{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20260912-pythagorean-counterexample-0001", "pilot_number": "R0070-35636", "number": 35636, "family_id": "qf-pythagorean-counterexample", "title": { "zh-Hant": "第 35636 題：畢達哥拉斯關係反例辨識", "en": "Question 35636: Counterexample to the Pythagorean Relation" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20260912-pythagorean-counterexample-0001", "batch_id": "batch-20260912-geometry-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-euclid-pythagorean-theorem" ], "title_zh": "畢達哥拉斯關係反例辨識", "stem_zh": "下列哪一組正數不滿足平方關係 a²+b²=c²（各組最後一個數作為 c）？\n甲：3、4、7；乙：3、4、5；丙：3、4、5；丁：3、4、5", "answer": { "kind": "choice", "value": "甲", "choices": [ "甲", "乙", "丙", "丁" ], "explanation_zh": "甲 組的 a²+b² 不等於 c²，因此不滿足本題列出的平方關係。" }, "difficulty": { "band": "lower_secondary", "level": 2, "skills": [ "平方關係", "反例辨識" ] }, "variant": { "family_id": "qf-pythagorean-counterexample", "seed": 20260997, "parameters": { "mode": "counterexample", "options": [ [ 3, 4, 7 ], [ 3, 4, 5 ], [ 3, 4, 5 ], [ 3, 4, 5 ] ], "counterexample_index": 0 } }, "verification": { "status": "candidate", "checks": [ "schema_pending", "independent_recomputation_pending" ], "warning_zh": "本題僅對列出的有限參數重算；不構成相關定義或定理的普遍證明，且尚未人工審查。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "generated_at": "2026-09-12", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20260912-pythagorean-counterexample-0001", "batch_id": "batch-20260912-geometry-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "06809a825e94d636af6b739ac332e42e311f82b75dfbfe354bfbdfc66c656fea", "family_id": "qf-pythagorean-counterexample", "seed": 20260997, "semantic_template": "qf-pythagorean-counterexample/20260997" }, "fields": { "title": "Counterexample to the Pythagorean Relation", "stem": "Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?\nA: 3, 4, 7; B: 3, 4, 5; C: 3, 4, 5; D: 3, 4, 5", "answer": { "kind": "choice", "value": "A", "choices": [ "A", "B", "C", "D" ], "explanation": "For option A, a²+b² is not equal to c², so it is the counterexample to the stated squared relation." }, "skills": [ "squared relation", "counterexample identification" ], "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": "下列哪一組正數不滿足平方關係 a²+b²=c²（各組最後一個數作為 c）？\n甲：3、4、7；乙：3、4、5；丙：3、4、5；丁：3、4、5", "en": "Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c?\nA: 3, 4, 7; B: 3, 4, 5; C: 3, 4, 5; D: 3, 4, 5" } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-pythagorean-counterexample, [counterexample_index, mode, options]]", "after": "derived = [counter, null, null]", "relation": "implies", "rule_id": "R70_SOURCE_bank_specific_interface", "conditions": [], "explanation": { "zh-Hant": "只使用此bank真正明示的欄位、operation/domain與型別；BASE容許保留額外metadata，不把LDL SAFE/集合上限或其他bank的去重規則搬來。", "en": "Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[[3, 4, 7], [3, 4, 5], [3, 4, 5], [3, 4, 5]]]", "after": "derived = [[0: 甲 = A, 1: 乙 = B, 2: 丙 = C, 3: 丁 = D]]", "relation": "implies", "rule_id": "Pythagorean_OPTION_position_legend_keep_duplicate_options", "conditions": [], "explanation": { "zh-Hant": "四選項依原位置保留，即使三個有效triple相同也不去重；中英選項標籤只按位置對應。c是每列最後一項，不能排序偷換。", "en": "Keep all FOUR options in ORIGINAL positions,even repeated valid triples.Chinese 甲乙丙丁 corresponds POSITIONALLY to English ABCD.The LAST coordinate is c;do not sort and silently change that role." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Pyth option 0 a squared := mul(3, 3)", "after": "Pyth option 0 a squared = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 0 a squared，不以小數近似替代。", "en": "Recompute Pyth option 0 a squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "Pyth option 0 a squared" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Pyth option 0 b squared := mul(4, 4)", "after": "Pyth option 0 b squared = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 0 b squared，不以小數近似替代。", "en": "Recompute Pyth option 0 b squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "4" ], "output": "16", "label": "Pyth option 0 b squared" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Pyth option 0 LAST c squared := mul(7, 7)", "after": "Pyth option 0 LAST c squared = 49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 0 LAST c squared，不以小數近似替代。", "en": "Recompute Pyth option 0 LAST c squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "7" ], "output": "49", "label": "Pyth option 0 LAST c squared" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Pyth option 0 sum a2 b2 := sum(9, 