{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261010-advance-20261014-r07-circle-0095", "pilot_number": "R0059-30470", "number": 30470, "family_id": "qf-density14-circle-horizontal-section-candidates", "title": { "zh-Hant": "第 30470 題：固定水平位移的截線候選", "en": "Question 30470: Circle Squared Remainder at a Fixed Horizontal Offset" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261010-advance-20261014-r07-circle-0095", "batch_id": "batch-20261010-advance-20261014-r07-circle-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-circle" ], "title_zh": "固定水平位移的截線候選", "stem_zh": "圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=20、水平位移dx=0，候選點(dx,y)的有序ys=[0, 20, 21]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。", "answer": { "kind": "expression", "value": "(400, [-400, 0, 41], [否, 是, 否])", "explanation_zh": "水平截線平方餘量與候選=(400, [-400, 0, 41], [否, 是, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。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": 3, "skills": [ "固定水平位移的截線候選" ] }, "variant": { "family_id": "qf-density14-circle-horizontal-section-candidates", "seed": 202610147095, "parameters": { "r": 20, "dx": 0, "ys": [ 0, 20, 21 ], "frame": "orthogonal_equal_unit", "domain": "finite_integer_circle_boundary_protocol", "operation": "density14_circle_horizontal_section_candidates", "semantic_template": "qf-density14-circle-horizontal-section-candidates/[\"400\",[\"0\",\"400\",\"441\"]]" } }, "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-0095", "batch_id": "batch-20261010-advance-20261014-r07-circle-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "978c3ab0106ff443b3f56ec3bd0e10005c61a3e1f74936445453707c8b560023", "family_id": "qf-density14-circle-horizontal-section-candidates", "seed": 202610147095, "semantic_template": "qf-density14-circle-horizontal-section-candidates/[\"400\",[\"0\",\"400\",\"441\"]]" }, "fields": { "title": "Circle Squared Remainder at a Fixed Horizontal Offset", "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. r=20, horizontal displacement dx=0, ordered candidate y values=[0, 20, 21] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values).", "answer": { "kind": "expression", "value": "(400, [-400, 0, 41], [No, Yes, No])", "explanation": "Use the squared remainder to audit candidates; a negative required value has no real square root and must not create a real intersection point. The exact result, retaining the requested order, is (400, [-400, 0, 41], [No, Yes, No])." }, "skills": [ "Circle Squared Remainder at a Fixed Horizontal Offset" ], "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；r=20、水平位移dx=0，候選點(dx,y)的有序ys=[0, 20, 21]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。", "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. r=20, horizontal displacement dx=0, ordered candidate y values=[0, 20, 21] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density14-circle-horizontal-section-candidates, density14_circle_horizontal_section_candidates, finite_integer_circle_boundary_protocol]", "after": "derived = [[required_vertical_square, ALL_squared_residuals, 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(20, 20)", "after": "Circle ORIGINAL nonnegative radius squared = 400", "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": [ "20", "20" ], "output": "400", "label": "Circle ORIGINAL nonnegative radius squared" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [20, orthogonal_equal_unit]", "after": "derived = [400]", "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 horizontal displacement squared := mul(0, 0)", "after": "Circle ORIGINAL horizontal displacement squared = 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 horizontal displacement squared，不以小數近似替代。", "en": "Recompute Circle ORIGINAL horizontal displacement squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Circle ORIGINAL horizontal displacement squared" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Circle ORIGINAL required vertical square := sub(400, 0)", "after": "Circle ORIGINAL required vertical square = 400", "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 required vertical square，不以小數近似替代。", "en": "Recompute Circle ORIGINAL required vertical square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "400", "0" ], "output": "400", "label": "Circle ORIGINAL required vertical square" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Circle ORIGINAL vertical candidate0 squared := mul(0, 0)", "after": "Circle ORIGINAL vertical candidate0 squared = 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 vertical candidate0 squared，不以小數近似替代。", "en": "Recompute Circle ORIGINAL vertical candidate0 squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Circle ORIGINAL vertical candidate0 squared" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Circle ORIGINAL candidate0 square minus required := sub(0, 400)", "after": "Circle ORIGINAL candidate0 square minus required = -400", "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 candidate0 square minus required，不以小數近似替代。", "en": "Recompute Circle ORIGINAL candidate0 square minus required with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "400" ], "output": "-400", "label": "Circle ORIGINAL candidate0 square minus required" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Circle ORIGINAL vertical candidate1 squared := mul(20, 20)", "after": "Circle ORIGINAL vertical candidate1 squared = 400", "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 vertical candidate1 squared，不以小數近似替代。", "en": "Recompute Circle ORIGINAL vertical candidate1 squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "20", "20" ], "output": "400", "label": "Circle ORIGINAL vertical candidate1 squared" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Circle ORIGINAL candidate1 square minus required := sub(400, 400)", "after": "Circle ORIGINAL candidate1 square minus required = 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 candidate1 square minus required，不以小數近似替代。", "en": "Recompute Circle ORIGINAL candidate1 square minus required with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "400", "400" ], "output": "0", "label": "Circle ORIGINAL candidate1 square minus required" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Circle ORIGINAL vertical candidate2 squared := mul(21, 21)", "after": "Circle ORIGINAL vertical candidate2 squared = 441", "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 vertical candidate2 squared，不以小數近似替代。", "en": "Recompute Circle ORIGINAL vertical candidate2 squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "21", "21" ], "output": "441", "label": "Circle ORIGINAL vertical candidate2 squared" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Circle ORIGINAL candidate2 square minus required := sub(441, 400)", "after": "Circle ORIGINAL candidate2 square minus required = 41", "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 candidate2 square minus required，不以小數近似替代。", "en": "Recompute Circle ORIGINAL candidate2 square minus required with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "441", "400" ], "output": "41", "label": "Circle ORIGINAL candidate2 square minus required" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [20, 0, [0, 20, 21]]", "after": "derived = [400, [-400, 0, 41], [否, 是, 否]]", "relation": "implies", "rule_id": "Circle_FIXED_horizontal_OFFSET_ALL_vertical_candidates_NEGATIVE_required_LEGAL", "conditions": [], "explanation": { "zh-Hant": "固定的是水平位移dx，candidate點為(dx,y)，由dx²+y²=r²驗y²-required。required可負且合法，表示不能生成實交點；仍保三候選的signed殘差／位置，不開負根、不clamp0、不加未給圓心平移。", "en": "The FIXED quantity is horizontal OFFSET dx;candidate points(dx,y) satisfy dx²+y²=r² IFF y²-required=0.A negative required value is LEGAL and gives no real intersection.STILL retain all three signed residuals/positions;do not take a negative square root,clamp0 or add an unspecified center translation." }, "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": [ "400", [ "-400", "0", "41" ], [ false, true, false ] ] }, "conclusion": { "canonical_value": "(400, [-400, 0, 41], [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": 95, "original_line_sha256": "d2425cb50683f6f162fa81ed1e7ccea024e38c76c9ac1dab931eb01dc8892d5a", "question_compact_sha256": "978c3ab0106ff443b3f56ec3bd0e10005c61a3e1f74936445453707c8b560023", "english_row_compact_sha256": "5375184841c30f7310206c3f917abb06f4df5d723c8a7030e449f07745b93f97", "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": "6405248b87f0aa23bdfec52e28b67c61c104cc83e0676ff4fa796dddad3ac2d2", "original_candidate_line_sha256": "6405248b87f0aa23bdfec52e28b67c61c104cc83e0676ff4fa796dddad3ac2d2" }, "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": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "106864a8e8485e563f18a9695a3bf6c3184eaef49d0f94fc5f2ba47a83e97e04", "arithmetic_steps": 9 }, "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." } }