{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r04-proof-0094",
  "pilot_number": "R0011-06294",
  "number": 6294,
  "family_id": "qf-density13-proof-horn-rule-removal",
  "title": {
    "zh-Hant": "第 6294 題：刪除原規則列前後的深度",
    "en": "Question 6294: Goal Depths Before and After Removing Original Rules"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r04-proof-0094",
    "batch_id": "batch-20261009-advance-20261013-r04-proof-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-proof"
    ],
    "title_zh": "刪除原規則列前後的深度",
    "stem_zh": "給定事實與單前提MP；索引從0，n=5，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 2]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[4]，goal=4。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。",
    "answer": {
      "kind": "expression",
      "value": "(3, 4, 否)",
      "explanation_zh": "刪規則前後深度=(3, 4, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "刪除原規則列前後的深度"
      ]
    },
    "variant": {
      "family_id": "qf-density13-proof-horn-rule-removal",
      "seed": 202610134094,
      "parameters": {
        "n": 5,
        "facts": [
          0
        ],
        "rules": [
          [
            0,
            1
          ],
          [
            1,
            2
          ],
          [
            2,
            3
          ],
          [
            3,
            4
          ],
          [
            0,
            2
          ]
        ],
        "remove_indices": [
          4
        ],
        "goal": 4,
        "operation": "density13_proof_horn_rule_removal",
        "domain": "finite_given_MP_protocol",
        "semantic_template": "qf-density13-proof-horn-rule-removal/[3,4]"
      }
    },
    "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": "2026-10-09",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261009-advance-20261013-r04-proof-0094",
    "batch_id": "batch-20261009-advance-20261013-r04-proof-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9df0b58f6917b5b174bcfd0dc1a5b9f5936fed9018823509fa4d529ec4941582",
      "family_id": "qf-density13-proof-horn-rule-removal",
      "seed": 202610134094,
      "semantic_template": "qf-density13-proof-horn-rule-removal/[3,4]"
    },
    "fields": {
      "title": "Goal Depths Before and After Removing Original Rules",
      "stem": "Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[4], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?).",
      "answer": {
        "kind": "expression",
        "value": "(3, 4, No)",
        "explanation": "Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 4, No)."
      },
      "skills": [
        "Goal Depths Before and After Removing Original Rules"
      ],
      "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": "給定事實與單前提MP；索引從0，n=5，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 2]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[4]，goal=4。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。",
        "en": "Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[4], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "n=5;given facts=[0];original indexed rules=[[0, 0, 1], [1, 1, 2], [2, 2, 3], [3, 3, 4], [4, 0, 2]]",
      "after": "use only given facts and single-premise MP;not-derived does not mean False",
      "relation": "implies",
      "rule_id": "given_unary_MP_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原子以0起算索引表示，規則i→j只在i已得時可用。沒有事實的循環不能自行產生結論；未得不等於假，有限核對不是一般邏輯完備性。",
        "en": "Atoms use0-based indices and a rule i→j requires an already derived i.A cycle without facts cannot generate conclusions.Not-derived is not False,and this finite check is not general logical completeness."
      },
      "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": "phase=before;given facts=[0]",
      "after": "phase=before;active_rules=[0,1,2,3,4]",
      "relation": "implies",
      "rule_id": "horn_closure_start",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從事實多源開始，保留原始規則列索引；還沒有前提的原子保持未得。",
        "en": "Start from all given facts,retaining original rule-row indices.Atoms with no derived premises remain not-derived."
      },
      "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": "phase=before;given fact=0",
      "after": "phase=before;derive atom=0;depth=0;basis=given",
      "relation": "implies",
      "rule_id": "horn_given_fact",
      "conditions": [],
      "explanation": {
        "zh-Hant": "題目給定事實的深度0，不需要另找MP前提，也不把其他原子一併當成事實。",
        "en": "A supplied fact has depth0 and needs no MP antecedent;other atoms are not silently made facts."
      },
      "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": "before MP depth for atom1 := sum(0, 1)",
      "after": "before MP depth for atom1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用一條MP，深度在已知前提深度上加1。",
        "en": "One MP application adds1 to the already known antecedent depth."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 before MP depth for atom1，不以小數近似替代。",
          "en": "Recompute before MP depth for atom1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "before MP depth for atom1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "phase=before;rule[0]=0→1;antecedent depth=0",
      "after": "phase=before;derive atom=1;depth=1;from=0;rule=0",
      "relation": "implies",
      "rule_id": "horn_MP_derivation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已得前提加題目給定的這一條規則，才首次導出目標；按佇列的非減深度順序擴展。",
        "en": "An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order."
      },
      "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": "before MP depth for atom2 := sum(0, 1)",
      "after": "before MP depth for atom2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用一條MP，深度在已知前提深度上加1。",
        "en": "One MP application adds1 to the already known antecedent depth."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 before MP depth for atom2，不以小數近似替代。",
          "en": "Recompute before MP depth for atom2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "before MP depth for atom2"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "phase=before;rule[4]=0→2;antecedent depth=0",
      "after": "phase=before;derive atom=2;depth=1;from=0;rule=4",
      "relation": "implies",
      "rule_id": "horn_MP_derivation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已得前提加題目給定的這一條規則，才首次導出目標；按佇列的非減深度順序擴展。",
        "en": "An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order."
