{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261104-r01-rank-nullity-theorem-0078",
  "pilot_number": "R0035-18378",
  "number": 18378,
  "family_id": "qf-mod35-ranknull-sampled-boundary",
  "title": {
    "zh-Hant": "第 18378 題：秩零度定理",
    "en": "Question 18378: Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261104-r01-rank-nullity-theorem-0078",
    "batch_id": "batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-rank-nullity-theorem"
    ],
    "title_zh": "秩零度定理",
    "stem_zh": "明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,2],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)",
      "explanation_zh": "Wave5第四層核對=([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)；依完整明示資料重算；n等於kernel dimension加rank且nullity不是解的個數；kernel image rank皆須symbolic certificate；sample不足。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "秩零度定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod35-ranknull-sampled-boundary",
      "seed": 202611041078,
      "parameters": {
        "matrix": [
          [
            1,
            0,
            2
          ],
          [
            0,
            1,
            1
          ]
        ],
        "samples": [
          [
            0,
            0,
            0
          ],
          [
            1,
            0,
            0
          ]
        ],
        "domain": "Exact_finite_matrix_rank_nullity",
        "operation": "ranknull-sampled-boundary",
        "semantic_template": "qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,2],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261104-r01-rank-nullity-theorem-0078",
    "batch_id": "batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "352329891de61b6f14fb8a8490e3ffad0ef880e482003cbd5858b50a4e397a48",
      "family_id": "qf-mod35-ranknull-sampled-boundary",
      "seed": 202611041078,
      "semantic_template": "qf-mod35-ranknull-sampled-boundary/{\"matrix\":[[1,0,2],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}"
    },
    "fields": {
      "title": "Sampled Vectors Cannot Replace the Full Kernel/Image Certificates",
      "stem": "Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)",
        "explanation": "The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)."
      },
      "skills": [
        "Sampled Vectors Cannot Replace the Full Kernel/Image Certificates"
      ],
      "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": "明示m×n矩陣代表T:R^n到R^m並使用exact RREF。操作=ranknull-sampled-boundary；資料={\"matrix\":[[1,0,2],[0,1,1]],\"samples\":[[0,0,0],[1,0,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=ranknull-sampled-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[samples, sufficient, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題的歷史介面標記保留在快照；本頁直接列原始欄位順序，不需JavaScript或外部表。欄位描述不自動成為數學證據，No標記須依本題的聲稱範圍解讀。",
        "en": "Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "complete original matrix=[[1, 0, 2], [0, 1, 1]];shape=2 rows by3 columns",
      "after": "usual real map T:R^3 to R^2;SOURCE dimension=3;TARGET dimension=2",
      "relation": "implies",
      "rule_id": "ranknull_rectangular_map_direction_source_columns",
      "conditions": [],
      "explanation": {
        "zh-Hant": "行數是target維度，欄數才source維度；不把本題矩形矩陣硬套平方或前輪R²特例。",
        "en": "Rows give target dimension and columns source dimension.Do not force this rectangular map into a square or earlier R²-only case."
      },
      "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": "ranknull pivot0 column0 normalized entry0 := div(1, 1)",
      "after": "ranknull pivot0 column0 normalized entry0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull pivot0 column0 normalized entry0，不以小數近似替代。",
          "en": "Recompute ranknull pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "ranknull pivot0 column0 normalized entry0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ranknull pivot0 column0 normalized entry1 := div(0, 1)",
      "after": "ranknull pivot0 column0 normalized entry1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull pivot0 column0 normalized entry1，不以小數近似替代。",
          "en": "Recompute ranknull pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "ranknull pivot0 column0 normalized entry1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ranknull pivot0 column0 normalized entry2 := div(2, 1)",
      "after": "ranknull pivot0 column0 normalized entry2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull pivot0 column0 normalized entry2，不以小數近似替代。",
          "en": "Recompute ranknull pivot0 column0 normalized entry2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "ranknull pivot0 column0 normalized entry2"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "pivot row=0;column=0;nonzero exact pivot=1;old row=[1, 0, 2]",
      "after": "normalized complete row=[1, 0, 2]",
      "relation": "implies",
      "rule_id": "ranknull_exact_pivot_scaling_all_entries",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列同除非零pivot，保精確有理數，不用浮點。",
        "en": "Divide the entire row by its nonzero pivot with exact rational values,not 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": "ranknull pivot1 column1 normalized entry0 := div(0, 1)",
      "after": "ranknull pivot1 column1 normalized entry0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull pivot1 column1 normalized entry0，不以小數近似替代。",
