{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261113-r02-real-exponential-function-0005", "pilot_number": "R0047-24230", "number": 24230, "family_id": "qf-mod44-exp-finite-partial-sum", "title": { "zh-Hant": "第 24230 題：實指數函數", "en": "Question 24230: Exact Finite Exponential Sums Are Not General Infinite-Value Certificates" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261113-r02-real-exponential-function-0005", "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-finite-partial-sum；資料={\"x\":[5,6],\"cutoff\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, 否, finite_partial_sum_is_not_the_infinite_exponential_value)", "explanation_zh": "Wave6第七層核對=(5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；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-finite-partial-sum", "seed": 202611132005, "parameters": { "x": [ 5, 6 ], "cutoff": 9, "domain": "Exact_real_exponential_series", "operation": "exp-finite-partial-sum", "semantic_template": "qf-mod44-exp-finite-partial-sum/{\"cutoff\":9,\"x\":[5,6]}" } }, "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-0005", "batch_id": "batch-20261015-advance-20261113-r02-real-exponential-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "08aaae32003954ee91b75ca09c9b6f0176e2737a22efc8db33628e2aed240b98", "family_id": "qf-mod44-exp-finite-partial-sum", "seed": 202611132005, "semantic_template": "qf-mod44-exp-finite-partial-sum/{\"cutoff\":9,\"x\":[5,6]}" }, "fields": { "title": "Exact Finite Exponential Sums Are Not General Infinite-Value Certificates", "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 exact x=5/6, cutoff=9, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).", "answer": { "kind": "expression", "value": "(5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, No, finite_partial_sum_is_not_the_infinite_exponential_value)", "explanation": "Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, No, finite_partial_sum_is_not_the_infinite_exponential_value)." }, "skills": [ "Exact Finite Exponential Sums Are Not General Infinite-Value Certificates" ], "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-finite-partial-sum；資料={\"x\":[5,6],\"cutoff\":9}。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 exact x=5/6, cutoff=9, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [exp-finite-partial-sum]", "after": "derived = [[x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary]]", "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": "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": "S003", "uses": [ "G001", "S002" ], "before": "(5/6)^0", "after": "Exp ORIGINAL 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": [ "5/6", "0" ], "output": "1", "label": "Exp ORIGINAL x power0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Exp term0 power over factorial := div(1, 1)", "after": "Exp term0 power over factorial = 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 term0 power over factorial，不以小數近似替代。", "en": "Recompute Exp term0 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term0 power over factorial" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Exp cumulative sum through0 := sum(0, 1)", "after": "Exp cumulative sum through0 = 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 cumulative sum through0，不以小數近似替代。", "en": "Recompute Exp cumulative sum through0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1" ], "output": "1", "label": "Exp cumulative sum through0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Exp term1 factorial step1 := mul(1, 1)", "after": "Exp term1 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 term1 factorial step1，不以小數近似替代。", "en": "Recompute Exp term1 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term1 factorial step1" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "(5/6)^1", "after": "Exp ORIGINAL x power1 = 5/6", "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": [ "5/6", "1" ], "output": "5/6", "label": "Exp ORIGINAL x power1" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Exp term1 power over factorial := div(5/6, 1)", "after": "Exp term1 power over factorial = 5/6", "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 term1 power over factorial，不以小數近似替代。", "en": "Recompute Exp term1 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "5/6", "1" ], "output": "5/6", "label": "Exp term1 power over factorial" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Exp cumulative sum through1 := sum(1, 5/6)", "after": "Exp cumulative sum through1 = 11/6", "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 cumulative sum through1，不以小數近似替代。", "en": "Recompute Exp cumulative sum through1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "5/6" ], "output": "11/6", "label": "Exp cumulative sum through1" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Exp term2 factorial step1 := mul(1, 1)", "after": "Exp term2 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 term2 factorial step1，不以小數近似替代。", "en": "Recompute Exp term2 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term2 factorial