{"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-20261015","title":{"zh-Hant":"七輪型別資料與選取：分數、比例、映射、合成與座標","en":"Typed Data and Selection: Fractions, Ratios, Maps, Compositions, and Coordinates"},"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-rational-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-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-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-set","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-real-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-counting-principle","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-factorial","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-fraction","position":2,"requirement":"optional","rationale":{"zh-Hant":"合法整數分數表示，原分母非零且負分母可用。真加法對照分子分母分別加的候選、四值相鄰差鏈、倒數先核零、兩次原始縮放表示與零縮放不合法；表示與數值分開，不把有限算術當商集或良定性形式證明。","en":"Keep legal original integer fractions with nonzero denominators, allowing negative denominators. Compare true addition with naive numerator/denominator addition, ordered difference chains, zero-guarded reciprocals, and two raw scaled representations. Zero scaling is illegal; representation/value separation is not a quotient well-definedness formal proof."}},{"id":"mko-domain-codomain-range","position":2,"requirement":"optional","rationale":{"zh-Hant":"Fin(n)到Fin(m)完整表的域／陪域／值域、兩個候選陪域值的原像、單格替換與值域增刪、一列協議候選表稽核。未使用格也須合型；重複同值資料列違反一列協議但不代表多值函數，不混同資料表示稽核與一般函數證明。","en":"Use complete Fin(n)→Fin(m) data for domain/codomain/range inventory, two typed-candidate preimages, one-cell replacement with range differences, and one-row protocol audits. Even unused cells must type-check; repeated equal-value rows violate the representation protocol without implying a multivalued function. Data audits are not general function proofs."}},{"id":"mko-sequence","position":2,"requirement":"optional","rationale":{"zh-Hant":"N從0起的明示有限索引、給定仿射規則的窗口、同四項前綴的兩個有限延伸、五項前綴和與有序差。次序與重複保留；空窗口不灌水、相同候選不算不同延伸，有限前綴不推定唯一無限規則、收斂或無窮級數。","en":"Use explicit finite zero-based indices, declared affine-recurrence windows, two finite extensions of a common four-term prefix, and five prefix sums/ordered differences. Keep order/repetition and genuinely empty windows; equal candidates are not different extensions. Finite prefixes imply neither a unique infinite rule nor convergence or an infinite series."}},{"id":"mko-permutation-combination","position":2,"requirement":"optional","rationale":{"zh-Hant":"相異標記Fin(n)物件無放回。有序／無序計數及k!與空選取、三個候選取樣資格、含／排物件0的組合分割、固定前r個物件順序後餘量與完成計數比。空選取只有一種，不套重複、不可區分、環形模型或擅稱概率。","en":"Select distinct labeled Fin(n) objects without replacement: ordered/unordered counts, k! and the unique empty selection, three candidate masks, inclusion/exclusion of label 0, and completion counts after a fixed ordered prefix. Do not use repetition, indistinguishable, or circular models, or silently call a counting ratio a probability."}},{"id":"mko-coordinate-plane","position":2,"requirement":"optional","rationale":{"zh-Hant":"同一正交有向xy框架中的整數有序對：四點欄位／軸／象限、逐分量差與相等、交換欄位候選、三對的首個不等。原點同屬兩軸，軸點不屬象限；交換欄位不作幾何反射形式橋，分量差不當距離或图形尺寸。","en":"Use ordered integer pairs in one orthogonal directed xy frame: four-point columns/axis/quadrant masks, component defects/equality, swapped-field candidates, and the first unequal pair. The origin lies on both axes and axis points are not in quadrants. A field swap is not a formal geometric reflection bridge; components are not distances or rendered dimensions."}},{"id":"mko-ratio-proportion","position":3,"requirement":"optional","rationale":{"zh-Hant":"Q數值比先核後項非零：正反向比、同k的完整三點律、同單位非負總量的正權分配、候選比例雙後項與交乘殘差。零總量的0:0仍非法，交乘零不恢復非法比；有限模型不外推因果、所有輸入或幾何相似。","en":"Numeric ratios in Q require nonzero second components: forward/reverse guards, a complete three-point common-k law, positive-weight same-unit nonnegative-total partitions, and candidate proportions with both denominator guards. Zero-total 0:0 remains illegal despite zero cross defects. Finite models imply no causality, universal law, or geometric similarity."}},{"id":"mko-composition","position":3,"requirement":"optional","rationale":{"zh-Hant":"先f後g且先核中間域；完整表路徑、g在明示子域的總表限制、兩序Q仿射合成與常數差、三總表的兩種括號路徑。未入g域不求值，未使用的g/h格仍須合型；合成不當乘法，不作全部函數等式形式證明。","en":"Evaluate f before g and guard the intermediate type/domain first. Check complete paths, g restricted to a declared subdomain while retaining original pairings, both Q affine orders and constant differences, and two parenthesizations of three maps. Affine output pairs are [slope,constant], not ascending coefficients. Outside g’s domain is not evaluated and unused g/h cells remain typed; no universal function-identity formal proof."}}]},"title":"Typed Data and Selection: Fractions, Ratios, Maps, Compositions, and Coordinates"}