The band-parity tower and its limit modulo eight, formalized in Lean

Let p be an odd prime with 2^a exactly dividing p - 1, and put m = (p - 1)/2^a. Let H_a be the subgroup of (Z/p)^x of order m, and let s(x) = (-1)^<x> be the parity of the least residue. The sums f_a = Σ_{u in H_a} s(u)s(u²) and e_a = Σ_{u in 2H_a} s(u)s(u²) are coordinates of the band-parity sum Σ_{0<n