{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20260921-proof-counterexample-universal-0018", "pilot_number": "R0067-34318", "number": 34318, "family_id": "qf-proof-counterexample-universal", "title": { "zh-Hant": "第 34318 題：全稱命題的反例", "en": "Question 34318: Counterexample to a Universal Proposition" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20260921-proof-counterexample-universal-0018", "batch_id": "batch-20260921-proof-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-proof" ], "title_zh": "全稱命題的反例", "stem_zh": "論域明示為整數。全稱命題是「每個整數都是偶數」。明示 witness 為 n=9。這個 witness 是否反駁該全稱命題？", "answer": { "kind": "choice", "value": "是", "choices": [ "是", "否" ], "explanation_zh": "n=9 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "全稱命題", "反例 witness" ] }, "variant": { "family_id": "qf-proof-counterexample-universal", "seed": 20261239, "parameters": { "domain": "integers", "universal_statement": "每個整數都是偶數", "predicate_id": "even", "bound": null, "witness": 9, "counterexample_role": "falsifies_the_stated_universal_only", "semantic_template": "qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"even\",\"universal_statement\":\"每個整數都是偶數\",\"witness\":9}" } }, "verification": { "status": "candidate", "checks": [ "schema_pending", "independent_recomputation_pending" ], "warning_zh": "本題是 candidate 練習；機械重算僅核對明示條件與指定實例，並非形式證明或人工審查。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "generated_at": "2026-09-21", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20260921-proof-counterexample-universal-0018", "batch_id": "batch-20260921-proof-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "042a070c38c11ce36b1152c6ffff9004025a1612d126acb29f4fe7cbdd1ae01b", "family_id": "qf-proof-counterexample-universal", "seed": 20261239, "semantic_template": "qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"even\",\"universal_statement\":\"每個整數都是偶數\",\"witness\":9}" }, "fields": { "title": "Counterexample to a Universal Proposition", "stem": "The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=9. Does this witness refute that universal proposition?", "answer": { "kind": "choice", "value": "Yes", "choices": [ "Yes", "No" ], "explanation": "n=9 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only." }, "skills": [ "universal propositions", "counterexample witness" ], "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": "論域明示為整數。全稱命題是「每個整數都是偶數」。明示 witness 為 n=9。這個 witness 是否反駁該全稱命題？", "en": "The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=9. Does this witness refute that universal proposition?" } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-proof-counterexample-universal, [bound, counterexample_role, domain, predicate_id, universal_statement, witness]]", "after": "derived = [BASE-no-required-operation, integers]", "relation": "implies", "rule_id": "R67_SOURCE_specific_keys_operation_domain_and_numeric_bank", "conditions": [], "explanation": { "zh-Hant": "按本bank核keys與operation/domain：legacy為各表underscore，MOD50用local hyphen；BASE參數較寬不加未知欄限制。Number.isInteger與SAFE兩種來源界分開，原JSON數字表示仍完整保存。", "en": "Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [even, null, 9]", "after": "derived = [[9, 2, 1], 否]", "relation": "implies", "rule_id": "Proof_UNIVERSAL_stated_predicate_and_INTEGER_witness_substitution", "conditions": [], "explanation": { "zh-Hant": "先核文字的指定predicate/bound與整數論域，再將這個整數witness代入原predicate，顯示它失敗；even/multiple用確切非零餘數，其餘保原嚴格/非嚴格不等式。不把標記當答案oracle。", "en": "FIRST match the stated predicate/bound and INTEGER domain.THEN substitute THIS integer witness and show failure.Use an exact nonzero remainder for even/multiple and retain strict/non-strict inequalities elsewhere.Do not use a metadata flag as an answer oracle." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [even, null, 9, 否]", "after": "derived = [exists integer n with not predicate(n), not forall integers predicate(n), 是]", "relation": "implies", "rule_id": "Proof_ONE_actual_failure_refutes_ONLY_stated_forall", "conditions": [], "explanation": { "zh-Hant": "一個論域內實際反例證∃n ¬P(n)，所以否定這一句∀n P(n)。不推出每個n都失敗、其他命題為假或一個有限樣本能證全稱。", "en": "ONE actual counterexample IN the domain proves∃n ¬P(n),therefore refuting THIS∀n P(n).It does not prove every n fails,refute other propositions or make finite successful samples a universal proof." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null } ], "math_model": { "result": "是" }, "conclusion": { "canonical_value": "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-20260921-proof-foundations", "batch_path": "public/data/questions/batches/batch-20260921-proof-foundations/questions.jsonl", "batch_sha256": "16755575d0aefd29143eb27fb780816bbf67cc3ca2da5ecfa277cea483cf877f", "original_line": 93, "original_line_sha256": "6d3fc67a274853b525b199c18503424fbafb368dfa8dc4f563389a48c767201e", "question_compact_sha256": "042a070c38c11ce36b1152c6ffff9004025a1612d126acb29f4fe7cbdd1ae01b", "english_row_compact_sha256": "d0538f57599c3387247e07e74ca55faf82bc75bc7c4cc3b765a85b770d646302", "english_manifest_path": "public/data/i18n/editions/batch-20260921-proof-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20260921-proof-foundations.en.jsonl" }, "mko_bindings": { "mko-proof": { "sha256": "72f6009b912caa3c05a2948a97ca422367574a19ad5fc665964a540875417b91", "english_sha256": "1c8ec335b48f7d913d3057b2d632c371b092ca869f799b056b72464d948504fe", "original_path": "public/data/mko/proof.json", "english_path": "public/data/i18n/editions/mko-proof.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "6bb156cec7bdb6baf2497762702fc01f94fd1db55a5926bc575b12ff9237a4de", "original_candidate_sha256": "0f7d7516a30ddb2a4a85690b616b141a03b9af94ea8558fff0995e6b2f7733db", "original_candidate_line_sha256": "0f7d7516a30ddb2a4a85690b616b141a03b9af94ea8558fff0995e6b2f7733db" }, "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-p68", "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": "c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c", "question_first_witness_sha256": "12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "04d3f7b1b53a2175e0fd8707acf3c0dbdc2b84c660bca89bf8f716e25f5ad050", "arithmetic_steps": 0 }, "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." } }