Relaxation Timescale and Scale-Dependent Structure Formation in the Verlinde–Bose–Einstein Framework

In a companion paper (Paper III), we confronted the Verlinde-Bose-Einstein (VBE) framework with cosmological observables and found that the Bose-Einstein enhancement, regulated by the cosmological external field effect, produces a perturbation growth factor only ~3x above the baryon-only prediction -- insufficient to match the ~6.4x enhancement provided by cold dark matter. The self-consistent effective matter density Omega_m,eff = 0.113 yields sigma_8 = 0.338, far below the Planck value of 0.811. Here we resolve this shortcoming by deriving a finite relaxation timescale tau = sqrt(L/a_0) for the entanglement strain response, arising from the BEC healing dynamics of the de Sitter medium. This timescale introduces a scale-dependent low-pass filter epsilon(k) = 1/sqrt(1 + omega^2 tau^2) that interpolates between baryon tracking at large scales and CDM-like behavior at small scales. The resulting transfer function T_VBE(k) = epsilon(k) T_b(k) + [1 - epsilon(k)] T_c(k) yields sigma_8 = 0.830 with zero free parameters, or sigma_8 = 0.810-0.811 when the geometric coefficient alpha_tau = 1/sqrt(pi) or 1/sqrt(3) is included. We show that the transient correction to the filter amplitude is exactly zero, that post-recombination relaxation is negligible by self-consistency, and that CMB phase shifts from the dark component lag are sub-percent effects. The BAO wiggle amplitude is predicted to be 0.51x the LCDM value, testable with DESI and Euclid. Definitive tests will require a full modified Boltzmann code treatment. This is Paper IV in the VBE series. Paper I derived the Bose-Einstein interpolation function and acceleration scale from the radial acceleration relation. Paper II extended the framework to galaxy clusters via the NFW potential ratio. Paper III confronted VBE with cosmological observables, identifying the structure formation gap that this paper resolves. This paper is a supplement to: https://doi.org/10.5281/zenodo.18881878 (Morris 2026) With references to: https://doi.org/10.21468/SciPostPhys.2.3.016 (Verlinde 2017) https://doi.org/10.1086/305424 (Eisenstein & Hu 1998)