A conservation law for the first moment in Graham's rearrangement problem

Let G be a finite abelian group of order n and A ⊆ G \0\ with $|A| = t$. For g ∈ G let Ug(A) be the expected number of pairs of partial sums of a random ordering of A differing by $-g$; $g = 0$ gives the first moment for Graham's rearrangement problem, and U₀(A) < 1 certifies a valid ordering. We prove the total first moment is an invariant: ∑g ∈ G Ug(A) = t-12 for every A and G, independently of n and of A. At least n - t-12 targets are certified, recovering the observed √2p threshold as an identity. Feeding Pham and Sauermann's fixed-size anticoncentration theorem into the identity shows the first moment is equidistributed over Zₚ, maxg Ug(A) = 1/pt-12 + O(√t log t), so O(√t log t) when t = o(√ p); the Kneserized estimate of Costa, Della Fiore, Feng and Liu extends this to composite cyclic moduli, subject to their dichotomy. Two upgrades are incorporated: Nguyen, Tran and Vu's announced two-sided estimate improves the error to O(√ t) (proof deferred), and Berger, Berkowitz, Devlin and Vu's fixed-size estimate over the reals settles the order of magnitude in our open slice-anticoncentration question, leaving the sharp constant and extremiser open. Combining z Nz(A,g) ≤ tz-1 with complementation gives U₀(A) ≤ B(t) = (log 2 + o(1))t, improving the recorded worst case t - 1 - Hₜ₋₁, trapping the extremal constant in [1/2, log 2]; we ask whether it is $1/2$. Each inequality behind $B(t)$ is sharp, attained at $z = 3$, A = F₂ᵐ \0\. Two corrections: the method fails at $t = 5$ over general abelian groups by strict excess (Z₆ \0\ gives U₀ = 6/5), and \± 1, , ± k\ is extremal only in its minimal ambient group: the obstruction is density, not symmetry.