{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261021-r06-linear-programming-0039",
  "pilot_number": "R0003-02439",
  "number": 2439,
  "family_id": "qf-app21-lp-feasible-inventory",
  "title": {
    "zh-Hant": "第 2439 題：有限線性規劃候選",
    "en": "Question 2439: Feasible Indices in Canonical Candidate-Grid Order"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261021-r06-linear-programming-0039",
    "batch_id": "batch-20261013-advance-20261021-r06-linear-programming-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-programming"
    ],
    "title_zh": "有限線性規劃候選",
    "stem_zh": "完整明示有限候選網格。操作=lp-feasible-inventory；資料={\"points\":[[0,0],[1,1],[2,3],[1,2]],\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]]}。請依 [constraints,point_rows,feasible_indices] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [是, 是, 是], 是, null], [1, 1, 1, [是, 是, 是], 是, null], [2, 1, 2, [是, 是, 是], 是, null], [3, 2, 3, [是, 是, 是], 是, null]], [0, 1, 2, 3])",
      "explanation_zh": "應用擴展核對=([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [是, 是, 是], 是, null], [1, 1, 1, [是, 是, 是], 是, null], [2, 1, 2, [是, 是, 是], 是, null], [3, 2, 3, [是, 是, 是], 是, null]], [0, 1, 2, 3])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；只稱所給候選中最佳；不宣稱一般LP全域最優。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限線性規劃候選"
      ]
    },
    "variant": {
      "family_id": "qf-app21-lp-feasible-inventory",
      "seed": 202610216039,
      "parameters": {
        "points": [
          [
            0,
            0
          ],
          [
            1,
            1
          ],
          [
            2,
            3
          ],
          [
            1,
            2
          ]
        ],
        "constraints": [
          [
            -1,
            0,
            0
          ],
          [
            0,
            -1,
            0
          ],
          [
            1,
            1,
            9
          ]
        ],
        "domain": "Finite_linear_program_candidate_grid",
        "operation": "lp-feasible-inventory",
        "semantic_template": "qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"points\":[[0,0],[1,1],[1,2],[2,3]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261021-r06-linear-programming-0039",
    "batch_id": "batch-20261013-advance-20261021-r06-linear-programming-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ab4245a9f412249ee5a740e06e29a0ed27cc8144285956075d2fcc10427ad191",
      "family_id": "qf-app21-lp-feasible-inventory",
      "seed": 202610216039,
      "semantic_template": "qf-app21-lp-feasible-inventory/{\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]],\"points\":[[0,0],[1,1],[1,2],[2,3]]}"
    },
    "fields": {
      "title": "Feasible Indices in Canonical Candidate-Grid Order",
      "stem": "Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 3], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.",
      "answer": {
        "kind": "expression",
        "value": "([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 3, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])",
        "explanation": "Keep the per-constraint truth list as well as overall feasibility. Canonical indices are not the original input positions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 1, [Yes, Yes, Yes], Yes, null], [2, 1, 2, [Yes, Yes, Yes], Yes, null], [3, 2, 3, [Yes, Yes, Yes], Yes, null]], [0, 1, 2, 3])."
      },
      "skills": [
        "Feasible Indices in Canonical Candidate-Grid Order"
      ],
      "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": "完整明示有限候選網格。操作=lp-feasible-inventory；資料={\"points\":[[0,0],[1,1],[2,3],[1,2]],\"constraints\":[[-1,0,0],[0,-1,0],[1,1,9]]}。請依 [constraints,point_rows,feasible_indices] 欄序回答。",
        "en": "Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 1], [2, 3], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "explicit finite candidate grid and linear rows a·x+b·y≤c",
      "after": "keep candidate-grid optimization separate from continuous LP",
      "relation": "implies",
      "rule_id": "finite_LP_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題只在列出的有限候選中核對；未提供完整頂點或全可行域證書，不宣稱一般連續LP全域最優。",
        "en": "Audit only the listed finite candidates;without a complete-vertex or full-feasible-domain certificate,this is not a general continuous-LP optimum claim."
      },
      "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": "[[0, 0], [1, 1], [2, 3], [1, 2]]",
      "after": "deduplicated canonical indexed points=[[0, 0, 0], [1, 1, 1], [2, 1, 2], [3, 2, 3]]",
      "relation": "implies",
      "rule_id": "candidate_index_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原資料保留；輸出索引依原契約對去重後的點排序，不冒充原輸入位置。",
        "en": "Keep the original data;output indices follow the source contract's sorted distinct points,not original input positions."
