{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261008-advance-20261012-r06-euclidean-length-0040", "pilot_number": "R0059-30165", "number": 30165, "family_id": "qf-density12-length-integer-distance-pair", "title": { "zh-Hant": "第 30165 題：兩段有序整數主距離", "en": "Question 30165: Ordered Integer Principal Segment Lengths" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261008-advance-20261012-r06-euclidean-length-0040", "batch_id": "batch-20261008-advance-20261012-r06-euclidean-length-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-euclidean-length" ], "title_zh": "兩段有序整數主距離", "stem_zh": "等單位正交L2；同一實平面與長度單位；點P、Q、R依序為 [[0, 0], [36, 48], [41, 60]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。", "answer": { "kind": "expression", "value": "(60, 13)", "explanation_zh": "兩段整數主距離=(60, 13)。分別求精確平方和及非負主根，保留兩段次序；重合點給零，不改用有號位移。" }, "difficulty": { "band": "upper_secondary", "level": 2, "skills": [ "兩段有序整數主距離" ] }, "variant": { "family_id": "qf-density12-length-integer-distance-pair", "seed": 202610126040, "parameters": { "points": [ [ 0, 0 ], [ 36, 48 ], [ 41, 60 ] ], "frame": "orthogonal_equal_unit", "operation": "density12_length_integer_distance_pair", "domain": "integer_euclidean_plane", "semantic_template": "qf-density12-length-integer-distance-pair/[\"3600\",\"169\"]" } }, "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-0040", "batch_id": "batch-20261008-advance-20261012-r06-euclidean-length-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3ee6f66562fe80f040b8dbcac88be8781bd819cc570d65fd47e0bcaab29e550a", "family_id": "qf-density12-length-integer-distance-pair", "seed": 202610126040, "semantic_template": "qf-density12-length-integer-distance-pair/[\"3600\",\"169\"]" }, "fields": { "title": "Ordered Integer Principal Segment Lengths", "stem": "Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [36, 48], [41, 60]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.", "answer": { "kind": "expression", "value": "(60, 13)", "explanation": "Compute each exact squared norm and its nonnegative principal root; coincidence gives zero and does not replace distance with a signed displacement. The exact result, retaining the requested order, is (60, 13)." }, "skills": [ "Ordered Integer Principal 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；點P、Q、R依序為 [[0, 0], [36, 48], [41, 60]]，兩段平方和均為完全平方。依序求 (d(P,Q), d(Q,R)) 的整數主距離。", "en": "Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [36, 48], [41, 60]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density12-length-integer-distance-pair, density12_length_integer_distance_pair, integer_euclidean_plane]", "after": "derived = [[nonnegative_distance_PQ, nonnegative_distance_QR]]", "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 segment0 ORIGINAL Q minus P coordinate0 := sub(36, 0)", "after": "Length segment0 ORIGINAL Q minus P coordinate0 = 36", "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 segment0 ORIGINAL Q minus P coordinate0，不以小數近似替代。", "en": "Recompute Length segment0 ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "36", "0" ], "output": "36", "label": "Length segment0 ORIGINAL Q minus P coordinate0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Length segment0 ORIGINAL Q minus P coordinate1 := sub(48, 0)", "after": "Length segment0 ORIGINAL Q minus P coordinate1 = 48", "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 segment0 ORIGINAL Q minus P coordinate1，不以小數近似替代。", "en": "Recompute Length segment0 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "48", "0" ], "output": "48", "label": "Length segment0 ORIGINAL Q minus P coordinate1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Length segment0 ORIGINAL square coordinate0 := mul(36, 36)", "after": "Length segment0 ORIGINAL square coordinate0 = 1296", "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 segment0 ORIGINAL square coordinate0，不以小數近似替代。", "en": "Recompute Length segment0 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "36", "36" ], "output": "1296", "label": "Length segment0 ORIGINAL square coordinate0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Length segment0 ORIGINAL square coordinate1 := mul(48, 48)", "after": "Length segment0 ORIGINAL square coordinate1 = 2304", "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 segment0 ORIGINAL square coordinate1，不以小數近似替代。", "en": "Recompute Length segment0 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "48", "48" ], "output": "2304", "label": "Length segment0 ORIGINAL square coordinate1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Length segment0 ORIGINAL squared norm sum := sum(1296, 2304)", "after": "Length segment0 ORIGINAL squared norm sum = 3600", "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 segment0 ORIGINAL squared norm sum，不以小數近似替代。", "en": "Recompute Length segment0 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1296", "2304" ], "output": "3600", "label": "Length segment0 ORIGINAL squared norm sum" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Length segment0, [0, 0], [36, 48]]", "after": "derived = [[36, 48], 3600]", "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": "Length segment0 candidate nonnegative root squared := mul(60, 60)", "after": "Length segment0 candidate nonnegative root squared = 3600", "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 segment0 candidate nonnegative root squared，不以小數近似替代。", "en": "Recompute Length segment0 candidate nonnegative root squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "60", "60" ], "output": "3600", "label": "Length segment0 candidate nonnegative root squared" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Length segment0 candidate next integer := sum(60, 1)", "after": "Length segment0 candidate next integer = 61", "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 segment0 candidate next integer，不以小數近似替代。", "en": "Recompute Length segment0 candidate next integer with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "60", "1" ], "output": "61", "label": "Length segment0 candidate next integer" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Length segment0 next integer squared := mul(61, 61)", "after": "Length segment0 next integer squared = 3721", "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 segment0 next integer squared，不以小數近似替代。", "en": "Recompute Length segment0 next integer squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "61", "61" ], "output": "3721", "label": "Length segment0 next integer squared" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [Length segment0, 3600]", "after": "derived = [60, 3600, 3721, 是]", "relation": "implies", "rule_id": "Length_EXACT_perfect_square_certificate_NONNEGATIVE_principal_root", "conditions": [], "explanation": { "zh-Hant": "給r>=0、r²=N且(r+1)²>N的精確certificate，確定唯一非負主根r；這個slice要求N真perfect square。不是浮點round，也不把±r都當距離；衍生r不限SAFE。", "en": "The exact certificate r>=0,r²=N and(r+1)²>N identifies the UNIQUE nonnegative principal root.This slice requires an actual perfect square.Do not round a float or return both±r as distances;derived r is not SAFE-capped." }, "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": "S012", "uses": [ "G001", "S011" ], "before": "Length segment1 ORIGINAL Q minus P coordinate0 := sub(41, 36)", "after": "Length segment1 ORIGINAL Q minus P coordinate0 = 5", "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 segment1 ORIGINAL Q minus P coordinate0，不以小數近似替代。", "en": "Recompute Length segment1 ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "41", "36" ], "output": "5", "label": "Length segment1 ORIGINAL Q minus P coordinate0" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Length segment1 ORIGINAL Q minus P coordinate1 := sub(60, 48)", "after": "Length segment1 ORIGINAL Q minus P coordinate1 = 12", "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 segment1 ORIGINAL Q minus P coordinate1，不以小數近似替代。", "en": "Recompute Length segment1 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "60", "48" ], "output": "12", "label": "Length segment1 ORIGINAL Q minus P coordinate1" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Length segment1 ORIGINAL square coordinate0 := mul(5, 5)", "after": "Length segment1 ORIGINAL square 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 segment1 ORIGINAL square coordinate0，不以小數近似替代。", "en": "Recompute Length segment1 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "5" ], "output": "25", "label": "Length segment1 ORIGINAL square coordinate0" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Length segment1 ORIGINAL square coordinate1 := mul(12, 12)", "after": "Length segment1 ORIGINAL square coordinate1 = 144", "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 segment1 ORIGINAL square coordinate1，不以小數近似替代。", "en": "Recompute Length segment1 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12", "12" ], "output": "144", "label": "Length segment1 ORIGINAL square coordinate1" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Length segment1 ORIGINAL squared norm sum := sum(25, 144)", "after": "Length segment1 ORIGINAL squared norm sum = 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 segment1 ORIGINAL squared norm sum，不以小數近似替代。", "en": "Recompute Length segment1 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "25", "144" ], "output": "169", "label": "Length segment1 ORIGINAL squared norm sum" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [Length segment1, [36, 48], [41, 60]]", "after": "derived = [[5, 12], 169]", "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": "S018", "uses": [ "G001", "S017" ], "before": "Length segment1 candidate nonnegative root squared := mul(13, 13)", "after": "Length segment1 candidate nonnegative root squared = 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 segment1 candidate nonnegative root squared，不以小數近似替代。", "en": "Recompute Length segment1 candidate nonnegative root squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "13" ], "output": "169", "label": "Length segment1 candidate nonnegative root squared" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Length segment1 candidate next integer := sum(13, 1)", "after": "Length segment1 candidate next integer = 14", "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 segment1 candidate next integer，不以小數近似替代。", "en": "Recompute Length segment1 candidate next integer with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "13", "1" ], "output": "14", "label": "Length segment1 candidate next integer" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Length segment1 next integer squared := mul(14, 14)", "after": "Length segment1 next integer squared = 196", "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 segment1 next integer squared，不以小數近似替代。", "en": "Recompute Length segment1 next integer squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "14", "14" ], "output": "196", "label": "Length segment1 next integer squared" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "source roles = [Length segment1, 169]", "after": "derived = [13, 169, 196, 是]", "relation": "implies", "rule_id": "Length_EXACT_perfect_square_certificate_NONNEGATIVE_principal_root", "conditions": [], "explanation": { "zh-Hant": "給r>=0、r²=N且(r+1)²>N的精確certificate，確定唯一非負主根r；這個slice要求N真perfect square。不是浮點round，也不把±r都當距離；衍生r不限SAFE。", "en": "The exact certificate r>=0,r²=N and(r+1)²>N identifies the UNIQUE nonnegative principal root.This slice requires an actual perfect square.Do not round a float or return both±r as distances;derived r is not SAFE-capped." }, "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": "S022", "uses": [ "G001", "S021" ], "before": "source roles = [[[0, 0], [36, 48], [41, 60]]]", "after": "derived = [[60, 13]]", "relation": "implies", "rule_id": "Length_TWO_original_segment_roots_IN_ORDER_not_sum_or_endpoint", "conditions": [], "explanation": { "zh-Hant": "按原P,Q,R順序回兩段距離(d(P,Q),d(Q,R))，可有coincident points且距離0；不是兩段和，也不是d(P,R)。", "en": "Return the TWO segment distances(d(P,Q),d(Q,R)) in the ORIGINAL order.Coincident points and distance0 are legal;do not substitute the sum or d(P,R)." }, "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": [ "60", "13" ] }, "conclusion": { "canonical_value": "(60, 13)", "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": 40, "original_line_sha256": "beacfad38a861f4ef1a0f90db8e701fab512ce8718fb694fcf6bb83c3c03ce2b", "question_compact_sha256": "3ee6f66562fe80f040b8dbcac88be8781bd819cc570d65fd47e0bcaab29e550a", "english_row_compact_sha256": "220d3e0e80f283c7a17c33b1b6e666f2b8202b64a371a5454abccd6ac44f2eb3", "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": "2937f254c92a75b39f7e5bbffe4017e28e469bc12dce4d7aab124a2193d5a6b7", "original_candidate_line_sha256": "2937f254c92a75b39f7e5bbffe4017e28e469bc12dce4d7aab124a2193d5a6b7" }, "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": 22, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "47171b2a6d8315096a165622b6e6189105c6be909980cb1eb6ee17dac436d269", "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." } }