{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261115-r03-bias-variance-decomposition-0078", "pilot_number": "R0050-25503", "number": 25503, "family_id": "qf-mod46-biasvar-biased-unbiased-boundary", "title": { "zh-Hant": "第 25503 題：偏誤－變異數分解", "en": "Question 25503: Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261115-r03-bias-variance-decomposition-0078", "batch_id": "batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-bias-variance-decomposition" ], "title_zh": "偏誤－變異數分解", "stem_zh": "逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-10,\"support\":[-13,-10,-7],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(-10, -13, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)", "explanation_zh": "Wave6第九層核對=(-10, -13, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "偏誤－變異數分解" ] }, "variant": { "family_id": "qf-mod46-biasvar-biased-unbiased-boundary", "seed": 202611153078, "parameters": { "theta": -10, "support": [ -13, -10, -7 ], "probabilities": [ [ 1, 4 ], [ 1, 2 ], [ 1, 4 ] ], "observed": -13, "domain": "Exact_finite_bias_variance_decomposition", "operation": "biasvar-biased-unbiased-boundary", "semantic_template": "qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-10,1],[-13,1],[-7,1]],\"theta\":[-10,1]}" } }, "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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0078", "batch_id": "batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "0824d5f132340a974d0299127ef63eacf3d0a2d9e11c0c69a393a08b8d1ab9cd", "family_id": "qf-mod46-biasvar-biased-unbiased-boundary", "seed": 202611153078, "semantic_template": "qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-10,1],[-13,1],[-7,1]],\"theta\":[-10,1]}" }, "fields": { "title": "Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency", "stem": "Use scalar theta=-10, complete estimator values=[-13, -10, -7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).", "answer": { "kind": "expression", "value": "(-10, -13, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)", "explanation": "Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-10, -13, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)." }, "skills": [ "Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency" ], "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": "逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-10,\"support\":[-13,-10,-7],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use scalar theta=-10, complete estimator values=[-13, -10, -7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [biasvar-biased-unbiased-boundary]", "after": "derived = [[theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "直接列完整來源欄序，不需JavaScript。LP weak為13欄、zero與strong為15欄，前提與角色依原operation／mode，不補欄。", "en": "List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Biasvar ORIGINAL single theta ORIGINAL mass sum := sum(1/4, 1/2, 1/4)", "after": "Biasvar ORIGINAL single theta ORIGINAL mass sum = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta ORIGINAL mass sum，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta ORIGINAL mass sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/4", "1/2", "1/4" ], "output": "1", "label": "Biasvar ORIGINAL single theta ORIGINAL mass sum" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [Biasvar ORIGINAL single theta, -10, [-13, -10, -7], [[1, 4], [1, 2], [1, 4]]]", "after": "derived = [[(-13, 1/4), (-10, 1/2), (-7, 1/4)], 1]", "relation": "implies", "rule_id": "Biasvar_complete_SAME_law_numeric_sorted_ALIGNED_null_rows", "conditions": [], "explanation": { "zh-Hant": "先核同一P_theta全law，原序mass非負總和1，再連value/mass一起數值排序；null rows保留。與前一MSE節點不同，這裡明示要證bias-variance分解。", "en": "Qualify this same complete P_theta law,with nonnegative original-order mass summing to1,and sort values numerically with their masses.Keep null rows.Unlike the earlier MSE node,this node explicitly proves the decomposition." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Biasvar ORIGINAL single theta mean product0 := mul(-13, 1/4)", "after": "Biasvar ORIGINAL single theta mean product0 = -13/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta mean product0，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta mean product0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-13", "1/4" ], "output": "-13/4", "label": "Biasvar ORIGINAL single theta mean product0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Biasvar ORIGINAL single theta mean product1 := mul(-10, 1/2)", "after": "Biasvar ORIGINAL single theta mean product1 = -5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta mean product1，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta mean product1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-10", "1/2" ], "output": "-5", "label": "Biasvar ORIGINAL single theta mean product1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Biasvar ORIGINAL single theta mean product2 := mul(-7, 1/4)", "after": "Biasvar ORIGINAL single theta mean product2 = -7/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta mean product2，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta mean product2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-7", "1/4" ], "output": "-7/4", "label": "Biasvar ORIGINAL single theta mean product2" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Biasvar ORIGINAL single theta mean sum := sum(-13/4, -5, -7/4)", "after": "Biasvar ORIGINAL single theta mean sum = -10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta mean sum，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta mean sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-13/4", "-5", "-7/4" ], "output": "-10", "label": "Biasvar ORIGINAL single theta mean sum" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Biasvar ORIGINAL single theta bias mean minus theta := sub(-10, -10)", "after": "Biasvar ORIGINAL single