Spectral determinants, prime orbits, and the Riemann zeta function in Universal Coset Field Theory

We construct the zeta sector of Universal Coset Field Theory (UCFT) using the finite adelic Albert--EIII completion and its associated one-dimensional zeta automorphic character. The resulting finite orbit quotient is identified with Af^/ Z^, , yielding a canonical prime-orbit correspondence and the usual Euler product trace for '/. Combining this finite arithmetic trace with the archimedean zeta oscillator gives the logarithmic derivative of the completed zeta factor. After passage to the shifted-square coordinate = (s-12) ², the Riemann hypothesis is reformulated as the assertion that all zeros of the corresponding function lie on R₀. Under the zeta determinant realization, this condition is equivalent to the absence of tachyonic or complex mass-squared modes in the zeta sector.