One entropy formula unifies gravity's bulk and boundary behavior

Starting from Padmanabhan's entropy functional — originally a tool for deriving Einstein's equations from thermodynamic principles — the authors show it naturally produces the Brown-York tensor, the standard measure of quasi-local gravitational energy, on any surface, whether timelike or null. The approach also explains why the null version of this tensor is asymmetric, a structural quirk that had lacked a clean interpretation. Extended to scalar-tensor gravity, the same framework accounts for why the boundary stress tensor fails to be conserved when a scalar field couples non-minimally to curvature.