16)", "after": "Pyth option 0 sum a2 b2 = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 0 sum a2 b2，不以小數近似替代。", "en": "Recompute Pyth option 0 sum a2 b2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "16" ], "output": "25", "label": "Pyth option 0 sum a2 b2" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [0, [3, 4, 7]]", "after": "derived = [是, 25, 49, 否]", "relation": "implies", "rule_id": "Pythagorean_EACH_positive_triple_square_relation", "conditions": [], "explanation": { "zh-Hant": "逐選項先查正長度且a²+b²=c²；這是題給平方關係的直接核對，不從草圖或largest-sort判斷，也不新增無需的幾何逆定理。", "en": "For EACH option,check positive entries AND a²+b²=c².This directly checks the STATED squared relation,not a drawing/largest-sort inference or an unneeded geometric converse." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Pyth option 1 a squared := mul(3, 3)", "after": "Pyth option 1 a squared = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 1 a squared，不以小數近似替代。", "en": "Recompute Pyth option 1 a squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "Pyth option 1 a squared" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Pyth option 1 b squared := mul(4, 4)", "after": "Pyth option 1 b squared = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 1 b squared，不以小數近似替代。", "en": "Recompute Pyth option 1 b squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "4" ], "output": "16", "label": "Pyth option 1 b squared" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Pyth option 1 LAST c squared := mul(5, 5)", "after": "Pyth option 1 LAST c squared = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 1 LAST c squared，不以小數近似替代。", "en": "Recompute Pyth option 1 LAST c squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "5" ], "output": "25", "label": "Pyth option 1 LAST c squared" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Pyth option 1 sum a2 b2 := sum(9, 16)", "after": "Pyth option 1 sum a2 b2 = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 1 sum a2 b2，不以小數近似替代。", "en": "Recompute Pyth option 1 sum a2 b2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "16" ], "output": "25", "label": "Pyth option 1 sum a2 b2" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [1, [3, 4, 5]]", "after": "derived = [是, 25, 25, 是]", "relation": "implies", "rule_id": "Pythagorean_EACH_positive_triple_square_relation", "conditions": [], "explanation": { "zh-Hant": "逐選項先查正長度且a²+b²=c²；這是題給平方關係的直接核對，不從草圖或largest-sort判斷，也不新增無需的幾何逆定理。", "en": "For EACH option,check positive entries AND a²+b²=c².This directly checks the STATED squared relation,not a drawing/largest-sort inference or an unneeded geometric converse." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Pyth option 2 a squared := mul(3, 3)", "after": "Pyth option 2 a squared = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 2 a squared，不以小數近似替代。", "en": "Recompute Pyth option 2 a squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "Pyth option 2 a squared" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Pyth option 2 b squared := mul(4, 4)", "after": "Pyth option 2 b squared = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 2 b squared，不以小數近似替代。", "en": "Recompute Pyth option 2 b squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "4" ], "output": "16", "label": "Pyth option 2 b squared" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Pyth option 2 LAST c squared := mul(5, 5)", "after": "Pyth option 2 LAST c squared = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 2 LAST c squared，不以小數近似替代。", "en": "Recompute Pyth option 2 LAST c squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "5" ], "output": "25", "label": "Pyth option 2 LAST c squared" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Pyth option 2 sum a2 b2 := sum(9, 16)", "after": "Pyth option 2 sum a2 b2 = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 2 sum a2 b2，不以小數近似替代。", "en": "Recompute Pyth option 2 sum a2 b2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "16" ], "output": "25", "label": "Pyth option 2 sum a2 b2" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [2, [3, 4, 5]]", "after": "derived = [是, 25, 25, 是]", "relation": "implies", "rule_id": "Pythagorean_EACH_positive_triple_square_relation", "conditions": [], "explanation": { "zh-Hant": "逐選項先查正長度且a²+b²=c²；這是題給平方關係的直接核對，不從草圖或largest-sort判斷，也不新增無需的幾何逆定理。", "en": "For EACH option,check positive entries AND a²+b²=c².This directly checks the STATED squared relation,not a drawing/largest-sort inference or an unneeded geometric converse." