{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r02-matrix-0034",
  "pilot_number": "R0014-07634",
  "number": 7634,
  "family_id": "qf-density16-matrix-sum-transpose",
  "title": {
    "zh-Hant": "第 7634 題：同尺寸和的轉置兩路",
    "en": "Question 7634: Same-Shape Matrix Sum and Both Transpose Routes"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r02-matrix-0034",
    "batch_id": "batch-20261012-advance-20261016-r02-matrix-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-matrix"
    ],
    "title_zh": "同尺寸和的轉置兩路",
    "stem_zh": "外部索引從1開始；每格完整整數資料。同尺寸M=[[-4, -5, -6, -7], [-3, -4, -5, -6]]、N=[[0, 1, 2, 3], [-1, 0, 1, 2]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。",
    "answer": {
      "kind": "expression",
      "value": "([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])",
      "explanation_zh": "有限結構核對=([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])。先核同尺寸，逐格相加後轉置，另一條路轉置後相加。有限精確核對不是一般形式證明。轉置不當求逆，尺寸不合先拒絕求值。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "同尺寸和的轉置兩路"
      ]
    },
    "variant": {
      "family_id": "qf-density16-matrix-sum-transpose",
      "seed": 202610162034,
      "parameters": {
        "M": [
          [
            -4,
            -5,
            -6,
            -7
          ],
          [
            -3,
            -4,
            -5,
            -6
          ]
        ],
        "N": [
          [
            0,
            1,
            2,
            3
          ],
          [
            -1,
            0,
            1,
            2
          ]
        ],
        "domain": "Z_rect_1based",
        "operation": "matrix-sum-transpose",
        "semantic_template": "qf-density16-matrix-sum-transpose/{\"M\":[[-4,-5,-6,-7],[-3,-4,-5,-6]],\"N\":[[0,1,2,3],[-1,0,1,2]]}"
      }
    },
    "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-20261016-r02-matrix-0034",
    "batch_id": "batch-20261012-advance-20261016-r02-matrix-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e8ff436d9052dc41abc05c5575a85f935592ba5c33959284e86931b322f3b70c",
      "family_id": "qf-density16-matrix-sum-transpose",
      "seed": 202610162034,
      "semantic_template": "qf-density16-matrix-sum-transpose/{\"M\":[[-4,-5,-6,-7],[-3,-4,-5,-6]],\"N\":[[0,1,2,3],[-1,0,1,2]]}"
    },
    "fields": {
      "title": "Same-Shape Matrix Sum and Both Transpose Routes",
      "stem": "Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-4, -5, -6, -7], [-3, -4, -5, -6]], N=[[0, 1, 2, 3], [-1, 0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).",
      "answer": {
        "kind": "expression",
        "value": "([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])",
        "explanation": "Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])."
      },
      "skills": [
        "Same-Shape Matrix Sum and Both Transpose Routes"
      ],
      "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": "外部索引從1開始；每格完整整數資料。同尺寸M=[[-4, -5, -6, -7], [-3, -4, -5, -6]]、N=[[0, 1, 2, 3], [-1, 0, 1, 2]]。求 (M尺寸, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)。",
        "en": "Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-4, -5, -6, -7], [-3, -4, -5, -6]], N=[[0, 1, 2, 3], [-1, 0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "M=[[-4, -5, -6, -7], [-3, -4, -5, -6]];shape=[2, 4]",
      "after": "external indices start at1;transpose is not inverse",
      "relation": "implies",
      "rule_id": "rectangular_matrix_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先確認非空矩形尺寸與完整整數資料。轉置對矩形矩陣也可定義，不要求可逆，更不是求逆。",
        "en": "First confirm a nonempty rectangular shape and complete integer entries.Transposition is defined for rectangular matrices without invertibility and is not matrix inversion."
