{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261116-r05-finite-sufficient-statistic-0081", "pilot_number": "P1000-0681", "number": 681, "family_id": "qf-mod47-suff-nonsufficient-zero-boundary", "title": { "zh-Hant": "第 681 題：有限模型充分統計量", "en": "Question 681: Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261116-r05-finite-sufficient-statistic-0081", "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-nonsufficient-zero-boundary；資料={\"mode\":\"zero_mass\",\"theta_values\":[\"a5\",\"b5\"],\"samples\":[50,51,52,53],\"probability_rows\":[[[1,2],0,[1,2],0],[0,[1,2],0,[1,2]]],\"statistic\":[0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(zero_mass, [[0, [[a5, [[50, 1/2], [52, 1/2]]]], 是], [1, [[b5, [[51, 1/2], [53, 1/2]]]], 是]], 是, [[a5, 1, 0, null], [b5, 0, 0, null]], null, 否, 否, 否, 是, 否, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)", "explanation_zh": "Wave6第十層核對=(zero_mass, [[0, [[a5, [[50, 1/2], [52, 1/2]]]], 是], [1, [[b5, [[51, 1/2], [53, 1/2]]]], 是]], 是, [[a5, 1, 0, null], [b5, 0, 0, null]], null, 否, 否, 否, 是, 否, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)；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": 1, "skills": [ "有限模型充分統計量" ] }, "variant": { "family_id": "qf-mod47-suff-nonsufficient-zero-boundary", "seed": 202611165081, "parameters": { "mode": "zero_mass", "theta_values": [ "a5", "b5" ], "samples": [ 50, 51, 52, 53 ], "probability_rows": [ [ [ 1, 2 ], 0, [ 1, 2 ], 0 ], [ 0, [ 1, 2 ], 0, [ 1, 2 ] ] ], "statistic": [ 0, 1, 0, 1 ], "domain": "Exact_finite_sufficient_statistic", "operation": "suff-nonsufficient-zero-boundary", "semantic_template": "qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"zero_mass\",\"probability_rows\":[[[1,2],[0,1],[1,2],[0,1]],[[0,1],[1,2],[0,1],[1,2]]],\"samples\":[50,51,52,53],\"statistic\":[0,1,0,1],\"theta_values\":[\"a5\",\"b5\"]}" } }, "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-0081", "batch_id": "batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "110494fa232e3f2123ef775dc51cd119c52589afc8f66a86fdd807535c8f8095", "family_id": "qf-mod47-suff-nonsufficient-zero-boundary", "seed": 202611165081, "semantic_template": "qf-mod47-suff-nonsufficient-zero-boundary/{\"mode\":\"zero_mass\",\"probability_rows\":[[[1,2],[0,1],[1,2],[0,1]],[[0,1],[1,2],[0,1],[1,2]]],\"samples\":[50,51,52,53],\"statistic\":[0,1,0,1],\"theta_values\":[\"a5\",\"b5\"]}" }, "fields": { "title": "Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks", "stem": "Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).", "answer": { "kind": "expression", "value": "(zero_mass, [[0, [[a5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, [[a5, 1, 0, null], [b5, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)", "explanation": "Retain every invariance row, zero-mass cells and an explicit failing fiber for nonsufficient. An injective identity statistic is sufficient. A constant statistic is sufficient exactly when all theta laws agree. The constant-equivalence slot checks that equivalence only in constant mode; other modes retain the source true placeholder, not an additional theorem. Neither sufficiency nor data reduction by itself certifies estimator quality. The exact result, retaining the requested order, is (zero_mass, [[0, [[a5, [[50, 1/2], [52, 1/2]]]], Yes], [1, [[b5, [[51, 1/2], [53, 1/2]]]], Yes]], Yes, [[a5, 1, 0, null], [b5, 0, 0, null]], null, No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)." }, "skills": [ "Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks" ], "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-nonsufficient-zero-boundary；資料={\"mode\":\"zero_mass\",\"theta_values\":[\"a5\",\"b5\"],\"samples\":[50,51,52,53],\"probability_rows\":[[[1,2],0,[1,2],0],[0,[1,2],0,[1,2]]],\"statistic\":[0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use complete theta labels=[a5, b5], samples=[50, 51, 52, 53], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "Θ=[a5, b5]; Ω=[50, 51, 52, 53]", "after": "[[1/2, 0, 1/2, 0], [0, 1/2, 0, 1/2]]", "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_a5(Ω) := sum(1/2, 0, 1/2, 0)", "after": "P_a5(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_a5(Ω)，不以小數近似替代。", "en": "Recompute P_a5(Ω) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "0", "1/2", "0" ], "output": "1", "label": "P_a5(Ω)" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "P_b5(Ω) := sum(0, 