The gravitational coupling of the electron as a power of the fine-structure constant: the relation α_G = (4/3) α^21 exp(−5α/2), its B_3 heat-kernel form, and falsifiable predictions

Is the weakness of gravity fixed by the strength of electromagnetism? Standard physics treats Newton's constant G, the fine-structure constant α and the electron mass me as independent inputs. We study the parameter-free relation αG ≡ G me2/(ħc) = (4/3) α21 e−5α/2, with the zero-momentum α and the electron pole mass. With CODATA 2022 inputs it gives G = 6.674312278×10−11 m3 kg−1 s−2, +1.84 parts per million (0.08σ) from the CODATA 2022 value and 1.16σ above the 2026 NIST redetermination. No continuous parameter is fitted, but the discrete factors were chosen with G known; counted against the formula search behind them, the agreement alone is weak evidence (about 1–2σ). The right-hand side equals (4/3)(7/4π)21 times the inverse square of the Spin(7) heat-kernel density at the identity at time α/7, up to a relative remainder below 10−8000: dim B3 = 21 supplies the power and the squared Weyl-vector norm |ρ|2 = 35/4 the exponent; time and prefactor are not derived, so the relation remains phenomenological. For one proposed source, an O(7)×F4 conformal field theory in five dimensions, we assemble the five-loop renormalization group, validate it exactly against its known limits and obtain by calibrated resummation ΔΦ = 1.576 ± 0.099 at d = 5, conditional on the fixed point's existence; under stated premises its real branch ends between four and six dimensions, and weakly coupled scalar compactification cannot carry its exponent to four dimensions. Under further stated hypotheses the same structure gives dated, falsifiable predictions: normal neutrino ordering, a CP-conserving leptonic phase for real mass couplings, a neutrino mass sum of 87.76–89.02 meV (outside the 95% limits of DESI DR2 and CMB data within ΛCDM, about 2.8σ, with the 64.2 meV bound meeting its registered <80 meV imminent-falsification trigger, but allowed within w0waCDM), four CP-parity bands of the neutrinoless double-beta-decay mass, and an ultralight scalar at 4.8102×10−11 eV. The author is an independent researcher without formal physics training. This research is AI-generated under his direction; he is responsible for its claims and limitations. Files. The paper (PDF); its LaTeX source with an arithmetic reproduction script; the numerical supplement (exact five-loop renormalization-group polynomials, fixed-point series, frozen continuation outputs, existence-boundary expressions) with SHA-256 checksums; the look-elsewhere audit (expression population, random targets, script and results); a ledger of the internal registration records of the predictions. This paper continues the May 2026 preprint doi:10.5281/zenodo.20120946.