{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-strong-law-large-numbers-finite-valued-iid",
  "semantic_family_id": "sf-strong-law-large-numbers-finite-valued-iid",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限值IID強大數律"
  },
  "summary": {
    "zh-Hant": "有限值無限IID序列的樣本平均幾乎必然收斂至共同期望。"
  },
  "statement": {
    "zh-Hant": "以明示infinite IID process certificate及normalized finite marginal law定義mu。SLLN結論為P(lim_{n→∞} n^{-1}sum_{i=1}^n X_i=mu)=1。exact families重算marginal mean、固定prefix的sample mean與固定n deviation event probability，但明示這些都不是a.s. convergence證書。non-IID與nonfinite-support只是本節scoped theorem不適用；finite prefix不證極限，wrong target不等於共同mean。"
  },
  "formula": {
    "tex": "S(X,n,e)=b",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>S</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>e</mi><mo>)</mo></mrow><mo>=</mo><mi>b</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "S",
          "glyph": "S"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "X",
            "glyph": "X"
          },
          {
            "type": "symbol",
            "name": "n",
            "glyph": "n"
          },
          {
            "type": "symbol",
            "name": "e",
            "glyph": "e"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "b",
        "glyph": "b"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ea211047be36c05baddef794ae1d4264c654f623a80461c836115f810409186e"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "process certificate涵蓋無限索引、independence與共同finite law。"
    },
    {
      "id": "a2",
      "zh-Hant": "marginal carrier有限且probabilities exact、nonnegative、sum為1。"
    }
  ],
  "symbols": [
    {
      "token": "S",
      "role": "strong_law_operator",
      "role_zh": "強大數律算子",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "infinite_finite_valued_iid_sequence",
      "role_zh": "有限值無限IID序列",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "prefix_length",
      "role_zh": "前綴長度",
      "scope": "representative_formula"
    },
    {
      "token": "e",
      "role": "positive_deviation",
      "role_zh": "正偏差門檻",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "almost_sure_convergence_statement",
      "role_zh": "幾乎必然收斂結論",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-iid-sequence",
      "title_zh": "獨立同分布序列",
      "reason_zh": "無限IID假設。"
    },
    {
      "id": "mko-almost-sure-convergence",
      "title_zh": "幾乎必然收斂",
      "reason_zh": "結論語義。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "有限模型的期望值",
      "reason_zh": "共同mean。"
    },
    {
      "id": "mko-sigma-algebra-measure-space",
      "title_zh": "σ代數與測度空間",
      "reason_zh": "a.s. event。"
    },
    {
      "id": "mko-variance",
      "title_zh": "有限機率模型的變異數",
      "reason_zh": "finite-valued law的moment背景。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "finite marginal law不能代替infinite IID process certificate。",
      "prefix mean與deviation probability只屬固定n audit。",
      "不適用邊界不被誤寫成一般SLLN反例。"
    ],
    "common_misconception_zh": "看到幾個prefix接近mu就宣稱a.s. convergence，或把nonfinite support當成一般SLLN必然失敗。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite prefix enumeration不證probability-one limit。",
    "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integrable IID一般SLLN、Borel–Cantelli與ergodic boundaries。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-strong-law-large-numbers-finite-valued-iid.md 整理；2026-10-15 脈衝 R02（使用者手動接續 2026-11-18 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}