{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261017-r05-graph-0026",
  "pilot_number": "R0015-08026",
  "number": 8026,
  "family_id": "qf-density17-graph-complete-neighbor-partitions",
  "title": {
    "zh-Hant": "第 8026 題：每頂點鄰居與非鄰居",
    "en": "Question 8026: Every Vertex Neighbor Partition and Isolated Vertices"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261017-r05-graph-0026",
    "batch_id": "batch-20261012-advance-20261017-r05-graph-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-graph"
    ],
    "title_zh": "每頂點鄰居與非鄰居",
    "stem_zh": "簡單無向圖，Fin頂點從0起。V=Fin(0)=∅，無向邊列項=[]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。",
    "answer": {
      "kind": "expression",
      "value": "(0, [], [], [], [])",
      "explanation_zh": "離散結構核對=(0, [], [], [], [])；由邊集建立鄰居；在其他頂點中取補集，空鄰居即孤立。；有限精確核對不是一般形式證明；無自環端點已合型；反向或重複邊不增加邊；不由畫法推相鄰。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "每頂點鄰居與非鄰居"
      ]
    },
    "variant": {
      "family_id": "qf-density17-graph-complete-neighbor-partitions",
      "seed": 202610175026,
      "parameters": {
        "n": 0,
        "edges": [],
        "domain": "Fin_simple_undirected_graph",
        "operation": "graph-complete-neighbor-partitions",
        "semantic_template": "qf-density17-graph-complete-neighbor-partitions/{\"edges\":[],\"n\":0}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261017-r05-graph-0026",
    "batch_id": "batch-20261012-advance-20261017-r05-graph-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ee62368407076b3d1e851789f4f51d43d0f513d59390a504e36bf1d7924647bf",
      "family_id": "qf-density17-graph-complete-neighbor-partitions",
      "seed": 202610175026,
      "semantic_template": "qf-density17-graph-complete-neighbor-partitions/{\"edges\":[],\"n\":0}"
    },
    "fields": {
      "title": "Every Vertex Neighbor Partition and Isolated Vertices",
      "stem": "Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices).",
      "answer": {
        "kind": "expression",
        "value": "(0, [], [], [], [])",
        "explanation": "Partition the other vertices at each original label; isolated vertices have degree zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], [], [], [])."
      },
      "skills": [
        "Every Vertex Neighbor Partition and Isolated Vertices"
      ],
      "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": "簡單無向圖，Fin頂點從0起。V=Fin(0)=∅，無向邊列項=[]；各邊兩端皆合法且相異。答 (n, 正規邊集, 全部鄰居列表, 全部非鄰居列表, 孤立頂點)。非鄰居排除頂點自己，每列表遞增。",
        "en": "Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. Give (n, normalized edges, each vertex’s neighbors, each vertex’s distinct nonneighbors excluding itself, isolated vertices)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "n=0;raw edges=[]",
      "after": "canonical edges=[]",
      "relation": "implies",
      "rule_id": "simple_graph_edge_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整張簡單無向圖先查端點合型且無自環；邊兩端排序、反向與重複邊只算一次，不以畫法猜相鄰。",
        "en": "Validate every simple undirected edge for typed distinct endpoints.Sort endpoints and count reversed or repeated edges once;do not infer adjacency from a drawing."
      },
      "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": "vertices=[];neighbors=[]",
      "after": "nonneighbors excluding self=[];isolated=[]",
      "relation": "implies",
      "rule_id": "neighbor_complement_excludes_self",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個頂點的鄰居與非鄰居分割其他頂點，不把自己列為非鄰居；度0正是孤立。",
        "en": "For each vertex,neighbors and nonneighbors partition the other vertices;do not include the vertex itself among nonneighbors.Degree0 means isolated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      0,
      [],
      [],
      [],
      []
    ]
  },
  "conclusion": {
    "canonical_value": "(0, [], [], [], [])",
    "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-20261012-advance-20261017-r05-graph-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261017-r05-graph-foundations/questions.jsonl",
      "batch_sha256": "a86da678771d2977669cd5e9bbbbc7d4a9dde189d293af5c1f2efa0ba35b82ce",
      "original_line": 26,
      "original_line_sha256": "1e9342ec439cb05ca9deddfd12619eaf4442416d39718a2ac5d83fdc95285585",
      "question_compact_sha256": "ee62368407076b3d1e851789f4f51d43d0f513d59390a504e36bf1d7924647bf",
      "english_row_compact_sha256": "028903f7f0b7bd1817d23aa61e52030791d5b8c2898651860f67d8864a246937",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261017-r05-graph-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261017-r05-graph-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-graph": {
        "sha256": "47d40107cd3836bb878ae46e5c4766288abb1b91e480c38ed14f61f749d27d55",
        "english_sha256": "985d93ed708f51da0e885228658ca76c7075f5bf67bb09b35d10745db33dc847",
        "original_path": "public/data/mko/graph.json",
        "english_path": "public/data/i18n/editions/mko-graph.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "ca1078dc467a0f3769865919aadc46c87712d098f1dfbde90f787471e22bcfec",
    "original_candidate_sha256": "85099de386bf25c310b65d802b7e3f184b17356d740825292607c2a6925d9127",
    "original_candidate_line_sha256": "85099de386bf25c310b65d802b7e3f184b17356d740825292607c2a6925d9127"
  },
  "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-20261026-005-p16",
    "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": "6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18",
      "question_first_witness_sha256": "e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 2,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4a0c0bcfd3dd6cb805b9c3daf48d3a0efefc28bf6dd459da54b4a27835a55376",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}