Prime polynomials in Legendre intervals over F2^k[t], IV: point counts on hexad modular curves, Brown invariants of Witt sheaves, and the constant X

Let q = 2^k with k odd. Part III (DOI 10.5281/zenodo.23290724) reduced the closed form X = q^6 + q^5 of the second constant of the degree-6 Legendre variance to two statements about the sextic family B_λ = λ5² u⁶ + λ3² u² + λ5 u + 1, whose roots form the hexad of ker L'_λ: the polynomial frequencies of its eleven factorisation types over F_q (Target Theorem 1.1), and the signs of the Gauss sums s_4, s_6 of explicit Z/4-valued quadratic forms on Fq^4, Fq^6 (Target Theorem 1.2). This paper proves three structural theorems about these Gauss sums. The hexad formula expresses s₂ₘ, a sum over F_{q²ᵐ}, as q^m times a signed sum of at most sixteen fourth roots of unity indexed by the Fq^m-rational points of the coset T'⁻¹(1) (the six roots and the ten synthematic totals). The trace parity theorem determines the absolute trace of every such point, Tr(w) ≡ mk + dim(K_λ ∩ Fq^m) (mod 2); for m = 1 it fixes the trace of every rational root of B_λ by Frobenius class. A sixteen-point lemma on the configuration of roots and totals then determines the signs in nine of the eleven classes: s_4 = +4q² throughout the classes 1^6 and 2^3; a half-half law in λ1 in the classes 2 1^4 and 2^2 1^2; s_4 ∈ {0, +4q²}, respectively {0, −4q²}, nonzero exactly on an explicit affine hyperplane, in the classes 4 1^2 and 4 2, hence W' = +2q^7 and −2q^7 as conjectured since Part II; and S'' = +4q^6 throughout the classes 3 1^3 and 3^2. The class 6 is settled under a single Witt identity verified for q ≤ 32; the class 3 2 1 remains open. The hyperplane parameter κ0 of the order-4 classes is given in closed form; contrary to what the data at q = 8 suggested, it is not constant on the class. For the factorisation counts, every moment Σ_λ N_m(B_λ) is expressed through Gauss sums of Gold-type forms on the trace-zero hyperplane of Fq^m; the identity is proved for all m, the third moment 2q² − 3q − 1 is verified for q ≤ 128, and an explicit nonsingular 11 × 11 system of pure and mixed moments, whose unique solution is the conjectured table of class counts, is written down. All numerical statements are reproducible with the self-contained Python code in the companion repository (https://github.com/Ruqing1963/legendre-intervals-part4), with the raw outputs for q = 8, 32, 128 included. 23 pages, 2 figures, 3 tables. Related records: Part I, DOI 10.5281/zenodo.23225896; Part II, DOI 10.5281/zenodo.23252762; Part III, DOI 10.5281/zenodo.23290724.