{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261007-advance-20261009-r05-antiderivative-0054", "pilot_number": "R0058-29929", "number": 29929, "family_id": "qf-density-primitive-anchor-constant", "title": { "zh-Hant": "第 29929 題：由明示錨點解原函數常數", "en": "Question 29929: Primitive Constant Selected by an Explicit Anchor" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261007-advance-20261009-r05-antiderivative-0054", "batch_id": "batch-20261007-advance-20261009-r05-antiderivative-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-antiderivative" ], "title_zh": "由明示錨點解原函數常數", "stem_zh": "全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-9, 4, -1, 4]，F0常數項固定零。令 F=F0+C 且 F(-4)=-9/1，求C。", "answer": { "kind": "expression", "value": "-1063/3", "explanation_zh": "錨點所需常數=-1063/3。求指定錨點的精確F0值再相減；保留實際點與目標值，C不由求導資格單獨決定。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "由明示錨點解原函數常數" ] }, "variant": { "family_id": "qf-density-primitive-anchor-constant", "seed": 202610095054, "parameters": { "f": [ -9, 4, -1, 4 ], "a": -4, "value_n": -9, "value_d": 1, "operation": "density_primitive_anchor_constant", "domain": "reals", "representation": "ascending_powers", "semantic_template": "qf-density-primitive-anchor-constant/[[-9,4,-1,4],-4,\"-9/1\"]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-07", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261007-advance-20261009-r05-antiderivative-0054", "batch_id": "batch-20261007-advance-20261009-r05-antiderivative-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "01ee21bb9a816388b1c593d7863683f7bbe1c183d1b4d442ef57f4aac84126d7", "family_id": "qf-density-primitive-anchor-constant", "seed": 202610095054, "semantic_template": "qf-density-primitive-anchor-constant/[[-9,4,-1,4],-4,\"-9/1\"]" }, "fields": { "title": "Primitive Constant Selected by an Explicit Anchor", "stem": "On all R, f has ascending coefficients [-9, 4, -1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=-9/1. Use reduced positive-denominator fractions, retaining /1 for integers.", "answer": { "kind": "expression", "value": "-1063/3", "explanation": "Compute C=V−F0(a) exactly to obtain -1063/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation." }, "skills": [ "Primitive Constant Selected by an Explicit Anchor" ], "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": "全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-9, 4, -1, 4]，F0常數項固定零。令 F=F0+C 且 F(-4)=-9/1，求C。", "en": "On all R, f has ascending coefficients [-9, 4, -1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=-9/1. Use reduced positive-denominator fractions, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density-primitive-anchor-constant, density_primitive_anchor_constant]", "after": "derived = [[required_additive_constant_fraction_retaining1]]", "relation": "implies", "rule_id": "R58_EXACT_source_interface_and_answer_ROLE", "conditions": [], "explanation": { "zh-Hant": "依本題銀行的keys、domain或frame、operation與輸出角色。STR22/App20 tuple與legacy scalar不同；分數是否保/1按原契約，不能用共同外觀改題。", "en": "Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [integrand f, [-9, 4, -1, 4], 否]", "after": "derived = [[-9/1, 4/1, -1/1, 4/1]]", "relation": "implies", "rule_id": "Primitive_RAW_arity_SAFE_inputs_POSITIVE_rational_den_trim_KEEPzero", "conditions": [], "explanation": { "zh-Hant": "整數f原長1..4、rational候選F原長1..5；每個raw數SAFE，rational den嚴格正（不是先normalize負den）。升冪只trim尾端0但zero polynomial保一項；衍生運算不套1e6或degree cap。", "en": "Raw integer f has length1..4 and rational candidate F length1..5.Every raw number is SAFE;rational denominators are STRICTLY positive,not repaired from negatives.Trim only trailing zeros,keeping a singleton zero polynomial.Derived algebra has no1e6 or raw-degree cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Primitive ORIGINAL coefficient degree1 := div(-9, 1)", "after": "Primitive ORIGINAL coefficient degree1 = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive ORIGINAL coefficient degree1，不以小數近似替代。", "en": "Recompute Primitive ORIGINAL coefficient degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-9", "1" ], "output": "-9", "label": "Primitive ORIGINAL coefficient degree1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Primitive ORIGINAL coefficient degree2 := div(4, 2)", "after": "Primitive ORIGINAL coefficient degree2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive ORIGINAL coefficient degree2，不以小數近似替代。", "en": "Recompute Primitive ORIGINAL coefficient degree2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "4", "2" ], "output": "2", "label": "Primitive ORIGINAL coefficient degree2" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Primitive ORIGINAL coefficient degree3 := div(-1, 3)", "after": "Primitive ORIGINAL coefficient degree3 = -1/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive ORIGINAL coefficient degree3，不以小數近似替代。", "en": "Recompute Primitive ORIGINAL coefficient degree3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-1", "3" ], "output": "-1/3", "label": "Primitive ORIGINAL coefficient degree3" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Primitive ORIGINAL coefficient degree4 := div(4, 4)", "after": "Primitive ORIGINAL coefficient degree4 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive ORIGINAL coefficient degree4，不以小數近似替代。", "en": "Recompute Primitive ORIGINAL coefficient degree4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "4", "4" ], "output": "1", "label": "Primitive ORIGINAL coefficient