Symmetry, Defects, and Diffusion in Continuous-Memory Recurrent Networks

Recurrent networks that store continuous variables must preserve differences between remembered values despite imperfect updates and state noise. We separate three problems: identifying memory directions, bounding their deterministic distortion, and controlling noise in the decoded coordinates. For ideal dynamics, exact equivariance for each input and uniform nondegeneracy along the trajectory prevent exponential growth or decay in the directions generated by continuous symmetry. For perturbed dynamics, local and finite-horizon bounds distinguish direct memory damage from leakage out of and back into its tangent space. With a fixed transverse contraction gap and no direct damage, the leakage contribution is second order. For noisy dynamics, a decoder-aware local covariance bound yields geometry design rules under a fixed coordinate metric and representation budget: equally scaled orthogonal directions are optimal for isotropic noise with least-squares decoding, whereas unequal noise can favor unequal allocation. In twenty fresh paired training units, covariance regularization improves noisy two-angle memory while meeting a specified clean-error equivalence margin; direct noise training incurs a clean-error tradeoff. In coupled four- and eight-angle integrators, the preference between anisotropic and isotropic regularization reverses when evaluation noise changes. These results connect symmetry-protected memory to measurable perturbation bounds and noise-specific representation design.