Unified Heat Kernel Chain from E8: Gravity, Inflation and Hawking Corrections

The 1-loop heat kernel of the 248 gauge bosons of E8 produces Seeley–DeWitt coefficients that organize the three observational pillars of the TTOE framework: a0 → cosmological constant (external input), a2 → Einstein–Hilbert with G_N = g*_Sak/M_UV² (derived, Sakharov route, g*_Sak = 6π/496), a4 → R² → Starobinsky → n_s = 0.9649, r = 0.0037 (conditional), and a4 → α_Hawking ≈ 0.20 (conditional). The functional form of each surviving term is derived; the absolute values of the scales are external inputs. Version 2 — correction notice. Central correction: the one-loop a4 of an E8 gauge multiplet contains NO independent R² invariant — one adjoint vector plus its two ghosts gives (1/10)C² − (31/180)E4, with zero R² part, as required for a conformally invariant field (method validated against the textbook anomaly ratios c_vec/c_sc = 12 and a_vec/a_sc = 62). The Starobinsky inflation chain therefore requires a non-conformal sector, and the R² coefficient is an input, not a consequence of the 248 gauge bosons; the v1 value (248/180)·ln(M_UV²/μ²) is withdrawn. Also corrected: α_Hawking = 1/30 + C_A·ln(248)/(2·496) = 0.200, not 0.167 (the 1/30 term had been dropped; the ratio to the log(248) ≈ 5.5 heuristic is 28×); the heat-kernel expansion is a_{2n}·tⁿ (as printed it carried a dimensionful term); the R² coefficient is dimensionless and logarithmic (a spurious 1/M_UV² removed); N_e = 57.1 is reclassified as an output of the observed n_s, not an E8 input; and a complete bibliography has been added (v1 had none).