{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261113-r02-real-exponential-function-0057", "pilot_number": "R0047-24207", "number": 24207, "family_id": "qf-mod44-exp-addition-convolution", "title": { "zh-Hant": "第 24207 題：實指數函數", "en": "Question 24207: One Convolution Degree Is Not the Whole Exponential Addition Proof" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261113-r02-real-exponential-function-0057", "batch_id": "batch-20261015-advance-20261113-r02-real-exponential-function-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-real-exponential-function" ], "title_zh": "實指數函數", "stem_zh": "E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={\"x\":[7,8],\"y\":[-8,9],\"degree\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)", "explanation_zh": "Wave6第七層核對=(7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "實指數函數" ] }, "variant": { "family_id": "qf-mod44-exp-addition-convolution", "seed": 202611132057, "parameters": { "x": [ 7, 8 ], "y": [ -8, 9 ], "degree": 6, "domain": "Exact_real_exponential_series", "operation": "exp-addition-convolution", "semantic_template": "qf-mod44-exp-addition-convolution/{\"degree\":6,\"x\":[7,8],\"y\":[-8,9]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261113-r02-real-exponential-function-0057", "batch_id": "batch-20261015-advance-20261113-r02-real-exponential-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "71e0c3319e03601e9bd982a2098c1b7a3e3f2d6837746df3a6e900dfcb3e8b13", "family_id": "qf-mod44-exp-addition-convolution", "seed": 202611132057, "semantic_template": "qf-mod44-exp-addition-convolution/{\"degree\":6,\"x\":[7,8],\"y\":[-8,9]}" }, "fields": { "title": "One Convolution Degree Is Not the Whole Exponential Addition Proof", "stem": "Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=7/8, y=-8/9, declared degree=6. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).", "answer": { "kind": "expression", "value": "(7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)", "explanation": "Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)." }, "skills": [ "One Convolution Degree Is Not the Whole Exponential Addition Proof" ], "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": "E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={\"x\":[7,8],\"y\":[-8,9],\"degree\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=7/8, y=-8/9, declared degree=6. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [exp-addition-convolution]", "after": "derived = [[x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。", "en": "List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "(7/8)^0", "after": "Exp convolution x power0 = 1", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "7/8", "0" ], "output": "1", "label": "Exp convolution x power0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "(-8/9)^6", "after": "Exp convolution y power6 = 262144/531441", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "-8/9", "6" ], "output": "262144/531441", "label": "Exp convolution y power6" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Exp convolution numerator0 := mul(1, 262144/531441)", "after": "Exp convolution numerator0 = 262144/531441", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution numerator0，不以小數近似替代。", "en": "Recompute Exp convolution numerator0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "262144/531441" ], "output": "262144/531441", "label": "Exp convolution numerator0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [0]", "after": "derived = [1]", "relation": "implies", "rule_id": "empty_factorial_is_multiplicative_unit", "conditions": [], "explanation": { "zh-Hant": "0!是空乘積1，不是0。", "en": "The empty product0! is1,not0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Exp convolution right0 factorial step1 := mul(1, 1)", "after": "Exp convolution right0 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right0 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution right0 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution right0 factorial step1" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Exp convolution right0 factorial step2 := mul(1, 2)", "after": "Exp convolution right0 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right0 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution right0 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution right0 factorial step2" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Exp convolution right0 factorial step3 := mul(2, 3)", "after": "Exp convolution right0 factorial step3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right0 factorial step3，不以小數近似替代。", "en": "Recompute Exp convolution right0 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp convolution right0 factorial step3" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Exp convolution right0 factorial step4 := mul(6, 4)", "after": "Exp convolution right0 factorial step4 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right0 factorial step4，不以小數近似替代。", "en": "Recompute Exp convolution right0 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp convolution right0 factorial step4" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Exp convolution right0 factorial step5 := mul(24, 5)", "after": "Exp convolution right0 factorial step5 = 120", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right0 factorial step5，不以小數近似替代。", "en": "Recompute Exp convolution right0 factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp convolution right0 factorial