{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-uniform-convergence-of-functions-01",
  "mko_id": "mko-uniform-convergence-of-functions",
  "title": {
    "zh-Hant": "三點域的精確上確界誤差",
    "en": "Exact supremum error on a three-point domain"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-uniform-convergence-of-functions-01",
    "source": {
      "mko_id": "mko-uniform-convergence-of-functions",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "c28f03cdaf36d8b06ca1718e4253a1d705745496ed7c655d09f55c2a82148ac0",
      "english_sha256": "f3588cebfd521dd72a1db18c1ae10af42c4b9398d451eb7680dac158b73a4462"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "epsilon",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "證 g_n(a)=1/(n+2),g_n(b)=-4/(n+2),g_n(c)=2/(n+2) 一致趨零。",
      "en": "Prove g_n(a)=1/(n+2),g_n(b)=-4/(n+2),g_n(c)=2/(n+2) converges uniformly to zero."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "D={a,b,c}, g=0；三個公式對每個 n≥0 定義全部 g_n:D→R。",
          "en": "D={a,b,c}, g=0; the three formulas define all of g_n:D to R for every n≥0."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "n\\ge0",
        "after_tex": "n+2>0",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "所有點共用正分母，絕對值計算無除零問題。",
          "en": "All points share a positive denominator, so absolute-error calculations avoid division by zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "共同 n 與共同域保持不變。",
            "en": "The common index and domain remain fixed."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "(|g_n(a)|,|g_n(b)|,|g_n(c)|)",
        "after_tex": "(1,4,2)/(n+2)",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐點取絕對值，保留座標 b 的大小 4。",
          "en": "Take each absolute value, retaining magnitude 4 at b."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "有號最大值不是絕對誤差最大值。",
            "en": "The maximum signed value is not the maximum absolute error."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "S02"
        ],
        "before_tex": "E_n=\\sup_{x\\in D}|g_n(x)-g(x)|",
        "after_tex": "E_n=\\max\\{1,4,2\\}/(n+2)=4/(n+2)",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "有限完整域的上確界就是三個誤差的最大值，且在 b 取到。",
          "en": "The complete finite domain makes the supremum the maximum of its three errors, attained at b."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02"
          ],
          "explanation": {
            "zh-Hant": "不是只用若干點的最大值估未知無限域。",
            "en": "No sampled maximum is used to estimate an unknown infinite-domain supremum."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S04",
        "uses": [
          "S03"
        ],
        "before_tex": "\\varepsilon>0,\\quad N=\\lfloor4/\\varepsilon\\rfloor+1",
        "after_tex": "N>4/\\varepsilon",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "用 Archimedean 整數選擇，N 只依 epsilon。",
          "en": "An Archimedean integer choice gives N depending only on epsilon."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "N 的公式沒有 x。",
            "en": "The formula for N contains no x."
          }
        },
        "leaf_basis": "epsilon"
      },
      {
        "id": "S05",
        "uses": [
          "S03",
          "S04"
        ],
        "before_tex": "n\\ge N",
        "after_tex": "|g_n(x)|\\le E_n=4/(n+2)<\\varepsilon\\quad\\forall x\\in D",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [
          "x\\in D,\\ n\\ge N"
        ],
        "explanation": {
          "zh-Hant": "全部尾端的分母均大於 4/epsilon，因此同時控制三點。",
          "en": "Every later denominator exceeds 4/epsilon and simultaneously controls all three points."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "對所有 n≥N 而非僅 n=N。",
            "en": "The statement is for all n≥N, not only n=N."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "S05"
        ],
        "before_tex": "\\forall\\varepsilon>0\\ \\exists N\\ \\forall n\\ge N\\ \\forall x\\in D",
        "after_tex": "g_n\\to0\\text{ uniformly on }D",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "量詞順序正好是共同 N 的一致收斂定義。",
          "en": "The quantifier order is exactly uniform convergence with a common N."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "不能改成先給 x 才選 N 來證一般一致收斂。",
            "en": "Choosing N only after x would not prove general uniform convergence."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S07",
        "uses": [
          "G1"
        ],
        "before_tex": "N_x=\\lfloor |c_x|/\\varepsilon\\rfloor+1,\\quad(c_a,c_b,c_c)=(1,-4,2)",
        "after_tex": "\\max_{x\\in D}N_x=\\lfloor4/\\varepsilon\\rfloor+1=N",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另一條有限最大值證法與精確 sup 誤差給同一證書。",
          "en": "The finite-maximum proof agrees with the exact-supremum certificate."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "最大存在是因域完整且有限。",
            "en": "The maximum exists because the entire domain is finite."
