{"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-20261009","title":{"zh-Hant":"Core80最後種子與五輪既有題族密度","en":"Final Core80 Seeds and Five Rounds of Existing-Family Depth"},"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-derivative","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-continuity","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for 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 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 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 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 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 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 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 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 this pulse."}},{"id":"mko-linear-transformation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-determinant","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-quadratic-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for this pulse."}},{"id":"mko-derivative-rules","position":2,"requirement":"optional","rationale":{"zh-Hant":"原有至多3次全R多項式切片：二階導數點值、乘積導數候選值的資格、兩個有序合成導數，以及和的導數完整升冪係數。合成可能升次，不截去高項。","en":"Deepen the full-real degree-at-most-three polynomial slice with second derivatives, product-value eligibility, ordered composition derivatives, and complete derivative coefficients without truncation."}},{"id":"mko-tree","position":2,"requirement":"optional","rationale":{"zh-Hant":"非空有限簡單無向圖同時連通且無環才是樹；保留單點、度1葉及加一條新邊的唯一環。","en":"A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions."}},{"id":"mko-function-limit","position":2,"requirement":"optional","rationale":{"zh-Hant":"在既有去心實數模型內加厚：m≠0且|m|δ>ε時取h=ε/|m|，得到0<h<δ但誤差=ε；點值與極限成對輸出；左右極限保持順序；商只在x≠a求值。","en":"Deepen the punctured model with a failing δ witness h=ε/|m|, center-versus-limit pairs, ordered one-sided limits, and quotient values only at x≠a."}},{"id":"mko-eigenvalue","position":2,"requirement":"optional","rationale":{"zh-Hant":"原有實二階方陣整數切片：降冪特徵多項式(1,−tr,det)、精確判別式、Mv−λv有序殘差、非零向量第一失配座標。沒有對角化、Jordan或複數譜擴充。","en":"Deepen real 2×2 integer matrices with descending characteristic coefficients, exact discriminants, ordered kernel residuals, and a nonzero vector’s first mismatch; no diagonalization or complex-spectrum expansion."}},{"id":"mko-antiderivative","position":3,"requirement":"optional","rationale":{"zh-Hant":"原有全R多項式與正分母有理係數切片：指定原函數係數、共享F0的兩常數差、由F(a)=V解C，以及候選F的導數係數。保持升冪、最簡分數與常數0規約。","en":"Deepen the rational-coefficient slice with a specified primitive coefficient, shared-primitive constant differences, anchor-selected constants, and candidate derivative coefficients."}},{"id":"mko-definite-integral","position":4,"requirement":"optional","rationale":{"zh-Hant":"原有多項式矩與有向有限區間切片：任意端點順序的三段積分；奇函數資格與對稱積分成對輸出；有序端點仿射左端有限和的精確誤差；非零常數函數積分的端點反解。","en":"Deepen oriented moments with three-segment additivity, paired oddness/symmetric-integral output, signed finite-left-sum error, and endpoint recovery for a nonzero constant integrand."}},{"id":"mko-fundamental-theorem-calculus","position":5,"requirement":"optional","rationale":{"zh-Hant":"在全實數連續前提下，累積定積分的導數還原被積函數；合法原函數給有向積分端點差。","en":"With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference."}}]},"title":"Final Core80 Seeds and Five Rounds of Existing-Family Depth"}