{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261106-r01-chromatic-number-0077",
  "pilot_number": "R0037-19427",
  "number": 19427,
  "family_id": "qf-mod37-chromatic-minimum-boundary",
  "title": {
    "zh-Hant": "第 19427 題：色數",
    "en": "Question 19427: Equality of k with the Minimum Is Not Validity of This Coloring"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261106-r01-chromatic-number-0077",
    "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-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,1,2],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)",
      "explanation_zh": "Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；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-minimum-boundary",
      "seed": 202611061077,
      "parameters": {
        "n": 6,
        "edges": [
          [
            0,
            1
          ],
          [
            0,
            2
          ],
          [
            0,
            5
          ],
          [
            1,
            2
          ],
          [
            2,
            3
          ],
          [
            3,
            4
          ],
          [
            4,
            5
          ]
        ],
        "colors": [
          0,
          1,
          2,
          0,
          1,
          2
        ],
        "k": 4,
        "domain": "Complete_finite_simple_graph_coloring",
        "operation": "chromatic-minimum-boundary",
        "semantic_template": "qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"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-0077",
    "batch_id": "batch-20261014-advance-20261106-r01-chromatic-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8e88aecacb5206244a7c7e2a3467292c822c5e85b850c9e9a6161153c7e6ad55",
      "family_id": "qf-mod37-chromatic-minimum-boundary",
      "seed": 202611061077,
      "semantic_template": "qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"
    },
    "fields": {
      "title": "Equality of k with the Minimum Is Not Validity of This Coloring",
      "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], [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, 2, 0, 1, 2] 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. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)",
        "explanation": "Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)."
      },
      "skills": [
        "Equality of k with the Minimum Is Not Validity of This Coloring"
      ],
      "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-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,1,2],\"k\":4}。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], [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, 2, 0, 1, 2] 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. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=chromatic-minimum-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, 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, 2], [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, 2];ORIGINAL palette labels=[0, 2]",
      "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, 2]",
      "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, 2]",
      "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=[2, 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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "original edge index=5;actual endpoint vertices=[3, 4];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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "original edge index=6;actual endpoint vertices=[4, 5];ORIGINAL palette labels=[1, 2]",
      "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, 1)",
      "after": "chromatic distinct used label count = 3",
      "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",
          "1"
        ],
        "output": "3",
        "label": "chromatic distinct used label count"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ORIGINAL complete color array=[0, 1, 2, 0, 1, 2];ALL edge properness flags=[是, 是, 是, 是, 是, 是, 是]",
      "after": "increasingly ordered DISTINCT used labels=[0, 1, 2];distinct count=3;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[0,4);ORIGINAL labels=[0, 1, 2, 0, 1, 2];per-vertex label<k flags=[是, 是, 是, 是, 是, 是];proper=是",
      "after": "within requested palette=是;THIS candidate upper certificate=是;distinct used-label count=3 is a DIFFERENT test",
      "relation": "implies",
      "rule_id": "chromatic_original_palette_range_not_semantic_renaming",
      "conditions": [],
      "explanation": {
        "zh-Hant": "即使用色數量≤k，若原標籤超出[0,k)仍不能讓這個指定palette證書通過。semantic改名不能換掉原資料；失敗只否決此候選，不宣稱圖無k-coloring。",
        "en": "Even if at most k distinct labels are used,the supplied palette certificate fails when original labels lie outside[0,k).Semantic relabeling must not replace source data.Failure rejects this candidate,not existence of another k-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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ACTUAL graph Fin(6);try palette[0,1);original vertex order0..5",
      "after": "search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune",
      "relation": "implies",
      "rule_id": "chromatic_complete_lex_search_not_greedy_optimality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個k依序完整搜尋，不把一次greedy失敗當不可著色；每層原vertex順序與遞增顏色決定來源第一個lex witness。實際邊衝突可剪枝，否則必繼續或完整回退。",
        "en": "Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack."
      },
      "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": "palette=1;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "palette=1;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[0];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "palette=1;original vertex=1;original fixed prefix=[0];ALL colors0..0 attempted",
      "after": "no valid completion of this prefix in this palette",
      "relation": "implies",
      "rule_id": "chromatic_full_palette_exhaustion_at_prefix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此vertex所有合法palette標籤都已試完，故固定prefix沒有完成解；從根節點的此結論才排除整個k palette。",
        "en": "All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette."
