{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-random-variable-independence",
  "semantic_family_id": "sf-random-variable-independence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "隨機變數獨立"
  },
  "summary": {
    "zh-Hant": "同一有限joint law下，隨機變數獨立要求全部值對的joint mass分解為邊際質量乘積。"
  },
  "statement": {
    "zh-Hant": "有限值隨機變數X,Y獨立，若對每個值對(x,y)皆有P(X=x,Y=y)=P(X=x)P(Y=y)。等價地，所有值事件逆像獨立。本輪完整核有限joint table；零協方差只作必要後果而非反向證書。"
  },
  "formula": {
    "tex": "K(X,Y)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>K</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "K",
        "glyph": "K"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        },
        {
          "type": "symbol",
          "name": "Y",
          "glyph": "Y"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "46b1dd9230068df248377cc4c2d9bc7c3052fb551235fe71a8bd4d0cb42cbab3"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限joint support與精確有理質量屬同一機率模型。"
    },
    {
      "id": "a2",
      "zh-Hant": "本輪主要處理兩變數；不由pairwise推三個以上變數mutual independence。"
    }
  ],
  "symbols": [
    {
      "token": "K",
      "role": "random_variable_independence",
      "role_zh": "隨機變數獨立紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "random_variable",
      "role_zh": "第一變數",
      "scope": "representative_formula"
    },
    {
      "token": "Y",
      "role": "random_variable",
      "role_zh": "第二變數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-event-independence",
      "title_zh": "事件獨立",
      "reason_zh": "值事件逆像須滿足事件乘積責任。"
    },
    {
      "id": "mko-finite-joint-distribution",
      "title_zh": "有限聯合分布",
      "reason_zh": "全部joint與marginal來自同一完整law。"
    },
    {
      "id": "mko-random-variable",
      "title_zh": "隨機變數",
      "reason_zh": "X與Y是同一機率空間上的變數。"
    },
    {
      "id": "mko-measurable-function",
      "title_zh": "可測函數",
      "reason_zh": "事件逆像須為可測事件。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先由joint table聚合兩個marginal。",
      "全部值對皆須通過factorization。",
      "獨立可推出有限二階矩下零協方差，但零協方差通常不能反推獨立。"
    ],
    "common_misconception_zh": "只核一個值對、只看邊際，或由零協方差直接宣稱獨立／IID。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核同一完整有限joint law的兩變數；不推出同分布、IID、Markov或三變數mutual independence。",
    "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": "加入獨立同分布序列。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-random-variable-independence.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-10-28 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"
  }
}