Simple and distinct zeros in a prime-modulus Dirichlet family at mesoscopic shrinking height

Fix $0<\alpha <1$. Let $q$ tend to infinity through odd primes, put $Q=\log q$, and set $I=(Q^{-\alpha },2Q^{-\alpha }]$. Averaging without weights over the $q-2$ nonprincipal characters modulo $q$, let ${\mathcal{N}}_q$ be the number of nontrivial zeros in $I$, counted with multiplicity; let ${\mathcal{N}}_{0,q}^s$ and ${\mathcal{N}}_{0,q}^{\ast }$ count, respectively, simple and distinct critical-line zero points; and let ${\mathcal{N}}_{d,q}$ count all distinct zero points, without a location restriction. We prove unconditionally that $${\mathcal{N}}_q=\frac{q(\log q{)}^{1-\alpha }}{2\pi }\{1+o_{\alpha }(1)\},$$ $$\underset{q\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{{\mathcal{N}}_{0,q}^s}{{\mathcal{N}}_q}\ge C_{\mathrm{M}\mathrm{T}},\underset{q\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{{\mathcal{N}}_{0,q}^{\ast }}{{\mathcal{N}}_q}\ge C_{\mathrm{M}\mathrm{T}},\underset{q\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{{\mathcal{N}}_{d,q}}{{\mathcal{N}}_q}\ge C_d,$$ where $$C_{\mathrm{M}\mathrm{T}}=\frac{3}{2}-\frac{1}{\sqrt{2}}\cot \left(\frac{1}{\sqrt{2}}\right)=0.672500703679\dots ,C_d=\frac{1+C_{\mathrm{M}\mathrm{T}}}{2}=0.836250351839\dots \mathrm{.}$$ The physical height tends to zero, whereas $I$ contains $\asymp (\log q{)}^{1-\alpha }$ local mean spacings. The proof combines small-height counting and zero-density deletion with a finite smooth Gabor compression of Weil’s Hermitian form. First and second matrix moments, together with an inertia-based rank–trace inequality, yield the three counting bounds. No form of the generalized Riemann hypothesis is assumed.