step1" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Exp term2 factorial step2 := mul(1, 2)", "after": "Exp term2 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 term2 factorial step2，不以小數近似替代。", "en": "Recompute Exp term2 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp term2 factorial step2" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "(5/6)^2", "after": "Exp ORIGINAL x power2 = 25/36", "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": [ "5/6", "2" ], "output": "25/36", "label": "Exp ORIGINAL x power2" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Exp term2 power over factorial := div(25/36, 2)", "after": "Exp term2 power over factorial = 25/72", "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 term2 power over factorial，不以小數近似替代。", "en": "Recompute Exp term2 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "25/36", "2" ], "output": "25/72", "label": "Exp term2 power over factorial" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Exp cumulative sum through2 := sum(11/6, 25/72)", "after": "Exp cumulative sum through2 = 157/72", "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 cumulative sum through2，不以小數近似替代。", "en": "Recompute Exp cumulative sum through2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "11/6", "25/72" ], "output": "157/72", "label": "Exp cumulative sum through2" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Exp term3 factorial step1 := mul(1, 1)", "after": "Exp term3 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 term3 factorial step1，不以小數近似替代。", "en": "Recompute Exp term3 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term3 factorial step1" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Exp term3 factorial step2 := mul(1, 2)", "after": "Exp term3 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 term3 factorial step2，不以小數近似替代。", "en": "Recompute Exp term3 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp term3 factorial step2" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Exp term3 factorial step3 := mul(2, 3)", "after": "Exp term3 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 term3 factorial step3，不以小數近似替代。", "en": "Recompute Exp term3 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp term3 factorial step3" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "(5/6)^3", "after": "Exp ORIGINAL x power3 = 125/216", "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": [ "5/6", "3" ], "output": "125/216", "label": "Exp ORIGINAL x power3" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Exp term3 power over factorial := div(125/216, 6)", "after": "Exp term3 power over factorial = 125/1296", "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 term3 power over factorial，不以小數近似替代。", "en": "Recompute Exp term3 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "125/216", "6" ], "output": "125/1296", "label": "Exp term3 power over factorial" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Exp cumulative sum through3 := sum(157/72, 125/1296)", "after": "Exp cumulative sum through3 = 2951/1296", "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 cumulative sum through3，不以小數近似替代。", "en": "Recompute Exp cumulative sum through3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "157/72", "125/1296" ], "output": "2951/1296", "label": "Exp cumulative sum through3" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Exp term4 factorial step1 := mul(1, 1)", "after": "Exp term4 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 term4 factorial step1，不以小數近似替代。", "en": "Recompute Exp term4 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term4 factorial step1" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Exp term4 factorial step2 := mul(1, 2)", "after": "Exp term4 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 term4 factorial step2，不以小數近似替代。", "en": "Recompute Exp term4 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp term4 factorial step2" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Exp term4 factorial step3 := mul(2, 3)", "after": "Exp term4 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 term4 factorial step3，不以小數近似替代。", "en": "Recompute Exp term4 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp term4 factorial step3" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "Exp term4 factorial step4 := mul(6, 4)", "after": "Exp term4 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 term4 factorial step4，不以小數近似替代。", "en": "Recompute Exp term4 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp term4 factorial step4" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "(5/6)^4", "after": "Exp ORIGINAL x power4 = 625/1296", "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": [ "5/6", "4" ], "output": "625/1296", "label": "Exp ORIGINAL x power4" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "Exp term4 power over factorial := div(625/1296, 24)", "after": "Exp term4 power over factorial = 625/31104", "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 term4 power over factorial，不以小數近似替代。", "en": "Recompute Exp term4 