{"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-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."}}]},"title":"Structural Depth Wave4 Fourth Layer: Finite Fields, Hilbert Spaces, Martingales, Quotients, Cardinality, CDFs and Convergence in Probability"}