      },
      "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": "before MP depth for atom3 := sum(1, 1)",
      "after": "before MP depth for atom3 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用一條MP，深度在已知前提深度上加1。",
        "en": "One MP application adds1 to the already known antecedent depth."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 before MP depth for atom3，不以小數近似替代。",
          "en": "Recompute before MP depth for atom3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "before MP depth for atom3"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "phase=before;rule[2]=2→3;antecedent depth=1",
      "after": "phase=before;derive atom=3;depth=2;from=2;rule=2",
      "relation": "implies",
      "rule_id": "horn_MP_derivation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已得前提加題目給定的這一條規則，才首次導出目標；按佇列的非減深度順序擴展。",
        "en": "An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order."
      },
      "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": "before MP depth for atom4 := sum(2, 1)",
      "after": "before MP depth for atom4 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用一條MP，深度在已知前提深度上加1。",
        "en": "One MP application adds1 to the already known antecedent depth."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 before MP depth for atom4，不以小數近似替代。",
          "en": "Recompute before MP depth for atom4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "before MP depth for atom4"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "phase=before;rule[3]=3→4;antecedent depth=2",
      "after": "phase=before;derive atom=4;depth=3;from=3;rule=3",
      "relation": "implies",
      "rule_id": "horn_MP_derivation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已得前提加題目給定的這一條規則，才首次導出目標；按佇列的非減深度順序擴展。",
        "en": "An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order."
      },
      "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": "phase=before;edge closure/depth audits=[[0, 0, 1, 0, 1, 是], [1, 1, 2, 1, 1, 是], [2, 2, 3, 1, 2, 是], [3, 3, 4, 2, 3, 是], [4, 0, 2, 0, 1, 是]]",
      "after": "phase=before;depths=[0,1,1,2,3]",
      "relation": "implies",
      "rule_id": "horn_closed_depth_certificate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個已得原子都有上述長度的實際鏈；全部給定邊對可達集合閉合且滿足深度界，任何更短鏈也會迫使較小深度，故此標號最短。null表示此規則模型未得，不表示假。",
        "en": "Every derived atom has a displayed chain of that length.The reached set is closed under every given edge and satisfies the depth bounds,so any shorter chain would force a smaller label.These labels are shortest;null means not-derived in this rule model,not False."
      },
      "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": "remove original rule rows=[4]",
      "after": "retained original indices=[0, 1, 2, 3]",
      "relation": "implies",
      "rule_id": "horn_original_rule_index_removal",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始列索引刪除，重複內容的不同列仍有不同索引；不重新編號後才解讀刪除清單。",
        "en": "Remove by original row index;duplicate-content rows still have distinct indices.Do not renumber before interpreting the removal list."
      },
      "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": "phase=after;given facts=[0]",
      "after": "phase=after;active_rules=[0,1,2,3]",
      "relation": "implies",
      "rule_id": "horn_closure_start",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從事實多源開始，保留原始規則列索引；還沒有前提的原子保持未得。",
        "en": "Start from all given facts,retaining original rule-row indices.Atoms with no derived premises remain not-derived."
      },
      "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": "phase=after;given fact=0",
      "after": "phase=after;derive atom=0;depth=0;basis=given",
      "relation": "implies",
      "rule_id": "horn_given_fact",
      "conditions": [],
      "explanation": {
        "zh-Hant": "題目給定事實的深度0，不需要另找MP前提，也不把其他原子一併當成事實。",
        "en": "A supplied fact has depth0 and needs no MP antecedent;other atoms are not silently made facts."
      },
      "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": "after MP depth for atom1 := sum(0, 1)",
      "after": "after MP depth for atom1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用一條MP，深度在已知前提深度上加1。",
        "en": "One MP application adds1 to the already known antecedent depth."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 after MP depth for atom1，不以小數近似替代。",
          "en": "Recompute after MP depth for atom1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "after MP depth for atom1"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "phase=after;rule[0]=0→1;antecedent depth=0",
      "after": "phase=after;derive atom=1;depth=1;from=0;rule=0",
      "relation": "implies",
      "rule_id": "horn_MP_derivation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已得前提加題目給定的這一條規則，才首次導出目標；按佇列的非減深度順序擴展。",
        "en": "An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order."
      },
      "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": "after MP depth for atom2 := sum(1, 1)",
      "after": "after MP depth for atom2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用一條MP，深度在已知前提深度上加1。",
        "en": "One MP application adds1 to the already known antecedent depth."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 after MP depth for atom2，不以小數近似替代。",
          "en": "Recompute after MP depth for atom2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "after MP depth for atom2"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "phase=after;rule[1]=1→2;antecedent depth=1",
      "after": "phase=after;derive atom=2;depth=2;from=1;rule=1",
      "relation": "implies",
      "rule_id": "horn_MP_derivation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已得前提加題目給定的這一條規則，才首次導出目標；按佇列的非減深度順序擴展。",
        "en": "An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order."
