{"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-20261013","title":{"zh-Hant":"七輪有限結構：畢氏、數系、推理、事件與標記三角","en":"Finite Structures: Pythagoras, Number Systems, Inference, Events, and Labeled Triangles"},"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-right-triangle","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-euclidean-length","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-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":"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-sample-space","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-subset","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-point-line-plane","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-euclid-pythagorean-theorem","position":2,"requirement":"optional","rationale":{"zh-Hant":"正整數兩股與明示歐氏直角邊長協議。平方欄位有序，完全平方才取正整數斜邊或缺股；候選負c即使平方殘差0也不合非退化長度。保留既有FELRA、向量核心及宣告side model證據，不宣稱任意仿射橋已完成。","en":"Use positive integer legs in an explicit Euclidean right-side protocol. Keep square fields ordered; only perfect squares yield positive integer hypotenuses or missing legs. A negative candidate c fails nondegenerate-length eligibility despite zero square residual. Preserve prior FELRA, vector-core, and declared side-model Evidence; no arbitrary affine bridge is completed."}},{"id":"mko-integer","position":2,"requirement":"optional","rationale":{"zh-Hant":"明示Z及有理候選。歐式整除要求正b及0≤r<b，負a不用截斷餘數；淨位移與步量分開；相反數及雙向差有序；負整數、零與非整數分數的資格和實值分組，有限切片不當環公理。","en":"Use explicit Z and rational candidates. Euclidean division requires b>0 and 0≤r<b, not truncated negative remainders. Separate net displacement from step magnitude, retain ordered opposites and directional differences, and classify zero/negative integers versus noninteger fractions; finite slices are not ring axioms."}},{"id":"mko-logical-connectives","position":2,"requirement":"optional","rationale":{"zh-Hant":"兩份完整八行0/1古典檔表逐行做固定合取／析取、材料蘊涵／雙條件、兩路德摩根與含括／排他或。材料蘊涵不作因果，逐行真假不當所有推理規則證明或通用邏輯語言。","en":"Evaluate two complete eight-row 0/1 classical tables for conjunction/disjunction, material implication/biconditional, both De Morgan routes, and inclusive/exclusive disjunction. Material implication is not causation; rowwise truths do not prove all inference rules or establish a general logic language."}},{"id":"mko-event","position":2,"requirement":"optional","rationale":{"zh-Hant":"明示非空有限S，區分F=P(S)與分塊聯集族。補事件／聯交外補、分塊的部分相交見證及最小閉包、候選觀測資格與首發生；有限S不自動讓全部子集可測，沒有概率或獨立假設。","en":"Declare a nonempty finite S and distinguish F=P(S) from partition-union families. Compute complement/union/intersection, partial-block witnesses and least containing events, typed observations, and first occurrence. Finite S does not automatically make every subset measurable, and probabilities/independence are not supplied."}},{"id":"mko-triangle","position":2,"requirement":"optional","rationale":{"zh-Hant":"整數標記A/B/C的歐氏頂點協議。相異／共線見證、合格三角形的對邊平方、循環及交換標記的座標行列式、對邊中點與三段平方；候選退化資料不冒稱三角形，有限座標核對不作一般幾何構造證明。","en":"Use integer-labeled Euclidean A/B/C vertex data: distinctness/noncollinearity witnesses, qualified opposite-side squares, cyclic/swapped coordinate determinants, opposite midpoints, and three squared segments. Degenerate candidates are not triangles and finite coordinates are not general construction proofs."}},{"id":"mko-rational-number","position":3,"requirement":"optional","rationale":{"zh-Hant":"分子與非零整數分母，包含負分母表示，答案正分母精確規約。和差、次序及精確差、除法資格與回乘、缺失加數回代，不用浮點或近似，不聲稱完整Rat構造或域公理。","en":"Represent rationals by integer numerators and nonzero denominators, including negative input denominators; outputs are exact reduced positive-denominator fractions. Check sum/difference, order and signed difference, guarded division/multiplication back, and missing-addend substitution. No floating-point approximation or complete Rat/field construction is claimed."}},{"id":"mko-proof","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限Pi標籤、給定前提與單前提MP邊。最短推導深度、型別資格、候選序列逐行審查與刪規則影響；無種子循環不自證，錯行不加入可用集。只是明示玩具推理協議，不當通用邏輯完備性、人審或形式Evidence。","en":"Use finite proposition labels, given facts, and single-premise MP edges. Check shortest depths, typing, candidate traces row by row, and rule-removal effects. Unseeded cycles cannot prove themselves and invalid rows never become premises. This toy protocol is neither general logical completeness nor human review or formal Evidence."}}]},"title":"Finite Structures: Pythagoras, Number Systems, Inference, Events, and Labeled Triangles"}