{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261008-advance-20261010-r01-polynomial-0079", "pilot_number": "R0007-04254", "number": 4254, "family_id": "qf-density10-poly-shift-coefficients", "title": { "zh-Hant": "第 4254 題：變數平移後的完整係數", "en": "Question 4254: Complete Polynomial Translation Coefficients" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261008-advance-20261010-r01-polynomial-0079", "batch_id": "batch-20261008-advance-20261010-r01-polynomial-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-polynomial" ], "title_zh": "變數平移後的完整係數", "stem_zh": "升冪係數 f=[-9, 4, -2, 1]，c=1。求 f(x+c) 的完整升冪係數，不只求某點值。", "answer": { "kind": "expression", "value": "[-6, 3, 1, 1]", "explanation_zh": "平移後升冪係數=[-6, 3, 1, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "變數平移後的完整係數" ] }, "variant": { "family_id": "qf-density10-poly-shift-coefficients", "seed": 202610101079, "parameters": { "f": [ -9, 4, -2, 1 ], "c": 1, "representation": "ascending_powers", "operation": "density10_poly_shift_coefficients", "domain": "integer_polynomial_slice", "semantic_template": "qf-density10-poly-shift-coefficients/[[-9,4,-2,1],1]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-08", "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-0079", "batch_id": "batch-20261008-advance-20261010-r01-polynomial-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b749b7164650fc379b6311d8b0528b936dc0b87589e3138cdaa919bdef86b997", "family_id": "qf-density10-poly-shift-coefficients", "seed": 202610101079, "semantic_template": "qf-density10-poly-shift-coefficients/[[-9,4,-2,1],1]" }, "fields": { "title": "Complete Polynomial Translation Coefficients", "stem": "Integer polynomials have ascending coefficients f=[-9, 4, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.", "answer": { "kind": "expression", "value": "[-6, 3, 1, 1]", "explanation": "Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-6, 3, 1, 1]." }, "skills": [ "Complete Polynomial Translation 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=[-9, 4, -2, 1]，c=1。求 f(x+c) 的完整升冪係數，不只求某點值。", "en": "Integer polynomials have ascending coefficients f=[-9, 4, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "f ascending coefficients=[-9, 4, -2, 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": "term -9(x+(1))^0", "after": "expand (-9)·Σ_j C(0,j)x^j(1)^(0−j)", "relation": "implies", "rule_id": "shifted_power_binomial_identity", "conditions": [], "explanation": { "zh-Hant": "保留原項係數後做二項式展開，再按每個新次冪j合併；平移不只是改常數項。", "en": "Retain the original coefficient during binomial expansion and collect each resulting power j;translation does not merely change the constant coefficient." }, "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": "(1)^0", "after": "shift power0,0 = 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": [ "1", "0" ], "output": "1", "label": "shift power0,0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "shift scaled binomial0,0 := mul(-9, 1)", "after": "shift scaled binomial0,0 = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial0,0，不以小數近似替代。", "en": "Recompute shift scaled binomial0,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-9", "1" ], "output": "-9", "label": "shift scaled binomial0,0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "shift contribution0,0 := mul(-9, 1)", "after": "shift contribution0,0 = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution0,0，不以小數近似替代。", "en": "Recompute shift contribution0,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-9", "1" ], "output": "-9", "label": "shift contribution0,0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "term 4(x+(1))^1", "after": "expand (4)·Σ_j C(1,j)x^j(1)^(1−j)", "relation": "implies", "rule_id": "shifted_power_binomial_identity", "conditions": [], "explanation": { "zh-Hant": "保留原項係數後做二項式展開，再按每個新次冪j合併；平移不只是改常數項。", "en": "Retain the original coefficient during binomial expansion and collect each resulting power j;translation does not merely change the constant coefficient." }, "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": "S007", "uses": [ "G001", "S006" ], "before": "(1)^1", "after": "shift power1,0 = 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": [ "1", "1" ], "output": "1", "label": "shift power1,0" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "shift scaled binomial1,0 := mul(4, 1)", "after": "shift scaled binomial1,0 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial1,0，不以小數近似替代。", "en": "Recompute shift scaled binomial1,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1" ], "output": "4", "label": "shift scaled binomial1,0" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "shift contribution1,0 := mul(4, 1)", "after": "shift contribution1,0 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution1,0，不以小數近似替代。", "en": "Recompute shift contribution1,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1" ], "output": "4", "label": "shift contribution1,0" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "(1)^0", "after": "shift power1,1 = 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": [ "1", "0" ], "output": "1", "label": "shift power1,1" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "shift scaled binomial1,1 := mul(4, 1)", "after": "shift scaled binomial1,1 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial1,1，不以小數近似替代。", "en": "Recompute shift scaled binomial1,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1" ], "output": "4", "label": "shift scaled binomial1,1" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "shift contribution1,1 := mul(4, 1)", "after": "shift contribution1,1 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution1,1，不以小數近似替代。", "en": "Recompute shift contribution1,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1" ], "output": "4", "label": "shift contribution1,1" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "term -2(x+(1))^2", "after": "expand (-2)·Σ_j C(2,j)x^j(1)^(2−j)", "relation": "implies", "rule_id": "shifted_power_binomial_identity", "conditions": [], "explanation": { "zh-Hant": "保留原項係數後做二項式展開，再按每個新次冪j合併；平移不只是改常數項。", "en": "Retain the original coefficient during binomial expansion and collect each resulting power j;translation does not merely change the constant coefficient." }, "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": "(1)^2", "after": "shift power2,0 = 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": [ "1", "2" ], "output": "1", "label": "shift power2,0" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "shift scaled binomial2,0 := mul(-2, 1)", "after": "shift scaled binomial2,0 = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial2,0，不以小數近似替代。", "en": "Recompute shift scaled binomial2,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "shift scaled binomial2,0" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "shift contribution2,0 := mul(-2, 1)", "after": "shift contribution2,0 = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution2,0，不以小數近似替代。", "en": "Recompute shift contribution2,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "shift contribution2,0" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "(1)^1", "after": "shift power2,1 = 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": [ "1", "1" ], "output": "1", "label": "shift power2,1" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "shift scaled binomial2,1 := mul(-2, 2)", "after": "shift scaled binomial2,1 = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial2,1，不以小數近似替代。", "en": "Recompute shift scaled binomial2,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "2" ], "output": "-4", "label": "shift scaled binomial2,1" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "shift contribution2,1 := mul(-4, 1)", "after": "shift contribution2,1 = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution2,1，不以小數近似替代。", "en": "Recompute shift contribution2,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-4", "1" ], "output": "-4", "label": "shift contribution2,1" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "(1)^0", "after": "shift power2,2 = 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": [ "1", "0" ], "output": "1", "label": "shift power2,2" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "shift scaled binomial2,2 := mul(-2, 1)", "after": "shift scaled binomial2,2 = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial2,2，不以小數近似替代。", "en": "Recompute shift scaled binomial2,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "shift scaled binomial2,2" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "shift contribution2,2 := mul(-2, 1)", "after": "shift contribution2,2 = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution2,2，不以小數近似替代。", "en": "Recompute shift contribution2,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "shift contribution2,2" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "term 1(x+(1))^3", "after": "expand (1)·Σ_j C(3,j)x^j(1)^(3−j)", "relation": "implies", "rule_id": "shifted_power_binomial_identity", "conditions": [], "explanation": { "zh-Hant": "保留原項係數後做二項式展開，再按每個新次冪j合併；平移不只是改常數項。", "en": "Retain the original coefficient during binomial expansion and collect each resulting power j;translation does not merely change the constant coefficient." }, "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": "(1)^3", "after": "shift power3,0 = 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": [ "1", "3" ], "output": "1", "label": "shift power3,0" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "shift scaled binomial3,0 := mul(1, 1)", "after": "shift scaled binomial3,0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial3,0，不以小數近似替代。", "en": "Recompute shift scaled binomial3,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "shift scaled binomial3,0" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "shift contribution3,0 := mul(1, 1)", "after": "shift contribution3,0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution3,0，不以小數近似替代。", "en": "Recompute shift contribution3,0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "shift contribution3,0" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "(1)^2", "after": "shift power3,1 = 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": [ "1", "2" ], "output": "1", "label": "shift power3,1" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "shift scaled binomial3,1 := mul(1, 3)", "after": "shift scaled binomial3,1 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial3,1，不以小數近似替代。", "en": "Recompute shift scaled binomial3,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "3" ], "output": "3", "label": "shift scaled binomial3,1" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "shift contribution3,1 := mul(3, 1)", "after": "shift contribution3,1 