We propose a unified 4D Effective Field Theory (EFT) where the string dilaton's evolution is governed by the spectral geometry of its S-duality moduli space, $\mathcal{M}= SL(2,\mathbb{Z})\backslash\mathbb{H}$. Utilizing the Selberg Trace Formula as a geometric proxy for the 1-loop effective action, we derive a leading-order exponential potential $V(\phi) \propto e^{-2\phi}$ through Weyl conformal rescaling. We demonstrate a "Spectral-Observational Bridge": large kinetic couplings $\xi$ compress high-frequency modular fluctuations into log-periodic features within the observable CMB window ($\ell \sim 10-1000$). In the late universe, this framework naturally yields thawing quintessence that alleviates the $H_0$ tension via a parametric degeneracy in the acoustic scale. Finally, we show that local gravity constraints are satisfied through a Chameleon mechanism with a characteristic $\rho^{4/5}$ mass scaling, ensuring consistency with Cassini-era bounds.