{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261012-advance-20261017-r07-counterexample-0003", "pilot_number": "R0061-31053", "number": 31053, "family_id": "qf-density17-counter-full-failure-min", "title": { "zh-Hant": "第 31053 題：全有限失敗集與最小見證", "en": "Question 31053: Complete Failure Set and Least Counterexample" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261012-advance-20261017-r07-counterexample-0003", "batch_id": "batch-20261012-advance-20261017-r07-counterexample-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-counterexample" ], "title_zh": "全有限失敗集與最小見證", "stem_zh": "明示有限論域，完整布林述詞表。U=Fin(3)=[0, 1, 2]，依索引列P的完整布林述詞表=[否, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。", "answer": { "kind": "expression", "value": "([[0, 否], [1, 否], [2, 否]], [0, 1, 2], 否, 0)", "explanation_zh": "離散結構核對=([[0, 否], [1, 否], [2, 否]], [0, 1, 2], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "全有限失敗集與最小見證" ] }, "variant": { "family_id": "qf-density17-counter-full-failure-min", "seed": 202610177003, "parameters": { "n": 3, "truth": [ false, false, false ], "domain": "Fin_predicate_table", "operation": "counter-full-failure-min", "semantic_template": "qf-density17-counter-full-failure-min/{\"n\":3,\"truth\":[false,false,false]}" } }, "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": "2026-10-12", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261012-advance-20261017-r07-counterexample-0003", "batch_id": "batch-20261012-advance-20261017-r07-counterexample-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3cfd32c4867380328ee419cf3f14180c484190d4f37ee1230424be72729ad7a7", "family_id": "qf-density17-counter-full-failure-min", "seed": 202610177003, "semantic_template": "qf-density17-counter-full-failure-min/{\"n\":3,\"truth\":[false,false,false]}" }, "fields": { "title": "Complete Failure Set and Least Counterexample", "stem": "The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).", "answer": { "kind": "expression", "value": "([[0, No], [1, No], [2, No]], [0, 1, 2], No, 0)", "explanation": "Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No]], [0, 1, 2], No, 0)." }, "skills": [ "Complete Failure Set and Least Counterexample" ], "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": "明示有限論域，完整布林述詞表。U=Fin(3)=[0, 1, 2]，依索引列P的完整布林述詞表=[否, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。", "en": "The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density17-counter-full-failure-min, counter-full-failure-min, domain, Fin_predicate_table]", "after": "derived = [[indexed_table, fail, universal, least_or_tag]]", "relation": "implies", "rule_id": "R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles", "conditions": [], "explanation": { "zh-Hant": "先核本題欄位與來源協議：distance只用完整座標frame，其他用domain；離散運算local-hyphen，舊bank用underscore。輸出欄序及型別依題族，保留原始中英資料與狀態。", "en": "Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [3, 0, 否]", "after": "derived = [[0, 否], 是]", "relation": "implies", "rule_id": "Counter_EACH_original_label_EXACT_Boolean_and_failure_test", "conditions": [], "explanation": { "zh-Hant": "表中值必須是Boolean，不以0/1代替。原0-based標記i的False才是反例，不可重編號或由缺資料推為False。", "en": "The table value must be EXACTLY Boolean,not0/1.At ORIGINAL zero-based label i,ONLY False gives a counterexample.Do not relabel or infer False from missing data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [3, 1, 否]", "after": "derived = [[1, 否], 是]", "relation": "implies", "rule_id": "Counter_EACH_original_label_EXACT_Boolean_and_failure_test", "conditions": [], "explanation": { "zh-Hant": "表中值必須是Boolean，不以0/1代替。