A Super-Flag Construction on Witten's Super-Twistor Space CP^(3|4)

Abstract

The inverse fine-structure constant satisfies $1/\alpha \approx \sum_{n=1}^{3}(2n-2)!\,\mathrm{Vol}(\mathbb{CP}^n) = \pi + \pi^2 + 4\pi^3$ to 2.2 ppm. The factorial weights $(2n-2)! = (1, 2, 24)$ lack a derivation from known mathematics on $\mathbb{CP}^3$. The measured value is $1/\alpha = 137.035\,999\,177(21)$ [Morel et al. 2020]. We show that the standard Berezinian measure on Witten’s super-twistor space $\mathbb{CP}^{3|4}$ does not produce these weights: all three integrals $\int_{\mathbb{CP}^{n|4}} d\mu_{\mathrm{Ber}}$ vanish because the Calabi–Yau condition makes the Berezinian density constant. However, a specific super-flag

with fermionic rank $k_n = 2n - 2$ per stratum and top fermion monomial insertion re-expresses the weights in super-geometric language and reproduces them numerically. The fermionic rank rule $k_n = 2n - 2 = \dim_{\mathbb{C}}\,G(2,n+1)$ equals the complex dimension of the Grassmannian of lines in $\mathbb{CP}^n$ — a fundamental object in twistor geometry. For $n = 3$, this is the 4-dimensional moduli space of twistor lines, i.e., complexified spacetime.

The same super-flag accommodates the mass sector: replacing the symmetric monomial with a nearest-neighbor chain determinant produces Fibonacci weights $F(2n-1) = (1, 2, 5)$ instead of factorials. The distinction between coupling constants (large, factorial-weighted) and masses (tiny, Fibonacci-weighted) reduces to the distinction between permutations and tilings of the same fermionic moduli.

The framework also accommodates the gravitational sector: existing empirical formulas $G \propto \alpha^8 y_e^5$ and $\Lambda \propto \alpha^4 (m_e/m_{\rm Pl})^5$ have a unified structural form $Q_s \propto \mathcal{I}[1]^{-2(s+2)} \cdot y_e^{F(5)}$, where the spin-dependent factorial-sector exponent is multiplied by the Fibonacci factor $F(5) = 5$ from the dominant $n = 3$ stratum (a reading that is structurally selected in the lepton-mass sector via the chain-determinant identity $\det A_r = F_{r+1}$ but is consistent with a canonical dimensional 4+1 decomposition for cosmology and gravity). The vocabulary further absorbs Nambu’s 74-year-old empirical relation $m_\pi/m_e \approx 2/\alpha$ as $F(3) \cdot \mathcal{I}[1]$, activating the previously-unused $n = 2$ Fibonacci weight in physics applications.

We report all four sectors and their limitations: four postulates are required, the insertions (P3 and P4) remain ad hoc, and the gravitational and hadronic extensions are structural observations, not derivations.

DOI

https://doi.org/10.5281/zenodo.19981089