{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-filtration",
  "semantic_family_id": "sf-finite-filtration",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限filtration"
  },
  "summary": {
    "zh-Hant": "有限時間filtration是在同一機率空間上隨時間資訊遞增的sigma代數序列。"
  },
  "statement": {
    "zh-Hant": "以同一有限樣本空間上的partition表示各時刻sigma代數。F_s包含於F_t等價於較晚partition細化較早partition。filtration只表資訊嵌套，不自動使過程adapted，也不宣稱natural filtration、獨立或Markov性。"
  },
  "formula": {
    "tex": "F(t)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>F</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "F",
        "glyph": "F"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "t",
          "glyph": "t"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8e63b590916a74c390ac2ecc4804e6fc24a7227c2db9165483e450450d0d60fa"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "所有partition使用同一0到n-1樣本載體，每列label只表示block等價。"
    },
    {
      "id": "a2",
      "zh-Hant": "逐相鄰時刻完整核refinement；有限抽樣事件不足以證sigma代數包含。"
    }
  ],
  "symbols": [
    {
      "token": "F",
      "role": "finite_filtration",
      "role_zh": "有限filtration",
      "scope": "representative_formula"
    },
    {
      "token": "t",
      "role": "time_index",
      "role_zh": "時間索引",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sigma-algebra-measure-space",
      "title_zh": "有限sigma代數與測度",
      "reason_zh": "每個時刻資訊是同一空間上的sigma代數。"
    },
    {
      "id": "mko-discrete-stochastic-process",
      "title_zh": "離散隨機過程",
      "reason_zh": "filtration提供時間資訊語義。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "sigma代數按時間排列。"
    },
    {
      "id": "mko-measurable-function",
      "title_zh": "可測函數",
      "reason_zh": "adaptedness另核每時刻變數可測。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "partition越細表示資訊越多。",
      "filtration要求每個較晚block包含於某個較早block。",
      "事件可測當且僅當它是當時blocks的聯集；adaptedness需另核。"
    ],
    "common_misconception_zh": "把任意partition序列叫filtration，或由filtration存在直接推出過程adapted、獨立或Markov。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限partition序列；不宣稱natural filtration或任何過程性質。",
    "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": "與條件期望共同加入Martingale。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-filtration.md 整理；2026-10-14 脈衝 R05（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"
  }
}