What question did this study set out to answer?

The research aims to extend Dense Associative Memory to the unit circle S¹, exploring its computational capabilities.

March 1, 20260 citationsOpen Access

Dense Associative Memory on S¹: Phase-Gate Computing and Superlinear Capacity in Circular Oscillator Networks

Key Points

The research aims to extend Dense Associative Memory to the unit circle S¹, exploring its computational capabilities.
Developed an energy function E for neurons with phase values instead of binary spins.
Analyzed fixed-point stability and empirically tested storage capacity with F=exp.
Performed extensive computational experiments using a 200 Hz phase oscillator architecture.
Achieved a storage capacity of alpha*=1.0 for network sizes N={32, 64, 128}.
Demonstrated a 7.2-fold increase over classical Hopfield networks.
Verified Turing completeness under specific conditions with high accuracy in Boolean gate implementations.

Abstract

Dense Associative Memory on S1 — v2. 0: Peer-Review Edition We present Dense Associative Memory (DAM) extended to the unit circle S1, where each neuron carries a phase in 0, 2pi) rather than a binary spin. The energy function E = -sumₘu F (sumᵢ cos (phiᵢ - xiᵢᵐu) ) generalizes the Krotov-Hopfield Dense AM from +-1N to S^1N. We prove fixed-point stability analytically and show empirically that F=exp achieves storage capacity alpha*=1. 0 for N in 32, 64, 128 -- a 7. 2-fold improvement over classical Hopfield (alpha*=0. 138). The F=exp update is formally equivalent to Transformer self-attention with circular inner products. The same dynamics implement universal Boolean gates at 100% accuracy. Physical substrate: 200 Hz-anchored phase oscillator arrays (REZON architecture). DOI: 10. 5281/zenodo. 18800042 WHAT'S NEW IN THIS VERSION Paper: - New Related Work section (rotor Hopfield, Modern Hopfield, Kuramoto 2025, NeurIPS 2024) - Theorem 1: full 3-step proof, assumption corrected to P 0 in 1 step at N=128, P=5 - CNOT: 100% pass rate, 20 seeds, Wilson 95% CI [0. 83, 1. 00 Tests: 66 tests passing, GitHub Actions CI green WHAT THE FULL PROJECT CONTAINS phasedenseₐm. py -- Dense AM: F=exp/poly2/poly3/linear, circular attention, capacity sweep phasegateᵤniversal. py -- NOT, AND, OR, XOR, NAND, NOR, half-adder via injection-locking ODE cnotₚhasegate. py -- CNOT with Wilson CI, noise sweep phasedlatch. py -- Bistable phase latch, robustness tests phaseₐutomaton. py -- 3-state FSM as phase attractors phasefullₐdder. py -- 1-bit adder with carry phaseₜuringdemo. py -- Cascaded memory + logic + sequential computation phaseₙeuralₙet. py -- PhaseNN: ODE-based classifier, 664 params vs MLP 5768 phaseₕopfield. py -- Baseline: validates alpha*=0. 138 paper. tex / paper. pdf -- Full scientific paper, 633 KB test_*. py (8 files) -- 66 tests, pytest CI reports/*. json -- N=32/64/128 capacity, CNOT, FSM results REPRODUCIBILITY. md -- 8-step protocol from scratch to results FORMALAPPENDIX. md -- Lemmas A1-D2, full proofs THREATSTOVALIDITY. md -- Threat analysis for reviewers Code: https: //github. com/krisss0mecom/REZON DOI: https: //doi. org/10. 5281/zenodo. 18800042

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