When Parameter Gradient Flows Close in Representation Space

Abstract

The represented gradient flow of a parameterized model depends on a mobility that can retain hidden parameter information. We study when one entire represented mobility map works for every autonomous loss from one fixed initialization. For two-factor matrix representations wider than both output dimensions, with individually full-row-rank factors, we classify all admissible joint ranks: closure occurs precisely at effective scalar-balanced minimal joint rank or a two-value signed spectral family at full joint rank. All intermediate ranks are excluded; positive represented flows belong to known square-root families. For general representations with constant ambient representation and gradient-Lie ranks, ordinary closure is equivalent to a local invariant-level metric in the analytic case and specified smooth regimes. An endpoint section forces the initially unrestricted mobility map to admit a locally smooth representative on the represented image. We also separate testing families: linear losses locally admit closure at every submersion, whereas one fixed translated strongly convex quadratic family fails it on a delayed-bracket example. A fixed output readout gives an ordinary squared-loss witness. These supporting results do not characterize quadratic-only closure in general. We retain the unresolved smooth output-rank-two case with Lie rank at least four. A frozen deterministic experiment checks a separate engineered-loss construction; it is distinguished from the squared-loss proof, universal theorems and empirical learning performance.