{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"paths","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{},"path":{"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."}}]},"title":"Structural Depth Wave4, First Slice: Abelian Groups, Norms, Cauchy Sequences, Joint Laws, Covariance, Graph Isomorphisms and Equivalence Relations"}