{"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-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."}}]},"title":"Wave6 Roots: Group Actions, Real Bounds, Topological Bases, Markov, Pigeonholes, Euler and Subsequences"}