      },
      "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": "after MP depth for atom3 := sum(2, 1)",
      "after": "after MP depth for atom3 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用一條MP，深度在已知前提深度上加1。",
        "en": "One MP application adds1 to the already known antecedent depth."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 after MP depth for atom3，不以小數近似替代。",
          "en": "Recompute after MP depth for atom3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "after MP depth for atom3"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "phase=after;rule[2]=2→3;antecedent depth=2",
      "after": "phase=after;derive atom=3;depth=3;from=2;rule=2",
      "relation": "implies",
      "rule_id": "horn_MP_derivation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已得前提加題目給定的這一條規則，才首次導出目標；按佇列的非減深度順序擴展。",
        "en": "An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order."
      },
      "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": "after MP depth for atom4 := sum(3, 1)",
      "after": "after MP depth for atom4 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用一條MP，深度在已知前提深度上加1。",
        "en": "One MP application adds1 to the already known antecedent depth."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 after MP depth for atom4，不以小數近似替代。",
          "en": "Recompute after MP depth for atom4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "1"
        ],
        "output": "4",
        "label": "after MP depth for atom4"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "phase=after;rule[3]=3→4;antecedent depth=3",
      "after": "phase=after;derive atom=4;depth=4;from=3;rule=3",
      "relation": "implies",
      "rule_id": "horn_MP_derivation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已得前提加題目給定的這一條規則，才首次導出目標；按佇列的非減深度順序擴展。",
        "en": "An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order."
      },
      "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": "phase=after;edge closure/depth audits=[[0, 0, 1, 0, 1, 是], [1, 1, 2, 1, 2, 是], [2, 2, 3, 2, 3, 是], [3, 3, 4, 3, 4, 是]]",
      "after": "phase=after;depths=[0,1,2,3,4]",
      "relation": "implies",
      "rule_id": "horn_closed_depth_certificate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個已得原子都有上述長度的實際鏈；全部給定邊對可達集合閉合且滿足深度界，任何更短鏈也會迫使較小深度，故此標號最短。null表示此規則模型未得，不表示假。",
        "en": "Every derived atom has a displayed chain of that length.The reached set is closed under every given edge and satisfies the depth bounds,so any shorter chain would force a smaller label.These labels are shortest;null means not-derived in this rule model,not False."
      },
      "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": "goal=4;before=3;after=4",
      "after": "derivability changed=否",
      "relation": "implies",
      "rule_id": "horn_reachability_vs_depth_change",
      "conditions": [],
      "explanation": {
        "zh-Hant": "可得性改變只指由未得到可得或由可得到未得；最短深度增長但仍可得，第三欄仍為否。刪規則不會憑空增加證明。",
        "en": "A derivability change concerns derived versus not-derived.If the shortest depth grows but the goal remains derived,the third field is still No.Removing rules cannot invent proofs."
      },
      "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": [
      3,
      4,
      false
    ],
    "before_depths": [
      0,
      1,
      1,
      2,
      3
    ],
    "after_depths": [
      0,
      1,
      2,
      3,
      4
    ],
    "retained_rule_indices": [
      0,
      1,
      2,
      3
    ]
  },
  "conclusion": {
    "canonical_value": "(3, 4, 否)",
    "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-20261009-advance-20261013-r04-proof-foundations",
      "batch_path": "public/data/questions/batches/batch-20261009-advance-20261013-r04-proof-foundations/questions.jsonl",
      "batch_sha256": "9d2b0a978c354c8317efdeb07f00669c8dabe08c14a9392d3101a946b73b4ac0",
      "original_line": 94,
      "original_line_sha256": "5e15f748ba1f0694375a5f25d42194467874de0783edf57c41419c328e98c37a",
      "question_compact_sha256": "9df0b58f6917b5b174bcfd0dc1a5b9f5936fed9018823509fa4d529ec4941582",
      "english_row_compact_sha256": "650379b9debfb57cee871a83c7901dccb305efef6a6e09d0efe0e19c6dfcf4ec",
      "english_manifest_path": "public/data/i18n/editions/batch-20261009-advance-20261013-r04-proof-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261009-advance-20261013-r04-proof-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-proof": {
        "sha256": "72f6009b912caa3c05a2948a97ca422367574a19ad5fc665964a540875417b91",
        "english_sha256": "1c8ec335b48f7d913d3057b2d632c371b092ca869f799b056b72464d948504fe",
        "original_path": "public/data/mko/proof.json",
        "english_path": "public/data/i18n/editions/mko-proof.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "5b2ce631281eb24b279908f47c6de66fa60d326520852879d324c4f3ac653970",
    "original_candidate_sha256": "e0d7ee5e4fde1dcd28f945e86e10c8a903cd274b993af26821eb1f166b82a757",
    "original_candidate_line_sha256": "e0d7ee5e4fde1dcd28f945e86e10c8a903cd274b993af26821eb1f166b82a757"
  },
  "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-p12",
    "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": "b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80",
      "question_first_witness_sha256": "93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 25,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "5ec6328c3bf423997c6dac843d56609d8c448d2d405a8e206192ab3b4b117dfc",
      "arithmetic_steps": 8
    },
    "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."
  }
}