Mathematical Structure of Baryonic Temperature-Dependent Gravity: Emergent Curvature from Long-Range Dependence and Relation-Dependent Hyperbolic Geometry

The Baryonic Temperature-Dependent Gravity (BTDG) framework introduced longrangedependent (LRD) state spaces and emergent curvature as functions of thermodynamicand radiative variables—prior to the appearance of relation-dependent hyperbolic heterogeneousgraph transformers (HypHGT). HypHGT learns multiple hyperbolic geometries withrelation-specific curvatures directly from data.Here, I (i) establish a concise mathematical correspondence between BTDG and HypHGT,and (ii) provide a minimal derivational bridge from power-law memory (LRD) toeffective negative curvature underlying both approaches. I also document the chronologicalprecedence of the analytic BTDG formulation with respect to the subsequent numericalrealization in HypHGT.