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution3,1，不以小數近似替代。", "en": "Recompute shift contribution3,1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1" ], "output": "3", "label": "shift contribution3,1" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "(1)^1", "after": "shift power3,2 = 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": [ "1", "1" ], "output": "1", "label": "shift power3,2" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "shift scaled binomial3,2 := mul(1, 3)", "after": "shift scaled binomial3,2 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial3,2，不以小數近似替代。", "en": "Recompute shift scaled binomial3,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "3" ], "output": "3", "label": "shift scaled binomial3,2" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "shift contribution3,2 := mul(3, 1)", "after": "shift contribution3,2 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution3,2，不以小數近似替代。", "en": "Recompute shift contribution3,2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1" ], "output": "3", "label": "shift contribution3,2" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "(1)^0", "after": "shift power3,3 = 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": [ "1", "0" ], "output": "1", "label": "shift power3,3" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "shift scaled binomial3,3 := mul(1, 1)", "after": "shift scaled binomial3,3 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘本項係數與二項係數。", "en": "Multiply this term's coefficient by its binomial coefficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift scaled binomial3,3，不以小數近似替代。", "en": "Recompute shift scaled binomial3,3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "shift scaled binomial3,3" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "shift contribution3,3 := mul(1, 1)", "after": "shift contribution3,3 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再乘c^(k−j)，作為x^j的貢獻。", "en": "Multiply by c^(k−j) to obtain the contribution to x^j." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift contribution3,3，不以小數近似替代。", "en": "Recompute shift contribution3,3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "shift contribution3,3" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "shift coefficient0 := sum(-9, 4, -2, 1)", "after": "shift coefficient0 = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同次冪貢獻全部相加。", "en": "Add all contributions to the same power." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift coefficient0，不以小數近似替代。", "en": "Recompute shift coefficient0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-9", "4", "-2", "1" ], "output": "-6", "label": "shift coefficient0" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "shift coefficient1 := sum(4, -4, 3)", "after": "shift coefficient1 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同次冪貢獻全部相加。", "en": "Add all contributions to the same power." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift coefficient1，不以小數近似替代。", "en": "Recompute shift coefficient1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "-4", "3" ], "output": "3", "label": "shift coefficient1" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "shift coefficient2 := sum(-2, 3)", "after": "shift coefficient2 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同次冪貢獻全部相加。", "en": "Add all contributions to the same power." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift coefficient2，不以小數近似替代。", "en": "Recompute shift coefficient2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "3" ], "output": "1", "label": "shift coefficient2" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "shift coefficient3 := sum(1)", "after": "shift coefficient3 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "同次冪貢獻全部相加。", "en": "Add all contributions to the same power." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 shift coefficient3，不以小數近似替代。", "en": "Recompute shift coefficient3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1" ], "output": "1", "label": "shift coefficient3" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "[-6, 3, 1, 1]", "after": "canonical ascending coefficient list", "relation": "implies", "rule_id": "polynomial_shift_coefficients", "conditions": [], "explanation": { "zh-Hant": "所得是完整f(x+c)係數恆等；常數或零多項式平移後不變。", "en": "The result is the full coefficient identity for f(x+c);constant and zero polynomials are unchanged by translation." }, "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": [ "-6", "3", "1", "1" ] }, "conclusion": { "canonical_value": "[-6, 3, 1, 1]", "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": 79, "original_line_sha256": "e4c2e3b2d80a8bb4481caf296ef67eacdf2e79a67305952fa8e54c124cb95457", "question_compact_sha256": "b749b7164650fc379b6311d8b0528b936dc0b87589e3138cdaa919bdef86b997", "english_row_compact_sha256": "d9e84c33f51d5a2dcfef52090c1673fc4020bd1016d253142a78cdef19d912b1", "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": "12b67097de39f8e1000a11ffa9859fd4e0c15a41d04366957f5bdd1ca816d479", "original_candidate_line_sha256": "12b67097de39f8e1000a11ffa9859fd4e0c15a41d04366957f5bdd1ca816d479" }, "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": 40, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "cc185685f5e3b851a910af3e4b3226b48d567dfbffe0ca642d1c8a4013fc95aa", "arithmetic_steps": 34 }, "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." } }