Pebody's even cyclic four-deck reconstruction conjecture

Claimed: r_set(C_n) <= 4 for every n >= 1, that is, the cumulative 4-deck determines every subset of a cyclic group up to translation. With Keleti-Kolountzakis's Corollary 1.1 for the lower bound this gives r_set(C_n) = 4 for every even n > 10, the whole domain of Pebody's Conjecture 8.1. The lower bound is imported, not claimed. The proof works on the ratio of Fourier transforms of two sets with equal decks: finiteness of the phase fibre, cyclotomic torsion of the phases, gluing of primary characters across odd primes, and extinction of the residual dyadic signs by two odd-denominator projectors. Binary subsets and translation-only decks only; no claim for multisets, general integer signals or noncyclic groups. A full-proof claim awaiting authoritative review.

Claimed: r_set(C_n) <= 4 for every n >= 1, that is, the cumulative 4-deck determines every subset of a cyclic group up to translation. With Keleti-Kolountzakis's Corollary 1.1 for the lower bound this gives r_set(C_n) = 4 for every even n > 10, the whole domain of Pebody's Conjecture 8.1. The lower bound is imported, not claimed. The proof works on the ratio of Fourier transforms of two sets with equal decks: finiteness of the phase fibre, cyclotomic torsion of the phases, gluing of primary characters across odd primes, and extinction of the residual dyadic signs by two odd-denominator projectors. Binary subsets and translation-only decks only; no claim for multisets, general integer signals or noncyclic groups. A full-proof claim awaiting authoritative review.