The shifted-prime Erdős-Wintner law for primitive-root determinant densities: extremal order, dimension zero, and Fourier decay
For a prime $p$, let $c(p)=\frac{φ(p-1)}{p-1}\prod_{j\ge1}(1-p^{-j})$, the limiting density of matrices over $\mathbb F_p$ with primitive-root determinant. Its limiting law over the primes is the classical continuous shifted-totient law on $[0,1/2]$. We prove Hausdorff dimension zero and vanishing lower and upper dyadic $L^q$ dimensions for $q>1$. Its image $μ_f$ under $x\mapsto-\log x$ is Rajchman. As $T\to\infty$, for $U_T$ uniform on $[0,T]$, $\log|\widehat{μ_f}(U_T)|/\log\log T\to-1$ in probability. For every $A>0$, $|\widehat{μ_f}(τ)|\le(\log\log T)^4/\log T$ outside a subset of $[0,T]$ of relative measure $O_A((\log T)^{-A})$. As $h\downarrow0$, $\sup_aμ_f([a,a+h])=\mathfrak S_2e^{-γ}/\log(1/h)+O(\log^{-2}(1/h))$, where $\mathfrak S_2$ is the twin-prime singular series; maximizing left endpoints lie within $h$ of $\log3$ for small $h$. We prove $\min_{p\le x}c(p)\sim e^{-γ}/\log\log x$ and $\limsup_{p\to\infty}(c(p)\log\log p)^{-1}=e^γ$. The limiting law of $\log(φ(p+1)/φ(p-1))$ has support $\mathbb R$ and Hausdorff dimension zero. For the classical law of $σ(p-1)/(p-1)$ on $[3/2,\infty)$, we prove dimension zero, a sharp left-endpoint asymptotic, and a Rajchman logarithmic image. Its odd-prime component has an entire Mellin transform of order one. Partial-factorization bounds yield certified asymptotic searches for fully splitting negacyclic number-theoretic transform primes with prescribed reciprocal-density bounds at fixed power-of-two length. We determine the second distinct squared norm of $A_{n_1}\otimes\cdots\otimes A_{n_k}$ for $k,n_i\ge2$, yielding exact cyclotomic codifferent shell gaps and a uniform smoothing asymptotic at $ε=2^{-cφ(m)}$ for $c>2\log_2(1+\sqrt6)$. These results are unconditional. An explicit unproved exponent-pair hypothesis yields $|\widehat{μ_f}(τ)|=O(1/\log\log|τ|)$.