For a prime $p$, let $c(p)=\frac{\varphi(p-1)}{p-1}\prod_{j\geq1}(1-p^{-j})$, the limiting density of matrices over $\mathbb F_p$ whose determinant is a primitive root. We determine its limiting law over the primes: a continuous prime-indexed Bernoulli product supported on $[0,\tfrac12]$. The limiting measure has Hausdorff dimension zero and vanishing lower and upper dyadic $L^q$ dimensions for every $q>1$. Its logarithmic push-forward $μ_f$ is nevertheless Rajchman, unconditionally. For every $A>0$, as $T\to\infty$, one has $|\widehat{μ_f}(τ)|\leq(\log T)^{-1+o(1)}$ outside a subset of $[0,T]$ of relative measure $O_A((\log\log T)^{-A})$. Its concentration function satisfies $\sup_aμ_f([a,a+h])\sim\mathfrak S_2e^{-γ}/\log(1/h)$, and every maximizing left endpoint lies near $\log 3$. We also prove $\min_{p\leq x}c(p)\asymp(\log\log x)^{-1}$ and $\limsup_{p\to\infty}(c(p)\log\log p)^{-1}=e^γ$. The limiting law of $\log(\varphi(p+1)/\varphi(p-1))$ has full support, is purely singular, and has Hausdorff dimension zero. We further prove a shifted-prime analogue of Gronwall's theorem and derive explicit bounds for $1/c(p)$ without complete factorization of $p-1$, yielding a certified asymptotic search for fully splitting NTT primes. Under an explicit unproved hypothesis on exponent-pair constants, $|\widehat{μ_f}(τ)|=O((\log\log|τ|)^{-1/2})$. Finally, exact shell gaps of cyclotomic codifferents yield a uniform smoothing asymptotic at $ε=2^{-c\varphi(m)}$ for $c>2\log_2(1+\sqrt6)$.

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