{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20260914-arithmetic-domain-boundary-0015", "pilot_number": "R0070-35515", "number": 35515, "family_id": "qf-arithmetic-domain-boundary", "title": { "zh-Hant": "第 35515 題：四則運算數系邊界", "en": "Question 35515: Arithmetic Number-System Boundary" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20260914-arithmetic-domain-boundary-0015", "batch_id": "batch-20260914-domain-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-arithmetic-operations" ], "title_zh": "四則運算數系邊界", "stem_zh": "在數系 N（包含 0）中，2-9 的結果應如何分類？", "answer": { "kind": "choice", "value": "在 N 中不封閉", "choices": [ "在 N 中不封閉", "除數為零，不能給一般數值答案" ], "explanation_zh": "2-9=-7，這個結果離開 N；這表示自然數對此減法不封閉，不是在所有數系都無定義。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "數系邊界", "四則運算" ] }, "variant": { "family_id": "qf-arithmetic-domain-boundary", "seed": 20260928, "parameters": { "number_system": "N", "operator": "-", "a": 2, "b": 9, "state": "not_closed_in_N", "semantic_template": "qf-arithmetic-domain-boundary/{\"a\":2,\"b\":9,\"number_system\":\"N\",\"operator\":\"-\"}" } }, "verification": { "status": "candidate", "checks": [ "schema_pending", "independent_recomputation_pending" ], "warning_zh": "本題僅依明示的有限參數重算或分類；不構成普遍證明、形式 Evidence、人工審查或發布。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "generated_at": "2026-09-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20260914-arithmetic-domain-boundary-0015", "batch_id": "batch-20260914-domain-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "016099d93b8f7a44d35f95ad05d63dcc39bf3b594d70d178630627487926f052", "family_id": "qf-arithmetic-domain-boundary", "seed": 20260928, "semantic_template": "qf-arithmetic-domain-boundary/{\"a\":2,\"b\":9,\"number_system\":\"N\",\"operator\":\"-\"}" }, "fields": { "title": "Arithmetic Number-System Boundary", "stem": "In the number system N, including 0, how should the result of 2-9 be classified?", "answer": { "kind": "choice", "value": "Not closed in N", "choices": [ "Not closed in N", "Zero divisor; no ordinary numerical result" ], "explanation": "2-9 is not a natural number in this instance, so the operation is not closed in N." }, "skills": [ "number-system boundaries", "arithmetic operations" ], "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": "在數系 N（包含 0）中，2-9 的結果應如何分類？", "en": "In the number system N, including 0, how should the result of 2-9 be classified?" } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-arithmetic-domain-boundary, [a, b, number_system, operator, state]]", "after": "derived = [arithmetic, null, null]", "relation": "implies", "rule_id": "R70_SOURCE_bank_specific_interface", "conditions": [], "explanation": { "zh-Hant": "只使用此bank真正明示的欄位、operation/domain與型別；BASE容許保留額外metadata，不把LDL SAFE/集合上限或其他bank的去重規則搬來。", "en": "Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [2, 9, -]", "after": "derived = [N={0,1,2,...}]", "relation": "implies", "rule_id": "Natural_arithmetic_GIVEN_N_includes0_and_operands", "conditions": [], "explanation": { "zh-Hant": "先固定本題N含0，兩操作數皆非負整數；問的是此系統內能否給數值，不是擅自換成整數或有理數。", "en": "FIRST fix THIS question’s N={0,1,2,...} and nonnegative integral operands.The question asks for a value WITHIN this system,not a silent change to integers/rationals." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Natural arithmetic ORIGINAL binary operation := sub(2, 9)", "after": "Natural arithmetic ORIGINAL binary operation = -7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Natural arithmetic ORIGINAL binary operation，不以小數近似替代。", "en": "Recompute Natural arithmetic ORIGINAL binary operation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2", "9" ], "output": "-7", "label": "Natural arithmetic ORIGINAL binary operation" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [-7]", "after": "derived = [not_closed_in_N, 在 N 中不封閉]", "relation": "implies", "rule_id": "Natural_arithmetic_classify_computed_integer_in_N", "conditions": [], "explanation": { "zh-Hant": "先算完整整數結果，再測非負；加法與乘法留在N，減法若負則在N中不封閉，不因知道負整數就更改題目數系。", "en": "Compute the COMPLETE integer result,then test nonnegativity.Addition/multiplication stay in N;negative subtraction is not closed in N.Knowing a negative integer does not change the declared number system." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [not_closed_in_N, not_closed_in_N]", "after": "derived = [在 N 中不封閉]", "relation": "implies", "rule_id": "Arithmetic_STATE_checked_AFTER_derivation", "conditions": [], "explanation": { "zh-Hant": "狀態metadata只在完成推導後一致性核對，從未拿它代替計算；除零、非封閉與N內數值三類分開。", "en": "Check state metadata ONLY AFTER deriving the result;never use it instead of calculation.Keep zero divisor,nonclosure and N-valued result distinct." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null } ], "math_model": { "result": "在 N 中不封閉" }, "conclusion": { "canonical_value": "在 N 中不封閉", "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-20260914-domain-foundations", "batch_path": "public/data/questions/batches/batch-20260914-domain-foundations/questions.jsonl", "batch_sha256": "726b3f94ccd05d8c91cc7e306b42fd71b6d9040e194db91739a250d488fd1de7", "original_line": 15, "original_line_sha256": "a92d8ae791f592893c57dc75c05cf74ac57f0e5950e5e961e52dbe82d6a8e5e2", "question_compact_sha256": "016099d93b8f7a44d35f95ad05d63dcc39bf3b594d70d178630627487926f052", "english_row_compact_sha256": "f079ca1599c21b5665db9de156196eeabc0ffd157564a3b5c3acb87ed8644cc6", "english_manifest_path": "public/data/i18n/editions/batch-20260914-domain-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20260914-domain-foundations.en.jsonl" }, "mko_bindings": { "mko-arithmetic-operations": { "sha256": "452d77028e2659b860c94200c1de159c4184dde86d6d73950e1998a0080e265a", "english_sha256": "cd55f6d985015e73e560232ec9e4f77f3559c06dc72f0cf006f7083206a54169", "original_path": "public/data/mko/arithmetic-operations.json", "english_path": "public/data/i18n/editions/mko-arithmetic-operations.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "20c37a7e64ee5969e7606c5cc3a00e575002ebebdfe82873521cc3d7c608caff", "original_candidate_sha256": "f98da8be1fd84c448e2b11201d06acbafa20f12ea4307ef6b7dedd51febd1945", "original_candidate_line_sha256": "f98da8be1fd84c448e2b11201d06acbafa20f12ea4307ef6b7dedd51febd1945" }, "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-20261026-012-p71", "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": "9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80", "question_first_witness_sha256": "c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "d3646e30b3dfd85f3e7d2d5e282b6b01bfa5f7f975234f5b753d168dbfcb3009", "arithmetic_steps": 1 }, "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." } }