          "en": "Recompute ranknull pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "ranknull pivot1 column1 normalized entry0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ranknull pivot1 column1 normalized entry1 := div(1, 1)",
      "after": "ranknull pivot1 column1 normalized entry1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull pivot1 column1 normalized entry1，不以小數近似替代。",
          "en": "Recompute ranknull pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "ranknull pivot1 column1 normalized entry1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ranknull pivot1 column1 normalized entry2 := div(1, 1)",
      "after": "ranknull pivot1 column1 normalized entry2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull pivot1 column1 normalized entry2，不以小數近似替代。",
          "en": "Recompute ranknull pivot1 column1 normalized entry2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "ranknull pivot1 column1 normalized entry2"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "pivot row=1;column=1;nonzero exact pivot=1;old row=[0, 1, 1]",
      "after": "normalized complete row=[0, 1, 1]",
      "relation": "implies",
      "rule_id": "ranknull_exact_pivot_scaling_all_entries",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列同除非零pivot，保精確有理數，不用浮點。",
        "en": "Divide the entire row by its nonzero pivot with exact rational values,not 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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ranknull free2 pivot0 solved coordinate := mul(-1, 2)",
      "after": "ranknull free2 pivot0 solved coordinate = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull free2 pivot0 solved coordinate，不以小數近似替代。",
          "en": "Recompute ranknull free2 pivot0 solved coordinate with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "2"
        ],
        "output": "-2",
        "label": "ranknull free2 pivot0 solved coordinate"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "ranknull free2 pivot1 solved coordinate := mul(-1, 1)",
      "after": "ranknull free2 pivot1 solved coordinate = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull free2 pivot1 solved coordinate，不以小數近似替代。",
          "en": "Recompute ranknull free2 pivot1 solved coordinate with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "ranknull free2 pivot1 solved coordinate"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "selected free column=2;ALL free columns=[2];same exact RREF=[[1, 0, 2], [0, 1, 1]]",
      "after": "set selected free coordinate1,other free coordinates0;solved kernel vector=[-2, -1, 1];NO primitive/sign renormalization",
      "relation": "implies",
      "rule_id": "ranknull_kernel_free_coordinate_source_convention",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個free欄給一基向量，該自由座標固定1後解pivot座標；保RREF原約定，不套前輪primitive整數正首座標的不同協定。",
        "en": "One basis vector per free column sets that free coordinate to1 and the others to0,then solves pivot coordinates.Keep this RREF convention rather than the earlier primitive-integer positive-leading convention."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "ranknull kernel free2 original row0 term0 := mul(1, -2)",
      "after": "ranknull kernel free2 original row0 term0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull kernel free2 original row0 term0，不以小數近似替代。",
          "en": "Recompute ranknull kernel free2 original row0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-2",
        "label": "ranknull kernel free2 original row0 term0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "ranknull kernel free2 original row0 term1 := mul(0, -1)",
      "after": "ranknull kernel free2 original row0 term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull kernel free2 original row0 term1，不以小數近似替代。",
          "en": "Recompute ranknull kernel free2 original row0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "ranknull kernel free2 original row0 term1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "ranknull kernel free2 original row0 term2 := mul(2, 1)",
      "after": "ranknull kernel free2 original row0 term2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull kernel free2 original row0 term2，不以小數近似替代。",
          "en": "Recompute ranknull kernel free2 original row0 term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "ranknull kernel free2 original row0 term2"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ranknull kernel free2 original row0 sum := sum(-2, 0, 2)",
      "after": "ranknull kernel free2 original row0 sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull kernel free2 original row0 sum，不以小數近似替代。",
          "en": "Recompute ranknull kernel free2 original row0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "0",
          "2"
        ],
        "output": "0",
        "label": "ranknull kernel free2 original row0 sum"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "ranknull kernel free2 original row1 term0 := mul(0, -2)",
      "after": "ranknull kernel free2 original row1 term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull kernel free2 original row1 term0，不以小數近似替代。",
          "en": "Recompute ranknull kernel free2 original row1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "0",
