Assuming proton decay is false, what's the ultimate fate of a neutron star or white dwarf? A still unproven theory that protons can decay, and have a halflife of $10^{30}$ years or so, meaning eventually all matter will dissolve because their constituent protons and therefore neutrons will decay. But this is yet to be proven. So assuming this is false, what would be the ultimate fate of a neutron star or white dwarf? What would become of it in $10^{50}$, $10^{100}$ years?
This is a classic question in physical eschatology, seeing what happens if we extrapolate current understanding of astrophysics forward. The classic papers are (Dyson 1979) and (Adams &amp; Laughlin 1997).
Obviously, over very long timescales white dwarfs cool down, crystallize. and become &quot;black dwarfs&quot;. This is fairly well-established from observation and modelling, although the final stages have not been studied much.
If weakly interacting dark matter can be captured, dense objects would acquire internal halos: in this case, if the dark matter is a mix of particles and antiparticles there would be some annihilation, heating the object for a long time. If it is non-annihilating in principle it could build up until the object imploded into a neutron star or black hole. This is highly dependent on the dark matter model, so this should be regarded as conjectural.
However, it is fairly well established that galaxies dissolve due to gravitational interactions over long timescales, and this will dump such objects into the central black hole before they plausibly could undergo collapse, or eject them into intergalactic space where they would no longer acquire dark matter.
The fate of intergalactic black dwarfs and neutron stars without proton decay depends on what other modes of decay and change are possible. In white dwarfs pycnonuclear fusion would continue until all fusible elements had fused. Dyson estimated the timescale until everything is iron to $10^{1500}$ years, although there are environmental effects in white dwarfs that likely speed things up. This can actually make heavier white dwarfs (above 1.2 solar masses) collapse into supernovas on a timescale of $10^{1100}$ years (Caplan 2020).
Dyson noted that over timescales of $10^{65}$ years matter behaves like a quantum fluid due to tunneling. But this doesn't change structure of remaining objects much. A more important issue may be tunneling into black hole states where a small part of the object tunnels together to form a small black hole that evaporates. Adams and Laughlin estimate timescales of $10^{45}$ years for neutron stars and $10^{336}$ years for white dwarfs to evaporate this way.
Even if this does not happen, there is an argument to be made that thermodynamic fluctuations eventually dissolve bound objects since this minimizes the Gibbs free energy $E-TS$: at a finite temperature (which is the standard assumption for accelerating expansion in $\Lambda$CDM) if there is enough space the entropy $S$ can be maximized by separating particles despite some binding energy: tunnelling will eventually dissolve everything. This is similar to the discussion about Herzfeld's paradox of the spontaneous ionization of hydrogen atoms. The mere presence of other matter &quot;outside the lab&quot; normally stabilizes bound systems but in the very far future isolated systems become destabilized. This assumes that (1) temperatures will stay finite (i.e. our understanding of horizon radiation and continued accelerated expansion are right), (2) there are no other limits on dissociation, (3) the Gibbs energy argument is valid in this context. All three can be debated.
Overall, the trend seems to be that entropy maximization will tend to dissolve objects into isolated particles while gravity either just keeps them together or causes implosion into black holes through some pathway, followed by evaporation.