{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261116-r05-finite-sufficient-statistic-0068", "pilot_number": "P1000-0668", "number": 668, "family_id": "qf-mod47-suff-invariance", "title": { "zh-Hant": "第 668 題：有限模型充分統計量", "en": "Question 668: Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261116-r05-finite-sufficient-statistic-0068", "batch_id": "batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-sufficient-statistic" ], "title_zh": "有限模型充分統計量", "stem_zh": "total statistic fibers與正質量conditional laws的完整theta-invariance責任。操作=suff-invariance；資料={\"theta_values\":[\"a17\",\"b17\"],\"samples\":[170,171,172,173],\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"statistic\":[0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], 是], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], 是]], 是, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)", "explanation_zh": "Wave6第十層核對=([[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], 是], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], 是]], 是, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)；依完整明示資料獨立重算；zero-mass conditional保持undefined，nonsufficient需明示witness；identity/constant boundaries分開且不使用factorization theorem或宣稱估計品質。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "有限模型充分統計量" ] }, "variant": { "family_id": "qf-mod47-suff-invariance", "seed": 202611165068, "parameters": { "theta_values": [ "a17", "b17" ], "samples": [ 170, 171, 172, 173 ], "probability_rows": [ [ [ 2, 5 ], [ 1, 10 ], [ 2, 5 ], [ 1, 10 ] ], [ [ 1, 10 ], [ 2, 5 ], [ 1, 10 ], [ 2, 5 ] ] ], "statistic": [ 0, 1, 0, 1 ], "domain": "Exact_finite_sufficient_statistic", "operation": "suff-invariance", "semantic_template": "qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[170,171,172,173],\"statistic\":[0,1,0,1],\"theta_values\":[\"a17\",\"b17\"]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261116-r05-finite-sufficient-statistic-0068", "batch_id": "batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "74a865e13553257b2d210a5c1c2a7cf323ef76daace35920f27536364aac0e09", "family_id": "qf-mod47-suff-invariance", "seed": 202611165068, "semantic_template": "qf-mod47-suff-invariance/{\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"samples\":[170,171,172,173],\"statistic\":[0,1,0,1],\"theta_values\":[\"a17\",\"b17\"]}" }, "fields": { "title": "Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber", "stem": "Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate).", "answer": { "kind": "expression", "value": "([[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)", "explanation": "For each statistic value compare the complete conditional laws only among theta values where that cell has positive mass. Every fiber must pass; zero-mass theta cells are not assigned invented conditional laws. No factorization theorem or estimator-quality claim is added. The exact result, retaining the requested order, is ([[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], Yes], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)." }, "skills": [ "Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "total statistic fibers與正質量conditional laws的完整theta-invariance責任。操作=suff-invariance；資料={\"theta_values\":[\"a17\",\"b17\"],\"samples\":[170,171,172,173],\"probability_rows\":[[[2,5],[1,10],[2,5],[1,10]],[[1,10],[2,5],[1,10],[2,5]]],\"statistic\":[0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use complete theta labels=[a17, b17], samples=[170, 171, 172, 173], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (fiber_invariance_rows, sufficient, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "Θ=[a17, b17]; Ω=[170, 171, 172, 173]", "after": "[[2/5, 1/10, 2/5, 1/10], [1/10, 2/5, 1/10, 2/5]]", "relation": "implies", "rule_id": "full_parametric_model", "conditions": [], "explanation": { "zh-Hant": "每列是一個固定參數的law，每欄是一個樣本結果；重排標籤時同時重排機率。", "en": "Each row is one parameter law and each column an outcome; reorder probabilities together with labels." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。", "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "P_a17(Ω) := sum(2/5, 1/10, 2/5, 1/10)", "after": "P_a17(Ω) = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先核對此law的完整總質量。", "en": "Check this law's complete total mass first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_a17(Ω)，不以小數近似替代。", "en": "Recompute P_a17(Ω) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2/5", "1/10", "2/5", "1/10" ], "output": "1", "label": "P_a17(Ω)" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "P_b17(Ω) := sum(1/10, 2/5, 1/10, 2/5)", "after": "P_b17(Ω) = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先核對此law的完整總質量。", "en": "Check this law's complete total mass first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_b17(Ω)，不以小數近似替代。", "en": "Recompute P_b17(Ω) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/10", "2/5", "1/10", "2/5" ], "output": "1", "label": "P_b17(Ω)" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "sample=170", "after": "T(170)=0", "relation": "implies", "rule_id": "total_statistic_map", "conditions": [], "explanation": { "zh-Hant": "每個樣本點都恰有一個統計量值。", "en": "Every sample has exactly one statistic value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。", "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "sample=171", "after": "T(171)=1", "relation": "implies", "rule_id": "total_statistic_map", "conditions": [], "explanation": { "zh-Hant": "每個樣本點都恰有一個統計量值。", "en": "Every sample has exactly one statistic value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。", "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "sample=172", "after": "T(172)=0", "relation": "implies", "rule_id": "total_statistic_map", "conditions": [], "explanation": { "zh-Hant": "每個樣本點都恰有一個統計量值。", "en": "Every