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 $A\subseteq G\setminus\{0\}$ with $|A|=t$; write $N_z(A,g)$ for the number of $z$-subsets of $A$ summing to $g$ and $U_g(A)=\sum_{z=2}^{t-1}(t-z)N_z(A,g)/\binom tz$. The previous notes established the conservation law $\sum_gU_g(A)=\binom{t-1}{2}$ and the bound $U_0(A)\le(\log 2+o(1))t$, leaving open whether the extremal constant $c^{*}=\limsup_t M(t)/t$, $M(t)=\sup_{|A|=t}U_0(A)$, equals $\tfrac12$ with the full groups extremal. We settle the two smallest deviation classes exactly. For $p$ an odd prime, $t=p-1$, $A=\mathbb{Z}_p\setminus\{0\}$: $U_0(A)=\frac{t(t-2)}{2(t+2)}$, so $c^{*}\ge\tfrac12$; and for $t=p-2$, $A=\mathbb{Z}_p\setminus\{0,e\}$, any $e\ne0$: $U_0(A)=\frac{(t-1)^2}{2(t+3)}$, rigid in $e$ — both by exact evaluation of alternating reciprocal-binomial sums (beta integral plus a parity involution giving $B_1(t)=0$ for $t$ odd). For general $A$ with deviation set $E=\langle A-A\rangle\setminus A$, $s=|E|$, $m=t+s$ prime, we prove the exact identity $U_0(A)=(\binom t2+S)/m$ with $S=\sum_z\frac{(t-z)(-1)^z}{\binom tz}(mM_z(0)-Q_z)$, $M_z(0)$ counting zero-sum multisets of size $\le z$ from the nonzero deviation elements $E^{*}=E\setminus\{0\}$; thus $U_0\le\tfrac t2$ is equivalent to a single explicit inequality $R\le\Psi(t,s)$ on an alternating sum of sub-Frobenius zero-sum composition counts. We formulate the maximality conjecture MC ($s\ge2$ never beats the full group), which implies $c^{*}=\tfrac12$ with the extremal family and exact rate identified; MC is certified with zero violations by exhaustive exact computation for $t\le14$, by DP for $t\le60$, and for prime moduli to $t=2500$, $s\le25$, with a measured margin of a factor $\approx4$ 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.Version 2 corrects errors found in a full external review: prop:MC is stated conditionally on the composite asymptotic (verified in range m<=3274, not proved); the false equivalence 'r=1 iff 0 in E' is removed (counterexample G=Z_8, A={1,3,5}); lem:B1 for even t is stated as conjectural with the odd-t case proved; the rearranged sum S is distinguished from the crude budget U_0 (explaining the apparent 1.02 ratio); the 'kick in' and 'exhaustive where feasible' claims are made precise; the abstract's c*_cyc = 1/2 claim carries the same restriction as the theorem.Version 2 corrects errors found in a full external review: prop:MC is stated conditionally on the composite asymptotic (verified in range m<=3274, not proved); the false equivalence 'r=1 iff 0 in E' is removed (counterexample G=Z_8, A={1,3,5}); lem:B1 for even t is stated as conjectural with the odd-t case proved; the rearranged sum S is distinguished from the crude budget U_0 (explaining the apparent 1.02 ratio); the 'kick in' and 'exhaustive where feasible' claims are made precise; the abstract's c*_cyc = 1/2 claim carries the same restriction as the theorem.