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

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.