{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261021-r04-initial-value-problem-0084",
  "pilot_number": "R0003-02259",
  "number": 2259,
  "family_id": "qf-app21-ivp-second-order-candidate-grid",
  "title": {
    "zh-Hant": "第 2259 題：多項式初值候選",
    "en": "Question 2259: A Second-Order Candidate with Two Initial Data"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261021-r04-initial-value-problem-0084",
    "batch_id": "batch-20261013-advance-20261021-r04-initial-value-problem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-initial-value-problem"
    ],
    "title_zh": "多項式初值候選",
    "stem_zh": "整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-4,-2,1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":3,\"xs\":[-1,0,1],\"x0\":1,\"y0\":-5,\"dy0\":0}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, 否], [0, -4, -2, 2, -1, 否], [1, -5, 0, 2, -1, 否]], 1, -5, 0, -5, 0, 是, 是, 否)",
      "explanation_zh": "應用擴展核對=([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, 否], [0, -4, -2, 2, -1, 否], [1, -5, 0, 2, -1, 否]], 1, -5, 0, -5, 0, 是, 是, 否)；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；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "多項式初值候選"
      ]
    },
    "variant": {
      "family_id": "qf-app21-ivp-second-order-candidate-grid",
      "seed": 202610214084,
      "parameters": {
        "coefficients": [
          -4,
          -2,
          1
        ],
        "a": 1,
        "b": 0,
        "c": 0,
        "rhs": 3,
        "xs": [
          -1,
          0,
          1
        ],
        "x0": 1,
        "y0": -5,
        "dy0": 0,
        "domain": "Polynomial_candidate_IVP",
        "operation": "ivp-second-order-candidate-grid",
        "semantic_template": "qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-4,-2,1],\"dy0\":0,\"rhs\":3,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-5}"
      }
    },
    "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-r04-initial-value-problem-0084",
    "batch_id": "batch-20261013-advance-20261021-r04-initial-value-problem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "82b2ca39055b6667c5b34f456c72d3b0726990e1941a2edf29eec8cbaf1d7dc9",
      "family_id": "qf-app21-ivp-second-order-candidate-grid",
      "seed": 202610214084,
      "semantic_template": "qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-4,-2,1],\"dy0\":0,\"rhs\":3,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-5}"
    },
    "fields": {
      "title": "A Second-Order Candidate with Two Initial Data",
      "stem": "Use the integer polynomial candidate with ascending-power coefficients=[-4, -2, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=3; input grid=[-1, 0, 1], x0=1, required y0=-5, required y_prime0=0. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).",
      "answer": {
        "kind": "expression",
        "value": "([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, No], [0, -4, -2, 2, -1, No], [1, -5, 0, 2, -1, No]], 1, -5, 0, -5, 0, Yes, Yes, No)",
        "explanation": "Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, No], [0, -4, -2, 2, -1, No], [1, -5, 0, 2, -1, No]], 1, -5, 0, -5, 0, Yes, Yes, No)."
      },
      "skills": [
        "A Second-Order Candidate with Two Initial Data"
      ],
      "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": "整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-4,-2,1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":3,\"xs\":[-1,0,1],\"x0\":1,\"y0\":-5,\"dy0\":0}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。",
        "en": "Use the integer polynomial candidate with ascending-power coefficients=[-4, -2, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=3; input grid=[-1, 0, 1], x0=1, required y0=-5, required y_prime0=0. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "candidate polynomial ascending coefficients=[-4, -2, 1]",
      "after": "initial point=1; required y₀=-5",
      "relation": "implies",
      "rule_id": "candidate_IVP_data",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留候選多項式及初始資料；網格點不一定包含初始點，需分開代入。",
        "en": "Retain the polynomial and initial data;the residual grid need not contain the initial point,so evaluate it separately."
