First Boltzmann-Level Einstein-Gap Sector in CLASS: Living Relay Field, Memory-Renormalized Closure, and the E_G(z) Discriminant — v2

Version 2 of the first Boltzmann-level Einstein-gap relay sector in CLASS. This release supersedes the previous constant-E_G suppression interpretation. The activation of the Einstein-gap relay source is now derived from a baryonic Sheth–Tormen structure-formation gate. The living relay field dynamically reaches the tracked branch across the full physical gate family. The primary observable prediction is a late-emerging, redshift-dependent suppression of E_G(z), robustly reaching E_G^{GRP}/E_G^{GR}\simeq0.93-0.94 at z\to0, while the slope at z\simeq0.2-0.6 discriminates the fraction-versus-amplitude normalisation of the source. The archive includes the v2 manuscript, Python scripts, JSON numerical outputs, figures, and the CLASS implementation patch. This release provides the first operational Boltzmann-level implementation of an Einstein-gap relay sector in the CLASS cosmological code. The package introduces a living relay perturbation field χ, its conjugate velocity πχ, and a memory register M as three dynamical scalar variables per Fourier mode. The source term is derived from the trace/Einstein separation of the Entangled Relativity perturbation equations and takes the asymmetric form: S = −(2/3)(k/𝓗)^2(Φ − ηχ). The release also includes the memory-renormalized closure bridge: η = (2 + F_q)/4, which connects the pre-memory linear source to the memory-tracked observable branch. With F_q = 1.0274747019, the bridge gives η = 0.7569 and A_tr ≈ −1.3457, leading to the effective clustering diagnostic μ ≈ 1.057. The CLASS implementation yields: σ₈^GRP / σ₈^ΛCDM = 1.003, and predicts the primary observational discriminant: E_G^GRP / E_G^GR ≈ 0.947, corresponding to a 5.3% suppression of the lensing-to-clustering statistic E_G relative to GR/ΛCDM. The package contains: - the final PDF and LaTeX manuscript; - a CLASS patch for the passive B2 field; - a CLASS patch for the B2 field plus effective v0μ projection; - standalone Python validators for B1.2 and B2; - a complete reproduction script generating the main analytical checks, convergence plots, crossover diagrams, activation sensitivity scans, σ₈ predictions, E_G predictions, and observational confrontation figures; - a README with reproduction instructions. This is a first-generation technical release. It does not claim a fully self-consistent final GRP cosmology. The observable response is currently represented by the memory-tracked effective μ projection, while the direct-slip route and the activation timescale remain open frontiers. No new fitted parameter is introduced in the CLASS implementation; all numerical coefficients are inherited from the Entangled Relativity / v10.1 memory-closure framework