{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-central-limit-theorem-finite-variance-iid-01",
  "mko_id": "mko-central-limit-theorem-finite-variance-iid",
  "title": {
    "zh-Hant": "有限variance IID的CLT資格與格點audit",
    "en": "CLT qualification and a finite lattice audit"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-central-limit-theorem-finite-variance-iid-01",
    "source": {
      "mko_id": "mko-central-limit-theorem-finite-variance-iid",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "c37700292553db11f9702a86ba165e9287d7830d81ee8351eff28de1c07ea9fe",
      "english_sha256": "951e71ab789a556b03eeb44c3472b5e89e3ac252d23992bc8abf538f39a494cb"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "清楚區分finite-n moments與named CLT應用。",
      "en": "Separate finite-n moments from the named CLT application."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "在一個已給定機率空間上，(X_i)_{i≥1}為無限IID序列，對任意有限不同indices的joint law都因子分解為Bernoulli(1/2)的乘積。此為全序列前提，不從有限資料估出。",
          "en": "On a supplied probability space, (X_i)_{i≥1} is an infinite IID sequence: every finite distinct-index joint law factors into Bernoulli(1/2) marginals. This is a whole-process premise, not inferred from finite data."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "X_i\\sim\\operatorname{Bernoulli}(1/2)",
        "after_tex": "\\mu=1/2,\\quad\\sigma^2=1/4\\in(0,\\infty)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "重算完整邊際law矩，核非退化。",
          "en": "Recompute the complete marginal moments and check nondegeneracy."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "零variance會使標準化失效，不能省正性。",
            "en": "Zero variance invalidates normalization, so positivity cannot be omitted."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "S_n=\\sum_{i=1}^nX_i",
        "after_tex": "E[S_n]=n/2,\\quad\\operatorname{Var}(S_n)=n/4",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "期望線性與獨立variance加法給和的矩。",
          "en": "Linearity and independent variance addition give moments of the sum."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "variance加法依已宣告的全序列獨立。",
            "en": "Variance addition uses the declared whole-sequence independence."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "Z_n=(S_n-n\\mu)/(\\sigma\\sqrt n)",
        "after_tex": "Z_n=\\frac{S_n-n/2}{\\sqrt n/2}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "σ=1/2，故分母是sqrt(n)/2。",
          "en": "Sigma=1/2, so the denominator is sqrt(n)/2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不是n/2；中心也必須扣n/2。",
            "en": "It is not n/2, and centering must subtract n/2."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "E[Z_n],\\quad\\operatorname{Var}(Z_n)",
        "after_tex": "E[Z_n]=0,\\quad\\operatorname{Var}(Z_n)=\\frac{n/4}{n/4}=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "標準化矩精確成立於每n≥1。",
          "en": "The standardized moments hold exactly for every n≥1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "矩為0、1不會單獨識別normal law。",
            "en": "Moments0 and 1 alone do not identify a normal law."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "n=4,\\quad k\\in\\{0,1,2,3,4\\}",
        "after_tex": "P(S_4=k)=\\binom4k/16=(1,4,6,4,1)_k/16",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整列五個格點機率，不省tail點。",
          "en": "List all five lattice probabilities without omitting tail points."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "係數總和16來自四次Bernoulli乘積。",
            "en": "The coefficients sum to 16 from the four-fold Bernoulli product."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "Z_4=S_4-2",
        "after_tex": "Z_4\\in\\{-2,-1,0,1,2\\},\\quad P(Z_4=0)=3/8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "n=4的標準化仍是離散law，有正單點質量。",
          "en": "At n=4 the standardized law is still discrete with positive singleton mass."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "standard normal無原子，所以不可能exact normal。",
            "en": "The standard normal is atomless, ruling out exact equality of laws."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\text{infinite IID},\\ E[X_i]=\\mu,\\ 0<\\sigma^2<\\infty",
        "after_tex": "\\forall z\\in\\mathbb R:\\lim_{n\\to\\infty}P(Z_n\\le z)=\\Phi(z)\\quad\\text{(named CLT)}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "假設named CLT並逐一對上IID與正有限variance；Φ處處連續。",
          "en": "Invoke the assumed named CLT after matching IID and positive finite variance; Phi is continuous everywhere."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這個極限來自named theorem，不來自五格點表。",
            "en": "This limit comes from the named theorem, not the five-point table."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "P(Z_4=0)=3/8,\\quad P(Z=0)=0\\ (Z\\sim N(0,1))",
        "after_tex": "Z_n\\Rightarrow N(0,1),\\quad Z_4\\not\\sim N(0,1)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "同時保留極限結論與finite-n不相等邊界。",
          "en": "Retain both the limit conclusion and the finite-n nonidentity boundary."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "未給normal approximation誤差率，也未形式化CLT證明。",
            "en": "No normal-approximation error rate or formal CLT proof is supplied."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "named CLT條件成立，故Z_n⇒N(0,1)；n=4的格點audit不證極限且不exact normal。",
      "en": "The named CLT applies, so Z_n⇒N(0,1); the n=4 lattice neither proves the limit nor equals a normal law."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "n=4的五點law就是normal。",
          "en": "The five-point n=4 law is normal."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "它在0有3/8原子，而normal單點質量零。",
          "en": "It has mass 3/8 at 0; a normal law has zero singleton mass."
