{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261008-advance-20261010-r01-polynomial-0050", "pilot_number": "R0007-04250", "number": 4250, "family_id": "qf-density10-poly-product-coefficients", "title": { "zh-Hant": "第 4250 題：多項式積的完整係數", "en": "Question 4250: Complete Polynomial Product Coefficients" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261008-advance-20261010-r01-polynomial-0050", "batch_id": "batch-20261008-advance-20261010-r01-polynomial-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-polynomial" ], "title_zh": "多項式積的完整係數", "stem_zh": "升冪係數 f=[12, 25, -24, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。", "answer": { "kind": "expression", "value": "[24, 38, -61, 75, 25, -47, 2]", "explanation_zh": "積的升冪係數=[24, 38, -61, 75, 25, -47, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。" }, "difficulty": { "band": "upper_secondary", "level": 3, "skills": [ "多項式積的完整係數" ] }, "variant": { "family_id": "qf-density10-poly-product-coefficients", "seed": 202610101050, "parameters": { "f": [ 12, 25, -24, 1 ], "g": [ 2, -1, 1, 2 ], "representation": "ascending_powers", "operation": "density10_poly_product_coefficients", "domain": "integer_polynomial_slice", "semantic_template": "qf-density10-poly-product-coefficients/[[12,25,-24,1],[2,-1,1,2]]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-08", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261008-advance-20261010-r01-polynomial-0050", "batch_id": "batch-20261008-advance-20261010-r01-polynomial-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "ccb71f014b681ad851896d8197d26d444e6d7b96106a4ebbf84acd4f1c297319", "family_id": "qf-density10-poly-product-coefficients", "seed": 202610101050, "semantic_template": "qf-density10-poly-product-coefficients/[[12,25,-24,1],[2,-1,1,2]]" }, "fields": { "title": "Complete Polynomial Product Coefficients", "stem": "Integer polynomials have ascending coefficients f=[12, 25, -24, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.", "answer": { "kind": "expression", "value": "[24, 38, -61, 75, 25, -47, 2]", "explanation": "Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [24, 38, -61, 75, 25, -47, 2]." }, "skills": [ "Complete Polynomial Product Coefficients" ], "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": "升冪係數 f=[12, 25, -24, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。", "en": "Integer polynomials have ascending coefficients f=[12, 25, -24, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "f ascending coefficients=[12, 25, -24, 1]", "after": "coefficient k multiplies x^k;keep all required degrees", "relation": "implies", "rule_id": "ascending_polynomial_identity", "conditions": [], "explanation": { "zh-Hant": "係數按升冪排列，常數在索引0；完整係數運算不是由單點相等推論恆等，原始係數清單保持不變。", "en": "Coefficients are in ascending powers with the constant at index0.Full coefficient operations are not identities inferred from one-point agreement;retain the original coefficient list." }, "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": "product term degrees0,0 := mul(12, 2)", "after": "product term degrees0,0 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees0,0，不以小數近似替代。", "en": "Recompute product term degrees0,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12", "2" ], "output": "24", "label": "product term degrees0,0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "degrees 0+0", "after": "contribute 24 to degree0", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "product term degrees0,1 := mul(12, -1)", "after": "product term degrees0,1 = -12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees0,1，不以小數近似替代。", "en": "Recompute product term degrees0,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12", "-1" ], "output": "-12", "label": "product term degrees0,1" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "degrees 0+1", "after": "contribute -12 to degree1", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "product term degrees0,2 := mul(12, 1)", "after": "product term degrees0,2 = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees0,2，不以小數近似替代。", "en": "Recompute product term degrees0,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12", "1" ], "output": "12", "label": "product term degrees0,2" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "degrees 0+2", "after": "contribute 12 to degree2", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "product term degrees0,3 := mul(12, 2)", "after": "product term degrees0,3 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees0,3，不以小數近似替代。", "en": "Recompute product term degrees0,3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12", "2" ], "output": "24", "label": "product term degrees0,3" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "degrees 0+3", "after": "contribute 24 to degree3", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "product term degrees1,0 := mul(25, 2)", "after": "product term degrees1,0 = 50", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees1,0，不以小數近似替代。", "en": "Recompute product term degrees1,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "25", "2" ], "output": "50", "label": "product term degrees1,0" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "degrees 1+0", "after": "contribute 50 to degree1", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "product term degrees1,1 := mul(25, -1)", "after": "product term degrees1,1 = -25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees1,1，不以小數近似替代。", "en": "Recompute product term degrees1,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "25", "-1" ], "output": "-25", "label": "product