{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261012-advance-20261017-r07-counterexample-0095", "pilot_number": "R0061-31045", "number": 31045, "family_id": "qf-density17-counter-domain-extension", "title": { "zh-Hant": "第 31045 題：增加一個明示述詞位置", "en": "Question 31045: One Declared New Predicate Value and Universal Truth" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261012-advance-20261017-r07-counterexample-0095", "batch_id": "batch-20261012-advance-20261017-r07-counterexample-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-counterexample" ], "title_zh": "增加一個明示述詞位置", "stem_zh": "明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 是, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。", "answer": { "kind": "expression", "value": "([4], [[0, 是], [1, 是], [2, 是], [3, 是], [4, 否], [5, 否]], [4, 5], 否, 否, 5)", "explanation_zh": "離散結構核對=([4], [[0, 是], [1, 是], [2, 是], [3, 是], [4, 否], [5, 否]], [4, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；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": 3, "skills": [ "增加一個明示述詞位置" ] }, "variant": { "family_id": "qf-density17-counter-domain-extension", "seed": 202610177095, "parameters": { "n": 5, "truth": [ true, true, true, true, false ], "extra_truth": false, "domain": "Fin_predicate_table", "operation": "counter-domain-extension", "semantic_template": "qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,true,true,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-0095", "batch_id": "batch-20261012-advance-20261017-r07-counterexample-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "61c1c24d16d49fd5d7d47b933df295f77f13b364ca5bf8ea15123cee2aa33f65", "family_id": "qf-density17-counter-domain-extension", "seed": 202610177095, "semantic_template": "qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,true,true,false]}" }, "fields": { "title": "One Declared New Predicate Value and Universal Truth", "stem": "The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, true, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).", "answer": { "kind": "expression", "value": "([4], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, No]], [4, 5], No, No, 5)", "explanation": "Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, No]], [4, 5], No, No, 5)." }, "skills": [ "One Declared New Predicate Value and Universal Truth" ], "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(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 是, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。", "en": "The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, true, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density17-counter-domain-extension, counter-domain-extension, domain, Fin_predicate_table]", "after": "derived = [[old_fail, extended_table, new_fail, old_universal, new_universal, new_label]]", "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 = [5, 0, 是]", "after": "derived = [[0, 是], 否]", "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 = [5, 1, 是]", "after": "derived = [[1, 是], 否]", "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 = [5, 2, 是]", "after": "derived = [[2, 是], 否]", "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 = [5, 3, 是]", "after": "derived = [[3, 是], 否]", "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": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [5, 4, 否]", "after": "derived = [[4, 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": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [5, [是, 是, 是, 是, 否]]", "after": "derived = [[[0, 是], [1, 是], [2, 是], [3, 是], [4, 否]], [4], 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": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [5, [是, 是, 是, 是, 否], 否]", "after": "derived = [[4], [[0, 是], [1, 是], [2, 是], [3, 是], [4, 否], [5, 否]], [4, 5], 否, 否, 5]", "relation": "implies", "rule_id": "Counter_EXTENSION_declared_new_truth_at_NEW_label_n_not_raw_revalidation", "conditions": [], "explanation": { "zh-Hant": "新標記是n，新增真值必須明示而非猜測。原n=8可衍生9列表，不重套raw上界；空域原全稱真也可能因新False變假。", "en": "The NEW label is n and its truth value is explicitly GIVEN,not inferred.Raw n=8 may derive a9-row table without recapping;an empty original universal may become false after appending False." }, "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": [ [ 4 ], [ [ 0, true ], [ 1, true ], [ 2, true ], [ 3, true ], [ 4, false ], [ 5, false ] ], [ 4, 5 ], false, false, 5 ] }, "conclusion": { "canonical_value": "([4], [[0, 是], [1, 是], [2, 是], [3, 是], [4, 否], [5, 否]], [4, 5], 否, 否, 5)", "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": 95, "original_line_sha256": "a921781cabf9f796cde1d3edc2dbd0fb584830fe2602e28bb6813678df063e2d", "question_compact_sha256": "61c1c24d16d49fd5d7d47b933df295f77f13b364ca5bf8ea15123cee2aa33f65", "english_row_compact_sha256": "88be57b056d16afbe2a74e17decadd16fcd001e02b618de8066db7c923a80a0e", "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": "dd1f936a9ba0811ec4958a8bb975ac2d0f4ee66ead5c1b48aac9fb2902da23b3", "original_candidate_line_sha256": "dd1f936a9ba0811ec4958a8bb975ac2d0f4ee66ead5c1b48aac9fb2902da23b3" }, "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": 8, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "cf960d0dec3fa4dff1744ec74ee162683be0ce780dbd7dbbed565cbfaa294283", "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." } }