{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-improper-integral",
  "semantic_family_id": "sf-improper-integral",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "瑕積分"
  },
  "summary": {
    "zh-Hant": "無界區間或含奇點的積分以適當有限區間積分的one-sided limits定義，且每個必要分段須獨立收斂。"
  },
  "statement": {
    "zh-Hant": "一側無界積分int_a^infinity f由b→infinity的有限區間積分極限定義；本輪明示f(x)=scale/x^p於[lower,infinity)。兩側無界exact family明示f(x)=left_scale/|x|^p於x≤-1、f(x)=0於-1<x<1、f(x)=right_scale/x^p於x≥1，左右tails必須獨立收斂。內部奇點family在[c-1,c+1]明示左右piecewise powers並於c左右分拆，兩側皆收斂。不能只靠symmetric principal value；absolute convergence充分但非必要，有限cutoffs看似穩定不足以證收斂。"
  },
  "formula": {
    "tex": "I(f,a,b)=L(F,b)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>I</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>=</mo><mrow><mi>L</mi><mo>(</mo><mi>F</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "I",
          "glyph": "I"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          },
          {
            "type": "symbol",
            "name": "b",
            "glyph": "b"
          }
        ]
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "L",
          "glyph": "L"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "F",
            "glyph": "F"
          },
          {
            "type": "symbol",
            "name": "b",
            "glyph": "b"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "dfdbabb89d9d5e03c8bf037d67bdf14cf2199ee178713efe091df9ffc7105c19"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "一側tail固定f(x)=scale/x^exponent於[lower,infinity)；兩側tail固定上述zero-middle piecewise integrand。"
    },
    {
      "id": "a2",
      "zh-Hant": "內部奇點固定於c，左側f(x)=left_scale*(c-x)^(-alpha_L)、右側f(x)=right_scale*(x-c)^(-alpha_R)，區間為[c-1,c+1]且0≤alpha_L,alpha_R<1。"
    },
    {
      "id": "a3",
      "zh-Hant": "所有one-sided limits與奇點左右責任分開記錄；principal value不冒充improper integral。"
    }
  ],
  "symbols": [
    {
      "token": "I",
      "role": "improper_integral_record",
      "role_zh": "瑕積分紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "integrand",
      "role_zh": "被積函數",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "lower_cutoff",
      "role_zh": "下方cutoff",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "upper_cutoff",
      "role_zh": "上方cutoff",
      "scope": "representative_formula"
    },
    {
      "token": "L",
      "role": "limit_operator",
      "role_zh": "極限算子",
      "scope": "representative_formula"
    },
    {
      "token": "F",
      "role": "finite_cutoff_integral",
      "role_zh": "有限cutoff積分",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-definite-integral",
      "title_zh": "有限區間上的定積分",
      "reason_zh": "cutoff integrals。"
    },
    {
      "id": "mko-function-limit",
      "title_zh": "函數極限",
      "reason_zh": "cutoff趨界或無窮的極限。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "離散cutoff近似的界線。"
    },
    {
      "id": "mko-absolute-convergence-of-series",
      "title_zh": "絕對收斂級數",
      "reason_zh": "absolute convergence與comparison背景。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "每個無界端或奇點都產生獨立one-sided convergence obligation。",
      "兩側發散即使對稱抵銷也只有principal value。",
      "finite truncation evidence不能量化所有更遠cutoffs。"
    ],
    "common_misconception_zh": "把Cauchy principal value等同通常瑕積分，或因幾個large cutoffs接近就宣稱收斂。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪exact計算限明示冪函數；一般integrability與comparison仍需另證。",
    "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": "加入標準常態分布與Lebesgue improper integrals。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-improper-integral.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-11-12 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}