Emergent Gravity and the Scale of a0 in the TTOE Framework: The E8 -> Sakharov -> FRG Chain, the External Input rho_vac, and a Conditional Prefactor

This manuscript formulates the emergent-gravity block of the TTOE framework as a mixed chain: the E8 structure fixes the induced gravitational coupling (g*_Sak = 6π/496 ≈ 0.03800 from the heat kernel of the 248 gauge bosons, corrected by FRG back-reaction to g*_AS ≈ 0.0363, consistent within 2.4% with the external FRG reference), while the cosmological anchor ρ_vac = 5.924×10⁻²⁷ kg/m³ is an explicit external input (Planck 2018). The scale A = c·√(G·ρ_vac) then gives a0 = (2/π)·A = 1.200×10⁻¹⁰ m/s², within +0.01% of the observed MOND acceleration — given the external input. Both the scale A and the prefactor K = 2/π are classified Conditional: five arguments (sinusoidality from standard QFT, the mass-gap desert, temporal averaging, random phase, and the L1 observational functional) support the canonical single-mode choice, but the single-coherent-mode reading is itself the selecting hypothesis — a Gaussian (multi-mode) vacuum ensemble would give K = √(2/π) ≈ 0.798, a 25% difference, and every K ∈ (0,1] is realizable by some bounded profile. The phenomenological hypothesis a0(z) ∝ H(z) is kept separate and is testable with MIGHTEE-HI. Version 3 — correction notice. a0 = 1.200×10⁻¹⁰ m/s² (the earlier 1.204×10⁻¹⁰ carried a +0.33% arithmetic error; agreement with observation is +0.01%, not +0.5%). 6π/496 = 0.03800 (the paper had stated two different values). The abstract formula now includes the 2/π factor, consistently with Eq. (1). The symbol g*_AS now distinguishes the chain output (0.0363) from the external FRG reference (0.0372). The false chain arrow "→ Friedmann → ρ_vac" is removed: ρ_vac is a Planck 2018 input and no E8 quantity enters its value. The 2/π prefactor is reclassified Derived → Conditional throughout, including the subtitle.