      },
      "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": "M transpose:source(1,1)=-4",
      "after": "target(1,1)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "M transpose:source(1,2)=-5",
      "after": "target(2,1)=-5",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "M transpose:source(1,3)=-6",
      "after": "target(3,1)=-6",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "M transpose:source(1,4)=-7",
      "after": "target(4,1)=-7",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "M transpose:source(2,1)=-3",
      "after": "target(1,2)=-3",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "M transpose:source(2,2)=-4",
      "after": "target(2,2)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "M transpose:source(2,3)=-5",
      "after": "target(3,2)=-5",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "M transpose:source(2,4)=-6",
      "after": "target(4,2)=-6",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "M shape=[2, 4];N shape=[2, 4]",
      "after": "same shape;entrywise sum permitted",
      "relation": "implies",
      "rule_id": "matrix_sum_shape_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "矩陣加法先要求同尺寸；不靠截短、補零或廣播修改原題。",
        "en": "Matrix addition requires equal shapes first.Do not truncate,pad with zeros or broadcast to alter the original problem."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "M+N(1,1) := sum(-4, 0)",
      "after": "M+N(1,1) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M+N(1,1)，不以小數近似替代。",
          "en": "Recompute M+N(1,1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-4",
          "0"
        ],
        "output": "-4",
        "label": "M+N(1,1)"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "M+N(1,2) := sum(-5, 1)",
      "after": "M+N(1,2) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M+N(1,2)，不以小數近似替代。",
          "en": "Recompute M+N(1,2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "1"
        ],
        "output": "-4",
        "label": "M+N(1,2)"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "M+N(1,3) := sum(-6, 2)",
      "after": "M+N(1,3) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M+N(1,3)，不以小數近似替代。",
          "en": "Recompute M+N(1,3) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "2"
        ],
        "output": "-4",
        "label": "M+N(1,3)"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "M+N(1,4) := sum(-7, 3)",
      "after": "M+N(1,4) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M+N(1,4)，不以小數近似替代。",
          "en": "Recompute M+N(1,4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-7",
          "3"
        ],
        "output": "-4",
        "label": "M+N(1,4)"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "M+N(2,1) := sum(-3, -1)",
      "after": "M+N(2,1) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M+N(2,1)，不以小數近似替代。",
          "en": "Recompute M+N(2,1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-3",
          "-1"
        ],
        "output": "-4",
        "label": "M+N(2,1)"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "M+N(2,2) := sum(-4, 0)",
      "after": "M+N(2,2) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M+N(2,2)，不以小數近似替代。",
          "en": "Recompute M+N(2,2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-4",
          "0"
        ],
        "output": "-4",
        "label": "M+N(2,2)"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "M+N(2,3) := sum(-5, 1)",
      "after": "M+N(2,3) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M+N(2,3)，不以小數近似替代。",
          "en": "Recompute M+N(2,3) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "1"
        ],
        "output": "-4",
        "label": "M+N(2,3)"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "M+N(2,4) := sum(-6, 2)",
      "after": "M+N(2,4) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M+N(2,4)，不以小數近似替代。",
          "en": "Recompute M+N(2,4) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "2"
        ],
        "output": "-4",
        "label": "M+N(2,4)"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "sum transpose:source(1,1)=-4",
      "after": "target(1,1)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "sum transpose:source(1,2)=-4",
      "after": "target(2,1)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "sum transpose:source(1,3)=-4",
      "after": "target(3,1)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "sum transpose:source(1,4)=-4",
      "after": "target(4,1)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "sum transpose:source(2,1)=-4",
      "after": "target(1,2)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "sum transpose:source(2,2)=-4",
      "after": "target(2,2)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "sum transpose:source(2,3)=-4",
      "after": "target(3,2)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "sum transpose:source(2,4)=-4",
      "after": "target(4,2)=-4",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "N transpose:source(1,1)=0",
      "after": "target(1,1)=0",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "N transpose:source(1,2)=1",
      "after": "target(2,1)=1",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "N transpose:source(1,3)=2",
      "after": "target(3,1)=2",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "N transpose:source(1,4)=3",
      "after": "target(4,1)=3",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "N transpose:source(2,1)=-1",
      "after": "target(1,2)=-1",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "N transpose:source(2,2)=0",
      "after": "target(2,2)=0",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "N transpose:source(2,3)=1",
      "after": "target(3,2)=1",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "N transpose:source(2,4)=2",
      "after": "target(4,2)=2",
      "relation": "implies",
      "rule_id": "transpose_coordinate_swap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "轉置只交換行與列位置，數值不變；此處位置均以1起算。",
        "en": "Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based."
