose axioms follow universally from the norm axioms."}},{"id":"mko-complete-metric-space","position":4,"requirement":"optional","rationale":{"zh-Hant":"每個Cauchy序列都收斂到空間內某點的度量空間稱為完備。","en":"A metric space is complete when every Cauchy sequence converges to a point still belonging to that space."}},{"id":"mko-real-bilinear-form","position":5,"requirement":"optional","rationale":{"zh-Hant":"實向量空間上的雙線性形式分別對兩個輸入保持向量加法與實純量乘法。","en":"A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input."}},{"id":"mko-general-linear-map","position":6,"requirement":"optional","rationale":{"zh-Hant":"一般實向量空間間映射若對所有向量與實純量保持線性組合，則為線性映射。","en":"A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars."}},{"id":"mko-event-independence","position":7,"requirement":"optional","rationale":{"zh-Hant":"兩事件獨立以交集機率等於邊際機率乘積定義；此定義不依賴零機率條件式。","en":"Two events are independent when their intersection probability equals the product of their marginal probabilities; this definition does not rely on conditioning on a zero-probability event."}},{"id":"mko-graph-walk-simple-path","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限無向簡單圖的walk要求每對連續頂點相鄰；simple path另要求所有頂點互異。","en":"In a finite undirected simple graph, a walk requires each consecutive pair of vertices to be adjacent; a simple path additionally requires all vertices to be distinct."}}]},{"id":"path-pulse-foundations-20261028","title":{"zh-Hant":"結構深度Wave4第二層：交換環、理想、Banach、內積、算子、獨立與連通分量","en":"Structural Depth Wave4 Second Layer: Commutative Rings, Ideals, Banach and Inner-Product Spaces, Operators, Independence and Components"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-unital-ring","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-abelian-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-normed-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-norm-induced-metric","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-complete-metric-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-bilinear-form","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-general-linear-map","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-event-independence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-measurable-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph-walk-simple-path","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equivalence-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-commutative-ring","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限含一環若乘法對全部元素對交換，則為有限交換環；零因子仍可能存在。","en":"A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist."}},{"id":"mko-finite-two-sided-ideal","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限環子集若為加法子群且吸收父環的左右乘法，則為雙邊理想。","en":"A subset of a finite ring is a two-sided ideal when it is an additive subgroup and absorbs multiplication by parent-ring elements on both sides."}},{"id":"mko-real-banach-space","position":4,"requirement":"optional","rationale":{"zh-Hant":"在自身範數誘導度量下完備的實賦範向量空間稱為實Banach空間。","en":"A real normed vector space is a Banach space when it is complete in the metric induced by its own norm."}},{"id":"mko-real-inner-product-space","position":5,"requirement":"optional","rationale":{"zh-Hant":"對稱且正定的實雙線性形式構成實內積；半正定或抽樣為正不足。","en":"A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice."}},{"id":"mko-bounded-linear-operator","position":6,"requirement":"optional","rationale":{"zh-Hant":"賦範空間間線性映射若有單一非負常數統一控制所有輸出範數，則為有界線性算子。","en":"A linear map between normed spaces is bounded when one nonnegative constant uniformly controls all output norms relative to input norms."}},{"id":"mko-random-variable-independence","position":7,"requirement":"optional","rationale":{"zh-Hant":"同一有限joint law下，隨機變數獨立要求全部值對的joint mass分解為邊際質量乘積。","en":"Under one finite joint law, independence requires every value-pair mass to factor as the product of its marginal masses."}},{"id":"mko-connected-component","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限無向簡單圖中，由存在路徑所定義的頂點等價類恰為連通分量。","en":"In a finite undirected simple graph, connected components are exactly the vertex equivalence classes defined by the existence of a path."}}]},{"id":"path-pulse-foundations-20261029","title":{"zh-Hant":"結構深度Wave4第三層：整環、多項式、內積範數、條件期望、filtration、IID與等價類","en":"Structural Depth Wave4 Third Layer: Integral Domains, Polynomials, Inner Norms, Conditional Expectation, Filtrations, IID and Classes"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-commutative-ring","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-polynomial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-inner-product-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-measurable-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable-independence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equivalence-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-integral-domain","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限交換含一環若1不等於0且沒有非零零因子，則為有限整環。","en":"A finite commutative unital ring with 1 distinct from 0 is an integral domain when it has no nonzero zero divisors."}},{"id":"mko-polynomial-ring-over-finite-commutative-ring","position":3,"requirement":"optional","rationale":{"zh-Hant":"係數取自有限交換環的普通多項式以有限支撐係數序列表示，但所有任意有限次多項式合成的載體仍是無限。","en":"Ordinary polynomials over a finite commutative coefficient ring have finite-support coefficient sequences, but the collection over all finite degrees is still infinite."}},{"id":"mko-inner-product-induced-norm","position":4,"requirement":"optional","rationale":{"zh-Hant":"同一實內積以向量自內積的非負平方根定義範數；三角不等式依Cauchy–Schwarz全稱證書。","en":"On the same real inner-product space, the norm is the nonnegative square root of the self-inner-product; the triangle inequality uses the universal Cauchy–Schwarz certificate."}},{"id":"mko-conditional-expectation","position":5,"requirement":"optional","rationale":{"zh-Hant":"有限機率空間中，相對分割生成sigma代數的條件期望在每個正質量區塊取加權平均並保持區塊積分。","en":"Relative to a partition-generated sigma-algebra on a finite probability space, conditional expectation takes the weighted mean on each positive-mass block and preserves block integrals."}},{"id":"mko-finite-filtration","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限時間filtration是在同一機率空間上隨時間資訊遞增的sigma代數序列。","en":"A finite-time filtration is a sequence of sigma-algebras on the same probability space whose information increases with time."}},{"id":"mko-iid-sequence","position":7,"requirement":"optional","rationale":{"zh-Hant":"IID序列同時要求每個變數有相同分布，且每個有限子族滿足mutual independence。","en":"An IID sequence requires every variable to have the same distribution and every finite subfamily to be mutually independent."}},{"id":"mko-equivalence-class","position":8,"requirement":"optional","rationale":{"zh-Hant":"固定完整等價關係下，元素a的等價類收集全部與a相關的載體元素。","en":"For a fixed complete equivalence relation, the class of an element a consists of every carrier element related to a."}}]},{"id":"path-pulse-foundations-20261030","title":{"zh-Hant":"結構深度Wave4第四層：有限域、Hilbert、Martingale、商集、基數、CDF與依機率收斂","en":"Structural Depth Wave4 Fourth Layer: Finite Fields, Hilbert Spaces, Martingales, Quotients, Cardinality, CDFs and Convergence in Probability"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-integral-domain","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-unital-ring","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-abelian-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-inner-product-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-inner-product-induced-norm","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-norm-induced-metric","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-complete-metric-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-filtration","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-measurable-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equivalence-class","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equivalence-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-injective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-surjective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-bijective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-discrete-probability-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-metric-convergence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-field","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限交換含一環中每個非零元素皆有乘法逆元時構成有限域。","en":"A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse."}},{"id":"mko-real-hilbert-space","position":3,"requirement":"optional","rationale":{"zh-Hant":"在同一內積誘導範數與度量下完備的實內積空間為實Hilbert空間。","en":"A real inner-product space is Hilbert when it is complete in the norm and metric induced by that same inner product."