{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261012-advance-20261017-r07-counterexample-0061", "pilot_number": "R0061-31086", "number": 31086, "family_id": "qf-density17-counter-restriction-witness", "title": { "zh-Hant": "第 31086 題：限制論域後的原見證資格", "en": "Question 31086: Restriction Can Remove an Original Counterexample" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261012-advance-20261017-r07-counterexample-0061", "batch_id": "batch-20261012-advance-20261017-r07-counterexample-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-counterexample" ], "title_zh": "限制論域後的原見證資格", "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, 否, 是, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=2。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。", "answer": { "kind": "expression", "value": "([1, 3, 4], [0, 2, 4], [4], 否, 否, 否)", "explanation_zh": "離散結構核對=([1, 3, 4], [0, 2, 4], [4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；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": 2, "skills": [ "限制論域後的原見證資格" ] }, "variant": { "family_id": "qf-density17-counter-restriction-witness", "seed": 202610177061, "parameters": { "n": 5, "truth": [ true, false, true, false, false ], "subset": [ 4, 2, 0 ], "witness": 2, "domain": "Fin_predicate_table", "operation": "counter-restriction-witness", "semantic_template": "qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,true,false,false],\"witness\":2}" } }, "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-0061", "batch_id": "batch-20261012-advance-20261017-r07-counterexample-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3005266ab791331e321917f6bc08e6645eba18e74fa8af7577d9a9d90f4dbdf4", "family_id": "qf-density17-counter-restriction-witness", "seed": 202610177061, "semantic_template": "qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,true,false,false],\"witness\":2}" }, "fields": { "title": "Restriction Can Remove an Original Counterexample", "stem": "The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).", "answer": { "kind": "expression", "value": "([1, 3, 4], [0, 2, 4], [4], No, No, No)", "explanation": "Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 4], [0, 2, 4], [4], No, No, No)." }, "skills": [ "Restriction Can Remove an Original 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(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 是, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=2。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。", "en": "The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density17-counter-restriction-witness, counter-restriction-witness, domain, Fin_predicate_table]", "after": "derived = [[fail, subset, restricted_fail, original_witness, restricted_witness, restricted_universal]]", "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, 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 = [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, 否]", "after": "derived = [[3, 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": "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, 否, 是, 否, 否]]", "after": "derived = [[[0, 是], [1, 否], [2, 是], [3, 否], [4, 否]], [1, 3, 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, 否, 是, 否, 否], [4, 2, 0], 2, [1, 3, 4]]", "after": "derived = [[0, 2, 4], [4], 否, 否, 否]", "relation": "implies", "rule_id": "Counter_RESTRICTION_set_dedup_OLD_labels_witness_guards_SEPARATE", "conditions": [], "explanation": { "zh-Hant": "subset先驗raw長度及型別，再去重排序但不改原標記。原w是反例與限制後仍是反例要分開；刪掉False點可以使限制全稱為真，超域w不讀predicate。", "en": "Validate RAW subset length/types BEFORE set normalization,retaining OLD labels.Original and retained witness flags differ.Deleting False points may make the restricted universal true;outside w never evaluates the predicate." }, "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": [ [ 1, 3, 4 ], [ 0, 2, 4 ], [ 4 ], false, false, false ] }, "conclusion": { "canonical_value": "([1, 3, 4], [0, 2, 4], [4], 否, 否, 否)", "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": 61, "original_line_sha256": "443cf4eee4575f223f798498a331a44b9153a4d1e3cea6b07fd94d3ac9b5d1af", "question_compact_sha256": "3005266ab791331e321917f6bc08e6645eba18e74fa8af7577d9a9d90f4dbdf4", "english_row_compact_sha256": "cb9d42ddcec5f9df3c9a94e07fbd707b124acc44671078d25f547512035ec142", "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": "f99a277bd4bab8e44bf06a3ba3c2842d27353de1f657b600d0495b46692f4483", "original_candidate_line_sha256": "f99a277bd4bab8e44bf06a3ba3c2842d27353de1f657b600d0495b46692f4483" }, "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": "ead3a80c8c1a71f938ac32cdd7c9a3d44b7d121ba97c4c92985446633cdae963", "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." } }