Certified Multistability Monitor for Equilibrium Networks

1. Integration point: attach the monitor to a low-dimensional equilibrium state, preferably a 2-16 dimensional bottleneck in a DEQ, neural ODE controller, or recurrent world model. Run it periodically during training on representative inputs with small input or parameter boxes. Initially do not backpropagate through discrete branch decisions; use the interval result to log diagnostics and optionally add a detached width penalty or a smooth surrogate based on sampled Jacobian norms. 2. Pseudocode: for each (u,P), initialize a queue with the operating box X0; compute interval F and J; discard boxes excluded by interval residual tests; accept boxes satisfying Krawczyk inclusion; otherwise bisect the widest coordinate until a minimum width or budget is reached. Merge overlapping accepted boxes only when their separation cannot be certified, and report enclosure width, unresolved count, and separated branch count. Trigger an intervention when B>1 or unresolved width exceeds a threshold: reduce recurrent gain, increase damping, or add a consistency loss between independently initialized fixed-point solves. 3. Computed versus estimated: outer containment and exclusion are interval-computed; branch counts are lower bounds unless every remaining box is uniquely certified. The determinant test is only a warning because interval dependency can make it conservative. 4. First cheap experiment: construct a 2D tanh DEQ with tunable recurrent gain and symmetric bias, train it on a synthetic classification task, and sweep gain and input uncertainty. Compare one-start Newton, 100 random starts, and the interval monitor. The predicted signature is an increase from B=1 to B>=2 near the symmetry-breaking gain, with interval unresolved width peaking there; random starts should report a lower branch count. After the intervention, test whether certified enclosure width decreases and the operating regime moves away from the transition without task-loss degradation.