Fibre-to-Section Lifting over Measure-Preserving Dynamics: Variational Reciprocity and Memory-Reversal Rigidity

Abstract

Many nonlinear and variational problems are posed fibrewise over a probability space, whereas the dynamically relevant objects are measurable sections coupled by measure-preserving transformations. We study structural obstructions created by this fibre-to-section lifting. First, for a global continuously differentiable potential on a real Hilbert space, absence of influence between closed subspaces is reciprocal; for finite orthogonal decompositions this yields an additive decomposition over dependency components. Second, finite Koopman memory channels are identifiable exactly under a sharp orbit-separation condition. Resolving the classical Hilbert-space potentiality criterion channel by channel then gives a shifted-adjoint balance between forward and reverse memories. At the global finite-difference level, if a section map with arbitrary finite dynamical memory is the gradient of a global continuously differentiable potential, its genuinely active lags are invariant under time reversal. Thus globally variational memory is reversal-complete, and one-sided memory collapses to present-state locality. Continuous proximity operators inherit the same rigidity through their convex-potential representation; a discontinuous proximal selection shows that continuity cannot generally be removed. Finally, under pointwise nondegeneracy of two coherent branches, vanishing branchpasting residual is equivalent to asymptotic invariance of the branch labels. For finitely generated ergodic probability-preserving actions, strong ergodicity is therefore the exact threshold for relative compactness of all branch-pasting approximate-zero sequences. We also record a strong-compactness criterion for full measurable-selection lifts and a nonconvex Wasserstein minimizing-step illustration.