{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261027-r07-graph-walk-simple-path-0069",
  "pilot_number": "R0025-13319",
  "number": 13319,
  "family_id": "qf-mod27-walk-length-endpoints",
  "title": {
    "zh-Hant": "第 13319 題：圖walk與simple path",
    "en": "Question 13319: Separate Candidate Length and Endpoints from Walk Qualification"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261027-r07-graph-walk-simple-path-0069",
    "batch_id": "batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-graph-walk-simple-path"
    ],
    "title_zh": "圖walk與simple path",
    "stem_zh": "有限無向無loop簡單圖與完整頂點序列。操作=walk-length-endpoints；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"vertices\":[3,4,0,1,0]}。請依 [walk,edge_length,start,end,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(是, 4, 3, 0, edge_length_equals_vertex_count_minus_one)",
      "explanation_zh": "Wave4根節點核對=(是, 4, 3, 0, edge_length_equals_vertex_count_minus_one)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；每對連續頂點都須相鄰且長度是邊數；walk可重複但simple path不得重複頂點。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "圖walk與simple path"
      ]
    },
    "variant": {
      "family_id": "qf-mod27-walk-length-endpoints",
      "seed": 202610277069,
      "parameters": {
        "n": 6,
        "edges": [
          [
            0,
            1
          ],
          [
            0,
            2
          ],
          [
            0,
            3
          ],
          [
            0,
            4
          ],
          [
            0,
            5
          ],
          [
            1,
            2
          ],
          [
            1,
            3
          ],
          [
            1,
            4
          ],
          [
            1,
            5
          ],
          [
            2,
            3
          ],
          [
            2,
            4
          ],
          [
            2,
            5
          ],
          [
            3,
            4
          ],
          [
            3,
            5
          ],
          [
            4,
            5
          ]
        ],
        "vertices": [
          3,
          4,
          0,
          1,
          0
        ],
        "domain": "Complete_finite_undirected_walk",
        "operation": "walk-length-endpoints",
        "semantic_template": "qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,4,0,1,0]}"
      }
    },
    "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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式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-20261027-r07-graph-walk-simple-path-0069",
    "batch_id": "batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ea25e325d286190b6c814e03bdff710933cfa67faa5c1db9b4b84b24855f56b1",
      "family_id": "qf-mod27-walk-length-endpoints",
      "seed": 202610277069,
      "semantic_template": "qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,4,0,1,0]}"
    },
    "fields": {
      "title": "Separate Candidate Length and Endpoints from Walk Qualification",
      "stem": "Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.",
      "answer": {
        "kind": "expression",
        "value": "(Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)",
        "explanation": "Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)."
      },
      "skills": [
        "Separate Candidate Length and Endpoints from Walk Qualification"
      ],
      "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": "有限無向無loop簡單圖與完整頂點序列。操作=walk-length-endpoints；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"vertices\":[3,4,0,1,0]}。請依 [walk,edge_length,start,end,boundary] 欄序回答。",
        "en": "Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "carrier Fin(6);raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];original vertex sequence=[3, 4, 0, 1, 0]",
      "after": "canonical undirected simple edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];nonempty typed sequence retained",
      "relation": "implies",
      "rule_id": "walk_simple_graph_and_ordered_candidate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "圖邊正規化無向端點並拒絕loop與反向重複；頂點序列則保留順序與重複，重複不是輸入解析錯，而可能使walk或simple資格失敗。",
        "en": "Normalize undirected graph edges and reject loops/reversed duplicates.Preserve vertex-sequence order and repetitions.Repetition is not a parsing error in a candidate,though it may fail walk or simplicity qualification."
      },
      "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": "proposed step index=0;from=3;to=4;unordered step edge=[3, 4]",
      "after": "edge present=是",
      "relation": "implies",
      "rule_id": "walk_every_consecutive_pair_not_just_vertices",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每兩個連續頂點都須在原圖相鄰；立即重複頂點給對角步而圖無loop，所以該步失敗。方向不改無向邊身份。",
        "en": "Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity."
