{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-pulse-20261119-r05-0063", "pilot_number": "R0067-34013", "number": 34013, "family_id": "qf-mod50-numerr-conditioning-witness", "title": { "zh-Hant": "第 34013 題：數值近似誤差", "en": "Question 34013: The Same Residual Can Correspond to Different Forward Errors" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-pulse-20261119-r05-0063", "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-conditioning-witness；明示資料={\"residual\":13,\"slope_a\":1,\"slope_b\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。", "answer": { "kind": "expression", "value": "(13, 1, 13, 4, 13/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)", "explanation_zh": "Wave6第十三層核對=(13, 1, 13, 4, 13/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；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": 2, "skills": [ "數值近似誤差" ] }, "variant": { "family_id": "qf-mod50-numerr-conditioning-witness", "seed": 202611195063, "parameters": { "residual": 13, "slope_a": 1, "slope_b": 4, "domain": "Exact_numerical_error_density", "operation": "numerr-conditioning-witness", "semantic_template": "numerr-conditioning-witness|{\"residual\":[13,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}" } }, "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-r05-0063", "batch_id": "batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "59dc4495837661d78bb459910aa312faa5787c9d67aee1775bb15f6b640709fb", "family_id": "qf-mod50-numerr-conditioning-witness", "seed": 202611195063, "semantic_template": "numerr-conditioning-witness|{\"residual\":[13,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}" }, "fields": { "title": "The Same Residual Can Correspond to Different Forward Errors", "stem": "Use positive absolute residual=13, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).", "answer": { "kind": "expression", "value": "(13, 1, 13, 4, 13/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)", "explanation": "Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (13, 1, 13, 4, 13/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)." }, "skills": [ "The Same Residual Can Correspond to Different Forward Errors" ], "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-conditioning-witness；明示資料={\"residual\":13,\"slope_a\":1,\"slope_b\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。", "en": "Use positive absolute residual=13, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod50-numerr-conditioning-witness, [domain, operation, residual, slope_a, slope_b]]", "after": "derived = [numerr-conditioning-witness, Exact_numerical_error_density]", "relation": "implies", "rule_id": "R67_SOURCE_specific_keys_operation_domain_and_numeric_bank", "conditions": [], "explanation": { "zh-Hant": "按本bank核keys與operation/domain：legacy為各表underscore，MOD50用local hyphen；BASE參數較寬不加未知欄限制。Number.isInteger與SAFE兩種來源界分開，原JSON數字表示仍完整保存。", "en": "Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [conditioning ORIGINAL residual, 13, 1]", "after": "derived = [[13, 1], 13]", "relation": "implies", "rule_id": "Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "MOD50先驗原num/den各±1億與非零den，之後才約分；不能先cross-cancel以規避此來源界。超界是SOURCE切片不合，不是數學為假。", "en": "MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator := div(13, 1)", "after": "conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator = 13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator，不以小數近似替代。", "en": "Recompute conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "13", "1" ], "output": "13", "label": "conditioning ORIGINAL residual ORIGINAL numerator over nonzero denominator" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [conditioning ORIGINAL slope A, 1, 1]", "after": "derived = [[1, 1], 1]", "relation": "implies", "rule_id": "Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "MOD50先驗原num/den各±1億與非零den，之後才約分；不能先cross-cancel以規避此來源界。超界是SOURCE切片不合，不是數學為假。", "en": "MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator := div(1, 1)", "after": "conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator，不以小數近似替代。", "en": "Recompute conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "conditioning ORIGINAL slope A ORIGINAL numerator over nonzero denominator" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [conditioning ORIGINAL slope B, 4, 1]", "after": "derived = [[4, 1], 4]", "relation": "implies", "rule_id": "Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "MOD50先驗原num/den各±1億與非零den，之後才約分；不能先cross-cancel以規避此來源界。超界是SOURCE切片不合，不是數學為假。", "en": "MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator := div(4, 1)", "after": "conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator，不以小數近似替代。", "en": "Recompute conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "4", "1" ], "output": "4", "label": "conditioning ORIGINAL slope B ORIGINAL numerator over nonzero denominator" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [A, 1]", "after": "derived = [1, 是]", "relation": "implies", "rule_id": "Conditioning_NONzero_absolute_slope_before_division", "conditions": [], "explanation": { "zh-Hant": "先驗斜率非零再取abs，正residual對不同abs slopes才是辨別兩種forward errors的有效例；不能拿兩個相反但同絕對值斜率當此反例。", "en": "FIRST require a nonzero slope,then take its absolute value.A positive residual and DIFFERENT absolute slopes discriminate the two forward errors.Opposite slopes of equal magnitude do not qualify." