Weak-field frame dragging: One-parameter law for galaxy–cluster dynamics

A weak-field gravitomagnetic (GEM) enhancement of frame dragging is formulated as an \emph{additive} acceleration with a galaxy-wide factor $G_{\rm rot}\!\propto\!M^{-1}$ and a late-time, environment-gated term that activates on group/cluster scales. The real-space law follows from a Proca-type completion with Jeans/mean-field closure and Yukawa screening; a high-$g_N$ core filter keeps line-of-sight lensing near unity convergence. On galaxy scales the framework recovers RAR/BTFR and yields an inverse-mass trend that remains valid across $\sim$5 decades in baryonic mass, as indicated by an internal disk sample augmented by twelve external bin averages for central galaxies. Simple cluster estimates (A1795, A2204) indicate that a susceptibility amplitude $\eta=\mathcal O(1)$ reconciles hydrostatic and weak-lensing masses at $r_{500}$ within 10--15\%. A linear-regime embedding via $\mu(k,a)$, $\gamma(k,a)$, and $\Sigma(k,a)$ leaves the primary CMB/BAO intact and is compatible with current constraints on $f\sigma_8(z)$ and $C_L^{\phi\phi}$ under conservative choices. The framework is falsifiable through the universal scaling $M^{-1}$, RAR slopes in $[1/2,1]$, lensing--dynamics consistency in clusters, and late-time large-scale probes (curl/rotation shear, pairwise kSZ, environment-split GGL). The results are consistent with a $\Lambda$ plus weak-field GEM \emph{extension} acting at late times and in weak fields, rather than a replacement of $\Lambda$CDM. Independent 2025 results reinforce this picture: a non-parametric weak-lensing analysis of CLASH clusters recovers nearly flat $V_c(r)$ and shows that clusters can lie on the same BTFR/RAR as galaxies when a small positive baryonic term is added \cite{Lelli2025}. Complementary constraints at the galaxy-scale and joint mass-calibrations further corroborate the universality of $g_{\rm obs}$--$g_{\rm bar}$ mapping \cite{Ma2025MOND,Zhang2022GGL}.