A Hopfield network stores patterns as energy minima. Lower the temperature and the network settles into one of them. The selection depends on initial conditions — which basin of attraction the system starts in — but the mechanism is always the same: thermal fluctuations decrease, the system commits, a pattern is recalled.
Sasaki (arXiv:2602.20620) builds a network that recalls different patterns at different temperatures. Two patterns are embedded in different graph structures: the first into a fully connected graph, the second into a sparse graph. The fully connected embedding is resistant to thermal noise — many connections reinforce the pattern, so it persists to higher temperatures. The sparse embedding is fragile — fewer connections mean thermal fluctuations can overwhelm it sooner.
At low temperature, both patterns are stable, but the fully connected one dominates because its energy minimum is deeper. At higher temperature, the dense pattern destabilizes first — no, the opposite. The sparse pattern destabilizes first because it has less reinforcement. So as temperature rises, the system transitions from recalling the sparse pattern to recalling the dense one. The transition is first-order: discontinuous, with a free-energy barrier between the two recalled states.
This is not a network that forgets as it warms up. It is a network that remembers differently. The high-temperature memory and the low-temperature memory are both genuine recall — stable, pattern-specific attractors. What changes is which attractor the system occupies. Temperature selects memory, not just clarity.
The mechanism is graph topology. A pattern embedded in a dense graph has redundancy — each connection independently reinforces the whole, so thermal noise must overcome many links simultaneously to disrupt recall. A pattern embedded in a sparse graph has fragility — removing a few connections collapses the attractor. The density of the embedding is a knob that sets the thermal robustness of each memory independently.
The general principle: the same system can store qualitatively different information at different energy scales, and the information that survives depends on how much noise the environment provides. This is not degradation. It is selection. The system does not lose memories as temperature increases — it changes which memories it can access.