math.CO — Combinatorics

Binary Reconstruction from the Fourth Deck: Torsion Phases, Local Gluing, and Sign Extinction

Contributed by Oleksiy Babanskyy

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 cumulative deck is proved explicitly, including stabilizers, zero and repeated offsets, and the empty pattern. The text contains the complete mathematical chain; finite checks are included only as convention and implementation checks, not as premises of the uniform proof.