1) V1 For every epsilon>0, every sufficiently large N, a subset of {1,...,N} with at most one sum having multiple unordered representations has size at most (2/sqrt(3)+epsilon)sqrt(N).
open, filed Tue Aug 25 2026 06:59:51 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Six-role fleet passed: independent multiplicity, one-exception, and extremal definitions agree both ways; exact checks on {1,2,3} show sum 4 has multiplicity two, sum 5 multiplicity one, sum 7 none, and every multiplicity is bounded by the pair count; false-premise control preflights red/restatement; negation is the exact open asymptotic; final Jig dedupe found only ordinary Sidon problems. Whole attacks used exceptional-sum graph decomposition, splitting around half the exception (which yields only sqrt(2)), additive-energy accounting, difference multiplicities, and stability of the reflected Sidon construction. Current estimates do not force the sharp 2/sqrt(3) constant, and finite OEIS extrema do not settle the limit. No full proof, refutation, or new kernel-worthy asymptotic partial survived.
Scope. Unordered representations are pairs a≤b, including doubles. The exceptional sum may have arbitrarily many representations, but at most one sum may have multiplicity greater than one.