Prime polynomials in Legendre intervals over F_{2^k}[t], II: exact variances via quadratic towers and supersingular Artin–Schreier curves

Let q = 2^k and let I_f = {f² + s : deg s ≤ d} be the Legendre interval of a monic polynomial f of degree d over F_q[t]. Open Problem 8.2 of the companion paper (DOI 10.5281/zenodo.23219609) asks why the variance of the prime-power count Ψ(f) = Σ_{P ∈ I_f} Λ(P) exceeds the Keating–Rudnick prediction by factors as large as q²/3. This paper resolves the case d = 4 completely and unconditionally. Using the quadratic tower F_q ⊂ F_{q²} ⊂ F_{q⁴} ⊂ F_{q⁸} we obtain a closed formula for Ψ on every class of the Hayes group G₃, from which Var_{G²}(Ψ)/q⁵ = q(q−1), Var_G(Ψ)/q⁵ = 3 − 3/q, the full structure of the quadratic-twist correlations ρ(ε), and an elementary proof of Carlitz's cubic evaluation over F_{2^{8k}} follow. For d = 6 we prove that the Legendre profile takes exactly three values, governed by [c₂ = 0] and Tr(c₄/c₂²), which reduces the variance to three exponential sums A₆, X, Y on U = {α ∈ F_{q¹²} : Tr α = Tr α³ = Tr α⁵ = 0}. We show that A₆ − q⁹ and X are sums of Frobenius traces of the supersingular Artin–Schreier curves y² + y = λ₅x⁵ + λ₃x³ + λ₁x, equivalently of F₂-quadratic forms on F_{q¹²} whose radicals are kernels of explicit linearised polynomials; the conjectured closed forms A₆ = q⁹ + (−1)^{k+1}(q−1)q⁵ and X = (−1)^{k+1}q⁶ + q⁵ are verified exactly for q ≤ 128 by an algorithm independent of the explicit formula, and the pure cubic part is proved for all k. The third constant Y = q³m_k is a Frobenius trace on the nine-dimensional variety U that is not a character sum; its weight and rank are discussed. Consequently the Legendre variance at d = 6 satisfies Var_{G²}(Ψ)/q⁷ ≍ q², not q⁴. The deposit contains the paper (LaTeX source and PDF, 14 pages), the two figures, the Python verification scripts, and the exact numerical data (text and CSV) quoted in the paper. Code and data are also at https://github.com/Ruqing1963/legendre-intervals-part2 (the paper-I repository is included as a submodule). A status table in the last section states precisely which results are proved, which are verified by exact computation, and which are conjectural.