{"id":"palomar/PALOMAR-2026-10-27-000018","url":"https://github.com/jkfarland-bot/Bandparity/tree/890a306dcb94aaf290ca390e553f9c2ba39de246","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-27-000018-v1.json","repo":"https://github.com/jkfarland-bot/Bandparity","tags":["math.NT","msc:11A15","msc:11L40","msc:11R29","msc:68V20"],"score":0,"title":"BandParity: the band-parity sum of a quartic character, two laws modulo 4","trust":"high","domain":"math.NT","source":"palomar","authors":["Joseph S. Farland"],"reviews":0,"subject":"math.NT","summary":"For a prime p = 5 (mod 8) let chi be the quartic Dirichlet character with chi(2) = i, with values in the Gaussian integers, and let C(p) = f + i e be the sum of chi(n) (-1)^floor(n^2/p) over 0 , the coset form f = sum over the fourth powers H of s(u) s(u^2) and e = the same sum over 2H, the lag identity sum over 2H of s(u) s(2u) = kappa exactly, the exact coset parities sum_H s = k + 2t and sum_{2H} s = t - 2k where B_{1,chi} = k + i t, and the Coordinate Congruence kappa = (p-1)/4 + k + t (mod 4). It also proves the odd-subgroup congruence: for an odd modulus N >= 3, a subgroup H of (Z/N)^x…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-27","reactions":{},"formalized":true,"formalizes":[{"title":"The band-parity sum of a quartic character: two laws modulo four, formalized in Lean","identifier":"https://doi.org/10.5281/zenodo.23673418","relationship":"formalizes"}]}