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "625/1296", "24" ], "output": "625/31104", "label": "Exp term4 power over factorial" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "Exp cumulative sum through4 := sum(2951/1296, 625/31104)", "after": "Exp cumulative sum through4 = 71449/31104", "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 cumulative sum through4，不以小數近似替代。", "en": "Recompute Exp cumulative sum through4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2951/1296", "625/31104" ], "output": "71449/31104", "label": "Exp cumulative sum through4" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "Exp term5 factorial step1 := mul(1, 1)", "after": "Exp term5 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 term5 factorial step1，不以小數近似替代。", "en": "Recompute Exp term5 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term5 factorial step1" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "Exp term5 factorial step2 := mul(1, 2)", "after": "Exp term5 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 term5 factorial step2，不以小數近似替代。", "en": "Recompute Exp term5 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp term5 factorial step2" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "Exp term5 factorial step3 := mul(2, 3)", "after": "Exp term5 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 term5 factorial step3，不以小數近似替代。", "en": "Recompute Exp term5 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp term5 factorial step3" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "Exp term5 factorial step4 := mul(6, 4)", "after": "Exp term5 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 term5 factorial step4，不以小數近似替代。", "en": "Recompute Exp term5 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp term5 factorial step4" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "Exp term5 factorial step5 := mul(24, 5)", "after": "Exp term5 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 term5 factorial step5，不以小數近似替代。", "en": "Recompute Exp term5 factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp term5 factorial step5" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "(5/6)^5", "after": "Exp ORIGINAL x power5 = 3125/7776", "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": [ "5/6", "5" ], "output": "3125/7776", "label": "Exp ORIGINAL x power5" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "Exp term5 power over factorial := div(3125/7776, 120)", "after": "Exp term5 power over factorial = 625/186624", "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 term5 power over factorial，不以小數近似替代。", "en": "Recompute Exp term5 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "3125/7776", "120" ], "output": "625/186624", "label": "Exp term5 power over factorial" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "Exp cumulative sum through5 := sum(71449/31104, 625/186624)", "after": "Exp cumulative sum through5 = 429319/186624", "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 cumulative sum through5，不以小數近似替代。", "en": "Recompute Exp cumulative sum through5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "71449/31104", "625/186624" ], "output": "429319/186624", "label": "Exp cumulative sum through5" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "Exp term6 factorial step1 := mul(1, 1)", "after": "Exp term6 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 term6 factorial step1，不以小數近似替代。", "en": "Recompute Exp term6 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term6 factorial step1" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "Exp term6 factorial step2 := mul(1, 2)", "after": "Exp term6 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 term6 factorial step2，不以小數近似替代。", "en": "Recompute Exp term6 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp term6 factorial step2" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "Exp term6 factorial step3 := mul(2, 3)", "after": "Exp term6 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 term6 factorial step3，不以小數近似替代。", "en": "Recompute Exp term6 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp term6 factorial step3" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "Exp term6 factorial step4 := mul(6, 4)", "after": "Exp term6 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 term6 factorial step4，不以小數近似替代。", "en": "Recompute Exp term6 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp term6 factorial step4" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "Exp term6 factorial step5 := mul(24, 5)", "after": "Exp term6 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 term6 factorial step5，不以小數近似替代。", "en": "Recompute Exp term6 factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp term6 factorial step5" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "Exp term6 factorial step6 := mul(120, 6)", "after": "Exp term6 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 term6 factorial step6，不以小數近似替代。", "en": "Recompute Exp term6 factorial step6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "120", "6" ], "output": "720", "label": "Exp term6 factorial