Symmetry-Protected Lyapunov Neutral Modes in Equivariant Recurrent Networks

Recurrent networks that store position, phase, or other continuous variables need state-space directions that remain neutral over long horizons. We give a symmetry-based account of when such neutral directions are guaranteed rather than merely tuned. For a finite-dimensional autonomous $C^1$ vector field equivariant under a Lie group $G$, we prove that any compact invariant set carrying a uniformly nondegenerate group-orbit bundle with stabilizer type $H$ has, at points where the Lyapunov spectrum is defined, at least $\dim(G/H)$ zero Lyapunov exponents tangent to the group orbit. These symmetry-protected modes have zero group-tangent growth because of exact equivariance and orbit geometry. When this protection is explicitly broken, the formerly protected direction can acquire a pseudo-gap; in our controlled breaking experiments this pseudo-gap predicts finite memory lifetime. We verify the finite-dimensional consequences with normalized equivariance error, direct group-tangent exponents, principal-angle alignment, autonomous-flow-zero controls, and orbit-dimension scaling across $S^1$, $T^q$, $SO(n)$, $U(m)$, product-group, and coupled equivariant RNN-style systems. We also train an exactly equivariant recurrent cell on velocity-input $S^1$ path integration across six seeds and compare it with matched GRU, LSTM, and orthogonal-RNN baselines. The learned equivariant cell preserves step equivariance to $3.2\times10^{-8}$, has a near-zero group-tangent exponent under the zero-input autonomous restriction, and improves horizon, speed, and restricted-phase generalization in this matched protocol. The learned task results are consequence evidence; the theorem-level evidence remains exact equivariance, group-tangent exponents, orbit-dimension scaling, and tangent-subspace alignment.