URSF-R2: A Phenomenological Background-Relaxation Kernel for Late-Time Expansion Diagnostics

This standalone release presents URSF-R2, a phenomenological background-relaxation diagnostic branch designed to test late-time expansion shifts independently of growth-sector modifications. URSF-R2 modifies an effective background relaxation term, Ω_rel(z), while preserving the standard matter and radiation sectors. Negative relaxation amplitudes lower late-time H(z), increase BAO-style distance ratios D_M/r_d and D_H/r_d, and produce smooth positive luminosity-distance and distance-modulus shifts relative to ΛCDM. Interpreted through an effective equation of state, the negative-B branch gives w_eff(z) > -1, corresponding to a quintessence-like effective background. The primary smooth candidate is: R2-C1: B = -0.05, z_star = 0.7, q = 2 The comparison candidate is: R2-C2: B = -0.05, z_star = 1.0, q = 3 The stress-test branch is: R2-S1: B = -0.10, z_star = 0.7, q = 2 At z ≈ 0.51, R2-C1 gives ΔH = -0.646%, Δ(D_M/r_d) = +1.223%, Δ(D_H/r_d) = +0.650%, and w_eff = -0.97687. Around z ≈ 0.5, it gives Δμ ≈ +0.02656 mag. This release is part of the broader URSF research programme, but it is intentionally standalone. URSF-R1 tested a growth-sector suppression branch, while URSF-R2 tests a background-expansion / relaxation branch. R1 and R2 are not combined in this release. This release should be interpreted as a reproducible diagnostic manuscript and supporting package. It is not a DESI fit, supernova fit, CMB fit, derivation, completed dark-energy replacement, or confirmation of URSF. A standalone diagnostic manuscript introducing URSF-R2, a phenomenological background-relaxation kernel that lowers late-time H(z), increases BAO-style distance ratios, and behaves as an effective quintessence-like background with w_eff(z) > -1. This work was prepared by Malin Hess with assistance from AI tools. AI assistance was used for research support, mathematical checking, code/debugging guidance, result organization, manuscript drafting, editing, formatting, and report preparation. All conceptual claims, interpretive choices, final wording, numerical results, and responsibility for the contents of this work remain with the author. AI-generated suggestions were reviewed, edited, and accepted or rejected by the author. This disclosure is provided for transparency.