> nothing special about special relativity

What's special about special relativity is that the spacetime is flat, i.e., there is no gravitation at all.

This work is about the sun's gravity, i.e. the (weakly) curved spacetime sourced by the central mass of our solar system, and so is about general relativity and not special relativity.

> change any of the underlying rules, it's going to become stable

The relevant thing here is chaos.

[italicizations below are mine]

§3.1 in the authors' preprint at <https://arxiv.org/abs/2303.05567> makes the point: "because the systems are chaotic and we use slightly different general relativistic corrections in each simulation, the simulations diverge quickly. This is effectively the same as varying the initial conditions of one planet by a tiny amount".

Quoting Wikipedia's Chaos Theory article: "Chaos theory is [...] focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions".

(The initial conditions in question are the orbital parameters--or more broadly the momentum--of Mercury. The equivalence is fairly obvious, but I think I prefer to look it at similar to your wording, "... change any of the underlying rules ...", although not just any rule, they change a specific rule (or closely related family of rules, more on that below).)

In the paper's §3.2 they identify the most prominent post-Newtonian contribution ("... the precession from GR (the gravito electric effect) .. follows closest behind Newtonian planet-planet interactions and is more than three orders of magnitude more influential than any additional effects"), and the paper in general focuses on the question of whether simulated planet-planet interactions' tendencies to eject Mercury from the solar system are dampened by GR's influence on the perihelion precession of Mercury. (Their answer: yes - Mercury sticks around for many billions of years even if the gravito-electric effect is small, but if it's zero, simulated Mercury is typically yeeted right out of simulated solar system in relatively few billions of years).

More specifically they look at the evolution of Mercury's orbital eccentricity and use it as a proxy for Mercury's ejection <https://www.urbandictionary.com/define.php?term=Yeet>. Mercury's orbit becomes more elliptical if over long periods of time its perihelion coincides with the perihelion of Jupiter. With practically any perihelion or ascending node precession of Mercury or Jupiter, Mercury's orbit remains roughly circular.

Even more specifically in their eqn. (2) they parameterize the general relativity corrections to the planets' perihelion precessions, and vary the parameter (eqn (3)). Eqns. (1) & (2), with the parameter \alpha set to unity, are straightforwardly extracted from an approximate solution to the Einstein Field Equations (of General Relativity; the EFEs), but going the other way from a modification of the parameter \alpha to a solution of the EFEs is something reviewers or a paper in reply may question, since it's not as straightforward. (One might compare how (Milgrom's) MOND can be seen as a parametrization in the same spirit eqns (1) & (2), "simples", but nobody has completely reverse-engineered the parametrized equation back into a set of relativistic field equations. The authors touch on this at the bottom of the first page's second column.)