{"id":"palomar/PALOMAR-2026-11-02-000005","url":"https://github.com/jkfarland-bot/Bandparitytower/tree/5eb71672dcda927c4b76a76bd955eb80dc724a50","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-02-000005-v1.json","repo":"https://github.com/jkfarland-bot/Bandparitytower","tags":["math.NT","msc:11A15","msc:11L40","msc:11R29","msc:68V20"],"score":0,"title":"BandParityTower: the band-parity tower and its limit modulo eight","trust":"high","domain":"math.NT","source":"palomar","authors":["Joseph S. Farland"],"reviews":0,"subject":"math.NT","summary":"For a prime p let a = v_2(p-1), m = (p-1)/2^a, H_a the subgroup of (Z/p)^x of order m and s(x) = (-1)^ . The development proves the tower of band-parity laws of Paper I of the author's series: the cyclotomic tower (every coordinate of the band-parity sum of the character of order 2^a is congruent modulo 4 to an explicit count), the band-parity tower f_a = sum over H_a of s(u) s(u^2) = m (mod 4) at every odd prime, the doubling count sum over 2H_a of s(u) s(2u) = 2 N_a - m, the imaginary law e_a = 2 N_a - m (mod 4) at every prime p = 1 (mod 4) with its level-3 form, the collapse e_a = +-f_a…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-02","reactions":{},"formalized":true,"formalizes":[{"title":"The band-parity tower and its limit modulo eight, formalized in Lean","identifier":"https://doi.org/10.5281/zenodo.23752472","relationship":"formalizes"},{"title":"The band-parity companion of a quartic L-value: a congruence law, its tower, and its limit modulo eight","identifier":"","relationship":"formalizes"}]}