Prime polynomials in Legendre intervals over F2^k[t], III: the hexad of a quartic linearised polynomial, the symplectic group Sp_4(F_2), and the Witt sums behind the constant X

Third part of a series on the variance of prime polynomials in Legendre intervals {f^2 + s : deg s ≤ d} over F_q[t], q = 2^k (Open Problem 8.2 of Part I, DOI 10.5281/zenodo.23225896; Part II, DOI 10.5281/zenodo.23252762). Part II reduced the degree-6 constant X to sums W'(λ_3, λ_5) of order-4 characters of the Hayes group, governed by the kernel of the linearised polynomial L' = λ_5^4 x^16 + λ_3^4 x^8 + x^4 + λ_3^2 x^2 + λ_5 x, on which the Frobenius was observed to act through S_6 ≅ Sp_4(F_2) with exact polynomial class counts. This paper explains that structure. The kernel of L' is the set of pairwise sums of the six roots of the sextic B = λ_5^2 u^6 + λ_3^2 u^2 + λ_5 u + 1 = u·A(u) + 1, where A is the Bluher quintic of Part II; the proof is the identity L' = x^4 B^2 + B + 1. Hence the Frobenius permutes the six roots, its S_6-class is the factorisation type of B over F_q, the symplectic form is the duad–syntheme pairing of the generalised quadrangle W(2), and the one-parameter family L + εx^4 corresponds to uA(u) + √ε, interpolating between the quadratic sums of A_6 and the order-4 sums of X. The number of Frobenius-stable symplectic structures of each type is 28|t|/|C|. First moments of the family u^6 + a u^2 + b u + b^2 are proved; the eleven class counts are verified for q ≤ 512 (odd k) and new polynomials, depending on k mod 4, are given for even k (q ≤ 1024). The order-4 sums are treated as Frobenius traces of rank-4 Witt–Artin–Schreier sheaves. Using the hexad, a multiplicity formula for Witt traces along towers and the moduli of Gauss sums of Z/4-valued quadratic forms, it is proved that every individual sum S''(λ) is a rational integer, that the normalised Frobenius eigenvalues are roots of unity, that in the class of 5-cycles the L-polynomial is (1 − η^5 q5/2 T^5)/(1 − η q1/2 T) with η a primitive 8th root of unity, so that W' = −q^7 unconditionally (+q^7 for even k), that for the six classes whose kernel lies in Fq^4 the sum over Fq^12 equals q^4 times the sum over Fq^4, and that the remaining classes have all S'' = ±4q^6. The closed form X = q^6 + q^5 (odd k) is thereby reduced conditionally to the class counts and to the Brown invariants of explicit small-field Z/4-valued forms, both stated precisely and verified exactly. Contents: paper (LaTeX source and PDF, 30 pages, four figures), self-contained Python verification code (hexad search, sextic identity, class counts, Witt sums and L-polynomials, trace identities, stable symplectic structures), exact data for q ≤ 1024, and a results summary. Code: MIT; paper, data, figures: CC BY 4.0. Repository: https://github.com/Ruqing1963/legendre-intervals-part3