{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261026-r06-graph-isomorphism-0069", "pilot_number": "R0023-12494", "number": 12494, "family_id": "qf-mod26-graphiso-qualification", "title": { "zh-Hant": "第 12494 題：圖同構", "en": "Question 12494: Conjoin Bijectivity with the Complete Adjacency-Iff Audit" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261026-r06-graph-isomorphism-0069", "batch_id": "batch-20261013-advance-20261026-r06-graph-isomorphism-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-graph-isomorphism" ], "title_zh": "圖同構", "stem_zh": "完整有限簡單圖與明示頂點映射。操作=graphiso-qualification；資料={\"n\":6,\"left_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"right_edges\":[[0,1],[1,5],[2,5],[2,3],[3,4]],\"mapping\":[0,1,5,2,3,4]}。請依 [n,mapping,bijective,bad_pairs,isomorphism] 欄序回答。", "answer": { "kind": "expression", "value": "(6, [0, 1, 5, 2, 3, 4], 是, [], 是)", "explanation_zh": "結構深度核對=(6, [0, 1, 5, 2, 3, 4], 是, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；同構要求鄰接與非鄰接iff；不變量相同不足以證同構。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "圖同構" ] }, "variant": { "family_id": "qf-mod26-graphiso-qualification", "seed": 202610266069, "parameters": { "n": 6, "left_edges": [ [ 0, 1 ], [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 4, 5 ] ], "right_edges": [ [ 0, 1 ], [ 1, 5 ], [ 2, 5 ], [ 2, 3 ], [ 3, 4 ] ], "mapping": [ 0, 1, 5, 2, 3, 4 ], "domain": "Complete_finite_simple_graph_isomorphism", "operation": "graphiso-qualification", "semantic_template": "qf-mod26-graphiso-qualification/{\"left_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"mapping\":[0,1,5,2,3,4],\"n\":6,\"right_edges\":[[0,1],[1,5],[2,3],[2,5],[3,4]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261026-r06-graph-isomorphism-0069", "batch_id": "batch-20261013-advance-20261026-r06-graph-isomorphism-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "fb4488467b13792a05bf2a2d68f4ccd4704901bda2d6562f18d0a55eae954db4", "family_id": "qf-mod26-graphiso-qualification", "seed": 202610266069, "semantic_template": "qf-mod26-graphiso-qualification/{\"left_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"mapping\":[0,1,5,2,3,4],\"n\":6,\"right_edges\":[[0,1],[1,5],[2,3],[2,5],[3,4]]}" }, "fields": { "title": "Conjoin Bijectivity with the Complete Adjacency-Iff Audit", "stem": "Use complete simple undirected graphs on Fin(6)={0,…,5} with source edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]], target edges=[[0, 1], [1, 5], [2, 5], [2, 3], [3, 4]] and complete vertex map=[0, 1, 5, 2, 3, 4] in source-label order. Loops and duplicate undirected edges are forbidden. Isomorphism requires both bijectivity and adjacency iff on every unordered pair, including nonedges; neither obligation replaces the other. Give (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?).", "answer": { "kind": "expression", "value": "(6, [0, 1, 5, 2, 3, 4], Yes, [], Yes)", "explanation": "Evaluate the two obligations independently, retaining bad-pair records even if bijectivity fails. Only their conjunction certifies this declared map as an isomorphism. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 5, 2, 3, 4], Yes, [], Yes)." }, "skills": [ "Conjoin Bijectivity with the Complete Adjacency-Iff Audit" ], "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": "完整有限簡單圖與明示頂點映射。操作=graphiso-qualification；資料={\"n\":6,\"left_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"right_edges\":[[0,1],[1,5],[2,5],[2,3],[3,4]],\"mapping\":[0,1,5,2,3,4]}。請依 [n,mapping,bijective,bad_pairs,isomorphism] 欄序回答。", "en": "Use complete simple undirected graphs on Fin(6)={0,…,5} with source edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]], target edges=[[0, 1], [1, 5], [2, 5], [2, 3], [3, 4]] and complete vertex map=[0, 1, 5, 2, 3, 4] in source-label order. Loops and duplicate undirected edges are forbidden. Isomorphism requires both bijectivity and adjacency iff on every unordered pair, including nonedges; neither obligation replaces the other. Give (n,mapping,bijective?,bad [a,b,source adjacent?,mapped target adjacent?] records,isomorphism?