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.

翻译：存储位置、相位或其他连续变量的递归网络需要在长时间尺度上保持状态空间方向的中性。我们基于对称性提出了一种理论框架，论证此类中性方向为何是必然存在而非仅靠参数调节获得。对于李群\(G\)等变的有限维自治\(C^1\)向量场，我们证明：任何具有紧致不变集、承载均匀非退化群轨道丛且稳定子类型为\(H\)的系统，在Lyapunov谱可定义的点处，至少存在\(\dim(G/H)\)个与群轨道相切的零Lyapunov指数。这些对称保护模式因精确等变性与轨道几何特性而具有零群切向增长率。当对称保护被显式破缺时，受保护方向会形成赝能隙；在受控破缺实验中，该赝能隙可预测有限记忆寿命。我们通过归一化等变误差、直接群切向指数、主角度对齐、自治流零控制以及跨越\(S^1\)、\(T^q\)、\(SO(n)\)、\(U(m)\)、乘积群和耦合等变RNN风格系统的轨道维度缩放，验证了有限维度结论。此外，我们训练了一个精确等变递归神经单元，用于速度输入\(S^1\)路径积分任务（六次不同种子初始化），并与匹配的GRU、LSTM和正交RNN基线进行对比。学习到的等变单元将步进等变性保持至\(3.2\times10^{-8}\)，在零输入自治约束下具有近零群切向指数，并在匹配协议中改进了时间跨度、速度及受限相位泛化能力。学习任务结果提供证据支持；而定理层面的证据仍包括精确等变性、群切向指数、轨道维度缩放及切子空间对齐。