For a prime $p$, let $c(p)=\frac{\varphi(p-1)}{p-1}\prod_{j\ge1}(1-p^{-j})$, 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 that this law has Hausdorff dimension zero and is carried by the Liouville numbers; its lower and upper dyadic $L^q$ dimensions vanish for $q>1$. Its logarithmic image $μ_f$ under $x\mapsto-\log x$ is Rajchman and satisfies $|\widehat{μ_f}(τ)|\ll(\log|τ|)^{-1/3}(\log\log|τ|)^{1/3}$ with effective constants and threshold. For $U_T$ uniform on $[0,T]$, $(\log T)|\widehat{μ_f}(U_T)|$ converges in law to an explicit random Euler product $W_F$, and $T^{-1}\int_0^T|\widehat{μ_f}(τ)|^s dτ\sim\mathbb E[W_F^s]/(\log T)^s$ for every fixed real $s>-1$. 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; every maximizing left endpoint lies within $h$ of $\log3$ for small $h$. Classical arguments give $\min_{p\le x}c(p)\sim e^{-γ}/\log\log x$; we obtain effective additive remainders and extrema in fixed reduced progressions. The laws of $σ(p-1)/(p-1)$ on $[3/2,\infty)$ and $\log(\varphi(p+1)/\varphi(p-1))$ on $\mathbb R$ also have Hausdorff dimension zero; the logarithmic divisor-sum law is Rajchman. Rational bounds for $1/c(p)$ without complete factorization of $p-1$ yield certified searches for fully splitting negacyclic number-theoretic transform primes at fixed power-of-two length. We determine the second distinct squared norm of tensor products of root lattices $A_n$ ($n\ge1$), yielding cyclotomic codifferent shell gaps and a uniform smoothing asymptotic at $ε=2^{-c\varphi(m)}$ for $c>2\log_2(1+\sqrt6)$. These results are unconditional.

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