{"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-20261118","title":{"zh-Hant":"Wave6第十二層：CLT、SLLN、confidence coverage、Wilson、p-adic valuation與regular languages","en":"Wave6 Twelfth Layer: CLT, SLLN, Confidence Coverage, Wilson, p-adic Valuations 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-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-convergence-in-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-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-standard-normal-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-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-binomial-theorem","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-almost-sure-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-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-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-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-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-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-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-factorial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-modular-inverse","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 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-unique-prime-factorization","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 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-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-regular-language","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 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-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-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-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-pigeonhole-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-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-central-limit-theorem-finite-variance-iid","position":2,"requirement":"optional","rationale":{"zh-Hant":"共同law且0<σ²<∞的無限IID序列，其正確中心化與sqrt(n)標準化和依分布收斂至標準常態。","en":"An infinite IID sequence with common finite positive variance has centered sums scaled by sigma*sqrt(n) converging in distribution to the standard normal law."}},{"id":"mko-strong-law-large-numbers-finite-valued-iid","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限值無限IID序列的樣本平均幾乎必然收斂至共同期望。","en":"The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation."}},{"id":"mko-finite-confidence-interval-coverage","position":4,"requirement":"optional","rationale":{"zh-Hant":"有限模型中預先固定的interval-valued procedure逐參數覆蓋真值的機率，定義其frequentist coverage。","en":"Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter."}},{"id":"mko-wilson-theorem","position":5,"requirement":"optional","rationale":{"zh-Hant":"整數n>1為質數，恰當(n-1)!模n同餘於-1。","en":"An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n."}},{"id":"mko-p-adic-valuation-integers","position":6,"requirement":"optional","rationale":{"zh-Hant":"對質數p與非零整數a，v_p(a)是p整除|a|的最高次數；本節固定v_p(0)=infinity。","en":"For prime p and nonzero integer a, v_p(a) is the largest exponent of p dividing |a|; this node fixes v_p(0)=infinity."}},{"id":"mko-regular-expression","position":7,"requirement":"optional","rationale":{"zh-Hant":"正規表示式由empty language、epsilon、alphabet symbols、union、concatenation與Kleene star遞迴建構語言。","en":"Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star."}},{"id":"mko-pumping-lemma-regular-languages","position":8,"requirement":"optional","rationale":{"zh-Hant":"每個正規語言都有p，使每個長度至少p的成員word存在可反覆pumping且仍留在語言的分解。","en":"Every regular language has a positive pumping length such that each sufficiently long member word has a split that remains in the language for every pump count."}}]},"title":"Wave6 Twelfth Layer: CLT, SLLN, Confidence Coverage, Wilson, p-adic Valuations and Regular Languages"}