{"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":{"from":"/en/paths/"},"path":{"id":"path-pulse-foundations-20261012","title":{"zh-Hant":"七輪基礎推理與幾何：歸納、計數、命題、型別與長度","en":"Foundational Reasoning and Geometry: Induction, Counting, Typing, and Length"},"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":"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-proposition","position":1,"requirement":"optional","rationale":{"zh-Hant":"固定古典整數比較陳述與有號算術見證。未代入的自由變數標未封閉；相等與否定分開；候選自報真假只作待核資料，與實際真假和正式證明項分離。不建立通用邏輯語言或新形式Evidence。","en":"Use classical integer comparison statements and signed arithmetic witnesses. Mark unassigned free variables as open; separate equality and negation, candidate reports, actual truths, and proof terms. Do not create a universal logic language or new formal Evidence."}},{"id":"mko-arithmetic-operations","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-set-membership","position":1,"requirement":"optional","rationale":{"zh-Hant":"有限同型別集合與完整特徵表。Fin(n)外候選只標不合型別，不盲讀謂詞；隸屬判定與插入／刪除結果、有限明示論域上的謂詞集合外延。保留Set α→Prop既有Evidence，不把程式容器當全部集合論。","en":"Use finite typed sets and complete characteristic tables. Outside-Fin(n) candidates are ill-typed, not blindly evaluated predicate inputs; distinguish membership from insertion/deletion and finite predicate extension. Preserve existing Set α→Prop Evidence without treating containers as all of set theory."}},{"id":"mko-right-triangle","position":1,"requirement":"optional","rationale":{"zh-Hant":"候選整數角度先核正角與和180的協議，再檢90；兩個90加零角不能通過。合格直角資料標記頂點、對邊及銳角值；π有理倍數精確換算，不用容差、不取模、不把候選當已構造的非退化三角形。","en":"Qualify candidate positive integer angles and exact sum 180 before checking 90; two 90s and a zero fail. Preserve vertex/opposite-side labels and ordered acute angles; convert rational π multiples exactly, without tolerance or periodic wrapping. Candidate data are not a constructed nondegenerate triangle."}},{"id":"mko-euclidean-length","position":1,"requirement":"optional","rationale":{"zh-Hant":"同一等單位正交L2實平面的整數座標切片。保留有向座標差及長度平方；完全平方才給整數主距離；中點為精確有理數，兩段平方分別核對。軸上路徑與端點直線長分開，零距離不排除；不重播舊浮點參考Evidence或聲稱全部度量公理完成。","en":"Use integer coordinates in one equal-unit orthogonal Euclidean L2 plane. Preserve signed displacement and squared lengths; give integer principal distances only for perfect squares. Check exact rational midpoints and both half-segment squares; separate axis path sums from direct endpoint length, retain zero distances, and do not claim all metric axioms or replay old floating-point Evidence as new."}},{"id":"mko-mathematical-induction","position":2,"requirement":"optional","rationale":{"zh-Hant":"N從0起，分清歸納假設n與目標n+1。只核對明示整數遞推的有限前綴、仿射候選的基礎／步缺陷與有限檢查覆蓋；不把有限樣本當成任意n的歸納證明。","en":"N starts at 0; distinguish hypothesis index n from target n+1. Check explicit integer recurrence prefixes, affine-candidate base/step defects, and finite coverage only; finite samples are not induction for arbitrary n."}},{"id":"mko-counting-principle","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限選擇協議明示互斥標籤或完整有序直積，包含空分支與零選項；後續寬度可依前一步變化，候選等寬法核對資格和缺陷，不從總數巧合推得等寬或概率獨立。兩集合重疊由實際外延計數修正。","en":"Explicitly tagged disjoint choices or full ordered products include empty branches and zero options. Later widths may depend on earlier choices; check candidate equal-width eligibility separately from total defects. Matching totals imply neither equal widths nor probabilistic independence; correct overlaps by actual set extension."}},{"id":"mko-function-mapping","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整有限型別上的總映射，允許空定義域及唯一空表，非空源到空陪域不可宣稱總函數。圖形、實際值域與未用陪域分開；型別外查詢標不合型別；兩張同型別總表逐點核差異。保留既有總函數core Evidence，不混作部分、有副作用或會失敗的程式函數。","en":"Total maps on complete finite types allow the empty domain and its unique empty table, but no total map from a nonempty source into an empty target. Distinguish graph, range, and unused codomain; outside queries are ill-typed, and compare both complete tables pointwise. Preserve existing total-function core Evidence; exclude partial, effectful, or failing programs."}}]},"title":"Foundational Reasoning and Geometry: Induction, Counting, Typing, and Length"}