{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-sequence-limit",
  "semantic_family_id": "sf-sequence-limit",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "數列極限與精確尾界"
  },
  "summary": {
    "zh-Hant": "從0起算的實數列以任意精度、某一自然尾界及整個尾段定義極限；本輪精確處理倒數尾項模型。"
  },
  "statement": {
    "zh-Hant": "實數列 u:N0→R 趨向 L，指對每個 ε>0，都存在自然數 N≥0，使所有 n≥N 有 |u(n)−L|<ε。本輪具體模型 u(n)=L+c/(n+d)，其中 d≥1，L、c為實數；整數參數切片的誤差是 |c|/(n+d)，不靠有限樣本猜收斂。"
  },
  "formula": {
    "tex": "u(n)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>u</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "u",
        "glyph": "u"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "n",
          "glyph": "n"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "0fac6f9ec31efca51308b66dca219619fc32bf98e4cfd59ab8ae3af1fcbdd062"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "N0包含0，所有項分母n+d>0。題庫L、c、d為安全整數，ε=p/q且p、q>0，n及候選N為非負安全整數。"
    },
    {
      "id": "a2",
      "zh-Hant": "固定模型的全部尾項誤差不增，所以候選N的均勻尾界等價於 |c|q<p(N+d)。等號不符合嚴格誤差條件。"
    },
    {
      "id": "a3",
      "zh-Hant": "最小合法N為 max(0,floor(|c|q/p)−d+1)；c=0時N=0即可，L不影響誤差與尾界。"
    }
  ],
  "symbols": [
    {
      "token": "u",
      "role": "real_sequence",
      "role_zh": "指定實數列",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "natural_index_from_zero",
      "role_zh": "從0起算的自然索引",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sequence",
      "title_zh": "數列與索引",
      "reason_zh": "先固定數列及索引慣例。"
    },
    {
      "id": "mko-tends-to-relation",
      "title_zh": "趨近關係",
      "reason_zh": "極限不是直接取到該值。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞與明示論域",
      "reason_zh": "∀ε、∃N、∀n≥N的順序。"
    },
    {
      "id": "mko-rational-number",
      "title_zh": "有理數",
      "reason_zh": "精確正分母容差及誤差。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "尾界N可依ε改變，但選定後要管全部尾段，不是只檢查第N項或前幾項就能替任意數列證明。這個倒數模型另有單調誤差界，才可把均勻尾界化成一個精確不等式。",
      "嚴格誤差會影響最小N：例如c=1、d=1、ε=1/2時，N=1的誤差等於容差，最小合法N是2。負c與正c的絕對誤差相同。",
      "本輪核對的是已宣告倒數尾項模型的參數公式與資格，保留一般收斂證明及形式橋。改L不產生新誤差計算，改c、d不產生新極限L的題型。"
    ],
    "common_misconception_zh": "把∀ε後的N固定死、只看有限項，或把誤差等於ε當作小於ε。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "固定模型的精確參數與尾界可重算；一般實數列收斂未新增形式 Evidence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "在固定版綁定 Tendsto atTop、實數鄰域、自然索引與倒數尾界的全尾段義務。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-sequence-limit.md 整理；2026-10-07 脈衝 R01（使用者手動接續 2026-10-08 工作序列；實際日期仍為 2026-10-07），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-07"
  }
}