Weak irreducibility as a spectral criterion for phase coexistence

What the math gives to ML

The paper provides a spectral decomposition of finite-size phase coexistence into two independently measurable coordinates: sector imbalance determines where coexistence is balanced, while inter-sector connectivity determines whether competing sectors remain mixed or become effectively reducible. In a two-sector effective operator, the dominant-state splitting is controlled by both imbalance and off-diagonal connectivity; at balance, the splitting directly measures connectivity, which can decay exponentially with interface cost. This mechanism can transfer to mixture-of-experts routing or multimodal training by monitoring whether balanced sectors are genuinely separated or merely rapidly communicating, then using the measured spectral gap to trigger anti-collapse or specialization schedules.