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source graph:carrier Fin(6);raw undirected edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]", "after": "canonical simple edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]];loops and repeated undirected edges absent", "relation": "implies", "rule_id": "complete_simple_graph_canonical_edges", "conditions": [], "explanation": { "zh-Hant": "每條邊端點正規化小者在前再排序；反向重複仍是重複，不自行刪掉非法紀錄；保留所有孤立頂點。", "en": "Normalize each edge with its smaller endpoint first,then sort.Reversed duplicates remain invalid rather than being silently removed.Preserve every isolated vertex." }, "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": "target graph:carrier Fin(6);raw undirected edges=[[0, 1], [1, 5], [2, 5], [2, 3], [3, 4]]", "after": "canonical simple edges=[[0, 1], [1, 5], [2, 3], [2, 5], [3, 4]];loops and repeated undirected edges absent", "relation": "implies", "rule_id": "complete_simple_graph_canonical_edges", "conditions": [], "explanation": { "zh-Hant": "每條邊端點正規化小者在前再排序；反向重複仍是重複，不自行刪掉非法紀錄；保留所有孤立頂點。", "en": "Normalize each edge with its smaller endpoint first,then sort.Reversed duplicates remain invalid rather than being silently removed.Preserve every isolated vertex." }, "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": "complete mapping=[0, 1, 5, 2, 3, 4];target label=0", "after": "source fiber=[0];exactly one=是", "relation": "implies", "rule_id": "finite_vertex_map_each_target_fiber", "conditions": [], "explanation": { "zh-Hant": "同樣有限大小的完整總映射，要每個目標標籤恰有一個原像才雙射，不把合型當雙射。", "en": "For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity." }, "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": "S004", "uses": [ "G001", "S003" ], "before": "complete mapping=[0, 1, 5, 2, 3, 4];target label=1", "after": "source fiber=[1];exactly one=是", "relation": "implies", "rule_id": "finite_vertex_map_each_target_fiber", "conditions": [], "explanation": { "zh-Hant": "同樣有限大小的完整總映射，要每個目標標籤恰有一個原像才雙射，不把合型當雙射。", "en": "For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "complete mapping=[0, 1, 5, 2, 3, 4];target label=2", "after": "source fiber=[3];exactly one=是", "relation": "implies", "rule_id": "finite_vertex_map_each_target_fiber", "conditions": [], "explanation": { "zh-Hant": "同樣有限大小的完整總映射，要每個目標標籤恰有一個原像才雙射，不把合型當雙射。", "en": "For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity." }, "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": "S006", "uses": [ "G001", "S005" ], "before": "complete mapping=[0, 1, 5, 2, 3, 4];target label=3", "after": "source fiber=[4];exactly one=是", "relation": "implies", "rule_id": "finite_vertex_map_each_target_fiber", "conditions": [], "explanation": { "zh-Hant": "同樣有限大小的完整總映射，要每個目標標籤恰有一個原像才雙射，不把合型當雙射。", "en": "For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity." }, "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": "S007", "uses": [ "G001", "S006" ], "before": "complete mapping=[0, 1, 5, 2, 3, 4];target label=4", "after": "source fiber=[5];exactly one=是", "relation": "implies", "rule_id": "finite_vertex_map_each_target_fiber", "conditions": [], "explanation": { "zh-Hant": "同樣有限大小的完整總映射，要每個目標標籤恰有一個原像才雙射，不把合型當雙射。", "en": "For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "complete mapping=[0, 1, 5, 2, 3, 4];target label=5", "after": "source fiber=[2];exactly one=是", "relation": "implies", "rule_id": "finite_vertex_map_each_target_fiber", "conditions": [], "explanation": { "zh-Hant": "同樣有限大小的完整總映射，要每個目標標籤恰有一個原像才雙射，不把合型當雙射。", "en": "For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "n=6;all target fibers=[[0], [1], [3], [4], [5], [2]]", "after": "bijective=是;empty carrier uses unique empty map", "relation": "implies", "rule_id": "whole_bijection_including_empty_carrier", "conditions": [], "explanation": { "zh-Hant": "空載體唯一空映射雙射是合法邊界；此步只核映射責任，沒有核任何鄰接性。", "en": "The unique empty map is bijective on empty carriers.This checks the mapping obligation only,not adjacency." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "source unordered pair=[0, 1];mapped pair=[0, 1]", "after": "source adjacent=是;target adjacent=是;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "source unordered pair=[0, 2];mapped pair=[0, 5]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S012", "uses": [ "G001", "S011" ], "before": "source unordered pair=[0, 3];mapped pair=[0, 2]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "source unordered pair=[0, 4];mapped pair=[0, 3]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S014", "uses": [ "G001", "S013" ], "before": "source unordered pair=[0, 5];mapped pair=[0, 4]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S015", "uses": [ "G001", "S014" ], "before": "source unordered pair=[1, 2];mapped pair=[1, 5]", "after": "source adjacent=是;target adjacent=是;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S016", "uses": [ "G001", "S015" ], "before": "source