{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-markov-chain",
  "semantic_family_id": "sf-finite-markov-chain",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限狀態 Markov 鏈"
  },
  "summary": {
    "zh-Hant": "有限時間齊次Markov鏈以row-stochastic轉移核與需要時的初始律指定有限路徑機率。"
  },
  "statement": {
    "zh-Hant": "狀態按固定順序排列，P_{ij}=Pr(X_{n+1}=j|X_n=i)，每列非負且精確和為1。本輪只處理單一時間齊次有限轉移矩陣；時刻律與路徑機率另需初始分布。"
  },
  "formula": {
    "tex": "P(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "82bda20a28179ce1d5f24b4cb9405369004e46c8a2f65647e3c93bbe3f8fa6db"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "所有機率以精確有理數表示，rows=current state、columns=next state。"
    },
    {
      "id": "a2",
      "zh-Hant": "只由kernel與初始律計算有限步結果；不宣稱平穩、不可約、遍歷或長期收斂。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "transition_kernel",
      "role_zh": "轉移核",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "finite_markov_chain",
      "role_zh": "有限Markov鏈",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-discrete-stochastic-process",
      "title_zh": "離散隨機過程",
      "reason_zh": "Markov鏈是具有條件獨立結構的過程。"
    },
    {
      "id": "mko-matrix",
      "title_zh": "矩陣",
      "reason_zh": "有限轉移核以固定狀態順序矩陣表示。"
    },
    {
      "id": "mko-conditional-probability",
      "title_zh": "條件機率",
      "reason_zh": "Markov性以給定現在後的條件分布表述。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "row-stochastic資格先於任何步進計算。",
      "一步律使用row-vector convention muP。",
      "有限路徑機率為初始質量乘上逐步轉移因子。"
    ],
    "common_misconception_zh": "把欄和當列和，或在沒有初始律時聲稱時刻分布與路徑機率。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "單一有限kernel模型不證平穩、遍歷或收斂。",
    "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": "加入狀態分類、平穩分布與首達時間。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-markov-chain.md 整理；2026-10-13 脈衝 R03（使用者手動接續 2026-10-24 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}