Electroweak Boundary Conditions in Twistor Configuration Geometry

Abstract

Twistor Configuration Geometry (TCG), in its FPA realization, is a dimensionless framework: chamber counts, matching counts, dimension/area ratios, and chamber-weighted Fubini–Study sums all live at the dimensionless level. The framework’s original postulate P5 proposed a contact-scale identification $\mu_{\rm contact} \simeq M_Z$, intended to anchor the GeV scale. We show this is not derivable: any candidate Reeb-spectrum construction requires an external odd-dimensional auxiliary, an external contact-form normalization, and an external unit conversion to GeV — none of which the FPA stratification supplies. We therefore close P5 as a derivation target and introduce a dimensionless replacement,

$P5': \quad g_{2,W}^2 = \frac{4}{3\pi}, \qquad \text{equivalently} \qquad \frac{M_W}{v} = \frac{1}{\sqrt{3\pi}},$

where $g_{2,W} := 2 M_W/v$ is the effective weak coupling defined by the tree-level pole relation. The right-hand side is the FPA-native line-deformation density ratio $r_3 / [\dim_{\mathbb{C}}\mathbb{CP}^3 \cdot \mathrm{Area}(\mathbb{CP}^1)] = 4/(3\pi)$, with $r_3 = 4$ from the rank rule (the line-deformation cohomology of $\mathcal{O}(1)^{\oplus 2}$ on $\mathbb{CP}^3$), $\dim_{\mathbb{C}}\mathbb{CP}^3 = 3$, and $\mathrm{Area}(\mathbb{CP}^1) = \pi$. Numerically, $M_W^{\rm TCG} = v/\sqrt{3\pi} \approx 80.20$ GeV, in 0.21% agreement with the PDG world average $M_W^{\rm PDG} \approx 80.37$ GeV.

P5’ preserves the framework’s dimensionless character while connecting to a verified empirical relation. We classify it explicitly as a phenomenological boundary condition, not yet a theorem: $3\pi/4$ is a derivation target suggested by the dim/area ratio, not the result of a completed Chern–Weil or Yang–Mills normalization calculation. A candidate gauge-kinetic derivation is sketched on the line-deformation bundle $\mathcal{E} \to G(2,4)$ with fiber $H^0(L, N_{L/\mathbb{CP}^3})$; we identify four pieces a derivation must supply, including a canonical $U(2) \to SU(2)$ reduction (the determinant $\det(\mathcal{O}(1)^{\oplus 2}) \cong \mathcal{O}(2)$ is non-trivial on $\mathbb{CP}^1$), a canonical metric on a canonical cycle, a trace normalization, and a physical argument linking the resulting density ratio to $g_{2,W}^2$ rather than to an unrelated geometric coupling.

This note supersedes the original P5 formulation in the TCG construction paper and the contact-scale constraint in the Predictions/No-Go paper. The architecture of progress: old P5 (dimensionful contact scale) is closed negatively because FPA has no internal dimensionful scale; new P5’ (dimensionless weak-sector boundary condition) is empirically supported at 0.21% but not yet theorem-level. The framework’s open question is now whether a gauge-kinetic computation on the line-deformation bundle closes the gap from postulate to theorem, not where to find an external GeV scale.

DOI

https://doi.org/10.5281/zenodo.20075926