        "label": "ranknull kernel free2 original row1 term0"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ranknull kernel free2 original row1 term1 := mul(1, -1)",
      "after": "ranknull kernel free2 original row1 term1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull kernel free2 original row1 term1，不以小數近似替代。",
          "en": "Recompute ranknull kernel free2 original row1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "ranknull kernel free2 original row1 term1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "ranknull kernel free2 original row1 term2 := mul(1, 1)",
      "after": "ranknull kernel free2 original row1 term2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull kernel free2 original row1 term2，不以小數近似替代。",
          "en": "Recompute ranknull kernel free2 original row1 term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "ranknull kernel free2 original row1 term2"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "ranknull kernel free2 original row1 sum := sum(0, -1, 1)",
      "after": "ranknull kernel free2 original row1 sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull kernel free2 original row1 sum，不以小數近似替代。",
          "en": "Recompute ranknull kernel free2 original row1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-1",
          "1"
        ],
        "output": "0",
        "label": "ranknull kernel free2 original row1 sum"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "complete exact RREF=[[1, 0, 2], [0, 1, 1]];pivot columns=[0, 1];free columns=[2];verified kernel vectors=[[-2, -1, 1]]",
      "after": "for ANY real solution,its free coordinates are unique combination coefficients of these vectors;pivot equations determine the rest;free-coordinate identity also proves independence",
      "relation": "implies",
      "rule_id": "ranknull_full_real_kernel_both_inclusions_and_independence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "可逆列運算保Mx=0。每個解的自由座標就是這些向量的係數，pivot座標由方程唯一決定；反向線性組合都映零。自由座標identity亦排除非零線性關係，故完整basis而非樣本。",
        "en": "Invertible row operations preserve Mx=0.Every solution's free coordinates are the unique coefficients in this family;the pivot equations determine the rest.Conversely every combination maps to zero.The free-coordinate identity proves independence,so this is a full basis,not samples."
      },
      "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": "ranknull image column0 row0 pivot0 term := mul(1, 1)",
      "after": "ranknull image column0 row0 pivot0 term = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column0 row0 pivot0 term，不以小數近似替代。",
          "en": "Recompute ranknull image column0 row0 pivot0 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "ranknull image column0 row0 pivot0 term"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "ranknull image column0 row0 pivot1 term := mul(0, 0)",
      "after": "ranknull image column0 row0 pivot1 term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column0 row0 pivot1 term，不以小數近似替代。",
          "en": "Recompute ranknull image column0 row0 pivot1 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "ranknull image column0 row0 pivot1 term"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "ranknull image column0 row0 sum := sum(1, 0)",
      "after": "ranknull image column0 row0 sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column0 row0 sum，不以小數近似替代。",
          "en": "Recompute ranknull image column0 row0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "ranknull image column0 row0 sum"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "ranknull image column0 row1 pivot0 term := mul(0, 1)",
      "after": "ranknull image column0 row1 pivot0 term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column0 row1 pivot0 term，不以小數近似替代。",
          "en": "Recompute ranknull image column0 row1 pivot0 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "ranknull image column0 row1 pivot0 term"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "ranknull image column0 row1 pivot1 term := mul(1, 0)",
      "after": "ranknull image column0 row1 pivot1 term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column0 row1 pivot1 term，不以小數近似替代。",
          "en": "Recompute ranknull image column0 row1 pivot1 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "ranknull image column0 row1 pivot1 term"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "ranknull image column0 row1 sum := sum(0, 0)",
      "after": "ranknull image column0 row1 sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column0 row1 sum，不以小數近似替代。",
          "en": "Recompute ranknull image column0 row1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "ranknull image column0 row1 sum"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "original column index=0;ORIGINAL pivot-column basis=[[1, 0], [0, 1]];RREF coordinate coefficients=[1, 0]",
      "after": "reconstructed original column=[1, 0]",
      "relation": "implies",
      "rule_id": "ranknull_original_pivot_columns_not_rref_image_columns",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從原pivot欄重建每個原欄，故全部像都在此span；每個原pivot欄本身是M e_p，反向包含亦成立。不能拿RREF欄當原像basis。",
        "en": "Reconstruct every original column from the original pivot columns,so the full image lies in their span.Each pivot column itself equals M e_p,giving the reverse inclusion.Do not substitute RREF columns as the original image basis."