      },
      "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": "palette=1;original prefix=[];vertex=0;chosen color=0;ALL its later completions exhausted",
      "after": "restore this unassigned coordinate and try the next ascending color",
      "relation": "implies",
      "rule_id": "chromatic_exhausted_branch_backtracking_not_one_bad_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "子樹完整失敗才回退，此支所有後續選色都已被實際衝突或其完整子樹排除，不是測一次就換色。",
        "en": "Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt."
      },
      "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": "palette=1;original vertex=0;original fixed prefix=[];ALL colors0..0 attempted",
      "after": "no valid completion of this prefix in this palette",
      "relation": "implies",
      "rule_id": "chromatic_full_palette_exhaustion_at_prefix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此vertex所有合法palette標籤都已試完，故固定prefix沒有完成解；從根節點的此結論才排除整個k palette。",
        "en": "All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette."
      },
      "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": "palette[0,1) searched from EMPTY root prefix with full branch exclusions",
      "after": "NO proper coloring exists in this palette;any smaller palette is a subset of it",
      "relation": "implies",
      "rule_id": "chromatic_actual_lower_palette_exhaustion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整根樹耗盡才給真正lower證據；任意更少顏色可嵌入此palette，故也被排除，不依賴任一greedy路徑。",
        "en": "Exhausting the complete root tree supplies the lower certificate.Any smaller palette embeds in this one and is excluded as well;no greedy path establishes this conclusion."
      },
      "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": "ACTUAL graph Fin(6);try palette[0,2);original vertex order0..5",
      "after": "search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune",
      "relation": "implies",
      "rule_id": "chromatic_complete_lex_search_not_greedy_optimality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個k依序完整搜尋，不把一次greedy失敗當不可著色；每層原vertex順序與遞增顏色決定來源第一個lex witness。實際邊衝突可剪枝，否則必繼續或完整回退。",
        "en": "Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack."
      },
      "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": "palette=2;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "palette=2;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[0];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "palette=2;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=1",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "palette=2;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[0];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "palette=2;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=1",
      "after": "ALL earlier adjacent vertices already using this label=[1];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "palette=2;original vertex=2;original fixed prefix=[0, 1];ALL colors0..1 attempted",
      "after": "no valid completion of this prefix in this palette",
      "relation": "implies",
      "rule_id": "chromatic_full_palette_exhaustion_at_prefix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此vertex所有合法palette標籤都已試完，故固定prefix沒有完成解；從根節點的此結論才排除整個k palette。",
        "en": "All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette."
      },
      "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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "palette=2;original prefix=[0];vertex=1;chosen color=1;ALL its later completions exhausted",
      "after": "restore this unassigned coordinate and try the next ascending color",
      "relation": "implies",
      "rule_id": "chromatic_exhausted_branch_backtracking_not_one_bad_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "子樹完整失敗才回退，此支所有後續選色都已被實際衝突或其完整子樹排除，不是測一次就換色。",
        "en": "Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "palette=2;original vertex=1;original fixed prefix=[0];ALL colors0..1 attempted",
      "after": "no valid completion of this prefix in this palette",
      "relation": "implies",
      "rule_id": "chromatic_full_palette_exhaustion_at_prefix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此vertex所有合法palette標籤都已試完，故固定prefix沒有完成解；從根節點的此結論才排除整個k palette。",
        "en": "All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette."
      },
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "palette=2;original prefix=[];vertex=0;chosen color=0;ALL its later completions exhausted",
      "after": "restore this unassigned coordinate and try the next ascending color",
      "relation": "implies",
      "rule_id": "chromatic_exhausted_branch_backtracking_not_one_bad_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "子樹完整失敗才回退，此支所有後續選色都已被實際衝突或其完整子樹排除，不是測一次就換色。",
        "en": "Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt."
      },
      "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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "palette=2;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=1",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "palette=2;original vertex=1;assigned ORIGINAL prefix=[1];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "palette=2;original vertex=2;assigned ORIGINAL prefix=[1, 0];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[1];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "palette=2;original vertex=2;assigned ORIGINAL prefix=[1, 0];try next ascending color=1",
      "after": "ALL earlier adjacent vertices already using this label=[0];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "palette=2;original vertex=2;original fixed prefix=[1, 0];ALL colors0..1 attempted",
      "after": "no valid completion of this prefix in this palette",
      "relation": "implies",
      "rule_id": "chromatic_full_palette_exhaustion_at_prefix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此vertex所有合法palette標籤都已試完，故固定prefix沒有完成解；從根節點的此結論才排除整個k palette。",
        "en": "All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette."
