Gravitational Coupling, QED Vacuum, and the Electron Yukawa: α_G ≈ α⁸ y_e⁵

Abstract

We report a numerical relation connecting the gravitational fine structure constant $\alpha_G = G m_e^2/(\hbar c)$, the electromagnetic fine structure constant $\alpha$, and the electron Yukawa coupling $y_e = \sqrt{2}\,m_e/v$:

The relation predicts $G = 6.6727 \times 10^{-11}$ m³ kg⁻¹ s⁻², within 0.024% (240 ppm) of the CODATA 2022 value. All quantities are true constants — the relation is epoch-independent and valid at all times. It decomposes the 45 orders of magnitude separating gravity from the electron scale into two factors: $\alpha^8 \approx 10^{-17}$ (a fourth-order QED suppression) and $y_e^5 \approx 10^{-28}$ (the fifth power of the electron Yukawa coupling). The exponent structure echoes that of a recently reported formula for the cosmological constant, $\Lambda \cdot \ell_{\rm Pl}^2 = (\alpha^4/4\pi)(m_e/m_{\rm Pl})^5$ [Zhang 2026a], which shares both the power of 5 on the electron mass ratio and a QED loop-order suppression (here $\alpha^8 = (\alpha^4)^2$). Both formulas admit a unified reading on the super-flag inside Witten’s super-twistor space $\mathbb{CP}^{3|4}$ [Zhang 2026h], in which the present $\alpha_G$ formula sits at the spin-2 (tensor) slot of $Q_s \propto \mathcal{I}[1]^{-2(s+2)} \cdot y_e^{F(5)}$ and $\Lambda$ sits at the spin-0 (scalar) slot, with the universal Fibonacci factor $y_e^{F(5)} = y_e^5$ shared between them; whether this supplies a microscopic mechanism is examined separately [Zhang 2026 super-CY]. Among all Standard Model fermions, the electron is the only one that yields clean integer exponents. The relation was identified using the DAEDALUS dimensional-analysis engine [Zhang 2026g]. No derivation from known physics is offered, and the formula is in 11σ tension with the CODATA 2022 value of $G$.

The empirical relation reported here is also reviewed in the companion DAEDALUS review [Zhang 2026 review], where it is classified as a Cabibbo-scenario regularity; in the broader Twistor Configuration Geometry (TCG) framework [Zhang 2026 TCG], the spin-dependent structure used above is recorded as postulate P6.

DOI

https://doi.org/10.5281/zenodo.19980817