step5" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Exp convolution right0 factorial step6 := mul(120, 6)", "after": "Exp convolution right0 factorial step6 = 720", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right0 factorial step6，不以小數近似替代。", "en": "Recompute Exp convolution right0 factorial step6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "120", "6" ], "output": "720", "label": "Exp convolution right0 factorial step6" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Exp convolution denominator0 := mul(1, 720)", "after": "Exp convolution denominator0 = 720", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution denominator0，不以小數近似替代。", "en": "Recompute Exp convolution denominator0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "720" ], "output": "720", "label": "Exp convolution denominator0" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Exp convolution coefficient term0 := div(262144/531441, 720)", "after": "Exp convolution coefficient term0 = 16384/23914845", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution coefficient term0，不以小數近似替代。", "en": "Recompute Exp convolution coefficient term0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "262144/531441", "720" ], "output": "16384/23914845", "label": "Exp convolution coefficient term0" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "(7/8)^1", "after": "Exp convolution x power1 = 7/8", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "7/8", "1" ], "output": "7/8", "label": "Exp convolution x power1" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "(-8/9)^5", "after": "Exp convolution y power5 = -32768/59049", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "-8/9", "5" ], "output": "-32768/59049", "label": "Exp convolution y power5" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Exp convolution numerator1 := mul(7/8, -32768/59049)", "after": "Exp convolution numerator1 = -28672/59049", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution numerator1，不以小數近似替代。", "en": "Recompute Exp convolution numerator1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7/8", "-32768/59049" ], "output": "-28672/59049", "label": "Exp convolution numerator1" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Exp convolution left1 factorial step1 := mul(1, 1)", "after": "Exp convolution left1 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left1 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution left1 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution left1 factorial step1" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Exp convolution right1 factorial step1 := mul(1, 1)", "after": "Exp convolution right1 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right1 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution right1 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution right1 factorial step1" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Exp convolution right1 factorial step2 := mul(1, 2)", "after": "Exp convolution right1 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right1 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution right1 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution right1 factorial step2" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Exp convolution right1 factorial step3 := mul(2, 3)", "after": "Exp convolution right1 factorial step3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right1 factorial step3，不以小數近似替代。", "en": "Recompute Exp convolution right1 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp convolution right1 factorial step3" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Exp convolution right1 factorial step4 := mul(6, 4)", "after": "Exp convolution right1 factorial step4 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right1 factorial step4，不以小數近似替代。", "en": "Recompute Exp convolution right1 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp convolution right1 factorial step4" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Exp convolution right1 factorial step5 := mul(24, 5)", "after": "Exp convolution right1 factorial step5 = 120", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right1 factorial step5，不以小數近似替代。", "en": "Recompute Exp convolution right1 factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp convolution right1 factorial step5" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Exp convolution denominator1 := mul(1, 120)", "after": "Exp convolution denominator1 = 120", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution denominator1，不以小數近似替代。", "en": "Recompute Exp convolution denominator1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "120" ], "output": "120", "label": "Exp convolution denominator1" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "Exp convolution coefficient term1 := div(-28672/59049, 120)", "after": "Exp convolution coefficient term1 = -3584/885735", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution coefficient term1，不以小數近似替代。", "en": "Recompute Exp convolution coefficient term1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-28672/59049", "120" ], "output": "-3584/885735", "label": "Exp convolution coefficient term1" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "(7/8)^2", "after": "Exp convolution x power2 = 49/64", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "7/8", "2" ], "output": "49/64", "label": "Exp convolution x power2" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "(-8/9)^4", "after": "Exp convolution y power4 = 4096/6561", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "-8/9", "4" ], "output": "4096/6561", "label": "Exp convolution y power4" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "Exp convolution numerator2 := mul(49/64, 4096/6561)", "after": "Exp convolution numerator2 = 3136/6561", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution numerator2，不以小數近似替代。", "en": "Recompute Exp convolution numerator2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "49/64", "4096/6561" ], "output": "3136/6561", "label": "Exp convolution numerator2" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "Exp convolution left2 factorial step1 := mul(1, 1)", "after": "Exp convolution left2 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left2 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution left2 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution left2 factorial step1" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "Exp convolution left2 factorial step2 := mul(1, 2)", "after": "Exp convolution left2 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left2 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution left2 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution left2 factorial step2" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "Exp convolution right2 factorial step1 := mul(1, 1)", "after": "Exp convolution right2 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right2 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution right2 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution right2 factorial step1" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "Exp convolution right2 factorial step2 := mul(1, 2)", "after": "Exp convolution right2 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right2 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution right2 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution right2 factorial step2" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "Exp convolution right2 factorial step3 := mul(2, 3)", "after": "Exp convolution right2 factorial step3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right2 factorial step3，不以小數近似替代。", "en": "Recompute Exp convolution right2 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp convolution right2 factorial step3" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "Exp convolution right2 factorial step4 := mul(6, 4)", "after": "Exp convolution right2 factorial step4 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right2 factorial step4，不以小數近似替代。", "en": "Recompute Exp convolution right2 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp convolution right2 factorial step4" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "Exp convolution denominator2 := mul(2, 24)", "after": "Exp convolution denominator2 = 48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution denominator2，不以小數近似替代。", "en": "Recompute Exp convolution denominator2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "24" ], "output": "48", "label": "Exp convolution denominator2" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "Exp convolution coefficient term2 := div(3136/6561, 48)", "after": "Exp convolution coefficient term2 = 196/19683", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution coefficient term2，不以小數近似替代。", "en": "Recompute Exp convolution coefficient term2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "3136/6561", "48" ], "output": "196/19683", "label": "Exp convolution coefficient term2" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "(7/8)^3", "after": "Exp convolution x power3 = 343/512", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "7/8", "3" ], "output": "343/512", "label": "Exp convolution x power3" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "(-8/9)^3", "after": "Exp convolution y power3 = -512/729", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "-8/9", "3" ], "output": "-512/729", "label": "Exp convolution y power3" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "Exp convolution numerator3 := mul(343/512, -512/729)", "after": "Exp convolution numerator3 = -343/729", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution numerator3，不以小數近似替代。", "en": "Recompute Exp convolution numerator3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "343/512", "-512/729" ], "output": "-343/729", "label": "Exp convolution numerator3" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "Exp convolution left3 factorial step1 := mul(1, 1)", "after": "Exp convolution left3 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left3 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution left3 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution left3 factorial step1" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "Exp convolution left3 factorial step2 := mul(1, 2)", "after": "Exp convolution left3 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left3 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution left3 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution left3 factorial step2" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "Exp convolution left3 factorial step3 := mul(2, 3)", "after": "Exp convolution left3 factorial step3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left3 factorial step3，不以小數近似替代。", "en": "Recompute Exp convolution left3 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp convolution left3 factorial step3" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "Exp