      },
      "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": "ranknull image column1 row0 pivot0 term := mul(1, 0)",
      "after": "ranknull image column1 row0 pivot0 term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column1 row0 pivot0 term，不以小數近似替代。",
          "en": "Recompute ranknull image column1 row0 pivot0 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "ranknull image column1 row0 pivot0 term"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "ranknull image column1 row0 pivot1 term := mul(0, 1)",
      "after": "ranknull image column1 row0 pivot1 term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column1 row0 pivot1 term，不以小數近似替代。",
          "en": "Recompute ranknull image column1 row0 pivot1 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "ranknull image column1 row0 pivot1 term"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "ranknull image column1 row0 sum := sum(0, 0)",
      "after": "ranknull image column1 row0 sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column1 row0 sum，不以小數近似替代。",
          "en": "Recompute ranknull image column1 row0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "ranknull image column1 row0 sum"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "ranknull image column1 row1 pivot0 term := mul(0, 0)",
      "after": "ranknull image column1 row1 pivot0 term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column1 row1 pivot0 term，不以小數近似替代。",
          "en": "Recompute ranknull image column1 row1 pivot0 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "ranknull image column1 row1 pivot0 term"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "ranknull image column1 row1 pivot1 term := mul(1, 1)",
      "after": "ranknull image column1 row1 pivot1 term = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column1 row1 pivot1 term，不以小數近似替代。",
          "en": "Recompute ranknull image column1 row1 pivot1 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "ranknull image column1 row1 pivot1 term"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "ranknull image column1 row1 sum := sum(0, 1)",
      "after": "ranknull image column1 row1 sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column1 row1 sum，不以小數近似替代。",
          "en": "Recompute ranknull image column1 row1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "ranknull image column1 row1 sum"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "original column index=1;ORIGINAL pivot-column basis=[[1, 0], [0, 1]];RREF coordinate coefficients=[0, 1]",
      "after": "reconstructed original column=[0, 1]",
      "relation": "implies",
      "rule_id": "ranknull_original_pivot_columns_not_rref_image_columns",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從原pivot欄重建每個原欄，故全部像都在此span；每個原pivot欄本身是M e_p，反向包含亦成立。不能拿RREF欄當原像basis。",
        "en": "Reconstruct every original column from the original pivot columns,so the full image lies in their span.Each pivot column itself equals M e_p,giving the reverse inclusion.Do not substitute RREF columns as the original image basis."
      },
      "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": "ranknull image column2 row0 pivot0 term := mul(1, 2)",
      "after": "ranknull image column2 row0 pivot0 term = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column2 row0 pivot0 term，不以小數近似替代。",
          "en": "Recompute ranknull image column2 row0 pivot0 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "ranknull image column2 row0 pivot0 term"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "ranknull image column2 row0 pivot1 term := mul(0, 1)",
      "after": "ranknull image column2 row0 pivot1 term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column2 row0 pivot1 term，不以小數近似替代。",
          "en": "Recompute ranknull image column2 row0 pivot1 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "ranknull image column2 row0 pivot1 term"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "ranknull image column2 row0 sum := sum(2, 0)",
      "after": "ranknull image column2 row0 sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column2 row0 sum，不以小數近似替代。",
          "en": "Recompute ranknull image column2 row0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "ranknull image column2 row0 sum"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "ranknull image column2 row1 pivot0 term := mul(0, 2)",
      "after": "ranknull image column2 row1 pivot0 term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column2 row1 pivot0 term，不以小數近似替代。",
          "en": "Recompute ranknull image column2 row1 pivot0 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "ranknull image column2 row1 pivot0 term"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "ranknull image column2 row1 pivot1 term := mul(1, 1)",
      "after": "ranknull image column2 row1 pivot1 term = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column2 row1 pivot1 term，不以小數近似替代。",
          "en": "Recompute ranknull image column2 row1 pivot1 term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "ranknull image column2 row1 pivot1 term"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "ranknull image column2 row1 sum := sum(0, 1)",
      "after": "ranknull image column2 row1 sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull image column2 row1 sum，不以小數近似替代。",
          "en": "Recompute ranknull image column2 row1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "ranknull image column2 row1 sum"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "original column index=2;ORIGINAL pivot-column basis=[[1, 0], [0, 1]];RREF coordinate coefficients=[2, 1]",
      "after": "reconstructed original column=[2, 1]",
      "relation": "implies",
      "rule_id": "ranknull_original_pivot_columns_not_rref_image_columns",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從原pivot欄重建每個原欄，故全部像都在此span；每個原pivot欄本身是M e_p，反向包含亦成立。不能拿RREF欄當原像basis。",
        "en": "Reconstruct every original column from the original pivot columns,so the full image lies in their span.Each pivot column itself equals M e_p,giving the reverse inclusion.Do not substitute RREF columns as the original image basis."