1/2, 0, 1/2)", "after": "P_b5(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_b5(Ω)，不以小數近似替代。", "en": "Recompute P_b5(Ω) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1/2", "0", "1/2" ], "output": "1", "label": "P_b5(Ω)" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "sample=50", "after": "T(50)=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=51", "after": "T(51)=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=52", "after": "T(52)=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=53", "after": "T(53)=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": "[(50, 0), (51, 1), (52, 0), (53, 1)]", "after": "fibers=[[0, [50, 52]], [1, [51, 53]]]", "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_a5(T=0) := sum(1/2, 1/2)", "after": "P_a5(T=0) = 1", "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_a5(T=0)，不以小數近似替代。", "en": "Recompute P_a5(T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2" ], "output": "1", "label": "P_a5(T=0)" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "P_a5(50|T=0) := div(1/2, 1)", "after": "P_a5(50|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_a5(50|T=0)，不以小數近似替代。", "en": "Recompute P_a5(50|T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "P_a5(50|T=0)" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "reconstruct atom 50 under a5 := mul(1/2, 1)", "after": "reconstruct atom 50 under a5 = 1/2", "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 50 under a5，不以小數近似替代。", "en": "Recompute reconstruct atom 50 under a5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "reconstruct atom 50 under a5" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "P_a5(52|T=0) := div(1/2, 1)", "after": "P_a5(52|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_a5(52|T=0)，不以小數近似替代。", "en": "Recompute P_a5(52|T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "P_a5(52|T=0)" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "reconstruct atom 52 under a5 := mul(1/2, 1)", "after": "reconstruct atom 52 under a5 = 1/2", "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 52 under a5，不以小數近似替代。", "en": "Recompute reconstruct atom 52 under a5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "reconstruct atom 52 under a5" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "conditional total (a5,0) := sum(1/2, 1/2)", "after": "conditional total (a5,0) = 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 (a5,0)，不以小數近似替代。", "en": "Recompute conditional total (a5,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2" ], "output": "1", "label": "conditional total (a5,0)" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "P_a5(T=1) := sum(0, 0)", "after": "P_a5(T=1) = 0", "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_a5(T=1)，不以小數近似替代。", "en": "Recompute P_a5(T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "P_a5(T=1)" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "P_a5(T=1)=0", "after": "conditional=null (undefined)", "relation": "implies", "rule_id": "null_fiber_guard", "conditions": [], "explanation": { "zh-Hant": "本合約不對零質量cell作條件除法，也不捏造零分布。", "en": "This contract leaves zero-mass conditional cells undefined, rather than fabricating a zero distribution." }, "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": "S017", "uses": [ "G001", "S016" ], "before": "P_b5(T=0) := sum(0, 0)", "after": "P_b5(T=0) = 0", "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_b5(T=0)，不以小數近似替代。", "en": "Recompute P_b5(T=0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "P_b5(T=0)" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "P_b5(T=0)=0", "after": "conditional=null (undefined)", "relation": "implies", "rule_id": "null_fiber_guard", "conditions": [], "explanation": { "zh-Hant": "本合約不對零質量cell作條件除法，也不捏造零分布。", "en": "This contract leaves zero-mass conditional cells undefined, rather than fabricating a zero distribution." }, "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": "S019", "uses": [ "G001", "S018" ], "before": "P_b5(T=1) := sum(1/2, 1/2)", "after": "P_b5(T=1) = 1", "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_b5(T=1)，不以小數近似替代。", "en": "Recompute P_b5(T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2" ], "output": "1", "label": "P_b5(T=1)" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "P_b5(51|T=1) := div(1/2, 