Weil positivity, the Hodge index theorem, and the missing surface for ζ: an expository note with computed examples

For curves over finite fields the Riemann hypothesis is a theorem, and its proof rests on a positivity: on the surface C×C the intersection form has exactly one positive direction (Hodge index theorem), which, applied to the diagonal and the graph of Frobenius, gives Weil's bound |#C(Fq^n) − q^n − 1| ≤ 2g qn/2.The note recalls this argument and checks it on an elliptic curve over F_7: the Frobenius eigenvalues have modulus exactly √7, the intersection matrix has one positive and three negative eigenvalues, and the bound holds for n = 1 to 6. It then explains why the same route is not available for the Riemann zeta function: the surface "Spec Z × Spec Z" with an intersection theory does not exist in present mathematics (approaches by Deninger and by Connes–Consani are cited).The mirror exchanging the two factors of C×C is identified with the Rosati involution, whose positivity Tr(uu′) > 0 is equivalent to a² < 4q. Four modifications of this mirror (rotated, tilted, rough, parabolic) are tested; the parabolic case is the boundary a² = 4q. The note distinguishes the proved Hodge index theorem from the open Hodge conjecture, and records which local pieces of a spectral framework studied in a companion survey are of the finite-field type.No new theorem is claimed; the note is expository. It does not prove the Riemann hypothesis or the Hodge conjecture.