{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20260919-integer-membership-0016", "pilot_number": "R0064-32541", "number": 32541, "family_id": "qf-integer-membership", "title": { "zh-Hant": "第 32541 題：qf-integer-membership", "en": "Question 32541: Integer Membership" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20260919-integer-membership-0016", "batch_id": "batch-20260919-integer-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-integer" ], "title_zh": "qf-integer-membership", "stem_zh": "15.5 是否屬於整數集合 Z？", "answer": { "kind": "choice", "value": "否", "choices": [ "是", "否" ], "explanation_zh": "15.5 有小數表示，不能因外觀當作整數。" }, "difficulty": { "band": "upper_secondary", "level": 2, "skills": [ "整數成員" ] }, "variant": { "family_id": "qf-integer-membership", "seed": 20260935, "parameters": { "raw": "15.5", "integer": false, "semantic_template": "qf-integer-membership/{\"integer\":false,\"raw\":\"15.5\"}" } }, "verification": { "status": "candidate", "checks": [ "schema_pending", "independent_recomputation_pending" ], "warning_zh": "本題是有限候選重算；不構成普遍證明、形式 Evidence 或人工審查。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "generated_at": "2026-09-19", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20260919-integer-membership-0016", "batch_id": "batch-20260919-integer-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "297d76c1933e1982f7f13a6a9306482b41ee727b1e385f481385b27c23b72700", "family_id": "qf-integer-membership", "seed": 20260935, "semantic_template": "qf-integer-membership/{\"integer\":false,\"raw\":\"15.5\"}" }, "fields": { "title": "Integer Membership", "stem": "Does 15.5 belong to the integer set Z?", "answer": { "kind": "choice", "value": "No", "choices": [ "Yes", "No" ], "explanation": "15.5 is not an integer; decimal notation must not be classified by appearance alone." }, "skills": [ "integer 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": "15.5 是否屬於整數集合 Z？", "en": "Does 15.5 belong to the integer set Z?" } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-integer-membership, no-required-operation, no-required-domain, []]", "after": "derived = [[mathematical_membership_YesNo_STRING]]", "relation": "implies", "rule_id": "R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles", "conditions": [], "explanation": { "zh-Hant": "legacy核來源精確欄位及operation/domain；BASE整數保metadata但不當oracle。先分NUMBER、字串、fractionpair或vector等式，不強套同一格式；原中英／AST／status不變。", "en": "Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [15.5, 否]", "after": "derived = [否, 31/2, 否]", "relation": "implies", "rule_id": "Integer_RAW_ASCII_syntax_flag_verified_SEPARATE_exact_numeric_value", "conditions": [], "explanation": { "zh-Hant": "先驗raw字串與原Boolean旗標符合明示ASCII記法，再以精確值判整數。本輪No都是真正半整數；字串1.0的parser拒絕不能推成實數1.0不在Z。超出本凍結語義的歧義例不冒充數學No。", "en": "FIRST verify the raw string and Boolean flag against the declared ASCII syntax,THEN classify the exact value.The frozen No cases are genuine half-integers.A parser rejecting string1.0 does NOT prove real1.0 is outside Z.Ambiguous out-of-slice cases are not false mathematical No answers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Integer nonmember ORIGINAL floor times reduced denominator := mul(15, 2)", "after": "Integer nonmember ORIGINAL floor times reduced denominator = 30", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Integer nonmember ORIGINAL floor times reduced denominator，不以小數近似替代。", "en": "Recompute Integer nonmember ORIGINAL floor times reduced denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "2" ], "output": "30", "label": "Integer nonmember ORIGINAL floor times reduced denominator" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Integer nonmember ORIGINAL numerator minus floor product := sub(31, 30)", "after": "Integer nonmember ORIGINAL numerator minus floor product = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Integer nonmember ORIGINAL numerator minus floor product，不以小數近似替代。", "en": "Recompute Integer nonmember ORIGINAL numerator minus floor product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "31", "30" ], "output": "1", "label": "Integer nonmember ORIGINAL numerator minus floor product" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Integer nonmember ORIGINAL positive proper remainder fraction := div(1, 2)", "after": "Integer nonmember ORIGINAL positive proper remainder fraction = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Integer nonmember ORIGINAL positive proper remainder fraction，不以小數近似替代。", "en": "Recompute Integer nonmember ORIGINAL positive proper remainder fraction with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "2" ], "output": "1/2", "label": "Integer nonmember ORIGINAL positive proper remainder fraction" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Integer nonmember ORIGINAL value minus lower integer := sub(31/2, 15)", "after": "Integer nonmember ORIGINAL value minus lower integer = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Integer nonmember ORIGINAL value minus lower integer，不以小數近似替代。", "en": "Recompute Integer nonmember ORIGINAL value minus lower integer with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "31/2", "15" ], "output": "1/2", "label": "Integer nonmember ORIGINAL value minus lower integer" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Integer nonmember ORIGINAL upper integer minus value := sub(16, 31/2)", "after": "Integer nonmember ORIGINAL upper integer minus value = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Integer nonmember ORIGINAL upper integer minus value，不以小數近似替代。", "en": "Recompute Integer nonmember ORIGINAL upper integer minus value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "16", "31/2" ], "output": "1/2", "label": "Integer nonmember ORIGINAL upper integer minus value" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [15.5, 31/2, 15, 2]", "after": "derived = [30, 1, 1/2, 1/2, 1/2, k < value < k+1]", "relation": "implies", "rule_id": "Integer_NONmember_EXACT_between_consecutive_integers_NOT_all_decimals", "conditions": [], "explanation": { "zh-Hant": "寫成最簡n/d再作n=kd+r，0