{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-continuous-real-distribution-function",
  "semantic_family_id": "sf-continuous-real-distribution-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "連續實值分布函數"
  },
  "summary": {
    "zh-Hant": "本節點以CDF本身連續定義continuous distribution；CDF仍須nondecreasing、right-continuous且兩端極限0與1。"
  },
  "statement": {
    "zh-Hant": "continuous CDF沒有jumps，所以每個singleton probability為0。這不等於sample-space上的random-variable map連續，也不推出density或absolute continuity；affine Cantor distribution提供continuous atomless singular boundary。CDF可有flat intervals，不必strictly increasing。"
  },
  "formula": {
    "tex": "F(x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "F",
        "glyph": "F"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "331980589ed7140f6e496b92bcc006e298d08c14e56119f808b377a5bbae12a0"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "exact positive families使用knots間linear interpolation、左側0與右側1。"
    },
    {
      "id": "a2",
      "zh-Hant": "knot values由0到1 nondecreasing且knots嚴格遞增；audit query_points也必須依exact rational order嚴格遞增。"
    }
  ],
  "symbols": [
    {
      "token": "F",
      "role": "continuous_cumulative_distribution_function",
      "role_zh": "連續CDF",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "real_threshold",
      "role_zh": "實threshold",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-cumulative-distribution-function",
      "title_zh": "累積分布函數",
      "reason_zh": "CDF obligations。"
    },
    {
      "id": "mko-continuity",
      "title_zh": "連續性",
      "reason_zh": "no jumps。"
    },
    {
      "id": "mko-function-limit",
      "title_zh": "函數極限",
      "reason_zh": "endpoint limits。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "nondecreasing。"
    },
    {
      "id": "mko-sigma-algebra-measure-space",
      "title_zh": "σ-代數與測度空間",
      "reason_zh": "probability law。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "CDF continuous是distribution property。",
      "continuous implies zero point masses。",
      "continuous/atomless不保證density。"
    ],
    "common_misconception_zh": "把continuous CDF等同存在PDF，或要求CDF strictly increasing。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "piecewise-linear families是exact slice；singular boundary不作density計算。",
    "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standard normal distribution。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-continuous-real-distribution-function.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-11-11 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"
  }
}