{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-convergence-in-distribution-01",
  "mko_id": "mko-convergence-in-distribution",
  "title": {
    "zh-Hant": "Bernoulli參數尾端的CDF證書",
    "en": "A Bernoulli parameter-tail CDF certificate"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-convergence-in-distribution-01",
    "source": {
      "mko_id": "mko-convergence-in-distribution",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "082a2a414602a248555b80f0fa923b8b69256bdacb1fe1fe32b38bb0924de21a",
      "english_sha256": "f58586abe9ae537f8873304988774caf2d138a9df726d7ab04aa6d482bdae7b2"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "在極限CDF連續點證依分布收斂。",
      "en": "Prove distributional convergence at continuity points of the limiting CDF."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "n≥1，X_n為Bernoulli(p_n)，p_n=1/3+1/(n+2)；X為Bernoulli(1/3)。只宣告laws，不宣告coupling。",
          "en": "For n≥1, X_n has Bernoulli(p_n) law with p_n=1/3+1/(n+2); X has Bernoulli(1/3) law. Only laws, not a coupling, are declared."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "n\\ge1",
        "after_tex": "1/3<p_n\\le2/3<1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "參數整個尾族都在合法機率範圍。",
          "en": "Parameters lie in the probability range throughout the family."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "最大值在n=1，不是以幾個樣本推估。",
            "en": "The maximum occurs at n=1, rather than being estimated from samples."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "t<0",
        "after_tex": "F_n(t)=F(t)=0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "負門檻的累積事件都空。",
          "en": "All negative-threshold cumulative events are empty."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這是所有t<0的完整分支。",
            "en": "This covers every t<0."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "0\\le t<1",
        "after_tex": "F_n(t)=1-p_n=2/3-1/(n+2),\\quad F(t)=2/3",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "中間區間的CDF只包含Bernoulli零值質量。",
          "en": "In the middle interval the CDF includes only the mass at zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不可把p_n誤作此區間CDF。",
            "en": "Do not use p_n as the CDF on this interval."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "t\\ge1",
        "after_tex": "F_n(t)=F(t)=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "門檻至少1時完整law均被包含。",
          "en": "At thresholds at least1 the whole law is included."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "t=1屬右段，雖是極限CDF的jump點。",
            "en": "The point 1 belongs to the right branch, although it is a jump point of the limiting CDF."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "F\\text{ is Bernoulli}(1/3)\\text{ CDF}",
        "after_tex": "\\operatorname{Cont}(F)=\\mathbb R\\setminus\\{0,1\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "極限CDF只有0與1兩個jump，其他點都在常數開區間。",
          "en": "The limiting CDF jumps only at 0 and 1; all other points lie in constant open regions."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "定義不要求在這兩個jump點收斂。",
            "en": "The definition imposes no convergence requirement at those jumps."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "0<t<1",
        "after_tex": "|F_n(t)-F(t)|=1/(n+2)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在唯一非平凡連續區間，誤差有明示有理尾端。",
          "en": "On the only nontrivial continuity region, the error has an explicit rational tail."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "區間內每個t同受這個界，不是固定幾點。",
            "en": "The bound covers every t in the interval, not a few fixed points."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\eta>0,\\quad N=\\lfloor1/\\eta\\rfloor+1",
        "after_tex": "n\\ge N\\implies1/(n+2)<\\eta",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "N大於1/eta，使所有後續分母足夠大。",
          "en": "N exceeds 1/eta, making all later denominators sufficiently large."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "eta任意且嚴格正，選N在取極限前。",
            "en": "Eta is arbitrary positive and N is selected before the tail conclusion."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "F_n(t)\\to F(t)\\quad(t\\in\\operatorname{Cont}(F))",
        "after_tex": "X_n\\Rightarrow X",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "合併三區間完成CDF連續點定義。",
          "en": "Combining the three regions establishes the continuity-point CDF definition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "沒有共同空間，故不把此結論寫成依機率收斂。",
            "en": "No common space was specified, so this is not stated as convergence in probability."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "Bernoulli(1/3+1/(n+2))依分布收斂到Bernoulli(1/3)。",
      "en": "Bernoulli(1/3+1/(n+2)) converges in distribution to Bernoulli(1/3)."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "此例已指定所有變數共用一個coin。",
          "en": "All variables share one coin here."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "資料僅指定laws，沒有coupling。",
          "en": "Only laws, not a coupling, were specified."
        }
      },
      {
        "claim": {
          "zh-Hant": "須先核jump點才符合定義。",
          "en": "Jump points must be checked for the definition."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "定義只要求極限CDF連續點。",
          "en": "Only continuity points of the limiting CDF are required."
        }
      },
      {
        "claim": {
          "zh-Hant": "CDF有限多個n接近即可證收斂。",
          "en": "A finite set of close CDFs proves convergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本證明用任意eta的全稱尾端。",
          "en": "The proof uses a universal tail for arbitrary eta."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "不要求共同機率空間或coupling。有限prefix不證收斂，極限CDF不連續點不要求收斂；依分布收斂不一般推出依機率、a.s.或Lp。",
          "en": "A common probability space or chosen coupling is not required. A finite prefix does not prove convergence, and the definition imposes no convergence requirement at discontinuities of the limiting CDF. Convergence in distribution does not generally imply convergence in probability, almost surely or in Lp."
        },
        "source": {
          "mko_id": "mko-convergence-in-distribution",
          "sha256": "082a2a414602a248555b80f0fa923b8b69256bdacb1fe1fe32b38bb0924de21a",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "Bernoulli CDF分段；依分布收斂只要求在極限CDF每個連續點收斂；floor(u)+1>u。",
          "en": "The Bernoulli CDF formula; distributional convergence tests each continuity point of the limit; floor(u)+1>u."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1/3+1/(1+2)",
        "rhs": "2/3"
      },
      {
        "id": "N02",
        "lhs": "1/(100+2)",
        "rhs": "1/100",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "不要求共同coupling；不推出更強模式。",
      "en": "No common coupling is required; no stronger mode is inferred."
    },
    "md_symbols": [
      [
        "p_n",
        "Bernoulli參數",
        "Bernoulli parameter"
      ],
      [
        "F_n,F",
        "各自CDF",
        "respective CDFs"
      ]
    ],
    "md_error_controls": [
      {
        "title": "合法參數 / Legal parameters",
        "zh": "對全部n≥1證p_n≤2/3。",
        "en": "Prove p_n≤2/3 for every n≥1."
      },
      {
        "title": "模式邊界 / Mode boundary",
        "zh": "law收斂不自行提供同空間差值。",
        "en": "Law convergence does not supply same-space differences."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "original_source_sha256": "3f7025055c3497aa6b82b00d019258a96a7e7d544c185df0da1bf15436bbdbd0",
    "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": [],
    "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."
  }
}