{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-newton-method-01",
  "mko_id": "mko-newton-method",
  "title": {
    "zh-Hant": "合法 Newton 更新與有限 trace",
    "en": "Qualified Newton updates and a finite trace"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-newton-method-01",
    "source": {
      "mko_id": "mko-newton-method",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "893923a37c1598a5dbd5ccc0d20d475225dd3942fcbef402eb50c9fa3b35f945",
      "english_sha256": "e89ccbf082a1fe4f6390c01d07f10fce767439cb5ede84dafe31c8b0c60d7429"
    },
    "mode": "algorithm_trace",
    "audience_prerequisites": [
      "algebra",
      "polynomial",
      "newton_update"
    ],
    "goal": {
      "zh-Hant": "對 x^2-2 從 1 做兩步精確 Newton，並核 root-first 與循環邊界。",
      "en": "Take two exact Newton steps for x^2-2 from one, and audit root-first stopping and a cycle boundary."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "f(x)=x^2-2，f'(x)=2x，x_0=1，全部資料 exact rationals；本例目標有限兩步。",
          "en": "f(x)=x^2-2, f'(x)=2x, x_0=1, using exact rationals; the requested trace has two steps."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "f(x)=x^2-2,\\quad f'(x)=2x",
        "after_tex": "\\operatorname{dom}f=\\operatorname{dom}f'=\\mathbb R",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整多項式給每個有理 iterate 的 domain 資格。",
          "en": "The full polynomial supplies domain qualification for every rational iterate."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "有 domain 不表示導數一定非零，仍逐步核。",
            "en": "Domain membership does not ensure a nonzero derivative, which remains a per-step check."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "x_0=1,\\quad f(x_0)=-1\\ne0,\\quad f'(x_0)=2\\ne0",
        "after_tex": "x_1=1-(-1)/2=3/2",
        "relation": "implies",
        "rule": "newton_update",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先 root test，再 derivative test，全部通過才除法。",
          "en": "Perform the root test before the derivative test; divide only after both qualify."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "不是先除後才發現分母為零。",
            "en": "Division does not precede checking its denominator."
          }
        },
        "leaf_basis": "newton_update"
      },
      {
        "id": "S03",
        "uses": [
          "S02"
        ],
        "before_tex": "f(3/2)=1/4\\ne0,\\quad f'(3/2)=3\\ne0",
        "after_tex": "x_2=3/2-(1/4)/3=17/12",
        "relation": "implies",
        "rule": "newton_update",
        "conditions": [],
        "explanation": {
          "zh-Hant": "重算新點的函數與導數，不能沿用上一步值。",
          "en": "Recompute the new function and derivative values instead of reusing the previous step's data."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02"
          ],
          "explanation": {
            "zh-Hant": "精確分母是 12，非浮點近似。",
            "en": "The denominator is exactly twelve, not a floating approximation."
          }
        },
        "leaf_basis": "newton_update"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S02",
          "S03"
        ],
        "before_tex": "f(x_0)=-1,\\quad f(x_1)=1/4",
        "after_tex": "f(x_2)=289/144-2=1/144",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "列出殘差大小下降，但這只是有限 trace；沒有未提供真根的 forward error。",
          "en": "The residual magnitudes decrease in this finite trace, but no forward error to an unprovided root is asserted."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "residual 不是 x 與真根的距離。",
            "en": "Residual is not the distance from x to a true root."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S05",
        "uses": [
          "G1"
        ],
        "before_tex": "x>0",
        "after_tex": "x-\\frac{x^2-2}{2x}=\\tfrac12(x+2/x)>0",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [
          "x>0"
        ],
        "explanation": {
          "zh-Hant": "平方根特殊型的正 iterate 保持正，因此下一步導數不零；仍未證收斂。",
          "en": "The square-root special form preserves positive iterates and hence nonzero derivatives, without yet proving convergence."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "正性只保合法更新，不自動給局部率或 global basin。",
            "en": "Positivity qualifies updates but does not supply a local rate or global basin."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "G1"
        ],
        "before_tex": "g(x)=x^3,\\quad x=0,\\quad g(0)=g'(0)=0",
        "after_tex": "\\text{root_found; do not evaluate }0-0/0",
        "relation": "implies",
        "rule": "newton_update",
        "conditions": [],
        "explanation": {
          "zh-Hant": "根先停止避免在重根執行未定義 0/0。",
          "en": "Stopping at the root first avoids an undefined 0/0 update at a multiple root."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "零導數不應蓋過已驗證 root 的合法停止。",
            "en": "A zero derivative must not override an already verified root stop."
          }
        },
        "leaf_basis": "newton_update"
      },
      {
        "id": "S07",
        "uses": [
          "G1"
        ],
        "before_tex": "p(x)=x^3-2x+2,\\quad p'(x)=3x^2-2,\\quad z_0=0",
        "after_tex": "z_1=0-2/(-2)=1,\\quad z_2=1-1/1=0",
        "relation": "implies",
        "rule": "newton_update",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩步都合法卻回到原點，完整確定性更新因此永遠在 0,1 間循環。",
          "en": "Both updates are legal but return to the start; deterministic repetition therefore cycles forever between zero and one."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "兩步本身加上確定性重複證全循環，而非僅有限觀察猜測。",
            "en": "The two steps plus deterministic repetition prove the whole cycle, not a guess from a finite observation."
