{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"paths","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},"filters":{},"kind":"paths","total":59,"pages":3,"page":1,"entries":[{"id":"path-geometry-foundations","title":{"zh-Hant":"幾何基礎到畢達哥拉斯定理","en":"Geometry Foundations to the Pythagorean Theorem"},"summary":{"zh-Hant":"由點、線、面的基礎語言，經三角形、角、長度與直角三角形進入畢達哥拉斯定理。","en":"Move from the language of points, lines, and planes through triangles, angles, length, and right triangles to the Pythagorean theorem."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-point-line-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"先區分幾何位置、延伸對象與圖示。","en":"First distinguish geometric positions, extended objects, and their diagrams."}},{"id":"mko-triangle","position":2,"requirement":"required","rationale":{"zh-Hant":"由三個相異不共線點建立三角形、頂點與邊。","en":"Build a triangle, its vertices, and its sides from three distinct noncollinear points."}},{"id":"mko-angle","position":2,"requirement":"required","rationale":{"zh-Hant":"建立共同端點兩射線與角量的語言。","en":"Establish the language of two rays with a common endpoint and angle measure."}},{"id":"mko-euclidean-length","position":2,"requirement":"required","rationale":{"zh-Hant":"平行建立邊長與距離的意義。","en":"In parallel, establish the meaning of side length and distance."}},{"id":"mko-right-triangle","position":3,"requirement":"required","rationale":{"zh-Hant":"再以直角與邊確定定理適用的幾何對象。","en":"Use a right angle and sides to identify the geometric object to which the theorem applies."}},{"id":"mko-euclid-pythagorean-theorem","position":4,"requirement":"required","rationale":{"zh-Hant":"最後理解平方關係與證明。","en":"Finally understand the squared relation and its proof boundaries."}}]},{"id":"path-function-to-limits","title":{"zh-Hant":"集合、函數到趨近關係","en":"From Sets and Functions to the Tends-To Relation"},"summary":{"zh-Hant":"建立集合隸屬、總函數與趨近語義的基礎鏈。","en":"Establish a foundational chain from set membership through total functions to convergence semantics."},"audience":["upper_secondary","undergraduate"],"review":"reviewed","nodes":[{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"建立元素與集合之間的基本關係。","en":"Establish the basic relation between an element and a set."}},{"id":"mko-function-mapping","position":2,"requirement":"required","rationale":{"zh-Hant":"理解定義域到陪域的總函數。","en":"Understand a total function from a domain to a codomain."}},{"id":"mko-tends-to-relation","position":3,"requirement":"required","rationale":{"zh-Hant":"在函數關係上引入趨近與濾子核心。","en":"Introduce approach and the filter core on top of function relations."}}]},{"id":"path-modern-general-literacy","title":{"zh-Hant":"現代通識數學樣本路徑","en":"Modern General Mathematics Literacy"},"summary":{"zh-Hant":"以六個現有物件示範通識入口的多分支學習方式。","en":"Use six current objects to demonstrate a branching entry path into general mathematical literacy."},"audience":["general_learner"],"review":"reviewed","nodes":[{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"先接觸現代數學常用的集合語言。","en":"Begin with the set language used throughout modern mathematics."}},{"id":"mko-function-mapping","position":2,"requirement":"required","rationale":{"zh-Hant":"建立函數作為映射的基本概念。","en":"Establish the basic concept of a function as a mapping."}},{"id":"mko-euclidean-length","position":2,"requirement":"optional","rationale":{"zh-Hant":"平行進入幾何量與距離。","en":"Enter geometric quantity and distance in parallel."}},{"id":"mko-right-triangle","position":3,"requirement":"optional","rationale":{"zh-Hant":"由長度進入直角幾何。","en":"Move from length into right-angle geometry."}},{"id":"mko-euclid-pythagorean-theorem","position":4,"requirement":"optional","rationale":{"zh-Hant":"觀察幾何、代數與證明的連接。","en":"Observe the connection among geometry, algebra, and proof."}},{"id":"mko-tends-to-relation","position":4,"requirement":"optional","rationale":{"zh-Hant":"由函數進入分析中的趨近語義。","en":"Move from functions into the convergence language of analysis."}}]},{"id":"path-formalized-mathematics-entry","title":{"zh-Hant":"形式化數學入門樣本","en":"Entry Path to Formalized Mathematics"},"summary":{"zh-Hant":"從集合與函數語義進入 Lean／Mathlib 的精確聲明與證據物件。","en":"Move from set and function semantics into precise Lean and Mathlib statements and content-addressed evidence objects."},"audience":["undergraduate","formalizer"],"review":"reviewed","nodes":[{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"對齊 Set 作為謂詞的形式語義。","en":"Align membership with the formal semantics of Set as a predicate."}},{"id":"mko-function-mapping","position":2,"requirement":"required","rationale":{"zh-Hant":"對齊總函數型別與唯一輸出。","en":"Align total-function types with the existence of a unique output."}},{"id":"mko-tends-to-relation","position":3,"requirement":"required","rationale":{"zh-Hant":"理解 Filter.Tendsto 的形式核心。","en":"Understand the formal filter core of Filter.Tendsto."}},{"id":"mko-euclid-pythagorean-theorem","position":4,"requirement":"optional","rationale":{"zh-Hant":"觀察百科語義到向量及邊長模型的形式橋。","en":"Inspect the formal bridge from encyclopedia semantics to vector and side-length models."}}]},{"id":"path-proof-and-evidence-literacy","title":{"zh-Hant":"證明與證據素養","en":"Proof and Evidence Literacy"},"summary":{"zh-Hant":"先辨識證明義務，再區分人工證明、有限計算證據與精確形式證明。","en":"Identify proof obligations before distinguishing human proof, finite computational evidence, and exact formal proof."},"audience":["general_learner","undergraduate"],"review":"reviewed","nodes":[{"id":"mko-proof","position":1,"requirement":"required","rationale":{"zh-Hant":"先建立前提、推理、結論與非證明證據的區分。","en":"First distinguish premises, inference, conclusions, and evidence that is not a proof."}},{"id":"mko-euclid-pythagorean-theorem","position":2,"requirement":"required","rationale":{"zh-Hant":"同一物件同時具有人工、Python、FELRA 與 Lean 證據。","en":"Inspect one object carrying human, Python, FELRA, and Lean evidence at the same time."}},{"id":"mko-right-triangle","position":3,"requirement":"optional","rationale":{"zh-Hant":"比較定義物件與定理物件的證據需求。","en":"Compare the evidence needs of a definition object and a theorem object."}},{"id":"mko-tends-to-relation","position":3,"requirement":"optional","rationale":{"zh-Hant":"比較形式核心與完整分析語義的邊界。","en":"Compare a formal core with the boundary of complete analysis semantics."}}]},{"id":"path-foundational-language-candidate","title":{"zh-Hant":"數學基礎語言候選路徑","en":"Foundational Mathematical Language"},"summary":{"zh-Hant":"從計數的具體語言前進到集合、命題與邏輯連接詞，建立閱讀後續數學物件所需的基礎語言。","en":"Move from the concrete language of counting to sets, propositions, and logical connectives, establishing the vocabulary needed to read later mathematical 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."}}]}],"title":"Learning paths"}