to zero as $p\to\infty$ from Lemma \[lem:1\], since $f$ is semi-admissible and $$\beta\left\| \psi({\mathbf{y}})\right\| \geq\beta \sqrt{p } = |C|^{1/4m} p^{-1/2} = p^{k/m-1/2} \to \infty  \mbox{ as } p \to \infty.
    \label{eq:boundPHurwitz}$$ From Lemma \[lem:balancedRings\] we conclude that the second term is upper bounded by the right-hand side of as $p \to \infty$.

For the maximal Hurwitz order $$\mathcal{H} = \left\{a+bi+cj+d(-1+i+j+ij)/2 : a, b, c, d \in \mathbb{Z}\right\},$$ the theorem follows by considering reductions from left-prime ideals $\mathcal{P} \triangleleft \mathcal{H}$. For any rational prime $p$, there exist isomorphisms $\mathcal{H}/p\mathcal{H} \sim \mathbb{F}_p(i,j,k) \sim \mathcal{M}_2(\mathbb{F}_p)$ (e.g. Wedderburn’s Theorem [@Voight Thm. 6.16 Lem 9.2.1]), where non-invertible elements in $\mathcal{H}/p\mathcal{H}$ have reduced norm (determinant) proportional to $p$. Notice that in this case we obtain a reduction $$\phi_p : D_4 \to \mathcal{M}_2(\mathbb{F}_p),$$ where $D_4$ is the checkerboard lattice in dimension four [@Splag Sec. 7.2]. An explicit realization of ring isomorphism is obtained by setting $$\phi(1) = \left(\begin{array}{cc}1 & 0\\0 & 1\end{array}\right), \phi(i) = \left(\begin{array}{cc}0 & -1\\1 & 0\end{array}\right) \mbox{ and } \phi(j) = \left(\begin{array}{cc}a & b\\b & -a\end{array}\right),$$ where $a$ and $b$ are two integers such that $a^2+b^2 \equiv -1 \text{ (mod } p)$. Notice that such an isomorphism preserves the residue class of the reduced 