A Stratified Volume Functional on Twistor Space and Fundamental Coupling Constants

Abstract

We define a linear functional on complex projective space $\mathbb{CP}^N$ — the stratified volume functional — which sums the Fubini–Study volumes of the subspaces $\mathbb{CP}^1 \subset \mathbb{CP}^2 \subset \cdots \subset \mathbb{CP}^N$, each weighted by the factorial of the real dimension of the preceding subspace:

For the constant function $f = 1$ on Penrose’s twistor space $\mathbb{CP}^3$, this evaluates to $\mathcal{I}[1] = \pi + \pi^2 + 4\pi^3 \approx 137.036$, within 2.2 ppm of the inverse fine-structure constant $1/\alpha$. We postulate $1/\alpha = \mathcal{I}[1]$ and explore the consequences. Evaluating the functional for $f(n) = n^2$ produces a non-trivial prediction: $\mathcal{I}[n^2]/\mathcal{I}[1] = 8.456$, matching the inverse strong coupling constant $1/\alpha_s(M_Z) = 8.482$ to 0.3%. Additionally, the complex dimension $N = 3$ of $\mathbb{CP}^3$ yields $\sin^2\theta_W(0) = 3/(4\pi)$, matching the Thomson-limit weak mixing angle to 0.03%; the ratio of the SU(5) GUT value $3/8$ to this is exactly $\pi/2$. Neither result was used as input. Combined with recently reported empirical formulas for Newton’s constant $G$ and the cosmological constant $\Lambda$, all four Standard Model gauge couplings and gravity become expressible through the geometry of $\mathbb{CP}^3$. The weights $(2n-2)!$ equal the factorial of the complex dimension of the Grassmannian of lines in each stratum: $2n - 2 = \dim_{\mathbb{C}}\,G(2,n+1)$. For $n = 3$, this is the 4-dimensional moduli space of twistor lines — i.e., complexified spacetime. The weights can be realized through a specific super-flag inside Witten’s super-twistor space $\mathbb{CP}^{3|4}$. We present this as a speculative framework, not a derivation. The axiom is approximate, and the weights are postulated.

DOI

https://doi.org/10.5281/zenodo.19981555