The Geometric Relay as Dark Matter: Exact CMB Decoupling, Identical P(k), and the Reinterpretation of Ωcdm

This working paper presents a consolidated interpretation of the geometric relay σ as an effective dark matter component within the Geometric Relay Programme (GRP), built on the Entangled Relativity framework of O. Minazzoli. The central proposal is that the cold dark matter density inferred by ΛCDM, usually written as Ωcdm = Ωm − Ωb, can be reinterpreted as a geometric relay tension rather than as a particle substance. In this formulation, the relay tension R* scales as (1+z)^3 in the CRT V3 mixed regime and is therefore algebraically indistinguishable from pressureless cold dark matter at the background level. This gives the same Friedmann evolution H(z) as ΛCDM when Ωrelay = Ωm − Ωb. The paper combines three main results. First, it uses the CMB decoupling theorem: at recombination, the microscopic electromagnetic Lagrangian of thermal radiation satisfies <L_EM> = 0 by E↔B symmetry, while baryons and electrons satisfy L_m ≈ T. Therefore Entangled Relativity reduces exactly to General Relativity at z ≈ 1100, allowing the CMB to remain effectively standard. Second, it identifies Ωm − Ωb as the relay tension required by the GRP closure chain Ωm → m_0 = sqrt(Ωm)H_0 → a_0 = m_0c/3 → MOND fixed point. Third, it introduces a two-scale interpretation of the radion σ = g_55: a slow cosmological mode m_0 controlling the background and the Milgrom constant, and a fast Kaluza-Klein stabilisation mode m_radion controlling perturbations. If m_radion >> H_0, relay perturbations are frozen and can act as a non-oscillating, CDM-like gravitational component. At the background and growth-equation level, the model reproduces the same H(z), D(z), BAO scale