The Readout-Gauge Conflict: Anchored Equation Discovery with Learned Koopman Closure

Learned Koopman representations can improve equation discovery, yet their flexibility can make the reported equation depend on the chosen coordinates: an auxiliary closure coordinate may absorb a named library term without changing the predicted vector field, even as the coefficient assigned to that term changes. We call this failure of the scientific readout the readout-gauge conflict. The Resolvent-based Tail-Residual Network (ReTRN) addresses this conflict by declaring the candidate equation library before fitting, adapting only auxiliary closure coordinates, and reading the equation from the same generator used to fit the lifted dynamics. We characterize the closure transformations that preserve this anchored readout and train the adaptive closure from sparse ordered observations through multiscale temporal moments, without numerical differentiation. Controlled experiments show that anchoring, rather than neural capacity alone, preserves coefficient recovery: across six systems, ReTRN gives the best result in 11 of 12 primary comparisons and all 180 paired rollout comparisons, and transfers to seven additional equation families.