{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-continuous-real-distribution-function-01",
  "mko_id": "mko-continuous-real-distribution-function",
  "title": {
    "zh-Hant": "有平坦段的連續CDF",
    "en": "A continuous CDF with a flat interval"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-continuous-real-distribution-function-01",
    "source": {
      "mko_id": "mko-continuous-real-distribution-function",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "1dede2c575b28e28d08e5c509fe02847a319df1bd2e2069c0c6163ebfa3ebfac",
      "english_sha256": "bfbdc0d972f8b891c5b2fb6f469ee997dab5a660e6ada2c36c2b8f63bda3eae1"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "從knots完整構造CDF並核atomless性。",
      "en": "Construct a complete knot-based CDF and audit atomlessness."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "knots為(0,0)、(1,1/2)、(2,1/2)、(3,1)；相鄰間線性插值，左側0、右側1。",
          "en": "The knots are(0,0),(1,1/2),(2,1/2),(3,1); interpolate linearly, with left tail 0 and right tail 1."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "0<1<2<3,\\quad0\\le1/2\\le1/2\\le1",
        "after_tex": "\\text{strict knot order and nondecreasing levels}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先核knots順序與CDF值合法範圍。",
          "en": "First audit strict knot order and legal nondecreasing CDF levels."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "相同level允許，只有x座標必須嚴格遞增。",
            "en": "Equal levels are allowed; only x-coordinates must be strictly ordered."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "t\\le0\\text{ or }t\\ge3",
        "after_tex": "F(t)=0\\ (t\\le0),\\quad F(t)=1\\ (t\\ge3)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "指定兩個尾端，完成全實軸定義的外部區間。",
          "en": "Specify both tails to cover the exterior real-line regions."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "端點0與3將和內插值一致。",
            "en": "The values at endpoints t=0 and t=3 must agree with the interpolated pieces."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "0\\le t\\le1",
        "after_tex": "F(t)=t/2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第一段斜率(1/2-0)/(1-0)=1/2。",
          "en": "The first slope is(1/2-0)/(1-0)=1/2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "t=0、1代回得到正確兩knots。",
            "en": "Substitution at 0 and 1 recovers both knots."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "1\\le t\\le2",
        "after_tex": "F(t)=1/2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "中間段兩端值相同，為合法平坦區間。",
          "en": "Equal endpoint levels produce a valid flat middle interval."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "CDF不必嚴格遞增。",
            "en": "A CDF need not be strictly increasing."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "2\\le t\\le3",
        "after_tex": "F(t)=1/2+(t-2)/2=(t-1)/2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "末段也有斜率1/2，且終點為1。",
          "en": "The last slope is also1/2 and reaches level1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "t=2要給1/2，不能用t/2接錯。",
            "en": "At t=2 it must give 1/2, not the incorrect continuation t/2."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "t\\in\\{0,1,2,3\\}",
        "after_tex": "F(t^-)=F(t)=F(t^+)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "各線性段與尾端在四個knots接合，內部各自連續。",
          "en": "All pieces and tails agree at four knots; their interiors are continuous."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "逐接點核對才證全域連續，不是只取樣。",
            "en": "Auditing every junction proves global continuity, not sampling."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "F\\text{ nondecreasing and continuous, with tails }0,1",
        "after_tex": "\\exists X:\\ F(t)=P(X\\le t);\\quad P(X=t)=F(t)-F(t^-)=0\\quad(t\\in\\mathbb R)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "由各段非負斜率、尾端恆0與1及已證連續性，先套明示CDF存在定理前提取得law X，再用jump公式對所有t得零singleton mass。",
          "en": "Nonnegative piece slopes, tails identically0 and 1, and proved continuity first invoke the explicitly assumed CDF-existence theorem to obtain a law X; then the jump identity gives zero singleton mass for every t."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "先取得law才談P(X=t)；無原子不等於區間機率零，例如P(0<X≤1)=1/2。",
            "en": "Obtain the law before writing P(X=t); atomless does not mean zero interval mass, since P(0<X≤1)=1/2."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "\\lim_{t\\to-\\infty}F(t)=0,\\quad\\lim_{t\\to\\infty}F(t)=1",
        "after_tex": "F\\text{ is a continuous CDF, not strictly increasing}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "單調、連續與尾端極限完成CDF資格；平坦段保留。",
          "en": "Monotonicity, continuity and tail limits qualify a continuous CDF, retaining its flat interval."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "一般continuous CDF可能singular，如Cantor law；不能由本例外推density。",
            "en": "General continuous CDFs may be singular, as with a Cantor law; this example cannot imply densities generally."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "這是一個有[1,2]平坦段的連續無原子CDF。",
      "en": "This is a continuous atomless CDF with flat interval[1,2]."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "所有continuous CDF都strictly increasing。",
          "en": "Every continuous CDF is strictly increasing."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "此例[1,2]恆為1/2。",
          "en": "This example is constant 1/2 on[1,2]."
        }
      },
      {
        "claim": {
          "zh-Hant": "continuous CDF必有density。",
          "en": "A continuous CDF must have a density."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "Cantor型singular law是邊界反例。",
          "en": "A singular Cantor-type law is a counterexample."
        }
      },
      {
        "claim": {
          "zh-Hant": "P(X=1)=F(1)=1/2。",
          "en": "P(X=1)=F(1)=1/2."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "singleton mass為jump，此處為零。",
          "en": "Singleton mass is the jump, which is zero here."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "A continuous CDF has no jumps, so every singleton has probability 0. This is not continuity of a random-variable map on the sample space and does not imply a density or absolute continuity. An affine Cantor distribution supplies a continuous, atomless, singular boundary. Flat CDF intervals are allowed; strict increase is not required."
        },
        "source": {
          "mko_id": "mko-continuous-real-distribution-function",
          "sha256": "1dede2c575b28e28d08e5c509fe02847a319df1bd2e2069c0c6163ebfa3ebfac",
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        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "分段線性函數在每段連續；接點左右值一致使全域連續；CDF存在定理與jump identity為前提。",
          "en": "Piecewise-linear functions are continuous on each piece; matching junctions ensure global continuity; CDF existence and the jump identity are prerequisites."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "(1/2)/2",
        "rhs": "1/4"
      },
      {
        "id": "N02",
        "lhs": "((5/2)-1)/2",
        "rhs": "3/4"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "piecewise-linear exact family；不由連續性推一般density存在。",
      "en": "An exact piecewise-linear family; continuity alone will not imply general density existence."
    },
    "md_symbols": [
      [
        "F",
        "實數CDF",
        "real CDF"
      ],
      [
        "t",
        "任意實數門檻",
        "arbitrary real threshold"
      ]
    ],
    "md_error_controls": [
      {
        "title": "接點 / Junctions",
        "zh": "逐核0、1、2、3的左值與右值。",
        "en": "Check both sides at 0,1,2,3."
      },
      {
        "title": "查詢排序 / Query ordering",
        "zh": "例如1/2、3/2、5/2嚴格遞增，值為1/4、1/2、3/4。",
        "en": "Queries1/2,3/2,5/2 are strictly ordered and yield1/4,1/2,3/4."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
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    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}