convolution right3 factorial step1 := mul(1, 1)", "after": "Exp convolution right3 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right3 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution right3 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution right3 factorial step1" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "Exp convolution right3 factorial step2 := mul(1, 2)", "after": "Exp convolution right3 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right3 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution right3 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution right3 factorial step2" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "Exp convolution right3 factorial step3 := mul(2, 3)", "after": "Exp convolution right3 factorial step3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right3 factorial step3，不以小數近似替代。", "en": "Recompute Exp convolution right3 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp convolution right3 factorial step3" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "Exp convolution denominator3 := mul(6, 6)", "after": "Exp convolution denominator3 = 36", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution denominator3，不以小數近似替代。", "en": "Recompute Exp convolution denominator3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "6" ], "output": "36", "label": "Exp convolution denominator3" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "Exp convolution coefficient term3 := div(-343/729, 36)", "after": "Exp convolution coefficient term3 = -343/26244", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution coefficient term3，不以小數近似替代。", "en": "Recompute Exp convolution coefficient term3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-343/729", "36" ], "output": "-343/26244", "label": "Exp convolution coefficient term3" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "(7/8)^4", "after": "Exp convolution x power4 = 2401/4096", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "7/8", "4" ], "output": "2401/4096", "label": "Exp convolution x power4" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "(-8/9)^2", "after": "Exp convolution y power2 = 64/81", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "-8/9", "2" ], "output": "64/81", "label": "Exp convolution y power2" } }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "Exp convolution numerator4 := mul(2401/4096, 64/81)", "after": "Exp convolution numerator4 = 2401/5184", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution numerator4，不以小數近似替代。", "en": "Recompute Exp convolution numerator4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2401/4096", "64/81" ], "output": "2401/5184", "label": "Exp convolution numerator4" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "Exp convolution left4 factorial step1 := mul(1, 1)", "after": "Exp convolution left4 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left4 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution left4 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution left4 factorial step1" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "Exp convolution left4 factorial step2 := mul(1, 2)", "after": "Exp convolution left4 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left4 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution left4 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution left4 factorial step2" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "Exp convolution left4 factorial step3 := mul(2, 3)", "after": "Exp convolution left4 factorial step3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left4 factorial step3，不以小數近似替代。", "en": "Recompute Exp convolution left4 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp convolution left4 factorial step3" } }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "Exp convolution left4 factorial step4 := mul(6, 4)", "after": "Exp convolution left4 factorial step4 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left4 factorial step4，不以小數近似替代。", "en": "Recompute Exp convolution left4 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp convolution left4 factorial step4" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "Exp convolution right4 factorial step1 := mul(1, 1)", "after": "Exp convolution right4 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right4 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution right4 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution right4 factorial step1" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "Exp convolution right4 factorial step2 := mul(1, 2)", "after": "Exp convolution right4 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right4 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution right4 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution right4 factorial step2" } }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "Exp convolution denominator4 := mul(24, 2)", "after": "Exp convolution denominator4 = 48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution denominator4，不以小數近似替代。", "en": "Recompute Exp convolution denominator4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "2" ], "output": "48", "label": "Exp convolution denominator4" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "Exp convolution coefficient term4 := div(2401/5184, 48)", "after": "Exp convolution coefficient term4 = 2401/248832", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution coefficient term4，不以小數近似替代。", "en": "Recompute Exp