      },
      "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": "y′ coefficient at degree0 := mul(1, -2)",
      "after": "y′ coefficient at degree0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ coefficient at degree0，不以小數近似替代。",
          "en": "Recompute y′ coefficient at degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-2",
        "label": "y′ coefficient at degree0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "y′ coefficient at degree1 := mul(2, 1)",
      "after": "y′ coefficient at degree1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ coefficient at degree1，不以小數近似替代。",
          "en": "Recompute y′ coefficient at degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "y′ coefficient at degree1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "y″ coefficient at degree0 := mul(1, 2)",
      "after": "y″ coefficient at degree0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y″ coefficient at degree0，不以小數近似替代。",
          "en": "Recompute y″ coefficient at degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "y″ coefficient at degree0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "initial y ascending coefficients=[-4, -2, 1]",
      "after": "evaluate at x=1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "initial y Horner multiply before degree1 := mul(1, 1)",
      "after": "initial y Horner multiply before degree1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 initial y Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute initial y Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "initial y Horner multiply before degree1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "initial y Horner add degree1 := sum(1, -2)",
      "after": "initial y Horner add degree1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 initial y Horner add degree1，不以小數近似替代。",
          "en": "Recompute initial y Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-1",
        "label": "initial y Horner add degree1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "initial y Horner multiply before degree0 := mul(-1, 1)",
      "after": "initial y Horner multiply before degree0 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 initial y Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "initial y Horner multiply before degree0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "initial y Horner add degree0 := sum(-1, -4)",
      "after": "initial y Horner add degree0 = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 initial y Horner add degree0，不以小數近似替代。",
          "en": "Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "-4"
        ],
        "output": "-5",
        "label": "initial y Horner add degree0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "initial y′ ascending coefficients=[-2, 2]",
      "after": "evaluate at x=1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "initial y′ Horner multiply before degree0 := mul(2, 1)",
      "after": "initial y′ Horner multiply before degree0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 initial y′ Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute initial y′ Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "initial y′ Horner multiply before degree0"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "initial y′ Horner add degree0 := sum(2, -2)",
      "after": "initial y′ Horner add degree0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 initial y′ Horner add degree0，不以小數近似替代。",
          "en": "Recompute initial y′ Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "-2"
        ],
        "output": "0",
        "label": "initial y′ Horner add degree0"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "y(1)=-5; required=-5",
      "after": "value condition=是",
      "relation": "implies",
      "rule_id": "initial_value_condition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接比較初始點的實際函數值，不借用其他網格點。",
        "en": "Compare the actual function value at the initial point,not another grid point."
      },
      "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": "y′(1)=0; required=0",
      "after": "derivative condition=是",
      "relation": "implies",
      "rule_id": "initial_derivative_condition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "二階初始資料還要核對導數，兩個條件不能互相替代。",
        "en": "Second-order initial data also require the derivative condition;the two conditions cannot replace one another."
      },
      "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": "[-1, 0, 1]",
      "after": "distinct ordered audit grid=[-1, 0, 1]",
      "relation": "implies",
      "rule_id": "finite_grid_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "去重與排序只固定呈現；之後逐一核對所有列出的點。",
        "en": "Deduplication and sorting fix presentation;every listed distinct point is then audited."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "y at-1 ascending coefficients=[-4, -2, 1]",
      "after": "evaluate at x=-1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "y at-1 Horner multiply before degree1 := mul(1, -1)",
      "after": "y at-1 Horner multiply before degree1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at-1 Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute y at-1 Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "y at-1 Horner multiply before degree1"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "y at-1 Horner add degree1 := sum(-1, -2)",
      "after": "y at-1 Horner add degree1 = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at-1 Horner add degree1，不以小數近似替代。",
          "en": "Recompute y at-1 Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "-2"
        ],
        "output": "-3",
        "label": "y at-1 Horner add degree1"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "y at-1 Horner multiply before degree0 := mul(-3, -1)",
      "after": "y at-1 Horner multiply before degree0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at-1 Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute y at-1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "-1"
        ],
        "output": "3",
        "label": "y at-1 Horner multiply before degree0"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "y at-1 Horner add degree0 := sum(3, -4)",
      "after": "y at-1 Horner add degree0 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at-1 Horner add degree0，不以小數近似替代。",
          "en": "Recompute y at-1 Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "-4"
        ],
        "output": "-1",
        "label": "y at-1 Horner add degree0"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "y′ at-1 ascending coefficients=[-2, 2]",
      "after": "evaluate at x=-1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "y′ at-1 Horner multiply before degree0 := mul(2, -1)",