sample has exactly one statistic value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。", "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "sample=173", "after": "T(173)=1", "relation": "implies", "rule_id": "total_statistic_map", "conditions": [], "explanation": { "zh-Hant": "每個樣本點都恰有一個統計量值。", "en": "Every sample has exactly one statistic value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。", "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "[(170, 0), (171, 1), (172, 0), (173, 1)]", "after": "fibers=[[0, [170, 172]], [1, [171, 173]]]", "relation": "implies", "rule_id": "fiber_partition", "conditions": [], "explanation": { "zh-Hant": "依取值分組，全部fibers兩兩不交且聯集為完整樣本域。", "en": "Grouping by value produces disjoint fibers whose union is the complete carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。", "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "P_a17(T=0) := sum(2/5, 2/5)", "after": "P_a17(T=0) = 4/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先求條件事件的完整質量。", "en": "First compute the conditioning event's full mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_a17(T=0)，不以小數近似替代。", "en": "Recompute P_a17(T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2/5", "2/5" ], "output": "4/5", "label": "P_a17(T=0)" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "P_a17(170|T=0) := div(2/5, 4/5)", "after": "P_a17(170|T=0) = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同一law的原子質量除以正fiber質量。", "en": "Divide the same law's atom mass by its positive fiber mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_a17(170|T=0)，不以小數近似替代。", "en": "Recompute P_a17(170|T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2/5", "4/5" ], "output": "1/2", "label": "P_a17(170|T=0)" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "reconstruct atom 170 under a17 := mul(1/2, 4/5)", "after": "reconstruct atom 170 under a17 = 2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "乘回fiber質量核對原子機率。", "en": "Multiply by fiber mass to check the original atom probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 reconstruct atom 170 under a17，不以小數近似替代。", "en": "Recompute reconstruct atom 170 under a17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "4/5" ], "output": "2/5", "label": "reconstruct atom 170 under a17" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "P_a17(172|T=0) := div(2/5, 4/5)", "after": "P_a17(172|T=0) = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同一law的原子質量除以正fiber質量。", "en": "Divide the same law's atom mass by its positive fiber mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_a17(172|T=0)，不以小數近似替代。", "en": "Recompute P_a17(172|T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2/5", "4/5" ], "output": "1/2", "label": "P_a17(172|T=0)" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "reconstruct atom 172 under a17 := mul(1/2, 4/5)", "after": "reconstruct atom 172 under a17 = 2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "乘回fiber質量核對原子機率。", "en": "Multiply by fiber mass to check the original atom probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 reconstruct atom 172 under a17，不以小數近似替代。", "en": "Recompute reconstruct atom 172 under a17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "4/5" ], "output": "2/5", "label": "reconstruct atom 172 under a17" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "conditional total (a17,0) := sum(1/2, 1/2)", "after": "conditional total (a17,0) = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "完整條件分布的質量必為1。", "en": "The complete conditional law must have total mass1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditional total (a17,0)，不以小數近似替代。", "en": "Recompute conditional total (a17,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2" ], "output": "1", "label": "conditional total (a17,0)" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "P_a17(T=1) := sum(1/10, 1/10)", "after": "P_a17(T=1) = 1/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先求條件事件的完整質量。", "en": "First compute the conditioning event's full mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_a17(T=1)，不以小數近似替代。", "en": "Recompute P_a17(T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/10", "1/10" ], "output": "1/5", "label": "P_a17(T=1)" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "P_a17(171|T=1) := div(1/10, 1/5)", "after": "P_a17(171|T=1) = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同一law的原子質量除以正fiber質量。", "en": "Divide the same law's atom mass by its positive fiber mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_a17(171|T=1)，不以小數近似替代。", "en": "Recompute P_a17(171|T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/10", "1/5" ], "output": "1/2", "label": "P_a17(171|T=1)" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "reconstruct atom 171 under a17 := mul(1/2, 1/5)", "after": "reconstruct atom 171 under a17 = 1/10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "乘回fiber質量核對原子機率。", "en": "Multiply by fiber mass to check the original atom probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 reconstruct atom 171 under a17，不以小數近似替代。", "en": "Recompute reconstruct atom 171 under a17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "1/5" ], "output": "1/10", "label": "reconstruct atom 171 under a17" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "P_a17(173|T=1) := div(1/10, 1/5)", "after": "P_a17(173|T=1) = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同一law的原子質量除以正fiber質量。", "en": "Divide the same law's atom mass by its positive fiber mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_a17(173|T=1)，不以小數近似替代。", "en": "Recompute