          }
        },
        "leaf_basis": "epsilon"
      },
      {
        "id": "S08",
        "uses": [
          "S04",
          "S05"
        ],
        "before_tex": "\\varepsilon=1/10,\\quad N=41",
        "after_tex": "E_n\\le4/43<1/10\\quad(n\\ge41)",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "代一個有理 epsilon 作算術回查，但一般結論已由符號尾端證明。",
          "en": "Substitute one rational epsilon as an arithmetic audit; the symbolic tail already proves the general conclusion."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "此數值核對沒有取代量化所有 epsilon 的證明。",
            "en": "This numerical audit does not replace the proof for every epsilon."
          }
        },
        "leaf_basis": "epsilon"
      }
    ],
    "result": {
      "zh-Hant": "sup 誤差恰為 4/(n+2)，共同 N=floor(4/epsilon)+1 證一致收斂。",
      "en": "The supremum error is exactly 4/(n+2); common N=floor(4/epsilon)+1 proves uniform convergence."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "sup 誤差是 2/(n+2)。",
          "en": "The supremum error is 2/(n+2)."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "漏掉 b 的絕對誤差。",
          "en": "It misses b's absolute error."
        }
      },
      {
        "claim": {
          "zh-Hant": "N 可以依點 x 而不需共同上界。",
          "en": "N may depend on x with no common bound."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "那只是逐點量詞。",
          "en": "That is only the pointwise quantifier order."
        }
      },
      {
        "claim": {
          "zh-Hant": "有限網格 sup 必等於整個連續區間 sup。",
          "en": "A finite-grid supremum must equal the supremum over the whole continuous interval."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "例如 h(x)=x(1-x) 在網格 {0,1} 最大值 0，但在 [0,1] 的 x=1/2 取值 1/4。",
          "en": "For h(x)=x*(1-x), grid {0,1} has maximum zero, while x=1/2 in [0,1] has value 1/4."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "使用有限domain及symbolic rational tails。逐點witness取有限最大值得該有限domain的一致witness；不可外推無限domain。另以sup gap與全稱tail核對，finite prefix不足。",
          "en": "Use a finite domain and symbolic rational tails. The maximum of all pointwise witnesses gives a uniform witness on this complete finite domain, not on arbitrary infinite domains. Check exact supremum gaps and the all-n tail; a finite prefix is insufficient."
        },
        "source": {
          "mko_id": "mko-uniform-convergence-of-functions",
          "sha256": "c28f03cdaf36d8b06ca1718e4253a1d705745496ed7c655d09f55c2a82148ac0",
          "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": "epsilon",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "收斂表示每個正 epsilon 有一個整數 N，使全部 n≥N 的誤差嚴格小於 epsilon；使用實數 Archimedean 性。",
          "en": "Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals."
        }
      },
      {
        "id": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "非空有限實數集合的 sup 是其最大值；一致收斂：每個正 epsilon 有一個共同 N，使全部 n≥N 及全部 x 的誤差小於 epsilon。",
          "en": "A nonempty finite set of real numbers has its maximum as its supremum; uniform convergence requires, for each positive epsilon, one common N making the error smaller than epsilon for every n≥N and every x."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "uniform-tail",
        "lhs": "4/43",
        "rhs": "1/10",
        "step": "S08",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限域及全 n≥0 的有理尾項；N 不得依 x。",
      "en": "A complete finite domain and rational tails for all n≥0; N must not depend on x."
    },
    "md_symbols": [
      [
        "E_n",
        "全域上確界誤差",
        "whole-domain supremum error"
      ],
      [
        "N",
        "共同尾端索引",
        "common tail index"
      ]
    ],
    "md_error_controls": [
      {
        "title": "先取絕對值 / Absolute values first",
        "zh": "先列各點誤差，才求最大值。",
        "en": "List absolute errors before taking their maximum."
      },
      {
        "title": "共同索引 / Common index",
        "zh": "N 只依 epsilon，並涵蓋全部後續 n。",
        "en": "N depends only on epsilon and controls every later n."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "original_source_sha256": "729e76f868ad91fc9a80212ab0f92d3ee8fb3af4d052e2d03b2201388c4fd253",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "definition_here"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "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."
  }
}