The Relation Survives the Jump; the Scale Does Not: The Radial Acceleration Relation in Galaxy Clusters

On the CLASH sample's reading, galaxy clusters obey a radial acceleration relation. In that sample (Tian et al. 2020), lensing-measured total acceleration tracks X-ray baryonic acceleration along a relation parallel to the galactic RAR, with intrinsic scatter of 15% — and a characteristic scale g‡ = (2.0 ± 0.1)×10⁻⁹ m/s², which is 16.7 times the galactic scale g† = (1.20 ± 0.02)×10⁻¹⁰ m/s². The separation is 18.8σ on the fits' own errors. The cluster mass–velocity-dispersion relation repeats the pattern in a second observable: M_bar ∝ σ⁴.¹ ± ⁰.¹ over the HIFLUGCS clusters — the slope-4 form of the baryonic Tully–Fisher/Faber–Jackson family, intact, at a characteristic acceleration an order of magnitude larger. So the one-scale demand — the demand this program has been enforcing on table after table — fails at cluster scale in a specific, surgical way: the functional form of the acceleration relation is scale-free across three decades of system mass, and its constant is not. In the deep low-acceleration regime, clusters at a given baryonic acceleration pull √16.7 ≈ 4.1 times harder than the galactic scale permits; the classical statement of the same wound is the residual missing mass of roughly a factor of two that modified-dynamics analyses have always carried in cluster cores (Sanders 1999, 2003; Pointecouteau & Silk 2005), where accelerations are transitional rather than deep. Verdict: one scale does not pay beyond galaxies. A universal acceleration constant of nature is dead at cluster scale at 18.8σ. Two things this verdict is not: it is not a wound to ΛCDM, inside which a tight cluster acceleration relation is an emergent baryon–halo coupling (the CLASH authors themselves note consistency with a ΛCDM semi-analytic model), and it is not uncontested — a counter-analysis (Chan & Del Popolo 2020) disputes the cluster relation's existence, and the cluster baryon census itself runs short of the cosmic fraction, worse at low mass, which is a second defect feeding the first. All of it is printed. What this program takes home is narrower and harder than any of the camps' headlines: form, yes; constant, no.