Conjecture: zero‑temperature non‑ergodic behavior is fully determined by topological symmetry breaking

Prove or refute that, at zero temperature, the non‑ergodic dynamics of systems with antagonistic interactions are fully captured by the topological symmetry‑breaking mechanism identified via the signed Laplacian; in particular, ascertain that the emergence of multiple stationary states is strictly linked to the percolation of clusters defined by the signed‑Laplacian structural analysis (e.g., sign partitions of the lowest‑eigenvalue eigenvector).

Background

After introducing a signed‑Laplacian framework and showing percolation of topological defects in lattices and networks, the authors propose a direct connection between structural percolation and dynamic multistability at vanishing thermal noise.

The conjecture asserts that the zero‑temperature landscape of metastable states is dictated by a topological symmetry‑breaking mechanism visible in the signed‑Laplacian spectrum, with percolating sign‑defined clusters controlling the onset of non‑ergodicity.