{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-central-limit-theorem-finite-variance-iid",
  "semantic_family_id": "sf-central-limit-theorem-finite-variance-iid",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限變異數IID中央極限定理"
  },
  "summary": {
    "zh-Hant": "共同law且0<σ²<∞的無限IID序列，其正確中心化與sqrt(n)標準化和依分布收斂至標準常態。"
  },
  "statement": {
    "zh-Hant": "明示無限IID process certificate、共同finite law、mu與0<sigma^2<infinity。令S_n=sum_{i=1}^n X_i，標準化Z_n=(S_n-n*mu)/(sigma*sqrt(n))，則對標準常態CDF的連續點z有P(Z_n≤z)→Phi(z)。exact families完整重算finite law moments、Z_n的mean 0／variance 1與Bernoulli sum lattice；另有完整hypotheses-to-named-theorem symbolic application。有限binomial table不是normal approximation或convergence evidence；non-IID只表示本定理不可套用，zero variance使標準化失效，wrong scaling與finite histogram均不得冒充CLT。"
  },
  "formula": {
    "tex": "C(X,n,z)=b",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>C</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>b</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "C",
          "glyph": "C"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "X",
            "glyph": "X"
          },
          {
            "type": "symbol",
            "name": "n",
            "glyph": "n"
          },
          {
            "type": "symbol",
            "name": "z",
            "glyph": "z"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "b",
        "glyph": "b"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5a968e7a5f678355ed4b5d0fd42f17d82edd023fea8c33eefb7d0ac5f68acae7"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "序列是明示的無限IID process，不由有限sample推定。"
    },
    {
      "id": "a2",
      "zh-Hant": "共同law具有有限mean與嚴格正、有限variance。"
    },
    {
      "id": "a3",
      "zh-Hant": "標準化使用sigma*sqrt(n)，不是n*sigma或未中心化的和。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "central_limit_operator",
      "role_zh": "中央極限定理算子",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "infinite_iid_sequence",
      "role_zh": "無限IID序列",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "sample_size",
      "role_zh": "樣本大小",
      "scope": "representative_formula"
    },
    {
      "token": "z",
      "role": "normal_cdf_threshold",
      "role_zh": "標準常態CDF門檻",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "distributional_convergence_statement",
      "role_zh": "分布收斂結論",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-iid-sequence",
      "title_zh": "獨立同分布序列",
      "reason_zh": "無限IID process。"
    },
    {
      "id": "mko-convergence-in-distribution",
      "title_zh": "依分布收斂",
      "reason_zh": "CLT結論。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "有限模型的期望值",
      "reason_zh": "共同mean。"
    },
    {
      "id": "mko-variance",
      "title_zh": "有限機率模型的變異數",
      "reason_zh": "共同positive finite variance。"
    },
    {
      "id": "mko-standard-normal-distribution",
      "title_zh": "標準常態分布",
      "reason_zh": "極限law與CDF。"
    },
    {
      "id": "mko-sigma-algebra-measure-space",
      "title_zh": "σ代數與測度空間",
      "reason_zh": "無限process與distribution語義。"
    },
    {
      "id": "mko-binomial-theorem",
      "title_zh": "二項式定理",
      "reason_zh": "Bernoulli sum exact lattice probabilities。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整無限IID與moment資格先於named theorem。",
      "finite standardized moments與binomial lattice可重算，但不是收斂證明。",
      "zero variance、non-IID、wrong scaling與finite histogram分層拒絕。"
    ],
    "common_misconception_zh": "由一個看似鐘形的有限histogram宣稱CLT，或把sqrt(n)誤換成n。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite laws與lattice tables只證指定n；一般distributional limit未形式化。",
    "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Lindeberg條件、Berry–Esseen誤差與multivariate CLT。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-central-limit-theorem-finite-variance-iid.md 整理；2026-10-15 脈衝 R01（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"
  }
}