{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261008-advance-20261012-r06-euclidean-length-0063", "pilot_number": "R0059-30188", "number": 30188, "family_id": "qf-density12-length-midpoint-equidistance", "title": { "zh-Hant": "第 30188 題：中點座標及兩段平方", "en": "Question 30188: Exact Midpoint and Two Squared Segment Lengths" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261008-advance-20261012-r06-euclidean-length-0063", "batch_id": "batch-20261008-advance-20261012-r06-euclidean-length-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-euclidean-length" ], "title_zh": "中點座標及兩段平方", "stem_zh": "等單位正交L2；同一實平面與長度單位；P=[-12, 1]、Q=[13, 14]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。", "answer": { "kind": "expression", "value": "(1/2, 15/2, 397/2, 397/2)", "explanation_zh": "中點與兩段平方=(1/2, 15/2, 397/2, 397/2)。座標平均給中點，每段平方為全段平方的四分之一；平方量不是距離本身，實際長度取非負主根。" }, "difficulty": { "band": "upper_secondary", "level": 2, "skills": [ "中點座標及兩段平方" ] }, "variant": { "family_id": "qf-density12-length-midpoint-equidistance", "seed": 202610126063, "parameters": { "P": [ -12, 1 ], "Q": [ 13, 14 ], "frame": "orthogonal_equal_unit", "operation": "density12_length_midpoint_equidistance", "domain": "integer_euclidean_plane", "semantic_template": "qf-density12-length-midpoint-equidistance/[\"1/2\",\"15/2\",\"794\"]" } }, "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-20261012-r06-euclidean-length-0063", "batch_id": "batch-20261008-advance-20261012-r06-euclidean-length-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "e77fd36122308248effbe816c5276002f7bb8d3fd314e77a8391a8d4e856af5e", "family_id": "qf-density12-length-midpoint-equidistance", "seed": 202610126063, "semantic_template": "qf-density12-length-midpoint-equidistance/[\"1/2\",\"15/2\",\"794\"]" }, "fields": { "title": "Exact Midpoint and Two Squared Segment Lengths", "stem": "Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-12, 1], Q=[13, 14], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers.", "answer": { "kind": "expression", "value": "(1/2, 15/2, 397/2, 397/2)", "explanation": "Average each coordinate for the midpoint; each half-segment square is a quarter of the full segment square. Squared quantities are not the distances themselves. The exact result, retaining the requested order, is (1/2, 15/2, 397/2, 397/2)." }, "skills": [ "Exact Midpoint and Two Squared Segment Lengths" ], "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": "等單位正交L2；同一實平面與長度單位；P=[-12, 1]、Q=[13, 14]，M是中點。依序求 (Mx, My, d(P,M)², d(M,Q)²)，四項均用最簡正分母分數，後兩項明示為平方。", "en": "Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-12, 1], Q=[13, 14], with midpoint M. Give (Mx, My, d(P,M)², d(M,Q)²). Use exact reduced fractions with positive denominators, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density12-length-midpoint-equidistance, density12_length_midpoint_equidistance, integer_euclidean_plane]", "after": "derived = [[midpoint_x, midpoint_y, SQUARED_PM, SQUARED_MQ]]", "relation": "implies", "rule_id": "R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles", "conditions": [], "explanation": { "zh-Hant": "保原bank的完整operation前綴、domain及輸出欄序；frame只用orthogonal_equal_unit，角度unit只用degrees。分數整數保/1；不增不存在的參數，不改原中英來源。", "en": "Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。", "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Length original PQ ORIGINAL Q minus P coordinate0 := sub(13, -12)", "after": "Length original PQ ORIGINAL Q minus P coordinate0 = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Length original PQ ORIGINAL Q minus P coordinate0，不以小數近似替代。", "en": "Recompute Length original PQ ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "13", "-12" ], "output": "25", "label": "Length original PQ ORIGINAL Q minus P coordinate0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Length original PQ ORIGINAL Q minus P coordinate1 := sub(14, 1)", "after": "Length original PQ ORIGINAL Q minus P coordinate1 = 13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Length original PQ ORIGINAL Q minus P coordinate1，不以小數近似替代。", "en": "Recompute Length original PQ ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "14", "1" ], "output": "13", "label": "Length original PQ ORIGINAL Q minus P coordinate1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Length original PQ ORIGINAL square coordinate0 := mul(25, 25)", "after": "Length original PQ ORIGINAL square coordinate0 = 625", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Length original PQ ORIGINAL square coordinate0，不以小數近似替代。", "en": "Recompute Length original PQ ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "25", "25" ], "output": "625", "label": "Length original PQ ORIGINAL square coordinate0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Length original PQ ORIGINAL square coordinate1 := mul(13, 13)", "after": "Length original PQ ORIGINAL square coordinate1 = 169", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Length original PQ ORIGINAL square coordinate1，不以小數近似替代。", "en": "Recompute Length original PQ ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "13" ], "output": "169", "label": "Length original PQ ORIGINAL square coordinate1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Length original PQ ORIGINAL squared norm sum := sum(625, 169)", "after": "Length original PQ ORIGINAL squared norm sum = 794", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Length original PQ ORIGINAL squared norm sum，不以小數近似替代。", "en": "Recompute Length original PQ ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "625", "169" ], "output": "794", "label": "Length original PQ ORIGINAL squared norm sum" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Length original PQ, [-12, 1], [13, 14]]", "after": "derived = [[25, 13], 794]", "relation": "implies", "rule_id": "Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm", "conditions": [], "explanation": { "zh-Hant": "同一等單位正交L2 frame，位移是Q-P保符號；兩平方和才是非負長度平方。純衍生coordinate可超SAFE，不重新驗raw界。", "en": "In the SAME equal-unit orthogonal L2 frame,the directed displacement is Q-P with signs retained.The sum of coordinate squares is the nonnegative SQUARED length.Derived coordinates may exceed SAFE without being revalidated as raw inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。", "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Midpoint ORIGINAL endpoint sum coordinate0 := sum(-12, 13)", "after": "Midpoint ORIGINAL endpoint sum coordinate0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint ORIGINAL endpoint sum coordinate0，不以小數近似替代。", "en": "Recompute Midpoint ORIGINAL endpoint sum coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-12", "13" ], "output": "1", "label": "Midpoint ORIGINAL endpoint sum coordinate0" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Midpoint ORIGINAL average coordinate0 := div(1, 2)", "after": "Midpoint ORIGINAL average coordinate0 = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint ORIGINAL average coordinate0，不以小數近似替代。", "en": "Recompute Midpoint ORIGINAL average coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "2" ], "output": "1/2", "label": "Midpoint ORIGINAL average coordinate0" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Midpoint ORIGINAL endpoint sum coordinate1 := sum(1, 14)", "after": "Midpoint ORIGINAL endpoint sum coordinate1 = 15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint ORIGINAL endpoint sum coordinate1，不以小數近似替代。", "en": "Recompute Midpoint ORIGINAL endpoint sum coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "14" ], "output": "15", "label": "Midpoint ORIGINAL endpoint sum coordinate1" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Midpoint ORIGINAL average coordinate1 := div(15, 2)", "after": "Midpoint ORIGINAL average coordinate1 = 15/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint ORIGINAL average coordinate1，不以小數近似替代。", "en": "Recompute Midpoint ORIGINAL average coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "15", "2" ], "output": "15/2", "label": "Midpoint ORIGINAL average coordinate1" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Midpoint ORIGINAL full squared length over four := div(794, 4)", "after": "Midpoint ORIGINAL full squared length over four = 397/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint ORIGINAL full squared length over four，不以小數近似替代。", "en": "Recompute Midpoint ORIGINAL full squared length over four with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "794", "4" ], "output": "397/2", "label": "Midpoint ORIGINAL full squared length over four" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Midpoint reverse PM displacement coordinate0 := sub(1/2, -12)", "after": "Midpoint reverse PM displacement coordinate0 = 25/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse PM displacement coordinate0，不以小數近似替代。", "en": "Recompute Midpoint reverse PM displacement coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1/2", "-12" ], "output": "25/2", "label": "Midpoint reverse PM displacement coordinate0" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Midpoint reverse PM displacement coordinate1 := sub(15/2, 1)", "after": "Midpoint reverse PM displacement coordinate1 = 13/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse PM displacement coordinate1，不以小數近似替代。", "en": "Recompute Midpoint reverse PM displacement coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "15/2", "1" ], "output": "13/2", "label": "Midpoint reverse PM displacement coordinate1" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Midpoint reverse PM coordinate0 square := mul(25/2, 25/2)", "after": "Midpoint reverse PM coordinate0 square = 625/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse PM coordinate0 square，不以小數近似替代。", "en": "Recompute Midpoint reverse PM coordinate0 square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "25/2", "25/2" ], "output": "625/4", "label": "Midpoint reverse PM coordinate0 square" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Midpoint reverse PM coordinate1 square := mul(13/2, 13/2)", "after": "Midpoint reverse PM coordinate1 square = 169/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse PM coordinate1 square，不以小數近似替代。", "en": "Recompute Midpoint reverse PM coordinate1 square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13/2", "13/2" ], "output": "169/4", "label": "Midpoint reverse PM coordinate1 square" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Midpoint reverse PM squared sum := sum(625/4, 169/4)", "after": "Midpoint reverse PM squared sum = 397/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse PM squared sum，不以小數近似替代。", "en": "Recompute Midpoint reverse PM squared sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "625/4", "169/4" ], "output": "397/2", "label": "Midpoint reverse PM squared sum" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Midpoint reverse MQ displacement coordinate0 := sub(13, 1/2)", "after": "Midpoint reverse MQ displacement coordinate0 = 25/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse MQ displacement coordinate0，不以小數近似替代。", "en": "Recompute Midpoint reverse MQ displacement coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "13", "1/2" ], "output": "25/2", "label": "Midpoint reverse MQ displacement coordinate0" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Midpoint reverse MQ displacement coordinate1 := sub(14, 15/2)", "after": "Midpoint reverse MQ displacement coordinate1 = 13/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse MQ displacement coordinate1，不以小數近似替代。", "en": "Recompute Midpoint reverse MQ displacement coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "14", "15/2" ], "output": "13/2", "label": "Midpoint reverse MQ displacement coordinate1" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Midpoint reverse MQ coordinate0 square := mul(25/2, 25/2)", "after": "Midpoint reverse MQ coordinate0 square = 625/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse MQ coordinate0 square，不以小數近似替代。", "en": "Recompute Midpoint reverse MQ coordinate0 square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "25/2", "25/2" ], "output": "625/4", "label": "Midpoint reverse MQ coordinate0 square" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Midpoint reverse MQ coordinate1 square := mul(13/2, 13/2)", "after": "Midpoint reverse MQ coordinate1 square = 169/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse MQ coordinate1 square，不以小數近似替代。", "en": "Recompute Midpoint reverse MQ coordinate1 square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13/2", "13/2" ], "output": "169/4", "label": "Midpoint reverse MQ coordinate1 square" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Midpoint reverse MQ squared sum := sum(625/4, 169/4)", "after": "Midpoint reverse MQ squared sum = 397/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Midpoint reverse MQ squared sum，不以小數近似替代。", "en": "Recompute Midpoint reverse MQ squared sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "625/4", "169/4" ], "output": "397/2", "label": "Midpoint reverse MQ squared sum" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "source roles = [[-12, 1], [13, 14], [1/2, 15/2], 794]", "after": "derived = [397/2, 397/2, 397/2]", "relation": "implies", "rule_id": "Length_midpoint_TWO_HALFsegment_SQUARES_not_distances", "conditions": [], "explanation": { "zh-Hant": "M逐coordinate平均，可半整數；M-P與Q-M皆原位移的一半，平方長皆N/4，並已分兩方向回算。輸出的是平方，不再開根；P=Q也合法。", "en": "Average EACH coordinate to obtain M,possibly half-integral.Both M-P and Q-M are half the original displacement,so EACH squared length is N/4,also recomputed separately.These are SQUARES,not square roots;P=Q is legal." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。", "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null } ], "math_model": { "result": [ "1/2", "15/2", "397/2", "397/2" ] }, "conclusion": { "canonical_value": "(1/2, 15/2, 397/2, 397/2)", "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-20261012-r06-euclidean-length-foundations", "batch_path": "public/data/questions/batches/batch-20261008-advance-20261012-r06-euclidean-length-foundations/questions.jsonl", "batch_sha256": "404bd34842cbef0a7d9ef8c3c52040107d820a1b12d1d7ff2d46ada7e5a732a7", "original_line": 63, "original_line_sha256": "86a6f93a7203792f1da8769ad76c0c6a818ffc40eb589c0f5690eff817415891", "question_compact_sha256": "e77fd36122308248effbe816c5276002f7bb8d3fd314e77a8391a8d4e856af5e", "english_row_compact_sha256": "f8bb47422b12aeca70772dfdf105723d72996a5412654e00468f2f36048b562a", "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261012-r06-euclidean-length-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261012-r06-euclidean-length-foundations.en.jsonl" }, "mko_bindings": { "mko-euclidean-length": { "sha256": "485948172a6b39e7ec2cf3170ab03489fed98190d0c868161fbbd037f7151e91", "english_sha256": "b3ddf0a703643ec48198e001b53296e8991e8855e011f8b9ac9462cd47d53435", "original_path": "public/data/mko/euclidean-length.json", "english_path": "public/data/i18n/editions/mko-euclidean-length.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "17d593a551933064f195bd9277d5085fe2b6c98e11cbda0c8aa27d11b7b07949", "original_candidate_sha256": "6135af40f98593764913ccae085ba4882bb48207fff2132c57d9b6e12b54c81f", "original_candidate_line_sha256": "6135af40f98593764913ccae085ba4882bb48207fff2132c57d9b6e12b54c81f" }, "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-20261026-011-p60", "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": "4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5", "question_first_witness_sha256": "a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf" }, "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": "8ac7cce0e82bc5d481fd8bdd66e79e889cff5ccc728baa71cab0fb015eedf6b2", "arithmetic_steps": 20 }, "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." } }