{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261022-r01-harmonic-oscillator-model-0068", "pilot_number": "R0057-29493", "number": 29493, "family_id": "qf-str22-oscillator-polynomial-residual", "title": { "zh-Hant": "第 29493 題：簡諧振子精確切片", "en": "Question 29493: A Polynomial Oscillator Residual at One Specified Point" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261022-r01-harmonic-oscillator-model-0068", "batch_id": "batch-20261013-advance-20261022-r01-harmonic-oscillator-model-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-harmonic-oscillator-model" ], "title_zh": "簡諧振子精確切片", "stem_zh": "無阻尼線性振子模型m x_second+k x=0。操作=oscillator-polynomial-residual；資料={\"coefficients\":[5,0,3],\"mass\":[2,1],\"spring\":[6,1],\"x\":0}。請依 [coefficients,x,position,acceleration,mass,spring,residual,zero_at_x] 欄序回答。", "answer": { "kind": "expression", "value": "([5, 0, 3], 0, 5, 6, 2, 6, 42, 否)", "explanation_zh": "結構基礎核對=([5, 0, 3], 0, 5, 6, 2, 6, 42, 否)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；不宣稱sin/cos閉式解；單點能量不證沿軌道守恆。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "簡諧振子精確切片" ] }, "variant": { "family_id": "qf-str22-oscillator-polynomial-residual", "seed": 202610221068, "parameters": { "coefficients": [ 5, 0, 3 ], "mass": [ 2, 1 ], "spring": [ 6, 1 ], "x": 0, "domain": "Exact_harmonic_oscillator_model", "operation": "oscillator-polynomial-residual", "semantic_template": "qf-str22-oscillator-polynomial-residual/{\"coefficients\":[5,0,3],\"mass\":[2,1],\"spring\":[6,1],\"x\":0}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261022-r01-harmonic-oscillator-model-0068", "batch_id": "batch-20261013-advance-20261022-r01-harmonic-oscillator-model-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "bef2106903ffde4d1b30e9a57139cfcb013d3cd681a5dbdc11149d820c93ae10", "family_id": "qf-str22-oscillator-polynomial-residual", "seed": 202610221068, "semantic_template": "qf-str22-oscillator-polynomial-residual/{\"coefficients\":[5,0,3],\"mass\":[2,1],\"spring\":[6,1],\"x\":0}" }, "fields": { "title": "A Polynomial Oscillator Residual at One Specified Point", "stem": "Use the undamped linear oscillator model m*x''+k*x=0, with mass m=2/1 and spring coefficient k=6/1. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Positive m and k qualify this model; this slice establishes no sin/cos closed-form solution or stability theorem. Ascending-power coefficients=[5, 0, 3], point=0. Give (coefficients,point,x(point),x''(point),mass,spring,m*x''+k*x,zero at this point?).", "answer": { "kind": "expression", "value": "([5, 0, 3], 0, 5, 6, 2, 6, 42, No)", "explanation": "Differentiate twice and recompute the signed residual. A zero at one point is not an interval-wide solution or a stability proof. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0, 3], 0, 5, 6, 2, 6, 42, No)." }, "skills": [ "A Polynomial Oscillator Residual at One Specified Point" ], "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": "無阻尼線性振子模型m x_second+k x=0。操作=oscillator-polynomial-residual；資料={\"coefficients\":[5,0,3],\"mass\":[2,1],\"spring\":[6,1],\"x\":0}。請依 [coefficients,x,position,acceleration,mass,spring,residual,zero_at_x] 欄序回答。", "en": "Use the undamped linear oscillator model m*x''+k*x=0, with mass m=2/1 and spring coefficient k=6/1. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Positive m and k qualify this model; this slice establishes no sin/cos closed-form solution or stability theorem. Ascending-power coefficients=[5, 0, 3], point=0. Give (coefficients,point,x(point),x''(point),mass,spring,m*x''+k*x,zero at this point?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-str22-oscillator-polynomial-residual, oscillator-polynomial-residual]", "after": "derived = [[coefficients, x, position, acceleration, mass, spring, residual, zero_at_x]]", "relation": "implies", "rule_id": "STR22_exact_interface_and_source_ORDERED_tuple_roles", "conditions": [], "explanation": { "zh-Hant": "此銀行依原tuple欄序與hyphen operation。fraction約分、正分母且整數省略/1，與前輪舊契約不同；不改原中英快照或題意。", "en": "Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [Oscillator mass, [2, 1]]", "after": "derived = [2, 2, 1]", "relation": "implies", "rule_id": "STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization", "conditions": [], "explanation": { "zh-Hant": "原rational必須二元pair且raw分子／分母各在±1e6、分母非0。先正分母與gcd約分；不能接受scalar或先約分再繞過raw輸入界。", "en": "A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [Oscillator spring, [6, 1]]", "after": "derived = [6, 6, 1]", "relation": "implies", "rule_id": "STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization", "conditions": [], "explanation": { "zh-Hant": "原rational必須二元pair且raw分子／分母各在±1e6、分母非0。先正分母與gcd約分；不能接受scalar或先約分再繞過raw輸入界。", "en": "A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [2, 6]", "after": "derived = [[是, 是], 是]", "relation": "implies", "rule_id": "Oscillator_BOTH_strict_model_qualifications_not_repaired", "conditions": [], "explanation": { "zh-Hant": "無阻尼線性振子m*x_second+k*x=0先要求m,k皆嚴格正；qualification可有0／負值作No，不先abs或替換。", "en": "For the undamped linear oscillator m*x_second+k*x=0 require BOTH m,k strictly positive.Qualification may legally contain0/negative candidates;do not repair them with abs or replacement." