Symmetry-Restoring Rectified Update Rules for Hopfield Neural Networks: Dynamics, Theorems, and Hamming-Distance Awareness

Hopfield neural networks (HNNs) provide a classical framework for associative memory, but the binary dynamics are not fully specified when a neuron's local field is zero. Common fixed-sign tie-breaking conventions can introduce an artificial sign preference and break the global sign symmetry inherent in the energy. We show that retaining the neuron's current state at zero field restores this symmetry and yields a principled Rectified Update Rule. This symmetry restoration exposes a Hamming-distance geometry underlying the resulting dynamics. For a single stored message, we obtain an exact characterization of convergence and symmetric two-cycles. For two stored messages, we formulate a Hamming-Distance-Aware Rectified representation and characterize the complete set of dynamical regimes, including convergence, self-cycles, hetero-cycles, and symmetric-cycles. These results further provide exact state counts and convergence-domain characterizations as functions of the network size and inter-memory Hamming distance. Exhaustive enumeration and Monte Carlo experiments agree with the theoretical predictions across a range of network sizes and memory separations.