Certified Coordinate Adaptation for Finite-Capacity Latent Dynamics Learners

A representation can retain sufficient information for prediction yet remain poorly matched to a finite-capacity downstream learner, limiting how effectively that information can be used. To address this mismatch, we introduce Profiled Certified Exact Gauge Adaptation (PCEGA), which learns an invertible coordinate map and refits a dynamics predictor while keeping the encoder and decoder frozen. Invertibility preserves encoded information, and exact inverse compensation preserves the original reconstruction. A computable global bi-Lipschitz bound controls coordinate distortion, while coordinate optimization uses decoded prediction error and accounts for predictor refitting and ridge selection. Theoretically, for any fixed integer $p\ge 1$, we prove that invertible affine maps are exactly the globally bi-Lipschitz transformations that preserve the full class of polynomial dynamics of degree at most $p$ under conjugacy. Experimentally, the complete default global-budget procedure reduces normalized mean squared error by 48.82% and 45.11% on independently trained Lorenz and R\"ossler charts, respectively. On an analytic-chart benchmark covering four dynamical systems, validation-selected maps from a learned coordinate library reduce normalized mean absolute error by 15.39%–71.11% across linear, quadratic polynomial, random-feature, and radial basis function learners. Both comparisons use independently tuned baselines in the original coordinates. Crossed-learner experiments further reveal learner-specific coordinate preferences, supporting adaptation tailored to the downstream learners.