Emergent Gravity and the Scale of a0 in the TTOE Framework: The E8 -> Sakharov -> FRG -> Friedmann Chain with a Conditional Prefactor

This manuscript formulates the emergent-gravity block of the TTOE framework as a mixed chain that fixes the scale A = c sqrt(G rho_vac) from the E8 structure, the Sakharov induced coupling, an FRG correction, and the observed cosmological input rho_vac. The numerical value a0 = (2/pi) A is presented only as a conditional result: it requires a sinusoidal time profile and a rectified L^1 observational functional. The possible evolution a0_eff(z) proportional to H(z) is kept separate as a testable phenomenological hypothesis, not as a direct verification of the full chain.