Frobenius Eigenvalues and Gauss Sums from Witt Vector Carry Arithmetic

We construct a transfer operator on the carry state space of truncated Witt vector multiplication over F_p and prove that its characteristic polynomial contains exact Frobenius polynomials of algebraic varieties. For p=3, the quadratic factor mu^2+3mu+3 is the Frobenius of a supersingular elliptic curve (Theorem A). For p=7, the genus-3 factor (mu^2-7)^2(mu^2+7) has supersingular roots equal to the quadratic Gauss sum g(chi^3) of the Legendre symbol (Theorem B). The carry exponent is equidistributed on F_p^*, and Weil eigenvectors are orthogonal to the all-ones vector, consistent with an Artin-Schreier interpretation. Dirichlet character twists produce Weil eigenvalues at every prime tested (p=3 through 23). We conjecture that the carry operator computes the Frobenius on the Artin-Schreier variety z^p - z = c_2(x,y).