1/α as a Sum of Stratified Fubini–Study Volumes on Penrose's CP³

Abstract

The well-known approximation $1/\alpha \approx \pi + \pi^2 + 4\pi^3$ (accurate to 2.2 ppm, ruled out as exact at $> 27{,}000\sigma$) can be rewritten as

where $\mathrm{Vol}(\mathbb{CP}^n) = \pi^n/n!$ is the Fubini–Study volume of complex projective $n$-space. The three terms correspond to the subspaces in the natural inclusion $\mathbb{CP}^1 \subset \mathbb{CP}^2 \subset \mathbb{CP}^3$, each weighted by the factorial of the real dimension of the previous subspace. The sum has exactly three terms because Penrose’s twistor space $\mathbb{CP}^3$ has three complex dimensions. We present this as an observation about an approximate numerical identity, not a derivation. The formula is experimentally excluded as exact, and the weights are reverse-engineered from the known coefficients. The observation is motivated by recently reported empirical formulas expressing both Newton’s constant $G$ and the cosmological constant $\Lambda$ in terms of $\alpha$ and the electron Yukawa coupling: if $\alpha$ is determined by the geometry of $\mathbb{CP}^3$, then $G$ and $\Lambda$ would also follow from twistor geometry. We note the connection to Penrose’s twistor program and to Atiyah’s unsuccessful 2018 attempt to derive $\alpha$ from related geometric structures.

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 chamber-weighted Fubini–Study sum reading proposed here is developed as the coupling-sector observable of the FPA model.

DOI

https://doi.org/10.5281/zenodo.19980960