term degrees1,1" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "degrees 1+1", "after": "contribute -25 to degree2", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "product term degrees1,2 := mul(25, 1)", "after": "product term degrees1,2 = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees1,2，不以小數近似替代。", "en": "Recompute product term degrees1,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "25", "1" ], "output": "25", "label": "product term degrees1,2" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "degrees 1+2", "after": "contribute 25 to degree3", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "product term degrees1,3 := mul(25, 2)", "after": "product term degrees1,3 = 50", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees1,3，不以小數近似替代。", "en": "Recompute product term degrees1,3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "25", "2" ], "output": "50", "label": "product term degrees1,3" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "degrees 1+3", "after": "contribute 50 to degree4", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "product term degrees2,0 := mul(-24, 2)", "after": "product term degrees2,0 = -48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees2,0，不以小數近似替代。", "en": "Recompute product term degrees2,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "2" ], "output": "-48", "label": "product term degrees2,0" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "degrees 2+0", "after": "contribute -48 to degree2", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "product term degrees2,1 := mul(-24, -1)", "after": "product term degrees2,1 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees2,1，不以小數近似替代。", "en": "Recompute product term degrees2,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "-1" ], "output": "24", "label": "product term degrees2,1" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "degrees 2+1", "after": "contribute 24 to degree3", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "product term degrees2,2 := mul(-24, 1)", "after": "product term degrees2,2 = -24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees2,2，不以小數近似替代。", "en": "Recompute product term degrees2,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "1" ], "output": "-24", "label": "product term degrees2,2" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "degrees 2+2", "after": "contribute -24 to degree4", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "product term degrees2,3 := mul(-24, 2)", "after": "product term degrees2,3 = -48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees2,3，不以小數近似替代。", "en": "Recompute product term degrees2,3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "2" ], "output": "-48", "label": "product term degrees2,3" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "degrees 2+3", "after": "contribute -48 to degree5", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "product term degrees3,0 := mul(1, 2)", "after": "product term degrees3,0 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees3,0，不以小數近似替代。", "en": "Recompute product term degrees3,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "product term degrees3,0" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "degrees 3+0", "after": "contribute 2 to degree3", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "product term degrees3,1 := mul(1, -1)", "after": "product term degrees3,1 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees3,1，不以小數近似替代。", "en": "Recompute product term degrees3,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-1" ], "output": "-1", "label": "product term degrees3,1" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "degrees 3+1", "after": "contribute -1 to degree4", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "product term degrees3,2 := mul(1, 1)", "after": "product term degrees3,2 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees3,2，不以小數近似替代。", "en": "Recompute product term degrees3,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "product term degrees3,2" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "degrees 3+2", "after": "contribute 1 to degree5", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "product term degrees3,3 := mul(1, 2)", "after": "product term degrees3,3 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "分配律給c_i d_j x^(i+j)，先乘係數。", "en": "Distributivity gives c_i d_j x^(i+j);multiply the coefficients first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product term degrees3,3，不以小數近似替代。", "en": "Recompute product term degrees3,3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "product term degrees3,3" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "degrees 3+3", "after": "contribute 2 to degree6", "relation": "implies", "rule_id": "polynomial_convolution_term", "conditions": [], "explanation": { "zh-Hant": "次數相加後歸到同次冪桶，最高保留到兩個次數的和。", "en": "Add degrees and assign the term to that coefficient bucket,retaining the full summed 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 }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "product coefficient0 := sum(24)", "after": "product coefficient0 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "加總所有滿足i+j=k的係數乘積，不截斷高次。", "en": "Sum every coefficient product with i+j=k,without truncating high powers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product coefficient0，不以小數近似替代。", "en": "Recompute product coefficient0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "24" ], "output": "24", "label": "product