}},{"id":"mko-martingale","position":4,"requirement":"optional","rationale":{"zh-Hant":"相對同一filtration適應且可積的有限過程，若每一步下一值條件期望等於現在值a.s.，則為Martingale。","en":"An integrable finite process adapted to the same filtration is a martingale when the next value's conditional expectation equals the current value almost surely at every step."}},{"id":"mko-quotient-set","position":5,"requirement":"optional","rationale":{"zh-Hant":"完整等價關係的全部互異等價類組成商集，並分割原載體。","en":"The distinct equivalence classes of a complete equivalence relation form its quotient set and partition the original carrier."}},{"id":"mko-cardinality-comparison","position":6,"requirement":"optional","rationale":{"zh-Hant":"明示source與target間的單射、滿射、雙射分別提供有限集合基數比較證書。","en":"An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate."}},{"id":"mko-cumulative-distribution-function","position":7,"requirement":"optional","rationale":{"zh-Hant":"實值隨機變數的CDF以F(t)=P(X小於等於t)定義，為單調右連續且有限離散跳躍等於點質量。","en":"For a real-valued random variable, F(t)=P(X≤t) is nondecreasing and right-continuous; in a finite discrete law, its jump is the point mass."}},{"id":"mko-convergence-in-probability","position":8,"requirement":"optional","rationale":{"zh-Hant":"X_n依機率收斂到X，若對每個epsilon>0，偏差超過epsilon的機率趨近0。","en":"X_n converges in probability to X when, for every epsilon>0, the probability of deviation greater than epsilon tends to zero."}}]},{"id":"path-pulse-foundations-20261031","title":{"zh-Hant":"Wave4收束：可數、二分圖、分布收斂、商環、著色與兩項深度密度","en":"Wave4 Closure: Countability, Bipartite Graphs, Distributional Convergence, Quotient Rings, Coloring and Two Depth Extensions"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-injective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-surjective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cumulative-distribution-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-unital-ring","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-two-sided-ideal","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equivalence-class","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quotient-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-integral-domain","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-abelian-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-filtration","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-measurable-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-at-most-countable-set","position":2,"requirement":"optional","rationale":{"zh-Hant":"集合若有限或存在到自然數的明示單射，則為至多可數；不等同可數無限。","en":"A set is at most countable if it is finite or has an explicit injection into the natural numbers; this is not the same as countably infinite."}},{"id":"mko-bipartite-graph","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限無向簡單圖若全體頂點可分為兩個互斥部分且每條邊跨部，則為二分圖。","en":"A finite undirected simple graph is bipartite when all vertices split into two disjoint parts and every edge crosses between the parts."}},{"id":"mko-convergence-in-distribution","position":2,"requirement":"optional","rationale":{"zh-Hant":"極限CDF每個連續點上F_n逐點收斂到F時，稱依分布收斂。","en":"Convergence in distribution means that F_n tends pointwise to F at every continuity point of the limiting CDF."}},{"id":"mko-finite-quotient-ring","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限環依proper雙邊理想的加法cosets形成商環，運算須與代表元選擇無關。","en":"Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice."}},{"id":"mko-graph-coloring","position":2,"requirement":"optional","rationale":{"zh-Hant":"proper k-coloring使用至多k種顏色並使每條邊兩端顏色不同。","en":"A proper k-coloring uses at most k colors and gives different colors to the endpoints of every edge."}},{"id":"mko-finite-field","position":2,"requirement":"optional","rationale":{"zh-Hant":"深化完整有限域表上的characteristic、非零乘法群、仿射方程與Frobenius恆等。","en":"Deepen characteristic, nonzero multiplicative groups, affine equations and Frobenius identities in the complete finite-field model."}},{"id":"mko-martingale","position":2,"requirement":"optional","rationale":{"zh-Hant":"深化有限filtration上martingale difference、affine stability、公平partial sum與常期望反例。","en":"Deepen martingale differences, affine stability, fair partial sums and constant-expectation counterexamples under finite filtrations."}}]},{"id":"path-pulse-foundations-20261101","title":{"zh-Hant":"Wave5根節點：子空間、拓撲覆蓋、幾乎必然收斂、匹配、φ函數與有限自動機","en":"Wave5 Roots: Subspaces, Topological Covers, Almost-Sure Convergence, Matching, Totient and Finite Automata"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-open-closed-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-convergence-in-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-prime-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-gcd-coprime","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-counting-principle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-computational-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-discrete-dynamical-system","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector-subspace","position":2,"requirement":"optional","rationale":{"zh-Hant":"實向量空間的非空子集若含零向量並對向量加法與每個實純量乘法封閉，則為向量子空間。","en":"A nonempty subset of a real vector space is a subspace when it contains zero and is closed under vector addition and multiplication by every real scalar."}},{"id":"mko-closure-and-interior","position":3,"requirement":"optional","rationale":{"zh-Hant":"集合閉包是所有閉超集的交，內部是所有開子集的聯；兩者相對於指定拓撲。","en":"The closure of a set is the intersection of all closed supersets, and its interior is the union of all open subsets, relative to the specified topology."}},{"id":"mko-open-cover-and-subcover","position":4,"requirement":"optional","rationale":{"zh-Hant":"開覆蓋由指定拓撲的開集組成且聯集包含目標集合；子覆蓋須取自原覆蓋並仍覆蓋同一目標。","en":"An open cover consists of open sets in the specified topology whose union contains the target; a subcover selects from that original cover and still covers the same target."}},{"id":"mko-almost-sure-convergence","position":5,"requirement":"optional","rationale":{"zh-Hant":"同一機率空間上的隨機變數序列若在零機率例外集外逐點收斂，則幾乎必然收斂。","en":"Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero."}},{"id":"mko-graph-matching","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限圖的matching是彼此不共享端點的邊子集；空matching合法。","en":"A matching in a finite graph is an edge subset whose edges share no endpoints; the empty matching is allowed."}},{"id":"mko-euler-totient-function","position":7,"requirement":"optional","rationale":{"zh-Hant":"Euler φ(n)計數一個完整剩餘代表系中與n互質的代表；約定φ(1)=1。","en":"Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1."}},{"id":"mko-finite-automaton","position":8,"requirement":"optional","rationale":{"zh-Hant":"DFA由有限狀態集、有限字母表、總轉移函數、唯一初始狀態與接受狀態子集組成。","en":"A DFA consists of finite states, a finite alphabet, a total transition function, one initial state and a subset of accepting states."}}]},{"id":"path-pulse-foundations-20261102","title":{"zh-Hant":"Wave5第二層：維度、核像、拓撲邊界、緊緻、Bayes與正規語言","en":"Wave5 Second Layer: Dimension, Kernels, Images, Topological Boundary, Compactness, Bayes and Regular Languages"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-independence-basis","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-general-linear-map","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector-subspace","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-closure-and-interior","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-open-cover-and-subcover","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-event","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-automaton","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-decision-problem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector-space-dimension","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限維向量空間的維度是任一有限基底的元素數；基底不同但基數不變。","en":"The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality."}},{"id":"mko-linear-map-kernel","position":3,"requirement":"optional","rationale":{"zh-Hant":"線性映射T的核是全部滿足T(x)=0的來源向量，並形成來源子空間。","en":"The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source."}},{"id":"mko-linear-map-image","position":4,"requirement":"optional","rationale":{"zh-Hant":"線性映射的像是所有輸出T(x)的集合；矩陣映射的像等於column span。","en":"The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span."}},{"id":"mko-topological-boundary","position":5,"requirement":"optional","rationale":{"zh-Hant":"集合A的拓撲邊界為closure(A)減去interior(A)，亦等於closure(A)交closure(complement A)。","en":"The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement."}},{"id":"mko-compact-topological-space","position":6,"requirement":"optional","rationale":{"zh-Hant":"拓撲空間緊緻，若每個開覆蓋都有有限子覆蓋；每個有限拓撲空間皆緊緻。","en":"A space is compact when every open cover has a finite subcover; every finite topological space is compact."