Paper10_S8 tension under B framework

The S8 tension is a late-time structure problem: baseline CMB fits in ΛCDM prefer asomewhat larger clustering amplitude than several low-redshift weak-lensing and large-scale-structure analyses. This paper develops a formal route by which the α–β–ϕ framework can movethe late-time clustering amplitude downward while remaining connected to the same visibleaction used in the broader program. Starting from the visible Lagrangian, the action is variedstep by step to obtain modified Einstein equations and, after homogeneous reduction, modifiedFriedmann equations. The same non-minimal coupling F (ϕ)R then yields a background effectivegravitational response and motivates a controlled perturbative completion for the growth sector.The quantity