{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","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":{"page":"2"},"kind":"paths","total":59,"pages":3,"page":2,"entries":[{"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."}}]}],"title":"學習路徑"}