        }
      },
      {
        "claim": {
          "zh-Hant": "分母用nσ也給variance1。",
          "en": "Dividing by n sigma also gives variance 1."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "會得1/n而不是1。",
          "en": "It gives variance 1/n instead of 1."
        }
      },
      {
        "claim": {
          "zh-Hant": "格點表驗過即證CLT。",
          "en": "Checking the lattice table proves the CLT."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "finite-n資料不給無限CDF極限；此處依named theorem。",
          "en": "Finite-n data does not establish the infinite CDF limit; that uses the named theorem."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "明示無限IID process certificate、共同finite law、mu與0<sigma^2<infinity。令S_n=sum_{i=1}^n X_i，標準化Z_n=(S_n-n*mu)/(sigma*sqrt(n))，則對標準常態CDF的連續點z有P(Z_n≤z)→Phi(z)。exact families完整重算finite law moments、Z_n的mean 0／variance 1與Bernoulli sum lattice；另有完整hypotheses-to-named-theorem symbolic application。有限binomial table不是normal approximation或convergence evidence；non-IID只表示本定理不可套用，zero variance使標準化失效，wrong scaling與finite histogram均不得冒充CLT。",
          "en": "Assume an infinite IID process with common mean mu and 0<sigma^2<infinity. For S_n=sum_(i=1)^n X_i define Z_n=(S_n−n*mu)/(sigma*sqrt(n)). Then P(Z_n≤z) tends to Phi(z) at every continuity point of the standard normal CDF. The exact families use a finite common marginal law, recalculate its moments, the fixed-n standardized moments0 and 1, and a complete Bernoulli sum lattice. A symbolic branch conditionally invokes the named theorem under declared hypotheses. The infinite_iid token declares a premise; it does not construct or prove a whole process. A finite binomial table or histogram is not a normal approximation or convergence proof. Non-IID excludes this particular theorem; zero variance and n*sigma scaling do not qualify this normalization."
        },
        "source": {
          "mko_id": "mko-central-limit-theorem-finite-variance-iid",
          "sha256": "c37700292553db11f9702a86ba165e9287d7830d81ee8351eff28de1c07ea9fe",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "明示named theorem前提：無限IID且0<σ²<∞時，中心化和除σ√n依分布收斂到standard normal；另假設獨立variance加法與binomial law。",
          "en": "Explicit named theorem prerequisite: infinite IID with 0<sigma²<infinity gives convergence of centered sums divided by sigma sqrt(n) to standard normal; also assume independent variance addition and the binomial law."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1/2-(1/2)**2",
        "rhs": "1/4"
      },
      {
        "id": "N02",
        "lhs": "(1+4+6+4+1)/16",
        "rhs": "1"
      },
      {
        "id": "N03",
        "lhs": "6/16",
        "rhs": "3/8"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "CLT為明示named theorem前提；不宣稱此處完整證明。",
      "en": "The CLT is an explicit named theorem prerequisite, not proved here."
    },
    "md_symbols": [
      [
        "S_n",
        "部分和",
        "partial sum"
      ],
      [
        "Z_n",
        "正確標準化",
        "correct standardization"
      ],
      [
        "Φ",
        "standard normal CDF",
        "standard-normal CDF"
      ]
    ],
    "md_error_controls": [
      {
        "title": "前提層次 / Premise layer",
        "zh": "無限IID明列為前提，不把token當構造證明。",
        "en": "Declare infinite IID as a premise, not a process-construction proof."
      },
      {
        "title": "非退化與scale / Nondegeneracy and scale",
        "zh": "σ²=1/4>0且分母σ√n；finite-n表保留離散性。",
        "en": "Use sigma²=1/4>0 and denominator sigma sqrt(n); retain finite-n discreteness."
      }
    ],
    "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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      "work"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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      "BR142-S237",
      "BR142-S238",
      "BR142-S232",
      "BR142-S233",
      "BR142-S234",
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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."
  }
}