}},{"id":"mko-bayes-theorem","position":7,"requirement":"optional","rationale":{"zh-Hant":"正分母條件下，Bayes定理以共同intersection與priors/likelihoods精確反轉條件機率。","en":"With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods."}},{"id":"mko-regular-language","position":8,"requirement":"optional","rationale":{"zh-Hant":"正規語言是存在一個完整DFA可辨識其全部字串的語言。","en":"A language is regular when a complete DFA recognizes all and exactly its words."}}]},{"id":"path-pulse-foundations-20261103","title":{"zh-Hant":"Wave5第三層：矩陣秩、函數列逐點收斂、級數、統計模型、連通、最大匹配與模反元素","en":"Wave5 Third Layer: Matrix Rank, Pointwise Function Limits, Series, Statistical Models, Connectedness, Maximum Matching and Modular Inverses"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-matrix","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-combination-span","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector-space-dimension","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-open-closed-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph-matching","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-optimization-problem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-gcd-coprime","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-modular-congruence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-matrix-rank","position":2,"requirement":"optional","rationale":{"zh-Hant":"矩陣秩是row/column space維度，可由精確rank-preserving消去的pivot數計算。","en":"Matrix rank is the dimension of the row or column space and can be computed as the number of pivots after exact rank-preserving elimination."}},{"id":"mko-pointwise-convergence-of-functions","position":3,"requirement":"optional","rationale":{"zh-Hant":"同一domain上的函數列逐點收斂，若每個固定x的值序列收斂至f(x)，且N可依x與epsilon改變。","en":"A function sequence on one domain converges pointwise when its value sequence converges at each fixed x; the witness N may depend on x and epsilon."}},{"id":"mko-infinite-series","position":4,"requirement":"optional","rationale":{"zh-Hant":"無窮級數以partial-sum sequence的極限定義；項趨零必要但不充分。","en":"An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence."}},{"id":"mko-parametric-statistical-model","position":5,"requirement":"optional","rationale":{"zh-Hant":"參數統計模型為每個theta指定同一有限可測sample space上的完整機率law P_theta。","en":"A parametric statistical model assigns each parameter theta a complete probability law P_theta on the same finite measurable sample space."}},{"id":"mko-connected-topological-space","position":6,"requirement":"optional","rationale":{"zh-Hant":"拓撲空間連通，若不存在由兩個不交非空開集組成的separation；等價於無非平凡clopen subset。","en":"A space is connected when it has no separation into two disjoint nonempty open sets, equivalently no nontrivial clopen subset."}},{"id":"mko-maximum-matching","position":7,"requirement":"optional","rationale":{"zh-Hant":"最大匹配是在完整有限圖全部合法matching中邊數最大者；可有多個且不必perfect。","en":"A maximum matching has the greatest edge count among all valid matchings of the complete finite graph; it may be nonunique and need not be perfect."}},{"id":"mko-modular-inverse","position":8,"requirement":"optional","rationale":{"zh-Hant":"n≥2時，a模n有乘法反元素恰當gcd(a,n)=1；反元素在模n意義唯一。","en":"For n≥2, a has a multiplicative inverse modulo n exactly when gcd(a,n)=1; the inverse is unique as a residue class modulo n."}}]},{"id":"path-pulse-foundations-20261104","title":{"zh-Hant":"Wave5第四層：rank-nullity、一致收斂、絕對收斂、估計量、CRT、生成樹與Lp收斂","en":"Wave5 Fourth Layer: Rank-Nullity, Uniform and Absolute Convergence, Estimators, CRT, Spanning Trees and Lp Limits"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-linear-map-kernel","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-map-image","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-matrix-rank","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector-space-dimension","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-pointwise-convergence-of-functions","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-metric-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-infinite-series","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sample-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-integer","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-modular-congruence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-gcd-coprime","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-tree","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-connected-component","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-rank-nullity-theorem","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限維線性映射T:V→W滿足dim V=dim ker T+rank T。","en":"A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T)."}},{"id":"mko-uniform-convergence-of-functions","position":3,"requirement":"optional","rationale":{"zh-Hant":"一致收斂要求同一N(epsilon)同時控制domain全部x。","en":"Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously."}},{"id":"mko-absolute-convergence-of-series","position":4,"requirement":"optional","rationale":{"zh-Hant":"sum |a_n|收斂時sum a_n絕對收斂；絕對收斂推出收斂，反向不一般成立。","en":"A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general."}},{"id":"mko-statistical-estimator","position":5,"requirement":"optional","rationale":{"zh-Hant":"估計量是固定且不依未知theta的可測total statistic；觀察後數值是estimate。","en":"An estimator is a fixed measurable total statistic independent of the unknown parameter; its observed numerical value is an estimate."}},{"id":"mko-chinese-remainder-theorem","position":6,"requirement":"optional","rationale":{"zh-Hant":"正兩兩互質模數的同餘系統在模乘積意義下存在唯一解。","en":"For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product."}},{"id":"mko-spanning-tree","position":7,"requirement":"optional","rationale":{"zh-Hant":"生成樹是包含父圖全部頂點的連通無圈子圖。","en":"A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph."}},{"id":"mko-lp-convergence","position":8,"requirement":"optional","rationale":{"zh-Hant":"共同機率空間上固定整數p≥1時，E|X_n−X|^p→0稱Lp收斂。","en":"On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero."}}]},{"id":"path-pulse-foundations-20261105","title":{"zh-Hant":"Wave5第五層：直和、對偶、Euclidean演算法、Euler trail、同胚、抽樣分布與弱大數律","en":"Wave5 Fifth Layer: Direct Sums, Duals, Euclid, Euler Trails, Homeomorphisms, Sampling Laws and the WLLN"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-vector-subspace","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-independence-basis","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-general-linear-map","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-gcd-coprime","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-divisibility","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-algorithm-correctness","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph-walk-simple-path","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-connected-component","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-bijective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-inverse-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-statistical-estimator","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-discrete-probability-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-iid-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-convergence-in-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-direct-sum","position":2,"requirement":"optional","rationale":{"zh-Hant":"二個子空間之和若每個向量有唯一分解便是內直和；要宣稱覆蓋母空間還須另證生成。","en":"The sum of two subspaces is an internal direct sum when every vector in that sum has a unique decomposition; covering an ambient space is a separate spanning obligation."}},{"id":"mko-dual-vector-space","position":3,"requirement":"optional","rationale":{"zh-Hant":"實向量空間的對偶空間由所有實值線性泛函組成；座標covector表示依賴基底。","en":"The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis."}},{"id":"mko-euclidean-algorithm","position":4,"requirement":"optional","rationale":{"zh-Hant":"Euclidean演算法以精確帶餘除法、gcd invariant與嚴格餘數下降求最大公因數。","en":"The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders."}},{"id":"mko-eulerian-trail","position":5,"requirement":"optional","rationale":{"zh-Hant":"Euler trail恰用每條邊一次；非零邊支撐連通且奇度數為0或2時分別有circuit或open trail。","en":"An Euler trail uses each edge once; on connected nonzero-edge support, zero odd-degree vertices give a circuit and two give an open Euler trail."}},{"id":"mko-homeomorphism","position":6,"requirement":"optional","rationale":{"zh-Hant":"同胚是雙射、連續且逆映射亦連續的映射；一般continuous bijection仍可能不是同胚。","en":"A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism."}},{"id":"mko-sampling-distribution","position":7,"requirement":"optional","rationale":{"zh-Hant":"固定theta時，估計量把樣本分布pushforward成統計量的抽樣分布；全部theta形成一族laws。","en":"At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws."}},{"id":"mko-weak-law-large-numbers","position":8,"requirement":"optional","rationale":{"zh-Hant":"IID且具有有限variance時，Chebyshev界給樣本平均依機率收斂到共同mean。","en":"For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean."}}]},{"id":"path-pulse-foundations-20261106","title":{"zh-Hant":"Wave5封閉層：色數、Clique與獨立集、估計量偏誤與變異數、Fermat小定理、Diophantine方程與商空間","en":"Wave5 Closure: Chromatic Minima, Cliques, Estimator Bias and Variance, Fermat, Diophantine Equations and Quotients"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-graph-coloring","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-optimization-problem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sampling-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-prime-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-modular-congruence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-integer","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-gcd-coprime","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-euclidean-algorithm","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector-subspace","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quotient-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equivalence-class","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector-space-dimension","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-chromatic-number","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限簡單圖的色數是proper頂點著色所需顏色數的最小值，需同時具備上界與下界證書。","en":"The chromatic number of a finite simple graph is the minimum number of colors in a proper vertex coloring; equality needs both upper and lower certificates."}},{"id":"mko-clique-and-independent-set","position":3,"requirement":"optional","rationale":{"zh-Hant":"Clique內每對不同頂點相鄰，獨立集內每對不同頂點不相鄰；最大與極大不可混同。","en":"Every pair of distinct vertices of a clique is adjacent; every pair of an independent set is nonadjacent. Inclusion-maximal and maximum-cardinality are different."}},{"id":"mko-estimator-bias","position":4,"requirement":"optional","rationale":{"zh-Hant":"估計量在theta的偏誤是其theta抽樣分布期望減去目標參數；無偏要求對所有theta成立。","en":"Bias at theta is the estimator expectation under its theta sampling law minus the target parameter; unbiasedness requires zero bias at every theta."}},{"id":"mko-estimator-variance","position":5,"requirement":"optional","rationale":{"zh-Hant":"估計量變異數逐theta衡量sampling law相對自身期望的二次離差，不等同偏誤或MSE。","en":"At each theta, estimator variance is the second central moment of its sampling law around its own mean; it is not bias or MSE by itself."}},{"id":"mko-fermat-little-theorem","position":6,"requirement":"optional","rationale":{"zh-Hant":"p為質數時a^p同餘a；若p不整除a則可化為a^(p-1)同餘1，逆命題不成立。","en":"For a prime p, a^p is congruent to a modulo p; if p does not divide a, a^(p−1) is congruent to1 modulo p. The converse fails."}},{"id":"mko-linear-diophantine-equation","position":7,"requirement":"optional","rationale":{"zh-Hant":"非退化整數方程ax+by=c有解恰當gcd(a,b)整除c，全部解由一個整數參數給出。","en":"A nondegenerate integer equation a*x+b*y=c has a solution exactly when gcd(a,b) divides c; all solutions have one integer parameter."}},{"id":"mko-quotient-vector-space","position":8,"requirement":"optional","rationale":{"zh-Hant":"V依子空間W的cosets形成商向量空間，運算須與代表元選擇無關。","en":"Cosets of a subspace W form the quotient vector space V/W, whose operations must be independent of representative choices."}}]},{"id":"path-pulse-foundations-20261107","title":{"zh-Hant":"Wave6根層：有限群作用、上下確界、拓撲基底、Markov不等式、鴿籠原理、Euler定理與子數列","en":"Wave6 Roots: Group Actions, Real Bounds, Topological Bases, Markov, Pigeonholes, Euler and Subsequences"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-open-closed-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-random-variable","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-counting-principle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-euler-totient-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-gcd-coprime","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-modular-congruence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-injective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-action","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群在有限集合上的作用是滿足單位與乘法相容律的total action table。","en":"An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws."}},{"id":"mko-supremum-and-infimum","position":3,"requirement":"optional","rationale":{"zh-Hant":"非空且有界實數集合的supremum是最小上界，infimum是最大下界，兩者未必屬於集合。","en":"For a nonempty appropriately bounded real set, the supremum is its least upper bound and the infimum its greatest lower bound; either need not belong to the set."}},{"id":"mko-topological-basis","position":4,"requirement":"optional","rationale":{"zh-Hant":"拓撲基底以覆蓋carrier與逐點局部交細化兩條公理生成拓撲。","en":"A topological basis generates a topology through carrier coverage and pointwise local intersection refinement."}},{"id":"mko-markov-inequality","position":5,"requirement":"optional","rationale":{"zh-Hant":"非負且有限期望的隨機變數在正閾值上的尾機率不超過E[X]/a。","en":"For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold."}},{"id":"mko-pigeonhole-principle","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限total map的domain比codomain大時必有兩個不同元素共享像。","en":"A total map from a larger finite domain into a smaller finite codomain sends two distinct source elements to the same value."}},{"id":"mko-euler-theorem-number-theory","position":7,"requirement":"optional","rationale":{"zh-Hant":"n>=2且gcd(a,n)=1時，a的phi(n)次方模n同餘於1。","en":"For n≥2 and gcd(a,n)=1, a^phi(n) is congruent to1 modulo n."}},{"id":"mko-subsequence","position":8,"requirement":"optional","rationale":{"zh-Hant":"子數列由嚴格遞增的自然數索引映射選取原數列項並保留次序。","en":"A subsequence selects terms by a strictly increasing natural-index map and preserves their original order."}}]},{"id":"path-pulse-foundations-20261108","title":{"zh-Hant":"Wave6第二層：群軌道、單調收斂、子空間拓撲、Chebyshev、容斥、唯一質因數分解與有向圖","en":"Wave6 Second Layer: Orbits, Monotone Limits, Subspace Topology, Chebyshev, Inclusion-Exclusion, Prime Factorization and Digraphs"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group-action","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equivalence-class","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-basis","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-markov-inequality","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-counting-principle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-mathematical-induction","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-prime-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-divisibility","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-euclidean-algorithm","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-gcd-coprime","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-orbit","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群作用下，一點被全部群元素送到的像集合形成軌道，全部軌道分割carrier。","en":"The images of one point under all group elements form its orbit; all orbits partition the point carrier."}},{"id":"mko-monotone-sequence-convergence-theorem","position":3,"requirement":"optional","rationale":{"zh-Hant":"非遞減且有上界的實數列收斂至其項集合supremum；非遞增版本對偶收斂至infimum。","en":"A nondecreasing real sequence bounded above converges to the supremum of its terms; the nonincreasing bounded-below version converges to the infimum."}},{"id":"mko-subspace-topology","position":4,"requirement":"optional","rationale":{"zh-Hant":"Y⊆X的相對拓撲由全部母空間open sets與Y的交集組成。","en":"For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y."}},{"id":"mko-chebyshev-inequality","position":5,"requirement":"optional","rationale":{"zh-Hant":"有限均值與變異數的隨機變數偏離均值至少epsilon的機率不超過variance/epsilon²。","en":"For a variable with finite mean and variance, the probability of absolute deviation at least epsilon is at most variance/epsilon^2."}},{"id":"mko-inclusion-exclusion-principle","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限indexed set family的union cardinality是全部非空index subsets交集大小的交替和。","en":"The union cardinality of a finite indexed set family is the alternating sum of intersection sizes over every nonempty index subset."}},{"id":"mko-unique-prime-factorization","position":7,"requirement":"optional","rationale":{"zh-Hant":"每個n>1有prime factorization且prime multiset唯一，只允許因數次序差異。","en":"Every positive integer n>1 has a prime factorization whose prime multiset is unique up to factor order."}},{"id":"mko-finite-directed-graph","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限loop-free simple digraph由有限vertices與不含loop、無平行同向重邊的ordered-pair edge set組成。","en":"Under the loop-free simple convention, a finite digraph has a finite vertex carrier and an ordered-pair edge set without loops or duplicate same-direction edges."