# CLT qualification and a finite lattice audit

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This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-central-limit-theorem-finite-variance-iid-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

Separate finite-n moments from the named CLT application.

The CLT is an explicit named theorem prerequisite, not proved here.





## Givens and symbols


- work


- G1 — 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.


- S_n — partial sum
- Z_n — correct standardization
- Φ — standard-normal CDF





## Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.


- mko_definition · cited — 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. · mko-central-limit-theorem-finite-variance-iid
- work · prerequisite — 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.





## Main derivation

S01 · impliesBefore

X_i\sim\operatorname{Bernoulli}(1/2)

After

\mu=1/2,\quad\sigma^2=1/4\in(0,\infty)

Conditions: none

Rule: work · Uses: G1, work

Leaf basis: work

Recompute the complete marginal moments and check nondegeneracy.



S02 · impliesBefore

S_n=\sum_{i=1}^nX_i

After

E[S_n]=n/2,\quad\operatorname{Var}(S_n)=n/4

Conditions: none

Rule: work · Uses: G1, S01, work

Leaf basis: work

Linearity and independent variance addition give moments of the sum.



S03 · impliesBefore

Z_n=(S_n-n\mu)/(\sigma\sqrt n)

After

Z_n=\frac{S_n-n/2}{\sqrt n/2}

Conditions: none

Rule: work · Uses: G1, S02, work

Leaf basis: work

Sigma=1/2, so the denominator is sqrt(n)/2.



S04 · impliesBefore

E[Z_n],\quad\operatorname{Var}(Z_n)

After

E[Z_n]=0,\quad\operatorname{Var}(Z_n)=\frac{n/4}{n/4}=1

Conditions: none

Rule: work · Uses: G1, S03, work

Leaf basis: work

The standardized moments hold exactly for every n≥1.



S05 · impliesBefore

n=4,\quad k\in\{0,1,2,3,4\}

After

P(S_4=k)=\binom4k/16=(1,4,6,4,1)_k/16

Conditions: none

Rule: work · Uses: G1, S04, work

Leaf basis: work

List all five lattice probabilities without omitting tail points.



S06 · impliesBefore

Z_4=S_4-2

After

Z_4\in\{-2,-1,0,1,2\},\quad P(Z_4=0)=3/8

Conditions: none

Rule: work · Uses: G1, S05, work

Leaf basis: work

At n=4 the standardized law is still discrete with positive singleton mass.



S07 · impliesBefore

\text{infinite IID},\ E[X_i]=\mu,\ 0<\sigma^2<\infty

After

\forall z\in\mathbb R:\lim_{n\to\infty}P(Z_n\le z)=\Phi(z)\quad\text{(named CLT)}

Conditions: none

Rule: work · Uses: G1, S06, work

Leaf basis: work

Invoke the assumed named CLT after matching IID and positive finite variance; Phi is continuous everywhere.



S08 · impliesBefore

P(Z_4=0)=3/8,\quad P(Z=0)=0\ (Z\sim N(0,1))

After

Z_n\Rightarrow N(0,1),\quad Z_4\not\sim N(0,1)

Conditions: none

Rule: work · Uses: G1, S07, work

Leaf basis: work

Retain both the limit conclusion and the finite-n nonidentity boundary.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · G1, S07, work — No normal-approximation error rate or formal CLT proof is supplied.
- S07 · premise_audit · G1, S06, work — This limit comes from the named theorem, not the five-point table.
- S06 · premise_audit · G1, S05, work — The standard normal is atomless, ruling out exact equality of laws.
- S05 · premise_audit · G1, S04, work — The coefficients sum to 16 from the four-fold Bernoulli product.
- S04 · premise_audit · G1, S03, work — Moments0 and 1 alone do not identify a normal law.
- S03 · premise_audit · G1, S02, work — It is not n/2, and centering must subtract n/2.
- S02 · premise_audit · G1, S01, work — Variance addition uses the declared whole-sequence independence.
- S01 · premise_audit · G1, work — Zero variance invalidates normalization, so positivity cannot be omitted.





## Error controls and negative controls

Premise layerDeclare infinite IID as a premise, not a process-construction proof.

Nondegeneracy and scaleUse sigma²=1/4>0 and denominator sigma sqrt(n); retain finite-n discreteness.


- The five-point n=4 law is normal. → reject — It has mass 3/8 at 0; a normal law has zero singleton mass.
- Dividing by n sigma also gives variance 1. → reject — It gives variance 1/n instead of 1.
- Checking the lattice table proves the CLT. → reject — Finite-n data does not establish the infinite CDF limit; that uses the named theorem.





## Result and limitations

The named CLT applies, so Z_n⇒N(0,1); the n=4 lattice neither proves the limit nor equals a normal law.







## Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 3b87490489b4be576acff8199037a9cb5b933e355624ff89027db6b289b31c18
- original source SHA-256: 45953c182abd08cc1f8022fde9a4c32399f9ee79d64b005a7d3c12e73d5f5add
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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