convolution coefficient term4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2401/5184", "48" ], "output": "2401/248832", "label": "Exp convolution coefficient term4" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "(7/8)^5", "after": "Exp convolution x power5 = 16807/32768", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "7/8", "5" ], "output": "16807/32768", "label": "Exp convolution x power5" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "(-8/9)^1", "after": "Exp convolution y power1 = -8/9", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "-8/9", "1" ], "output": "-8/9", "label": "Exp convolution y power1" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "Exp convolution numerator5 := mul(16807/32768, -8/9)", "after": "Exp convolution numerator5 = -16807/36864", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution numerator5，不以小數近似替代。", "en": "Recompute Exp convolution numerator5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16807/32768", "-8/9" ], "output": "-16807/36864", "label": "Exp convolution numerator5" } }, { "id": "S061", "uses": [ "G001", "S060" ], "before": "Exp convolution left5 factorial step1 := mul(1, 1)", "after": "Exp convolution left5 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left5 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution left5 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution left5 factorial step1" } }, { "id": "S062", "uses": [ "G001", "S061" ], "before": "Exp convolution left5 factorial step2 := mul(1, 2)", "after": "Exp convolution left5 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left5 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution left5 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution left5 factorial step2" } }, { "id": "S063", "uses": [ "G001", "S062" ], "before": "Exp convolution left5 factorial step3 := mul(2, 3)", "after": "Exp convolution left5 factorial step3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left5 factorial step3，不以小數近似替代。", "en": "Recompute Exp convolution left5 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp convolution left5 factorial step3" } }, { "id": "S064", "uses": [ "G001", "S063" ], "before": "Exp convolution left5 factorial step4 := mul(6, 4)", "after": "Exp convolution left5 factorial step4 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left5 factorial step4，不以小數近似替代。", "en": "Recompute Exp convolution left5 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp convolution left5 factorial step4" } }, { "id": "S065", "uses": [ "G001", "S064" ], "before": "Exp convolution left5 factorial step5 := mul(24, 5)", "after": "Exp convolution left5 factorial step5 = 120", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left5 factorial step5，不以小數近似替代。", "en": "Recompute Exp convolution left5 factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp convolution left5 factorial step5" } }, { "id": "S066", "uses": [ "G001", "S065" ], "before": "Exp convolution right5 factorial step1 := mul(1, 1)", "after": "Exp convolution right5 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution right5 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution right5 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution right5 factorial step1" } }, { "id": "S067", "uses": [ "G001", "S066" ], "before": "Exp convolution denominator5 := mul(120, 1)", "after": "Exp convolution denominator5 = 120", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution denominator5，不以小數近似替代。", "en": "Recompute Exp convolution denominator5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "120", "1" ], "output": "120", "label": "Exp convolution denominator5" } }, { "id": "S068", "uses": [ "G001", "S067" ], "before": "Exp convolution coefficient term5 := div(-16807/36864, 120)", "after": "Exp convolution coefficient term5 = -16807/4423680", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution coefficient term5，不以小數近似替代。", "en": "Recompute Exp convolution coefficient term5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-16807/36864", "120" ], "output": "-16807/4423680", "label": "Exp convolution coefficient term5" } }, { "id": "S069", "uses": [ "G001", "S068" ], "before": "(7/8)^6", "after": "Exp convolution x power6 = 117649/262144", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "7/8", "6" ], "output": "117649/262144", "label": "Exp convolution x power6" } }, { "id": "S070", "uses": [ "G001", "S069" ], "before": "(-8/9)^0", "after": "Exp convolution y power0 = 1", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "-8/9", "0" ], "output": "1", "label": "Exp convolution y power0" } }, { "id": "S071", "uses": [ "G001", "S070" ], "before": "Exp convolution numerator6 := mul(117649/262144, 1)", "after": "Exp convolution numerator6 = 117649/262144", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution numerator6，不以小數近似替代。", "en": "Recompute Exp convolution numerator6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "117649/262144", "1" ], "output": "117649/262144", "label": "Exp convolution numerator6" } }, { "id": "S072", "uses": [ "G001", "S071" ], "before": "Exp convolution left6 factorial step1 := mul(1, 1)", "after": "Exp convolution left6 factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left6 factorial step1，不以小數近似替代。", "en": "Recompute Exp convolution left6 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp convolution left6 factorial step1" } }, { "id": "S073", "uses": [ "G001", "S072" ], "before": "Exp convolution left6 factorial step2 := mul(1, 