      "after": "y′ at-1 Horner multiply before degree0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ at-1 Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute y′ at-1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "-2",
        "label": "y′ at-1 Horner multiply before degree0"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "y′ at-1 Horner add degree0 := sum(-2, -2)",
      "after": "y′ at-1 Horner add degree0 = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ at-1 Horner add degree0，不以小數近似替代。",
          "en": "Recompute y′ at-1 Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-2"
        ],
        "output": "-4",
        "label": "y′ at-1 Horner add degree0"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "y″ at-1 ascending coefficients=[2]",
      "after": "evaluate at x=-1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "a·y″ at-1 := mul(1, 2)",
      "after": "a·y″ at-1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各導數乘其自己的方程係數。",
        "en": "Multiply each derivative by its own equation coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a·y″ at-1，不以小數近似替代。",
          "en": "Recompute a·y″ at-1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "a·y″ at-1"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "b·y′ at-1 := mul(0, -4)",
      "after": "b·y′ at-1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各導數乘其自己的方程係數。",
        "en": "Multiply each derivative by its own equation coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b·y′ at-1，不以小數近似替代。",
          "en": "Recompute b·y′ at-1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-4"
        ],
        "output": "0",
        "label": "b·y′ at-1"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "c·y at-1 := mul(0, -1)",
      "after": "c·y at-1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各導數乘其自己的方程係數。",
        "en": "Multiply each derivative by its own equation coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 c·y at-1，不以小數近似替代。",
          "en": "Recompute c·y at-1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "c·y at-1"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "left side at-1 := sum(2, 0, 0)",
      "after": "left side at-1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加總完整方程左邊。",
        "en": "First sum the complete left side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left side at-1，不以小數近似替代。",
          "en": "Recompute left side at-1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "0",
          "0"
        ],
        "output": "2",
        "label": "left side at-1"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "residual at-1 := sub(2, 3)",
      "after": "residual at-1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以左邊減右邊定義殘差。",
        "en": "Define the residual as left side minus right side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 residual at-1，不以小數近似替代。",
          "en": "Recompute residual at-1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "3"
        ],
        "output": "-1",
        "label": "residual at-1"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "residual at-1=-1",
      "after": "point equation match=否",
      "relation": "implies",
      "rule_id": "point_equation_check",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此旗標只回答目前這個網格點。",
        "en": "This flag answers only for the current grid point."
      },
      "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": "y at0 ascending coefficients=[-4, -2, 1]",
      "after": "evaluate at x=0",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "y at0 Horner multiply before degree1 := mul(1, 0)",
      "after": "y at0 Horner multiply before degree1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at0 Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute y at0 Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "y at0 Horner multiply before degree1"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "y at0 Horner add degree1 := sum(0, -2)",
      "after": "y at0 Horner add degree1 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at0 Horner add degree1，不以小數近似替代。",
          "en": "Recompute y at0 Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "-2",
        "label": "y at0 Horner add degree1"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "y at0 Horner multiply before degree0 := mul(-2, 0)",
      "after": "y at0 Horner multiply before degree0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at0 Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute y at0 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "0",
        "label": "y at0 Horner multiply before degree0"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "y at0 Horner add degree0 := sum(0, -4)",
      "after": "y at0 Horner add degree0 = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at0 Horner add degree0，不以小數近似替代。",
          "en": "Recompute y at0 Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-4"
        ],
        "output": "-4",
        "label": "y at0 Horner add degree0"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "y′ at0 ascending coefficients=[-2, 2]",
      "after": "evaluate at x=0",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "y′ at0 Horner multiply before degree0 := mul(2, 0)",
      "after": "y′ at0 Horner multiply before degree0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ at0 Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute y′ at0 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "y′ at0 Horner multiply before degree0"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "y′ at0 Horner add degree0 := sum(0, -2)",
      "after": "y′ at0 Horner add degree0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ at0 Horner add degree0，不以小數近似替代。",
          "en": "Recompute y′ at0 Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "-2",
        "label": "y′ at0 Horner add degree0"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "y″ at0 ascending coefficients=[2]",
      "after": "evaluate at x=0",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "a·y″ at0 := mul(1, 2)",
      "after": "a·y″ at0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各導數乘其自己的方程係數。",
        "en": "Multiply each derivative by its own equation coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a·y″ at0，不以小數近似替代。",
          "en": "Recompute a·y″ at0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "a·y″ at0"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "b·y′ at0 := mul(0, -2)",
      "after": "b·y′ at0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各導數乘其自己的方程係數。",
        "en": "Multiply each derivative by its own equation coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b·y′ at0，不以小數近似替代。",
          "en": "Recompute b·y′ at0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "0",
        "label": "b·y′ at0"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "c·y at0 := mul(0, -4)",
