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No, here "entropic" is as in the entropic force that returns a stretched rubber band to its unstretched condition, which (as it tends to be scrunched a bit) is at a higher entropy.

https://en.wikipedia.org/wiki/Rubber_band_experiment

"The stretching of the rubber band is an isobaric expansion (A → B) that increases the energy but reduces the entropy"

[apologies for any reversed signs below, I think I caught them all]

In Verlinde' entropic gravity, there is a gravitational interaction that "unstretches" the connection between a pair of masses. When they are closer together they are at higher entropy than when they are further apart. There is a sort of tension that drags separated objects together. In Carney et al's approach there is a "pressure mediated by a microscopic system which is driven towards extremization of its free energy", which means that when objects are far apart there is a lower entropy condition than when they are closer together, and this entropy arises from a gas with a pressure which is lower when objects are closer together than when objects are further apart. Pressure is just the inverse of tension, so at a high enough level, in both entropic gravity theories, you just have a universal law -- comparable to Newton's -- where objects are driven (whether "pulled" or "pushed") together by an entropic force.

This entropic force is not fundamental - it arises from the statistical behaviour of quantum (or otherwise microscopic) degrees of freedom in a holographic setting (i.e., with more dimensions than 3+1). It's a very string-theory idea.

The approach is very hard to make it work unless the entropic force is strictly radial, and so it's hard to see how General Relativity (in the regime where it has been very well tested) can emerge.

The local theory part of the Carney et al paper (preprint <https://arxiv.org/abs/2502.17575>) is interesting in that it isn't obviously related to string theory / holographic entropic gravity. Instead masses induce a spin polarization near them which is a lower entropy state. Two masses with two polarized spin-clouds will attract each other as the system tries to thermalize to a higher-entropy state. With careful choices of parameters, they can generate any central force, and they explore a particular choice which corresponds to Newtons 1/r^2 mutual attraction.

The paper cannot deal with fast-moving masses at all: it's not just the relativstic regime (where speeds are significant fractions of c) but rather the masses must move more slowly than the thermalization. This is hugely restrictive.

Finally, comparing themselves to the traditional approach of quantizing perturbations (e.g. turning classical (General Relativity) gravitational waves into lots of spin-2 gravitons) the authors write:

  The gravitational interactions we observe at accessible
  length scales could in principle emerge in many ways from
  physics at the Planck scale ρ ∼ mPl/ℓ3 Pl ∼ 10104 J/cm3.
  Perhaps the simplest is that gravitational perturbations
  are quantized as gravitons, i.e., as another quantum field
  theory like the gauge bosons of the other fundamental
  forces in nature. This is a perfectly good effective quan-
  tum field theory; nothing in principle forces us to aban-
  don this picture until energies near the Planck scale.

They also say that while their starting point was being very different from the holographic picture:

  we find that the models have a range of free parameters,
  and in some parameter regimes become indistinguishable
  from standard virtual graviton exchange

Some of this will necessarily by driven by the need to be compatible with General Relativity in the weak field limit. They are not compatible with strong gravity in General Relativity at present.

So while the idea is kinda interesting, I think they are putting the cart before the horse in asking what their model says about things like the interaction between gravitation and entanglement. That's simply unmeasurable by experiment right now whereas the very-well-understood relativistic precession of Mercury's perihelion is completely out of scope for this initial paper.