On the extremal constant of the rearrangement moment: exact values for full groups and a reduction of the general case

Let G be a finite abelian group and A⊆ G\0\ with $|A|=t$; write Nz(A,g) for the number of z-subsets of A summing to g and Ug(A)=∑z=2ᵗ⁻¹(t-z)Nz(A,g)/ tz. The previous notes established the conservation law ∑gUg(A)=t-12 and the bound U₀(A)≤(log 2+o(1))t, leaving open whether the extremal constant c*=ₜ M(t)/t, M(t)=|A|=tU₀(A), equals 12 with the full groups extremal. We settle the two smallest deviation classes exactly. For p an odd prime, $t=p-1$, A=Zₚ\0\: U₀(A)=t(t-2)/2(t+2), so c*≥12; and for $t=p-2$, A=Zₚ\0,e\, any e≠0: U₀(A)=(t-1)²/2(t+3), rigid in e — both by exact evaluation of alternating reciprocal-binomial sums (beta integral plus a parity involution giving B₁(t)=0 for t odd). For general A with deviation set E= A-A A, $s=|E|$, $m=t+s$ prime, we prove the exact identity U₀(A)=( t2+S)/m with S=∑z(t-z)(-1)ᶻ/ tz(mMz(0)-Qz), Mz(0) counting zero-sum multisets of size ≤ z from the nonzero deviation elements E*=E\0\; thus U₀≤ t2 is equivalent to a single explicit inequality R≤Ψ(t,s) on an alternating sum of sub-Frobenius zero-sum composition counts. We formulate the maximality conjecture MC (s≥2 never beats the full group), which implies c*=12 with the extremal family and exact rate identified; MC is certified with zero violations by exhaustive exact computation for t≤14, by DP for t≤60, and for prime moduli to $t=2500$, s≤25, with a measured margin of a factor ≈4 in the reduced inequality. Four natural strengthenings (monotonicity in s, a universal gap law, a pointwise conjecture, the identity over composite moduli) are refuted by machine search, and the structural obstruction is isolated: every relevant weight lies below the Frobenius threshold, so only the global sign structure of S can close the argument. All claims are certified by exact-arithmetic scripts with non-emptiness gates.