{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261115-r07-many-one-reduction-0029", "pilot_number": "R0051-26154", "number": 26154, "family_id": "qf-mod46-mreduction-iff-audit", "title": { "zh-Hant": "第 26154 題：多對一歸約", "en": "Question 26154: A Many-One Reduction Preserves Both Yes and No Membership" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261115-r07-many-one-reduction-0029", "batch_id": "batch-20261015-advance-20261115-r07-many-one-reduction-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-many-one-reduction" ], "title_zh": "多對一歸約", "stem_zh": "明示decision predicates與complete total map的雙向membership equivalence。操作=mreduction-iff-audit；資料={\"source_carrier\":[0,1,2,3,4,5],\"target_carrier\":[0,1],\"source_yes\":[1,3,5],\"target_yes\":[1],\"mapping\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([1, 3, 5], [1], [[0, 否, 0, 否, 是], [1, 是, 1, 是, 是], [2, 否, 0, 否, 是], [3, 是, 1, 是, 是], [4, 否, 0, 否, 是], [5, 是, 1, 是, 是]], 是, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)", "explanation_zh": "Wave6第九層核對=([1, 3, 5], [1], [[0, 否, 0, 否, 是], [1, 是, 1, 是, 是], [2, 否, 0, 否, 是], [3, 是, 1, 是, 是], [4, 否, 0, 否, 是], [5, 是, 1, 是, 是]], 是, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)；依完整明示資料獨立重算；composition與decider transfer保持A≤mB的方向；partial map與yes-only檢查無效；finite table只證finite domain instance，不證無限domain computability；不混同Turing reduction或反向decidability transfer。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "多對一歸約" ] }, "variant": { "family_id": "qf-mod46-mreduction-iff-audit", "seed": 202611157029, "parameters": { "source_carrier": [ 0, 1, 2, 3, 4, 5 ], "target_carrier": [ 0, 1 ], "source_yes": [ 1, 3, 5 ], "target_yes": [ 1 ], "mapping": [ 0, 1, 0, 1, 0, 1 ], "domain": "Complete_finite_many_one_reduction", "operation": "mreduction-iff-audit", "semantic_template": "qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5],\"source_yes\":[1,3,5],\"target_carrier\":[0,1],\"target_yes\":[1]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261115-r07-many-one-reduction-0029", "batch_id": "batch-20261015-advance-20261115-r07-many-one-reduction-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "6c398dfea79d5be9a2662fb57f64b5321505293be32bb633d4a84383a2ea6dd9", "family_id": "qf-mod46-mreduction-iff-audit", "seed": 202611157029, "semantic_template": "qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5],\"source_yes\":[1,3,5],\"target_carrier\":[0,1],\"target_yes\":[1]}" }, "fields": { "title": "A Many-One Reduction Preserves Both Yes and No Membership", "stem": "Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).", "answer": { "kind": "expression", "value": "([1, 3, 5], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)", "explanation": "Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)." }, "skills": [ "A Many-One Reduction Preserves Both Yes and No 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": "明示decision predicates與complete total map的雙向membership equivalence。操作=mreduction-iff-audit；資料={\"source_carrier\":[0,1,2,3,4,5],\"target_carrier\":[0,1],\"source_yes\":[1,3,5],\"target_yes\":[1],\"mapping\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [mreduction-iff-audit]", "after": "derived = [[source_yes, target_yes, membership_rows, reduction, certificate]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "本頁直接列完整來源欄序，不需JavaScript；contract的6/4/6/6與IVT的8/7/8由原mode選，不補欄也不由答案形狀猜。", "en": "List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [Reduction ORIGINAL map, [0, 1, 2, 3, 4, 5], [0, 1], [0, 1, 0, 1, 0, 1]]", "after": "derived = [[0, 1, 2, 3, 4, 5], [0, 1], [0, 1, 0, 1, 0, 1], [[0, 0, 是], [1, 1, 是], [2, 0, 是], [3, 1, 是], [4, 0, 是], [5, 1, 是]], 是, null]", "relation": "implies", "rule_id": "Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types", "conditions": [], "explanation": { "zh-Hant": "source數值排序要連image一起移動；target也數值排序。