A Cosmological-Background Effective Gravitational Kernel: Unifying disk galaxies, galaxy groups, galaxy clusters, and gravitational lensing under a single effective law

The paper presents a q-deformed kernel K_eff that describes the ratio of observed to baryonic gravitational acceleration across four independent observational regimes — 171 SPARC disk galaxies, 23 Sun+2009 galaxy groups, 13 X-COP hydrostatic clusters, and 23 CLASH lensing clusters — using two globally calibrated parameters and a single exponent-law coefficient A = 0.4068 fitted on 59 systems. The central results are: — The Baryonic Tully-Fisher Relation emerges as a structural theorem (n = 1/2 derived from flat rotation curves, not fitted) — The transition exponent follows n(ξ) = 1/2 + 0.4068 ξ, where ξ = g₅₀₀/g†(z) spans disk galaxies to massive clusters with r = 0.97 across four decades in mass — The cosmological scale g†(z) = cH(z)/(2π) is empirically detected in CLASH at 4.1σ (permutation test), LOO-CLASH predictive RMS = 0.006 dex — A first radial profile test against X-COP mass data (39 points, 13 clusters × 3 radii) yields RMS = 0.040 dex — 6.3× better than MOND on the full set, 7.8× on the genuinely predictive subset — Three falsified intermediate hypotheses are documented (Ramanujan mock theta functions, n = n(β), polytropic closure) The program is explicitly phenomenological — no Lagrangian, no covariant theory — and the paper is candid about what remains semi-predictive versus fully predictive. The validation pipeline and data files are publicly available at: https://github.com/maia-karlosmarden/rgd-kernel.