      },
      "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": "proposed step index=1;from=4;to=0;unordered step edge=[0, 4]",
      "after": "edge present=是",
      "relation": "implies",
      "rule_id": "walk_every_consecutive_pair_not_just_vertices",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每兩個連續頂點都須在原圖相鄰；立即重複頂點給對角步而圖無loop，所以該步失敗。方向不改無向邊身份。",
        "en": "Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity."
      },
      "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": "proposed step index=2;from=0;to=1;unordered step edge=[0, 1]",
      "after": "edge present=是",
      "relation": "implies",
      "rule_id": "walk_every_consecutive_pair_not_just_vertices",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每兩個連續頂點都須在原圖相鄰；立即重複頂點給對角步而圖無loop，所以該步失敗。方向不改無向邊身份。",
        "en": "Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity."
      },
      "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": "proposed step index=3;from=1;to=0;unordered step edge=[0, 1]",
      "after": "edge present=是",
      "relation": "implies",
      "rule_id": "walk_every_consecutive_pair_not_just_vertices",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每兩個連續頂點都須在原圖相鄰；立即重複頂點給對角步而圖無loop，所以該步失敗。方向不改無向邊身份。",
        "en": "Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity."
      },
      "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": "all bad proposed steps=[]",
      "after": "walk=是",
      "relation": "implies",
      "rule_id": "complete_walk_candidate_qualification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整步清單全通過才walk；單頂點無步清單空泛通過，仍是長度0的合法walk。",
        "en": "The whole step list must pass.A singleton has no proposed steps and passes vacuously as a valid length-zero walk."
      },
      "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": "proposed edge length := sub(5, 1)",
      "after": "proposed edge length = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。",
        "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 proposed edge length，不以小數近似替代。",
          "en": "Recompute proposed edge length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "1"
        ],
        "output": "4",
        "label": "proposed edge length"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "candidate vertices=[3, 4, 0, 1, 0];walk=是",
      "after": "proposed edge length=4;start=3;end=0",
      "relation": "implies",
      "rule_id": "candidate_length_endpoints_even_if_walk_invalid",
      "conditions": [],
      "explanation": {
        "zh-Hant": "即使某步不合格，仍按協定保留候選步數與端點，不把這些數值冒稱已證walk。",
        "en": "Even if a step fails,retain the proposed length and endpoints as specified.These values do not certify the candidate as a walk."
      },
      "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": [
      true,
      4,
      3,
      0,
      "edge_length_equals_vertex_count_minus_one"
    ]
  },
  "conclusion": {
    "canonical_value": "(是, 4, 3, 0, edge_length_equals_vertex_count_minus_one)",
    "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-20261027-r07-graph-walk-simple-path-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations/questions.jsonl",
      "batch_sha256": "67197261606ced0d60fcec689034546dca308648cd0928f12df5674ba363e5bb",
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      "question_compact_sha256": "ea25e325d286190b6c814e03bdff710933cfa67faa5c1db9b4b84b24855f56b1",
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      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations.en.jsonl"
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        "original_path": "public/data/mko/graph-walk-simple-path.json",
        "english_path": "public/data/i18n/editions/mko-graph-walk-simple-path.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a02aa2dccf790869cf864b821d0e60119686214561153daba232d921e07e7924",
    "original_candidate_sha256": "f7172416c44b83222149e81d288c6c0f25bce233ff76c36a666e764a06dece97",
    "original_candidate_line_sha256": "f7172416c44b83222149e81d288c6c0f25bce233ff76c36a666e764a06dece97"
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  "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-20261026-007-p26",
    "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": "0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5",
      "question_first_witness_sha256": "ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f583a61e65507ddbdd9b5a1d3dcc0414973af0e1da1a5953cd8c1b7b1c223fb9",
      "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."
  }
}