Replica-symmetry-breaking threshold at the satisfiability threshold

Establish that the replica-symmetry-breaking transition for H_n(π,k,m) occurs at density d_k/k under the stated reducibility or symmetry assumptions, with asymptotic independence of two uniformly sampled variable spins below the threshold and failure of that independence above the threshold when constant solutions exist.

Background

The paper proves the clustering transition and satisfiability threshold for broad classes of random UE-SAT models. It also notes that, for random linear equations over finite fields, the condensation or replica-symmetry-breaking threshold coincides with the satisfiability threshold.

For general spin-set sizes and distributions, the authors were unable to prove replica symmetry up to the satisfiability threshold and therefore formulate an explicit conjecture describing the transition at d_k/k.

References

We expect that the phase transition of RSB occurs when $m/n$ is around $d_k/k$ when $supp(\pi)$ satisfies the same conditions as in Theorem~\ref{thm:clustering}. Indeed, it follows easily from our results in this paper (see Remark~\ref{r:RSB} below) when $r$ is the product of distinct prime numbers. However, for general $r$ we did not manage to prove that the replica symmetry holds until reaching the satisfiability threshold, and thus we leave it as a conjecture; see Conjecture~\ref{conj:RSB}.

— The satisfiability threshold and solution space of random uniquely extendable constraint satisfaction problems  (2512.13819 - Gao et al., 15 Dec 2025) in Conjecture~\ref{conj:RSB}, Section “More conjectures and future work”