{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-iid-sequence",
  "semantic_family_id": "sf-iid-sequence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "獨立同分布序列"
  },
  "summary": {
    "zh-Hant": "IID序列同時要求每個變數有相同分布，且每個有限子族滿足mutual independence。"
  },
  "statement": {
    "zh-Hant": "本輪用Bernoulli(p)無限product-law／cylinder symbolic certificate建立IID序列：任意有限0-1 word的機率為p的一數次方乘(1-p)的零數次方。有限prefix只能稱finite IID family，不能獨自證無限序列。"
  },
  "formula": {
    "tex": "I(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>I</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "I",
        "glyph": "I"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "189717dae8abc99c892be562673e41d5ab2dffcf2773d54700341c5e59fe9e06"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "p為0到1的精確有理數；cylinder word為有限0-1序列。"
    },
    {
      "id": "a2",
      "zh-Hant": "無限IID聲明必須明示symbolic product-law certificate；相同期望或一段prefix不足。"
    }
  ],
  "symbols": [
    {
      "token": "I",
      "role": "iid_sequence",
      "role_zh": "IID序列紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "random_variable_sequence",
      "role_zh": "隨機變數序列",
      "scope": "representative_formula"
    },
    {
      "token": "p",
      "role": "bernoulli_parameter",
      "role_zh": "共同Bernoulli參數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-random-variable-independence",
      "title_zh": "隨機變數獨立",
      "reason_zh": "每個有限子族須mutually independent。"
    },
    {
      "id": "mko-discrete-stochastic-process",
      "title_zh": "離散隨機過程",
      "reason_zh": "變數依離散時間索引。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "無限變數族以序列表示。"
    },
    {
      "id": "mko-finite-joint-distribution",
      "title_zh": "有限聯合分布",
      "reason_zh": "每個finite cylinder有精確joint law。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "期望值",
      "reason_zh": "共同mean是後果但非同分布證書。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "identically distributed核相同完整marginal，不是只看mean。",
      "independence量化每個有限子族。",
      "Bernoulli product-law對任意finite word給統一cylinder公式。"
    ],
    "common_misconception_zh": "用相同期望、pairwise independence或檢查固定prefix就宣稱無限IID。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "無限聲明只依明示Bernoulli product-law symbolic certificate；不推出LLN、CLT或Markov額外結論。",
    "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": "加入弱大數律與中心極限定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-iid-sequence.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-10-29 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"
  }
}