Emergent Gravity and the Scale of a0 in the TTOE Framework: The E8 -> Sakharov -> FRG -> Friedmann Chain with a Derived 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 prefactor K=2/pi is Derived (0 arbitrary hypotheses): sinusoidality from standard QFT, mass gap desert from the E8 gap, random phase from standard QM, and L1 functional from MOND physics (ANA_TEO_794). The numerical value a0=(2/pi)A = 1.204×10⁻¹⁰ m/s² agrees with observation at +0.5%, given the external input rho_vac (Planck 2018). 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.