A trained physical network's function is readable from the soft modes of its Hessian, tying conserved protein regions to slow collective modes

"Physical networks become what they learn" (arXiv:2406.09689) argues that after a physical network is trained, its function can be recovered from its physical structure alone: the task-critical edges become measurable soft modes of the network's Hessian, so in principle one can reverse-engineer what a network was trained to do without knowing the task. The paper connects this structure–function correspondence to biology — per the capture's hop read, "conserved regions in proteins were associated with slow collective modes."

This reframes learning as literal structural reorganization: training does not merely set parameters, it reshapes the network's low-energy geometry, and that geometry is where the function now lives. It generalizes the specific result that allosteric response is cheap to induce (claim-removing-one-percent-of-bonds-makes-a-random-network-allosteric) into a principle — the same soft-mode geometry that carries engineered long-range response also carries evolved function in proteins — and it helps explain why a network trained by a purely local physical rule (claim-coupled-learning-elastic-networks-compute-without-a-processor) can end up structurally indistinguishable from a system that was deliberately designed. It is the structure-function hinge of the vault's emerging physical-learning cluster and a sibling to backpropagation-gap's question of whether shared representations imply a shared mechanism.

The load-bearing formalism — the physical-Hessian ↔ cost-Hessian correspondence and the "reverse-engineer function from the Hessian" mechanism — is recorded here from a hop read of the paper, not a verbatim primary extraction; only the phrase "conserved regions... slow collective modes" is preserved verbatim. The note is held seedling under [unverified-mechanism] and the formalism re-read is routed to question-verify-physical-networks-hessian-cost-formalism.