{"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-20261011","title":{"zh-Hant":"七輪基礎補缺：自然數、表達式、集合與有限趨近模型","en":"Foundation Gaps: Natural Numbers, Expressions, Sets, and Finite Limit Models"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-natural-number","position":1,"requirement":"optional","rationale":{"zh-Hant":"固定N包含0。列有限後繼窗、雙點的部分自然前驅、有限閉區間大小與三個有理候選的N資格；0無自然前驅，不混用截斷pred、負數或分數。","en":"Fix N including 0; list successor windows, two partial predecessors, closed-interval sizes, and rational candidates for N. Zero has no natural predecessor in this protocol; do not conflate it with truncated pred, negative values, or fractions."}},{"id":"mko-set","position":1,"requirement":"optional","rationale":{"zh-Hant":"明示整數型別的有限集合。去重排序、相等及雙向差、候選元素的內外分割、至多4元素的全部子集合列舉；空集合的冪集只有空集合，不宣稱無限制概括或新集合論公理。","en":"Use finite sets of an explicit integer type: deduplication/sorting, equality with both differences, candidate membership partitions, and complete powersets of at most four elements. The empty set’s powerset contains the empty set; no unrestricted comprehension or new axiom is asserted."}},{"id":"mko-set-membership","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-arithmetic-operations","position":2,"requirement":"optional","rationale":{"zh-Hant":"固定有理數或N輸入的Q結果切片。四則有序結果、括號雙式、逆元候選有號缺陷、自然數差商的資格；除零未定義，負差不截成0，商不是整除。","en":"Use rational arithmetic, or natural inputs with Q results: ordered four operations, parenthesized pairs, inverse candidates with signed defects, and natural eligibility of differences/quotients. Division by zero is undefined, negative differences are not truncated to zero, and quotients are not integer division."}},{"id":"mko-subset","position":2,"requirement":"optional","rationale":{"zh-Hant":"同一整數型別的有限集合。全部A不在B的元素、非嚴格／嚴格包含及雙向見證、三段包含鏈的阻擋元素、三個候選反例的域內資格；空集包含與相等不是嚴格包含，有限檢查不當普遍集合證明。","en":"Compute all A-outside-B obstructions, non-strict/strict inclusion with both witnesses, three chain-obstruction sets, and typed candidate masks. Empty-set inclusion and equality are not strict inclusion, and finite checks are not universal set proofs."}},{"id":"mko-sample-space","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限且明示的可能結果協議。完整有序直積、記錄映射像及原結果纖維、候選清單的漏列／域外項、基本結果和事件隸屬的雙mask；不指定概率，不假設均勻、獨立或由未知試驗推得完整性。","en":"Use explicitly declared possible-outcome protocols: ordered products, recording images and original fibers, missing/outside candidate outcomes, and separate outcome/event masks. Assign no probabilities and infer neither uniformity, independence, nor completeness of an unknown experiment."}},{"id":"mko-tends-to-relation","position":2,"requirement":"optional","rationale":{"zh-Hant":"來源及目標均明示為有限主濾子，映射在完整有限型別上為總函數。核對像生成集、f(S)不在T的阻擋點、反例候選資格與兩條合成像計算；允許空生成集的bottom濾子。只關閉明示有限模型，不把裸箭頭當一般數列／點極限，不新增形式Evidence。","en":"Both filters are explicit finite principal filters, with a total map on complete finite types. Check image generators, f(S)-outside-T obstructions, typed candidates, and two composition-image routes; empty generators and the bottom filter are allowed. This finite model does not interpret a bare arrow as a general sequence/point limit or create new formal Evidence."}},{"id":"mko-variable-expression","position":3,"requirement":"optional","rationale":{"zh-Hant":"只指定Q中的四種小表達式，不建立通用AST。比較兩環境值、同時交換兩變數、內外層詞法遮蔽及a/(x-b)的原式定義域；零分子也不消除排除點，代數式不當方程式。","en":"Specify four small expressions over Q, not a universal AST: two environments, simultaneous variable exchange, lexical shadowing, and the original domain of a/(x−b). A zero numerator does not remove an excluded point; expressions are not equations."}}]},"title":"Foundation Gaps: Natural Numbers, Expressions, Sets, and Finite Limit Models"}