Emergent Gravity from QED and the Electroweak Sector

Abstract

Two recently reported numerical relations — $\Lambda \cdot \ell_{\rm Pl}^2 = (\alpha^4/4\pi)(m_e/m_{\rm Pl})^5$ for the cosmological constant [Zhang 2026a] and $\alpha_G = \alpha^8 y_e^5$ for the gravitational coupling [Zhang 2026b, building on Kalinski 2021] — can be combined to express both Newton’s constant $G$ and the cosmological constant $\Lambda$ entirely in terms of electromagnetic and electroweak quantities. Specifically,

where $\alpha$ is the fine structure constant, $y_e = \sqrt{2}\,m_e/v$ the electron Yukawa coupling, and $\bar{\lambda}_e = \hbar/(m_e c)$ the electron Compton wavelength. No gravitational quantities ($G$, $m_{\rm Pl}$, $\ell_{\rm Pl}$) appear on the right-hand side. If these relations are physical, gravity would not be an independent interaction — its coupling strength and its vacuum energy contribution are both determined by QED and the electron’s coupling to the Higgs field. The hierarchy problem (why is $G$ so small?) and the cosmological constant problem (why is $\Lambda$ so small?) reduce to a single question: why does the electron have the mass it has?

Five testable predictions follow, including a specific value of Newton’s constant ($G = 6.6727 \times 10^{-11}$ m³ kg⁻¹ s⁻², in 11σ tension with CODATA 2022), a strictly constant dark energy equation of state $w = -1$ (testable by DESI and Euclid within 2–3 years), and a unique correlation $\delta G/G = 8\,\delta\alpha/\alpha$ between variations of the gravitational and fine structure constants. The two input formulas admit a unified reading on the super-flag inside Witten’s super-twistor space $\mathbb{CP}^{3|4}$ [Zhang 2026h], in which both share the universal Fibonacci factor $y_e^{F(5)} = y_e^5$ on the dominant $n = 3$ stratum and the spin-dependent factorial-sector exponent $\alpha^{2(s+2)}$ carries the only spin information; under this reading the spin-1 fifth-force prediction is $\alpha_Y \approx 1.88 \times 10^4$ relative to gravity (updated from earlier versions of this paper), allowed in the surviving $\lambda \lesssim 7$–8 µm window by current short-range bounds — lying ~500× below the binding bound at $\lambda = 5$ µm [Venugopalan et al. 2026] while sitting at the boundary of Geraci et al. 2008 at $\lambda \approx 10$ µm — and falsifiable by two to three further iterations of optomechanical vector sensing.

The input formulas were identified using the DAEDALUS dimensional-analysis engine [Zhang 2026g]. No first-principles derivation is available, and the $\alpha_G$ formula is in 11σ tension with the CODATA 2022 value of $G$. The empirical relations reported here are also reviewed in the companion DAEDALUS review [Zhang 2026 review], where they are classified as Cabibbo-scenario regularities; in the broader Twistor Configuration Geometry (TCG) framework [Zhang 2026 TCG], the spin-1 prediction discussed above is the principal forward-extrapolation.

DOI

https://doi.org/10.5281/zenodo.19980880