Radion Protection from Neutrino Sourcing in KK-Reduced Entangled Relativity

This work addresses a specific consistency problem in Entangled Relativity: non-relativistic neutrinos satisfy \(L_m = 0\) on shell while having a non-zero stress-energy trace \(T \neq 0\). In a purely four-dimensional scalar interpretation, this can source the ER dilaton at the level \(\delta\vartheta/\vartheta \sim \Omega_\nu \sim 10^{-3}\). If this scalar were directly identified with the electromagnetic fine-structure sector, it would imply a variation of \(\alpha_{\rm EM}\) in tension with atomic-clock and quasar absorption constraints. We show that the Geometric Relay Theory (GRT), defined as the Kaluza-Klein reduction of Entangled Relativity from five to four dimensions, resolves this problem by separating the relevant physical channels. The compact radion \(\sigma = g_{55}\), which controls the electromagnetic gauge coupling through \(\alpha_{\rm EM} \propto 1/\sigma\), is governed by an equation proportional to \(L_m^2\). Since the on-shell neutrino matter Lagrangian vanishes, \(L_{m,\nu}=0\), neutrinos do not directly source the radion. The direct protection is therefore exact: \((\delta\alpha/\alpha)_{\nu,\mathrm{direct}}=0\). The remaining indirect channel, in which neutrinos modify \(R_4\) through their stress-energy tensor and thereby affect the coefficient of the radion equation, is bounded by a self-consistency argument. The indirect effect is at most a fraction \(\Omega_\nu/\Omega_m\) of the already constrained dust-driven radion response. This gives \((\delta\alpha/\alpha)_{\nu,\mathrm{indirect}} < 5 \times 10^{-8}\) for the minimal neutrino mass sum, safely below current observational bounds. The robust observable signature is therefore not a neutrino-induced variation of the fine-structure constant, but a small correction to the expansion history: \[ \nu \rightarrow \delta H(z), \qquad \nu \nrightarrow \delta\alpha_{\rm EM}. \] The predicted correction is of order \(\delta H/H \sim \Omega_\nu \sim 10^{-3}\), potentially testable with DESI, Euclid, and future cosmological surveys. A Python reproduction script i Ens provided.