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

We study odd-prime-dimensional pure states whose amplitudes carry the reciprocal phase $x \to c/x$. Their discrete Wigner functions reduce, away from one exceptional phase-space row, to a two-parameter family of finite-field exponential sums with phase $\frac{A t}{t^2 - 1} + B t$. The generic compactly supported cohomology has rank $4$. The fourth Wigner moment admits an exact pairing decomposition. Besides three pair-contraction components, the residual component is an explicit algebraic surface $Q = e_2 - e_4 + 3$ whose quadratic discriminant factors as $(ab - 1)(ab + 3)(a + b - 2)(a + b + 2)$. A natural compactification is a singular $K3$ surface. Its elliptic fibration has fibers $2 I_2^* + 2 I_2 + I_4$, Mordell-Weil group $\mathbb{Z} + (\mathbb{Z}/2\mathbb{Z})^2$, generator height $\frac{3}{4}$, and transcendental lattice $A_2(2)$. Using a finite Livné comparison, the transcendental Galois representation is identified with the weight-three, level-twelve CM newform $\eta(2\tau)^3 \eta(6\tau)^3$. If $a_p$ denotes its $p$-th Fourier coefficient and $b_p$ denotes the Frobenius trace of a fixed genus-one boundary curve, we derive an exact finite-$p$ formula for the normalized Wigner fourth moment. In particular, $$E[Z^4] = 3 - \frac{6}{p} + \frac{a_p}{p^2} + O\left(p^{-3/2}\right)$$ Thus the leading finite-size fluctuation of this quantum quasiprobability observable contains a normalized Hecke eigenvalue and is arithmetic rather than numerical noise. Higher numerical moments agree with the standard trace moments of $\mathrm{USp}(4)$; this paper records the resulting Sato-Tate/monodromy interpretation as a conjectural direction, not as a proved monodromy theorem. Conditional on $\mathrm{USp}(4)$ equidistribution, the mana constant is $$E_{\mathrm{USp}(4)} \vert{}\operatorname{Tr} U\vert{} = \frac{4096}{525 \pi^2}$$