A Late-Time Growth-Only Modification of Gravity with CMB-Lensing Consistency

This work presents a numerically robust, late-time, growth-only modification of gravityimplemented within the hi_class Boltzmann solver. The modification is introduced at a single, well-defined choke point: the Newtonian-gaugeEinstein–Poisson closure for the scalar potential ψ. The background expansion, radiationhierarchy, neutrino sector, and photon geodesics are left completely unmodified. As aresult, all early-Universe observables and Cosmic Microwave Background (CMB) constraintsare preserved by construction. A smooth and numerically stable activation function μ(a) = 1 + α g(a) is employed, withg(a) = 1 / (1 + (a_t / a)^p), ensuring exact recovery of General Relativity at early timesand controlled deviations only at late times. Extensive audit tests demonstrate that themodification affects exclusively the matter growth channel. The implementation passes a stringent CMB lensing null test, yieldingΔC_L^{φφ} = 0 relative to ΛCDM within numerical precision over the full multipole range8 ≤ L ≤ 400. Late-time effects are quantified through the matter power spectrum P(k),redshift-space distortions, and fσ₈(z), showing smooth, sub-percent-level deviations forbenchmark parameter values. The model makes a clear, falsifiable prediction for the bias-independent E_G(z) statistic:because lensing remains unchanged while growth is modified, E_G(z) is predicted to deviatesystematically from its ΛCDM value at low redshift. This provides a direct observationaltarget for upcoming surveys such as Euclid. All numerical results are fully reproducible, and the archive includes the final,forensics-verified implementation together with the configuration files required toreproduce the reported tests.