{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261108-r07-finite-directed-graph-0090",
  "pilot_number": "R0040-20865",
  "number": 20865,
  "family_id": "qf-mod39-digraph-loop-boundary",
  "title": {
    "zh-Hant": "第 20865 題：有限有向圖",
    "en": "Question 20865: A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261108-r07-finite-directed-graph-0090",
    "batch_id": "batch-20261014-advance-20261108-r07-finite-directed-graph-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-directed-graph"
    ],
    "title_zh": "有限有向圖",
    "stem_zh": "完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 0]], 否, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)",
      "explanation_zh": "Wave6第二層核對=([[0, 0]], 否, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限有向圖"
      ]
    },
    "variant": {
      "family_id": "qf-mod39-digraph-loop-boundary",
      "seed": 202611087090,
      "parameters": {
        "n": 5,
        "edges": [
          [
            0,
            0
          ],
          [
            0,
            1
          ],
          [
            1,
            2
          ],
          [
            1,
            3
          ],
          [
            2,
            3
          ],
          [
            2,
            4
          ],
          [
            3,
            0
          ],
          [
            3,
            4
          ],
          [
            4,
            0
          ]
        ],
        "domain": "Complete_finite_loopless_simple_digraph",
        "operation": "digraph-loop-boundary",
        "semantic_template": "qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"
      }
    },
    "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、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、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-20261108-r07-finite-directed-graph-0090",
    "batch_id": "batch-20261014-advance-20261108-r07-finite-directed-graph-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "843a068949dfdb1a13ec51868dd13d0f4d0d144868a7b819e667dad0a3556639",
      "family_id": "qf-mod39-digraph-loop-boundary",
      "seed": 202611087090,
      "semantic_template": "qf-mod39-digraph-loop-boundary/{\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]],\"n\":5}"
    },
    "fields": {
      "title": "A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention",
      "stem": "Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([[0, 0]], No, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)",
        "explanation": "Require at least one loop in this explicit boundary. Report those loops, No qualification under the declared main model and the supplied edge-list in/out counts. A loop contributes once to each directed count here; other loop/multigraph conventions require a different model. The exact result, retaining the requested order, is ([[0, 0]], No, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)."
      },
      "skills": [
        "A Loop Fails This Loop-Free Model, Not Every Possible Digraph Convention"
      ],
      "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ordered-pair edge set與loop-free simple digraph convention。操作=digraph-loop-boundary；資料={\"n\":5,\"edges\":[[0,0],[0,1],[1,2],[1,3],[2,3],[2,4],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (loops, declared_model_qualified, out_degrees, in_degrees, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=digraph-loop-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[loops, declared_model_qualified, out_degrees, in_degrees, 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；本頁直接列來源欄位順序，不需JavaScript或外部表。標籤不自動是數學證據，No保持來源的聲稱範圍。",
        "en": "Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source 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": "ORIGINAL complete directed carrier Fin(5);ORIGINAL lexicographically canonical ORDERED arcs=[[0, 0], [0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 0], [3, 4], [4, 0]]",
      "after": "all endpoints in range and same-direction arcs unique;actual loops=[[0, 0]];temporary loop-boundary extended data,NOT main loop-free qualification",
      "relation": "implies",
      "rule_id": "digraph_ordered_pairs_not_undirected_endpoint_sorting",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只核整張arc列表的from/to排序，不把各arc端點排序。(u,v)與(v,u)不同且可同時存在；main禁止loop，boundary暫收loop用來比對慣例，不作一般digraph不存在的判決。",
        "en": "Check outer from/to order without sorting endpoints inside each arc.(u,v) and(v,u)are distinct and may coexist.The main convention forbids loops;the boundary temporarily accepts them to audit that convention,not to deny other digraph models."
      },
      "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": "digraph vertex0 exact outgoing count := sum(1, 1)",
      "after": "digraph vertex0 exact outgoing count = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex0 exact outgoing count，不以小數近似替代。",
          "en": "Recompute digraph vertex0 exact outgoing count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "digraph vertex0 exact outgoing count"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "digraph vertex0 exact incoming count := sum(1, 1, 1)",
      "after": "digraph vertex0 exact incoming count = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex0 exact incoming count，不以小數近似替代。",
          "en": "Recompute digraph vertex0 exact incoming count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1",
          "1"
        ],
        "output": "3",
        "label": "digraph vertex0 exact incoming count"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ORIGINAL vertex=0;ALL outgoing original-arc indices=[0, 1];ALL incoming indices=[0, 6, 8]",
      "after": "outdegree=2;indegree=3;a loop at this vertex contributes ONCE to EACH separate count",
      "relation": "implies",
      "rule_id": "digraph_every_vertex_separate_in_out_incidence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條arc只有一個source、一個target；loop也各占一次，不套undirected loop degree規則或把兩種degree合併。孤立點兩欄都保0。",
        "en": "Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates."
