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 Lₘ²/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 Cₗ^kappa kappa is identical to Lambda-CDM at the linear level, because the relay tension R* = Omegaₘ - Omegab 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: Mₗens = Mbar x nu (gbar/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 gₘunu. This resolves the standard lensing problem of non-covariant MOND-like theories. At the cluster scale, the total gravitational mass satisfies: Mₜotal / Mbar = Omegaₘ / Omegab ≈ 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 Omegaₘ - Omegab. 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.