{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-pulse-20261119-r05-0022", "pilot_number": "R0066-33972", "number": 33972, "family_id": "qf-mod50-numerr-affine-residual-forward", "title": { "zh-Hant": "第 33972 題：數值近似誤差", "en": "Question 33972: An Affine Residual Equals Absolute Slope Times Forward Error" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-pulse-20261119-r05-0022", "batch_id": "batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-numerical-approximation-error" ], "title_zh": "數值近似誤差", "stem_zh": "exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":17,\"intercept\":22,\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。", "answer": { "kind": "expression", "value": "(-3, 22, 17, 22/3, 29, 29/3, 29, 是, residual_and_forward_error_are_distinct_quantities)", "explanation_zh": "Wave6第十三層核對=(-3, 22, 17, 22/3, 29, 29/3, 29, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "數值近似誤差" ] }, "variant": { "family_id": "qf-mod50-numerr-affine-residual-forward", "seed": 202611195022, "parameters": { "slope": -3, "intercept": 22, "approximation": 17, "domain": "Exact_numerical_error_density", "operation": "numerr-affine-residual-forward", "semantic_template": "numerr-affine-residual-forward|{\"approximation\":[17,1],\"intercept\":[22,1],\"slope\":[-3,1]}" } }, "verification": { "status": "candidate", "checks": [ "producer_contract_recomputation", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "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-pulse-20261119-r05-0022", "batch_id": "batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "0facaf5cf254515d6f7c4c85b88c2d50d1dae58e83e64b064dd4ad147d2e6d10", "family_id": "qf-mod50-numerr-affine-residual-forward", "seed": 202611195022, "semantic_template": "numerr-affine-residual-forward|{\"approximation\":[17,1],\"intercept\":[22,1],\"slope\":[-3,1]}" }, "fields": { "title": "An Affine Residual Equals Absolute Slope Times Forward Error", "stem": "Use nonzero slope=-3, intercept=22, approximation=17 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).", "answer": { "kind": "expression", "value": "(-3, 22, 17, 22/3, 29, 29/3, 29, Yes, residual_and_forward_error_are_distinct_quantities)", "explanation": "Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 22, 17, 22/3, 29, 29/3, 29, Yes, residual_and_forward_error_are_distinct_quantities)." }, "skills": [ "An Affine Residual Equals Absolute Slope Times Forward Error" ], "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": "exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":17,\"intercept\":22,\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。", "en": "Use nonzero slope=-3, intercept=22, approximation=17 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod50-numerr-affine-residual-forward, [approximation, domain, intercept, operation, slope]]", "after": "derived = [numerr-affine-residual-forward, Exact_numerical_error_density]", "relation": "implies", "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain", "conditions": [], "explanation": { "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。", "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [error ORIGINAL slope, 100000000, -3, 1]", "after": "derived = [[-3, 1], -3]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "error ORIGINAL slope ORIGINAL numerator over denominator := div(-3, 1)", "after": "error ORIGINAL slope ORIGINAL numerator over denominator = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 error ORIGINAL slope ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute error ORIGINAL slope ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-3", "1" ], "output": "-3", "label": "error ORIGINAL slope ORIGINAL numerator over denominator" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [error ORIGINAL intercept, 100000000, 22, 1]", "after": "derived = [[22, 1], 22]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "error ORIGINAL intercept ORIGINAL numerator over denominator := div(22, 1)", "after": "error ORIGINAL intercept ORIGINAL numerator over denominator = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 error ORIGINAL intercept ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute error ORIGINAL intercept ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "22", "1" ], "output": "22", "label": "error ORIGINAL intercept ORIGINAL numerator over denominator" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [error ORIGINAL approximation, 100000000, 17, 1]", "after": "derived = [[17, 1], 17]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "error ORIGINAL approximation ORIGINAL numerator over denominator := div(17, 1)", "after": "error ORIGINAL approximation ORIGINAL numerator over denominator = 17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 error ORIGINAL approximation ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute error ORIGINAL approximation ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "17", "1" ], "output": "17", "label": "error ORIGINAL approximation ORIGINAL numerator over denominator" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [Affine exact root SOURCE naive div, 100000000, -22, -3]", "after": "derived = [[22, 3], 22/3]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Affine exact root SOURCE naive div ORIGINAL numerator over denominator := div(-22, -3)", "after": "Affine exact root SOURCE naive div ORIGINAL numerator over denominator = 22/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine exact root SOURCE naive div ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute Affine exact root SOURCE naive div ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-22", "-3" ], "output": "22/3", "label": "Affine exact root SOURCE naive div ORIGINAL numerator over denominator" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Affine exact root ORIGINAL rational div := div(-22, -3)", "after": "Affine exact root ORIGINAL rational div = 22/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine exact root ORIGINAL rational div，不以小數近似替代。", "en": "Recompute Affine exact root ORIGINAL rational div with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-22", "-3" ], "output": "22/3", "label": "Affine exact root ORIGINAL rational div" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [Affine residual original slope times approximation SOURCE naive mul, 100000000, -51, 1]", "after": "derived = [[-51, 1], -51]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator := div(-51, 1)", "after": "Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator = -51", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-51", "1" ], "output": "-51", "label": "Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Affine residual original slope times approximation ORIGINAL rational mul := mul(-3, 17)", "after": "Affine residual original slope times approximation ORIGINAL rational mul = -51", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine residual original slope times approximation ORIGINAL rational mul，不以小數近似替代。", "en": "Recompute Affine residual original slope times approximation ORIGINAL rational mul with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "17" ], "output": "-51", "label": "Affine residual original slope times approximation ORIGINAL rational mul" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "source roles = [Affine residual add original intercept SOURCE naive sum, 100000000, -29, 1]", "after": "derived = [[-29, 1], -29]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator := div(-29, 1)", "after": "Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator = -29", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-29", "1" ], "output": "-29", "label": "Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Affine residual add original intercept ORIGINAL rational sum := sum(-51, 22)", "after": "Affine residual add original intercept ORIGINAL rational sum = -29", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine residual add original intercept ORIGINAL rational sum，不以小數近似替代。", "en": "Recompute Affine residual add original intercept ORIGINAL rational sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-51", "22" ], "output": "-29", "label": "Affine residual add original intercept ORIGINAL rational sum" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [Affine forward original approximation minus exact root SOURCE naive sub, 100000000, 29, 3]", "after": "derived = [[29, 3], 29/3]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator := div(29, 3)", "after": "Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator = 29/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "29", "3" ], "output": "29/3", "label": "Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Affine forward original approximation minus exact root ORIGINAL rational sub := sub(17, 22/3)", "after": "Affine forward original approximation minus exact root ORIGINAL rational sub = 29/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine forward original approximation minus exact root ORIGINAL rational sub，不以小數近似替代。", "en": "Recompute Affine forward original approximation minus exact root ORIGINAL rational sub with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "17", "22/3" ], "output": "29/3", "label": "Affine forward original approximation minus exact root ORIGINAL rational sub" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "source roles = [Affine absolute slope times forward error SOURCE naive mul, 100000000, 87, 3]", "after": "derived = [[29, 1], 29]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator := div(87, 3)", "after": "Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator = 29", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "87", "3" ], "output": "29", "label": "Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Affine absolute slope times forward error ORIGINAL rational mul := mul(3, 29/3)", "after": "Affine absolute slope times forward error ORIGINAL rational mul = 29", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine absolute slope times forward error ORIGINAL rational mul，不以小數近似替代。", "en": "Recompute Affine absolute slope times forward error ORIGINAL rational mul with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "29/3" ], "output": "29", "label": "Affine absolute slope times forward error ORIGINAL rational mul" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "source roles = [-3, 22, 17]", "after": "derived = [22/3, -29, 29, 29/3, 29/3, 29, 是]", "relation": "implies", "rule_id": "Error_AFFINE_exact_root_residual_forward_scale_identity", "conditions": [], "explanation": { "zh-Hant": "m非0先求真root=−b/m；分別計算|m*xhat+b|與|xhat−root|。因m*root+b=0，residual=|m|*forward。數值有時相同也不表示兩量的定義相同；原model才供應真root。", "en": "For m≠0 first solve the TRUE root−b/m.Calculate |m*xhat+b| and |xhat−root| SEPARATELY.Since m*root+b=0,residual=|m|*forward.Occasional numerical equality does not identify the definitions;the supplied model gives the true root." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null } ], "math_model": { "result": [ "-3", "22", "17", "22/3", "29", "29/3", "29", true, "residual_and_forward_error_are_distinct_quantities" ] }, "conclusion": { "canonical_value": "(-3, 22, 17, 22/3, 29, 29/3, 29, 是, residual_and_forward_error_are_distinct_quantities)", "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-20261119-r05-numerical-approximation-error-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations/questions.jsonl", "batch_sha256": "932ee14b8b2a8fed6d8cebd6df3641eed4560176b36945b28664b2208604fe14", "original_line": 22, "original_line_sha256": "2c3ff8cdc1996ace3eec7b042ee9a1bb5d885caa8610be86ee0d5ccc24014507", "question_compact_sha256": "0facaf5cf254515d6f7c4c85b88c2d50d1dae58e83e64b064dd4ad147d2e6d10", "english_row_compact_sha256": "2234372cbd3a111386abac576174f128721def9f3397f6df29c053627785d07e", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations.en.jsonl" }, "mko_bindings": { "mko-numerical-approximation-error": { "sha256": "ced94d430d6417b9e607be1f09be7234591c3138507fecd536d3d7dc024ecbe0", "english_sha256": "fe552b17f866ceee13097202f2ac4dd54a72ef64059402444b22146b41d82149", "original_path": "public/data/mko/numerical-approximation-error.json", "english_path": "public/data/i18n/editions/mko-numerical-approximation-error.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "df58fcce25614251047f7c41d31ff93b5bfec19cd9bbcaa5fbb42a4daceb0310", "original_candidate_sha256": "afdd31f0596df4844acf49620eb24f62903a64c7ecfec67e76a71b6b61e2c045", "original_candidate_line_sha256": "afdd31f0596df4844acf49620eb24f62903a64c7ecfec67e76a71b6b61e2c045" }, "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-20261026-012-p67", "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": "99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267", "question_first_witness_sha256": "d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 23, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "55ad0087155bd285a28e58cf44d8affb92c2e0949d3b6fec4729cde4d63f320a", "arithmetic_steps": 13 }, "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." } }