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 $$\vert{}R_c^{(2)}\rangle=\frac{1}{\sqrt{p-1}} \sum_{x\in\mathbf F_p^\times}e_p(c/x^2)\vert{}x\rangle,$$ for odd primes $p$. Their discrete Wigner functions produce, away from one exceptional row, the two-parameter rational exponential sums $$T_p^{(2)}(A,B)=\sum_{t\ne\pm1} e_p\!\left(A\frac{t}{(t^2-1)^2}+Bt\right).$$ 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 $x=e_2$, $y=e_3$, $z=e_4$, the residual quotient is the cubic surface $$(z+3)y^2+(x+z+1)(x^2-3xz+3x-10z+6)=0.$$ Its projective closure is a split singular cubic of type $A_3+2A_1$ and is purely Tate after resolution; in particular, the second family does not repeat the K3 mechanism of the reciprocal $1/x$ family. The arithmetic correction instead decomposes into several distinct pieces. The pairing–residual boundary has a genus-four normalization with $$\mathrm{Jac}(C)\sim_{\mathbb{Q}}E_{45}^{\,3}\times E_{30},$$ where both elliptic isogeny classes are non-CM. The physical quadratic-coset restriction introduces a Kummer threefold whose anti-invariant cohomology has a natural $\mathbb{Q}(i)$-induction structure and whose trace vanishes identically for $p\equiv3\pmod4$. Further $S_4$-isotypic analysis identifies an $X_0(2)$-type bielliptic packet in the $V_2$ sector and a rational-elliptic-surface pencil in the $V_3$ sector. The latter has generic fiber configuration $3I_2+6I_1$, Mordell–Weil lattice $D_4^*\oplus A_1^*$, a proper-support $E_{45}(-1)$ term, and a remaining rank-four finite-monodromy lattice local system with image in $\mathrm{Aut}(D_4)\simeq W(F_4)$. We isolate the statements that are fully proved from open monodromy questions; in particular, no claim of $\mathrm{Sp}_6$ geometric monodromy is made here.