[guided]We now rebuild the lower-bound construction from the beginning, because Assouad's lemma needs an explicit finite cube of alternatives inside $\mathcal F$. Choose a nonzero bump
\begin{align*}
\psi:[0,1]^d\to\mathbb R
\end{align*}
in $C_c^\infty((0,1)^d)$ such that
\begin{align*}
\int_{[0,1]^d}\psi(x)\,d\mathcal L^d(x)=0.
\end{align*}
For a resolution $h\in(0,1]$, choose $M$ coordinate-aligned cubes $Q_1,\dots,Q_M\subset[0,1]^d$ of side length comparable to $h$ and separated by gaps comparable to $h$. A regular grid inside a fixed interior subcube gives constants $c_M,C_M>0$, depending only on $d$, such that
\begin{align*}
c_Mh^{-d}\le M\le C_Mh^{-d}.
\end{align*}
Let $x_j\in Q_j$ be the coordinatewise minimal vertex of $Q_j$, and define
\begin{align*}
\psi_j:[0,1]^d\to\mathbb R
\end{align*}
by
\begin{align*}
\psi_j(x)=h^s\psi\left(\frac{x-x_j}{h}\right).
\end{align*}
For each sign vector $\theta=(\theta_1,\dots,\theta_M)\in\{-1,1\}^M$, define
\begin{align*}
f_\theta:[0,1]^d\to\mathbb R
\end{align*}
by
\begin{align*}
f_\theta(x)=1+a\sum_{j=1}^M\theta_j\psi_j(x),
\end{align*}
where $a>0$ is a small constant to be fixed.
We verify that these alternatives are densities in $\mathcal F$. The supports of the $\psi_j$ are disjoint, and the change of variables $u=(x-x_j)/h$ gives
\begin{align*}
\int_{[0,1]^d}\psi_j(x)\,d\mathcal L^d(x)=h^{s+d}\int_{[0,1]^d}\psi(u)\,d\mathcal L^d(u)=0.
\end{align*}
Therefore
\begin{align*}
\int_{[0,1]^d}f_\theta(x)\,d\mathcal L^d(x)=1.
\end{align*}
The separated supports ensure that cross-support Hölder quotients are bounded by the same scale as the local quotients. The Hölder scaling, with $m=\lfloor s\rfloor$ and the Hölder-Zygmund convention when $s$ is an integer, gives a constant $C_3=C_3(s,d,\psi)>0$ such that
\begin{align*}
\left\|\sum_{j=1}^M\theta_j\psi_j\right\|_{\mathcal H^s}\le C_3.
\end{align*}
The uniform bound gives a constant $C_4=C_4(\psi)>0$ such that
\begin{align*}
\left|a\sum_{j=1}^M\theta_j\psi_j(x)\right|\le aC_4h^s\le aC_4.
\end{align*}
Choose
\begin{align*}
a\le \min\left\{\frac{\rho}{C_3},\frac{1-b}{C_4},\frac{B-1}{C_4}\right\}.
\end{align*}
Then the perturbation $g_\theta=a\sum_j\theta_j\psi_j$ has $\|g_\theta\|_{\mathcal H^s([0,1]^d)}\le\rho$, so the interior-radius hypothesis gives $f_\theta=1+g_\theta\in\mathcal H^s(L;[0,1]^d)$. The strict inequalities $b<1<B$ make the remaining two bounds positive, so $b\le f_\theta\le B$, and $f_\theta$ integrates to $1$. Therefore $f_\theta\in\mathcal F$ for every $\theta\in\{-1,1\}^M$.
Now compare adjacent vertices. Let $\theta,\theta'\in\{-1,1\}^M$ differ only in coordinate $j$. Then $f_\theta-f_{\theta'}=2a\theta_j\psi_j$ on $Q_j$ and is zero off $Q_j$. Hence
\begin{align*}
\int_{[0,1]^d}|f_\theta(x)-f_{\theta'}(x)|^2\,d\mathcal L^d(x)=4a^2h^{2s+d}\int_{[0,1]^d}|\psi(u)|^2\,d\mathcal L^d(u).
\end{align*}
This quantity is the coordinatewise squared-loss separation
\begin{align*}
\delta^2=4a^2h^{2s+d}\int_{[0,1]^d}|\psi(u)|^2\,d\mathcal L^d(u).
\end{align*}
We also need adjacent statistical experiments to remain close. Let $\mathbb P_{\theta,n}$ denote the $n$-fold product law generated by $f_\theta$. For probability measures $P$ and $Q$ dominated by a measure $\mu$, define
\begin{align*}
H^2(P,Q)=\int \left(\sqrt{\frac{dP}{d\mu}}-\sqrt{\frac{dQ}{d\mu}}\right)^2\,d\mu.
\end{align*}
Since every $f_\theta$ is bounded below by $b$, the pointwise inequality
\begin{align*}
(\sqrt{u}-\sqrt{v})^2\le \frac{(u-v)^2}{b}
\end{align*}
for $u,v\ge b$ gives a one-sample Hellinger bound bounded by a constant multiple of $a^2h^{2s+d}$. [Tensorization of Hellinger affinity](/theorems/5908) for independent samples then gives a constant $C_5=C_5(b,\psi)>0$ such that
\begin{align*}
H^2(\mathbb P_{\theta,n},\mathbb P_{\theta',n})\le C_5na^2h^{2s+d}.
\end{align*}
Choose
\begin{align*}
h_n=n^{-1/(2s+d)}.
\end{align*}
After reducing $a$ if necessary, this gives
\begin{align*}
H^2(\mathbb P_{\theta,n},\mathbb P_{\theta',n})\le 2-\eta
\end{align*}
for some $\eta>0$ and every adjacent pair. Since the Hellinger affinity equals $1-H^2/2$ under this normalization, every adjacent affinity is at least $\eta/2$.
We apply the finite-hypercube form of Assouad's lemma: if the coordinatewise squared separation is at least $\delta^2$ and adjacent affinities are at least $\kappa>0$, then
\begin{align*}
\inf_{\tilde f_n}\sup_\theta\mathbb E_\theta\left[\int_{[0,1]^d}|\tilde f_n(x)-f_\theta(x)|^2\,d\mathcal L^d(x)\right]\ge \frac{\kappa}{4}M\delta^2.
\end{align*}
For completeness, this form follows by assigning to any estimator $\tilde f_n$ a sign estimate for each coordinate according to which of the two adjacent alternatives is closer on the block $Q_j$. On the event that the $j$th sign is misclassified, the disjoint-block $L^2$ loss is at least $\delta^2/4$. The two-point testing lower bound gives misclassification probability at least one half of the Hellinger affinity of the adjacent laws, hence at least $\kappa/2$. Summing the block losses over disjoint $Q_j$ gives the displayed constant after weakening constants.
Here $\kappa=\eta/2$, and the hypercube lies inside $\mathcal F$, so
\begin{align*}
R_n(\mathcal F)\ge c_1M h_n^{2s+d}
\end{align*}
for a constant $c_1=c_1(s,d,L,b,B,\psi)>0$. Finally $M\asymp h_n^{-d}$, so
\begin{align*}
R_n(\mathcal F)\ge c h_n^{2s}=c n^{-2s/(2s+d)}.
\end{align*}[/guided]