}}]},{"id":"path-pulse-foundations-20261109","title":{"zh-Hant":"Wave6第三層：Stabilizer、Bolzano-Weierstrass、緊緻子集、全期望、二項式、Möbius與CFG","en":"Wave6 Third Layer: Stabilizers, Bolzano-Weierstrass, Compact Subsets, Total Expectation, Binomial Theorem, Mobius Function and CFGs"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group-action","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subsequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-compact-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subspace-topology","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-event","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-permutation-combination","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-mathematical-induction","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-polynomial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-prime-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-unique-prime-factorization","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-computational-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-stabilizer","position":2,"requirement":"optional","rationale":{"zh-Hant":"指定點的stabilizer是有限群作用中固定該點的全部群元素，並形成子群。","en":"The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup."}},{"id":"mko-bolzano-weierstrass-theorem-real","position":3,"requirement":"optional","rationale":{"zh-Hant":"每個整體有界的實數列都有一個以嚴格遞增無限索引取得的收斂子數列。","en":"Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map."}},{"id":"mko-compact-subset","position":4,"requirement":"optional","rationale":{"zh-Hant":"K⊆X在subspace topology下compact，等價地每個relative-open cover有有限subcover。","en":"A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover."}},{"id":"mko-law-of-total-expectation","position":5,"requirement":"optional","rationale":{"zh-Hant":"同一有限機率空間上，完整partition各cell的conditional mean按cell probability加權後等於E[X]。","en":"On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X]."}},{"id":"mko-binomial-theorem","position":6,"requirement":"optional","rationale":{"zh-Hant":"交換乘法環境中，natural exponent n的二項式冪以binomial coefficients完整展開。","en":"With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients."}},{"id":"mko-mobius-function","position":7,"requirement":"optional","rationale":{"zh-Hant":"Möbius函數在positive integers上由完整prime multiplicities決定：mu(1)=1、含平方質因數為0、squarefree取parity sign。","en":"On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign."}},{"id":"mko-context-free-grammar","position":8,"requirement":"optional","rationale":{"zh-Hant":"CFG由有限且不交的nonterminals、terminals、start與單一nonterminal左側的finite productions生成語言。","en":"A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side."}}]},{"id":"path-pulse-foundations-20261110","title":{"zh-Hant":"Wave6第四層：Orbit-Stabilizer、MVT、緊緻像、全變異數、遞迴、Möbius反演與PDA","en":"Wave6 Fourth Layer: Orbit-Stabilizer, MVT, Compact Images, Total Variance, Recurrences, Mobius Inversion and PDAs"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group-orbit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-stabilizer","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-rate-of-change","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-compact-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-law-of-total-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-mobius-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-divisibility","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-counting-principle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-computational-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-automaton","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-context-free-grammar","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-orbit-stabilizer-theorem","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整有限群作用中，同一點的orbit大小乘以stabilizer大小等於群階。","en":"For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order."}},{"id":"mko-mean-value-theorem","position":3,"requirement":"optional","rationale":{"zh-Hant":"若a<b，f在[a,b]連續且在(a,b)可微，則某個內點c的導數等於端點割線斜率。","en":"If a<b, f is continuous on [a,b] and differentiable on (a,b), some interior c has derivative equal to the endpoint secant slope."}},{"id":"mko-continuous-image-of-compact","position":4,"requirement":"optional","rationale":{"zh-Hant":"若K在X中compact且f:X→Y連續，則image f(K)在Y中compact。","en":"If K is compact in X and f:X→Y is continuous, then f(K) is compact in Y."}},{"id":"mko-law-of-total-variance","position":5,"requirement":"optional","rationale":{"zh-Hant":"同一有限law與完整partition生成的G下，總變異數等於cell內變異數期望加cell均值變異數。","en":"For the same finite law and partition-generated G, total variance is the expected within-cell variance plus the variance of the cell means."}},{"id":"mko-recurrence-relation","position":6,"requirement":"optional","rationale":{"zh-Hant":"order-r recurrence以明示initial values與total single-valued update rule逐項決定sequence。","en":"An order-r recurrence uses explicit initial values and a total single-valued state update to determine successive sequence terms."}},{"id":"mko-mobius-inversion-finite-divisors","position":7,"requirement":"optional","rationale":{"zh-Hant":"若F(n)是所有正因數d上的f(d)總和，則以Möbius權重可精確反演f(n)。","en":"If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n)."}},{"id":"mko-pushdown-automaton","position":8,"requirement":"optional","rationale":{"zh-Hant":"nondeterministic PDA以有限控制、input與stack transitions運算，本輪固定讀完input且到final state接受。","en":"A nondeterministic PDA uses finite control, input and stack transitions; this slice accepts only after reading the whole input in a final state."}}]},{"id":"path-pulse-foundations-20261111","title":{"zh-Hant":"Wave6第五層：Lagrange、高階導數、Hausdorff、連續CDF、Catalan、二次剩餘與NFA","en":"Wave6 Fifth Layer: Lagrange, Higher Derivatives, Hausdorff Spaces, Continuous CDFs, Catalan Numbers, Quadratic Residues and NFAs"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-divisibility","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quotient-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative-rules","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-mathematical-induction","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-neighborhood","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cumulative-distribution-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-recurrence-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-binomial-theorem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-counting-principle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-modular-congruence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-prime-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-automaton","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-lagrange-theorem-finite-groups","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群G的子群H以等大左陪集分割G，因此|G|=[G:H]|H|且|H|整除|G|。","en":"Equal-size left cosets of a subgroup H partition the finite group G, so |G|=[G:H]*|H| and |H| divides |G|."}},{"id":"mko-higher-order-derivative","position":3,"requirement":"optional","rationale":{"zh-Hant":"以f^(0)=f及f^(n+1)=(f^(n))'遞迴定義每個存在的有限階導數。","en":"Define each existing finite derivative recursively by f^(0)=f and f^(n+1)=(f^(n))'."}},{"id":"mko-hausdorff-space","position":4,"requirement":"optional","rationale":{"zh-Hant":"拓撲空間X若每對不同點都有不交open neighborhoods即為Hausdorff。","en":"A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods."}},{"id":"mko-continuous-real-distribution-function","position":5,"requirement":"optional","rationale":{"zh-Hant":"本節點以CDF本身連續定義continuous distribution；CDF仍須nondecreasing、right-continuous且兩端極限0與1。","en":"Here a continuous distribution means that its CDF is continuous; the CDF must still be nondecreasing, right-continuous and have endpoint limits0 and1."}},{"id":"mko-catalan-numbers","position":6,"requirement":"optional","rationale":{"zh-Hant":"Catalan數採C_0=1，並以convolution recurrence或兩個exact closed forms定義同一zero-based sequence。","en":"With C_0=1, the convolution recurrence and two exact closed forms define the same zero-based Catalan sequence."}},{"id":"mko-quadratic-residue","position":7,"requirement":"optional","rationale":{"zh-Hant":"本節點採odd prime p上的nonzero convention：a mod p非零且等於某個nonzero square時為quadratic residue。","en":"Under this odd-prime nonzero convention, a class is a quadratic residue when its normalized nonzero value is a square of a nonzero class modulo p."}},{"id":"mko-nondeterministic-finite-automaton","position":8,"requirement":"optional","rationale":{"zh-Hant":"epsilon-NFA以狀態、input symbol或epsilon對應可能後繼集合，存在一條full-input final path即接受。","en":"An epsilon-NFA relates a state and input symbol or epsilon to possible successor states; one full-input final path suffices for acceptance."