DAEDALUS: A Review of Nine Numerical Observations on Fundamental Constants and a Twistor Configuration

Abstract

We present a consolidated review of nine numerical relations among fundamental constants and the Twistor Configuration Geometry (TCG) framework that captures their combinatorial structure, both developed by the author across a sequence of recent papers. The relations are: $\Lambda\,\ell_{\rm Pl}^2 = (\alpha^4/4\pi)(m_e/m_{\rm Pl})^5$ (1.9%), $\alpha_G \equiv G m_e^2/(\hbar c) = \alpha^8 y_e^5$ (0.024%), $1/\alpha \approx \pi+\pi^2+4\pi^3$ (2.2 ppm), $\alpha_s(M_Z)^{-1} \approx 8.456$ (0.22%), $\sin^2\theta_W \approx 3/(4\pi)$ at low $Q^2$ (0.05%), $y_e \approx (1-1/(2\pi))\,e^{-4\pi}$ (0.09%), $m_\mu/m_e$ and $m_\tau/m_e$ via golden-ratio scaling on $L_\ell = \ln(1/y_\ell)$ (0.24%, 0.54%), $m_\pi/m_e \approx 2(\pi+\pi^2+4\pi^3)$ (0.4%), and $M_W/v \approx 1/\sqrt{3\pi}$ (0.21%) where $v = (\sqrt{2}\, G_F)^{-1/2}$ is the Higgs vacuum expectation value.

Twistor Configuration Geometry (TCG) is presented as a candidate framework class — a research program rather than a single proposal — that reproduces the combinatorial structure of these relations. The specific realization studied here is the FPA model (Framed Permuto–Associahedral), built from a stratification $\bigsqcup_{n=1}^{3} \mathbb{CP}^n \times \overline{\operatorname{Conf}}^{\rm lab}_{r_n}(I)$ over the Witten–Penrose twistor strata, with chamberwise Fulton–MacPherson/Axelrod–Singer compactifications. The framework derives four combinatorial quantities (rank $r_n = 2n-2$, chamber count $r_n!$, matching count $F_{r_n+1}$, and chamber-weighted Fubini–Study sum $\pi + \pi^2 + 4\pi^3$); that nine independent few-percent-or-better empirical relations (eight sub-percent or sharper, with the $\Lambda$ relation at 1.9%) admit coordinated reproduction within these four quantities is, on its face, a striking conjunction whose status is left open.

Seven structural postulates (P0–P6) remain, supplemented in applications by additional phenomenological identifications. Several constructions previously considered have been ruled out and are recorded as negative results, including the even-dimensional obstruction to a canonical Reeb structure on the framework, the over-specification $\chi(\text{tree}) = 1$ in tree-Euler boundary corrections, an unspecified transformed scaling rule needed to translate a Perron eigenvalue of the Fibonacci transfer matrix into a charged-lepton mass ratio, and the absence of a characteristic scale in $\mathbb{CP}^3$ alone for $\alpha_s$. The nine relations satisfy a Cabibbo-scenario classification: empirical regularities surviving as inputs to phenomenology rather than consequences of derivation, in the same status as the Cabibbo angle (1963 to present) and the Koide formula (1981 to present). Five concrete upgrade paths and four falsifiable experimental consequences are stated.

DOI

https://doi.org/10.5281/zenodo.19984246