{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-almost-sure-convergence-01",
  "mko_id": "mko-almost-sure-convergence",
  "title": {
    "zh-Hant": "逐原子尾端與零質量例外",
    "en": "Atomwise tails and a null exception"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-almost-sure-convergence-01",
    "source": {
      "mko_id": "mko-almost-sure-convergence",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "a848e75d5e3643fc87b620fc61c2385c8ed080ba4eac3a8573d9cb5e575f0b70",
      "english_sha256": "7fe83946a8db77910d7a5c8f69cd807c4ac79a2b6c29e835673de60af2a605ef"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "完整列出每個原子的收斂或失敗證書。",
      "en": "Give a convergence or failure certificate for every atom."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Ω={a,b,z}，質量(1/3,2/3,0)，X=(1,-2,0)。n≥1時X_n(a)=1+1/(n+1)，X_n(b)=-1若n<4、否則-2，X_n(z)=(-1)^n。",
          "en": "Omega={a,b,z} has masses(1/3,2/3,0), target X=(1,-2,0). For n≥1, X_n(a)=1+1/(n+1); X_n(b)=-1 if n<4 and -2 otherwise; X_n(z)=(-1)^n."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "1/3+2/3+0=1",
        "after_tex": "X_n,X\\text{ share a complete finite space}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "包含零質量原子z，所有函數都在同一carrier定義。",
          "en": "Include null atom z; every function is defined on the same carrier."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "零質量不表示可把函數定義域省掉z。",
            "en": "Null mass does not remove z from the function domain."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "|X_n(a)-X(a)|=1/(n+1)",
        "after_tex": "N_a(\\varepsilon)=\\lfloor1/\\varepsilon\\rfloor+1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對任意正epsilon，取Na使n≥Na時誤差小於epsilon。",
          "en": "For arbitrary positive epsilon choose Na so every n≥Na has error below epsilon."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "明示全n公式，非觀察有限下降值。",
            "en": "This is a formula for all n, not observed finite decreases."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "n\\ge4",
        "after_tex": "|X_n(b)-X(b)|=0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "b有固定settling index4，之後完全相等。",
          "en": "Atom b has settling index 4 and exact equality afterward."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "n<4的誤差不影響極限。",
            "en": "Errors before 4 do not affect the limit."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "X_n(z)=(-1)^n,\\quad X(z)=0",
        "after_tex": "|X_n(z)-X(z)|=1\\quad(n\\ge1)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "z的偏差永為1，取epsilon=1/2即可否證收斂。",
          "en": "The deviation at z is always1; epsilon=1/2 disproves convergence there."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不是只靠兩個交替prefix值判失敗，而是全稱幅度公式。",
            "en": "Failure follows from the all-index amplitude formula, not two alternating prefix values."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "C=\\{\\omega:\\lim_nX_n(\\omega)=X(\\omega)\\}",
        "after_tex": "C=\\{a,b\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐原子的證書完整確定收斂事件。",
          "en": "The atomwise certificates completely determine the convergence event."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "三個原子均已分類，沒有未檢情況。",
            "en": "All three atoms are classified with none left unchecked."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "P(C)=P(a)+P(b)",
        "after_tex": "P(C)=1/3+2/3=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "正質量原子都收斂，失敗原子總質量零。",
          "en": "All positive-mass atoms converge and the failure atom has total mass zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "a.s.要求事件質量一，不要求每一原子成功。",
            "en": "Almost sure means event mass one, not success at every atom."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\omega\\in\\{a,b\\},\\quad\\varepsilon>0",
        "after_tex": "N=\\max(4,N_a(\\varepsilon))\\implies\\forall n\\ge N:\\ |X_n(\\omega)-X(\\omega)|<\\varepsilon",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "有限正質量原子可取settling indices最大值。",
          "en": "For finitely many positive-mass atoms take the maximum of their tail indices."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "N不需要控制null原子z。",
            "en": "N need not control null atom z."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "P(C)=1",
        "after_tex": "X_n\\to X\\quad\\text{almost surely, but not at every atom}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依定義得a.s.結論，同時保留z的點態失敗。",
          "en": "The definition gives almost-sure convergence while retaining pointwise failure at z."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "一般空間依機率不必推a.s.；本證明未使用那種反推。",
            "en": "On general spaces probability convergence need not imply a.s.; no such reverse inference was used."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "X_n幾乎必然收斂到X，但在z不收斂到X(z)。",
      "en": "X_n converges to X almost surely, but X_n(z) does not converge to X(z)."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "z失敗，所以不a.s.收斂。",
          "en": "Failure at z disproves a.s. convergence."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "z的質量為零。",
          "en": "z has zero mass."
        }
      },
      {
        "claim": {
          "zh-Hant": "a的前十項下降就足夠。",
          "en": "Ten decreasing terms at a suffice."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "須有所有n的尾端公式。",
          "en": "An all-index tail formula is required."
        }
      },
      {
        "claim": {
          "zh-Hant": "任何正質量失敗原子也可忽略。",
          "en": "A failing positive-mass atom may also be ignored."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "會使收斂事件機率小於1。",
          "en": "It would make the convergence event have mass below 1."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "本輪用完整有限原子機率空間與每個原子的 symbolic eventual certificate。所有正質量原子都須收斂；零質量原子失敗可忽略。有限 prefix 本身不能證明無限尾端收斂；一般機率空間中，單獨依機率收斂不保證 a.s. 收斂。這並非否認固定有限原子空間中可證得的額外等價關係。",
          "en": "This slice uses a complete finite atomic probability space with a symbolic eventual certificate for each atom. Every positive-mass atom must converge, while a zero-mass failure may be ignored. A finite prefix does not certify a full tail; the warning that convergence in probability alone is insufficient refers to the general convergence-mode implication, not a denial of additional finite-atom results."
        },
        "source": {
          "mko_id": "mko-almost-sure-convergence",
          "sha256": "a848e75d5e3643fc87b620fc61c2385c8ed080ba4eac3a8573d9cb5e575f0b70",
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        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "a.s.收斂定義為收斂原子集合的機率為1；有理尾端1/(n+1)趨零；有限原子事件機率可加。",
          "en": "Almost-sure convergence means the convergence set has probability 1; the rational tail 1/(n+1) tends to 0; finite atomic event probabilities add."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1/3+2/3",
        "rhs": "1"
      },
      {
        "id": "N02",
        "lhs": "1/(101+1)",
        "rhs": "1/100",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限原子空間；不否認此空間可有更強模式等價。",
      "en": "A finite atomic space; no denial of stronger finite-space equivalences."
    },
    "md_symbols": [
      [
        "C",
        "收斂事件",
        "convergence event"
      ],
      [
        "z",
        "零質量例外",
        "null exception"
      ]
    ],
    "md_error_controls": [
      {
        "title": "原子完整 / Complete atoms",
        "zh": "明列a、b、z各自證書與質量。",
        "en": "List certificates and masses for a,b,z."
      },
      {
        "title": "零例外 / Null exception",
        "zh": "只用P(z)=0，不把z改成成功。",
        "en": "Use P(z)=0 without pretending z converges."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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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": [
      "mko_definition",
      "work"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    "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."
  }
}