{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261103-r01-matrix-rank-0035",
  "pilot_number": "R0034-17685",
  "number": 17685,
  "family_id": "qf-mod34-rank-minor-lower-bound",
  "title": {
    "zh-Hant": "第 17685 題：矩陣秩",
    "en": "Question 17685: A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261103-r01-matrix-rank-0035",
    "batch_id": "batch-20261014-advance-20261103-r01-matrix-rank-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-matrix-rank"
    ],
    "title_zh": "矩陣秩",
    "stem_zh": "2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,9],[0,10]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0], [0], [[1]], 1, 是, rank_at_least_1)",
      "explanation_zh": "Wave5第三層核對=([0], [0], [[1]], 1, 是, rank_at_least_1)；依完整明示資料重算；非零minor只給下界且上界需完整RREF或全部較大minor；det等於零只排除full rank；row-rank等於column-rank依定理。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "矩陣秩"
      ]
    },
    "variant": {
      "family_id": "qf-mod34-rank-minor-lower-bound",
      "seed": 202611031035,
      "parameters": {
        "matrix": [
          [
            1,
            9
          ],
          [
            0,
            10
          ]
        ],
        "rows": [
          0
        ],
        "cols": [
          0
        ],
        "domain": "Exact_square_matrix_rank",
        "operation": "rank-minor-lower-bound",
        "semantic_template": "qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,9],[0,10]],\"rows\":[0]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261103-r01-matrix-rank-0035",
    "batch_id": "batch-20261014-advance-20261103-r01-matrix-rank-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "82ec1f8b332486cfad153ae2cb5c706b7da84e8950ac68a96d77b6438d58046d",
      "family_id": "qf-mod34-rank-minor-lower-bound",
      "seed": 202611031035,
      "semantic_template": "qf-mod34-rank-minor-lower-bound/{\"cols\":[0],\"matrix\":[[1,9],[0,10]],\"rows\":[0]}"
    },
    "fields": {
      "title": "A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank",
      "stem": "Use the complete ordered integer matrix A=[[1, 9], [0, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).",
      "answer": {
        "kind": "expression",
        "value": "([0], [0], [[1]], 1, Yes, rank_at_least_1)",
        "explanation": "Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1)."
      },
      "skills": [
        "A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank"
      ],
      "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": "2×2或3×3完整矩陣與精確rank-preserving消去。操作=rank-minor-lower-bound；資料={\"matrix\":[[1,9],[0,10]],\"rows\":[0],\"cols\":[0]}。請依 [rows,cols,minor,determinant,nonzero,lower_bound] 欄序回答。",
        "en": "Use the complete ordered integer matrix A=[[1, 9], [0, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "rank pivot0 column0 normalized entry0 := div(1, 1)",
      "after": "rank pivot0 column0 normalized entry0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank pivot0 column0 normalized entry0，不以小數近似替代。",
          "en": "Recompute rank pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "rank pivot0 column0 normalized entry0"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "rank pivot0 column0 normalized entry1 := div(9, 1)",
      "after": "rank pivot0 column0 normalized entry1 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank pivot0 column0 normalized entry1，不以小數近似替代。",
          "en": "Recompute rank pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "9",
          "1"
        ],
        "output": "9",
        "label": "rank pivot0 column0 normalized entry1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "pivot row=0;pivot column=0;nonzero pivot=1;old row=[1, 9]",
      "after": "normalized full row=[1, 9]",
      "relation": "implies",
      "rule_id": "rank_nonzero_pivot_entire_row_scaling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列同除非零pivot，逐格保留精確有理值；乘回可逆，禁止浮點近似。",
        "en": "Divide every entry by the same nonzero pivot,retaining exact rationals.The inverse operation multiplies back;do not approximate by floats."