theta bias mean minus theta = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta bias mean minus theta，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta bias mean minus theta with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-10", "-10" ], "output": "0", "label": "Biasvar ORIGINAL single theta bias mean minus theta" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Biasvar ORIGINAL single theta centered deviation0 := sub(-13, -10)", "after": "Biasvar ORIGINAL single theta centered deviation0 = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered deviation0，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered deviation0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-13", "-10" ], "output": "-3", "label": "Biasvar ORIGINAL single theta centered deviation0" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Biasvar ORIGINAL single theta centered square0 := mul(-3, -3)", "after": "Biasvar ORIGINAL single theta centered square0 = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered square0，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered square0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "-3" ], "output": "9", "label": "Biasvar ORIGINAL single theta centered square0" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Biasvar ORIGINAL single theta variance weighted term0 := mul(1/4, 9)", "after": "Biasvar ORIGINAL single theta variance weighted term0 = 9/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta variance weighted term0，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta variance weighted term0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "9" ], "output": "9/4", "label": "Biasvar ORIGINAL single theta variance weighted term0" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Biasvar ORIGINAL single theta centered deviation1 := sub(-10, -10)", "after": "Biasvar ORIGINAL single theta centered deviation1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered deviation1，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered deviation1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-10", "-10" ], "output": "0", "label": "Biasvar ORIGINAL single theta centered deviation1" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Biasvar ORIGINAL single theta centered square1 := mul(0, 0)", "after": "Biasvar ORIGINAL single theta centered square1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered square1，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered square1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Biasvar ORIGINAL single theta centered square1" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Biasvar ORIGINAL single theta variance weighted term1 := mul(1/2, 0)", "after": "Biasvar ORIGINAL single theta variance weighted term1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta variance weighted term1，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta variance weighted term1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "0" ], "output": "0", "label": "Biasvar ORIGINAL single theta variance weighted term1" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Biasvar ORIGINAL single theta centered deviation2 := sub(-7, -10)", "after": "Biasvar ORIGINAL single theta centered deviation2 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered deviation2，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered deviation2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-7", "-10" ], "output": "3", "label": "Biasvar ORIGINAL single theta centered deviation2" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Biasvar ORIGINAL single theta centered square2 := mul(3, 3)", "after": "Biasvar ORIGINAL single theta centered square2 = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered square2，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered square2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "Biasvar ORIGINAL single theta centered square2" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Biasvar ORIGINAL single theta variance weighted term2 := mul(1/4, 9)", "after": "Biasvar ORIGINAL single theta variance weighted term2 = 9/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta variance weighted term2，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta variance weighted term2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "9" ], "output": "9/4", "label": "Biasvar ORIGINAL single theta variance weighted term2" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Biasvar ORIGINAL single theta variance sum := sum(9/4, 0, 9/4)", "after": "Biasvar ORIGINAL single theta variance sum = 9/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta variance sum，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta variance sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9/4", "0", "9/4" ], "output": "9/2", "label": "Biasvar ORIGINAL single theta variance sum" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Biasvar ORIGINAL single theta theta error0 := sub(-13, -10)", "after": "Biasvar ORIGINAL single theta theta error0 = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta theta error0，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta theta error0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-13", "-10" ], "output": "-3", "label": "Biasvar ORIGINAL single theta theta error0" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Biasvar ORIGINAL single theta theta square0 := mul(-3, -3)", "after": "Biasvar ORIGINAL single theta theta square0 = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta theta square0，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta theta square0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "-3" ], "output": "9", "label": "Biasvar ORIGINAL single theta theta square0" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Biasvar ORIGINAL single theta direct MSE term0 := mul(1/4, 9)", "after": "Biasvar ORIGINAL single theta direct MSE term0 = 9/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta direct MSE term0，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta direct MSE term0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "9" ], "output": "9/4", "label": "Biasvar ORIGINAL single theta direct MSE term0" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Biasvar ORIGINAL single theta theta error1 := sub(-10, -10)", "after": "Biasvar ORIGINAL single theta theta error1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta theta error1，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta theta error1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-10", "-10" ], "output": "0", "label": "Biasvar ORIGINAL single theta theta error1" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Biasvar ORIGINAL single theta theta square1 := mul(0, 0)", "after": "Biasvar ORIGINAL single theta theta square1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta theta square1，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta theta square1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Biasvar ORIGINAL single theta theta square1" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "Biasvar ORIGINAL single theta direct MSE term1 := mul(1/2, 0)", "after": "Biasvar ORIGINAL single theta direct MSE term1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta direct MSE term1，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta direct MSE term1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "0" ], "output": "0", "label": "Biasvar ORIGINAL single theta direct MSE term1" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "Biasvar ORIGINAL single theta theta error2 := sub(-7, -10)", "after": "Biasvar ORIGINAL single theta theta error2 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta theta error2，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta theta error2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-7", "-10" ], "output": "3", "label": "Biasvar ORIGINAL single theta theta error2" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "Biasvar ORIGINAL single theta theta square2 := mul(3, 3)", "after": "Biasvar ORIGINAL single theta theta square2 = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta theta square2，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta theta square2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "Biasvar ORIGINAL single theta theta square2" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "Biasvar ORIGINAL single theta direct MSE term2 := mul(1/4, 9)", "after": "Biasvar ORIGINAL single theta direct MSE term2 = 9/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta direct MSE term2，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta direct MSE term2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "9" ], "output": "9/4", "label": "Biasvar ORIGINAL single theta direct MSE term2" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "Biasvar ORIGINAL single theta direct MSE sum := sum(9/4, 0, 9/4)", "after": "Biasvar ORIGINAL single theta direct MSE sum = 9/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta direct MSE sum，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta direct MSE sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9/4", "0", "9/4" ], "output": "9/2", "label": "Biasvar ORIGINAL single theta direct MSE sum" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "Biasvar ORIGINAL single theta centered mean product0 := mul(1/4, -3)", "after": "Biasvar ORIGINAL single theta centered mean product0 = -3/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered mean product0，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered mean product0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "-3" ], "output": "-3/4", "label": "Biasvar ORIGINAL single theta centered mean product0" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "Biasvar ORIGINAL single theta centered mean product1 := mul(1/2, 0)", "after": "Biasvar ORIGINAL single theta centered mean product1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered mean product1，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered mean product1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "0" ], "output": "0", "label": "Biasvar ORIGINAL single theta centered mean product1" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "Biasvar ORIGINAL single theta centered mean product2 := mul(1/4, 3)", "after": "Biasvar ORIGINAL single theta centered mean product2 = 3/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered mean product2，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered mean product2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "3" ], "output": "3/4", "label": "Biasvar ORIGINAL single theta centered mean product2" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "Biasvar ORIGINAL single theta centered mean sum := sum(-3/4, 0, 3/4)", "after": "Biasvar ORIGINAL single theta centered mean sum = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta centered mean sum，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta centered mean sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-3/4", "0", "3/4" ], "output": "0", "label": "Biasvar ORIGINAL single theta centered mean sum" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "Biasvar ORIGINAL single theta bias times centered mean := mul(0, 0)", "after": "Biasvar ORIGINAL single theta bias times centered mean = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta bias times centered mean，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta bias times centered mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Biasvar ORIGINAL single theta bias times centered mean" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "Biasvar ORIGINAL single theta cross twice inner := mul(2, 