          }
        },
        "leaf_basis": "newton_update"
      },
      {
        "id": "S08",
        "uses": [
          "S04",
          "S06",
          "S07"
        ],
        "before_tex": "\\text{finite exact Newton traces and a nonconvergent cycle}",
        "after_tex": "\\text{valid updates do not by themselves prove convergence}",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "主例可報兩個精確 iterate 與殘差；循環例否定無條件收斂，保留所有未測的率與穩定性界線。",
          "en": "Report the two exact iterates and residuals for the main example; the cycle refutes unconditional convergence while rates and stability remain unclaimed."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04",
            "S06",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "不因算法 trace 成功就升為一般定理。",
            "en": "A successful trace is not promoted to a general theorem."
          }
        },
        "leaf_basis": "algebra"
      }
    ],
    "result": {
      "zh-Hant": "兩步為 3/2、17/12，殘差大小 1/4、1/144；root-first 合法停止，另有合法循環反例。",
      "en": "The two iterates are 3/2 and 17/12 with residual magnitudes 1/4 and 1/144; root-first stopping is qualified and a separate legal cycle exists."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "f=f'=0 時仍更新 0/0。",
          "en": "At f=f'=0 still update via 0/0."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "已找到根先停止。",
          "en": "Stop first at the verified root."
        }
      },
      {
        "claim": {
          "zh-Hant": "殘差就是 forward error。",
          "en": "Residual is forward error."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "未給真根及殘差到距離的界。",
          "en": "No true root or residual-to-distance bound is provided."
        }
      },
      {
        "claim": {
          "zh-Hant": "合法 Newton 必收斂。",
          "en": "Legal Newton updates must converge."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "S07 之 0↔1 循環不收斂。",
          "en": "The 0-to-1 cycle at S07 does not converge."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "對可微實函數f，只有在目前iterate x_k位於domain且f'(x_k)不等於0時，Newton update x_{k+1}=x_k-f(x_k)/f'(x_k)才有定義。本輪exact slice使用低次整係數多項式與有理iterate，逐步精確重算f、f'及更新；若已到f(x_k)=0，先以root_found停止，不再對可能同時為0的導數做0/0。有限orbit、residual下降或某一步接近根都不證一般收斂、二次收斂或global root；residual |f(x)|也不是未給定真根時的forward error。",
          "en": "For a differentiable real function, the Newton update x_(k+1)=x_k−f(x_k)/f_prime(x_k) requires the current iterate in the domain and a nonzero derivative. The exact polynomial families use coefficients in low-to-high degree order and bounded rational iterates, recalculating f, its derivative and every update. If f(x_k)=0, stop as root_found before any division, even if the derivative also vanishes. A finite orbit, decreasing residual or one iterate near a root does not prove general or quadratic convergence, a global root basin or root uniqueness. Residual |f(x)| is not forward error to an unprovided true root."
        },
        "source": {
          "mko_id": "mko-newton-method",
          "sha256": "893923a37c1598a5dbd5ccc0d20d475225dd3942fcbef402eb50c9fa3b35f945",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "polynomial",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "多項式在實數上連續且可逐項微分；d(x^k)/dx=k x^(k-1)，常數導數為零。",
          "en": "Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero."
        }
      },
      {
        "id": "newton_update",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "每步先核 f(x)=0 則 root_found 停止；否則須在函數及導數域且 f'(x)≠0，才可更新 x-f(x)/f'(x)。",
          "en": "At each step first stop as root_found when f(x)=0; otherwise require the function/derivative domain and f'(x)≠0 before updating x-f(x)/f'(x)."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "newton-first",
        "lhs": "1-(-1)/2",
        "rhs": "3/2",
        "step": "S02"
      },
      {
        "id": "newton-second",
        "lhs": "3/2-(1/4)/3",
        "rhs": "17/12",
        "step": "S03"
      },
      {
        "id": "newton-residual",
        "lhs": "(17/12)**2-2",
        "rhs": "1/144",
        "step": "S04"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只做有限有理 orbit，不宣稱一般收斂、二次率、唯一根或浮點穩定。",
      "en": "A finite rational orbit only, without general convergence, quadratic rate, root uniqueness or floating-point stability."
    },
    "md_symbols": [
      [
        "x_k",
        "精確有限 iterate",
        "exact finite iterate"
      ],
      [
        "|f(x_k)|",
        "殘差大小",
        "residual magnitude"
      ]
    ],
    "md_error_controls": [
      {
        "title": "停止順序 / Stopping order",
        "zh": "domain 後先 root，再 derivative 非零，再除法。",
        "en": "After domain qualification test root first, then nonzero derivative, then divide."
      },
      {
        "title": "有限 trace 邊界 / Finite-trace boundary",
        "zh": "精確 orbit 不等於收斂或二次率證明。",
        "en": "An exact orbit is not a convergence or quadratic-rate proof."
      }
    ],
    "md_open_items": []
  },
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    "formal_Lean": "NOT_RUN",
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    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}