Gravitational Lensing in the Geometric Relay Programme: Three-Scale Closure from Cosmic Shear to Cluster Convergence

Description This document develops the gravitational-lensing pillar of the Geometric Relay Programme (GRP), built on the Entangled Relativity action L_m^2/R and its Kaluza–Klein reduced scalar relay structure. The central claim is that gravitational lensing does not constitute a fatal objection to the GRP. Instead, the programme provides a three-scale account of lensing phenomena without invoking dark matter particles and without introducing new free parameters beyond those already fixed in the framework. At the cosmological scale, the weak-lensing convergence spectrum C_l^{kappa kappa} is identical to Lambda-CDM at the linear level, because the relay tension R* = Omega_m - Omega_b scales as (1+z)^3 and reproduces the same H(z), P(k), D(z), and linear growth history. The gravitational slip Phi = Psi is preserved at the linear level by the Minazzoli decoupling mechanism. At the galactic scale, the lensing mass is given by: M_lens = M_bar x nu(g_bar/a0), where nu is the exact stable fixed point of the dual relay. Since Entangled Relativity is a covariant metric theory, photon deflection follows the modified metric g_munu. This resolves the standard lensing problem of non-covariant MOND-like theories. At the cluster scale, the total gravitational mass satisfies: M_total / M_bar = Omega_m / Omega_b ≈ 6.4, which is algebraically identical to the Lambda-CDM cluster mass ratio once the relay tension R* is identified with the cold geometric component Omega_m - Omega_b. The non-linear regime remains discriminating: the GRP predicts S8 compression, core profiles instead of NFW cusps, and small offsets in dissociative clusters linked to the relay activation mechanism. The paper therefore upgrades the gravitational-lensing pillar of the GRP from an open challenge to a structured A/B-level sector: linear lensing is equivalent to Lambda-CDM under the relay-tension identification, while the non-linear regime provides testable departures. This document is part of the broader Geometric Relay Programme and is intended as a technical internal synthesis of the lensing sector.