      "after": "c·y at0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各導數乘其自己的方程係數。",
        "en": "Multiply each derivative by its own equation coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 c·y at0，不以小數近似替代。",
          "en": "Recompute c·y at0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-4"
        ],
        "output": "0",
        "label": "c·y at0"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "left side at0 := sum(2, 0, 0)",
      "after": "left side at0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加總完整方程左邊。",
        "en": "First sum the complete left side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left side at0，不以小數近似替代。",
          "en": "Recompute left side at0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "0",
          "0"
        ],
        "output": "2",
        "label": "left side at0"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "residual at0 := sub(2, 3)",
      "after": "residual at0 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以左邊減右邊定義殘差。",
        "en": "Define the residual as left side minus right side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 residual at0，不以小數近似替代。",
          "en": "Recompute residual at0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "3"
        ],
        "output": "-1",
        "label": "residual at0"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "residual at0=-1",
      "after": "point equation match=否",
      "relation": "implies",
      "rule_id": "point_equation_check",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此旗標只回答目前這個網格點。",
        "en": "This flag answers only for the current grid point."
      },
      "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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "y at1 ascending coefficients=[-4, -2, 1]",
      "after": "evaluate at x=1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "y at1 Horner multiply before degree1 := mul(1, 1)",
      "after": "y at1 Horner multiply before degree1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at1 Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute y at1 Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "y at1 Horner multiply before degree1"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "y at1 Horner add degree1 := sum(1, -2)",
      "after": "y at1 Horner add degree1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at1 Horner add degree1，不以小數近似替代。",
          "en": "Recompute y at1 Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-1",
        "label": "y at1 Horner add degree1"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "y at1 Horner multiply before degree0 := mul(-1, 1)",
      "after": "y at1 Horner multiply before degree0 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at1 Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute y at1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "y at1 Horner multiply before degree0"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "y at1 Horner add degree0 := sum(-1, -4)",
      "after": "y at1 Horner add degree0 = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y at1 Horner add degree0，不以小數近似替代。",
          "en": "Recompute y at1 Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "-4"
        ],
        "output": "-5",
        "label": "y at1 Horner add degree0"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "y′ at1 ascending coefficients=[-2, 2]",
      "after": "evaluate at x=1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "y′ at1 Horner multiply before degree0 := mul(2, 1)",
      "after": "y′ at1 Horner multiply before degree0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ at1 Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute y′ at1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "y′ at1 Horner multiply before degree0"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "y′ at1 Horner add degree0 := sum(2, -2)",
      "after": "y′ at1 Horner add degree0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ at1 Horner add degree0，不以小數近似替代。",
          "en": "Recompute y′ at1 Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "-2"
        ],
        "output": "0",
        "label": "y′ at1 Horner add degree0"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "y″ at1 ascending coefficients=[2]",
      "after": "evaluate at x=1",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "a·y″ at1 := mul(1, 2)",
      "after": "a·y″ at1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各導數乘其自己的方程係數。",
        "en": "Multiply each derivative by its own equation coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a·y″ at1，不以小數近似替代。",
          "en": "Recompute a·y″ at1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "a·y″ at1"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "b·y′ at1 := mul(0, 0)",
      "after": "b·y′ at1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各導數乘其自己的方程係數。",
        "en": "Multiply each derivative by its own equation coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b·y′ at1，不以小數近似替代。",
          "en": "Recompute b·y′ at1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "b·y′ at1"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "c·y at1 := mul(0, -5)",
      "after": "c·y at1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各導數乘其自己的方程係數。",
        "en": "Multiply each derivative by its own equation coefficient."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 c·y at1，不以小數近似替代。",
          "en": "Recompute c·y at1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-5"
        ],
        "output": "0",
        "label": "c·y at1"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "left side at1 := sum(2, 0, 0)",
      "after": "left side at1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先加總完整方程左邊。",
        "en": "First sum the complete left side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left side at1，不以小數近似替代。",
          "en": "Recompute left side at1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "0",
          "0"
        ],
        "output": "2",
        "label": "left side at1"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "residual at1 := sub(2, 3)",
      "after": "residual at1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以左邊減右邊定義殘差。",
        "en": "Define the residual as left side minus right side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 residual at1，不以小數近似替代。",
          "en": "Recompute residual at1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "3"
        ],
        "output": "-1",
        "label": "residual at1"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "residual at1=-1",
      "after": "point equation match=否",
      "relation": "implies",
      "rule_id": "point_equation_check",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此旗標只回答目前這個網格點。",
        "en": "This flag answers only for the current grid point."
