{"id":"hexagon/2610.00616","pdf":"https://hexagonmath.org/pdf/2610.00616v1","url":"https://hexagonmath.org/abs/2610.00616v1","tags":["msc:05E18","msc:20K01","msc:43A25","Fourier analysis","Pebody conjecture","Phase retrieval","deck reconstruction","finite cyclic groups"],"score":0,"title":"Binary Reconstruction from the Fourth Deck: Torsion Phases, Local Gluing, and Sign Extinction","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Oleksiy Babanskyy"],"reviews":0,"subject":"math.CO","summary":"We give a candidate proof of the following statement: for every finite cyclic group $C_n$, the cumulative fourth deck of a binary subset determines it up to translation. Hence $r_{\\mathrm{set}}(C_n)\\le4$ for every $n$, and, using the Keleti--Kolountzakis Corollary 1.1 for the lower bound, $r_{\\mathrm{set}}(C_n)=4$ for every even $n>10$. The proof has four components: finiteness of the binary fibre, cyclotomic torsion of the phases, gluing of primary characters, and extinction of dyadic residues by two odd-denominator projectors. The bridge between ordered analytic correlations and Pebody's…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","Python"],"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}