degree4" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Primitive reverse ORIGINAL derivative coefficient0 := mul(-9, 1)", "after": "Primitive reverse ORIGINAL derivative coefficient0 = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive reverse ORIGINAL derivative coefficient0，不以小數近似替代。", "en": "Recompute Primitive reverse ORIGINAL derivative coefficient0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-9", "1" ], "output": "-9", "label": "Primitive reverse ORIGINAL derivative coefficient0" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Primitive reverse ORIGINAL derivative coefficient1 := mul(2, 2)", "after": "Primitive reverse ORIGINAL derivative coefficient1 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive reverse ORIGINAL derivative coefficient1，不以小數近似替代。", "en": "Recompute Primitive reverse ORIGINAL derivative coefficient1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Primitive reverse ORIGINAL derivative coefficient1" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Primitive reverse ORIGINAL derivative coefficient2 := mul(-1/3, 3)", "after": "Primitive reverse ORIGINAL derivative coefficient2 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive reverse ORIGINAL derivative coefficient2，不以小數近似替代。", "en": "Recompute Primitive reverse ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1/3", "3" ], "output": "-1", "label": "Primitive reverse ORIGINAL derivative coefficient2" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Primitive reverse ORIGINAL derivative coefficient3 := mul(1, 4)", "after": "Primitive reverse ORIGINAL derivative coefficient3 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive reverse ORIGINAL derivative coefficient3，不以小數近似替代。", "en": "Recompute Primitive reverse ORIGINAL derivative coefficient3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "4" ], "output": "4", "label": "Primitive reverse ORIGINAL derivative coefficient3" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [[-9/1, 4/1, -1/1, 4/1]]", "after": "derived = [[0/1, -9/1, 2/1, -1/3, 1/1], [-9/1, 4/1, -1/1, 4/1], 是]", "relation": "implies", "rule_id": "Primitive_ZEROconstant_ALL_monomial_coefficients_AND_reverse_identity", "conditions": [], "explanation": { "zh-Hant": "由d(x^(j+1)/(j+1))/dx=x^j，逐每項求零constant代表F0，再把所有係數微分回原f確認identity。zero polynomial保[0/1]，不靠幾個sample吻合。", "en": "Since d(x^(j+1)/(j+1))/dx=x^j,integrate EACH monomial into the zero-constant representative F0,and differentiate ALL coefficients back to the original f as an identity.Keep[0/1] for zero;do not rely on sampled agreement." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Primitive anchor ORIGINAL Horner product degree4 := mul(0, -4)", "after": "Primitive anchor ORIGINAL Horner product degree4 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner product degree4，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner product degree4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-4" ], "output": "0", "label": "Primitive anchor ORIGINAL Horner product degree4" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Primitive anchor ORIGINAL Horner sum degree4 := sum(0, 1)", "after": "Primitive anchor ORIGINAL Horner sum degree4 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner sum degree4，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner sum degree4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1" ], "output": "1", "label": "Primitive anchor ORIGINAL Horner sum degree4" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Primitive anchor ORIGINAL Horner product degree3 := mul(1, -4)", "after": "Primitive anchor ORIGINAL Horner product degree3 = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner product degree3，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner product degree3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-4" ], "output": "-4", "label": "Primitive anchor ORIGINAL Horner product degree3" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Primitive anchor ORIGINAL Horner sum degree3 := sum(-4, -1/3)", "after": "Primitive anchor ORIGINAL Horner sum degree3 = -13/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner sum degree3，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner sum degree3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-4", "-1/3" ], "output": "-13/3", "label": "Primitive anchor ORIGINAL Horner sum degree3" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Primitive anchor ORIGINAL Horner product degree2 := mul(-13/3, -4)", "after": "Primitive anchor ORIGINAL Horner product degree2 = 52/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner product degree2，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner product degree2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-13/3", "-4" ], "output": "52/3", "label": "Primitive anchor ORIGINAL Horner product degree2" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Primitive anchor ORIGINAL Horner sum degree2 := sum(52/3, 2)", "after": "Primitive anchor ORIGINAL Horner sum degree2 = 58/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner sum degree2，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner sum degree2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "52/3", "2" ], "output": "58/3", "label": "Primitive anchor ORIGINAL Horner sum degree2" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Primitive anchor ORIGINAL Horner product degree1 := mul(58/3, -4)", "after": "Primitive anchor ORIGINAL Horner product degree1 = -232/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner product degree1，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner product degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "58/3", "-4" ], "output": "-232/3", "label": "Primitive anchor ORIGINAL Horner product degree1" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Primitive anchor ORIGINAL Horner sum degree1 := sum(-232/3, -9)", "after": "Primitive anchor ORIGINAL Horner sum degree1 = -259/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner sum degree1，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner sum degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-232/3", "-9" ], "output": "-259/3", "label": "Primitive anchor ORIGINAL Horner sum degree1" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Primitive anchor ORIGINAL Horner product degree0 := mul(-259/3, -4)", "after": "Primitive anchor ORIGINAL Horner product degree0 = 1036/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner product degree0，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner product degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-259/3", "-4" ], "output": "1036/3", "label": "Primitive anchor ORIGINAL Horner product degree0" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Primitive anchor ORIGINAL Horner sum degree0 := sum(1036/3, 0)", "after": "Primitive anchor ORIGINAL Horner sum degree0 = 1036/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL Horner sum degree0，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL Horner sum degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1036/3", "0" ], "output": "1036/3", "label": "Primitive anchor ORIGINAL Horner sum degree0" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Primitive anchor ORIGINAL target minus F0 at anchor := sub(-9, 1036/3)", "after": "Primitive anchor ORIGINAL target minus F0 at anchor = -1063/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor ORIGINAL target minus F0 at anchor，不以小數近似替代。", "en": "Recompute Primitive anchor ORIGINAL target minus F0 at anchor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-9", "1036/3" ], "output": "-1063/3", "label": "Primitive anchor ORIGINAL target minus F0 at anchor" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Primitive anchor reverse SAME F0 value plus C := sum(1036/3, -1063/3)", "after": "Primitive anchor reverse SAME F0 value plus C = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Primitive anchor reverse SAME F0 value plus C，不以小數近似替代。", "en": "Recompute Primitive anchor reverse SAME F0 value plus C with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1036/3", "-1063/3" ], "output": "-9", "label": "Primitive anchor reverse SAME F0 value plus C" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "source roles = [[-9/1, 4/1, -1/1, 4/1], -4, -9/1, [0/1, -9/1, 2/1, -1/3, 1/1]]", "after": "derived = [1036/3, -1063/3, -9/1, 是]", "relation": "implies", "rule_id": "Primitive_anchor_ORIGINAL_target_C_changes_values_NOT_derivative", "conditions": [], "explanation": { "zh-Hant": "在明定族F=F0+C代原anchor a，F(a)=F0(a)+C=V，所以C=V-F0(a)。同anchor回代得V且(F0+C)_prime=f；C不影響導數卻影響值，不能drop C或改anchor。只求本代表族所需常數。", "en": "In the DECLARED family F=F0+C,substitute the ORIGINAL anchor a:F(a)=F0(a)+C=V,so C=V-F0(a).Reverse substitution at the SAME anchor gives V while(F0+C)_prime=f.C changes values although not derivatives;do not drop it or change the anchor.This asks for the constant in this representative family." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null } ], "math_model": { "result": "-1063/3" }, "conclusion": { "canonical_value": "-1063/3", "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-20261007-advance-20261009-r05-antiderivative-foundations", "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r05-antiderivative-foundations/questions.jsonl", "batch_sha256": "085ade26180bd6f2bb25667f7334717d6fd8127fddfdf06e74e67bdbe897b44e", "original_line": 54, "original_line_sha256": "62a3aee7f2129968731f42f3d47355e59507bbe56230412a17f952315482058b", "question_compact_sha256": "01ee21bb9a816388b1c593d7863683f7bbe1c183d1b4d442ef57f4aac84126d7", "english_row_compact_sha256": "a76e4cf4d7a2abcbc4bb08e1b43d3ad6266dca847949ff86b46a5ef297f5b31d", "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r05-antiderivative-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r05-antiderivative-foundations.en.jsonl" }, "mko_bindings": { "mko-antiderivative": { "sha256": "5f2ecff6866e471d1f8da7fea1fa5e121364ae2d24b2fea405d2300fe9ff1cae", "english_sha256": "f983dd8b898bc9eac9d5237c7895e5f7dc182ac46efb5f9b58b925fc6c2ecb9d", "original_path": "public/data/mko/antiderivative.json", "english_path": "public/data/i18n/editions/mko-antiderivative.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "623c23f45586b08629c5a347ffbc994cf8dbff2133c4f4cdb227b851d74ea6b6", "original_candidate_sha256": "cb1f978e79a05a5aacf4c8ad5d6405d2e32e1c4d9a04afdbc41394064ec77f23", "original_candidate_line_sha256": "cb1f978e79a05a5aacf4c8ad5d6405d2e32e1c4d9a04afdbc41394064ec77f23" }, "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-20261026-011-p59", "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": "66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df", "question_first_witness_sha256": "b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 24, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "4069a1a163e67412c31c24d68dbec51b347b9541ffe88686abd470c5bb6698bb", "arithmetic_steps": 20 }, "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." } }