      },
      "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": "digraph vertex1 exact outgoing count := sum(1, 1)",
      "after": "digraph vertex1 exact outgoing count = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex1 exact outgoing count，不以小數近似替代。",
          "en": "Recompute digraph vertex1 exact outgoing count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "digraph vertex1 exact outgoing count"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "digraph vertex1 exact incoming count := sum(1)",
      "after": "digraph vertex1 exact incoming count = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex1 exact incoming count，不以小數近似替代。",
          "en": "Recompute digraph vertex1 exact incoming count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "digraph vertex1 exact incoming count"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ORIGINAL vertex=1;ALL outgoing original-arc indices=[2, 3];ALL incoming indices=[1]",
      "after": "outdegree=2;indegree=1;a loop at this vertex contributes ONCE to EACH separate count",
      "relation": "implies",
      "rule_id": "digraph_every_vertex_separate_in_out_incidence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條arc只有一個source、一個target；loop也各占一次，不套undirected loop degree規則或把兩種degree合併。孤立點兩欄都保0。",
        "en": "Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates."
      },
      "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": "digraph vertex2 exact outgoing count := sum(1, 1)",
      "after": "digraph vertex2 exact outgoing count = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex2 exact outgoing count，不以小數近似替代。",
          "en": "Recompute digraph vertex2 exact outgoing count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "digraph vertex2 exact outgoing count"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "digraph vertex2 exact incoming count := sum(1)",
      "after": "digraph vertex2 exact incoming count = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex2 exact incoming count，不以小數近似替代。",
          "en": "Recompute digraph vertex2 exact incoming count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "digraph vertex2 exact incoming count"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ORIGINAL vertex=2;ALL outgoing original-arc indices=[4, 5];ALL incoming indices=[2]",
      "after": "outdegree=2;indegree=1;a loop at this vertex contributes ONCE to EACH separate count",
      "relation": "implies",
      "rule_id": "digraph_every_vertex_separate_in_out_incidence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條arc只有一個source、一個target；loop也各占一次，不套undirected loop degree規則或把兩種degree合併。孤立點兩欄都保0。",
        "en": "Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates."
      },
      "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": "digraph vertex3 exact outgoing count := sum(1, 1)",
      "after": "digraph vertex3 exact outgoing count = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex3 exact outgoing count，不以小數近似替代。",
          "en": "Recompute digraph vertex3 exact outgoing count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "digraph vertex3 exact outgoing count"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "digraph vertex3 exact incoming count := sum(1, 1)",
      "after": "digraph vertex3 exact incoming count = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex3 exact incoming count，不以小數近似替代。",
          "en": "Recompute digraph vertex3 exact incoming count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "digraph vertex3 exact incoming count"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "ORIGINAL vertex=3;ALL outgoing original-arc indices=[6, 7];ALL incoming indices=[3, 4]",
      "after": "outdegree=2;indegree=2;a loop at this vertex contributes ONCE to EACH separate count",
      "relation": "implies",
      "rule_id": "digraph_every_vertex_separate_in_out_incidence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條arc只有一個source、一個target；loop也各占一次，不套undirected loop degree規則或把兩種degree合併。孤立點兩欄都保0。",
        "en": "Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates."
      },
      "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": "digraph vertex4 exact outgoing count := sum(1)",
      "after": "digraph vertex4 exact outgoing count = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex4 exact outgoing count，不以小數近似替代。",
          "en": "Recompute digraph vertex4 exact outgoing count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "digraph vertex4 exact outgoing count"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "digraph vertex4 exact incoming count := sum(1, 1)",
      "after": "digraph vertex4 exact incoming count = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph vertex4 exact incoming count，不以小數近似替代。",
          "en": "Recompute digraph vertex4 exact incoming count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "digraph vertex4 exact incoming count"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ORIGINAL vertex=4;ALL outgoing original-arc indices=[8];ALL incoming indices=[5, 7]",
      "after": "outdegree=1;indegree=2;a loop at this vertex contributes ONCE to EACH separate count",
      "relation": "implies",
      "rule_id": "digraph_every_vertex_separate_in_out_incidence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條arc只有一個source、一個target；loop也各占一次，不套undirected loop degree規則或把兩種degree合併。孤立點兩欄都保0。",
        "en": "Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates."
