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 ζ'/ζ. 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 λ=(s-12)², the Riemann hypothesis is reformulated as the assertion that all zeros of the corresponding function Φ lie on R≤0. Under the zeta determinant realization, this condition is equivalent to the absence of tachyonic or complex mass-squared modes in the zeta sector.