P_a17(173|T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/10", "1/5" ], "output": "1/2", "label": "P_a17(173|T=1)" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "reconstruct atom 173 under a17 := mul(1/2, 1/5)", "after": "reconstruct atom 173 under a17 = 1/10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "乘回fiber質量核對原子機率。", "en": "Multiply by fiber mass to check the original atom probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 reconstruct atom 173 under a17，不以小數近似替代。", "en": "Recompute reconstruct atom 173 under a17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "1/5" ], "output": "1/10", "label": "reconstruct atom 173 under a17" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "conditional total (a17,1) := sum(1/2, 1/2)", "after": "conditional total (a17,1) = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "完整條件分布的質量必為1。", "en": "The complete conditional law must have total mass1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditional total (a17,1)，不以小數近似替代。", "en": "Recompute conditional total (a17,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2" ], "output": "1", "label": "conditional total (a17,1)" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "P_b17(T=0) := sum(1/10, 1/10)", "after": "P_b17(T=0) = 1/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先求條件事件的完整質量。", "en": "First compute the conditioning event's full mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_b17(T=0)，不以小數近似替代。", "en": "Recompute P_b17(T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/10", "1/10" ], "output": "1/5", "label": "P_b17(T=0)" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "P_b17(170|T=0) := div(1/10, 1/5)", "after": "P_b17(170|T=0) = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同一law的原子質量除以正fiber質量。", "en": "Divide the same law's atom mass by its positive fiber mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_b17(170|T=0)，不以小數近似替代。", "en": "Recompute P_b17(170|T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/10", "1/5" ], "output": "1/2", "label": "P_b17(170|T=0)" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "reconstruct atom 170 under b17 := mul(1/2, 1/5)", "after": "reconstruct atom 170 under b17 = 1/10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "乘回fiber質量核對原子機率。", "en": "Multiply by fiber mass to check the original atom probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 reconstruct atom 170 under b17，不以小數近似替代。", "en": "Recompute reconstruct atom 170 under b17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "1/5" ], "output": "1/10", "label": "reconstruct atom 170 under b17" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "P_b17(172|T=0) := div(1/10, 1/5)", "after": "P_b17(172|T=0) = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同一law的原子質量除以正fiber質量。", "en": "Divide the same law's atom mass by its positive fiber mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_b17(172|T=0)，不以小數近似替代。", "en": "Recompute P_b17(172|T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/10", "1/5" ], "output": "1/2", "label": "P_b17(172|T=0)" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "reconstruct atom 172 under b17 := mul(1/2, 1/5)", "after": "reconstruct atom 172 under b17 = 1/10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "乘回fiber質量核對原子機率。", "en": "Multiply by fiber mass to check the original atom probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 reconstruct atom 172 under b17，不以小數近似替代。", "en": "Recompute reconstruct atom 172 under b17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "1/5" ], "output": "1/10", "label": "reconstruct atom 172 under b17" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "conditional total (b17,0) := sum(1/2, 1/2)", "after": "conditional total (b17,0) = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "完整條件分布的質量必為1。", "en": "The complete conditional law must have total mass1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditional total (b17,0)，不以小數近似替代。", "en": "Recompute conditional total (b17,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2" ], "output": "1", "label": "conditional total (b17,0)" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "P_b17(T=1) := sum(2/5, 2/5)", "after": "P_b17(T=1) = 4/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先求條件事件的完整質量。", "en": "First compute the conditioning event's full mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_b17(T=1)，不以小數近似替代。", "en": "Recompute P_b17(T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2/5", "2/5" ], "output": "4/5", "label": "P_b17(T=1)" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "P_b17(171|T=1) := div(2/5, 4/5)", "after": "P_b17(171|T=1) = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同一law的原子質量除以正fiber質量。", "en": "Divide the same law's atom mass by its positive fiber mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_b17(171|T=1)，不以小數近似替代。", "en": "Recompute P_b17(171|T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2/5", "4/5" ], "output": "1/2", "label": "P_b17(171|T=1)" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "reconstruct atom 171 under b17 := mul(1/2, 4/5)", "after": "reconstruct atom 171 under b17 = 2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "乘回fiber質量核對原子機率。", "en": "Multiply by fiber mass to check the original atom probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 reconstruct atom 171 under b17，不以小數近似替代。", "en": "Recompute reconstruct atom 171 under b17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "4/5" ], "output": "2/5", "label": "reconstruct atom 171 under b17" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "P_b17(173|T=1) := div(2/5, 4/5)", "after": "P_b17(173|T=1) = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同一law的原子質量除以正fiber質量。", "en": "Divide the same law's atom mass by its positive fiber mass." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P_b17(173|T=1)，不以小數近似替代。", "en": "Recompute P_b17(173|T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2/5", "4/5" ], "output": "1/2", "label": "P_b17(173|T=1)" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "reconstruct atom 173 under b17 := mul(1/2, 4/5)", "after": "reconstruct atom 173 under b17 = 2/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "乘回fiber質量核對原子機率。", "en": "Multiply by fiber mass to check the original atom probability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 reconstruct atom 173 under b17，不以小數近似替代。", "en": "Recompute reconstruct atom 173 under b17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "4/5" ], "output": "2/5", "label": "reconstruct atom 173 under b17" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "conditional total (b17,1) := sum(1/2, 1/2)", "after": "conditional total (b17,1) = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "完整條件分布的質量必為1。", "en": "The complete conditional law must have total mass1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditional total (b17,1)，不以小數近似替代。", "en": "Recompute conditional total (b17,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2" ], "output": "1", "label": "conditional total (b17,1)" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "valid positive-mass laws for T=0: [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]]", "after": "parameter-invariant: 是", "relation": "implies", "rule_id": "conditional_law_comparison", "conditions": [], "explanation": { "zh-Hant": "只比較有定義的完整條件law；逐樣本機率都相同才算參數無關。", "en": "Compare full defined conditional laws; parameter invariance requires every sample probability to agree." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。", "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima." } }, "arithmetic": null }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "valid positive-mass laws for T=1: [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]]", "after": "parameter-invariant: 是", "relation": "implies", "rule_id": "conditional_law_comparison", "conditions": [], "explanation": { "zh-Hant": "只比較有定義的完整條件law；逐樣本機率都相同才算參數無關。", "en": "Compare full defined conditional laws; parameter invariance requires every sample probability to agree." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。", "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima." } }, "arithmetic": null }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "[[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], 是], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], 是]]", "after": "all fibers invariant: 是", "relation": "implies", "rule_id": "finite_sufficiency_definition", "conditions": [], "explanation": { "zh-Hant": "本有限合約的充分性要求每個fiber都通過，不能只核一個。", "en": "Sufficiency under this finite contract requires every fiber to pass, not just one." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。", "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima." } }, "arithmetic": null } ], "math_model": { "result": [ [ [ 0, [ [ "a17", [ [ 170, "1/2" ], [ 172, "1/2" ] ] ], [ "b17", [ [ 170, "1/2" ], [ 172, "1/2" ] ] ] ], true ], [ 1, [ [ "a17", [ [ 171, "1/2" ], [ 173, "1/2" ] ] ], [ "b17", [ [ 171, "1/2" ], [ 173, "1/2" ] ] ] ], true ] ], true, "conditional_law_must_be_theta_independent_on_every_positive_mass_cell" ] }, "conclusion": { "canonical_value": "([[0, [[a17, [[170, 1/2], [172, 1/2]]], [b17, [[170, 1/2], [172, 1/2]]]], 是], [1, [[a17, [[171, 1/2], [173, 1/2]]], [b17, [[171, 1/2], [173, 1/2]]]], 是]], 是, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)", "explanation": { "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。", "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings." } }, "definition_note": { "zh-Hant": "只在正質量 fiber 上定義條件律；每個 fiber 的條件律都與參數無關才算充分。", "en": "Conditional laws are defined only on positive-mass fibers; sufficiency needs every fiber to be parameter-invariant." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations/questions.jsonl", "batch_sha256": "8ce239dc6b93456c36b291b3ff0d0d055f7f5a0896790e1605f94f4266893893", "original_line": 68, "original_line_sha256": "e969a8584fa21fc7728328c6d0527a2fb99d940601ed498e722d907294e5346a", "question_compact_sha256": "74a865e13553257b2d210a5c1c2a7cf323ef76daace35920f27536364aac0e09", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations.en.jsonl", "english_row_compact_sha256": "33e1ee1d7943e3fbe3ca6b54883171a8125e816a108f113b11a47bdf262eef5e" }, "mko_bindings": { "mko-finite-sufficient-statistic": { "sha256": "5b80cc27ea20e17864cb5068dedde0919001d6163cabe652473150401c131231", "english_sha256": "cd5defd0a6ff5e44e92db46ae4d8f1c71413883dfb875e6539fbcd7cac62fef6", "original_path": "public/data/mko/finite-sufficient-statistic.json", "english_path": "public/data/i18n/editions/mko-finite-sufficient-statistic.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e", "original_candidate_sha256": "0198323d7d32b21025d3f5d3c926d18c19d5dd9e2e3874a7fd144a1b647abcec", "original_candidate_line_sha256": "0198323d7d32b21025d3f5d3c926d18c19d5dd9e2e3874a7fd144a1b647abcec" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。", "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI review or general formal proof. Original questions, answers, MKOs, formulas, ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-003-02", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772", "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 35, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "4777623a1f69a405f6b5f51f8f83c87bf7ae9875daa7d46d1d1126bb18a0f741", "arithmetic_steps": 26 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }