{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261023-r07-multivariable-real-function-0042", "pilot_number": "R0020-10667", "number": 10667, "family_id": "qf-mod23-multivar-rectangle-domain-audit", "title": { "zh-Hant": "第 10667 題：二變數實函數", "en": "Question 10667: Inclusive Membership in a Declared Closed Rectangle" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261023-r07-multivariable-real-function-0042", "batch_id": "batch-20261013-advance-20261023-r07-multivariable-real-function-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-multivariable-real-function" ], "title_zh": "二變數實函數", "stem_zh": "明示R2到R多項式與座標角色。操作=multivar-rectangle-domain-audit；資料={\"bounds\":[-18,18,-17,19],\"point\":[-1,-1]}。請依 [bounds,point,in_domain] 欄序回答。", "answer": { "kind": "expression", "value": "([-18, 18, -17, 19], [-1, -1], 是)", "explanation_zh": "現代橋梁核對=([-18, 18, -17, 19], [-1, -1], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；未聲明全域連續或可微；不外推向量值函數。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "二變數實函數" ] }, "variant": { "family_id": "qf-mod23-multivar-rectangle-domain-audit", "seed": 202610237042, "parameters": { "bounds": [ -18, 18, -17, 19 ], "point": [ -1, -1 ], "domain": "Polynomial_R2_to_R_slice", "operation": "multivar-rectangle-domain-audit", "semantic_template": "qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-18,18,-17,19],\"point\":[-1,-1]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261023-r07-multivariable-real-function-0042", "batch_id": "batch-20261013-advance-20261023-r07-multivariable-real-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "16d458e5e63c8c9899c9de2f64c83a70dfc0bc4766e8401f71382649e7f3d20b", "family_id": "qf-mod23-multivar-rectangle-domain-audit", "seed": 202610237042, "semantic_template": "qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-18,18,-17,19],\"point\":[-1,-1]}" }, "fields": { "title": "Inclusive Membership in a Declared Closed Rectangle", "stem": "Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-18, 18, -17, 19] and point [x,y]=[-1, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.", "answer": { "kind": "expression", "value": "([-18, 18, -17, 19], [-1, -1], Yes)", "explanation": "Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-18, 18, -17, 19], [-1, -1], Yes)." }, "skills": [ "Inclusive Membership in a Declared Closed Rectangle" ], "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": "明示R2到R多項式與座標角色。操作=multivar-rectangle-domain-audit；資料={\"bounds\":[-18,18,-17,19],\"point\":[-1,-1]}。請依 [bounds,point,in_domain] 欄序回答。", "en": "Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-18, 18, -17, 19] and point [x,y]=[-1, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "rectangle coordinate0 lower slack := sub(-1, -18)", "after": "rectangle coordinate0 lower slack = 17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 rectangle coordinate0 lower slack，不以小數近似替代。", "en": "Recompute rectangle coordinate0 lower slack with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-18" ], "output": "17", "label": "rectangle coordinate0 lower slack" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "rectangle coordinate0 upper slack := sub(18, -1)", "after": "rectangle coordinate0 upper slack = 19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 rectangle coordinate0 upper slack，不以小數近似替代。", "en": "Recompute rectangle coordinate0 upper slack with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "18", "-1" ], "output": "19", "label": "rectangle coordinate0 upper slack" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "coordinate index=0;value=-1;closed bounds=[-18, 18]", "after": "signed slacks=[17, 19];inside inclusive interval=是", "relation": "implies", "rule_id": "rectangle_both_coordinate_closed_guards", "conditions": [], "explanation": { "zh-Hant": "上下界相等也合法，端點用非嚴格比較；兩個座標都需通過，域限制不由多項式可能有全域定義而消失。", "en": "Equal bounds are valid and endpoints are included.Both coordinates must pass;the domain restriction is not erased by a polynomial's possible global definition." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "rectangle coordinate1 lower slack := sub(-1, -17)", "after": "rectangle coordinate1 lower slack = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 rectangle coordinate1 lower slack，不以小數近似替代。", "en": "Recompute rectangle coordinate1 lower slack with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-17" ], "output": "16", "label": "rectangle coordinate1 lower slack" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "rectangle coordinate1 upper slack := sub(19, -1)", "after": "rectangle coordinate1 upper slack = 20", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 rectangle coordinate1 upper slack，不以小數近似替代。", "en": "Recompute rectangle coordinate1 upper slack with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "19", "-1" ], "output": "20", "label": "rectangle coordinate1 upper slack" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "coordinate index=1;value=-1;closed bounds=[-17, 19]", "after": "signed slacks=[16, 20];inside inclusive interval=是", "relation": "implies", "rule_id": "rectangle_both_coordinate_closed_guards", "conditions": [], "explanation": { "zh-Hant": "上下界相等也合法，端點用非嚴格比較；兩個座標都需通過，域限制不由多項式可能有全域定義而消失。", "en": "Equal bounds are valid and endpoints are included.Both coordinates must pass;the domain restriction is not erased by a polynomial's possible global definition." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ [ -18, 18, -17, 19 ], [ -1, -1 ], true ] }, "conclusion": { "canonical_value": "([-18, 18, -17, 19], [-1, -1], 是)", "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-20261013-advance-20261023-r07-multivariable-real-function-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261023-r07-multivariable-real-function-foundations/questions.jsonl", "batch_sha256": "bbcd1bbf9a79f292228317e6265f305ff43d9aef65cddda0b0ef95dd02563ab4", "original_line": 42, "original_line_sha256": "083dbe2b4c99d2e853d2aa8fa1001911bad6df1a2708f4241f166e91fdb4c425", "question_compact_sha256": "16d458e5e63c8c9899c9de2f64c83a70dfc0bc4766e8401f71382649e7f3d20b", "english_row_compact_sha256": "6d1a6ce6033e94175dbab27724ff100ee446f5f4e65248450bb535451e296979", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261023-r07-multivariable-real-function-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261023-r07-multivariable-real-function-foundations.en.jsonl" }, "mko_bindings": { "mko-multivariable-real-function": { "sha256": "f96ebe8babf93fa8de6b32cb4156ad2b710b938550218a6aff16be79ba4ef928", "english_sha256": "3bc9e3338d7c9026e1405d2719cc08c58786f3e4c4480e9803ca45cf10591239", "original_path": "public/data/mko/multivariable-real-function.json", "english_path": "public/data/i18n/editions/mko-multivariable-real-function.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "bd67681fffd9e29972c322d497cd945a668a794dd25e0c8cd26f1a6ce22e4dfb", "original_candidate_sha256": "415f47a5c812b4687218ff1dd2b2b17265c136333a547f87ffb0c35983b6a68e", "original_candidate_line_sha256": "415f47a5c812b4687218ff1dd2b2b17265c136333a547f87ffb0c35983b6a68e" }, "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-20261026-006-p21", "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": "b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f", "question_first_witness_sha256": "3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 6, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "40c7a25a4578b975ab9678cca5a29d5ff960230dd6d3428336674268d1c2b95b", "arithmetic_steps": 4 }, "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." } }