      },
      "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": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "ORIGINAL pivot-column family=[[1, 0], [0, 1]];pivot indices=[0, 1];same RREF pivot columns=[[1, 0], [0, 1]]",
      "after": "ANY zero combination stays zero under invertible row operations;distinct target-coordinate identity columns force EVERY coefficient0;family independent,and previous reconstructions prove full image span",
      "relation": "implies",
      "rule_id": "ranknull_image_basis_independence_via_pivot_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原pivot欄的零線性組合左乘同一可逆列運算後，變成target座標標準向量的零組合，各係數只能0。配合上面完整生成性才是像basis；空族獨立且span為零空間。",
        "en": "Apply the same invertible row operations to any zero combination of original pivot columns.The resulting distinct target-coordinate identity columns force all coefficients to zero.With the full-span reconstructions this is an image basis.The empty family is independent and spans the zero space."
      },
      "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": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "ranknull source dimension identity sum := sum(1, 2)",
      "after": "ranknull source dimension identity sum = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ranknull source dimension identity sum，不以小數近似替代。",
          "en": "Recompute ranknull source dimension identity sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "2"
        ],
        "output": "3",
        "label": "ranknull source dimension identity sum"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "original matrix shape=2x3;rank=2;FULL kernel basis count=1;FULL original-pivot image basis count=2",
      "after": "SOURCE column dimension=3=nullity+rank=3;target row dimension=2 retained separately",
      "relation": "implies",
      "rule_id": "ranknull_dimension_identity_not_solution_count",
      "conditions": [],
      "explanation": {
        "zh-Hant": "pivot/free欄劃分source座標，完整kernel與image basis數分別為nullity與rank；nullity不是零解或格點個數，亦不拿target行數代替source欄數。",
        "en": "Pivot/free columns partition source coordinates,and the full kernel/image bases give nullity and rank.Nullity is not the number of solutions or grid points,and target rows do not replace source columns."
      },
      "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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "original supplied finite samples retained=[[0, 0, 0], [1, 0, 0]];SEPARATE whole-matrix certificates available",
      "after": "samples-alone sufficiency=否;sampled_vectors_do_not_prove_kernel_image_or_rank_nullity",
      "relation": "implies",
      "rule_id": "ranknull_samples_no_not_denial_of_symbolic_certificate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "不假設samples已在kernel或image，也不拿有限樣本個數當dimension；No只扣住樣本自身的證明力，不否定已由完整矩陣得的symbolic證書。",
        "en": "Do not presume samples lie in the kernel/image or count them as dimension.The No withholds inference from samples alone,not the separate symbolic certificates derived from the complete matrix."
      },
      "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,
          0
        ],
        [
          1,
          0,
          0
        ]
      ],
      false,
      "sampled_vectors_do_not_prove_kernel_image_or_rank_nullity"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 0, 0], [1, 0, 0]], 否, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)",
    "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-20261104-r01-rank-nullity-theorem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations/questions.jsonl",
      "batch_sha256": "d3eac7e1e3687bf898001b6c32145204b9d57a708fef7de1ef7f623d9c902b20",
      "original_line": 78,
      "original_line_sha256": "cf70aff222b718dea683f2d068c341ee13cbc10801ab834196fd72f3917392b6",
      "question_compact_sha256": "352329891de61b6f14fb8a8490e3ffad0ef880e482003cbd5858b50a4e397a48",
      "english_row_compact_sha256": "cfd4e947ebc62f058c546683a566adef5128933f586b25af6f185dd274a0d50d",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-rank-nullity-theorem": {
        "sha256": "e567ac132f3923949776cbd2866741a3564ba2c4d3e79d654abe99ba42b5ee0a",
        "english_sha256": "c3ab90f3622c1f66f461358b65aaa251f413745f3f1300b780355f32f83b5c59",
        "original_path": "public/data/mko/rank-nullity-theorem.json",
        "english_path": "public/data/i18n/editions/mko-rank-nullity-theorem.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9adff0bbee39d9c3dec346304161ea9633576ccd5ca954f6b64ad482d2533a8a",
    "original_candidate_sha256": "77fa9b298b609ecbbb40c6deb0659dd579113c011ffd1b9047bd65892d0a503b",
    "original_candidate_line_sha256": "77fa9b298b609ecbbb40c6deb0659dd579113c011ffd1b9047bd65892d0a503b"
  },
  "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-p36",
    "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": "713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc",
      "question_first_witness_sha256": "0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2"
    },
    "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": "f698f986312bf24a1cc332e3653331041e5e973f0d069436f9af0d70c22ad1e7",
      "arithmetic_steps": 35
    },
    "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."
  }
}