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