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

This paper should be read as a three-scale lensing closure proposal within the Geometric Relay Programme. It explores how the relay response may connect cosmic shear, galactic lensing and cluster convergence. It is complementary to the later GRP mu-Sigma CLASS implementation, which provides the perturbative dynamical-lensing split. The document should not be read as a final proof that the full cluster-lensing problem is solved.

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 structured three-scale account of lensing phenomena, in which part of the effect usually attributed to dark matter is reinterpreted as an effective geometric relay response.

At the cosmological scale, the weak-lensing sector is expected to remain close to Lambda-CDM in the background and effective linear limit, because the relay tension R* = Omega_m - Omega_b reproduces the same effective matter-density scale. The later GRP mu-Sigma CLASS implementation refines this statement by showing that the relay need not be exactly equivalent to a CDM fluid: it may instead generate a mild dynamical-lensing split.

At the galactic scale, the lensing mass is modeled as:

M_lens = M_bar x nu(g_bar/a0),

where nu is the stable fixed-point response of the dual relay. Since Entangled Relativity is a covariant metric framework, photon deflection follows the effective metric structure rather than a purely non-relativistic MOND-like prescription.

At the cluster scale, the effective gravitational mass ratio is organized around:

M_total / M_bar approximately Omega_m / Omega_b approximately 6.4,

which matches the usual Lambda-CDM cluster mass-ratio scale once the relay tension R* is identified with the cold geometric component Omega_m - Omega_b. The non-linear regime remains the discriminating sector: the GRP predicts possible S8 compression, core-like profiles instead of NFW-like cusps, and small offsets in dissociative clusters linked to relay activation and saturation mechanisms.

The paper therefore frames gravitational lensing as a structured A/B-level sector of the GRP. The background and effective linear regimes can reproduce Lambda-CDM-like lensing scales, while the non-linear and cluster regimes remain open but testable.

This document is part of the broader Geometric Relay Programme and is intended as a technical internal synthesis of the lensing sector.