step6" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "(5/6)^6", "after": "Exp ORIGINAL x power6 = 15625/46656", "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": [ "5/6", "6" ], "output": "15625/46656", "label": "Exp ORIGINAL x power6" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "Exp term6 power over factorial := div(15625/46656, 720)", "after": "Exp term6 power over factorial = 3125/6718464", "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 term6 power over factorial，不以小數近似替代。", "en": "Recompute Exp term6 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "15625/46656", "720" ], "output": "3125/6718464", "label": "Exp term6 power over factorial" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "Exp cumulative sum through6 := sum(429319/186624, 3125/6718464)", "after": "Exp cumulative sum through6 = 15458609/6718464", "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 cumulative sum through6，不以小數近似替代。", "en": "Recompute Exp cumulative sum through6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "429319/186624", "3125/6718464" ], "output": "15458609/6718464", "label": "Exp cumulative sum through6" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "Exp term7 factorial step1 := mul(1, 1)", "after": "Exp term7 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 term7 factorial step1，不以小數近似替代。", "en": "Recompute Exp term7 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term7 factorial step1" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "Exp term7 factorial step2 := mul(1, 2)", "after": "Exp term7 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 term7 factorial step2，不以小數近似替代。", "en": "Recompute Exp term7 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp term7 factorial step2" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "Exp term7 factorial step3 := mul(2, 3)", "after": "Exp term7 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 term7 factorial step3，不以小數近似替代。", "en": "Recompute Exp term7 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp term7 factorial step3" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "Exp term7 factorial step4 := mul(6, 4)", "after": "Exp term7 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 term7 factorial step4，不以小數近似替代。", "en": "Recompute Exp term7 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp term7 factorial step4" } }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "Exp term7 factorial step5 := mul(24, 5)", "after": "Exp term7 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 term7 factorial step5，不以小數近似替代。", "en": "Recompute Exp term7 factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp term7 factorial step5" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "Exp term7 factorial step6 := mul(120, 6)", "after": "Exp term7 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 term7 factorial step6，不以小數近似替代。", "en": "Recompute Exp term7 factorial step6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "120", "6" ], "output": "720", "label": "Exp term7 factorial step6" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "Exp term7 factorial step7 := mul(720, 7)", "after": "Exp term7 factorial step7 = 5040", "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 term7 factorial step7，不以小數近似替代。", "en": "Recompute Exp term7 factorial step7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "720", "7" ], "output": "5040", "label": "Exp term7 factorial step7" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "(5/6)^7", "after": "Exp ORIGINAL x power7 = 78125/279936", "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": [ "5/6", "7" ], "output": "78125/279936", "label": "Exp ORIGINAL x power7" } }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "Exp term7 power over factorial := div(78125/279936, 5040)", "after": "Exp term7 power over factorial = 15625/282175488", "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 term7 power over factorial，不以小數近似替代。", "en": "Recompute Exp term7 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "78125/279936", "5040" ], "output": "15625/282175488", "label": "Exp term7 power over factorial" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "Exp cumulative sum through7 := sum(15458609/6718464, 15625/282175488)", "after": "Exp cumulative sum through7 = 649277203/282175488", "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 cumulative sum through7，不以小數近似替代。", "en": "Recompute Exp cumulative sum through7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "15458609/6718464", "15625/282175488" ], "output": "649277203/282175488", "label": "Exp cumulative sum through7" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "Exp term8 factorial step1 := mul(1, 1)", "after": "Exp term8 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 term8 factorial step1，不以小數近似替代。", "en": "Recompute Exp term8 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term8 factorial step1" } }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "Exp term8 factorial step2 := mul(1, 2)", "after": "Exp term8 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 