Arithmetic Magic in Reciprocal Qudit States: Exact Wigner Moments, a Singular K3 Surface, and CM Modularity

Abstract

We study a family of odd-prime-dimensional pure states whose amplitudes carry the reciprocal phase $x\mapsto c/x$. Their discrete Wigner functions reduce, away from one exceptional phase-space row, to the two-parameter finite-field exponential sums $$T_p(A,B)=\sum_{t\ne\pm1} e_p\!\left(A\frac{t}{t^2-1}+Bt\right), \qquad A\in\mathbb{F}_p^\times,\ B\in\mathbb{F}_p.$$ The generic cohomology of this family has rank four. We then determine the fourth Wigner moment exactly. The moment variety factors into three pairing components and a residual surface $$Q=e_2-e_4+3=0,$$ whose quadratic discriminant is $(ab-1)(ab+3)(a+b-2)(a+b+2)$. A natural compactification of the residual surface is a singular K3 surface. An elliptic fibration has fibers $2I_2^*+2I_2+I_4$, Mordell--Weil group $\mathbf Z\oplus(\mathbf Z/2\mathbf Z)^2$, generator height $3/4$, and transcendental lattice $A_2(2)$. Its transcendental Galois representation is identified, by a finite Livné comparison, with the weight-three CM newform $$f(\tau)=\eta(2\tau)^3\eta(6\tau)^3=\sum_{n\ge1}a_nq^n.$$ Consequently a Hecke eigenvalue appears directly in the finite-size correction to a quantum quasiprobability statistic. If $Z=p^{3/2}W$ denotes the normalized Wigner value, then for every prime $p>3$ we obtain an explicit finite-$p$ formula and in particular $$\mathbb{E} Z^4=3-\frac{6}{p}+\frac{a_p}{p^2}+O(p^{-3/2}).$$ Thus the leading non-smooth finite-size fluctuation is arithmetic rather than numerical noise. Numerically, higher Wigner moments agree with the standard trace moments of $\mathrm{USp}(4)$; we record this as a monodromy/Sato--Tate direction, not as a proved identification. Under that equidistribution hypothesis, the mana has the explicit asymptotic constant $\mathbb{E}_{\mathrm{USp}(4)}\vert{}\mathrm{Tr}\, U\vert{}=4096/(525\pi^2)$.