{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-convergence-in-probability-01",
  "mko_id": "mko-convergence-in-probability",
  "title": {
    "zh-Hant": "eventual equality的全稱尾端證書",
    "en": "A universal tail certificate from eventual equality"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-convergence-in-probability-01",
    "source": {
      "mko_id": "mko-convergence-in-probability",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "f6a36c4c48518f81f1515a2208543968a39ea38fccfb6c38d7f3613607756354",
      "english_sha256": "662ebc13877df7163257281afba7044949cf6e065b78a7bd62b854e3ad9fbbff"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "以共同有限空間明證依機率收斂。",
      "en": "Prove convergence in probability on a common finite space."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Ω={a,b}，P(a)=1/3、P(b)=2/3；目標X=0。X1=(2,-1)，X2=(1,4)，對所有n≥3設X_n=(0,0)。",
          "en": "Omega={a,b}, with masses 1/3,2/3 and target X=0. X1=(2,-1), X2=(1,4), and for every n≥3 set X_n=(0,0)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "P(a)=1/3,\\ P(b)=2/3",
        "after_tex": "P(\\Omega)=1,\\quad X_n,X:\\Omega\\to\\mathbb R",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "所有變數有共同兩原子carrier與同一law。",
          "en": "All variables have the same two-atom carrier and law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不另為各n換空間。",
            "en": "The space is not changed with n."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "|X_1-X|=(2,1),\\quad\\varepsilon=1",
        "after_tex": "P(|X_1-X|>1)=1/3",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "嚴格偏差事件只含a。",
          "en": "The strict deviation event contains only a."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "b的偏差等於1，不是大於1。",
            "en": "The deviation at b equals 1 and is not greater than 1."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "|X_2-X|=(1,4)",
        "after_tex": "P(|X_2-X|>1)=2/3",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二項偏差機率反而增加，仍不否定尾端收斂。",
          "en": "The second deviation probability increases, without refuting a tail limit."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不能把prefix單調性當定義。",
            "en": "Prefix monotonicity is not part of the definition."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "n\\ge3",
        "after_tex": "X_n(\\omega)-X(\\omega)=0\\quad(\\omega\\in\\Omega)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "由給定的全稱公式，不由兩個觀測值，得到eventual equality。",
          "en": "The declared universal formula, not two observations, gives eventual equality."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "n≥3的量詞覆蓋無限尾端。",
            "en": "The quantifier n≥3 covers the infinite tail."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "\\varepsilon>0,\\quad n\\ge3",
        "after_tex": "\\{|X_n-X|>\\varepsilon\\}=\\varnothing",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "零誤差不會超過任何正epsilon。",
          "en": "Zero error exceeds no positive epsilon."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "epsilon是任意正數，不只已算的1。",
            "en": "Epsilon is arbitrary positive, not merely the tested value 1."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "n\\ge3,\\quad\\varepsilon>0",
        "after_tex": "P(|X_n-X|>\\varepsilon)=0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空事件的機率為零，得到所有尾端項的界。",
          "en": "The empty event has probability zero for every tail index."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這不是單一n的查表結論。",
            "en": "This is not a lookup for one n."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\varepsilon>0,\\ \\delta>0",
        "after_tex": "N=3\\implies\\forall n\\ge N:\\ P(|X_n-X|>\\varepsilon)=0<\\delta",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "同一N=3滿足任意epsilon、delta的全稱尾端責任。",
          "en": "The same N=3 meets the universal tail obligation for arbitrary epsilon and delta."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "必須先任取epsilon、delta再提供N。",
            "en": "Choose arbitrary epsilon and delta before supplying N."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "\\forall\\varepsilon>0\\ \\forall\\delta>0\\ \\exists N\\ \\forall n\\ge N",
        "after_tex": "X_n\\xrightarrow{P}X=0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "精確量詞次序完成依機率收斂。",
          "en": "The exact quantifier order proves convergence in probability."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "本證書較強，但本節結論不依賴未宣告跨mode定理。",
            "en": "This certificate is stronger, but the stated conclusion uses no undeclared mode-implication theorem."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "雖然prefix偏差機率增加，X_n仍依機率收斂到0。",
      "en": "Despite an increasing prefix deviation probability, X_n converges in probability to 0."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "第二項機率增加，故不收斂。",
          "en": "The increased second probability disproves convergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "有限prefix不決定尾端。",
          "en": "A finite prefix does not determine the tail."
        }
      },
      {
        "claim": {
          "zh-Hant": "只核epsilon=1即可。",
          "en": "Checking epsilon=1 suffices."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "定義須涵蓋所有正epsilon。",
          "en": "The definition requires every positive epsilon."
        }
      },
      {
        "claim": {
          "zh-Hant": "找到一個很大的n即可。",
          "en": "One large n suffices."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "必須控制所有n≥N。",
          "en": "Every n≥N must be controlled."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "全部X_n與X須在同一機率空間，並保留forall epsilon>0、forall delta>0、exists N、forall n>=N的tail責任。本輪使用明示c/(n+s) deviation-probability profile或eventual equality證書；有限prefix不構成收斂。",
          "en": "All X_n and X must belong to the same probability space. Retain the tail obligations: for every epsilon>0 and every delta>0 there exists N such that every n≥N has deviation probability below delta. This slice uses an explicitly declared c/(n+s) deviation-probability profile on a valid probability domain, or an eventual-equality certificate. A finite prefix does not establish convergence."
        },
        "source": {
          "mko_id": "mko-convergence-in-probability",
          "sha256": "f6a36c4c48518f81f1515a2208543968a39ea38fccfb6c38d7f3613607756354",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "依機率收斂定義為任意epsilon>0、delta>0可控制全部n≥N的偏差機率；空事件機率零。",
          "en": "Convergence in probability means controlling every n≥N for arbitrary epsilon>0 and delta>0; the empty event has probability zero."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1/3+2/3",
        "rhs": "1"
      },
      {
        "id": "N02",
        "lhs": "0",
        "rhs": "1/100",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "prefix可不單調；證明來自全稱尾端公式。",
      "en": "The prefix may be nonmonotone; proof comes from a universal tail formula."
    },
    "md_symbols": [
      [
        "ε,δ",
        "任意正容忍值",
        "arbitrary positive tolerances"
      ],
      [
        "N",
        "尾端起點",
        "tail index"
      ]
    ],
    "md_error_controls": [
      {
        "title": "嚴格事件 / Strict event",
        "zh": "固定使用>epsilon，區分偏差等於1的原子。",
        "en": "Keep>epsilon and distinguish equality at 1."
      },
      {
        "title": "尾端來源 / Tail source",
        "zh": "全稱公式n≥3為證據；prefix只供audit。",
        "en": "The universal n≥3 formula is evidence; the prefix is only an audit."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "ad6c0a4edec6e457a60c1142e494599b7c1c21a43a35d0576b5fb99866768bb6",
    "original_source_sha256": "b0e2eaaf1242436e89c41152d920ec68ff2a8ba30a6bfe51f2b2f7c9c8ee9196",
    "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": [
      "BR142-S017",
      "BR142-S021",
      "BR142-S020",
      "BR142-S018",
      "BR142-S019"
    ],
    "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."
  }
}