0)", "after": "Biasvar ORIGINAL single theta cross twice inner = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta cross twice inner，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta cross twice inner with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "0" ], "output": "0", "label": "Biasvar ORIGINAL single theta cross twice inner" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "Biasvar ORIGINAL single theta squared bias := mul(0, 0)", "after": "Biasvar ORIGINAL single theta squared bias = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta squared bias，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta squared bias with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Biasvar ORIGINAL single theta squared bias" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "Biasvar ORIGINAL single theta variance plus squared bias := sum(9/2, 0)", "after": "Biasvar ORIGINAL single theta variance plus squared bias = 9/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL single theta variance plus squared bias，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL single theta variance plus squared bias with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9/2", "0" ], "output": "9/2", "label": "Biasvar ORIGINAL single theta variance plus squared bias" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "source roles = [Biasvar ORIGINAL single theta, -10, -10, 0, [(-13, 1/4), (-10, 1/2), (-7, 1/4)]]", "after": "derived = [[[-13, 1/4, -3, 0, -3, 9, 0, 0, 9], [-10, 1/2, 0, 0, 0, 0, 0, 0, 0], [-7, 1/4, 3, 0, 3, 9, 0, 0, 9]], 0, 0, 9/2, 9/2, 9/2, 是]", "relation": "implies", "rule_id": "Biasvar_EVERY_value_square_expansion_and_GENERAL_cross_cancellation", "conditions": [], "explanation": { "zh-Hant": "逐值T−theta=(T−mean)+bias，平方成centered²+2*bias*centered+bias²，再按同一mass加權。E[T−mean]=E[T]−mean*1=0使cross消失，不要求bias0或每個deviation0；mass1保bias²。逐點cross是額外精確教學量，不套原source未執行的新profile界。", "en": "For every value,T−theta=(T−mean)+bias.Square to centered²+2*bias*centered+bias² and weight under the same law.E[T−mean]=E[T]−mean*1=0 cancels the cross term,without requiring bias0 or every deviation0.Total mass1 leaves bias².Pointwise cross terms are extra exact teaching values,not new source-profile restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "Biasvar ORIGINAL observed minus theta := sub(-13, -10)", "after": "Biasvar ORIGINAL observed minus theta = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL observed minus theta，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL observed minus theta with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-13", "-10" ], "output": "-3", "label": "Biasvar ORIGINAL observed minus theta" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "Biasvar ORIGINAL observed square := mul(-3, -3)", "after": "Biasvar ORIGINAL observed square = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。", "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Biasvar ORIGINAL observed square，不以小數近似替代。", "en": "Recompute Biasvar ORIGINAL observed square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "-3" ], "output": "9", "label": "Biasvar ORIGINAL observed square" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "source roles = [-13, 9, 0, 9/2, 9/2]", "after": "derived = [是, 是, 否, 否]", "relation": "implies", "rule_id": "Biasvar_observed_error_unbiased_variance_risk_consistency_separate", "conditions": [], "explanation": { "zh-Hant": "unbiased只把bias²去掉，給MSE=variance，未給variance0或consistency；biased時MSE>=variance。observed loss另算，可能偶合MSE但不是它的定義。", "en": "Unbiasedness only removes bias²,giving MSE=variance,not variance0 or consistency.When biased,MSE>=variance.One observed loss is computed separately and may coincide with MSE without defining it." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ "-10", "-13", "9", "0", true, "9/2", "9/2", true, false, false, "unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE" ] }, "conclusion": { "canonical_value": "(-10, -13, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations/questions.jsonl", "batch_sha256": "8bb4a1c7244564cd7b74f9fc3d033d2ccf9e13b062ca7e18233eec11de9da585", "original_line": 78, "original_line_sha256": "ff323903e410bfed0567ee5633bc1d1c651fe33f071ba73691c78a02b921e171", "question_compact_sha256": "0824d5f132340a974d0299127ef63eacf3d0a2d9e11c0c69a393a08b8d1ab9cd", "english_row_compact_sha256": "047087d234ba7fe870255e1b487d661d6886932e269892e6e640cd95d294a798", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations.en.jsonl" }, "mko_bindings": { "mko-bias-variance-decomposition": { "sha256": "03c45bb05c65b40d9f88d66e9009cd0182117446dd67211eee759fa9e0b2b377", "english_sha256": "f96f4ef0ee398bbc7afc47fb6083de94a60ab677ee02c79749ac9b4e06a44499", "original_path": "public/data/mko/bias-variance-decomposition.json", "english_path": "public/data/i18n/editions/mko-bias-variance-decomposition.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "4624c29c8cf22048332daea19f430f922e032c30410aa690ba3f1924d8b9ee45", "original_candidate_sha256": "aff8fcf701863f6e64eaf6a27db544413b10f0a6bdafa7fa30b70c5011b30a6f", "original_candidate_line_sha256": "aff8fcf701863f6e64eaf6a27db544413b10f0a6bdafa7fa30b70c5011b30a6f" }, "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": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human 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-20261026-010-p51", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings 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": "74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82", "question_first_witness_sha256": "6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 40, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "91038a7000fff88f43b560025610bcf33950083ecce03c63a2c7a8234ba0d63a", "arithmetic_steps": 36 }, "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." } }