{"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-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."}}]},"title":"Structural Depth Wave4 Roots: Rings, Metric Completeness, Linear Structures, Independence and Graph Paths"}