      },
      "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": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "all listed grid points pass=否; initial conditions pass=是",
      "after": "combined finite check=否",
      "relation": "implies",
      "rule_id": "finite_grid_initial_conjunction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "題目要求的網格與初始條件都通過才算本有限檢查通過，不能少核一點或一個初始條件。",
        "en": "Every requested grid and initial condition must pass;do not omit a point or an initial condition."
      },
      "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": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "finite grid agreement alone is not a general interval certificate",
      "after": "a polynomial coefficient/degree certificate is a separate possible strengthening",
      "relation": "implies",
      "rule_id": "grid_vs_interval_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對這些多項式，另用全係數恆等或已證次數界與足夠不同點可補強；本輸出旗標只記有限檢查，不否定那種額外證法。",
        "en": "For these polynomials,coefficient identity or a proved degree bound with enough distinct points could separately strengthen the evidence;this output flag records only the finite check,not a denial of that additional method."
      },
      "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": [
      [
        -4,
        -2,
        1
      ],
      1,
      0,
      0,
      3,
      [
        -1,
        0,
        1
      ],
      [
        [
          -1,
          "-1",
          "-4",
          "2",
          "-1",
          false
        ],
        [
          0,
          "-4",
          "-2",
          "2",
          "-1",
          false
        ],
        [
          1,
          "-5",
          "0",
          "2",
          "-1",
          false
        ]
      ],
      1,
      "-5",
      "0",
      -5,
      0,
      true,
      true,
      false
    ],
    "claim_scope": "finite_grid_and_initial_data"
  },
  "conclusion": {
    "canonical_value": "([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, 否], [0, -4, -2, 2, -1, 否], [1, -5, 0, 2, -1, 否]], 1, -5, 0, -5, 0, 是, 是, 否)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261021-r04-initial-value-problem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261021-r04-initial-value-problem-foundations/questions.jsonl",
      "batch_sha256": "510511592eeb41e0d6eb15d17354e824e2e9a97ffe1195123370f8395c6d3673",
      "original_line": 84,
      "original_line_sha256": "bc64f15f228815336539d9a4b61572796cef3af44ce0beaa16df9d9fb355692f",
      "question_compact_sha256": "82b2ca39055b6667c5b34f456c72d3b0726990e1941a2edf29eec8cbaf1d7dc9",
      "english_row_compact_sha256": "835048ffbaa90bf16aff7008dcc9e4351cc84befbe833187ddd38429f192a397",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261021-r04-initial-value-problem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261021-r04-initial-value-problem-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-initial-value-problem": {
        "sha256": "84ab0a91c84b74df8cfa9a3a8033b7ad83825d55e89f1d0ca847ed81f85e9925",
        "english_sha256": "9fff0f0029d4033fa0555d7c1431a25e5fe58386793836198e0241c073506dc0",
        "original_path": "public/data/mko/initial-value-problem.json",
        "english_path": "public/data/i18n/editions/mko-initial-value-problem.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "09b39eb68bdeb3e1e4277f359de31284852356add54b99a43cf1e38a33d69d15",
    "original_candidate_sha256": "3d83f60c19ba9e3b059f137d1d9e169413ad6b76d14bc95ff2e25b2055a5fd9f",
    "original_candidate_line_sha256": "3d83f60c19ba9e3b059f137d1d9e169413ad6b76d14bc95ff2e25b2055a5fd9f"
  },
  "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": 62,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "fe79f4ff4a66297586df2ca3ea513f6e9fe4a0271f3d442d787ebb4bd8a69599",
      "arithmetic_steps": 42
    },
    "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."
  }
}