The second layer of Wolstenholme determinants: derived linear symbols modulo p², the w-law, and the mirrored-adjugate structure in higher corank

We derive, rather than fit, the second layer of the Wolstenholme determinants of Notes 1–4: the mod-p² refinement Λ₂ of the leading-form congruence D_g(p−a)/p2κ₀ ≡ ρ*⁻¹ det(window). Conditionally on Z.-H. Sun's congruences for Bernoulli numbers (Discrete Appl. Math. 105 (2000), including its Remark 5.1, which we verify to modulus p⁴), we mechanize a symbolic engine that reproves amplification, null pairing and the symbol formula of Note 3, discovers a second null pairing (all third digits and all higher products cancel identically), and derives the second layer in closed linear form in the coordinates q_μ = Bp−1−2μ/(2μ+1) and the Kummer defects w_μ = (B2p−2−2μ − ((2μ+2)/(2μ+1)) Bp−1−2μ)/p. For corank κ₀ = 1 the result is a certified schema: the w-coefficients obey the w-law coef(w_μ) = −ρ*⁻¹ c_μ/(2μ+2) — the second-floor symbol is the first-floor symbol on the next Kummer rung — and the q-coefficients decompose with an explicit rational part and a measured residue σ_μ, tabulated exactly for five families and confirmed on a golden-test family never used in the fit (41 fresh primes, zero discrepancies). For κ₀ ≥ 2 we verify a mirrored-adjugate structure with verbatim coefficient transfer between families and the same w-law in every shift. The lattice-reduction (LLL) methodology that anchored the derivation, including a factor-50 negative control and a detection-ceiling lesson, is documented. All computations are exact; every labelled statement is reproducible from the attached scripts.