      },
      "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": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "palette=2;original prefix=[1];vertex=1;chosen color=0;ALL its later completions exhausted",
      "after": "restore this unassigned coordinate and try the next ascending color",
      "relation": "implies",
      "rule_id": "chromatic_exhausted_branch_backtracking_not_one_bad_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "子樹完整失敗才回退，此支所有後續選色都已被實際衝突或其完整子樹排除，不是測一次就換色。",
        "en": "Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt."
      },
      "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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "palette=2;original vertex=1;assigned ORIGINAL prefix=[1];try next ascending color=1",
      "after": "ALL earlier adjacent vertices already using this label=[0];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "palette=2;original vertex=1;original fixed prefix=[1];ALL colors0..1 attempted",
      "after": "no valid completion of this prefix in this palette",
      "relation": "implies",
      "rule_id": "chromatic_full_palette_exhaustion_at_prefix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此vertex所有合法palette標籤都已試完，故固定prefix沒有完成解；從根節點的此結論才排除整個k palette。",
        "en": "All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette."
      },
      "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": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "palette=2;original prefix=[];vertex=0;chosen color=1;ALL its later completions exhausted",
      "after": "restore this unassigned coordinate and try the next ascending color",
      "relation": "implies",
      "rule_id": "chromatic_exhausted_branch_backtracking_not_one_bad_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "子樹完整失敗才回退，此支所有後續選色都已被實際衝突或其完整子樹排除，不是測一次就換色。",
        "en": "Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt."
      },
      "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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "palette=2;original vertex=0;original fixed prefix=[];ALL colors0..1 attempted",
      "after": "no valid completion of this prefix in this palette",
      "relation": "implies",
      "rule_id": "chromatic_full_palette_exhaustion_at_prefix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此vertex所有合法palette標籤都已試完，故固定prefix沒有完成解；從根節點的此結論才排除整個k palette。",
        "en": "All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette."
      },
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "palette[0,2) searched from EMPTY root prefix with full branch exclusions",
      "after": "NO proper coloring exists in this palette;any smaller palette is a subset of it",
      "relation": "implies",
      "rule_id": "chromatic_actual_lower_palette_exhaustion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整根樹耗盡才給真正lower證據；任意更少顏色可嵌入此palette，故也被排除，不依賴任一greedy路徑。",
        "en": "Exhausting the complete root tree supplies the lower certificate.Any smaller palette embeds in this one and is excluded as well;no greedy path establishes this conclusion."
      },
      "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": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "ACTUAL graph Fin(6);try palette[0,3);original vertex order0..5",
      "after": "search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune",
      "relation": "implies",
      "rule_id": "chromatic_complete_lex_search_not_greedy_optimality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個k依序完整搜尋，不把一次greedy失敗當不可著色；每層原vertex順序與遞增顏色決定來源第一個lex witness。實際邊衝突可剪枝，否則必繼續或完整回退。",
        "en": "Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack."
      },
      "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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "palette=3;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "palette=3;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[0];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "palette=3;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=1",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "palette=3;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[0];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "palette=3;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=1",
      "after": "ALL earlier adjacent vertices already using this label=[1];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "palette=3;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=2",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "palette=3;original vertex=3;assigned ORIGINAL prefix=[0, 1, 2];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "palette=3;original vertex=4;assigned ORIGINAL prefix=[0, 1, 2, 0];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[3];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "palette=3;original vertex=4;assigned ORIGINAL prefix=[0, 1, 2, 0];try next ascending color=1",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "palette=3;original vertex=5;assigned ORIGINAL prefix=[0, 1, 2, 0, 1];try next ascending color=0",
      "after": "ALL earlier adjacent vertices already using this label=[0];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "palette=3;original vertex=5;assigned ORIGINAL prefix=[0, 1, 2, 0, 1];try next ascending color=1",
      "after": "ALL earlier adjacent vertices already using this label=[4];admissible extension=否",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "palette=3;original vertex=5;assigned ORIGINAL prefix=[0, 1, 2, 0, 1];try next ascending color=2",
      "after": "ALL earlier adjacent vertices already using this label=[];admissible extension=是",
      "relation": "implies",
      "rule_id": "chromatic_every_prefix_color_conflict_or_recursion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列出此prefix嘗試顏色的全部先前真鄰點衝突；有衝突整個擴充都不合法，無衝突則還須遞迴核完其餘頂點，不把局部通過當全解。",
        "en": "List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution."
