{"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-20261010","title":{"zh-Hant":"七輪跨域密度：代數、集合、距離、有限概率與量詞","en":"Cross-Domain Depth: Algebra, Sets, Distances, Finite Probability, and Quantifiers"},"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-variable-expression","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-power-root","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-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-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-coordinate-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-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-probability","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-random-variable","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-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-polynomial","position":2,"requirement":"optional","rationale":{"zh-Hant":"固定整數升冪係數至多3次的Q[x]切片；和與乘積输出完整正規化係數，乘積可升至6次；同一多項式雙點有序求值；p(x+c)完整係數以精確二項式展開核對。不把單點相同當多項式恆等。","en":"Use the integer-coefficient slice of Q[x] through input degree 3: complete normalized sum/product coefficients (products through degree 6), ordered evaluation at two points, and complete coefficients of p(x+c) by exact expansion. Equality at one point is not polynomial identity."}},{"id":"mko-set-operations","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限明示整數集合正規化重複與排序；聯／交大小對、(A\\B)∪(B\\A)、U\\(A∪B)與(U\\A)∩(U\\B)的完整結果、三段有序分割。補集先要求A、B⊆明示U。只封閉此有限实例，不新增一般集合論形式證明。","en":"Normalize finite explicit integer sets; compute union/intersection cardinalities, symmetric differences, both relative De Morgan sets, and ordered three-part partitions. Require A,B⊆the declared U before complements; finite instances do not create general set-theory formal Evidence."}},{"id":"mko-distance-formula","position":2,"requirement":"optional","rationale":{"zh-Hant":"所有向量是同一等單位正交實平面從共同原點出發的整數座標差；精確平方對、候選非負資格與平方缺陷、有三個具索引距離的含同距分組排序，以及兩個完全平方距離的整數和。不是非正交／不等尺度或Prod預設metric；不拿兩段和當另一段距離。","en":"Vectors are integer coordinate differences from a common origin in one equal-unit orthogonal real plane. Check ordered squares, candidate sign and square defect, indexed three-distance tie groups, and sums of two integer principal distances. Exclude nonorthogonal/unequal-scale and default product-metric interpretations; a two-segment sum is not another segment’s length."}},{"id":"mko-expectation","position":2,"requirement":"optional","rationale":{"zh-Hant":"同一有限合法歸一化整數值模型：逐點wX向量、E[X−Y]、(E[XY],E[X]E[Y])有序對與給定均值下的正質量未知整數值反解。只有共同S與權重，不假設X/Y獨立；數值相等也不推出独立。未知slot明示null、正權重且解為整數；不做樣本估計或一般積分存在性。","en":"On a shared normalized finite integer-valued model, compute wX contributions, E[X−Y], ordered (E[XY],E[X]E[Y]), and an integer value at an explicit null slot of positive mass from a known mean. No independence, sample-estimation, or general integrability claim; equal numerical expressions do not establish independence."}},{"id":"mko-quantifier","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限明示整數U及可判定大小／相等／非零整除述詞。完整真假元素分割、(¬∀P,¬∃P)、三個有序候選的域內反例资格mask，及有限U/V上(滿足∃y內條件的x數,滿足∀x內條件的y數)。空域全稱真、存在假；候選域外無资格；嵌套枚舉不交换量詞、不外推全部整數或一般一階逻輯證明。","en":"Enumerate finite explicit integer domains with order, equality, or nonzero-divisibility predicates: true/false partitions, (¬∀P,¬∃P), typed counterexample masks, and ordered nested existence/universality counts on U/V. Empty-domain universal is True and existential False. Do not exchange quantifiers or extrapolate to all integers or general first-order proofs."}},{"id":"mko-conditional-probability","position":3,"requirement":"optional","rationale":{"zh-Hant":"S=0至n−1、全冪集、非負整數mass與明示正分母D，Σmass=D才為歸一化模型。B質量嚴格正，雙向另要求A正。枚舉交集／B概率對、雙向條件值、對全部singleton在B條件下的概率向量、候選值與真正條件值的有號缺陷。非空但mass0不能條件化；不宣稱Bayes或独立性。","en":"Use explicit finite S, its full powerset, nonnegative integer masses, and positive D with total mass D. Require positive B mass, and positive A mass for both directions. Compute probability pairs, directional conditionals, every conditional singleton weight, and signed candidate defects. Nonempty zero-mass events cannot condition; no Bayes or independence proof is claimed."}},{"id":"mko-variance","position":3,"requirement":"optional","rationale":{"zh-Hant":"合法有限概率模型：逐項w(X−μ)²、(E[X²],μ²)精確對、先構造共模型X−Y再求其變異數，以及相對第一個正權重值的第一個正支持失配索引。不是n−1樣本估計；保留零權重異值；不把Var(X−Y)拆成兩個Var，也不聲稱協方差理論完成。","en":"Compute w(X−μ)², exact (E[X²],μ²), variance after forming the shared-model difference, and the first positive-support mismatch relative to the first positive-weight value. Preserve zero-weight outliers; do not use n−1 sample variance, split Var(X−Y) into two variances, or claim complete covariance theory."}}]},"title":"Cross-Domain Depth: Algebra, Sets, Distances, Finite Probability, and Quantifiers"}