GRP-CLASS: an .ini-controlled CLASS prototype for a late-activated geometric relay field

This updated version documents the GRP-CLASS v2. 1 Q00-QS addendum to the terminal CLASS implementation of a Geometric Relay Programme (GRP) relay field. The original GRP-CLASS prototype inserted the relay variables chi, pi = dchi/du, and M directly into the CLASS perturbation vector, with activation controlled through. ini parameters. The relay can be disabled to recover the LambdaCDM reference, or enabled to generate controlled late-time signatures in the Newtonian-gauge potentials and in external growth/lensing diagnostics. The main change in v2. 1 is the replacement of the previous phenomenological Poisson-relaxation channel, phi'GRP = phi'CLASS + epsilonP T (a) H CQ chi, by a quasi-static Q00 channel derived from the perturbative GRP closure: delta Q⁰₀ |QS = + (k²/a²) chi. Operationally, the v2. 1 implementation applies the effective Newtonian-gauge correction Psi -> Psi + epsilonₛlip T (a) chi - 1/2 epsilonQ00 T (a) chi. For the benchmark configuration, grpᵣelay = yes grpₑpsₛlip = 1e-2 grpₑpsₚoisson = 0 grpₑpsq00 = 1e-2 grpₐₛtar = 0. 6 grpₙₚower = 4 grpCQ = 1. 0 the net tested correction is PsiGRP = PsiGR + 5e-3 T (a) chi. The v2. 1 implementation compiles, runs, and preserves the controlled late-time diagnostic pattern previously found in the terminal GRP-CLASS prototype. At z = 0, over the conservative range 10^-4 <= k <= 1 Mpc^-1, the corrected external diagnostics give approximately: muₑff - 1 ≈ -0. 00623, Sigmaₑff - 1 ≈ -0. 00034, EG proxy - 1 ≈ +0. 00593. At z = 1, the signal is attenuated: muₑff - 1 ≈ -0. 00329, Sigmaₑff - 1 ≈ -0. 00112, EG proxy - 1 ≈ +0. 00217. At z = 10, the signal is essentially switched off, with median deviations of order 10^-6. This version therefore improves the theoretical status of the prototype: the earlier Poisson-relaxation placeholder is replaced by a derived quasi-static Q00 channel while preserving numerical stability and the late-time growth/lensing split. This release remains a technical prototype, not a final physical closure of the GRP/ER perturbation equations. The Q00-QS channel is quasi-static and effective. A complete non-QS Q⁰₀ derivation, gauge-consistency checks, native CLASS outputs for chi, pi and M, and full Cₗ / lensing-spectrum tests remain future work. The archive includes Markdown and LaTeX versions of the v2. 1 addendum, together with the GRP-CLASS v2. 1 Q00-QS outputs bundle containing patched source files,. ini configurations, diagnostic CSV files, plots, and validation summaries. The technical note is released under CC-BY-4. 0. The patched CLASS-related files are provided for reproducibility and should be used consistently with the original CLASS license.