{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-pointwise-convergence-of-functions-01",
  "mko_id": "mko-pointwise-convergence-of-functions",
  "title": {
    "zh-Hant": "有限定義域的逐點尾項證書",
    "en": "Pointwise tail certificates on a finite domain"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-pointwise-convergence-of-functions-01",
    "source": {
      "mko_id": "mko-pointwise-convergence-of-functions",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "dde2c47154f28c5cd5dac55ff9f8a148e7761d085014f6ee6d2f72acbfb6012c",
      "english_sha256": "fd5f5de5b9ebd60cc4bb9c9abc7007107dd5d3642bc91e1ce97dc06e4f06b6f2"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "epsilon",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "證 f_n(p)=1+2/(n+1), f_n(q)=-2-3/(n+1) 逐點趨於 f。",
      "en": "Prove pointwise convergence of f_n(p)=1+2/(n+1), f_n(q)=-2-3/(n+1) to f."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "D={p,q}, f(p)=1,f(q)=-2；所有 n≥0 都由上述完整公式定義 f_n。",
          "en": "D={p,q}, f(p)=1,f(q)=-2; the displayed full formulas define f_n for every n≥0."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "n\\ge0",
        "after_tex": "n+1>0,\\quad f_n:D\\to\\mathbb R",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "分母不零，共同定義域與值域明確。",
          "en": "The denominator is nonzero and all maps share the specified domain and codomain."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不是每個 n 換不同定義域。",
            "en": "The domain does not vary with n."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "f_n(p)-f(p)=2/(n+1)",
        "after_tex": "|f_n(p)-f(p)|=2/(n+1)",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先固定 p，絕對值因分母正與分子正可直接去除。",
          "en": "Fix p first; positivity of numerator and denominator removes the absolute value."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "符號核對依完整公式，不靠有限值表。",
            "en": "The sign check uses the full formula, not a finite table."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "S02"
        ],
        "before_tex": "\\varepsilon>0,\\quad N_p=\\lfloor2/\\varepsilon\\rfloor+1",
        "after_tex": "n\\ge N_p\\Longrightarrow n+1>2/\\varepsilon",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [
          "\\varepsilon\\in\\mathbb Q_{>0}"
        ],
        "explanation": {
          "zh-Hant": "取嚴格大於 2/epsilon 的整數，全部尾端分母更大。",
          "en": "Choose an integer strictly greater than 2/epsilon; every later denominator is larger."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02"
          ],
          "explanation": {
            "zh-Hant": "N_p 只依 p 與 epsilon，不依後來的 n。",
            "en": "N_p depends only on p and epsilon, not the later index n."
          }
        },
        "leaf_basis": "epsilon"
      },
      {
        "id": "S04",
        "uses": [
          "S02",
          "S03"
        ],
        "before_tex": "n+1>2/\\varepsilon",
        "after_tex": "|f_n(p)-f(p)|<\\varepsilon\\quad(n\\ge N_p)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "正數取倒數反向，得到嚴格 epsilon 界。",
          "en": "Reciprocating positive numbers reverses the inequality and yields the strict epsilon bound."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "未把 ≤epsilon 誤作所需嚴格界。",
            "en": "A non-strict epsilon bound is not silently substituted for the strict one."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "f_n(q)-f(q)=-3/(n+1)",
        "after_tex": "|f_n(q)-f(q)|=3/(n+1)",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "固定另一點 q，負號在絕對值中消失。",
          "en": "At q the negative sign disappears under absolute value."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "不是把原函數的負值當負誤差。",
            "en": "Negative function values do not make the absolute error negative."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "S05"
        ],
        "before_tex": "N_q=\\lfloor3/\\varepsilon\\rfloor+1",
        "after_tex": "n\\ge N_q\\Longrightarrow |f_n(q)-f(q)|<\\varepsilon",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [
          "\\varepsilon>0"
        ],
        "explanation": {
          "zh-Hant": "同一分母比較給 q 的全尾證書。",
          "en": "The same denominator comparison supplies q's all-tail certificate."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "逐點定義允許 N_q 與 N_p 不同。",
            "en": "Pointwise convergence permits different N_q and N_p."