term8 factorial step2，不以小數近似替代。", "en": "Recompute Exp term8 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp term8 factorial step2" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "Exp term8 factorial step3 := mul(2, 3)", "after": "Exp term8 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 term8 factorial step3，不以小數近似替代。", "en": "Recompute Exp term8 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp term8 factorial step3" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "Exp term8 factorial step4 := mul(6, 4)", "after": "Exp term8 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 term8 factorial step4，不以小數近似替代。", "en": "Recompute Exp term8 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp term8 factorial step4" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "Exp term8 factorial step5 := mul(24, 5)", "after": "Exp term8 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 term8 factorial step5，不以小數近似替代。", "en": "Recompute Exp term8 factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp term8 factorial step5" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "Exp term8 factorial step6 := mul(120, 6)", "after": "Exp term8 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 term8 factorial step6，不以小數近似替代。", "en": "Recompute Exp term8 factorial step6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "120", "6" ], "output": "720", "label": "Exp term8 factorial step6" } }, { "id": "S061", "uses": [ "G001", "S060" ], "before": "Exp term8 factorial step7 := mul(720, 7)", "after": "Exp term8 factorial step7 = 5040", "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 term8 factorial step7，不以小數近似替代。", "en": "Recompute Exp term8 factorial step7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "720", "7" ], "output": "5040", "label": "Exp term8 factorial step7" } }, { "id": "S062", "uses": [ "G001", "S061" ], "before": "Exp term8 factorial step8 := mul(5040, 8)", "after": "Exp term8 factorial step8 = 40320", "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 term8 factorial step8，不以小數近似替代。", "en": "Recompute Exp term8 factorial step8 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5040", "8" ], "output": "40320", "label": "Exp term8 factorial step8" } }, { "id": "S063", "uses": [ "G001", "S062" ], "before": "(5/6)^8", "after": "Exp ORIGINAL x power8 = 390625/1679616", "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": [ "5/6", "8" ], "output": "390625/1679616", "label": "Exp ORIGINAL x power8" } }, { "id": "S064", "uses": [ "G001", "S063" ], "before": "Exp term8 power over factorial := div(390625/1679616, 40320)", "after": "Exp term8 power over factorial = 78125/13544423424", "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 term8 power over factorial，不以小數近似替代。", "en": "Recompute Exp term8 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "390625/1679616", "40320" ], "output": "78125/13544423424", "label": "Exp term8 power over factorial" } }, { "id": "S065", "uses": [ "G001", "S064" ], "before": "Exp cumulative sum through8 := sum(649277203/282175488, 78125/13544423424)", "after": "Exp cumulative sum through8 = 31165383869/13544423424", "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 cumulative sum through8，不以小數近似替代。", "en": "Recompute Exp cumulative sum through8 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "649277203/282175488", "78125/13544423424" ], "output": "31165383869/13544423424", "label": "Exp cumulative sum through8" } }, { "id": "S066", "uses": [ "G001", "S065" ], "before": "Exp term9 factorial step1 := mul(1, 1)", "after": "Exp term9 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 term9 factorial step1，不以小數近似替代。", "en": "Recompute Exp term9 factorial step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Exp term9 factorial step1" } }, { "id": "S067", "uses": [ "G001", "S066" ], "before": "Exp term9 factorial step2 := mul(1, 2)", "after": "Exp term9 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 term9 factorial step2，不以小數近似替代。", "en": "Recompute Exp term9 factorial step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "Exp term9 factorial step2" } }, { "id": "S068", "uses": [ "G001", "S067" ], "before": "Exp term9 factorial step3 := mul(2, 3)", "after": "Exp term9 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 term9 factorial step3，不以小數近似替代。", "en": "Recompute Exp term9 factorial step3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Exp term9 factorial step3" } }, { "id": "S069", "uses": [ "G001", "S068" ], "before": "Exp term9 factorial step4 := mul(6, 4)", "after": "Exp term9 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 term9 factorial step4，不以小數近似替代。", "en": "Recompute Exp term9 factorial step4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "4" ], "output": "24", "label": "Exp term9 factorial step4" } }, { "id": "S070", "uses": [ "G001", "S069" ], "before": "Exp term9 factorial step5 := mul(24, 5)", "after": "Exp term9 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 term9 factorial step5，不以小數近似替代。", "en": "Recompute Exp term9 factorial