      },
      "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": "point0 row0 a·x := mul(-1, 0)",
      "after": "point0 row0 a·x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 row0 a·x，不以小數近似替代。",
          "en": "Recompute point0 row0 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "point0 row0 a·x"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "point0 row0 b·y := mul(0, 0)",
      "after": "point0 row0 b·y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 row0 b·y，不以小數近似替代。",
          "en": "Recompute point0 row0 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "point0 row0 b·y"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "point0 row0 LHS := sum(0, 0)",
      "after": "point0 row0 LHS = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 row0 LHS，不以小數近似替代。",
          "en": "Recompute point0 row0 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "point0 row0 LHS"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "row0: 0≤0",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "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": "point0 row1 a·x := mul(0, 0)",
      "after": "point0 row1 a·x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 row1 a·x，不以小數近似替代。",
          "en": "Recompute point0 row1 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "point0 row1 a·x"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "point0 row1 b·y := mul(-1, 0)",
      "after": "point0 row1 b·y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 row1 b·y，不以小數近似替代。",
          "en": "Recompute point0 row1 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "point0 row1 b·y"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "point0 row1 LHS := sum(0, 0)",
      "after": "point0 row1 LHS = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 row1 LHS，不以小數近似替代。",
          "en": "Recompute point0 row1 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "point0 row1 LHS"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "row1: 0≤0",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "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": "point0 row2 a·x := mul(1, 0)",
      "after": "point0 row2 a·x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 row2 a·x，不以小數近似替代。",
          "en": "Recompute point0 row2 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "point0 row2 a·x"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "point0 row2 b·y := mul(1, 0)",
      "after": "point0 row2 b·y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 row2 b·y，不以小數近似替代。",
          "en": "Recompute point0 row2 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "point0 row2 b·y"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "point0 row2 LHS := sum(0, 0)",
      "after": "point0 row2 LHS = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 row2 LHS，不以小數近似替代。",
          "en": "Recompute point0 row2 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "point0 row2 LHS"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "row2: 0≤9",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "point1 row0 a·x := mul(-1, 1)",
      "after": "point1 row0 a·x = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 row0 a·x，不以小數近似替代。",
          "en": "Recompute point1 row0 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "point1 row0 a·x"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "point1 row0 b·y := mul(0, 1)",
      "after": "point1 row0 b·y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 row0 b·y，不以小數近似替代。",
          "en": "Recompute point1 row0 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "point1 row0 b·y"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "point1 row0 LHS := sum(-1, 0)",
      "after": "point1 row0 LHS = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 row0 LHS，不以小數近似替代。",
          "en": "Recompute point1 row0 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "-1",
        "label": "point1 row0 LHS"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "row0: -1≤0",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "point1 row1 a·x := mul(0, 1)",
      "after": "point1 row1 a·x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 row1 a·x，不以小數近似替代。",
          "en": "Recompute point1 row1 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "point1 row1 a·x"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "point1 row1 b·y := mul(-1, 1)",
      "after": "point1 row1 b·y = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 row1 b·y，不以小數近似替代。",
          "en": "Recompute point1 row1 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "point1 row1 b·y"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "point1 row1 LHS := sum(0, -1)",
      "after": "point1 row1 LHS = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 row1 LHS，不以小數近似替代。",
          "en": "Recompute point1 row1 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "-1",
        "label": "point1 row1 LHS"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "row1: -1≤0",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "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": "point1 row2 a·x := mul(1, 1)",
      "after": "point1 row2 a·x = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 row2 a·x，不以小數近似替代。",
          "en": "Recompute point1 row2 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "point1 row2 a·x"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "point1 row2 b·y := mul(1, 1)",
      "after": "point1 row2 b·y = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 row2 b·y，不以小數近似替代。",
          "en": "Recompute point1 row2 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "point1 row2 b·y"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "point1 row2 LHS := sum(1, 1)",
      "after": "point1 row2 LHS = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 row2 LHS，不以小數近似替代。",
          "en": "Recompute point1 row2 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1"
        ],
        "output": "2",
        "label": "point1 row2 LHS"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "row2: 2≤9",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "point2 row0 a·x := mul(-1, 1)",
      "after": "point2 row0 a·x = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 row0 a·x，不以小數近似替代。",
          "en": "Recompute point2 row0 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "point2 row0 a·x"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "point2 row0 b·y := mul(0, 2)",
      "after": "point2 row0 b·y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 row0 b·y，不以小數近似替代。",
