Theory of Directional Associative Memories: Dense Hopfield Networks on the Unit Sphere S²

We extend Dense Associative Memory (DAM) from binary state spaces to the unit sphere S² ⊂ ℝ³, introducing Directional Associative Memory (DAM-S²). Each neuron carries a unit vector vᵢ ∈ S², patterns are stored as arbitrary unit vectors, and retrieval dynamics follow gradient flow projected onto the tangent plane of S² with renormalization after each Euler step. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ CONTEXT: ASSOCIATIVE MEMORIES ON SPHERES This is the companion paper to "Dense Associative Memory on S¹ with injection-locking dynamics" (DOI: 10.5281/zenodo.18800042). Together they form a family of associative memories on spheres Sⁿ: ● S¹ (circle) — Phase oscillators, α* = 1.0, 7.2× classical Hopfield capacity. Basis for the S1 memory layer in PGAN. ● S² (sphere, this work) — Directional patterns, α = 1.56 (all tested), √3 SNR advantage over binary. Basis for the S2 attention layer in PGAN, connecting Dense AM directly to Transformer attention via spherical softmax. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ KEY RESULTS ● Exponential capacity for F = eᵐ, verified empirically up to N = 256 ● Non-polar patterns: 100% recall at α = 1.56 (P=100, N=64) ● Noise robustness: 100% recall up to ~36% bit flips ● Spherical attention = one-step discrete update = softmax-weighted spherical interpolation ● Logic gates on S² with two-anchor framework; ROT(θ) gate unique to S²; connection to quaternion arithmetic ● Turing completeness inherited from S¹ for binary computation ● Generalization to Sⁿ; connection to von Mises-Fisher distributions ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ CONTENTS Paper (PDF + LaTeX source), Python implementation (DenseS2AM class), 14 unit tests, 7 benchmarks with reproducible results, and all figures.