Positive Dual Certificates for Real Separated Subset Sums: Chamber Optimality of the Conway–Guy and Bohman Sets, and Exact Values up to Eight

Let f_R(n) be the least possible largest element of a set of n positive reals whose 2^n subset sums are pairwise at least 1 apart, and let f_Z(n) ≥ f_R(n) be the corresponding integer quantity of the Erdős distinct subset sums problem.

Fixing the signs of all signed relations ε·x, ε ∈ {−1,0,1}^n, decomposes the real problem into polyhedral chambers, on each of which it is a linear program.

We prove that for every n the Conway–Guy set is the unique optimum of its chamber.

The proof is an explicit positive dual certificate consisting of n signed relations that are tight at the Conway–Guy set and unit lower triangular in gap coordinates; the relations are described by blocks of consecutive coordinates together with the positions n − k(k+1)/2, and tightness reduces to an identity of the Conway–Guy sequence proved by induction.

We then show, by a branch-and-bound over chambers in which every leaf carries an exact rational certificate, that f_R(n) = f_Z(n) for all n ≤ 8; in particular f_R(7) = 44 and f_R(8) = 84. We also prove chamber optimality for both of Bohman's families, which first improve on Conway–Guy at n = 12 (1159 < 1164): every Bohman set is the unique optimum of its chamber, with a block-triangular certificate given by explicit formulas. Finally we exhibit chambers whose optimal vertex is not an integer point.