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Oscillator ORIGINAL first derivative coefficient0 := mul(0, 1)", "after": "Oscillator ORIGINAL first derivative coefficient0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL first derivative coefficient0，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL first derivative coefficient0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "Oscillator ORIGINAL first derivative coefficient0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Oscillator ORIGINAL first derivative coefficient1 := mul(3, 2)", "after": "Oscillator ORIGINAL first derivative coefficient1 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL first derivative coefficient1，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL first derivative coefficient1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "2" ], "output": "6", "label": "Oscillator ORIGINAL first derivative coefficient1" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Oscillator ORIGINAL second derivative coefficient0 := mul(6, 1)", "after": "Oscillator ORIGINAL second derivative coefficient0 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL second derivative coefficient0，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL second derivative coefficient0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "1" ], "output": "6", "label": "Oscillator ORIGINAL second derivative coefficient0" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [[5, 0, 3]]", "after": "derived = [[0, 6], [6]]", "relation": "implies", "rule_id": "Oscillator_WHOLE_polynomial_differentiate_TWICE_ALL_coefficients", "conditions": [], "explanation": { "zh-Hant": "按升冪對每一項用d(c_i*x^i)=i*c_i*x^(i-1)兩次。保原coeff vector與trailing zeros；空derivative vector為零，不加constant/nonconstant限制或derived係數1000界。", "en": "Differentiate EVERY ascending-power monomial twice using d(c_i*x^i)=i*c_i*x^(i-1).Keep the original coefficient vector/trailing zeros;an empty derivative means0.Do not require nonconstancy or impose raw1000 bounds on derived coefficients." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Oscillator ORIGINAL position Horner multiply degree2 := mul(0, 0)", "after": "Oscillator ORIGINAL position Horner multiply degree2 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL position Horner multiply degree2，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL position Horner multiply degree2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Oscillator ORIGINAL position Horner multiply degree2" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Oscillator ORIGINAL position Horner add degree2 := sum(0, 3)", "after": "Oscillator ORIGINAL position Horner add degree2 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL position Horner add degree2，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL position Horner add degree2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "3" ], "output": "3", "label": "Oscillator ORIGINAL position Horner add degree2" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Oscillator ORIGINAL position Horner multiply degree1 := mul(3, 0)", "after": "Oscillator ORIGINAL position Horner multiply degree1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL position Horner multiply degree1，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL position Horner multiply degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "0" ], "output": "0", "label": "Oscillator ORIGINAL position Horner multiply degree1" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Oscillator ORIGINAL position Horner add degree1 := sum(0, 0)", "after": "Oscillator ORIGINAL position Horner add degree1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL position Horner add degree1，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL position Horner add degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "Oscillator ORIGINAL position Horner add degree1" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Oscillator ORIGINAL position Horner multiply degree0 := mul(0, 0)", "after": "Oscillator ORIGINAL position Horner multiply degree0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL position Horner multiply degree0，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL position Horner multiply degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Oscillator ORIGINAL position Horner multiply degree0" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Oscillator ORIGINAL position Horner add degree0 := sum(0, 5)", "after": "Oscillator ORIGINAL position Horner add degree0 = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL position Horner add degree0，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL position Horner add degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "5" ], "output": "5", "label": "Oscillator ORIGINAL position Horner add degree0" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Oscillator ORIGINAL acceleration Horner multiply degree0 := mul(0, 0)", "after": "Oscillator ORIGINAL acceleration Horner multiply degree0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL acceleration Horner multiply degree0，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL acceleration Horner multiply degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Oscillator ORIGINAL acceleration Horner multiply degree0" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Oscillator ORIGINAL acceleration Horner add degree0 := sum(0, 6)", "after": "Oscillator ORIGINAL acceleration Horner add degree0 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL acceleration Horner add degree0，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL acceleration Horner add degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "6" ], "output": "6", "label": "Oscillator ORIGINAL acceleration Horner add degree0" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [[5, 0, 3], [6], 0]", "after": "derived = [[5, 6]]", "relation": "implies", "rule_id": "Oscillator_ORIGINAL_Horner_position_AND_acceleration_at_SAMEpoint", "conditions": [], "explanation": { "zh-Hant": "同一指定x對原polynomial及二階導數完整Horner；整數evaluate本身無1e6 