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This research aims to present the first operational Boltzmann-level Einstein-gap relay sector in the CLASS code.

June 14, 2026Open Access

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

Key Points

This research aims to present the first operational Boltzmann-level Einstein-gap relay sector in the CLASS code.
Introduced a living relay perturbation field and derived a source term from the perturbation equations.
Developed a memory-renormalized closure bridge to connect linear sources to observable branches.
Implemented the CLASS patches for various field projections and provided validation scripts.
Predicted E_G^GRP/E_G^GR ≈ 0.947, indicating a 5.3% suppression of lensing-to-clustering statistics relative to GR/ΛCDM.
Achieved σ₈^GRP / σ₈^ΛCDM = 1.003, implying compatibility with existing models.
Demonstrated a late-emerging, redshift-dependent suppression in E_G(z) reaching E_G^{GRP}/E_G^{GR} ≈ 0.93-0.94 at z→0.

Abstract

Version 2 of the first Boltzmann-level Einstein-gap relay sector in CLASS. This release supersedes the previous constant-EG 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 EG (z) , robustly reaching EG^GRP/EG^GR0. 93-0. 94 at z0, while the slope at z0. 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 + Fq) /4, which connects the pre-memory linear source to the memory-tracked observable branch. With Fq = 1. 0274747019, the bridge gives η = 0. 7569 and Aₜr ≈ −1. 3457, leading to the effective clustering diagnostic μ ≈ 1. 057. The CLASS implementation yields: σ₈GRP / σ₈^ΛCDM = 1. 003, and predicts the primary observational discriminant: EGGRP / EGGR ≈ 0. 947, corresponding to a 5. 3% suppression of the lensing-to-clustering statistic EG 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, EG 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

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