Spurious states are labels: A multi-label code in the Hopfield-RBM duality

For forty years, the superpositions of stored patterns known as spurious states have been treated as the defining defect of the Hopfield model. We read them as an answer: a mixture of $L$ patterns is a composite signal, and identifying its constituents is multi-label classification. A Hopfield configuration cannot represent a set of labels; its dual restricted Boltzmann machine (RBM) can. We show that the Hopfield-RBM duality analytically fixes this representation: across $N$ visible neurons, active labels sit on the hidden layer at scale $\sqrt{N}$ with an amplitude matching the archetype-mixture overlap. Under this label assignment, the class averages of noisy training examples form an exact fixed point in mean of one-step supervised contrastive divergence. We prove a finite-size drift bound that decomposes into four vanishing error channels: thermal sampling, dataset noise, mini-batch consensus, and finite-dimensional fluctuations. For continuous mixtures combined prior to quantization (such as musical chords and hyperspectral pixels), this overlap amplitude transitions to an arcsine law via one-bit random projections. Across four benchmarks on three public datasets, the resulting configuration acts as an optimal analytical warm start: computed in a single pass without gradient updates, it surpasses random initializations by $0.08$ to $0.43$ in score, saving thousands of training steps. Finally, the induced parameter-free read-out rivals computationally intensive one-bit compressed sensing baselines at a fraction of their inference cost.