Hopfield networks store memories as energy minima. The fundamental limitation: capacity scales linearly with network size. Store more than about 0.14N patterns in N neurons and the memories corrupt each other. This ceiling has stood for four decades.

Replace neurons with oscillators. Instead of binary firing states, each unit has a continuous phase. Instead of symmetric couplings, use Kuramoto dynamics. Instead of a fully-connected graph, arrange oscillators in a honeycomb topology. The result: memory capacity that scales exponentially with network size.

The mechanism is elegant. Each honeycomb cycle stores multiple distinct phase-locked configurations — the stable states where all oscillators in the cycle maintain fixed phase differences. A cycle of n_c oscillators supports (2⌈n_c/4⌉ - 1) such configurations. Connect m cycles and the total capacity multiplies: (2⌈n_c/4⌉ - 1)^m patterns. The exponent is the number of cycles, so capacity grows exponentially with the modular structure of the network.

The basins of attraction — the regions of phase space from which the network reliably converges to a stored pattern — have guaranteed minimum sizes that don’t shrink with network scale. Bigger networks store exponentially more patterns without becoming less reliable at retrieving each one.

The topology does the work. A fully-connected network wastes coupling capacity on redundant connections. The honeycomb gives each oscillator exactly the neighbors it needs to define a phase-locked state, and no more. Structured sparsity creates capacity that density cannot.

The practical test: charge-density-wave oscillators in hardware confirm the theory. This isn’t just mathematical possibility — it’s physically realizable. Neuromorphic memory at exponential scale, without the linear ceiling that Hopfield hit in 1982.