}}]},{"id":"path-pulse-foundations-20261112","title":{"zh-Hant":"Wave6第六層：Cauchy、Taylor、緊緻閉集、瑕積分、收斂半徑、Legendre與子集合構造","en":"Wave6 Sixth Layer: Cauchy, Taylor, Compact Closed Subsets, Improper Integrals, Convergence Radii, Legendre and Subset Construction"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-lagrange-theorem-finite-groups","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-prime-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-action","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative-rules","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-polynomial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-factorial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-higher-order-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-hausdorff-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-compact-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-open-closed-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-definite-integral","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-convergence-of-series","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-infinite-series","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quadratic-residue","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-nondeterministic-finite-automaton","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-automaton","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cauchy-theorem-finite-groups","position":2,"requirement":"optional","rationale":{"zh-Hant":"若質數p整除有限群G的階，則G含有一個exact order為p的元素。","en":"If a prime p divides the order of a finite group G, G contains an element of exact order p."}},{"id":"mko-taylor-polynomial","position":3,"requirement":"optional","rationale":{"zh-Hant":"在中心a由f的零階到n階導數值依factorial縮放組成n階Taylor多項式。","en":"At center a, derivative values of orders0 through n, scaled by their factorials, form the order-n Taylor polynomial."}},{"id":"mko-compact-subset-closed-in-hausdorff","position":4,"requirement":"optional","rationale":{"zh-Hant":"若K是Hausdorff空間X的緊緻子集，則K在X中為閉集。","en":"If K is compact in the Hausdorff space X, then K is closed in X."}},{"id":"mko-improper-integral","position":5,"requirement":"optional","rationale":{"zh-Hant":"無界區間或含奇點的積分以適當有限區間積分的one-sided limits定義，且每個必要分段須獨立收斂。","en":"Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently."}},{"id":"mko-power-series-radius-of-convergence","position":6,"requirement":"optional","rationale":{"zh-Hant":"冪級數在中心c周圍具有內部絕對收斂、外部發散的半徑R，邊界點須逐案判定。","en":"Around its center c, a power series has absolute convergence inside radius R and divergence outside; endpoints require separate decisions."}},{"id":"mko-legendre-symbol","position":7,"requirement":"optional","rationale":{"zh-Hant":"對奇質數p，Legendre符號以0、1、-1分別編碼zero class、nonzero quadratic residue與nonresidue。","en":"For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively."}},{"id":"mko-subset-construction-nfa-to-dfa","position":8,"requirement":"optional","rationale":{"zh-Hant":"以epsilon-NFA狀態子集合為DFA狀態，start、transition與acceptance按epsilon closure定義即可保持語言。","en":"Use epsilon-NFA state subsets as DFA states, defining start, transitions and acceptance with epsilon closure to preserve the language."}}]},{"id":"path-pulse-foundations-20261113","title":{"zh-Hant":"Wave6第七層：正規子群、實指數、解析OGF、Euler判準、Turing判定、梯度與流網路","en":"Wave6 Seventh Layer: Normal Subgroups, Real Exponential, Analytic OGFs, Euler Criterion, Turing Decisions, Gradients and Flow Networks"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-infinite-series","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-series-radius-of-convergence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-factorial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-convergence-of-series","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-binomial-theorem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-recurrence-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-euler-theorem-number-theory","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quadratic-residue","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-legendre-symbol","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-computational-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-decision-problem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-automaton","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-algorithm-correctness","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-partial-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-multivariable-real-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-directed-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-programming","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-normal-subgroup","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群G的子群H若對每個g皆滿足gHg^{-1}=H，則H為正規子群；等價地每個左陪集等於相應右陪集。","en":"A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset."}},{"id":"mko-real-exponential-function","position":3,"requirement":"optional","rationale":{"zh-Hant":"實指數函數以對每個實數絕對收斂的冪級數E(x)=sum_{n≥0}x^n/n!定義，並滿足E(x+y)=E(x)E(y)與E(x)>0。","en":"Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0."}},{"id":"mko-analytic-ordinary-generating-function","position":4,"requirement":"optional","rationale":{"zh-Hant":"數列(a_n)在明示收斂域內以ordinary power series A(x)=sum_{n≥0}a_nx^n解析編碼。","en":"An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain."}},{"id":"mko-euler-criterion-quadratic-residue","position":5,"requirement":"optional","rationale":{"zh-Hant":"若p為奇質數且p不整除a，則a^((p-1)/2)模p等於Legendre符號(a/p)的模p代表。","en":"For odd prime p and a not divisible by p, a^((p−1)/2) modulo p is the modulo-p representative of the Legendre symbol (a/p)."}},{"id":"mko-turing-machine-decision-model","position":6,"requirement":"optional","rationale":{"zh-Hant":"固定deterministic single-tape Turing機語義；decider必須對每個合法輸入停機並給正確accept/reject答案。","en":"Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer."}},{"id":"mko-gradient-vector","position":7,"requirement":"optional","rationale":{"zh-Hant":"在固定standard Euclidean coordinates中，可微實值函數的一階偏導依座標排列成梯度向量。","en":"In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector."}},{"id":"mko-finite-flow-network","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限有向網路中的非負流須逐邊不超過capacity，並在source與sink以外每個頂點滿足流量守恆。","en":"A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink."}}]},{"id":"path-pulse-foundations-20261114","title":{"zh-Hant":"Wave6第八層：kernel、商群、共軛類、標準常態、MSE、LP標準形與頂點覆蓋","en":"Wave6 Eighth Layer: Kernels, Quotient Groups, Conjugacy, Standard Normal, MSE, LP Standard Form and Vertex Covers"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-normal-subgroup","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quotient-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-action","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-orbit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-continuous-real-distribution-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-improper-integral","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-exponential-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-estimator-bias","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-estimator-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sampling-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-programming","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-matrix","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-system-linear-equations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-homomorphism-kernel","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群同態f:G→Q映到目標identity的完整原像是G的正規子群，且同態fibers恰為kernel的陪集。","en":"For a finite group homomorphism f:G→Q, the full preimage of the target identity is a normal subgroup of G, and nonempty fibers are its cosets."}},{"id":"mko-finite-quotient-group","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限群G依正規子群N的陪集形成商群，元素是cosets且乘法(gN)(hN)=(gh)N必須與代表元無關。","en":"Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives."}},{"id":"mko-finite-group-conjugacy-class","position":4,"requirement":"optional","rationale":{"zh-Hant":"有限群G以conjugation g·x=gxg^{-1}作用於自身，其orbits形成共軛類並分割G。","en":"A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group."}},{"id":"mko-standard-normal-distribution","position":5,"requirement":"optional","rationale":{"zh-Hant":"標準常態Z以Gaussian density phi(x)=(2pi)^(-1/2)exp(-x^2/2)與其連續CDF Phi定義，並具有mean 0、variance 1。","en":"The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1."