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [conditioning error A SOURCE naive division, 13, 1]", "after": "derived = [[13, 1], 13]", "relation": "implies", "rule_id": "Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "MOD50先驗原num/den各±1億與非零den，之後才約分；不能先cross-cancel以規避此來源界。超界是SOURCE切片不合，不是數學為假。", "en": "MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator := div(13, 1)", "after": "conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator = 13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator，不以小數近似替代。", "en": "Recompute conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "13", "1" ], "output": "13", "label": "conditioning error A SOURCE naive division ORIGINAL numerator over nonzero denominator" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "conditioning error A ORIGINAL residual divided by absolute slope := div(13, 1)", "after": "conditioning error A ORIGINAL residual divided by absolute slope = 13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditioning error A ORIGINAL residual divided by absolute slope，不以小數近似替代。", "en": "Recompute conditioning error A ORIGINAL residual divided by absolute slope with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "13", "1" ], "output": "13", "label": "conditioning error A ORIGINAL residual divided by absolute slope" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [B, 4]", "after": "derived = [4, 是]", "relation": "implies", "rule_id": "Conditioning_NONzero_absolute_slope_before_division", "conditions": [], "explanation": { "zh-Hant": "先驗斜率非零再取abs，正residual對不同abs slopes才是辨別兩種forward errors的有效例；不能拿兩個相反但同絕對值斜率當此反例。", "en": "FIRST require a nonzero slope,then take its absolute value.A positive residual and DIFFERENT absolute slopes discriminate the two forward errors.Opposite slopes of equal magnitude do not qualify." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "source roles = [conditioning error B SOURCE naive division, 13, 4]", "after": "derived = [[13, 4], 13/4]", "relation": "implies", "rule_id": "Conditioning_SOURCE_naive_PREreduction_1e8_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "MOD50先驗原num/den各±1億與非零den，之後才約分；不能先cross-cancel以規避此來源界。超界是SOURCE切片不合，不是數學為假。", "en": "MOD50 bounds the PRE-reduction numerator/denominator by±100million and requires nonzero denominator BEFORE reduction.Do not cross-cancel first to evade THIS source envelope.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator := div(13, 4)", "after": "conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator = 13/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator，不以小數近似替代。", "en": "Recompute conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "13", "4" ], "output": "13/4", "label": "conditioning error B SOURCE naive division ORIGINAL numerator over nonzero denominator" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "conditioning error B ORIGINAL residual divided by absolute slope := div(13, 4)", "after": "conditioning error B ORIGINAL residual divided by absolute slope = 13/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 conditioning error B ORIGINAL residual divided by absolute slope，不以小數近似替代。", "en": "Recompute conditioning error B ORIGINAL residual divided by absolute slope with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "13", "4" ], "output": "13/4", "label": "conditioning error B ORIGINAL residual divided by absolute slope" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "source roles = [13, 1, 4]", "after": "derived = [13, 13/4, 是]", "relation": "implies", "rule_id": "Conditioning_SAME_positive_residual_DIFFERENT_affine_forward_errors", "conditions": [], "explanation": { "zh-Hant": "affine模型由residual=abs(slope)*forward得forward=residual/abs(slope)。相同正residual可有不同forward error，因model尺度不同；這個例不提供任意問題的condition number理論或收斂證書。", "en": "The affine identity residual=abs(slope)*forward gives forward=residual/abs(slope).The SAME positive residual has different forward errors under different model scales.This witness does not supply general condition-number theory or convergence." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null } ], "math_model": { "result": [ "13", "1", "13", "4", "13/4", true, "same_residual_can_have_different_forward_error_under_different_conditioning" ] }, "conclusion": { "canonical_value": "(13, 1, 13, 4, 13/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)", "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": 63, "original_line_sha256": "f7d82febdad59c8f50b9304ccba944156300b34d5d97a824536e0e5f58b33574", "question_compact_sha256": "59dc4495837661d78bb459910aa312faa5787c9d67aee1775bb15f6b640709fb", "english_row_compact_sha256": "805bb99c45c48960a8e627ed959cb0c1abcbe2105168748d94f4aa41e0daa431", "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": "6bb156cec7bdb6baf2497762702fc01f94fd1db55a5926bc575b12ff9237a4de", "original_candidate_sha256": "c59b91502b58309ca7611aa1535784c80c4a73b650c93019c4aa93e5c2d8a1c3", "original_candidate_line_sha256": "c59b91502b58309ca7611aa1535784c80c4a73b650c93019c4aa93e5c2d8a1c3" }, "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-p68", "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": "c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c", "question_first_witness_sha256": "12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 16, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "4e7744c71b8d56ada4b2be54cc9391970426e9ac71d7ce1550306e30161c82f5", "arithmetic_steps": 7 }, "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." } }