A Note on the Approximate Relation 1/α ≈ π + π² + 4π³ and the Volumes of Complex Projective Spaces

The well-known approximation 1/α ≈ π + π² + 4π³ (accurate to 2.2 ppm, ruled out as exact at >27{,}000σ) can be rewritten as

(1/α) ≈ \sum_(n=1)³ (2n-2)!\ (π^n/n!) = \sum_(n=1)³ (\dim_(R) CPⁿ⁻¹)!\ ×\ { Vol}(CP^n)

where { Vol}(CP^n) = π^n/n! is the Fubini-Study volume of complex projective n-space. The three terms correspond to the subspaces in the natural inclusion CP¹ ⊂ CP² ⊂ CP³, each weighted by the factorial of the real dimension of the previous subspace. The sum has exactly three terms because Penrose's twistor space CP³ 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 Λ in terms of α and the electron Yukawa coupling: if α is determined by the geometry of CP³, then G and Λ would also follow from twistor geometry. We note the connection to Penrose's twistor program and to Atiyah's unsuccessful 2018 attempt to derive α 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.