$c(p)$: limiting fraction of $n\times n$ matrices over $\mathbb F_p$ with primitive-root determinant; $1/c(p)$: rejection-sampling loss in PQ VSS; $c(p)=\tfrac{\varphi(p-1)}{p-1}\prod_{j\ge1}(1-p^{-j})$. Over primes, $c(p)$ follows $\varphi(p-1)/(p-1)$: continuous on $[0,1/2]$, $X=\tfrac12\prod{\ell\ge3,\mathrm{prime}}(1-1/\ell)^{B\ell}$, $B\ell$ independent, $\Pr(B\ell{=}1)=1/(\ell{-}1)$. Hence $\inf_p c(p)=0$, $\min{p\le x}c(p)\asymp1/\log\log x$, $\limsup_p 1/(c(p)\log\log p)=e^γ$ on a primorial progression. $μ_G$ is singular with $\dim_Hμ_G=0$, sharpening Erdős (1939) for $\varphi(n)/n$. $1-G(\tfrac12-ε)\sim\mathfrak S_2e^{-γ}/\log(1/ε)$, $\mathfrak S_2$ twin-prime singular series (infinitude unneeded); $μ_f=\logμ_G$ is Rajchman, with an explicit unconditional Fourier-decay rate from an effective bound for Graham-Kolesnik exponent-pair constants. $\mathbb E[X^{-1}]\approx2.83$ vs. worst case $(1+o(1))e^γ\log\log p$; deterministic poly$(\log p)$-time, factoring-free two-sided certificate for $1/c(p)$ of gap $1+o(1)$; Las Vegas generator of NTT-friendly primes $q\equiv1\pmod{2N}$; $1/c(p)>2$ for all $p$, $1/c(p)\le(e^γ+o(1))\log y$ on $y$-friable shifts. Micciancio-Regev smoothing $ηε(Λ)$ has kissing floor $F=\sqrt{\ln(K/ε)/π}/λ_1(Λ^*)$ ($K$ dual kissing number): at $ε=2^{-cn}$ every lattice has $F\leηε\le\sqrt{π/(\min(c,1)\ln2)},F$; at fixed $ε$ the ratio $ηε/F$ can diverge as $\sqrt{\log n}$. For cyclotomic $\mathbb Q(ζ_m)$ ((Ring-)LWE), an exact three-case law gives dual shell gap $g_m\in{\sqrt{3/2},\sqrt2,\sqrt3}$, infimum $\sqrt{3/2}$ on ${m:ω{\mathrm{odd}}(m)\ge2}$, discharging gap hypothesis unconditionally; at $ε=2^{-c\varphi(m)}$, $c>2\log_2(1+\sqrt6)$ pins $η_ε(Λ_m)$ to the floor within $1+O(1/(c\varphi(m)))$.

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