The band-parity sum of a quartic character: two laws modulo four, formalized in Lean
Year: 2026
Type: preprint
Abstract: For a prime p ≡ 5 (mod 8), let χ be the quartic character with χ(2) = i, and weight its half-range sum by the parity of ⌊n²/p⌋. The result is a Gaussian integer 𝒞(p) = f + ie. We prove f ≡ (p−1)/4 and e ≡ κ (mod 4), where κ = Σ_{0<n<p/4} (n/p), which is h(−4p)/2 by Dirichlet’s formula. The mechanism is a matching principle: modulo 4, the correlation of two families of signs does not depend on how they are matched. For the imaginary coordinate, matching by multiplication by 2 has correlation exactly κ. The same principle gives a congruence for every coset of every odd-order subgroup of the units modulo any odd integer. For p > 5, together with the coset parities of the quartic Bernoulli number B_{1,χ} = k + it, it gives κ ≡ (p−1)/4 + k + t (mod 4). Every numbered result of Sections 3 to 7 is formally verified in Lean 4 with Mathlib. The Lean development is archived at doi:10.5281/zenodo.23673068.
Author: Farland, Joseph S.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223653451