{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"practice","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":{"page":"6"},"kind":"batches","total":360,"pages":15,"page":6,"entries":[{"id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","title":{"zh-Hant":"貝氏定理基礎 · 脈衝 R06","en":"Exact Priors, Likelihoods, Partition Normalization and Zero Denominators"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-bayes-theorem"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","title":{"zh-Hant":"緊緻拓撲空間基礎 · 脈衝 R05","en":"Finite Universal Compactness and Separate Supplied-Cover Audits"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-compact-topological-space"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","title":{"zh-Hant":"拓撲邊界基礎 · 脈衝 R04","en":"Topology-Relative Boundaries, Complements and Empty/Whole Sets"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-topological-boundary"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261102-r03-linear-map-image-foundations","title":{"zh-Hant":"線性映射的像基礎 · 脈衝 R03","en":"Full Real Column Images, Exact Membership and Noninteger Preimages"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-linear-map-image"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","title":{"zh-Hant":"線性映射的核基礎 · 脈衝 R02","en":"Zero-Target Kernels, Full Real Bases and Bounded-Search Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-linear-map-kernel"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261102-r01-vector-space-dimension-foundations","title":{"zh-Hant":"有限維向量空間的維度基礎 · 脈衝 R01","en":"Basis Qualification, Exact Dimension and One-Way Family Bounds"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-vector-space-dimension"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261101-r07-finite-automaton-foundations","title":{"zh-Hant":"確定有限自動機基礎 · 脈衝 R07","en":"Complete DFA Tables, Ordered Word Traces and Empty-Word Acceptance"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-finite-automaton"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261101-r06-euler-totient-function-foundations","title":{"zh-Hant":"Euler φ函數基礎 · 脈衝 R06","en":"Complete Coprime Representatives, Phi of One and Coprime Multiplicativity"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-euler-totient-function"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261101-r05-graph-matching-foundations","title":{"zh-Hant":"圖匹配基礎 · 脈衝 R05","en":"Endpoint-Disjoint Matchings, Maximality and Larger-Witness Distinctions"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-graph-matching"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","title":{"zh-Hant":"幾乎必然收斂基礎 · 脈衝 R04","en":"Positive-Atom Tails, Null-Atom Exceptions and Finite-Prefix Evidence Limits"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-almost-sure-convergence"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","title":{"zh-Hant":"開覆蓋與子覆蓋基礎 · 脈衝 R03","en":"Original Open Covers, Indexed Subcovers and Inclusion-Minimality Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-open-cover-and-subcover"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261101-r02-closure-and-interior-foundations","title":{"zh-Hant":"拓撲閉包與內部基礎 · 脈衝 R02","en":"Complete Topology-Relative Closure, Interior and Membership"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-closure-and-interior"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261101-r01-vector-subspace-foundations","title":{"zh-Hant":"實向量子空間基礎 · 脈衝 R01","en":"Universal Homogeneous Subspaces, Affine Boundaries and Finite Counterexamples"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-vector-subspace"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261031-r07-martingale-foundations","title":{"zh-Hant":"Martingale密度：差分、affine stability、公平增量與反例基礎 · 脈衝 R07","en":"Constant-Expectation Counterexamples, Conditional Differences, Affine Stability and Fair Sums"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-martingale"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261031-r06-finite-field-foundations","title":{"zh-Hant":"有限域密度：characteristic、乘法循環、仿射方程與Frobenius基礎 · 脈衝 R06","en":"Prime-Field Characteristic, Multiplicative Powers, Affine Equations and Frobenius"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-finite-field"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","title":{"zh-Hant":"圖著色基礎 · 脈衝 R05","en":"Complete Coloring Audits, Nonoptimal Upper Bounds and Color-Name Invariance"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-graph-coloring"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations","title":{"zh-Hant":"有限環的商環基礎 · 脈衝 R04","en":"Proper-Ideal Cosets and Representative-Independent Quotient Operations"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-finite-quotient-ring"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations","title":{"zh-Hant":"依分布收斂基礎 · 脈衝 R03","en":"Bernoulli CDF Tails, Limit Continuity Points and Finite-Prefix Limits"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-convergence-in-distribution"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","title":{"zh-Hant":"二分圖基礎 · 脈衝 R02","en":"Complete Bipartitions, Actual Odd Cycles and Isolated-Vertex Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-bipartite-graph"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","title":{"zh-Hant":"至多可數集基礎 · 脈衝 R01","en":"Complete Finite Certificates, Universal Integer Encodings and Failed-Witness Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-at-most-countable-set"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261030-r07-convergence-in-probability-foundations","title":{"zh-Hant":"依機率收斂基礎 · 脈衝 R07","en":"Symbolic Deviation Tails, Finite-Prefix Limits and Eventual Equality"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-convergence-in-probability"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","title":{"zh-Hant":"累積分布函數基礎 · 脈衝 R06","en":"Exact Cumulative Masses, Jumps, Monotonicity and Right Continuity"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-cumulative-distribution-function"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","title":{"zh-Hant":"集合基數比較基礎 · 脈衝 R05","en":"Mapping-Bound Injection, Surjection, Bijection and Finite Pigeonhole Certificates"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-cardinality-comparison"],"state":"candidate_validated"},{"id":"batch-20261014-advance-20261030-r04-quotient-set-foundations","title":{"zh-Hant":"商集基礎 · 脈衝 R04","en":"Distinct Full Classes, Canonical Projection and Partition Boundaries"},"summary":{"zh-Hant":"","en":""},"date":"2026-10-14","count":100,"mko_ids":["mko-quotient-set"],"state":"candidate_validated"}],"title":"Question batches"}