coefficient0" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "product coefficient1 := sum(-12, 50)", "after": "product coefficient1 = 38", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "加總所有滿足i+j=k的係數乘積，不截斷高次。", "en": "Sum every coefficient product with i+j=k,without truncating high powers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product coefficient1，不以小數近似替代。", "en": "Recompute product coefficient1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-12", "50" ], "output": "38", "label": "product coefficient1" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "product coefficient2 := sum(12, -25, -48)", "after": "product coefficient2 = -61", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "加總所有滿足i+j=k的係數乘積，不截斷高次。", "en": "Sum every coefficient product with i+j=k,without truncating high powers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product coefficient2，不以小數近似替代。", "en": "Recompute product coefficient2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "12", "-25", "-48" ], "output": "-61", "label": "product coefficient2" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "product coefficient3 := sum(24, 25, 24, 2)", "after": "product coefficient3 = 75", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "加總所有滿足i+j=k的係數乘積，不截斷高次。", "en": "Sum every coefficient product with i+j=k,without truncating high powers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product coefficient3，不以小數近似替代。", "en": "Recompute product coefficient3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "24", "25", "24", "2" ], "output": "75", "label": "product coefficient3" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "product coefficient4 := sum(50, -24, -1)", "after": "product coefficient4 = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "加總所有滿足i+j=k的係數乘積，不截斷高次。", "en": "Sum every coefficient product with i+j=k,without truncating high powers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product coefficient4，不以小數近似替代。", "en": "Recompute product coefficient4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "50", "-24", "-1" ], "output": "25", "label": "product coefficient4" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "product coefficient5 := sum(-48, 1)", "after": "product coefficient5 = -47", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "加總所有滿足i+j=k的係數乘積，不截斷高次。", "en": "Sum every coefficient product with i+j=k,without truncating high powers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product coefficient5，不以小數近似替代。", "en": "Recompute product coefficient5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-48", "1" ], "output": "-47", "label": "product coefficient5" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "product coefficient6 := sum(2)", "after": "product coefficient6 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "加總所有滿足i+j=k的係數乘積，不截斷高次。", "en": "Sum every coefficient product with i+j=k,without truncating high powers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 product coefficient6，不以小數近似替代。", "en": "Recompute product coefficient6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2" ], "output": "2", "label": "product coefficient6" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "[24, 38, -61, 75, 25, -47, 2]", "after": "canonical ascending coefficient list", "relation": "implies", "rule_id": "polynomial_product_coefficients", "conditions": [], "explanation": { "zh-Hant": "完整有限卷積給乘積多項式，三次乘三次可到六次；零因子產生[0]。", "en": "The full finite convolution gives the product;two cubic factors may produce degree6,and a zero factor produces[0]." }, "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": [ "24", "38", "-61", "75", "25", "-47", "2" ] }, "conclusion": { "canonical_value": "[24, 38, -61, 75, 25, -47, 2]", "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-20261008-advance-20261010-r01-polynomial-foundations", "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r01-polynomial-foundations/questions.jsonl", "batch_sha256": "a405b95ddca6b0d773ef4f12f900f94620ea7ec880a98ae7ebae6688f5e23ad6", "original_line": 50, "original_line_sha256": "35d6e0014d6d4a0a451e8a5c717e59c5e8015141afac928ebb6c7216b0d4d777", "question_compact_sha256": "ccb71f014b681ad851896d8197d26d444e6d7b96106a4ebbf84acd4f1c297319", "english_row_compact_sha256": "96023ea797a0f62ff6dfee45b8a6fcbc8a2a8883d41c7156f7e162b0c31df577", "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r01-polynomial-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r01-polynomial-foundations.en.jsonl" }, "mko_bindings": { "mko-polynomial": { "sha256": "0a0d76b12363d10d9ecbe8b45d7bebb1136637db1c41e6308481c675c8db9e6c", "english_sha256": "4c5e7d9539a5612d401321e99d2e0f4667c558a65abb5f412bf6cb3518f9fbc8", "original_path": "public/data/mko/polynomial.json", "english_path": "public/data/i18n/editions/mko-polynomial.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95", "original_candidate_sha256": "195c50abc1ad1ec3658d4d57f1616c4ce52b667e0791eccf8ff3363d03dc3af0", "original_candidate_line_sha256": "195c50abc1ad1ec3658d4d57f1616c4ce52b667e0791eccf8ff3363d03dc3af0" }, "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-20261026-004-p8", "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": "faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a", "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 41, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "3bae46a9a4e99224a445d28e2851533659da989819e853f7463ce2dd29a6dda8", "arithmetic_steps": 23 }, "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." } }