          }
        },
        "leaf_basis": "epsilon"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S04",
          "S06"
        ],
        "before_tex": "\\forall x\\in D\\ \\forall\\varepsilon>0\\ \\exists N_x\\ \\forall n\\ge N_x",
        "after_tex": "f_n(x)\\to f(x)\\text{ pointwise on }D",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任給實數 epsilon>0，由已宣告的 Archimedean 性選正整數 k>1/epsilon，令 delta=1/k，則 0<delta<epsilon 且 delta 為有理數。把前述尾項證書的容差實例化為 delta，得到 N_p=2k+1、N_q=3k+1；取 N=3k+1。當 n≥N，兩點的誤差皆不超過 3/(n+1)<1/k<epsilon，故全部正實數容差的逐點量詞成立，並且這個兩點模型還有共同尾端。",
          "en": "Given any real epsilon>0, the declared Archimedean property supplies a positive integer k>1/epsilon. Set delta=1/k, so delta is rational and 0<delta<epsilon. Instantiate the preceding tail certificates at tolerance delta to obtain N_p=2k+1 and N_q=3k+1, and take N=3k+1. For every n≥N, both pointwise errors are at most 3/(n+1)<1/k<epsilon. Thus the pointwise quantifiers hold for every positive real tolerance; this two-point model additionally has a common tail."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "沒有交換 x 與 N 的量詞來定義逐點收斂。",
            "en": "The x and N quantifiers have not been interchanged in the definition."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S08",
        "uses": [
          "S03",
          "S06"
        ],
        "before_tex": "N=\\max(N_p,N_q)",
        "after_tex": "n\\ge N\\Longrightarrow\\forall x\\in D\\ |f_n(x)-f(x)|<\\varepsilon",
        "relation": "implies",
        "rule": "epsilon",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對任給正實數 epsilon，沿用 S07 的有理容差 delta=1/k；此處 N_p、N_q 是在 delta 處取得的索引，取其有限最大值 N=3k+1，便對全部 n≥N 與兩個 x 同時給出小於 epsilon 的誤差。這是有限域的額外推論，不外推到無限域。",
          "en": "For any positive real epsilon, use the rational tolerance delta=1/k chosen in S07. Here N_p and N_q are the indices instantiated at delta; their finite maximum N=3k+1 controls both points for every n≥N with error smaller than epsilon. This is an additional finite-domain inference, not an extrapolation to infinite domains."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "共同 N 是額外有限性推論，不是逐點定義自帶。",
            "en": "The common N is an additional finiteness inference, not part of pointwise convergence alone."
          }
        },
        "leaf_basis": "epsilon"
      }
    ],
    "result": {
      "zh-Hant": "兩點均逐點收斂，且此有限例另有共同 N=max(N_p,N_q)。",
      "en": "Both points converge pointwise, and this finite instance additionally has common N=max(N_p,N_q)."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "只核 n≤100 就證逐點收斂。",
          "en": "Checking n≤100 proves pointwise convergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "尚有所有 n≥N 的責任。",
          "en": "Every n≥N remains an obligation."
        }
      },
      {
        "claim": {
          "zh-Hant": "負號使 q 的絕對誤差為負。",
          "en": "The minus sign makes the q absolute error negative."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "誤差為 3/(n+1)>0。",
          "en": "The error is 3/(n+1)>0."
        }
      },
      {
        "claim": {
          "zh-Hant": "任意無限域逐點收斂亦可取有限最大 N。",
          "en": "Every infinite-domain pointwise limit permits a finite maximum N."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "無限多個逐點索引可能無共同上界。",
          "en": "Infinitely many pointwise indices may have no common bound."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "本輪用完整有限domain及f_n(x)=f(x)+c_x/(n+s)的symbolic tail。逐x給exact epsilon witness；有限domain可取所有N的最大值推出該有限模型的一致收斂，但不可外推無限domain。有限prefix不足。",
          "en": "This slice uses a complete finite domain and the symbolic tail f_n(x)=f(x)+c_x/(n+s). Supply an exact epsilon witness for each x. A maximum of the finitely many witnesses gives uniform convergence for this finite model, not for an arbitrary infinite domain. A finite prefix is insufficient."
        },
        "source": {
          "mko_id": "mko-pointwise-convergence-of-functions",
          "sha256": "dde2c47154f28c5cd5dac55ff9f8a148e7761d085014f6ee6d2f72acbfb6012c",
          "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": "逐點收斂：每個 x、每個正 epsilon 皆有 N(x,epsilon)，使全部 n≥N 的該點誤差小於 epsilon。",
          "en": "Pointwise convergence: for each x and positive epsilon there exists N(x,epsilon) making the error at that x smaller than epsilon for every n≥N."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "p-tail",
        "lhs": "2/22",
        "rhs": "1/10",
        "step": "S04",
        "relation": "<"
      },
      {
        "id": "q-tail",
        "lhs": "3/32",
        "rhs": "1/10",
        "step": "S06",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "共同有限域 {p,q}、R 值、n≥0；先以正有理容差建立精確尾項證書，再以 Archimedean 性銜接每個正實數 epsilon；不外推無限域。",
      "en": "The common finite domain {p,q}, real values and n≥0; exact rational-tolerance tail certificates are extended to every positive real epsilon using the Archimedean property, with no infinite-domain extrapolation."
    },
    "md_symbols": [
      [
        "D",
        "共同有限域",
        "common finite domain"
      ],
      [
        "N_p,N_q",
        "逐點尾端索引",
        "point-dependent tail indices"
      ]
    ],
    "md_error_controls": [
      {
        "title": "嚴格 epsilon / Strict epsilon",
        "zh": "取 floor(C/epsilon)+1 保留嚴格不等式。",
        "en": "Use floor(C/epsilon)+1 to preserve strict inequality."
      },
      {
        "title": "完整域 / Complete domain",
        "zh": "兩點已窮盡，但不把兩點測試當無限域證明。",
        "en": "The two points exhaust this domain, not an infinite one."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "original_source_sha256": "a911769c2c8e25b8fc40746c08867b7ff998c9f34caf51de52c5f65f95c6a20a",
    "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": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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      "BR142-M001-SCOPE-ZH",
      "BR142-M001-S07-EN",
      "BR142-M001-S07-ZH",
      "BR142-M001-S08-EN",
      "BR142-M001-S08-ZH"
    ],
    "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."
  }
}