α-EFC II: Full Non-Adiabatic Perturbative Treatment, Spontaneous Isocurvature Generation and Holographic Drag in Entropy Field Cosmology Addressing the Hubble and BAO Tensions via Covariant Entropy-Field Perturbation Theory

Abstract: We present the complete non-adiabatic perturbative framework of $\alpha$-Entropy Field Cosmology ($\alpha$-EFC). Starting from a covariant scalar-tensor action with a localized entropic source $\Gamma(t)$ derived from horizon thermodynamics, we derive the full set of linear perturbation equations for the photon-baryon plasma in synchronous gauge. A thermodynamically consistent modification of the effective baryon sound speed $c_{b,eff}^{2}(z)$, implemented natively in CLASS, contracts the sound horizon to $r_{drag} \approx 133\text{--}135$ Mpc. Spontaneous non-adiabatic pressure perturbations $\delta P_{nad} \propto \Gamma(t)S_{b\gamma}$ generate a baryon-photon isocurvature mode that induces a compensating acoustic phase shift, restoring exact agreement with Planck CMB peaks. We provide the complete perturbative implementation (tested via Runs XXIII-XXIV on Planck+Pantheon+SH0ES datasets), identifying the gap between background and perturbative implementations as the central open problem for the third installment of this series. This paper is the second in the $\alpha$-EFC series. It builds upon the thermodynamic foundations established in "$\alpha$-EFC: Entropy Field Cosmology - A Unified Solution to the Cosmic Tensions via Horizon Thermodynamics" (2026). doi 10.5281/zenodo.20429926