          "en": "Recompute point2 row0 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "point2 row0 b·y"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "point2 row0 LHS := sum(-1, 0)",
      "after": "point2 row0 LHS = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 row0 LHS，不以小數近似替代。",
          "en": "Recompute point2 row0 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "-1",
        "label": "point2 row0 LHS"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "row0: -1≤0",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "point2 row1 a·x := mul(0, 1)",
      "after": "point2 row1 a·x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 row1 a·x，不以小數近似替代。",
          "en": "Recompute point2 row1 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "point2 row1 a·x"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "point2 row1 b·y := mul(-1, 2)",
      "after": "point2 row1 b·y = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 row1 b·y，不以小數近似替代。",
          "en": "Recompute point2 row1 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "2"
        ],
        "output": "-2",
        "label": "point2 row1 b·y"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "point2 row1 LHS := sum(0, -2)",
      "after": "point2 row1 LHS = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 row1 LHS，不以小數近似替代。",
          "en": "Recompute point2 row1 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "-2",
        "label": "point2 row1 LHS"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "row1: -2≤0",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "point2 row2 a·x := mul(1, 1)",
      "after": "point2 row2 a·x = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 row2 a·x，不以小數近似替代。",
          "en": "Recompute point2 row2 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "point2 row2 a·x"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "point2 row2 b·y := mul(1, 2)",
      "after": "point2 row2 b·y = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 row2 b·y，不以小數近似替代。",
          "en": "Recompute point2 row2 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "point2 row2 b·y"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "point2 row2 LHS := sum(1, 2)",
      "after": "point2 row2 LHS = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 row2 LHS，不以小數近似替代。",
          "en": "Recompute point2 row2 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "2"
        ],
        "output": "3",
        "label": "point2 row2 LHS"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "row2: 3≤9",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "point3 row0 a·x := mul(-1, 2)",
      "after": "point3 row0 a·x = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 row0 a·x，不以小數近似替代。",
          "en": "Recompute point3 row0 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "2"
        ],
        "output": "-2",
        "label": "point3 row0 a·x"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "point3 row0 b·y := mul(0, 3)",
      "after": "point3 row0 b·y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 row0 b·y，不以小數近似替代。",
          "en": "Recompute point3 row0 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "3"
        ],
        "output": "0",
        "label": "point3 row0 b·y"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "point3 row0 LHS := sum(-2, 0)",
      "after": "point3 row0 LHS = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 row0 LHS，不以小數近似替代。",
          "en": "Recompute point3 row0 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "-2",
        "label": "point3 row0 LHS"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "row0: -2≤0",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "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": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "point3 row1 a·x := mul(0, 2)",
      "after": "point3 row1 a·x = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 row1 a·x，不以小數近似替代。",
          "en": "Recompute point3 row1 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "point3 row1 a·x"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "point3 row1 b·y := mul(-1, 3)",
      "after": "point3 row1 b·y = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 row1 b·y，不以小數近似替代。",
          "en": "Recompute point3 row1 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "3"
        ],
        "output": "-3",
        "label": "point3 row1 b·y"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "point3 row1 LHS := sum(0, -3)",
      "after": "point3 row1 LHS = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 row1 LHS，不以小數近似替代。",
          "en": "Recompute point3 row1 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-3"
        ],
        "output": "-3",
        "label": "point3 row1 LHS"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "row1: -3≤0",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "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": "point3 row2 a·x := mul(1, 2)",
      "after": "point3 row2 a·x = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一座標乘自己的列係數。",
        "en": "Multiply the first coordinate by its row coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 row2 a·x，不以小數近似替代。",
          "en": "Recompute point3 row2 a·x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "point3 row2 a·x"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "point3 row2 b·y := mul(1, 3)",
      "after": "point3 row2 b·y = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二座標也按同一欄序計算。",
        "en": "Compute the second coordinate in the same declared order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 row2 b·y，不以小數近似替代。",
          "en": "Recompute point3 row2 b·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "point3 row2 b·y"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "point3 row2 LHS := sum(2, 3)",
      "after": "point3 row2 LHS = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加後才比較整列不等式。",
        "en": "Add before comparing the complete row inequality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 row2 LHS，不以小數近似替代。",
          "en": "Recompute point3 row2 LHS with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "3"
        ],
        "output": "5",
        "label": "point3 row2 LHS"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "row2: 5≤9",
      "after": "satisfied=是",
      "relation": "implies",
      "rule_id": "inclusive_constraint",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等號允許，負係數已在乘法中保留；不用任意改不等號方向。",
        "en": "Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality."