2)", "after": "Exp convolution left6 factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left6 factorial step2，不以小數近似替代。", "en": "Recompute Exp convolution left6 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp convolution left6 factorial step2" } }, { "id": "S074", "uses": [ "G001", "S073" ], "before": "Exp convolution left6 factorial step3 := mul(2, 3)", "after": "Exp convolution left6 factorial step3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left6 factorial step3，不以小數近似替代。", "en": "Recompute Exp convolution left6 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp convolution left6 factorial step3" } }, { "id": "S075", "uses": [ "G001", "S074" ], "before": "Exp convolution left6 factorial step4 := mul(6, 4)", "after": "Exp convolution left6 factorial step4 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left6 factorial step4，不以小數近似替代。", "en": "Recompute Exp convolution left6 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp convolution left6 factorial step4" } }, { "id": "S076", "uses": [ "G001", "S075" ], "before": "Exp convolution left6 factorial step5 := mul(24, 5)", "after": "Exp convolution left6 factorial step5 = 120", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left6 factorial step5，不以小數近似替代。", "en": "Recompute Exp convolution left6 factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp convolution left6 factorial step5" } }, { "id": "S077", "uses": [ "G001", "S076" ], "before": "Exp convolution left6 factorial step6 := mul(120, 6)", "after": "Exp convolution left6 factorial step6 = 720", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution left6 factorial step6，不以小數近似替代。", "en": "Recompute Exp convolution left6 factorial step6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "120", "6" ], "output": "720", "label": "Exp convolution left6 factorial step6" } }, { "id": "S078", "uses": [ "G001", "S077" ], "before": "source roles = [0]", "after": "derived = [1]", "relation": "implies", "rule_id": "empty_factorial_is_multiplicative_unit", "conditions": [], "explanation": { "zh-Hant": "0!是空乘積1，不是0。", "en": "The empty product0! is1,not0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S079", "uses": [ "G001", "S078" ], "before": "Exp convolution denominator6 := mul(720, 1)", "after": "Exp convolution denominator6 = 720", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution denominator6，不以小數近似替代。", "en": "Recompute Exp convolution denominator6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "720", "1" ], "output": "720", "label": "Exp convolution denominator6" } }, { "id": "S080", "uses": [ "G001", "S079" ], "before": "Exp convolution coefficient term6 := div(117649/262144, 720)", "after": "Exp convolution coefficient term6 = 117649/188743680", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp convolution coefficient term6，不以小數近似替代。", "en": "Recompute Exp convolution coefficient term6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "117649/262144", "720" ], "output": "117649/188743680", "label": "Exp convolution coefficient term6" } }, { "id": "S081", "uses": [ "G001", "S080" ], "before": "Exp complete declared-degree convolution := sum(16384/23914845, -3584/885735, 196/19683, -343/26244, 2401/248832, -16807/4423680, 117649/188743680)", "after": "Exp complete declared-degree convolution = 1/100306130042880", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp complete declared-degree convolution，不以小數近似替代。", "en": "Recompute Exp complete declared-degree convolution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "16384/23914845", "-3584/885735", "196/19683", "-343/26244", "2401/248832", "-16807/4423680", "117649/188743680" ], "output": "1/100306130042880", "label": "Exp complete declared-degree convolution" } }, { "id": "S082", "uses": [ "G001", "S081" ], "before": "Exp ORIGINAL arguments sum := sum(7/8, -8/9)", "after": "Exp ORIGINAL arguments sum = -1/72", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp ORIGINAL arguments sum，不以小數近似替代。", "en": "Recompute Exp ORIGINAL arguments sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "7/8", "-8/9" ], "output": "-1/72", "label": "Exp ORIGINAL arguments sum" } }, { "id": "S083", "uses": [ "G001", "S082" ], "before": "(-1/72)^6", "after": "Exp sum argument declared power = 1/139314069504", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": { "operation": "integer_power", "inputs": [ "-1/72", "6" ], "output": "1/139314069504", "label": "Exp sum argument declared power" } }, { "id": "S084", "uses": [ "G001", "S083" ], "before": "Exp sum argument factorial step1 := mul(1, 1)", "after": "Exp sum argument factorial step1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp sum argument factorial step1，不以小數近似替代。", "en": "Recompute Exp sum argument factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp sum argument factorial step1" } }, { "id": "S085", "uses": [ "G001", "S084" ], "before": "Exp sum argument factorial step2 := mul(1, 2)", "after": "Exp sum argument factorial step2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp sum argument factorial step2，不以小數近似替代。", "en": "Recompute Exp sum argument factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp sum argument factorial step2" } }, { "id": "S086", "uses": [ "G001", "S085" ], "before": "Exp sum argument factorial step3 := mul(2, 3)", "after": "Exp sum argument factorial step3 = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp sum argument factorial step3，不以小數近似替代。", "en": "Recompute Exp sum argument factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp sum argument factorial step3" } }, { "id": "S087", "uses": [ "G001", "S086" ], "before": "Exp sum argument factorial step4 := mul(6, 4)", "after": "Exp sum argument factorial step4 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp sum argument factorial step4，不以小數近似替代。", "en": "Recompute Exp sum argument factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp sum argument factorial step4" } }, { "id": "S088", "uses": [ "G001", "S087" ], "before": "Exp sum argument factorial step5 := mul(24, 5)", "after": "Exp sum argument factorial step5 = 120", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp sum argument factorial step5，不以小數近似替代。", "en": "Recompute Exp sum argument factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp sum argument factorial step5" } }, { "id": "S089", "uses": [ "G001", "S088" ], "before": "Exp sum argument factorial step6 := mul(120, 6)", "after": "Exp sum argument factorial step6 = 720", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp sum argument factorial step6，不以小數近似替代。", "en": "Recompute Exp sum argument factorial step6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "120", "6" ], "output": "720", "label": "Exp sum argument factorial step6" } }, { "id": "S090", "uses": [ "G001", "S089" ], "before": "Exp sum argument coefficient := div(1/139314069504, 720)", "after": "Exp sum argument coefficient = 1/100306130042880", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Exp sum argument coefficient，不以小數近似替代。", "en": "Recompute Exp sum argument coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/139314069504", "720" ], "output": "1/100306130042880", "label": "Exp sum argument coefficient" } }, { "id": "S091", "uses": [ "G001", "S090" ], "before": "source roles = [[7/8, -8/9], a_n=1/n! for EVERY n>=0]", "after": "derived = [coefficient ratio=1/(n+1), for EVERY fixed real z,term absolute ratio=|z|/(n+1)->0, absolute convergence and real-valued E(z)]", "relation": "implies", "rule_id": "Exp_WHOLE_factorial_law_ALL_fixed_real_arguments", "conditions": [], "explanation": { "zh-Hant": "對任意固定實z選N+1>2|z|，之後項的絕對比值小於1/2，由幾何尾界保絕對收斂。不是由有限prefix或把係數比值漏乘|z|。", "en": "For any fixed real z,choose N+1>2|z|;later absolute term ratios are below1/2 and a geometric tail proves absolute convergence.This is not finite-prefix extrapolation or omission of |z|." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S092", "uses": [ "G001", "S091" ], "before": "source roles = [[7/8, -8/9, 6], E(x),E(y) absolutely convergent]", "after": "derived = [for EVERY n,sum_k x^k*y^(n-k)/(k!*(n-k)!)=(x+y)^n/n!, E(x)E(y)=E(x+y)]", "relation": "implies", "rule_id": "Exp_ALL_degree_binomial_and_absolute_Cauchy_product", "conditions": [], "explanation": { "zh-Hant": "逐n乘上n!，有限卷積成二項式展開（n個因子中選k個x）；這對任意n成立。兩級數絕對收斂使雙和可重排成Cauchy product，才得到整體加法律，不由單一degree外推。", "en": "Multiply the finite degree-n convolution by n!:choosing k of n factors gives the binomial expansion for arbitrary n.Absolute convergence permits regrouping the double sum into the Cauchy product,establishing the whole addition law,not extrapolating one degree." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ "7/8", "-8/9", 6, [ [ 0, "16384/23914845" ], [ 1, "-3584/885735" ], [ 2, "196/19683" ], [ 3, "-343/26244" ], [ 4, "2401/248832" ], [ 5, "-16807/4423680" ], [ 6, "117649/188743680" ] ], "1/100306130042880", "1/100306130042880", true, "binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement" ] }, "conclusion": { "canonical_value": "(7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)", "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-20261113-r02-real-exponential-function-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261113-r02-real-exponential-function-foundations/questions.jsonl", "batch_sha256": "e937e416efd79ec1e73c82a290ba61f2b60a0d2fc00f91f29ef00d18b9473f95", "original_line": 57, "original_line_sha256": "d90203611905bf58336cb9be4d918a8508aecd2c387edcb5f4c010c59ccfeb19", "question_compact_sha256": "71e0c3319e03601e9bd982a2098c1b7a3e3f2d6837746df3a6e900dfcb3e8b13", "english_row_compact_sha256": "9fef603fc74a1a2bd60ab7068f13fd570831fe6f9bb0dc044370e69b100d8dbd", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261113-r02-real-exponential-function-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261113-r02-real-exponential-function-foundations.en.jsonl" }, "mko_bindings": { "mko-real-exponential-function": { "sha256": "7b0736c512a2b1e27562551d8a9de463446e4b54c5998868cfe7573aee0b4f17", "english_sha256": "1fafde29fc55b9df97ad047683d8c6f01be2efa1ac103b831e7d23ec97eafb33", "original_path": "public/data/mko/real-exponential-function.json", "english_path": "public/data/i18n/editions/mko-real-exponential-function.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "b3aad195dad743751e4c8a4a6ce44f091c3824da1e52c21e9266d4ccd1dc1b68", "original_candidate_sha256": "de536e61d72314e672f5535fb5e1cbf13556e3b1055fe6baab2561099aa0657c", "original_candidate_line_sha256": "de536e61d72314e672f5535fb5e1cbf13556e3b1055fe6baab2561099aa0657c" }, "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-20261026-009-p48", "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": "9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386", "question_first_witness_sha256": "eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 92, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "d03ec4d47db8a5ada6b651eb0f200d9953d11d53e5aee3968d818321c25c5803", "arithmetic_steps": 87 }, "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." } }