1)", "after": "P_b5(51|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_b5(51|T=1)，不以小數近似替代。", "en": "Recompute P_b5(51|T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "P_b5(51|T=1)" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "reconstruct atom 51 under b5 := mul(1/2, 1)", "after": "reconstruct atom 51 under b5 = 1/2", "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 51 under b5，不以小數近似替代。", "en": "Recompute reconstruct atom 51 under b5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "reconstruct atom 51 under b5" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "P_b5(53|T=1) := div(1/2, 1)", "after": "P_b5(53|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_b5(53|T=1)，不以小數近似替代。", "en": "Recompute P_b5(53|T=1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "P_b5(53|T=1)" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "reconstruct atom 53 under b5 := mul(1/2, 1)", "after": "reconstruct atom 53 under b5 = 1/2", "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 53 under b5，不以小數近似替代。", "en": "Recompute reconstruct atom 53 under b5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "reconstruct atom 53 under b5" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "conditional total (b5,1) := sum(1/2, 1/2)", "after": "conditional total (b5,1) = 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 (b5,1)，不以小數近似替代。", "en": "Recompute conditional total (b5,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2" ], "output": "1", "label": "conditional total (b5,1)" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "valid positive-mass laws for T=0: [[a5, [[50, 1/2], [52, 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": "S026", "uses": [ "G001", "S025" ], "before": "valid positive-mass laws for T=1: [[b5, [[51, 1/2], [53, 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": "S027", "uses": [ "G001", "S026" ], "before": "[[0, [[a5, [[50, 1/2], [52, 1/2]]]], 是], [1, [[b5, [[51, 1/2], [53, 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 }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "[[a5, 1, 0, null], [b5, 0, 0, null]]", "after": "zero-mass cells do not supply a fabricated conditional law", "relation": "implies", "rule_id": "zero_mass_boundary", "conditions": [], "explanation": { "zh-Hant": "留下null並只比較正質量cell；null不是一個與其他分布不同的零向量。", "en": "Retain null and compare only positive-mass cells; null is not a zero vector to compare against distributions." }, "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": [ "zero_mass", [ [ 0, [ [ "a5", [ [ 50, "1/2" ], [ 52, "1/2" ] ] ] ], true ], [ 1, [ [ "b5", [ [ 51, "1/2" ], [ 53, "1/2" ] ] ] ], true ] ], true, [ [ "a5", 1, "0", null ], [ "b5", 0, "0", null ] ], null, false, false, false, true, false, "identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim" ] }, "conclusion": { "canonical_value": "(zero_mass, [[0, [[a5, [[50, 1/2], [52, 1/2]]]], 是], [1, [[b5, [[51, 1/2], [53, 1/2]]]], 是]], 是, [[a5, 1, 0, null], [b5, 0, 0, null]], null, 否, 否, 否, 是, 否, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)", "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": "一個正質量 fiber 的條件律不同即否定充分性；單射統計量充分；常數統計量充分恰當各參數律相同；零質量 cell 保持 null。", "en": "One positive-mass fiber with differing conditional laws refutes sufficiency. An injective statistic is sufficient; a constant one exactly when all laws agree; zero-mass cells stay null." }, "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": 81, "original_line_sha256": "bc491b7b3b8c5faa40eb94073521f932e9a41e75e1a2cd988b19c08323fd6165", "question_compact_sha256": "110494fa232e3f2123ef775dc51cd119c52589afc8f66a86fdd807535c8f8095", "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": "b4f884fa08eea0bdbf63f7ce287df3f15305201f320c6df13289111c38f58332" }, "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": "21854fdca6e8ccb861d67d54f291b623f21d868a45a3babec2b69c1226b35e6e", "original_candidate_line_sha256": "21854fdca6e8ccb861d67d54f291b623f21d868a45a3babec2b69c1226b35e6e" }, "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": 28, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "33654605d111be435757fd4c5bba32384409181b7c36f17574c19815a6d20787", "arithmetic_steps": 16 }, "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." } }