      },
      "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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "[[0, 0, 0, [是, 是, 是], 是, null], [1, 1, 1, [是, 是, 是], 是, null], [2, 1, 2, [是, 是, 是], 是, null], [3, 2, 3, [是, 是, 是], 是, null]]",
      "after": "all feasible listed indices=[0, 1, 2, 3]",
      "relation": "implies",
      "rule_id": "feasible_candidate_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整掃描列出的點，保留所有通過條件者，沒有要求它們全都最優。",
        "en": "Scan every listed point and retain every feasible one;they need not all be optimal."
      },
      "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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "finite candidates have been completely audited",
      "after": "no unlisted-point or general-LP optimality claim",
      "relation": "implies",
      "rule_id": "finite_grid_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限清單的最佳值不能排除未列出的可行點更好；此界線不靠程式PASS自動提升。",
        "en": "A best value in the finite list does not exclude a better unlisted feasible point;script PASS does not lift this boundary."
      },
      "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": [
      [
        [
          -1,
          0,
          0
        ],
        [
          0,
          -1,
          0
        ],
        [
          1,
          1,
          9
        ]
      ],
      [
        [
          0,
          0,
          0,
          [
            true,
            true,
            true
          ],
          true,
          null
        ],
        [
          1,
          1,
          1,
          [
            true,
            true,
            true
          ],
          true,
          null
        ],
        [
          2,
          1,
          2,
          [
            true,
            true,
            true
          ],
          true,
          null
        ],
        [
          3,
          2,
          3,
          [
            true,
            true,
            true
          ],
          true,
          null
        ]
      ],
      [
        0,
        1,
        2,
        3
      ]
    ],
    "claim_scope": "listed_candidates_only"
  },
  "conclusion": {
    "canonical_value": "([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [[0, 0, 0, [是, 是, 是], 是, null], [1, 1, 1, [是, 是, 是], 是, null], [2, 1, 2, [是, 是, 是], 是, null], [3, 2, 3, [是, 是, 是], 是, null]], [0, 1, 2, 3])",
    "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-20261013-advance-20261021-r06-linear-programming-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261021-r06-linear-programming-foundations/questions.jsonl",
      "batch_sha256": "4f359198d877f6deb65f90624f20b930560aaea99c5c0bb90a9f1cdd464ff288",
      "original_line": 39,
      "original_line_sha256": "920adb898854f9e10d3f9e672f4a4ac5844a7b3f397e0fa7607395f45dfec83b",
      "question_compact_sha256": "ab4245a9f412249ee5a740e06e29a0ed27cc8144285956075d2fcc10427ad191",
      "english_row_compact_sha256": "f8a1b3412b3b6474bf933a46a7e6be3b8ec9add6610644b685d0cb3364d78f06",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261021-r06-linear-programming-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261021-r06-linear-programming-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-linear-programming": {
        "sha256": "e954454f13d345f635417e689ccae76261e754c182baf343173c0279585e8ba4",
        "english_sha256": "b16ebfdf1e39849a91ad524e7052c80e4c5e21153a5faf6290fd437bb617a283",
        "original_path": "public/data/mko/linear-programming.json",
        "english_path": "public/data/i18n/editions/mko-linear-programming.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "09b39eb68bdeb3e1e4277f359de31284852356add54b99a43cf1e38a33d69d15",
    "original_candidate_sha256": "fdc9015ded2e9b244ba4f1af60e95efbff7e8ef6e23a61cafd2dbba450eb2d3b",
    "original_candidate_line_sha256": "fdc9015ded2e9b244ba4f1af60e95efbff7e8ef6e23a61cafd2dbba450eb2d3b"
  },
  "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-20261026-004-p4",
    "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": "5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f",
      "question_first_witness_sha256": "cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 52,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2205f5ffba573dcd0d3125241bd9af64b07b7ba3ad33a08deb15b9047890f7ab",
      "arithmetic_steps": 36
    },
    "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."
  }
}