      },
      "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": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "MT+NT(1,1) := sum(-4, 0)",
      "after": "MT+NT(1,1) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MT+NT(1,1)，不以小數近似替代。",
          "en": "Recompute MT+NT(1,1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-4",
          "0"
        ],
        "output": "-4",
        "label": "MT+NT(1,1)"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "MT+NT(1,2) := sum(-3, -1)",
      "after": "MT+NT(1,2) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MT+NT(1,2)，不以小數近似替代。",
          "en": "Recompute MT+NT(1,2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-3",
          "-1"
        ],
        "output": "-4",
        "label": "MT+NT(1,2)"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "MT+NT(2,1) := sum(-5, 1)",
      "after": "MT+NT(2,1) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MT+NT(2,1)，不以小數近似替代。",
          "en": "Recompute MT+NT(2,1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "1"
        ],
        "output": "-4",
        "label": "MT+NT(2,1)"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "MT+NT(2,2) := sum(-4, 0)",
      "after": "MT+NT(2,2) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MT+NT(2,2)，不以小數近似替代。",
          "en": "Recompute MT+NT(2,2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-4",
          "0"
        ],
        "output": "-4",
        "label": "MT+NT(2,2)"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "MT+NT(3,1) := sum(-6, 2)",
      "after": "MT+NT(3,1) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MT+NT(3,1)，不以小數近似替代。",
          "en": "Recompute MT+NT(3,1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "2"
        ],
        "output": "-4",
        "label": "MT+NT(3,1)"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "MT+NT(3,2) := sum(-5, 1)",
      "after": "MT+NT(3,2) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MT+NT(3,2)，不以小數近似替代。",
          "en": "Recompute MT+NT(3,2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "1"
        ],
        "output": "-4",
        "label": "MT+NT(3,2)"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "MT+NT(4,1) := sum(-7, 3)",
      "after": "MT+NT(4,1) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MT+NT(4,1)，不以小數近似替代。",
          "en": "Recompute MT+NT(4,1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-7",
          "3"
        ],
        "output": "-4",
        "label": "MT+NT(4,1)"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "MT+NT(4,2) := sum(-6, 2)",
      "after": "MT+NT(4,2) = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MT+NT(4,2)，不以小數近似替代。",
          "en": "Recompute MT+NT(4,2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "2"
        ],
        "output": "-4",
        "label": "MT+NT(4,2)"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "left=[[-4, -4], [-4, -4], [-4, -4], [-4, -4]];right=[[-4, -4], [-4, -4], [-4, -4], [-4, -4]]",
      "after": "same value at every transposed coordinate",
      "relation": "implies",
      "rule_id": "transpose_sum_two_routes",
      "conditions": [],
      "explanation": {
        "zh-Hant": "一條路先相加再交換位置，另一條路先各自交換位置再相加，核的是同一對原格。",
        "en": "One route adds before swapping positions;the other swaps each matrix first and then adds.Both use the same original entry pair."
      },
      "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": [
      [
        2,
        4
      ],
      [
        [
          "-4",
          "-4",
          "-4",
          "-4"
        ],
        [
          "-4",
          "-4",
          "-4",
          "-4"
        ]
      ],
      [
        [
          "-4",
          "-4"
        ],
        [
          "-4",
          "-4"
        ],
        [
          "-4",
          "-4"
        ],
        [
          "-4",
          "-4"
        ]
      ],
      [
        [
          "-4",
          "-4"
        ],
        [
          "-4",
          "-4"
        ],
        [
          "-4",
          "-4"
        ],
        [
          "-4",
          "-4"
        ]
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])",
    "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-20261016-r02-matrix-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r02-matrix-foundations/questions.jsonl",
      "batch_sha256": "2cfe67df8364350332cadfafbf7125a57390570620e1132ca9237e218f8c4cf8",
      "original_line": 34,
      "original_line_sha256": "6f569de8894264b218a16b1657babf7e5edc9043f897d565e8cbef5bd795809c",
      "question_compact_sha256": "e8ff436d9052dc41abc05c5575a85f935592ba5c33959284e86931b322f3b70c",
      "english_row_compact_sha256": "567f271045257100b0a9422d38b52d7494ff15d1c7e2da3993a6a950fe13ce65",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r02-matrix-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r02-matrix-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-matrix": {
        "sha256": "c0233ecc6ee07427af1589feb16108500eb2e48b0b2fb84b5b39f0f38b02f109",
        "english_sha256": "18504021d52173d8f2f8aa663d83dc058f5205bea2c677a9722daba770776432",
        "original_path": "public/data/mko/matrix.json",
        "english_path": "public/data/i18n/editions/mko-matrix.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d7860245c70062705fa4858c13821db6ff286678dec63f22464364f5d24cfcc1",
    "original_candidate_sha256": "faec1e487add923a9b05bdc85abc77c2f1728a1ca95bf8be7976f4a64ea353f7",
    "original_candidate_line_sha256": "faec1e487add923a9b05bdc85abc77c2f1728a1ca95bf8be7976f4a64ea353f7"
  },
  "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-p15",
    "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": "23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77",
      "question_first_witness_sha256": "17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 43,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "70c6741dd69b0ee3308f9d6546b75fc1904561d33f3cf47814e77d5235784293",
      "arithmetic_steps": 16
    },
    "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."
  }
}