step5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "5" ], "output": "120", "label": "Exp term9 factorial step5" } }, { "id": "S071", "uses": [ "G001", "S070" ], "before": "Exp term9 factorial step6 := mul(120, 6)", "after": "Exp term9 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 term9 factorial step6，不以小數近似替代。", "en": "Recompute Exp term9 factorial step6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "120", "6" ], "output": "720", "label": "Exp term9 factorial step6" } }, { "id": "S072", "uses": [ "G001", "S071" ], "before": "Exp term9 factorial step7 := mul(720, 7)", "after": "Exp term9 factorial step7 = 5040", "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 term9 factorial step7，不以小數近似替代。", "en": "Recompute Exp term9 factorial step7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "720", "7" ], "output": "5040", "label": "Exp term9 factorial step7" } }, { "id": "S073", "uses": [ "G001", "S072" ], "before": "Exp term9 factorial step8 := mul(5040, 8)", "after": "Exp term9 factorial step8 = 40320", "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 term9 factorial step8，不以小數近似替代。", "en": "Recompute Exp term9 factorial step8 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5040", "8" ], "output": "40320", "label": "Exp term9 factorial step8" } }, { "id": "S074", "uses": [ "G001", "S073" ], "before": "Exp term9 factorial step9 := mul(40320, 9)", "after": "Exp term9 factorial step9 = 362880", "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 term9 factorial step9，不以小數近似替代。", "en": "Recompute Exp term9 factorial step9 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "40320", "9" ], "output": "362880", "label": "Exp term9 factorial step9" } }, { "id": "S075", "uses": [ "G001", "S074" ], "before": "(5/6)^9", "after": "Exp ORIGINAL x power9 = 1953125/10077696", "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": [ "5/6", "9" ], "output": "1953125/10077696", "label": "Exp ORIGINAL x power9" } }, { "id": "S076", "uses": [ "G001", "S075" ], "before": "Exp term9 power over factorial := div(1953125/10077696, 362880)", "after": "Exp term9 power over factorial = 390625/731398864896", "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 term9 power over factorial，不以小數近似替代。", "en": "Recompute Exp term9 power over factorial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1953125/10077696", "362880" ], "output": "390625/731398864896", "label": "Exp term9 power over factorial" } }, { "id": "S077", "uses": [ "G001", "S076" ], "before": "Exp cumulative sum through9 := sum(31165383869/13544423424, 390625/731398864896)", "after": "Exp cumulative sum through9 = 1682931119551/731398864896", "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 cumulative sum through9，不以小數近似替代。", "en": "Recompute Exp cumulative sum through9 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "31165383869/13544423424", "390625/731398864896" ], "output": "1682931119551/731398864896", "label": "Exp cumulative sum through9" } }, { "id": "S078", "uses": [ "G001", "S077" ], "before": "source roles = [5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896]]", "after": "derived = [1682931119551/731398864896, 否, finite sum is not a general infinite-value certificate]", "relation": "implies", "rule_id": "Exp_finite_sum_No_is_not_strict_inequality_at_every_x", "conditions": [], "explanation": { "zh-Hant": "此計算只到cutoff且含0階。固定No不由有限和宣稱一般E(x) exact值；x=0時有限和可真等E(0)=1，因此不寫一律不等。", "en": "This calculation includes degree0 through the cutoff.No withholds a general exact infinite-value certificate.At x=0 the finite sum can equal E(0)=1,so do not assert universal inequality." }, "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": [ "5/6", 9, [ "1", "5/6", "25/72", "125/1296", "625/31104", "625/186624", "3125/6718464", "15625/282175488", "78125/13544423424", "390625/731398864896" ], [ "1", "11/6", "157/72", "2951/1296", "71449/31104", "429319/186624", "15458609/6718464", "649277203/282175488", "31165383869/13544423424", "1682931119551/731398864896" ], "1682931119551/731398864896", false, "finite_partial_sum_is_not_the_infinite_exponential_value" ] }, "conclusion": { "canonical_value": "(5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, 否, finite_partial_sum_is_not_the_infinite_exponential_value)", "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": 5, "original_line_sha256": "e9693ff89537844d2494449713874dc23a1539fd9cd3b22e7f90a7ed79a5e0ca", "question_compact_sha256": "08aaae32003954ee91b75ca09c9b6f0176e2737a22efc8db33628e2aed240b98", "english_row_compact_sha256": "c61290dd3b743da7e3c518d5e0b6dd74c11c48c7e8387a7dacd11b0713f8adfb", "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": "85ddcf294428eba3f348a4206862fbb527c0aaf950fc8e6a1da76ec94c401c75", "original_candidate_line_sha256": "85ddcf294428eba3f348a4206862fbb527c0aaf950fc8e6a1da76ec94c401c75" }, "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": 78, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "83d4a122fce9181abec580a6703f9ab20d80e44de9b64bd1502425f73c0c0560", "arithmetic_steps": 75 }, "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." } }