{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r04-determinant-0058",
  "pilot_number": "R0013-07333",
  "number": 7333,
  "family_id": "qf-density16-det-product-two-orders",
  "title": {
    "zh-Hant": "第 7333 題：矩陣乘積兩序及行列式",
    "en": "Question 7333: Both Matrix Product Orders, Determinants, and Equality"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r04-determinant-0058",
    "batch_id": "batch-20261012-advance-20261016-r04-determinant-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-determinant"
    ],
    "title_zh": "矩陣乘積兩序及行列式",
    "stem_zh": "R上2×2矩陣，D為有號行列式。M=[[-5, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。",
    "answer": {
      "kind": "expression",
      "value": "([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, 否)",
      "explanation_zh": "有限結構核對=([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, 否)。兩序各按行乘列求矩陣；行列式以兩個原D的乘積計算，再逐格比較矩陣。有限精確核對不是一般形式證明。通常行乘列不當逐項乘法，不取絕對值當有號行列式。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "矩陣乘積兩序及行列式"
      ]
    },
    "variant": {
      "family_id": "qf-density16-det-product-two-orders",
      "seed": 202610164058,
      "parameters": {
        "M": [
          [
            -5,
            1
          ],
          [
            0,
            1
          ]
        ],
        "N": [
          [
            1,
            0
          ],
          [
            2,
            1
          ]
        ],
        "domain": "R_2x2_integer_slice",
        "operation": "det-product-two-orders",
        "semantic_template": "qf-density16-det-product-two-orders/{\"M\":[[-5,1],[0,1]],\"N\":[[1,0],[2,1]]}"
      }
    },
    "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-r04-determinant-0058",
    "batch_id": "batch-20261012-advance-20261016-r04-determinant-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "47fc18ceba8cd6b148247b1c60146396fa3f5958c1bef1e9d2e18f9acea9e82e",
      "family_id": "qf-density16-det-product-two-orders",
      "seed": 202610164058,
      "semantic_template": "qf-density16-det-product-two-orders/{\"M\":[[-5,1],[0,1]],\"N\":[[1,0],[2,1]]}"
    },
    "fields": {
      "title": "Both Matrix Product Orders, Determinants, and Equality",
      "stem": "Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).",
      "answer": {
        "kind": "expression",
        "value": "([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, No)",
        "explanation": "Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, No)."
      },
      "skills": [
        "Both Matrix Product Orders, Determinants, and Equality"
      ],
      "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": "R上2×2矩陣，D為有號行列式。M=[[-5, 1], [0, 1]]、N=[[1, 0], [2, 1]]。求 (MN, NM, D(MN), D(NM), D(M)D(N), MN與NM相同資格)。",
        "en": "Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "[[-5, 1], [0, 1]]",
      "after": "real2×2 integer-entry slice;determinants are signed",
      "relation": "implies",
      "rule_id": "signed_two_by_two_determinant_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "行列式按有號ad−bc計算，不以絕對值取代；矩陣乘法按行乘列，不當逐項乘法。此有限核對不是一般形式證明。",
        "en": "Compute the signed determinant ad−bc,not its absolute value;matrix multiplication uses row-by-column,not entrywise products.This finite audit is not a general formal proof."
