{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r01-0029",
  "pilot_number": "R0055-28029",
  "number": 28029,
  "family_id": "qf-mod50-newton-finite-orbit",
  "title": {
    "zh-Hant": "第 28029 題：Newton法",
    "en": "Question 28029: A Finite Newton Orbit Records Steps without Certifying Convergence"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r01-0029",
    "batch_id": "batch-20261015-advance-20261119-r01-newton-method-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-newton-method"
    ],
    "title_zh": "Newton法",
    "stem_zh": "Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-4,0,1],\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "([-4, 0, 1], 1, [[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)",
      "explanation_zh": "Wave6第十三層核對=([-4, 0, 1], 1, [[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "Newton法"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-newton-finite-orbit",
      "seed": 202611191029,
      "parameters": {
        "coefficients": [
          -4,
          0,
          1
        ],
        "x0": 5,
        "steps": 1,
        "domain": "Exact_Newton_iteration_boundary",
        "operation": "newton-finite-orbit",
        "semantic_template": "newton-finite-orbit|{\"coefficients\":[[-4,1],[0,1],[1,1]],\"steps\":1,\"x0\":[5,1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_contract_recomputation",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-pulse-20261119-r01-0029",
    "batch_id": "batch-20261015-advance-20261119-r01-newton-method-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6827475c53dd450ec22685526cb53a19de22e193f01c93d2e8d9a9036553d721",
      "family_id": "qf-mod50-newton-finite-orbit",
      "seed": 202611191029,
      "semantic_template": "newton-finite-orbit|{\"coefficients\":[[-4,1],[0,1],[1,1]],\"steps\":1,\"x0\":[5,1]}"
    },
    "fields": {
      "title": "A Finite Newton Orbit Records Steps without Certifying Convergence",
      "stem": "Coefficients in low-to-high degree order are=[-4, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=5 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).",
      "answer": {
        "kind": "expression",
        "value": "([-4, 0, 1], 1, [[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)",
        "explanation": "Recalculate f, f_prime and every qualified update. Each record gives index, current x, f, f_prime, status and next or null. Exact roots stop early; a nonroot zero derivative is rejected. A finite residual pattern proves neither a general limit nor quadratic convergence, a global basin or a forward-error bound. The exact result, retaining the requested order, is ([-4, 0, 1], 1, [[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)."
      },
      "skills": [
        "A Finite Newton Orbit Records Steps without Certifying Convergence"
      ],
      "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": "Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-4,0,1],\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Coefficients in low-to-high degree order are=[-4, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=5 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-mod50-newton-finite-orbit, newton-finite-orbit]",
      "after": "derived = [[coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope]]",
      "relation": "implies",
      "rule_id": "source_answer_roles_or_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁明列本題scalar／label／set／point或mode欄序，不強套現代tuple。舊題domain與operation底線格式按原資料，分數整數亦保/1；原中英快照不改。",
        "en": "List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient1 := mul(1, 0)",
      "after": "Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient1，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient2 := mul(2, 1)",
      "after": "Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient2，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [Newton ORIGINAL orbit polynomial, [-4, 0, 1]]",
      "after": "derived = [[-4, 0, 1], [0, 2]]",
      "relation": "implies",
      "rule_id": "Newton_polynomial_raw_arity_trim_nonconstant_AND_ALL_derivative_coefficients",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原vector先要求2..6，再只刪trailing zeros且仍須非constant。逐x^i微分得i*c_i*x^(i-1)，來源derivative係數亦先過1e8 profile，即使稍後root停止或steps0也不跳過。",
        "en": "Require raw length2..6 before trimming trailing zeros,and remain nonconstant.Differentiate every monomial to i*c_i*x^(i-1).Source derivative coefficients must pass their1e8 profile even if a root later stops or steps=0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2 := mul(0, 5)",
      "after": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "5"
        ],
        "output": "0",
        "label": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2 := sum(0, 1)",
      "after": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1 := mul(1, 5)",
      "after": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "5"
        ],
        "output": "5",
        "label": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1 := sum(5, 0)",
      "after": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "5",
          "0"
        ],
        "output": "5",
        "label": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0 := mul(5, 5)",
      "after": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0 = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "5"
        ],
        "output": "25",
        "label": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0 := sum(25, -4)",
