Let p be a prime, 2 ≤ a ≤ g, and let Dg (p−a) = det (E₂₈−₉) g×g, where Eₖ = eₖ (1, 1/2, …, 1/ (p−a) ), be the Wolstenholme determinants of Notes 1–2 of this program. Those notes established, for p ≥ 2g+3, the unconditional amplification vₚ (Dg) ≥ 2κ₀ with κ₀ = ⌈ (g+1−a) /2⌉, and computed the leading forms Lg, a with Dg/p^2κ₀ ≡ L₆, ₀ (mod p) for the 14 families g ≤ 6 as explicit polynomials in Bernoulli numbers, together with 22 "sporadic" primes where the valuation jumps. Here we prove — conditionally only on the published inputs of Notes 1–2 and on Glaisher's classical congruence, with p ≥ 2g+3 throughout — that this entire phenomenology is governed by a single closed formula. Writing q_μ = B−₁−₂⏛/ (2μ+1) and P (y) = ∏l=1^a−1 (y − l^−2) = Σᵢ Pᵢ yⁱ, the second-order Schur layer K of Dg on the canonical kernels is a Toeplitz window, universal in g (the universal window theorem), of the explicit symbol c^ (a) −j = (−1) ᵃ (a−1) ! Σᵢ Pᵢ (j+i) q₉+₈ (mod p) (the symbol formula), so that Dg (p−a) /p^2κ₀ ≡ ρ^−1 det (c^ (a) ₓ−₌) (mod p), with the normalization ρ = ρ* (g, a) given by a conjectural closed formula, verified for all 65 families g ≤ 12. The formula predicted, before measurement, the previously unknown leading-form tables for 7 ≤ g ≤ 10 and 40 new sporadic primes (bringing the total to 62, all with excess exactly +1), including two below the historical p ≥ 17 scanning floor. We give a verified criterion for the exact excess, and a rigid probabilistic model whose local density is exactly 1/p by classical counts of singular Toeplitz/Hankel matrices (Daykin; Kaltofen–Lobo; Anzis et al. ), with intra-prime correlations computed exactly. All statements labelled "verified" are reproducible from the attached exact-arithmetic scripts.
José Cláudio da Silva (Sun,) studied this question.