      },
      "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 diagonal product := mul(-5, 1)",
      "after": "M diagonal product = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘主對角線a·d。",
        "en": "Multiply the main diagonal entries a·d."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M diagonal product，不以小數近似替代。",
          "en": "Recompute M diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "1"
        ],
        "output": "-5",
        "label": "M diagonal product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "M off-diagonal product := mul(1, 0)",
      "after": "M off-diagonal product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另乘副對角線b·c，減號不可漏。",
        "en": "Multiply the off-diagonal entries b·c separately,retaining subtraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M off-diagonal product，不以小數近似替代。",
          "en": "Recompute M off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "M off-diagonal product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "M signed determinant := sub(-5, 0)",
      "after": "M signed determinant = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減保留符號。",
        "en": "Subtract the terms,retaining the sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M signed determinant，不以小數近似替代。",
          "en": "Recompute M signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-5",
          "0"
        ],
        "output": "-5",
        "label": "M signed determinant"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "MN entry0,0 term0 := mul(-5, 1)",
      "after": "MN entry0,0 term0 = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry0,0 term0，不以小數近似替代。",
          "en": "Recompute MN entry0,0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "1"
        ],
        "output": "-5",
        "label": "MN entry0,0 term0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "MN entry0,0 term1 := mul(1, 2)",
      "after": "MN entry0,0 term1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry0,0 term1，不以小數近似替代。",
          "en": "Recompute MN entry0,0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "MN entry0,0 term1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "MN entry0,0 := sum(-5, 2)",
      "after": "MN entry0,0 = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個行列乘積相加，得到這一格。",
        "en": "Add the two row-column products to obtain this cell."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry0,0，不以小數近似替代。",
          "en": "Recompute MN entry0,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "2"
        ],
        "output": "-3",
        "label": "MN entry0,0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "product=MN;row=0;column=0;terms=[-5, 2]",
      "after": "entry=-3",
      "relation": "implies",
      "rule_id": "row_column_matrix_product_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列與行索引位置固定，不把MN與NM的因子次序或逐項乘法混用。",
        "en": "Retain row/column positions;do not interchange MN and NM or use entrywise multiplication."
      },
      "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": "MN entry0,1 term0 := mul(-5, 0)",
      "after": "MN entry0,1 term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry0,1 term0，不以小數近似替代。",
          "en": "Recompute MN entry0,1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "0"
        ],
        "output": "0",
        "label": "MN entry0,1 term0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "MN entry0,1 term1 := mul(1, 1)",
      "after": "MN entry0,1 term1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry0,1 term1，不以小數近似替代。",
          "en": "Recompute MN entry0,1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "MN entry0,1 term1"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "MN entry0,1 := sum(0, 1)",
      "after": "MN entry0,1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個行列乘積相加，得到這一格。",
        "en": "Add the two row-column products to obtain this cell."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry0,1，不以小數近似替代。",
          "en": "Recompute MN entry0,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "MN entry0,1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "product=MN;row=0;column=1;terms=[0, 1]",
      "after": "entry=1",
      "relation": "implies",
      "rule_id": "row_column_matrix_product_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列與行索引位置固定，不把MN與NM的因子次序或逐項乘法混用。",
        "en": "Retain row/column positions;do not interchange MN and NM or use entrywise multiplication."
      },
      "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": "MN entry1,0 term0 := mul(0, 1)",
      "after": "MN entry1,0 term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry1,0 term0，不以小數近似替代。",
          "en": "Recompute MN entry1,0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "MN entry1,0 term0"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "MN entry1,0 term1 := mul(1, 2)",
      "after": "MN entry1,0 term1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry1,0 term1，不以小數近似替代。",
          "en": "Recompute MN entry1,0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "MN entry1,0 term1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "MN entry1,0 := sum(0, 2)",
      "after": "MN entry1,0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個行列乘積相加，得到這一格。",
        "en": "Add the two row-column products to obtain this cell."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry1,0，不以小數近似替代。",
          "en": "Recompute MN entry1,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "MN entry1,0"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "product=MN;row=1;column=0;terms=[0, 2]",
      "after": "entry=2",
      "relation": "implies",
      "rule_id": "row_column_matrix_product_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列與行索引位置固定，不把MN與NM的因子次序或逐項乘法混用。",
        "en": "Retain row/column positions;do not interchange MN and NM or use entrywise multiplication."
