The shifted-prime Erd s-Wintner law for primitive-root determinant densities: extremal order, dimension zero, and Fourier decay

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For a prime p, let c(p)= φ(p-1) over p-1 product j at least1(1-p-j), the limiting density of matrices over 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, 12]. 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 infinity, one has (τ) at most(T)-1+o(1) outside a subset of [0,T] of relative measure O A((T)-A). Its concentration function satisfies aμ f([a,a+h]) about S 2e-γ/ (1/h), and every maximizing left endpoint lies near 3. We also prove p at most xc(p) (x)-1 and p to infinity(c(p) p)-1=e γ. The limiting law of (φ(p+1)/φ(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, (τ)=O((τ)-1/2). Finally, exact shell gaps of cyclotomic codifferents yield a uniform smoothing asymptotic at ε=2-cφ(m) for c>2 2(1+ sqrt6).