      },
      "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": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "FIRST completed vertex-order increasing-color search witness=[0, 1, 2, 0, 1, 2];palette size=3;ALL smaller palettes exhausted",
      "after": "every ORIGINAL edge endpoints different;constructive chi<= 3 and exhaustive chi>= 3;chi=3;first LEX witness retained",
      "relation": "implies",
      "rule_id": "chromatic_constructive_upper_and_exhausted_lower_tie_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "找到的完整著色逐原邊合法給上界，前面的完整失敗palette給下界；二者相等才是exact最小值。因DFS原順序，這是來源第一個lex witness，非任意同色數解。非空無邊時第一個k=1給全0，不需虛構0色搜尋。",
        "en": "The complete proper original-edge coloring gives the upper bound;the preceding exhaustive failures give the lower bound.Their equality establishes the exact minimum.Fixed DFS order preserves the source's first lexicographic witness,not an arbitrary optimum.A nonempty edgeless graph succeeds first at k=1 with all zeros;no fictitious0-color search is needed."
      },
      "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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "requested k=4;THIS supplied candidate validity=是;SEPARATE exact chi=3;SEPARATE canonical optimum witness=[0, 1, 2, 0, 1, 2]",
      "after": "k equals chromatic number=否;comparison does NOT certify supplied colors;at-most-k and exact-k-minimum are separate",
      "relation": "implies",
      "rule_id": "chromatic_k_equals_chi_independent_of_supplied_candidate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "k與真色數相等可能為Yes而給定colors仍不合法，或合法k palette只給chi≤k而等號False。這些欄位各答各的，不把一個成功flag複製到另一欄。",
        "en": "k may equal the true chromatic number while the supplied coloring fails.Or a valid k-palette may only give chi≤k with equality false.Keep these separate questions and do not copy one successful flag into another."
      },
      "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": [
      4,
      true,
      3,
      false,
      "k_colorable_means_at_most_k_and_greedy_not_optimality_proof"
    ]
  },
  "conclusion": {
    "canonical_value": "(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)",
    "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-r01-chromatic-number-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261106-r01-chromatic-number-foundations/questions.jsonl",
      "batch_sha256": "8aaec2a44fde857f773f9ca294e9c6462b5725323a2150b7c7adf52fce1046d0",
      "original_line": 77,
      "original_line_sha256": "25a48ba223e756349285785b4d9380d27f8b62e8340bcc81d85d89d4bfd318e8",
      "question_compact_sha256": "8e88aecacb5206244a7c7e2a3467292c822c5e85b850c9e9a6161153c7e6ad55",
      "english_row_compact_sha256": "fb780acb634a9f1cf0a4be34cb7389d46c538e6313866b7ad9f3344bdc23b985",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261106-r01-chromatic-number-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261106-r01-chromatic-number-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-chromatic-number": {
        "sha256": "c6461c8fe8f1d88f27aaea185ce8ec6d8d3e4318855946ae3b5ef9daf6f5f109",
        "english_sha256": "c01822b4c5a7c77a39857d06850f1e4afce84eeea3ca3c93ec2fdaab43fec31e",
        "original_path": "public/data/mko/chromatic-number.json",
        "english_path": "public/data/i18n/editions/mko-chromatic-number.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "7be57d88eceb13736454f82dd18bb3c4ec9d30c0367f974dcefdf8990dc4e3cb",
    "original_candidate_sha256": "d49abc06fe258d113179712ad2b0633f3eaf93f43c336f7b973ecd64f1d3af50",
    "original_candidate_line_sha256": "d49abc06fe258d113179712ad2b0633f3eaf93f43c336f7b973ecd64f1d3af50"
  },
  "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-p38",
    "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": "065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80",
      "question_first_witness_sha256": "7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 55,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f0ab8f06b7a63b70267e000bce4d19d1308173e90aae2fa6c38be0b8b2364d5a",
      "arithmetic_steps": 1
    },
    "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."
  }
}