      },
      "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": "digraph complete outgoing-degree sum := sum(2, 2, 2, 2, 1)",
      "after": "digraph complete outgoing-degree sum = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph complete outgoing-degree sum，不以小數近似替代。",
          "en": "Recompute digraph complete outgoing-degree sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "2",
          "2",
          "2",
          "1"
        ],
        "output": "9",
        "label": "digraph complete outgoing-degree sum"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "digraph complete incoming-degree sum := sum(3, 1, 1, 2, 2)",
      "after": "digraph complete incoming-degree sum = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 digraph complete incoming-degree sum，不以小數近似替代。",
          "en": "Recompute digraph complete incoming-degree sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "1",
          "1",
          "2",
          "2"
        ],
        "output": "9",
        "label": "digraph complete incoming-degree sum"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "ALL vertex-order outdegrees=[2, 2, 2, 2, 1];indegrees=[3, 1, 1, 2, 2];complete directed arc count=9",
      "after": "each arc counted once at its source and once at its target;SUM(out)=9=|E| and SUM(in)=9=|E|;the SEPARATE counting rule is |E|,not an imported undirected2|E| rule (numbers may coincide when E=0)",
      "relation": "implies",
      "rule_id": "digraph_two_separate_degree_sum_certificates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對source計數是arc到source的完整分組，target亦然，所以兩個獨立總和各等|E|。沒有推出無向連通、DAG、強連通或Hamilton性質。",
        "en": "Grouping all arcs by their sources or by their targets gives two separate totals,each |E|.This supplies no undirected-connectivity,DAG,strong-connectivity or Hamiltonian 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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "ACTUAL loops in supplied ordered arc set=[[0, 0]];extended-data counts out=[2, 2, 2, 2, 1],in=[3, 1, 1, 2, 2]",
      "after": "declared MAIN loop-free model qualification=否;loops may exist under a DIFFERENT declared convention;not a denial of all digraph definitions",
      "relation": "implies",
      "rule_id": "digraph_loop_No_declared_convention_not_global_impossibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "False只按本題主模型的loop-free約定拒絕，不把loop說成任何圖論模型都不合法；這裡仍保明示loop資料的方向度數，loop各貢獻一入一出。",
        "en": "False rejects the declared loop-free main convention only,not every graph-theoretic model.Retain directed counts for the explicit loop data,where each loop contributes one incoming and one outgoing incidence."
      },
      "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,
          0
        ]
      ],
      false,
      [
        2,
        2,
        2,
        2,
        1
      ],
      [
        3,
        1,
        1,
        2,
        2
      ],
      "loops_rejected_by_declared_loop_free_simple_digraph_convention"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 0]], 否, [2, 2, 2, 2, 1], [3, 1, 1, 2, 2], loops_rejected_by_declared_loop_free_simple_digraph_convention)",
    "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-20261108-r07-finite-directed-graph-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations/questions.jsonl",
      "batch_sha256": "b76dd0131fe7faeb6a32b6d736c48955db141a0c42218e36d7e584617fde48c5",
      "original_line": 90,
      "original_line_sha256": "3b7cf89bdb89e0013b6fa17a343f5a679f95bda7979025f451da5abf2f976498",
      "question_compact_sha256": "843a068949dfdb1a13ec51868dd13d0f4d0d144868a7b819e667dad0a3556639",
      "english_row_compact_sha256": "eea28ee04c659e13a0d308c282977ec9b02d4813f1797c17dcf271e76107f608",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-directed-graph": {
        "sha256": "09311b0ec711e3f02a16b1c332e69c9f718fe2a288f2005683d393695a41161b",
        "english_sha256": "1a9289ec7bd86c2ec776a919d938ee1c37455b7d8c5ec5c81e2293906e548c98",
        "original_path": "public/data/mko/finite-directed-graph.json",
        "english_path": "public/data/i18n/editions/mko-finite-directed-graph.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "f1ed3239c9185524aa4f0839ab4cfbe8f7d74200b283ea09e850a792f0178a3b",
    "original_candidate_sha256": "50e67d1958bbf236fd889fd605cda3b8731bb60ab70a44c750b26fed72c3bb56",
    "original_candidate_line_sha256": "50e67d1958bbf236fd889fd605cda3b8731bb60ab70a44c750b26fed72c3bb56"
  },
  "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-p41",
    "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": "c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d",
      "question_first_witness_sha256": "e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 21,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f8432e6fa3943653ed8cc7a01ae1b429c97ce5b8146174b086365e22523d1471",
      "arithmetic_steps": 12
    },
    "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."
  }
}