Modular Spectral Cosmology: A Unified Proxy for Primordial Features and Late-Time Tensions

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, M= SL(2,Z). Utilizing the Selberg Trace Formula as a geometric proxy for the 1-loop effective action, we derive a leading-order exponential potential V(φ) ∝ e-2φ through Weyl conformal rescaling. We demonstrate a "Spectral-Observational Bridge": large kinetic couplings ξ compress high-frequency modular fluctuations into log-periodic features within the observable CMB window (~ 10-1000). In the late universe, this framework naturally yields thawing quintessence that alleviates the H₀ 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 ρ4/5 mass scaling, ensuring consistency with Cassini-era bounds.