}},{"id":"mko-estimator-mean-squared-error","position":6,"requirement":"optional","rationale":{"zh-Hant":"scalar parameter theta下，估計量T的MSE_theta(T)=E_theta[(T-theta)^2]由完整sampling distribution逐theta定義。","en":"For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately."}},{"id":"mko-linear-program-standard-form","position":7,"requirement":"optional","rationale":{"zh-Hant":"本節點固定唯一primal convention：maximize c^T x，subject to Ax≤b與x≥0。","en":"This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0."}},{"id":"mko-vertex-cover","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限simple undirected graph的頂點子集C若與每條edge至少一端相交，則C為vertex cover。","en":"A vertex subset C covers a finite simple undirected graph if every edge has at least one endpoint in C."}}]},{"id":"path-pulse-foundations-20261115","title":{"zh-Hant":"Wave6第九層：第一同構、群中心、bias–variance、LP對偶、max-flow/min-cut、最小頂點覆蓋與many-one reduction","en":"Wave6 Ninth Layer: First Isomorphism, Centers, Bias-Variance, LP Duality, Max-Flow Min-Cut, Minimum Covers and Many-One Reduction"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group-homomorphism-kernel","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-quotient-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-bijective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-group-conjugacy-class","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-normal-subgroup","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-estimator-mean-squared-error","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-estimator-bias","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-estimator-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-program-standard-form","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-optimization-problem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-flow-network","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-programming","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vertex-cover","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-decision-problem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-composition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-turing-machine-decision-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-first-isomorphism-theorem-finite-groups","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群同態f:G→Q由kernel cosets誘導出G/ker f到im f的群同構，而非自動同構到整個codomain。","en":"A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain."}},{"id":"mko-finite-group-center","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限群G的中心Z(G)由與每個群元素交換的全部元素構成，且是G的正規子群。","en":"The center consists of all elements commuting with every group element and is a normal subgroup."}},{"id":"mko-bias-variance-decomposition","position":4,"requirement":"optional","rationale":{"zh-Hant":"逐參數theta，scalar estimator的MSE等於其variance加上bias平方。","en":"For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias."}},{"id":"mko-linear-program-duality","position":5,"requirement":"optional","rationale":{"zh-Hant":"對固定primal max c^Tx、Ax≤b、x≥0，其dual為min b^Ty、A^Ty≥c、y≥0；feasible pair滿足弱對偶。","en":"For primal maximize c^T*x with A*x≤b,x≥0, the dual minimizes b^T*y with A^T*y≥c,y≥0; feasible pairs satisfy weak duality."}},{"id":"mko-max-flow-min-cut-theorem","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限容量網路的最大feasible flow value等於全部s–t cuts中的最小capacity。","en":"In a finite capacitated network the largest feasible flow value equals the least capacity among all source–sink cuts."}},{"id":"mko-minimum-vertex-cover","position":7,"requirement":"optional","rationale":{"zh-Hant":"有限simple graph中，cardinality在全部vertex covers裡最小的候選是minimum vertex cover。","en":"A minimum vertex cover has least cardinality among all vertex covers of the finite simple graph."}},{"id":"mko-many-one-reduction","position":8,"requirement":"optional","rationale":{"zh-Hant":"A≤mB表示存在total computable f，使每個x皆滿足x∈A恰當f(x)∈B。","en":"A≤mB means a total computable map f satisfies x∈A exactly when f(x)∈B for every x."}}]},{"id":"path-pulse-foundations-20261116","title":{"zh-Hant":"Wave6第十層：contraction、IVT、Taylor remainder、finite MLE、sufficiency、toposort與multiplicative order","en":"Wave6 Tenth Layer: Contraction, IVT, Taylor Remainders, Finite MLE, Sufficiency, Topological Orders and Modular Order"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-orbit-fixed-periodic","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-complete-metric-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-metric-convergence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-infinite-series","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-taylor-polynomial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-higher-order-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-mean-value-theorem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-statistical-estimator","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-optimization-problem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sampling-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-conditional-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-directed-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-algorithm-correctness","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-mathematical-induction","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-modular-inverse","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-gcd-coprime","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-divisibility","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-euler-theorem-number-theory","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-contraction-mapping-theorem","position":2,"requirement":"optional","rationale":{"zh-Hant":"非空完備度量空間上的self-map若以固定q<1壓縮所有距離，則有唯一fixed point且Picard orbit收斂。","en":"A self-map on a nonempty complete metric space with one global contraction constant q<1 has a unique fixed point and convergent Picard iterates."}},{"id":"mko-intermediate-value-theorem","position":3,"requirement":"optional","rationale":{"zh-Hant":"連續實函數在閉區間端點值之間取到每個中間值。","en":"A continuous real function on a closed interval attains each value between its endpoint values."}},{"id":"mko-taylor-remainder-theorem","position":4,"requirement":"optional","rationale":{"zh-Hant":"足夠高階可微函數的Taylor approximation error由Lagrange remainder與derivative bound控制。","en":"Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error."}},{"id":"mko-finite-maximum-likelihood-estimator","position":5,"requirement":"optional","rationale":{"zh-Hant":"有限參數模型中，MLE選取使觀察likelihood最大的參數，tie需固定且與theta無關。","en":"A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter."}},{"id":"mko-finite-sufficient-statistic","position":6,"requirement":"optional","rationale":{"zh-Hant":"finite model中，給定statistic value後sample conditional law不依theta時，T對theta充分。","en":"A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass."}},{"id":"mko-topological-ordering-dag","position":7,"requirement":"optional","rationale":{"zh-Hant":"有限有向圖存在topological ordering恰當其無directed cycle。","en":"A finite directed graph has a topological ordering exactly when it has no directed cycle."}},{"id":"mko-multiplicative-order-mod-n","position":8,"requirement":"optional","rationale":{"zh-Hant":"gcd(a,n)=1時，ord_n(a)是a^k≡1 mod n的最小正整數k。","en":"For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n."}}]},{"id":"path-pulse-foundations-20261117","title":{"zh-Hant":"Wave6第十一層：series tests、uniform limits、product／quotient topology、path connectivity與components","en":"Wave6 Eleventh Layer: Series Tests, Uniform Limits, Product and Quotient Topologies, Paths and Components"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-infinite-series","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence-limit","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-monotone-sequence-convergence-theorem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-convergence-of-series","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-uniform-convergence-of-functions","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-basis","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-composition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quotient-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-surjective","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-connected-topological-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-intermediate-value-theorem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equivalence-class","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subset","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-closure-and-interior","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-connected-component","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-subspace-topology","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-series-comparison-test","position":2,"requirement":"optional","rationale":{"zh-Hant":"非負項級數可由逐項或最終逐項上界與下界傳遞收斂或發散。","en":"Eventual termwise bounds transfer convergence from a nonnegative upper series or divergence from a nonnegative lower series."