      },
      "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": "rank pivot1 column1 normalized entry0 := div(0, 10)",
      "after": "rank pivot1 column1 normalized entry0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank pivot1 column1 normalized entry0，不以小數近似替代。",
          "en": "Recompute rank pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "10"
        ],
        "output": "0",
        "label": "rank pivot1 column1 normalized entry0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "rank pivot1 column1 normalized entry1 := div(10, 10)",
      "after": "rank pivot1 column1 normalized entry1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank pivot1 column1 normalized entry1，不以小數近似替代。",
          "en": "Recompute rank pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "10",
          "10"
        ],
        "output": "1",
        "label": "rank pivot1 column1 normalized entry1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "pivot row=1;pivot column=1;nonzero pivot=10;old row=[0, 10]",
      "after": "normalized full row=[0, 1]",
      "relation": "implies",
      "rule_id": "rank_nonzero_pivot_entire_row_scaling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列同除非零pivot，逐格保留精確有理值；乘回可逆，禁止浮點近似。",
        "en": "Divide every entry by the same nonzero pivot,retaining exact rationals.The inverse operation multiplies back;do not approximate by floats."
      },
      "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": "rank eliminate row0 pivot1 column1 entry0 product := mul(9, 0)",
      "after": "rank eliminate row0 pivot1 column1 entry0 product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank eliminate row0 pivot1 column1 entry0 product，不以小數近似替代。",
          "en": "Recompute rank eliminate row0 pivot1 column1 entry0 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "0"
        ],
        "output": "0",
        "label": "rank eliminate row0 pivot1 column1 entry0 product"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "rank eliminate row0 pivot1 column1 entry0 difference := sub(1, 0)",
      "after": "rank eliminate row0 pivot1 column1 entry0 difference = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank eliminate row0 pivot1 column1 entry0 difference，不以小數近似替代。",
          "en": "Recompute rank eliminate row0 pivot1 column1 entry0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "rank eliminate row0 pivot1 column1 entry0 difference"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "rank eliminate row0 pivot1 column1 entry1 product := mul(9, 1)",
      "after": "rank eliminate row0 pivot1 column1 entry1 product = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank eliminate row0 pivot1 column1 entry1 product，不以小數近似替代。",
          "en": "Recompute rank eliminate row0 pivot1 column1 entry1 product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "1"
        ],
        "output": "9",
        "label": "rank eliminate row0 pivot1 column1 entry1 product"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "rank eliminate row0 pivot1 column1 entry1 difference := sub(9, 9)",
      "after": "rank eliminate row0 pivot1 column1 entry1 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rank eliminate row0 pivot1 column1 entry1 difference，不以小數近似替代。",
          "en": "Recompute rank eliminate row0 pivot1 column1 entry1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "9",
          "9"
        ],
        "output": "0",
        "label": "rank eliminate row0 pivot1 column1 entry1 difference"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "target row=0;old row=[1, 9];subtract factor=9 times pivot row=[0, 1]",
      "after": "new row=[1, 0];cleared pivot-column entry=0",
      "relation": "implies",
      "rule_id": "rank_exact_all_entry_elimination_not_partial_trace",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每格作原值減factor倍pivot格，加回即可反轉；把此pivot欄其他列都消零才繼續。",
        "en": "Subtract the factor times the pivot entry in every coordinate;adding back reverses the operation.Clear this pivot column in all other rows before continuing."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "complete original square matrix=[[1, 9], [0, 10]];final exact RREF=[[1, 0], [0, 1]];increasing pivot columns=[0, 1]",
      "after": "all rows retained inclzero;rank=2;row operations invertible;row-rank equals column-rank by the finite-dimensional theorem",
      "relation": "implies",
      "rule_id": "rank_full_rref_rank_preserving_certificate_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整消去保留所有列，非零pivot數給精確rank。列運算保rank；row-rank=column-rank依有限維定理，不以幾個成功樣本證一般定理，亦不宣稱實際欄空間逐點不變。",
        "en": "Keep every row and complete all available pivots for exact rank.Invertible row operations preserve rank.Row-rank equals column-rank by the finite-dimensional theorem,not a few successful samples;literal column spans need not remain unchanged."