      },
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "MN entry1,1 term0 := mul(0, 0)",
      "after": "MN entry1,1 term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry1,1 term0，不以小數近似替代。",
          "en": "Recompute MN entry1,1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "MN entry1,1 term0"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "MN entry1,1 term1 := mul(1, 1)",
      "after": "MN entry1,1 term1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry1,1 term1，不以小數近似替代。",
          "en": "Recompute MN entry1,1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "MN entry1,1 term1"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "MN entry1,1 := sum(0, 1)",
      "after": "MN entry1,1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個行列乘積相加，得到這一格。",
        "en": "Add the two row-column products to obtain this cell."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN entry1,1，不以小數近似替代。",
          "en": "Recompute MN entry1,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "MN entry1,1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "product=MN;row=1;column=1;terms=[0, 1]",
      "after": "entry=1",
      "relation": "implies",
      "rule_id": "row_column_matrix_product_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列與行索引位置固定，不把MN與NM的因子次序或逐項乘法混用。",
        "en": "Retain row/column positions;do not interchange MN and NM or use entrywise multiplication."
      },
      "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": "NM entry0,0 term0 := mul(1, -5)",
      "after": "NM entry0,0 term0 = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry0,0 term0，不以小數近似替代。",
          "en": "Recompute NM entry0,0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-5"
        ],
        "output": "-5",
        "label": "NM entry0,0 term0"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "NM entry0,0 term1 := mul(0, 0)",
      "after": "NM entry0,0 term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry0,0 term1，不以小數近似替代。",
          "en": "Recompute NM entry0,0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "NM entry0,0 term1"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "NM entry0,0 := sum(-5, 0)",
      "after": "NM entry0,0 = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個行列乘積相加，得到這一格。",
        "en": "Add the two row-column products to obtain this cell."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry0,0，不以小數近似替代。",
          "en": "Recompute NM entry0,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5",
          "0"
        ],
        "output": "-5",
        "label": "NM entry0,0"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "product=NM;row=0;column=0;terms=[-5, 0]",
      "after": "entry=-5",
      "relation": "implies",
      "rule_id": "row_column_matrix_product_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列與行索引位置固定，不把MN與NM的因子次序或逐項乘法混用。",
        "en": "Retain row/column positions;do not interchange MN and NM or use entrywise multiplication."
      },
      "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": "NM entry0,1 term0 := mul(1, 1)",
      "after": "NM entry0,1 term0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry0,1 term0，不以小數近似替代。",
          "en": "Recompute NM entry0,1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "NM entry0,1 term0"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "NM entry0,1 term1 := mul(0, 1)",
      "after": "NM entry0,1 term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry0,1 term1，不以小數近似替代。",
          "en": "Recompute NM entry0,1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "NM entry0,1 term1"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "NM entry0,1 := sum(1, 0)",
      "after": "NM entry0,1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個行列乘積相加，得到這一格。",
        "en": "Add the two row-column products to obtain this cell."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry0,1，不以小數近似替代。",
          "en": "Recompute NM entry0,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "NM entry0,1"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "product=NM;row=0;column=1;terms=[1, 0]",
      "after": "entry=1",
      "relation": "implies",
      "rule_id": "row_column_matrix_product_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列與行索引位置固定，不把MN與NM的因子次序或逐項乘法混用。",
        "en": "Retain row/column positions;do not interchange MN and NM or use entrywise multiplication."
      },
      "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": "NM entry1,0 term0 := mul(2, -5)",
      "after": "NM entry1,0 term0 = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry1,0 term0，不以小數近似替代。",
          "en": "Recompute NM entry1,0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-5"
        ],
        "output": "-10",
        "label": "NM entry1,0 term0"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "NM entry1,0 term1 := mul(1, 0)",
      "after": "NM entry1,0 term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry1,0 term1，不以小數近似替代。",
          "en": "Recompute NM entry1,0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "NM entry1,0 term1"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "NM entry1,0 := sum(-10, 0)",
      "after": "NM entry1,0 = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個行列乘積相加，得到這一格。",
        "en": "Add the two row-column products to obtain this cell."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry1,0，不以小數近似替代。",
          "en": "Recompute NM entry1,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-10",
          "0"
        ],
        "output": "-10",
        "label": "NM entry1,0"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "product=NM;row=1;column=0;terms=[-10, 0]",
      "after": "entry=-10",
      "relation": "implies",
      "rule_id": "row_column_matrix_product_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列與行索引位置固定，不把MN與NM的因子次序或逐項乘法混用。",
        "en": "Retain row/column positions;do not interchange MN and NM or use entrywise multiplication."
