Universal Toeplitz symbols for the leading forms of Wolstenholme determinants

Let p be a prime, 2 ≤ a ≤ g, and let D_g(p−a) = det(E2i−j){g×g}, where E_k = e_k(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 v_p(D_g) ≥ 2κ₀ with κ₀ = ⌈(g+1−a)/2⌉, and computed the leading forms L{g,a} with D_g/p2κ₀ ≡ Lg,a (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 = Bp−1−2μ/(2μ+1) and P(y) = ∏{l=1}ᵃ⁻¹(y − l⁻²) = Σ_i P_i y^i, the second-order Schur layer K of D_g 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)! Σ_i P_i (j+i) qⱼ₊ᵢ (mod p) (the symbol formula), so that D_g(p−a)/p2κ₀ ≡ ρ⁻¹ det(c⁽ᵃ⁾t−m) (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.