      },
      "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": "whole original matrix diagonal product := mul(1, 10)",
      "after": "whole original matrix diagonal product = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 whole original matrix diagonal product，不以小數近似替代。",
          "en": "Recompute whole original matrix diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "10"
        ],
        "output": "10",
        "label": "whole original matrix diagonal product"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "whole original matrix off-diagonal product := mul(9, 0)",
      "after": "whole original matrix off-diagonal product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 whole original matrix off-diagonal product，不以小數近似替代。",
          "en": "Recompute whole original matrix off-diagonal product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "0"
        ],
        "output": "0",
        "label": "whole original matrix off-diagonal product"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "whole original matrix determinant difference := sub(10, 0)",
      "after": "whole original matrix determinant difference = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。",
        "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 whole original matrix determinant difference，不以小數近似替代。",
          "en": "Recompute whole original matrix determinant difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "10",
          "0"
        ],
        "output": "10",
        "label": "whole original matrix determinant difference"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "original row indices in SUPPLIED order=[0];original column indices in SUPPLIED order=[0];same whole matrix=[[1, 9], [0, 10]]",
      "after": "selected exact square minor=[[1]]",
      "relation": "implies",
      "rule_id": "rank_selected_minor_order_not_semantic_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "行列索引只要求互異、不要求遞增；照原順序取minor，排列可能反轉det符號，不能拿semantic排序改寫原問題。",
        "en": "Indices must be distinct but need not be increasing.Extract the minor in supplied order;a permutation can change its determinant sign.Do not use semantic-template sorting to rewrite the question."
      },
      "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": "selected supplied-order minor:same1x1 minor=[[1]]",
      "after": "determinant is its sole entry=1",
      "relation": "implies",
      "rule_id": "rank_one_by_one_minor_entry",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1×1行列式就是該原格，不把空minor約定或其他格的值代入。",
        "en": "A1x1 determinant is its one original entry,not an empty-minor convention or another cell."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "supplied minor size=1;exact signed determinant=1;nonzero=是",
      "after": "certificate from THIS minor=rank_at_least_1",
      "relation": "implies",
      "rule_id": "rank_minor_lower_bound_not_whole_rank_from_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零r階minor才給rank≥r；這個minor為零只扣住它的證書，不推出整份矩陣rank為零，也不擅改本題輸出為另有的完整RREF結論。",
        "en": "A nonzero r-minor gives rank≥r.A zero selected minor withholds only its certificate;it does not give whole-matrix rank0 or authorize replacing this output with the separate full-RREF result."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        0
      ],
      [
        0
      ],
      [
        [
          1
        ]
      ],
      1,
      true,
      "rank_at_least_1"
    ]
  },
  "conclusion": {
    "canonical_value": "([0], [0], [[1]], 1, 是, rank_at_least_1)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261103-r01-matrix-rank-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261103-r01-matrix-rank-foundations/questions.jsonl",
      "batch_sha256": "4b2cd41178797ab59baab1d5c8de49ed79ffcafb129a100a24bcd595b80e2ffa",
      "original_line": 35,
      "original_line_sha256": "113b79aa1bc5e6618b8e25752d111938c72615c1150d7a80f32ab06c404ab6fe",
      "question_compact_sha256": "82ec1f8b332486cfad153ae2cb5c706b7da84e8950ac68a96d77b6438d58046d",
      "english_row_compact_sha256": "63ded881e9f6f7f3e067481bc3f87a2be880a905aec37f6fa6643042b872e83b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261103-r01-matrix-rank-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261103-r01-matrix-rank-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-matrix-rank": {
        "sha256": "fdddbb93e9e8a00070f3e18fbc7fb5a9406d8ddab249ec9f7798986636ffd45e",
        "english_sha256": "a23c09f0694655deaf0ccb46613f70ba13f76194f5d465dd35153804fabb5727",
        "original_path": "public/data/mko/matrix-rank.json",
        "english_path": "public/data/i18n/editions/mko-matrix-rank.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d5b0037cbaeef26e3828d04bf29e0e0f6e466926098b6b09e3237d1fdbdbd0e2",
    "original_candidate_sha256": "bd8d17137a703073f51feaa597706bd6215902c0f5a8070de3818f274bd154b5",
    "original_candidate_line_sha256": "bd8d17137a703073f51feaa597706bd6215902c0f5a8070de3818f274bd154b5"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-008-p35",
    "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": "f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336",
      "question_first_witness_sha256": "22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 18,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "97f28ca964644dc6a0fcee4c634bc2feabf5e9840388c3d225aa2a709c8d70ce",
      "arithmetic_steps": 11
    },
    "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."
  }
}