Looks super interesting! Will apply!

That’s the story! If you’re curious about generalised focusing, entropy, or horizon dynamics—let’s discuss!
Full paper: arxiv.org/abs/2509.00628 (8/8)

Next steps: extend these results to semiclassical & nonlinear regimes.
Excited to see how “entropic geometry” reshapes our view of spacetime & gravity. (7/8)

For higher-spin fields, the theorem holds conditionally. I propose a “higher-spin focusing condition” as a criterion for physical consistency. (E.g. 3D sl(3,R) higher-spin black holes satisfy it ✅) (6/8)

Key insight: Geometrically, light rays don’t always focus under positive energy in general gravity theories. But in entropic geometry—measuring separation via Wall entropy, not area—they do. Gravity remains attractive! (5/8)

This generalised expansion decreases monotonically under the null energy condition. Even more: it equals the change in Wall entropy density, which satisfies the 1st & 2nd laws. So it’s a natural higher-order/dynamical generalisation of BH/Wald entropy. (4/8)

We extend this to all diffeo-invariant gravities with bosonic matter.
On perturbed Killing horizons, the linearised equation of motion gives a generalised Raychaudhuri equation → a notion of “generalised expansion.” (3/8)

In GR, the focusing theorem says: with positive energy (NEC), light rays converge. This simple idea reveals the attractive nature of gravity, and underpins Hawking’s area theorem & Penrose’s singularity theorem. (2/8)

Thanks very much!

🥂🥂🥂