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Pyth option 3 a squared := mul(3, 3)", "after": "Pyth option 3 a squared = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 3 a squared，不以小數近似替代。", "en": "Recompute Pyth option 3 a squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "Pyth option 3 a squared" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Pyth option 3 b squared := mul(4, 4)", "after": "Pyth option 3 b squared = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 3 b squared，不以小數近似替代。", "en": "Recompute Pyth option 3 b squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "4" ], "output": "16", "label": "Pyth option 3 b squared" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Pyth option 3 LAST c squared := mul(5, 5)", "after": "Pyth option 3 LAST c squared = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 3 LAST c squared，不以小數近似替代。", "en": "Recompute Pyth option 3 LAST c squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "5" ], "output": "25", "label": "Pyth option 3 LAST c squared" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Pyth option 3 sum a2 b2 := sum(9, 16)", "after": "Pyth option 3 sum a2 b2 = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Pyth option 3 sum a2 b2，不以小數近似替代。", "en": "Recompute Pyth option 3 sum a2 b2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "16" ], "output": "25", "label": "Pyth option 3 sum a2 b2" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "source roles = [3, [3, 4, 5]]", "after": "derived = [是, 25, 25, 是]", "relation": "implies", "rule_id": "Pythagorean_EACH_positive_triple_square_relation", "conditions": [], "explanation": { "zh-Hant": "逐選項先查正長度且a²+b²=c²；這是題給平方關係的直接核對，不從草圖或largest-sort判斷，也不新增無需的幾何逆定理。", "en": "For EACH option,check positive entries AND a²+b²=c².This directly checks the STATED squared relation,not a drawing/largest-sort inference or an unneeded geometric converse." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "source roles = [[否, 是, 是, 是], [0]]", "after": "derived = [甲]", "relation": "implies", "rule_id": "Pythagorean_UNIQUE_invalid_after_ALL_four_checks", "conditions": [], "explanation": { "zh-Hant": "四項全查後只有一個違反，才選它原標籤；counterexample_index僅最後核對，不拿它跳過其他選項。", "en": "Select the ORIGINAL label ONLY after ALL FOUR checks establish exactly one failure.Check counterexample_index metadata LAST;it never substitutes for testing the options." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null } ], "math_model": { "result": "甲" }, "conclusion": { "canonical_value": "甲", "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-20260912-geometry-foundations", "batch_path": "public/data/questions/batches/batch-20260912-geometry-foundations/questions.jsonl", "batch_sha256": "818fc9bd154b15a97199c437c5345260e55307eada85586bf547e52c4ae518dc", "original_line": 86, "original_line_sha256": "caa7c2d5c825c339a1892faee456ea5a41111958976f9361457d47856c63cd94", "question_compact_sha256": "06809a825e94d636af6b739ac332e42e311f82b75dfbfe354bfbdfc66c656fea", "english_row_compact_sha256": "89a11e444b651d3c5793c615b27bb8ce7e9c3c176f9f96fe10726bacbf81157c", "english_manifest_path": "public/data/i18n/editions/batch-20260912-geometry-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20260912-geometry-foundations.en.jsonl" }, "mko_bindings": { "mko-euclid-pythagorean-theorem": { "sha256": "f91873f82805e9a6d9ee8358fa64afa78e7594fc6568999814e34c1205ad51c6", "english_sha256": "1ad53a259b7aa8574f8579d208fda5c28c60bbd7d60b814200022c5917b7a849", "original_path": "public/data/mko/pythagorean-theorem.json", "english_path": "public/data/i18n/editions/mko-euclid-pythagorean-theorem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "20c37a7e64ee5969e7606c5cc3a00e575002ebebdfe82873521cc3d7c608caff", "original_candidate_sha256": "edb2d5fe9898710d891b60419a719f5e9606f3704c7065113eb4bf89b627f9a4", "original_candidate_line_sha256": "edb2d5fe9898710d891b60419a719f5e9606f3704c7065113eb4bf89b627f9a4" }, "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-20261026-012-p71", "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": "9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80", "question_first_witness_sha256": "c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 23, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "328552ce599112dc887ad09b4fe75a5bdc6d47664d3901212259835f28abf055", "arithmetic_steps": 16 }, "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." } }