}},{"id":"mko-ratio-test-series","position":3,"requirement":"optional","rationale":{"zh-Hant":"非零項絕對值相鄰比極限小於一時絕對收斂，大於一時發散，等於一時不判定。","en":"For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive."}},{"id":"mko-uniform-limit-continuity-theorem","position":4,"requirement":"optional","rationale":{"zh-Hant":"同一domain上的連續函數列若一致收斂至f，則極限函數f連續。","en":"Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain."}},{"id":"mko-finite-product-topology","position":5,"requirement":"optional","rationale":{"zh-Hant":"有限多個拓撲空間的Cartesian product由open rectangles生成乘積拓撲。","en":"Open rectangles generate the topology of a finite Cartesian product of topological spaces."}},{"id":"mko-quotient-topology","position":6,"requirement":"optional","rationale":{"zh-Hant":"滿射q:X→Y使U⊆Y恰在q逆像開時為開，從而賦予Y商拓撲。","en":"For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open."}},{"id":"mko-path-connected-space","position":7,"requirement":"optional","rationale":{"zh-Hant":"空間中任意兩點若可由單位區間的連續映射連接，則空間道路連通。","en":"A space is path connected when every pair of its points is joined by a continuous map from the unit interval."}},{"id":"mko-topological-connected-component","position":8,"requirement":"optional","rationale":{"zh-Hant":"指定點的連通分支是包含該點的極大連通子集；所有分支分割空間且各自閉。","en":"The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed."}}]},{"id":"path-pulse-foundations-20261118","title":{"zh-Hant":"Wave6第十二層：CLT、SLLN、confidence coverage、Wilson、p-adic valuation與regular languages","en":"Wave6 Twelfth Layer: CLT, SLLN, Confidence Coverage, Wilson, p-adic Valuations and Regular Languages"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-iid-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-convergence-in-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-expectation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-variance","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-standard-normal-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-binomial-theorem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-almost-sure-convergence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sampling-distribution","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-statistical-estimator","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-probability","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-prime-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-modular-congruence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-factorial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-modular-inverse","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-divisibility","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-unique-prime-factorization","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-integer","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-regular-language","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-mathematical-induction","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-finite-automaton","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proof","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-pigeonhole-principle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quantifier","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-central-limit-theorem-finite-variance-iid","position":2,"requirement":"optional","rationale":{"zh-Hant":"共同law且0<σ²<∞的無限IID序列，其正確中心化與sqrt(n)標準化和依分布收斂至標準常態。","en":"An infinite IID sequence with common finite positive variance has centered sums scaled by sigma*sqrt(n) converging in distribution to the standard normal law."}},{"id":"mko-strong-law-large-numbers-finite-valued-iid","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限值無限IID序列的樣本平均幾乎必然收斂至共同期望。","en":"The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation."}},{"id":"mko-finite-confidence-interval-coverage","position":4,"requirement":"optional","rationale":{"zh-Hant":"有限模型中預先固定的interval-valued procedure逐參數覆蓋真值的機率，定義其frequentist coverage。","en":"Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter."}},{"id":"mko-wilson-theorem","position":5,"requirement":"optional","rationale":{"zh-Hant":"整數n>1為質數，恰當(n-1)!模n同餘於-1。","en":"An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n."}},{"id":"mko-p-adic-valuation-integers","position":6,"requirement":"optional","rationale":{"zh-Hant":"對質數p與非零整數a，v_p(a)是p整除|a|的最高次數；本節固定v_p(0)=infinity。","en":"For prime p and nonzero integer a, v_p(a) is the largest exponent of p dividing |a|; this node fixes v_p(0)=infinity."}},{"id":"mko-regular-expression","position":7,"requirement":"optional","rationale":{"zh-Hant":"正規表示式由empty language、epsilon、alphabet symbols、union、concatenation與Kleene star遞迴建構語言。","en":"Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star."}},{"id":"mko-pumping-lemma-regular-languages","position":8,"requirement":"optional","rationale":{"zh-Hant":"每個正規語言都有p，使每個長度至少p的成員word存在可反覆pumping且仍留在語言的分解。","en":"Every regular language has a positive pumping length such that each sufficiently long member word has a split that remains in the language for every pump count."}}]},{"id":"path-pulse-foundations-20261119","title":{"zh-Hant":"Wave6第十三層閉合：Newton、梯形積分、梯度下降、Lagrange與三輪核心密度","en":"Wave6 Thirteenth Layer: Newton, Trapezoids, Gradient Descent, Lagrange Conditions and Numerical-Optimization Depth"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete four new traceable knowledge objects and deepen three existing objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-algorithm-correctness","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-sequence","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-derivative-rules","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-definite-integral","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-vector-space","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-system-linear-equations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-matrix-rank","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-independence-basis","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-absolute-value","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-ratio-proportion","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-partial-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-vector","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-multivariable-real-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-numerical-approximation-error","position":2,"requirement":"optional","rationale":{"zh-Hant":"在既有誤差MKO上加厚affine equation的residual／forward error、誤差傳播、conditioning witness與tolerance邊界。","en":"Extend the existing error object with affine residual versus forward error, propagation, conditioning witnesses and tolerance boundaries."}},{"id":"mko-gradient-vector","position":2,"requirement":"optional","rationale":{"zh-Hant":"在既有standard Euclidean coordinate語義上加厚gradient線性、負梯度內積、symmetric quadratic差分與metric邊界。","en":"Extend fixed standard Euclidean semantics with gradient linearity, negative-gradient inner products, symmetric-quadratic differences and metric boundaries."}},{"id":"mko-optimization-problem","position":2,"requirement":"optional","rationale":{"zh-Hant":"在完整有限可行域與鄰接關係上分開local/global最小、strict improvement、lower-bound certificate及空域邊界。","en":"Separate finite neighborhood-local and global minima, strict improvement, attained lower bounds and empty-domain boundaries on the complete feasible structure."}},{"id":"mko-newton-method","position":3,"requirement":"optional","rationale":{"zh-Hant":"在導數非零的目前點，以切線零點更新x至x-f(x)/f'(x)；收斂是另需局部假設的責任。","en":"At a qualified point with nonzero derivative, Newton method moves to the tangent-line zero x−f(x)/f_prime(x); convergence requires separate local hypotheses."}},{"id":"mko-trapezoidal-quadrature","position":3,"requirement":"optional","rationale":{"zh-Hant":"在嚴格遞增partition上逐段以端點平均乘各自寬度，得到定積分的複合梯形近似。","en":"On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral."}},{"id":"mko-gradient-descent","position":3,"requirement":"optional","rationale":{"zh-Hant":"在固定Euclidean coordinates中以正步長沿負梯度更新；下降與收斂仍需目標及步長假設。","en":"In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses."}},{"id":"mko-lagrange-multiplier-condition","position":3,"requirement":"optional","rationale":{"zh-Hant":"正則等式約束局部極值點的目標梯度，必為constraint gradients的線性組合。","en":"At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients."}}]}]}