      "after": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0 = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "25",
          "-4"
        ],
        "output": "21",
        "label": "Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1 := mul(0, 5)",
      "after": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "5"
        ],
        "output": "0",
        "label": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1 := sum(0, 2)",
      "after": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0 := mul(2, 5)",
      "after": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "5"
        ],
        "output": "10",
        "label": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0 := sum(10, 0)",
      "after": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "0"
        ],
        "output": "10",
        "label": "Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime := div(21, 10)",
      "after": "Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime = 21/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "21",
          "10"
        ],
        "output": "21/10",
        "label": "Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update := sub(5, 21/10)",
      "after": "Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update = 29/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update，不以小數近似替代。",
          "en": "Recompute Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "21/10"
        ],
        "output": "29/10",
        "label": "Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [Newton ORIGINAL orbit attempt0, [-4, 0, 1], [0, 2], 5]",
      "after": "derived = [21, 10, updated, 29/10]",
      "relation": "implies",
      "rule_id": "Newton_EACH_f_AND_fprime_ROOT_FIRST_then_nonzero_derivative_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先完整計f與f_prime，再先驗f=0；exact root即停止，哪怕slope0也不作0/0。非根才需slope非0，解tangent線f(x)+f_prime(x)(z-x)=0得z=x-f/f_prime。",
        "en": "Evaluate BOTH f and f_prime completely,then test f=0 FIRST.An exact root stops even at slope0,without0/0.Only a nonroot requires nonzero slope;the tangent-line zero solves z=x-f/f_prime."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [Newton ORIGINAL orbit, 1, 否]",
      "after": "derived = [[[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10]",
      "relation": "implies",
      "rule_id": "Newton_attempt_budget_actual_records_status_NOT_guessed_from_final_x",
      "conditions": [],
      "explanation": {
        "zh-Hant": "records是實際attempts，可在root／允許的undefined提前停止。最後一次updated即使落在root，沒有下一次budgeted root-test也不把status改成root_found；steps0只作資格，不捏造record。有限trace不保一般convergence/rate。",
        "en": "Records are actual attempts and may stop early at a root/allowed undefined state.A last update may land on a root,but without another budgeted test status is not changed to root_found.At steps0 qualify inputs without inventing records.A finite trace is not general convergence/rate evidence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        "-4",
        "0",
        "1"
      ],
      1,
      [
        [
          0,
          "5",
          "21",
          "10",
          "updated",
          "29/10"
        ]
      ],
      "step_budget_exhausted",
      "29/10",
      false,
      "finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate"
    ]
  },
  "conclusion": {
    "canonical_value": "([-4, 0, 1], 1, [[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)",
    "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-20261015-advance-20261119-r01-newton-method-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r01-newton-method-foundations/questions.jsonl",
      "batch_sha256": "3c8902f38e2a90dbc1f6bcabe487d71a0a5b3448365cfe4d39bd4660b8a1531f",
      "original_line": 29,
      "original_line_sha256": "e115fc93b857990ae3a62bf5bc754c0860a453e13b802d466675b238349a5a88",
      "question_compact_sha256": "6827475c53dd450ec22685526cb53a19de22e193f01c93d2e8d9a9036553d721",
      "english_row_compact_sha256": "d3158e1a96be40e3d0f8c94fdbe079051eedee531a3b89eac73b417a09d3b4f5",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r01-newton-method-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r01-newton-method-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-newton-method": {
        "sha256": "893923a37c1598a5dbd5ccc0d20d475225dd3942fcbef402eb50c9fa3b35f945",
        "english_sha256": "e89ccbf082a1fe4f6390c01d07f10fce767439cb5ede84dafe31c8b0c60d7429",
        "original_path": "public/data/mko/newton-method.json",
        "english_path": "public/data/i18n/editions/mko-newton-method.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b5763df142e86ef7f9787b6ea86050afc60bd3c5c53538d4839606827cc44686",
    "original_candidate_sha256": "b3af2a7877bcf42b1d7daad411a1b41616c75b5f8279cfb12e3b85d17e4fde07",
    "original_candidate_line_sha256": "b3af2a7877bcf42b1d7daad411a1b41616c75b5f8279cfb12e3b85d17e4fde07"
  },
  "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-20261026-010-p56",
    "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": "6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683",
      "question_first_witness_sha256": "611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 18,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "aefbfc5b6df64eeaa167f678e447f84730cc2197e3aec5210eb3b52eeb111e38",
      "arithmetic_steps": 14
    },
    "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."
  }
}