null／outside／Boolean不是合法integer target值，False不能當0；qualification保實際rawimage與首invalid row，不先捏造total map。", "en": "Sort source values numerically with their aligned images,and target values numerically.Null,outside and Boolean images are not valid integer target values;False is not0.Qualification retains actual raw images and the first invalid row,not an invented total map." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [Reduction ORIGINAL map, [1, 3, 5], [1], [0, 1, 2, 3, 4, 5], [0, 1], [0, 1, 0, 1, 0, 1]]", "after": "derived = [[[0, 否, 0, 否, 是], [1, 是, 1, 是, 是], [2, 否, 0, 否, 是], [3, 是, 1, 是, 是], [4, 否, 0, 否, 是], [5, 是, 1, 是, 是]], 是, 是]", "relation": "implies", "rule_id": "Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only", "conditions": [], "explanation": { "zh-Hant": "每個x都驗x∈A iff f(x)∈B，包含yes與no兩方向；invalid image的target predicate是null而非False。totality與membership責任合取才是此有限reduction。", "en": "For every x audit x in A iff f(x) in B,in both yes and no directions.Invalid images have null target predicates,notFalse.Totality and full membership equivalence jointly qualify this finite reduction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ [ 1, 3, 5 ], [ 1 ], [ [ 0, false, 0, false, true ], [ 1, true, 1, true, true ], [ 2, false, 0, false, true ], [ 3, true, 1, true, true ], [ 4, false, 0, false, true ], [ 5, true, 1, true, true ] ], true, "x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions" ] }, "conclusion": { "canonical_value": "([1, 3, 5], [1], [[0, 否, 0, 否, 是], [1, 是, 1, 是, 是], [2, 否, 0, 否, 是], [3, 是, 1, 是, 是], [4, 否, 0, 否, 是], [5, 是, 1, 是, 是]], 是, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)", "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-20261015-advance-20261115-r07-many-one-reduction-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261115-r07-many-one-reduction-foundations/questions.jsonl", "batch_sha256": "79d665ac8c8fbc11cf7d58a248b5543f574c9747cf1e02b49b91740b2bee6198", "original_line": 29, "original_line_sha256": "15a489cdc8ba601e2432c8a3e561bacc0a2ee776c77de36b8bf50c7518c1cdee", "question_compact_sha256": "6c398dfea79d5be9a2662fb57f64b5321505293be32bb633d4a84383a2ea6dd9", "english_row_compact_sha256": "862f20a3f84d1ae337066da6bff65e8c59074f5c688e935bb3a249a857315393", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261115-r07-many-one-reduction-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261115-r07-many-one-reduction-foundations.en.jsonl" }, "mko_bindings": { "mko-many-one-reduction": { "sha256": "c1a3c40ef9e301b3c5bce75380905c6f44c8cfc3bdb5126ad469d6587f9af2c6", "english_sha256": "1f0bd01bf78477131defe2c7465aa0fc959339c884a40075d0373878e7a0cb78", "original_path": "public/data/mko/many-one-reduction.json", "english_path": "public/data/i18n/editions/mko-many-one-reduction.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "74f6f34e0a695e06f9d836b5f5c677f9f716d2291c75c8f2f5b809ff0935bcff", "original_candidate_sha256": "6cfa0f32cd0fccaf6157098ed8ad64fbeb9fa3d6586a7a0fcdffd77feaa56129", "original_candidate_line_sha256": "6cfa0f32cd0fccaf6157098ed8ad64fbeb9fa3d6586a7a0fcdffd77feaa56129" }, "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-20261026-010-p52", "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": "90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3", "question_first_witness_sha256": "ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c" }, "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": "caaa174a71244d110b8c5c3506071a3b11867537a9bfeea2f7fb882b12b5e641", "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." } }