Arithmetic Magic Beyond the Reciprocal Phase: Rank-Six Wigner Moments, a Rational Cubic, and Non-CM Motives

Abstract

We study the power-reciprocal qudit states $$\vert{}R_c^{(2)}\rangle = (p-1)^{-1/2} \sum_{x \in \mathbb{F}_p^*} e_p\left(\frac{c}{x^2}\right)\vert{}x\rangle$$ for odd primes $p$. Their discrete Wigner functions reduce, away from one exceptional row, to the two-parameter rational exponential sums $$T_p^{(2)}(A,B) = \sum_{t \neq \pm 1} e_p\left(\frac{A t}{(t^2-1)^2} + Bt\right).$$ We prove an exact Wigner-to-trace dictionary and show that the generic compactly supported first cohomology has rank six. The fourth-moment variety factors into three pairing components and a geometrically irreducible residual component. In symmetric coordinates the residual quotient is a split singular cubic surface of type $A_3 + 2A_1$ and has purely Tate second cohomology after resolution, so the second family does not repeat the $\mathrm{K3}$ mechanism of the reciprocal $1/x$ family. Instead, the arithmetic correction separates into several different pieces: The pairing-residual boundary has a genus-four normalization with Jacobian isogenous over $\mathbb{Q}$ to $E_{45}^3 \times E_{30}$, where both elliptic isogeny classes are non-CM. The physical quadratic-coset restriction introduces a Kummer threefold whose deck-anti-invariant cohomology has a natural $\mathbb{Q}(i)$-induction structure and whose trace vanishes at primes $p \equiv 3 \pmod 4$. An $S_4$-isotypic analysis further identifies a bielliptic $X_0(2)$-type packet in the $V_2$ sector and a rational-elliptic-surface pencil in the $V_3$ sector. The latter has generic fiber configuration $3\mathrm{I}_2 + 6\mathrm{I}_1$, Mordell–Weil lattice $D_4^* + A_1^*$, a proper-support $E_{45}(-1)$ contribution, and a remaining rank-four finite-monodromy lattice local system with image contained in $\mathrm{Aut}(D_4) \cong W(F_4)$. These results provide a second exact instance of Wigner-forced arithmetic geometry with a motivic architecture genuinely different from the first reciprocal family. We deliberately leave the geometric monodromy problem for the rank-six trace sheaf open and make no claim of $\mathrm{Sp}_6$ or $\mathrm{USp}(6)$ equidistribution in this paper.