A Super-Calabi–Yau Reading of the Cosmological-Constant Exponent: Vol(CP^(3|4)) = 0

Abstract

The empirical relations $\Lambda \ell_{\rm Pl}^2 = (\alpha^4/4\pi)(m_e/m_{\rm Pl})^5$ and $G\,m_e^2/(\hbar c) = \alpha^8 y_e^5$ have unexplained $\alpha^4$ and $\alpha^8$ exponents. We observe that the naked integral of the canonical Berezinian on Witten’s super-twistor space $\mathbb{CP}^{3|4}$, and the super-Fubini–Study volume of the same supermanifold, both vanish identically by the super-Calabi–Yau condition $m = n+1$. The vanishing is computed explicitly through Voronov’s volume formula

in which the super-CY locus appears as a $\Gamma$-function pole. The lower strata $\mathbb{CP}^{1|0}$ and $\mathbb{CP}^{2|2}$ in the natural super-flag chain have non-vanishing super-volumes, so the vanishing is specific to the super-CY top stratum.

The natural twistor encoding of the spacetime cosmological-constant operator $\int d^4x\sqrt{-g}$ is an integral over the moduli of holomorphic lines (chiral superspace $\mathbb{M}^{4|8} \simeq G(2|0, 4|4)$), not an ambient integral over $\mathbb{CP}^{3|4}$, so the Voronov vanishing does not directly apply to vacuum energy. The relevant curve-moduli super-volume $\mathrm{vol}(G(2|0, 4|4))$ is, however, also rigorously zero: Honey 2021 §5.5 proves that $\mathrm{vol}(\mathrm{Gr}_{r|s}(\mathbb{C}^{p|q})) = 0$ when $p = q$, by direct examination of the integrand. So both the ambient vanishing on $\mathbb{CP}^{3|4}$ and the curve-moduli vanishing on $G(2|0, 4|4)$ are theorems.

Earlier versions of this note proposed that these structural vanishings supply a Berezinian-saturation selection rule for the empirical $\alpha^4$ exponent. We retract that mechanism here. Section 4.4 systematically tests four concrete non-standard candidate constructions — twistor composite half-$D$-term projection (Koster–Mitev–Staudacher–Wilhelm style), massive two-twistor worldline with doubled fermionic zero-modes (Penrose–Perjés / Albonico–Geyer–Mason), graph-supermanifold rank filter (Marcolli–Rej style), and local-RG / Weyl-anomaly $\beta^2$ norm with gravitational tadpole (Osborn / Komargodski–Schwimmer) — and finds each fails at a specific load-bearing obstruction pinned to published formalism. The §4 discussion is retained as motivation; future derivations must address the four obstructions identified.

A spin-dependent extension $Q_s \propto \alpha^{2(s+2)}$ recovers both observed exponents and predicts a spin-1 partner with Yukawa coupling strength $\alpha_Y \equiv Q_{s=1}/Q_{s=2} \approx 1.88\times 10^4$ relative to gravity. Existing Yukawa-fifth-force bounds (Geraci et al. 2008) exclude the prediction for ranges $\lambda \gtrsim 10\,\mu$m and confine any surviving spin-1 mediator to $m \gtrsim 20$–40 meV. The spin-dependent extension is offered as a conjecture, not a derivation; the volume-vanishing observation is offered as a structural fact.

The empirical relations reported here are also reviewed in the companion DAEDALUS review, where they are classified as Cabibbo-scenario regularities; in the broader Twistor Configuration Geometry (TCG) framework, an alternative geometric setting reproduces the same factorial/Fibonacci combinatorics via real configuration spaces as the FPA model, rather than via super-twistor Berezinian integration.

DOI

https://doi.org/10.5281/zenodo.19981300