{"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":{},"path":{"id":"path-pulse-foundations-20261008","title":{"zh-Hant":"實數極限、連續、導數與多項式積分：七輪脈衝","en":"Real Limits, Continuity, Derivatives, and Polynomial Integration: 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-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-tends-to-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-rational-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-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-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-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-rate-of-change","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-polynomial","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-area","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-limit","position":2,"requirement":"optional","rationale":{"zh-Hant":"從0起算的實數列以任意精度、某一自然尾界及整個尾段定義極限；本輪精確處理倒數尾項模型。","en":"A zero-based real sequence converges when every precision admits a natural cutoff controlling the entire tail; this pulse treats an exact reciprocal-tail model."}},{"id":"mko-function-limit","position":3,"requirement":"optional","rationale":{"zh-Hant":"在a附近排除a本身，以ε-δ控制函數值；點值不屬於去心極限條件。","en":"An ε–δ limit controls values near a while excluding a itself; the center value is not part of the punctured-limit condition."}},{"id":"mko-continuity","position":4,"requirement":"optional","rationale":{"zh-Hant":"在全實數論域連續需要點值存在且與兩側極限一致；補點與原函數連續不可混用。","en":"Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct."}},{"id":"mko-derivative","position":5,"requirement":"optional","rationale":{"zh-Hant":"導數是非零步差商的兩側極限，不是任何一個有限步差商；絕對值尖點另保留不存在。","en":"A derivative is the two-sided limit of nonzero-step quotients, not any one finite-step quotient; an absolute-value corner retains nonexistence."}},{"id":"mko-derivative-rules","position":6,"requirement":"optional","rationale":{"zh-Hant":"四條微分規則先保留可微前提；本輪只在有限整數係數多項式上精確計算。","en":"Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials."}},{"id":"mko-antiderivative","position":7,"requirement":"optional","rationale":{"zh-Hant":"原函數須在宣告區間導回f；本輪全R多項式以常數項固定0的有理係數原函數作可重播代表。","en":"An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials."}},{"id":"mko-definite-integral","position":8,"requirement":"optional","rationale":{"zh-Hant":"連續多項式的有向定積分與精確矩公式；有限左端黎曼和只是有限和，不冒充一般積分或微積分基本定理完成。","en":"Continuous polynomials have oriented definite integrals with exact moment formulas; a finite left Riemann sum is not a general integral or a completed general FTC bridge."}}]},"title":"實數極限、連續、導數與多項式積分：七輪脈衝"}