Triangulation Test for Thermodynamic-Lensing Coupling: A Triangulation Test for Possible Thermodynamic Contributions to Gravitational Lensing in Galaxy Clusters

We present a triangulation methodology designed to test whether residual discrepancies between gravitational lensing and hydrostatic mass estimates in galaxy clusters trace the thermodynamic structure of the intracluster medium (ICM). The method combines three independent observables: (1) projected mass from weak and strong gravitational lensing, (2) dynamical mass inferred under the assumption of hydrostatic equilibrium (HSE) from X-ray temperature and density profiles, and (3) entropy gradients derived from the same X-ray data. We define a radial mass residual, R(r)=Mlens(r)−MHSE(r)R(r) = M_{\rm lens}(r) - M_{\rm HSE}(r)R(r)=Mlens​(r)−MHSE​(r), and evaluate its correlation with the magnitude of the entropy gradient ∣∇K(r)∣|\nabla K(r)|∣∇K(r)∣.This approach functions as a null test of the standard assumption that gravitational lensing responds only to the mass distribution. We outline order-of-magnitude consistency requirements that any phenomenological coupling between lensing convergence and thermodynamic structure would need to satisfy in order to remain compatible with existing precision tests of gravity and cosmology. The triangulation method is applied to two pilot systems: the merging Bullet Cluster and the relaxed cool-core cluster Abell 1835. In both cases, we detect no statistically significant correlation between mass residuals and entropy gradients, consistent with current data sensitivity and with conventional explanations based on non-thermal pressure support and dynamical state.The methodology is fully general and can be applied to larger cluster samples and future high-precision lensing and X-ray data sets. As a speculative extension, we briefly discuss a possible secondary observable involving line-of-sight redshift perturbations associated with strong thermodynamic gradients, emphasizing that this is not required for the triangulation test itself. Future surveys with improved statistical power will determine whether such couplings can be detected or tightly constrained.