{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-pulse-20261119-r01-0069",
  "pilot_number": "R0055-28069",
  "number": 28069,
  "family_id": "qf-mod50-newton-square-root-orbit",
  "title": {
    "zh-Hant": "第 28069 題：Newton法",
    "en": "Question 28069: Positive Square-Root Iterates Use Exact Rational Newton Steps"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-pulse-20261119-r01-0069",
    "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-square-root-orbit；明示資料={\"radicand\":20,\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。",
    "answer": {
      "kind": "expression",
      "value": "(20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)",
      "explanation_zh": "Wave6第十三層核對=(20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；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": 3,
      "skills": [
        "Newton法"
      ]
    },
    "variant": {
      "family_id": "qf-mod50-newton-square-root-orbit",
      "seed": 202611191069,
      "parameters": {
        "radicand": 20,
        "x0": 5,
        "steps": 1,
        "domain": "Exact_Newton_iteration_boundary",
        "operation": "newton-square-root-orbit",
        "semantic_template": "newton-square-root-orbit|{\"radicand\":[20,1],\"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-r01-0069",
    "batch_id": "batch-20261015-advance-20261119-r01-newton-method-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5dc376e062c37d1304f15ec3b229743e0865cf6dc57e796579961ffeecf64327",
      "family_id": "qf-mod50-newton-square-root-orbit",
      "seed": 202611191069,
      "semantic_template": "newton-square-root-orbit|{\"radicand\":[20,1],\"steps\":1,\"x0\":[5,1]}"
    },
    "fields": {
      "title": "Positive Square-Root Iterates Use Exact Rational Newton Steps",
      "stem": "Use positive radicand A=20, positive initial iterate=5, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).",
      "answer": {
        "kind": "expression",
        "value": "(20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)",
        "explanation": "At an exact root stop before dividing; otherwise update to (x+A/x)/2. Each record gives index, current x, signed residual, status and next or null. The final absolute residual is |x^2−A|, not an unprovided forward error to sqrt(A) or a general Newton convergence proof. The exact result, retaining the requested order, is (20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)."
      },
      "skills": [
        "Positive Square-Root Iterates Use Exact Rational Newton Steps"
      ],
      "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-square-root-orbit；明示資料={\"radicand\":20,\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。",
        "en": "Use positive radicand A=20, positive initial iterate=5, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-mod50-newton-square-root-orbit, newton-square-root-orbit]",
      "after": "derived = [[radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_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 sqrt attempt0 ORIGINAL square := mul(5, 5)",
      "after": "Newton sqrt attempt0 ORIGINAL square = 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 sqrt attempt0 ORIGINAL square，不以小數近似替代。",
          "en": "Recompute Newton sqrt attempt0 ORIGINAL square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "5"
        ],
        "output": "25",
        "label": "Newton sqrt attempt0 ORIGINAL square"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Newton sqrt attempt0 ORIGINAL signed residual := sub(25, 20)",
      "after": "Newton sqrt attempt0 ORIGINAL signed residual = 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 sqrt attempt0 ORIGINAL signed residual，不以小數近似替代。",
          "en": "Recompute Newton sqrt attempt0 ORIGINAL signed residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "25",
          "20"
        ],
        "output": "5",
        "label": "Newton sqrt attempt0 ORIGINAL signed residual"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Newton sqrt attempt0 ORIGINAL A over x := div(20, 5)",
      "after": "Newton sqrt attempt0 ORIGINAL A over x = 4",
      "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 sqrt attempt0 ORIGINAL A over x，不以小數近似替代。",
          "en": "Recompute Newton sqrt attempt0 ORIGINAL A over x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "20",
          "5"
        ],
        "output": "4",
        "label": "Newton sqrt attempt0 ORIGINAL A over x"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Newton sqrt attempt0 ORIGINAL x plus A over x := sum(5, 4)",
      "after": "Newton sqrt attempt0 ORIGINAL x plus A over x = 9",
      "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 sqrt attempt0 ORIGINAL x plus A over x，不以小數近似替代。",
          "en": "Recompute Newton sqrt attempt0 ORIGINAL x plus A over x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "5",
          "4"
        ],
        "output": "9",
        "label": "Newton sqrt attempt0 ORIGINAL x plus A over x"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Newton sqrt attempt0 ORIGINAL half update := div(9, 2)",
      "after": "Newton sqrt attempt0 ORIGINAL half update = 9/2",
      "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 sqrt attempt0 ORIGINAL half update，不以小數近似替代。",
          "en": "Recompute Newton sqrt attempt0 ORIGINAL half update with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "9",
          "2"
        ],
        "output": "9/2",
        "label": "Newton sqrt attempt0 ORIGINAL half update"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Newton sqrt ORIGINAL final square EVEN steps0 := mul(9/2, 9/2)",
      "after": "Newton sqrt ORIGINAL final square EVEN steps0 = 81/4",
      "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 sqrt ORIGINAL final square EVEN steps0，不以小數近似替代。",
          "en": "Recompute Newton sqrt ORIGINAL final square EVEN steps0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9/2",
          "9/2"
        ],
        "output": "81/4",
        "label": "Newton sqrt ORIGINAL final square EVEN steps0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Newton sqrt ORIGINAL final signed residual := sub(81/4, 20)",
      "after": "Newton sqrt ORIGINAL final signed residual = 1/4",
      "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 sqrt ORIGINAL final signed residual，不以小數近似替代。",
          "en": "Recompute Newton sqrt ORIGINAL final signed residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "81/4",
          "20"
        ],
        "output": "1/4",
        "label": "Newton sqrt ORIGINAL final signed residual"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [20, 5, 1]",
      "after": "derived = [[[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, 否]",
      "relation": "implies",
      "rule_id": "Newton_sqrt_positive_iteration_ROOT_FIRST_residual_NOT_forward_error",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正A與正x0使(x+A/x)/2為正且可除，exact root先停止。最後abs residual即使steps0仍按來源計；它是|x²-A|，不是未提供的|x-sqrt(A)|，有限軌道不作一般收斂證書。",
        "en": "Positive A/x0 keep(x+A/x)/2 positive and division defined;an exact root stops first.Always compute the final absolute residual,even at steps0.It is|x²-A|,not an unprovided forward error|x-sqrt(A)|,and a finite orbit is not a general convergence certificate."
      },
      "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": [
      "20",
      "5",
      1,
      [
        [
          0,
          "5",
          "5",
          "updated",
          "9/2"
        ]
      ],
      "step_budget_exhausted",
      "9/2",
      "1/4",
      false,
      "finite_square_root_orbit_does_not_prove_a_general_Newton_theorem"
    ]
  },
  "conclusion": {
    "canonical_value": "(20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)",
    "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": 69,
      "original_line_sha256": "f71b144166817d8f1baa88b3fa6b6f17bbbc6c81d53d68e3b3758ae0cf9f543d",
      "question_compact_sha256": "5dc376e062c37d1304f15ec3b229743e0865cf6dc57e796579961ffeecf64327",
      "english_row_compact_sha256": "082cdf10faaa39b7365b45454f5f779dfb0f2e4b7da5ed3bb4effdb4d57f10a1",
      "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": {
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        "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": "5ffaaa72208f20822070a80a3990063a38d26a12d1eb2cd1570633ee8f1b5ed2",
    "original_candidate_line_sha256": "5ffaaa72208f20822070a80a3990063a38d26a12d1eb2cd1570633ee8f1b5ed2"
  },
  "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": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "75b9bd79f511451c98bc8c97822a94420b9b02616c9268ffb4e7c52a25f5566f",
      "arithmetic_steps": 7
    },
    "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."
  }
}