{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","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":{"from":"/zh-Hant/paths/"},"path":{"id":"path-pulse-foundations-20261005","title":{"zh-Hant":"函數可逆性與有限機率矩數：每日七輪","en":"Function Invertibility and Finite Probability Moments: Seven-Round Workflow"},"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 top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"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-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-composition","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-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-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-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-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-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-injective","position":2,"requirement":"optional","rationale":{"zh-Hant":"單射要求不同合法輸入不能有同一輸出，不要求覆蓋整個陪域。","en":"Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain."}},{"id":"mko-surjective","position":3,"requirement":"optional","rationale":{"zh-Hant":"滿射要求陪域每個元素都有合法原像；相同輸出可以有多個原像。","en":"Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages."}},{"id":"mko-bijective","position":4,"requirement":"optional","rationale":{"zh-Hant":"雙射同時單射與滿射，等價於陪域每個元素恰有一個合法原像。","en":"A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage."}},{"id":"mko-inverse-function","position":5,"requirement":"optional","rationale":{"zh-Hant":"雙射 f:A→B 的反函數 g:B→A 須驗證兩個方向的還原，不能把交換資料當成無條件反函數。","en":"For a bijection f:A→B, an inverse g:B→A restores values in both directions; exchanging a table is not an unconditional inverse construction."}},{"id":"mko-random-variable","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限模型中的隨機變數是基本結果到實數的總函數；它不是基本結果，也不是權重本身。","en":"A random variable in a finite model is a total function from elementary outcomes to real numbers, distinct from an outcome or its probability weight."}},{"id":"mko-expectation","position":7,"requirement":"optional","rationale":{"zh-Hant":"有限期望是合法機率權重與隨機變數值的加權總和，線性性不要求獨立。","en":"Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence."}},{"id":"mko-variance","position":8,"requirement":"optional","rationale":{"zh-Hant":"本輪變異數為 E[(X−E[X])²]，不是 n−1 分母的樣本變異數；零權重離群值不影響它。","en":"This entry defines variance as E[(X−E[X])²], distinct from sample variance with an n−1 denominator. Zero-weight outliers do not affect it."}}]},"title":"函數可逆性與有限機率矩數：每日七輪"}