{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261010-advance-20261014-r07-circle-0003", "pilot_number": "R0059-30478", "number": 30478, "family_id": "qf-density14-circle-offset-residual-triple", "title": { "zh-Hant": "第 30478 題：三點殘差與圓邊界資格", "en": "Question 30478: Squared Circle Distances, Residuals, and Boundary Membership" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261010-advance-20261014-r07-circle-0003", "batch_id": "batch-20261010-advance-20261014-r07-circle-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-circle" ], "title_zh": "三點殘差與圓邊界資格", "stem_zh": "圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, -1]、r=2，points=[[1, -1], [3, -1], [4, 0]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。", "answer": { "kind": "expression", "value": "([0, 4, 10], [-4, 0, 6], [內, 圓上, 外], [否, 是, 否])", "explanation_zh": "三點平方距離殘差與圓資格=([0, 4, 10], [-4, 0, 6], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。" }, "difficulty": { "band": "lower_secondary", "level": 1, "skills": [ "三點殘差與圓邊界資格" ] }, "variant": { "family_id": "qf-density14-circle-offset-residual-triple", "seed": 202610147003, "parameters": { "O": [ 1, -1 ], "r": 2, "points": [ [ 1, -1 ], [ 3, -1 ], [ 4, 0 ] ], "frame": "orthogonal_equal_unit", "domain": "finite_integer_circle_boundary_protocol", "operation": "density14_circle_offset_residual_triple", "semantic_template": "qf-density14-circle-offset-residual-triple/[\"4\",[\"0\",\"4\",\"10\"]]" } }, "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": "2026-10-10", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261010-advance-20261014-r07-circle-0003", "batch_id": "batch-20261010-advance-20261014-r07-circle-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "a34c88ce5d69265e75a070277fc078d81ceeb35cacd5edd6e44bc6593bfbc38e", "family_id": "qf-density14-circle-offset-residual-triple", "seed": 202610147003, "semantic_template": "qf-density14-circle-offset-residual-triple/[\"4\",[\"0\",\"4\",\"10\"]]" }, "fields": { "title": "Squared Circle Distances, Residuals, and Boundary Membership", "stem": "Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -1], r=2, ordered points=[[1, -1], [3, -1], [4, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).", "answer": { "kind": "expression", "value": "([0, 4, 10], [-4, 0, 6], [inside, on-boundary, outside], [No, Yes, No])", "explanation": "Compare exact squared coordinate differences with r²; an inside point is not a boundary member. This is not a general formal circle-API bridge. The exact result, retaining the requested order, is ([0, 4, 10], [-4, 0, 6], [inside, on-boundary, outside], [No, Yes, No])." }, "skills": [ "Squared Circle Distances, Residuals, and Boundary Membership" ], "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": "圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, -1]、r=2，points=[[1, -1], [3, -1], [4, 0]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。", "en": "Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. Center O=[1, -1], r=2, ordered points=[[1, -1], [3, -1], [4, 0]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density14-circle-offset-residual-triple, density14_circle_offset_residual_triple, finite_integer_circle_boundary_protocol]", "after": "derived = [[ALL_squared_distances, ALL_signed_residuals, ALL_location_labels, ALL_boundary_flags]]", "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": "Circle ORIGINAL nonnegative radius squared := mul(2, 2)", "after": "Circle ORIGINAL nonnegative radius squared = 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 Circle ORIGINAL nonnegative radius squared，不以小數近似替代。", "en": "Recompute Circle ORIGINAL nonnegative radius squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Circle ORIGINAL nonnegative radius squared" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [2, orthogonal_equal_unit]", "after": "derived = [4]", "relation": "implies", "rule_id": "Circle_BOUNDARY_not_disk_and_zero_radius_SINGLEcenter", "conditions": [], "explanation": { "zh-Hant": "此處圓是L2距離平方等於r²的boundary，不是<=的圓盤。半徑非負且0合法；r0只有中心在boundary，衍生坐標／平方不限SAFE。", "en": "The circle is the BOUNDARY with squared L2 distance EQUAL to r²,not the disk defined by<=.The radius is nonnegative and0 is legal;at r=0 only the center is on the boundary.Derived coordinates/squares are 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": "S004", "uses": [ "G001", "S003" ], "before": "Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 := sub(1, 1)", "after": "Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 = 0", "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 Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "1" ], "output": "0", "label": "Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 := sub(-1, -1)", "after": "Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 = 0", "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 Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-1" ], "output": "0", "label": "Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Circle ORIGINAL center to point0 ORIGINAL square coordinate0 := mul(0, 0)", "after": "Circle ORIGINAL center to point0 ORIGINAL square coordinate0 = 0", "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 Circle ORIGINAL center to point0 ORIGINAL square coordinate0，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point0 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Circle ORIGINAL center to point0 ORIGINAL square coordinate0" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Circle ORIGINAL center to point0 ORIGINAL square coordinate1 := mul(0, 0)", "after": "Circle ORIGINAL center to point0 ORIGINAL square coordinate1 = 0", "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 Circle ORIGINAL center to point0 ORIGINAL square coordinate1，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point0 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Circle ORIGINAL center to point0 ORIGINAL square coordinate1" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Circle ORIGINAL center to point0 ORIGINAL squared norm sum := sum(0, 0)", "after": "Circle ORIGINAL center to point0 ORIGINAL squared norm sum = 0", "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 Circle ORIGINAL center to point0 ORIGINAL squared norm sum，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point0 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "Circle ORIGINAL center to point0 ORIGINAL squared norm sum" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [Circle ORIGINAL center to point0, [1, -1], [1, -1]]", "after": "derived = [[0, 0], 0]", "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": "S010", "uses": [ "G001", "S009" ], "before": "Circle ORIGINAL point0 square minus radius square := sub(0, 4)", "after": "Circle ORIGINAL point0 square minus radius square = -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 Circle ORIGINAL point0 square minus radius square，不以小數近似替代。", "en": "Recompute Circle ORIGINAL point0 square minus radius square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "4" ], "output": "-4", "label": "Circle ORIGINAL point0 square minus radius square" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [[1, -1], 2, [1, -1], 0, -4]", "after": "derived = [內, 否]", "relation": "implies", "rule_id": "Circle_EACH_signed_square_residual_location_AND_BOUNDARY_membership", "conditions": [], "explanation": { "zh-Hant": "殘差<0只分類inside，並非boundary成員；=0才on-boundary且Yes，>0outside且No。逐原point都驗，包含zero-radius中心。", "en": "A negative residual classifies inside,NOT boundary membership.ONLY zero is on-boundary with Yes;positive is outside with No.Check EVERY original point,including a zero-radius center." }, "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": "Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 := sub(3, 1)", "after": "Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 = 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 Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "3", "1" ], "output": "2", "label": "Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 := sub(-1, -1)", "after": "Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 = 0", "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 Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-1" ], "output": "0", "label": "Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Circle ORIGINAL center to point1 ORIGINAL square coordinate0 := mul(2, 2)", "after": "Circle ORIGINAL center to point1 ORIGINAL square coordinate0 = 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 Circle ORIGINAL center to point1 ORIGINAL square coordinate0，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point1 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Circle ORIGINAL center to point1 ORIGINAL square coordinate0" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Circle ORIGINAL center to point1 ORIGINAL square coordinate1 := mul(0, 0)", "after": "Circle ORIGINAL center to point1 ORIGINAL square coordinate1 = 0", "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 Circle ORIGINAL center to point1 ORIGINAL square coordinate1，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point1 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Circle ORIGINAL center to point1 ORIGINAL square coordinate1" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Circle ORIGINAL center to point1 ORIGINAL squared norm sum := sum(4, 0)", "after": "Circle ORIGINAL center to point1 ORIGINAL squared norm sum = 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 Circle ORIGINAL center to point1 ORIGINAL squared norm sum，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point1 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "0" ], "output": "4", "label": "Circle ORIGINAL center to point1 ORIGINAL squared norm sum" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [Circle ORIGINAL center to point1, [1, -1], [3, -1]]", "after": "derived = [[2, 0], 4]", "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": "Circle ORIGINAL point1 square minus radius square := sub(4, 4)", "after": "Circle ORIGINAL point1 square minus radius square = 0", "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 Circle ORIGINAL point1 square minus radius square，不以小數近似替代。", "en": "Recompute Circle ORIGINAL point1 square minus radius square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "4", "4" ], "output": "0", "label": "Circle ORIGINAL point1 square minus radius square" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "source roles = [[1, -1], 2, [3, -1], 4, 0]", "after": "derived = [圓上, 是]", "relation": "implies", "rule_id": "Circle_EACH_signed_square_residual_location_AND_BOUNDARY_membership", "conditions": [], "explanation": { "zh-Hant": "殘差<0只分類inside，並非boundary成員；=0才on-boundary且Yes，>0outside且No。逐原point都驗，包含zero-radius中心。", "en": "A negative residual classifies inside,NOT boundary membership.ONLY zero is on-boundary with Yes;positive is outside with No.Check EVERY original point,including a zero-radius center." }, "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": "S020", "uses": [ "G001", "S019" ], "before": "Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 := sub(4, 1)", "after": "Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 = 3", "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 Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "4", "1" ], "output": "3", "label": "Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 := sub(0, -1)", "after": "Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 = 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 Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "-1" ], "output": "1", "label": "Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Circle ORIGINAL center to point2 ORIGINAL square coordinate0 := mul(3, 3)", "after": "Circle ORIGINAL center to point2 ORIGINAL square coordinate0 = 9", "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 Circle ORIGINAL center to point2 ORIGINAL square coordinate0，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point2 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "Circle ORIGINAL center to point2 ORIGINAL square coordinate0" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Circle ORIGINAL center to point2 ORIGINAL square coordinate1 := mul(1, 1)", "after": "Circle ORIGINAL center to point2 ORIGINAL square coordinate1 = 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 Circle ORIGINAL center to point2 ORIGINAL square coordinate1，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point2 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Circle ORIGINAL center to point2 ORIGINAL square coordinate1" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "Circle ORIGINAL center to point2 ORIGINAL squared norm sum := sum(9, 1)", "after": "Circle ORIGINAL center to point2 ORIGINAL squared norm sum = 10", "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 Circle ORIGINAL center to point2 ORIGINAL squared norm sum，不以小數近似替代。", "en": "Recompute Circle ORIGINAL center to point2 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "1" ], "output": "10", "label": "Circle ORIGINAL center to point2 ORIGINAL squared norm sum" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "source roles = [Circle ORIGINAL center to point2, [1, -1], [4, 0]]", "after": "derived = [[3, 1], 10]", "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": "S026", "uses": [ "G001", "S025" ], "before": "Circle ORIGINAL point2 square minus radius square := sub(10, 4)", "after": "Circle ORIGINAL point2 square minus radius square = 6", "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 Circle ORIGINAL point2 square minus radius square，不以小數近似替代。", "en": "Recompute Circle ORIGINAL point2 square minus radius square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "10", "4" ], "output": "6", "label": "Circle ORIGINAL point2 square minus radius square" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "source roles = [[1, -1], 2, [4, 0], 10, 6]", "after": "derived = [外, 否]", "relation": "implies", "rule_id": "Circle_EACH_signed_square_residual_location_AND_BOUNDARY_membership", "conditions": [], "explanation": { "zh-Hant": "殘差<0只分類inside，並非boundary成員；=0才on-boundary且Yes，>0outside且No。逐原point都驗，包含zero-radius中心。", "en": "A negative residual classifies inside,NOT boundary membership.ONLY zero is on-boundary with Yes;positive is outside with No.Check EVERY original point,including a zero-radius center." }, "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": [ [ "0", "4", "10" ], [ "-4", "0", "6" ], [ "內", "圓上", "外" ], [ false, true, false ] ] }, "conclusion": { "canonical_value": "([0, 4, 10], [-4, 0, 6], [內, 圓上, 外], [否, 是, 否])", "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-20261010-advance-20261014-r07-circle-foundations", "batch_path": "public/data/questions/batches/batch-20261010-advance-20261014-r07-circle-foundations/questions.jsonl", "batch_sha256": "364c6501dc54730733b5b6b94cd2a95815a1b4419f33cafb01c9de044038395f", "original_line": 3, "original_line_sha256": "8a01246329e84b781f49cbcca2d32ec269a2bf3042404864f3db7f15df039199", "question_compact_sha256": "a34c88ce5d69265e75a070277fc078d81ceeb35cacd5edd6e44bc6593bfbc38e", "english_row_compact_sha256": "d7354c45c7afe8e16028b2109978f45958d27a686e6bd02d68de35917bb00beb", "english_manifest_path": "public/data/i18n/editions/batch-20261010-advance-20261014-r07-circle-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261010-advance-20261014-r07-circle-foundations.en.jsonl" }, "mko_bindings": { "mko-circle": { "sha256": "f5666d01288a4c7140be59abad5dbf093293c9136d7d60dc752a79a497d99d2e", "english_sha256": "e3c801529b17bbcfc6b73fef34a12a234a6dedbbaf8f1e283d3371fd2c7290d0", "original_path": "public/data/mko/circle.json", "english_path": "public/data/i18n/editions/mko-circle.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "17d593a551933064f195bd9277d5085fe2b6c98e11cbda0c8aa27d11b7b07949", "original_candidate_sha256": "77ada568721c46b35e187882a0a9c0e42564fabfe40d7e5e4b76577e34e6338e", "original_candidate_line_sha256": "77ada568721c46b35e187882a0a9c0e42564fabfe40d7e5e4b76577e34e6338e" }, "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": 27, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "108241470f9406b240c53ec4cb22ff7f6f47ba2ed15fbe1bb1df99d185c8fe18", "arithmetic_steps": 19 }, "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." } }