GRP Addendum — CMB Decoupling and Dark Matter Reinterpretation: Exact ER → GR Decoupling at Recombination

This addendum to the Geometric Relay Programme (GRP) addresses the CMB sector of the programme by showing why the Cosmic Microwave Background can remain effectively identical to the standard GR/ΛCDM description at recombination, while still allowing a geometric reinterpretation of dark matter at lower redshift and galactic scales. The central result is an exact ER → GR decoupling theorem at recombination. In Entangled Relativity, based on the action S = -|C| ∫ Lₘ²/R sqrt (-g) d⁴x, the matter Lagrangian Lₘ is physically active and cannot be replaced by a fluid shortcut without care. The addendum resolves the Lₘ ambiguity for radiation by distinguishing the microscopic electromagnetic Lagrangian from the Hamiltonian energy density. For thermal photons, the electromagnetic Lagrangian satisfies = 0 because of the E ↔ B symmetry of free radiation, while the energy density remains nonzero. Therefore, at z ≈ 1100, baryons and electrons contribute Lₘ ≈ T ≈ -rhob, while thermal photons contribute Lₘ = 0 and Tgamma = 0. This yields Lₘ = T and activates the Minazzoli decoupling condition: Entangled Relativity reduces exactly to General Relativity at recombination. This result is not treated as a weakness of the GRP but as a structural advantage. It explains why the CMB can measure Ωₘ in the standard way, without relay contamination. The Planck value of Ωₘ then feeds the GRP closure chain: Ωₘ → m₀ = sqrt (Ωₘ) H₀ → a₀ = m₀ c / 3 → MOND fixed point. The addendum further proposes that the quantity usually interpreted as cold dark matter, Ωₘ - Ωb ≈ 0. 266, corresponds in the GRP to the relay tension R* (z), scaling as (1+z) ³ in the CRT V3 mixed regime. Thus the same geometric relay field behaves as an effective CDM component at the CMB epoch and as a MOND-like non-linear relay at galactic scales. The work introduces no new free parameter. It does not claim a full replacement of the Boltzmann perturbation calculation. The next required step is a quantitative implementation of the relay perturbations in a modified Boltzmann framework such as hiCLASS or EFTCAMB, with the key failure criterion being whether R* reproduces the observed acoustic peak amplitude ratios, especially Dₗ3/Dₗ2, within observational tolerance. In short: the GRP does not modify the CMB at recombination; it explains why the CMB must look standard, then reinterprets the dark component measured by Planck as a geometric relay tension rather than a particle species.