{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261106-r02-clique-and-independent-set-0025", "pilot_number": "R0038-19525", "number": 19525, "family_id": "qf-mod37-clique-clique-audit", "title": { "zh-Hant": "第 19525 題：Clique與獨立集", "en": "Question 19525: Only a Valid Clique Size Certifies a Clique-Number Lower Bound" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261106-r02-clique-and-independent-set-0025", "batch_id": "batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-clique-and-independent-set" ], "title_zh": "Clique與獨立集", "stem_zh": "完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)", "explanation_zh": "Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "Clique與獨立集" ] }, "variant": { "family_id": "qf-mod37-clique-clique-audit", "seed": 202611062025, "parameters": { "n": 6, "edges": [ [ 0, 1 ], [ 0, 2 ], [ 0, 5 ], [ 1, 2 ], [ 1, 5 ], [ 3, 4 ] ], "vertices": [ 0, 1, 2 ], "domain": "Complete_finite_simple_graph_extrema", "operation": "clique-clique-audit", "semantic_template": "qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}" } }, "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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261106-r02-clique-and-independent-set-0025", "batch_id": "batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "c20dd4e1abfac5ca0461464e3b864d89ec5d38eb245d5a2c36f9ee147c817e3f", "family_id": "qf-mod37-clique-clique-audit", "seed": 202611062025, "semantic_template": "qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}" }, "fields": { "title": "Only a Valid Clique Size Certifies a Clique-Number Lower Bound", "stem": "Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).", "answer": { "kind": "expression", "value": "([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)", "explanation": "Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)." }, "skills": [ "Only a Valid Clique Size Certifies a Clique-Number Lower Bound" ], "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": "完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=clique-clique-audit;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[vertices, is_clique, candidate_size, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不自動成為數學證據，No依各題聲稱範圍解讀。", "en": "Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope." }, "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": "complete ORIGINAL finite carrier Fin(6);ALL canonical simple undirected original edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]", "after": "all vertices and edge indices retained;extrema slice n<=9 includes empty carrier;complement reverses ONLY distinct-vertex adjacency", "relation": "implies", "rule_id": "clique_complete_graph_carrier_not_sample_or_loop_complement", "conditions": [], "explanation": { "zh-Hant": "空圖與孤立點都保留；complement只在不同頂點間反轉有無邊，不新增自環。這是bounded完整圖證書，不從某個局部子圖推原圖極值。", "en": "Keep empty carriers and isolates.Complementation reverses adjacency only between distinct vertices,without adding loops.This is a bounded complete-graph certificate,not an extremal inference from a sampled subgraph." }, "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": "RAW original candidate vertices=[0, 1, 2];normalized SORTED vertices=[0, 1, 2];ALL distinct candidate pair adjacency rows=[[0, 1, 是], [0, 2, 是], [1, 2, 是]]", "after": "clique=是 iff ALL pair entries true;independent=否 iff ALL false;candidate size=3;duplicates were rejected,not deduplicated", "relation": "implies", "rule_id": "clique_every_pair_full_candidate_not_size_alone", "conditions": [], "explanation": { "zh-Hant": "逐候選全部不同頂點對核完整原邊表；空／單點候選的pair條件依全稱空真。size是候選大小，只有對應valid才給真正extremum下界，不因大小相等就修好無效候選。", "en": "Audit every distinct candidate pair against the complete original edge list.Empty/singleton pair conditions are vacuously true.Size is only the candidate count;it gives the corresponding extremal lower bound only when valid,and a matching size does not repair an invalid candidate." }, "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": [ [ 0, 1, 2 ], true, 3, "candidate_clique_only_lower_bound_on_omega" ] }, "conclusion": { "canonical_value": "([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)", "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-20261014-advance-20261106-r02-clique-and-independent-set-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations/questions.jsonl", "batch_sha256": "56b85f706a53096428b5d249690dca1095a7f84cdf47ee09960da50b16724001", "original_line": 25, "original_line_sha256": "14a07fee3ae96a44b7f27c97455f517d4c4dc362e425637863ed1d7afcba49bd", "question_compact_sha256": "c20dd4e1abfac5ca0461464e3b864d89ec5d38eb245d5a2c36f9ee147c817e3f", "english_row_compact_sha256": "abcf3896e4c774813438e40f0adf0806ca8da6b96d94f11b6b5a7bcb0f89b806", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations.en.jsonl" }, "mko_bindings": { "mko-clique-and-independent-set": { "sha256": "9477f7b7e5b3af0c1c270f6f42b508ec6142c458868b507c850d56c9a5b946fe", "english_sha256": "7d32b2cc13a3ecb7fe4cb5c14409d97ce2b9decc8176de4b5d780f0c601b22ef", "original_path": "public/data/mko/clique-and-independent-set.json", "english_path": "public/data/i18n/editions/mko-clique-and-independent-set.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "9ed24319f1e4df0dd58d2a2dfbf2846def03f020cb5ebf70613fab53c9165f11", "original_candidate_sha256": "d600c727be2ba5282c84ad0909661940512a2ed69800644cab9a9a5d680c7f91", "original_candidate_line_sha256": "d600c727be2ba5282c84ad0909661940512a2ed69800644cab9a9a5d680c7f91" }, "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-20261026-008-p39", "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": "d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612", "question_first_witness_sha256": "c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7" }, "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": "fae33f024aa642b1bd1305500768f30d34b39599921f3da1d2f0ce1fd0f713f8", "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." } }