原0-based標記i的False才是反例，不可重編號或由缺資料推為False。", "en": "The table value must be EXACTLY Boolean,not0/1.At ORIGINAL zero-based label i,ONLY False gives a counterexample.Do not relabel or infer False from missing data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [3, 2, 否]", "after": "derived = [[2, 否], 是]", "relation": "implies", "rule_id": "Counter_EACH_original_label_EXACT_Boolean_and_failure_test", "conditions": [], "explanation": { "zh-Hant": "表中值必須是Boolean，不以0/1代替。原0-based標記i的False才是反例，不可重編號或由缺資料推為False。", "en": "The table value must be EXACTLY Boolean,not0/1.At ORIGINAL zero-based label i,ONLY False gives a counterexample.Do not relabel or infer False from missing data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [3, [否, 否, 否]]", "after": "derived = [[[0, 否], [1, 否], [2, 否]], [0, 1, 2], 否]", "relation": "implies", "rule_id": "Counter_COMPLETE_finite_table_failures_EMPTY_universal_true", "conditions": [], "explanation": { "zh-Hant": "遍歷整個已給有限論域後，無反例才是此表的全稱真；空論域也為真。這個指定有限模型不等於任意無限命題的證明。", "en": "AFTER traversing the whole GIVEN finite domain,no failures means its universal statement is true,including the empty domain.This assigned finite model does not prove arbitrary infinite statements." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [[[0, 否], [1, 否], [2, 否]], [0, 1, 2]]", "after": "derived = [否, 0]", "relation": "implies", "rule_id": "Counter_LEAST_failure_zero_is_real_index_NOT_absence", "conditions": [], "explanation": { "zh-Hant": "反例按原標記遞增排列，第一個是最小反例；0是有效標記，不可當none。沒有反例時用不存在標籤而非數值0。", "en": "Failures are in increasing ORIGINAL label order;the first is the least counterexample.Index0 is valid,not absence.With no failure use the does-not-exist tag,not numerical0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null } ], "math_model": { "result": [ [ [ 0, false ], [ 1, false ], [ 2, false ] ], [ 0, 1, 2 ], false, 0 ] }, "conclusion": { "canonical_value": "([[0, 否], [1, 否], [2, 否]], [0, 1, 2], 否, 0)", "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-20261012-advance-20261017-r07-counterexample-foundations", "batch_path": "public/data/questions/batches/batch-20261012-advance-20261017-r07-counterexample-foundations/questions.jsonl", "batch_sha256": "01daceda609e9e1f875cd64e46167353d8c3207652b71bf97c9a5aa25e6c2b2f", "original_line": 3, "original_line_sha256": "3c9fccb5c7181565cfe2e35a5b2861b1d9477ff3f033286642b0cd04c23c3ece", "question_compact_sha256": "3cfd32c4867380328ee419cf3f14180c484190d4f37ee1230424be72729ad7a7", "english_row_compact_sha256": "bd29aec3d047a64b38dbd057ad89e4ff35075929780cac4c6492c6748a60871f", "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261017-r07-counterexample-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261017-r07-counterexample-foundations.en.jsonl" }, "mko_bindings": { "mko-counterexample": { "sha256": "c3528c0ad339526d31db8324f3a6cbcdc28e756c638ba57e2792a24d8cc5beef", "english_sha256": "8fb486a8d29e7dd741f3fc050973e1897157bfa5f03ee86c37e9b8e1b728a8e4", "original_path": "public/data/mko/counterexample.json", "english_path": "public/data/i18n/editions/mko-counterexample.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "912db26eb6c90aa6e256f4356cb5ee94cd0a352f04f667876ae6739f8181a17c", "original_candidate_sha256": "ec38f10e4b42ea0d77b206d97e382a9e865ca646e71f9f409e3621ac7d300a06", "original_candidate_line_sha256": "ec38f10e4b42ea0d77b206d97e382a9e865ca646e71f9f409e3621ac7d300a06" }, "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-p62", "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": "5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69", "question_first_witness_sha256": "c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495" }, "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": "07eb7572e98783931ca899990dde464090fdf7804c6456bbf1af59bc58491a80", "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." } }