cap，後續原rational乘加才按來源profile。", "en": "Evaluate the original polynomial AND its second derivative by complete Horner at the SAME given point.Integer evaluation itself has no1e6 cap;subsequent SOURCE rational products/sums retain their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Oscillator ORIGINAL mass times acceleration := mul(2, 6)", "after": "Oscillator ORIGINAL mass times acceleration = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL mass times acceleration，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL mass times acceleration with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "6" ], "output": "12", "label": "Oscillator ORIGINAL mass times acceleration" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "source roles = [Oscillator ORIGINAL mass times acceleration, mul, [2, 6]]", "after": "derived = [[1, 1], [12, 1], 12]", "relation": "implies", "rule_id": "STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate", "conditions": [], "explanation": { "zh-Hant": "來源加法先取分母gcd，乘法先cross-cancel，再驗本步raw numerator與denominator各<=1e6，最後才約分；除法另先驗非零及reciprocal。不能誤用未消約乘積界，也不能先最終約分才驗界。", "en": "SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Oscillator ORIGINAL spring times position := mul(6, 5)", "after": "Oscillator ORIGINAL spring times position = 30", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL spring times position，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL spring times position with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "5" ], "output": "30", "label": "Oscillator ORIGINAL spring times position" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "source roles = [Oscillator ORIGINAL spring times position, mul, [6, 5]]", "after": "derived = [[1, 1], [30, 1], 30]", "relation": "implies", "rule_id": "STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate", "conditions": [], "explanation": { "zh-Hant": "來源加法先取分母gcd，乘法先cross-cancel，再驗本步raw numerator與denominator各<=1e6，最後才約分；除法另先驗非零及reciprocal。不能誤用未消約乘積界，也不能先最終約分才驗界。", "en": "SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Oscillator ORIGINAL signed ODE residual := sum(12, 30)", "after": "Oscillator ORIGINAL signed ODE residual = 42", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Oscillator ORIGINAL signed ODE residual，不以小數近似替代。", "en": "Recompute Oscillator ORIGINAL signed ODE residual with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "12", "30" ], "output": "42", "label": "Oscillator ORIGINAL signed ODE residual" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "source roles = [Oscillator ORIGINAL signed ODE residual, sum, [12, 30]]", "after": "derived = [[1], [42, 1], 42]", "relation": "implies", "rule_id": "STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate", "conditions": [], "explanation": { "zh-Hant": "來源加法先取分母gcd，乘法先cross-cancel，再驗本步raw numerator與denominator各<=1e6，最後才約分；除法另先驗非零及reciprocal。不能誤用未消約乘積界，也不能先最終約分才驗界。", "en": "SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "source roles = [[5, 0, 3], 0, 2, 6]", "after": "derived = [5, 6, 42, 否]", "relation": "implies", "rule_id": "Oscillator_SINGLE_point_residual_zero_NOT_interval_solution", "conditions": [], "explanation": { "zh-Hant": "回代R(x)=m*p_second(x)+k*p(x)，保signed residual。R(x)=0只在本指定點吻合，非對整個interval的ODE solution，更不證stability。", "en": "Substitute R(x)=m*p_second(x)+k*p(x),retaining the SIGNED residual.R(x)=0 at THIS specified point does not establish an interval-wide ODE solution or stability." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null } ], "math_model": { "result": [ [ 5, 0, 3 ], 0, 5, 6, "2", "6", "42", false ] }, "conclusion": { "canonical_value": "([5, 0, 3], 0, 5, 6, 2, 6, 42, 否)", "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-20261013-advance-20261022-r01-harmonic-oscillator-model-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261022-r01-harmonic-oscillator-model-foundations/questions.jsonl", "batch_sha256": "5cc1e0de37c01b0a2562731df1fbc64244a3ad4d4497199476460cea35571ccd", "original_line": 68, "original_line_sha256": "284a263313cfea97c94edfe1cab88e8de5911b5c5c550ad0d8e948b89bbe0cff", "question_compact_sha256": "bef2106903ffde4d1b30e9a57139cfcb013d3cd681a5dbdc11149d820c93ae10", "english_row_compact_sha256": "35bb3bd3aead5b49b28164e8d78591a411ebb7984d0f97404ca6622487488b6a", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261022-r01-harmonic-oscillator-model-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261022-r01-harmonic-oscillator-model-foundations.en.jsonl" }, "mko_bindings": { "mko-harmonic-oscillator-model": { "sha256": "920969c8aeda483138db1c073acdd7c8acdf5a854c7cef34fb51509539f7ebb4", "english_sha256": "435d03c200d484abe1512ca6364253ffef8da54d800818ba816553870515ab66", "original_path": "public/data/mko/harmonic-oscillator-model.json", "english_path": "public/data/i18n/editions/mko-harmonic-oscillator-model.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "cd714b09e116fbc0088007895c4d956bc0f4fad5a61507e4a16f09649d05a3b0", "original_candidate_sha256": "f4cc7b72f8fe3b558805b4aa49e1897299263551e87ddc6f5d40e083e9509f2f", "original_candidate_line_sha256": "f4cc7b72f8fe3b558805b4aa49e1897299263551e87ddc6f5d40e083e9509f2f" }, "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-p58", "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": "db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6", "question_first_witness_sha256": "58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5" }, "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": "106dbd5410a5d0ff48a2fe50517dcceadce5e2677c8d6383504bd0b0ac65074f", "arithmetic_steps": 14 }, "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." } }