      },
      "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": "NM entry1,1 term0 := mul(2, 1)",
      "after": "NM entry1,1 term0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry1,1 term0，不以小數近似替代。",
          "en": "Recompute NM entry1,1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "NM entry1,1 term0"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "NM entry1,1 term1 := mul(1, 1)",
      "after": "NM entry1,1 term1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左矩陣第i行與右矩陣第j列的同k分量相乘。",
        "en": "Multiply the matching k-components of left row i and right column j."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry1,1 term1，不以小數近似替代。",
          "en": "Recompute NM entry1,1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "NM entry1,1 term1"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "NM entry1,1 := sum(2, 1)",
      "after": "NM entry1,1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個行列乘積相加，得到這一格。",
        "en": "Add the two row-column products to obtain this cell."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM entry1,1，不以小數近似替代。",
          "en": "Recompute NM entry1,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "1"
        ],
        "output": "3",
        "label": "NM entry1,1"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "product=NM;row=1;column=1;terms=[2, 1]",
      "after": "entry=3",
      "relation": "implies",
      "rule_id": "row_column_matrix_product_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "列與行索引位置固定，不把MN與NM的因子次序或逐項乘法混用。",
        "en": "Retain row/column positions;do not interchange MN and NM or use entrywise multiplication."
      },
      "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": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "MN diagonal product := mul(-3, 1)",
      "after": "MN diagonal product = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘主對角線a·d。",
        "en": "Multiply the main diagonal entries a·d."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN diagonal product，不以小數近似替代。",
          "en": "Recompute MN diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "1"
        ],
        "output": "-3",
        "label": "MN diagonal product"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "MN off-diagonal product := mul(1, 2)",
      "after": "MN off-diagonal product = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另乘副對角線b·c，減號不可漏。",
        "en": "Multiply the off-diagonal entries b·c separately,retaining subtraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN off-diagonal product，不以小數近似替代。",
          "en": "Recompute MN off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "MN off-diagonal product"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "MN signed determinant := sub(-3, 2)",
      "after": "MN signed determinant = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減保留符號。",
        "en": "Subtract the terms,retaining the sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 MN signed determinant，不以小數近似替代。",
          "en": "Recompute MN signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-3",
          "2"
        ],
        "output": "-5",
        "label": "MN signed determinant"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "NM diagonal product := mul(-5, 3)",
      "after": "NM diagonal product = -15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘主對角線a·d。",
        "en": "Multiply the main diagonal entries a·d."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM diagonal product，不以小數近似替代。",
          "en": "Recompute NM diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "3"
        ],
        "output": "-15",
        "label": "NM diagonal product"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "NM off-diagonal product := mul(1, -10)",
      "after": "NM off-diagonal product = -10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另乘副對角線b·c，減號不可漏。",
        "en": "Multiply the off-diagonal entries b·c separately,retaining subtraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM off-diagonal product，不以小數近似替代。",
          "en": "Recompute NM off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-10"
        ],
        "output": "-10",
        "label": "NM off-diagonal product"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "NM signed determinant := sub(-15, -10)",
      "after": "NM signed determinant = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減保留符號。",
        "en": "Subtract the terms,retaining the sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 NM signed determinant，不以小數近似替代。",
          "en": "Recompute NM signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-15",
          "-10"
        ],
        "output": "-5",
        "label": "NM signed determinant"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "N diagonal product := mul(1, 1)",
