{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261106-r01-chromatic-number-0028", "pilot_number": "R0037-19478", "number": 19478, "family_id": "qf-mod37-chromatic-upper-bound", "title": { "zh-Hant": "第 19478 題：色數", "en": "Question 19478: A Supplied k-Palette Is a Candidate Upper Certificate" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261106-r01-chromatic-number-0028", "batch_id": "batch-20261014-advance-20261106-r01-chromatic-number-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-chromatic-number" ], "title_zh": "色數", "stem_zh": "完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(3, 是, 是, 是, chi_at_most_k_not_equality)", "explanation_zh": "Wave5封閉層核對=(3, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "色數" ] }, "variant": { "family_id": "qf-mod37-chromatic-upper-bound", "seed": 202611061028, "parameters": { "n": 6, "edges": [ [ 0, 1 ], [ 0, 3 ], [ 0, 5 ], [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 4, 5 ] ], "colors": [ 0, 1, 0, 1, 0, 1 ], "k": 3, "domain": "Complete_finite_simple_graph_coloring", "operation": "chromatic-upper-bound", "semantic_template": "qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}" } }, "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-r01-chromatic-number-0028", "batch_id": "batch-20261014-advance-20261106-r01-chromatic-number-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "770361f06560c0dbbfaa4e037a72a8b2da7c9c3287baad6d1061094a64bf0734", "family_id": "qf-mod37-chromatic-upper-bound", "seed": 202611061028, "semantic_template": "qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}" }, "fields": { "title": "A Supplied k-Palette Is a Candidate Upper Certificate", "stem": "Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).", "answer": { "kind": "expression", "value": "(3, Yes, Yes, Yes, chi_at_most_k_not_equality)", "explanation": "Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality)." }, "skills": [ "A Supplied k-Palette Is a Candidate Upper Certificate" ], "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、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。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, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=chromatic-upper-bound;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[k, proper, palette_within_k, candidate_upper_bound_certificate, 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；本頁直接列原始欄位順序，不需JavaScript或查外部表。欄位名字本身不自動是數學證據，No依各題的聲稱範圍解讀。", "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific 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": "ACTUAL complete finite carrier Fin(6);ALL original canonical simple undirected edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]", "after": "use full original vertex/edge data;exact search domain n<=9;empty carrier and isolated vertices RETAINED", "relation": "implies", "rule_id": "chromatic_complete_graph_not_semantic_color_relabeling", "conditions": [], "explanation": { "zh-Hant": "頂點與邊完整且索引原封不動；此題族含n=0，不能套前題n至少1或把孤立點刪掉。後面實際palette標籤不能被semantic改名取代。", "en": "Retain every original vertex and edge.This family includes n=0 rather than importing a previous nonempty-carrier guard or deleting isolates.Actual palette labels must not be replaced by semantic relabeling." }, "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": "original edge index=0;actual endpoint vertices=[0, 1];ORIGINAL palette labels=[0, 1]", "after": "endpoint labels different=是", "relation": "implies", "rule_id": "chromatic_every_original_edge_properness", "conditions": [], "explanation": { "zh-Hant": "逐完整原邊查不同色，數字標籤只表示顏色的同異，不是色數。某次著色失敗不自動表示不存在別的合法著色。", "en": "Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring." }, "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": "original edge index=1;actual endpoint vertices=[0, 3];ORIGINAL palette labels=[0, 1]", "after": "endpoint labels different=是", "relation": "implies", "rule_id": "chromatic_every_original_edge_properness", "conditions": [], "explanation": { "zh-Hant": "逐完整原邊查不同色，數字標籤只表示顏色的同異，不是色數。某次著色失敗不自動表示不存在別的合法著色。", "en": "Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring." }, "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": "original edge index=2;actual endpoint vertices=[0, 5];ORIGINAL palette labels=[0, 1]", "after": "endpoint labels different=是", "relation": "implies", "rule_id": "chromatic_every_original_edge_properness", "conditions": [], "explanation": { "zh-Hant": "逐完整原邊查不同色，數字標籤只表示顏色的同異，不是色數。某次著色失敗不自動表示不存在別的合法著色。", "en": "Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring." }, "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": "original edge index=3;actual endpoint vertices=[1, 2];ORIGINAL palette labels=[1, 0]", "after": "endpoint labels different=是", "relation": "implies", "rule_id": "chromatic_every_original_edge_properness", "conditions": [], "explanation": { "zh-Hant": "逐完整原邊查不同色，數字標籤只表示顏色的同異，不是色數。某次著色失敗不自動表示不存在別的合法著色。", "en": "Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring." }, "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": "original edge index=4;actual endpoint vertices=[2, 3];ORIGINAL palette labels=[0, 1]", "after": "endpoint labels different=是", "relation": "implies", "rule_id": "chromatic_every_original_edge_properness", "conditions": [], "explanation": { "zh-Hant": "逐完整原邊查不同色，數字標籤只表示顏色的同異，不是色數。某次著色失敗不自動表示不存在別的合法著色。", "en": "Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring." }, "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": "original edge index=5;actual endpoint vertices=[3, 4];ORIGINAL palette labels=[1, 0]", "after": "endpoint labels different=是", "relation": "implies", "rule_id": "chromatic_every_original_edge_properness", "conditions": [], "explanation": { "zh-Hant": "逐完整原邊查不同色，數字標籤只表示顏色的同異，不是色數。某次著色失敗不自動表示不存在別的合法著色。", "en": "Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring." }, "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": "original edge index=6;actual endpoint vertices=[4, 5];ORIGINAL palette labels=[0, 1]", "after": "endpoint labels different=是", "relation": "implies", "rule_id": "chromatic_every_original_edge_properness", "conditions": [], "explanation": { "zh-Hant": "逐完整原邊查不同色，數字標籤只表示顏色的同異，不是色數。某次著色失敗不自動表示不存在別的合法著色。", "en": "Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring." }, "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": "chromatic distinct used label count := sum(1, 1)", "after": "chromatic distinct used label count = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留參數、樣本、目標與分母，不交換未授權的聲稱。", "en": "Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 chromatic distinct used label count，不以小數近似替代。", "en": "Recompute chromatic distinct used label count with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "chromatic distinct used label count" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "ORIGINAL complete color array=[0, 1, 0, 1, 0, 1];ALL edge properness flags=[是, 是, 是, 是, 是, 是, 是]", "after": "increasingly ordered DISTINCT used labels=[0, 1];distinct count=2;proper=是;not max_label+1", "relation": "implies", "rule_id": "chromatic_distinct_label_count_not_palette_range_or_minimum", "conditions": [], "explanation": { "zh-Hant": "色數量是distinct labels數，不是最大標籤加1；空陣列給0。proper coloring若有效只給一個上界，不因greedy或label小就證最小。", "en": "Count distinct labels rather than max_label+1,with0 for an empty array.A proper supplied coloring gives an upper bound,not minimum status from greediness or small labels." }, "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": "requested ACTUAL palette[0,3);ORIGINAL labels=[0, 1, 0, 1, 0, 1];per-vertex label