{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261105-r04-eulerian-trail-0094",
  "pilot_number": "R0036-18994",
  "number": 18994,
  "family_id": "qf-mod36-euler-zero-edge-convention",
  "title": {
    "zh-Hant": "第 18994 題：Euler trail與circuit",
    "en": "Question 18994: A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261105-r04-eulerian-trail-0094",
    "batch_id": "batch-20261014-advance-20261105-r04-eulerian-trail-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-eulerian-trail"
    ],
    "title_zh": "Euler trail與circuit",
    "stem_zh": "完整有限簡單無向圖、edge support與明示walk。操作=euler-zero-edge-convention；資料={\"n\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(19, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)",
      "explanation_zh": "Wave5第五層核對=(19, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；依完整明示資料獨立重算；trail可重複頂點但不可重複邊；0或2個奇度判準還要求非孤立頂點同一分量；孤立點不破壞非零邊support；零邊非空圖採length-zero circuit且不推Hamilton性質。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "Euler trail與circuit"
      ]
    },
    "variant": {
      "family_id": "qf-mod36-euler-zero-edge-convention",
      "seed": 202611054094,
      "parameters": {
        "n": 19,
        "domain": "Complete_finite_simple_undirected_graph",
        "operation": "euler-zero-edge-convention",
        "semantic_template": "qf-mod36-euler-zero-edge-convention/{\"n\":19}"
      }
    },
    "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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r04-eulerian-trail-0094",
    "batch_id": "batch-20261014-advance-20261105-r04-eulerian-trail-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2c7c9cf7446b82fbaaf9c7ef74c1fa2098638aa3bf5dae367cacc8fd54bcebdb",
      "family_id": "qf-mod36-euler-zero-edge-convention",
      "seed": 202611054094,
      "semantic_template": "qf-mod36-euler-zero-edge-convention/{\"n\":19}"
    },
    "fields": {
      "title": "A Nonempty Vertex Set Admits the Declared Length-Zero Circuit",
      "stem": "Use vertex carrier Fin(19)={0,…,18} and no edges, with n=19≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(19, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)",
        "explanation": "Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (19, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)."
      },
      "skills": [
        "A Nonempty Vertex Set Admits the Declared Length-Zero Circuit"
      ],
      "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": "完整有限簡單無向圖、edge support與明示walk。操作=euler-zero-edge-convention；資料={\"n\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use vertex carrier Fin(19)={0,…,18} and no edges, with n=19≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=euler-zero-edge-convention;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[vertex_count, edge_count, length_zero_circuit, convention, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。",
        "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence."
      },
      "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 nonempty original carrier Fin(19);NO edges",
      "after": "choose single-vertex walk[0],length0;all edges used exactly once vacuously;length_zero_circuit_on_nonempty_vertex_set",
      "relation": "implies",
      "rule_id": "euler_zero_edge_nonempty_carrier_explicit_convention",
      "conditions": [],
      "explanation": {
        "zh-Hant": "n≥1才可選頂點0。沒有邊時單頂點walk長0，依明示慣例是circuit；不要求全部孤立頂點在walk出現，也不納入空頂點圖或推出Hamilton。",
        "en": "Since n≥1 vertex0 is available.A one-vertex walk has length0 and is a circuit by the stated convention.It need not visit all isolated vertices,does not admit the empty-vertex graph,and gives no Hamiltonian claim."
      },
      "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": [
      19,
      0,
      true,
      "length_zero_circuit_on_nonempty_vertex_set",
      "empty_vertex_graph_out_of_scope"
    ]
  },
  "conclusion": {
    "canonical_value": "(19, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)",
    "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-20261105-r04-eulerian-trail-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r04-eulerian-trail-foundations/questions.jsonl",
      "batch_sha256": "290234a8be5114292ccdef5d758669877237520e20f4d7270260f7f29b4c745d",
      "original_line": 94,
      "original_line_sha256": "69d1b1d025bcf3d632a759f8e96e8257519c39da08c5e7123e6bf3c89f49f7a3",
      "question_compact_sha256": "2c7c9cf7446b82fbaaf9c7ef74c1fa2098638aa3bf5dae367cacc8fd54bcebdb",
      "english_row_compact_sha256": "5e93f3f760e6f3e577e7e19cfaddea3c05a6b1749d53158fb06870a96c889db5",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r04-eulerian-trail-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r04-eulerian-trail-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-eulerian-trail": {
        "sha256": "bd4c7accbd4091e00d7a37abe72e796dc663a93d6cdb230cfaa24a0ec125ad41",
        "english_sha256": "e03f13cd5f3980be232f919922b722c842b0d6d7a7dca8b3ba1e5ea56bf7f340",
        "original_path": "public/data/mko/eulerian-trail.json",
        "english_path": "public/data/i18n/editions/mko-eulerian-trail.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994",
    "original_candidate_sha256": "be907d1172be816cebe8225343d44cc6304e5ab963fc083dce4c3672d87fd8e5",
    "original_candidate_line_sha256": "be907d1172be816cebe8225343d44cc6304e5ab963fc083dce4c3672d87fd8e5"
  },
  "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-p37",
    "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": "efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301",
      "question_first_witness_sha256": "acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2"
    },
    "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": "87fdf47f01a9836501076d9ae3c3467dda7eb21ed51261322fa1a86efcc9af12",
      "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."
  }
}