Rank-Six Symplectic Monodromy, Horizontal USp(6), and Coherent-Spike Renormalization for Reciprocal-Square Qudit Wigner Functions

Abstract

We study the reciprocal-square qudit state $\psi_c(x) = (p-1)^{-1/2} e_p(c/x^2)$ for $x$ nonzero, with $\psi_c(0)=0$, and the rank-six exponential-sum family arising from its discrete Wigner function. The associated two-parameter $\ell$-adic local system has generic Swan ledger $(2,2,1)$. We prove that for every odd prime $p$ its geometric monodromy group is $\mathrm{Sp}_6$. The proof combines an alternating self-duality coming from the involution $t \to -t$, a Fourier-Deligne transvection obtained from the codimension-one rank drop at $B=0$, and a fourth-moment computation whose ordered residual cover has geometric Galois group $S_4$. A load-bearing omission in the earlier rank-six paper is corrected explicitly: irreducibility of the cubic resolvent and nonsquareness of the quartic discriminant do not by themselves exclude an $S_3$ point stabilizer. We supply a uniform odd-characteristic $z=1$ certificate proving quartic irreducibility, cubic-resolvent irreducibility, and discriminant nonsquareness. After a weight-zero normalization the arithmetic and geometric monodromy groups both equal $\mathrm{Sp}_6$. We then give a family-specific bridge to Sawin-Forey-Fresán-Kowalski, Quantitative Sheaf Theory, JAMS 36 (2023), Theorem 7.22. Uniform complexity is verified directly from the physical three-variable Artin-Schreier phase using Corollary 7.7, Theorem 6.8, and Proposition 6.24. Hence, for any sequence $c_p \in \mathbb{F}_p^\times$, the generic physical Wigner values converge horizontally to the trace law of Haar $\mathrm{USp}(6)$, with $O_\rho(p^{-1/2})$ character averages for every fixed nontrivial irreducible representation $\rho$. Finally, the exact coherent exceptional row produces a sharp separation between weak convergence and high moments. After subtracting an explicit spike counterterm, every fixed signed moment returns to the $\mathrm{USp}(6)$ Haar moment with square-root rate. For absolute moments the exact critical exponent is $4$: convergence holds for $0 < r < 4$, the fourth moment has defect one, and all moments of order $r > 4$ diverge at the coherent-spike scale. The $\mathrm{USp}(6)$ absolute-trace constant is represented by an exact Bessel integral and numerically equals $0.7984915095402\dots$, giving an explicit asymptotic for the Wigner mana.