Stable States as Block-Uniform Flips of the Symbolic Centroid in Binary Hopfield Neural Networks

Existing analyses of spurious attractors in binary Hopfield neural networks (HNNs) have largely been developed within statistical-mechanical frameworks, often under specific memory ensembles, loading regimes, or structured storage configurations. While these approaches provide important macroscopic insights, they do not directly yield an exact structural characterization of spurious attractors for a given memory realization and local field configuration, particularly with respect to the boundaries of collective block flips. In this work we show that, for $M\ge 3$ stored memories, stable states in binary HNNs are block-uniform flips of the symbolic centroid, the bit-wise majority of the stored memories. By partitioning neurons into local domains according to their memory-alignment patterns, we prove that all neurons in a block share the same alignment with the centroid, so stability is decided block by block rather than neuron by neuron. Reformulating the weight matrix accordingly, we derive a matrix-based stability criterion for identifying stable block configurations and characterizing spurious attractors. This reduces the analysis from $N$ neuron-level conditions to a finite set of conditions over the $P$ memory-induced blocks. Beyond this reduction, the block variables open a route toward a block-macrostate network in which variable-assignment methods may be able to determine basins of attraction exactly, laying the groundwork for the precise identification and elimination of spurious attractors.