{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-lp-convergence",
  "semantic_family_id": "sf-lp-convergence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Lp收斂"
  },
  "summary": {
    "zh-Hant": "共同機率空間上固定整數p≥1時，E|X_n−X|^p→0稱Lp收斂。"
  },
  "statement": {
    "zh-Hant": "本輪限定正整數p及完整有限原子law，使用symbolic error amplitudes c_i/(n+s)精確計算p-moment與epsilon tail。finite prefix不足；不無條件推a.s.、其他Lq或反向機率模式。"
  },
  "formula": {
    "tex": "L(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "L",
        "glyph": "L"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8c22400d21af14503c3522b6c0fb69e798532c6cf85af40594ed9817597be6df"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "所有X_n,X共享同一完整有限空間與精確有理masses。"
    },
    {
      "id": "a2",
      "zh-Hant": "p為正整數且每個p-moment有限。"
    }
  ],
  "symbols": [
    {
      "token": "L",
      "role": "lp_convergence",
      "role_zh": "Lp收斂",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "random_variable_sequence",
      "role_zh": "隨機變數列",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-random-variable",
      "title_zh": "隨機變數",
      "reason_zh": "共同空間。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "期望值",
      "reason_zh": "p-moment。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "誤差。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "moment→0。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "冪與根",
      "reason_zh": "p次方。"
    },
    {
      "id": "mko-sigma-algebra-measure-space",
      "title_zh": "測度空間",
      "reason_zh": "共同law。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "固定p及共同space。",
      "精確moment tail對所有n≥N。",
      "finite prefix與跨mode implications另需定理。"
    ],
    "common_misconception_zh": "從finite samples或其他mode無條件推出Lp。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限原子、整數p與symbolic rational amplitude。",
    "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mode implication theorems。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-lp-convergence.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-11-04 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"
  }
}