unordered pair=[1, 3];mapped pair=[1, 2]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S017", "uses": [ "G001", "S016" ], "before": "source unordered pair=[1, 4];mapped pair=[1, 3]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S018", "uses": [ "G001", "S017" ], "before": "source unordered pair=[1, 5];mapped pair=[1, 4]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S019", "uses": [ "G001", "S018" ], "before": "source unordered pair=[2, 3];mapped pair=[5, 2]", "after": "source adjacent=是;target adjacent=是;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S020", "uses": [ "G001", "S019" ], "before": "source unordered pair=[2, 4];mapped pair=[5, 3]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S021", "uses": [ "G001", "S020" ], "before": "source unordered pair=[2, 5];mapped pair=[5, 4]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S022", "uses": [ "G001", "S021" ], "before": "source unordered pair=[3, 4];mapped pair=[2, 3]", "after": "source adjacent=是;target adjacent=是;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S023", "uses": [ "G001", "S022" ], "before": "source unordered pair=[3, 5];mapped pair=[2, 4]", "after": "source adjacent=否;target adjacent=否;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S024", "uses": [ "G001", "S023" ], "before": "source unordered pair=[4, 5];mapped pair=[3, 4]", "after": "source adjacent=是;target adjacent=是;iff=是", "relation": "implies", "rule_id": "graph_every_edge_and_nonedge_iff", "conditions": [], "explanation": { "zh-Hant": "每個無序來源對都比兩側真假，包含來源非邊；映射碰撞時目標對角不會是簡單圖邊。即使雙射失敗仍保留完整鄰接失敗清單。", "en": "Compare both truth values on every unordered source pair,including source nonedges.A collision maps to a target diagonal,which is not an edge in a simple graph.Retain the full adjacency failure list even if bijectivity fails." }, "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": "S025", "uses": [ "G001", "S024" ], "before": "mapping bijective=是;all bad adjacency pairs=[]", "after": "adjacency iff=是;specified map isomorphism=是", "relation": "implies", "rule_id": "graph_bijection_and_adjacency_separate_conjunction", "conditions": [], "explanation": { "zh-Hant": "雙射與完整鄰接iff是不同責任；不讓只有其中一條就變成同構。此處只驗明示映射，不是搜尋所有映射。", "en": "Bijectivity and complete adjacency iff are separate obligations.Neither alone certifies isomorphism.This audits the declared map rather than searching all maps." }, "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": [ 6, [ 0, 1, 5, 2, 3, 4 ], true, [], true ] }, "conclusion": { "canonical_value": "(6, [0, 1, 5, 2, 3, 4], 是, [], 是)", "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-20261013-advance-20261026-r06-graph-isomorphism-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261026-r06-graph-isomorphism-foundations/questions.jsonl", "batch_sha256": "894a25ae8120f126dba0bbf73c146628348596e72b997643fe14c5a5779a1fee", "original_line": 69, "original_line_sha256": "107a98f9bbb67652853b801750bce0fdf10eb4781c872551f15704798aacdf1b", "question_compact_sha256": "fb4488467b13792a05bf2a2d68f4ccd4704901bda2d6562f18d0a55eae954db4", "english_row_compact_sha256": "c6a8c68034bb454b5c897ed6fad3040349b8e7bce49190a921a394ca3fc181f4", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261026-r06-graph-isomorphism-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261026-r06-graph-isomorphism-foundations.en.jsonl" }, "mko_bindings": { "mko-graph-isomorphism": { "sha256": "36b491efcb2f7e6297c71cc86792ccbd2cc61cace4f61f1b9324257b32a5d09f", "english_sha256": "ec4fb7d7ac3e76e9a33d43225389ae22fde71112b0fa77885b4a4282b89c7c54", "original_path": "public/data/mko/graph-isomorphism.json", "english_path": "public/data/i18n/editions/mko-graph-isomorphism.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "c6ae588be9bc8b3dd8461be32c073abfa25a0f827b4602cf28bf291ce532d507", "original_candidate_sha256": "3c49df31b1f325e97cf828abc07780a7be3090603ac8607a3f6a86e5f1588603", "original_candidate_line_sha256": "3c49df31b1f325e97cf828abc07780a7be3090603ac8607a3f6a86e5f1588603" }, "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-20261026-006-p24", "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": "f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff", "question_first_witness_sha256": "49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 25, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "bd1c4f4faa99a28eea501725a8d29ee2270aa441115ba351f9cf9d49c4d97ea9", "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." } }