An Approximate Closed Form for the Electron Yukawa Coupling: y_e ≈ (1 − 1/(2π)) e^(−4π)

Abstract

We report a numerical coincidence: the electron Yukawa coupling $y_e = \sqrt{2}\,m_e/v \approx 2.935 \times 10^{-6}$ is approximated to 0.09% by

The number $4\pi$ in the exponent is close to $\chi(\mathbb{CP}^3) \times \pi$, where $\chi(\mathbb{CP}^3) = 4$ is the Euler characteristic of Penrose’s twistor space. This initially suggested an instanton interpretation (degree-4 holomorphic curve on $\mathbb{CP}^3$). However, explicit computation shows that the standard Gromov–Witten machinery does not produce this formula: the no-insertion genus-0 degree-4 invariant of $\mathbb{CP}^3$ vanishes (dimension mismatch), and the normal-bundle determinant ratio is $1/8! \approx 2.5 \times 10^{-5}$ per factor (combined: $1/(8!)^2 \approx 6 \times 10^{-10}$), not $1 - 1/(2\pi)$. We report the numerical match honestly and the falsification of the proposed mechanism equally honestly. A separate geometric structure does connect $y_e$ to twistor space: the mass sector of the stratified volume functional [Zhang 2026f] uses Fibonacci weights $F(2n-1) = (1, 2, 5)$ — realized as fermionic chain determinants on the same super-flag that produces the coupling-constant weights [Zhang 2026h]. The exponential smallness of $y_e$ is consistent with a clockwork-like suppression from nearest-neighbor locality on the fermionic fiber.

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 boundary prefactor $1 - 1/(2\pi)$ appears as postulate P4.

DOI

https://doi.org/10.5281/zenodo.19981497