      "after": "N diagonal product = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘主對角線a·d。",
        "en": "Multiply the main diagonal entries a·d."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 N diagonal product，不以小數近似替代。",
          "en": "Recompute N diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "N diagonal product"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "N off-diagonal product := mul(0, 2)",
      "after": "N off-diagonal product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另乘副對角線b·c，減號不可漏。",
        "en": "Multiply the off-diagonal entries b·c separately,retaining subtraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 N off-diagonal product，不以小數近似替代。",
          "en": "Recompute N off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "N off-diagonal product"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "N signed determinant := sub(1, 0)",
      "after": "N signed determinant = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減保留符號。",
        "en": "Subtract the terms,retaining the sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 N signed determinant，不以小數近似替代。",
          "en": "Recompute N signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "N signed determinant"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "product of original determinants := mul(-5, 1)",
      "after": "product of original determinants = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原兩個有號行列式另行相乘。",
        "en": "Multiply the two original signed determinants separately."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product of original determinants，不以小數近似替代。",
          "en": "Recompute product of original determinants with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "1"
        ],
        "output": "-5",
        "label": "product of original determinants"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "MN=[[-3, 1], [2, 1]];NM=[[-5, 1], [-10, 3]];determinants=(-5,-5)",
      "after": "matrices equal=否;D(M)D(N)=-5",
      "relation": "implies",
      "rule_id": "determinant_product_not_matrix_commutation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此數值核對中兩序行列式同等於D(M)D(N)，仍須逐格比較矩陣；相同行列式不保證MN=NM。",
        "en": "Both product determinants here equal D(M)D(N),but matrix equality still requires every cell to agree.Equal determinants do not imply MN=NM."
      },
      "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",
          "1"
        ],
        [
          "2",
          "1"
        ]
      ],
      [
        [
          "-5",
          "1"
        ],
        [
          "-10",
          "3"
        ]
      ],
      "-5",
      "-5",
      "-5",
      false
    ]
  },
  "conclusion": {
    "canonical_value": "([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, 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-20261012-advance-20261016-r04-determinant-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r04-determinant-foundations/questions.jsonl",
      "batch_sha256": "e8a897b13857e0c229dad6a8865c2007e594f9c51f688f9113b353ff2562dd0a",
      "original_line": 58,
      "original_line_sha256": "016711ca1e1a0636d61e3229163e1fff366f50755fdfe973bd3406b78a7c8af2",
      "question_compact_sha256": "47fc18ceba8cd6b148247b1c60146396fa3f5958c1bef1e9d2e18f9acea9e82e",
      "english_row_compact_sha256": "f59b73b71340eb21fef745967023f14fd654a54a18c171cde60d8378d24ec41e",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r04-determinant-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r04-determinant-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-determinant": {
        "sha256": "d49050d70b58e0d9d2cab2f5ddc64133813f5a99418865b5941af4f1ec240093",
        "english_sha256": "c20e33f6fef226f483add30aa030f5fa68b5878d5ea16a6e3e75b0b360299311",
        "original_path": "public/data/mko/determinant.json",
        "english_path": "public/data/i18n/editions/mko-determinant.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "944e5a6e7ddc77eb28f9280c1b90da6c7083069b4790cbeb7488d009328efce8",
    "original_candidate_sha256": "2284b8e5e8075d696e1a64e40606a70f7e0a1a60f60bbef908cd518728956d2c",
    "original_candidate_line_sha256": "2284b8e5e8075d696e1a64e40606a70f7e0a1a60f60bbef908cd518728956d2c"
  },
  "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-p14",
    "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": "e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59",
      "question_first_witness_sha256": "0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 47,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "1c88a2931befadd345a2e3706d96ceba98e02e162f704355054ecda9a19a6f13",
      "arithmetic_steps": 37
    },
    "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."
  }
}