Graph-Kolmogorov-Arnold Networks with Dense Associative Memory

Michael I Aksman

Published Jul 12, 2026• 0 citations

Abstract

We present a unied architecture combining KolmogorovArnold Networks and Dense Associative Memory on discrete spectral graphs, in which computation is conned to a protected low-frequency subspace by a rigid spectral projectorreplacing active gate-based error correction with passive, hardware-enforced mode suppression. Four results carry the framework. (I)The graph Laplacian generates the native unitary evolution U = e−iL∆t, and an orthogonal spectral projector P=Pk≤M |φk⟩⟨φk| truncates the Hilbert space to the protected span. (II)The Projection Stability Theorem: under xn+1 = P F(xn) for any nonlinearity F, the dynamicsremain conned with exactly zero leakageby idempotency, (I −P)xn+1 = (I −P)P F(xn) = 0,an algebraic identity, not an approximation. (III) Graph-KAN layers place trainable B-splinefunctions on the edges rather than static weights on the nodes, and their local support suppliesa decisive inductive bias for localized graph signals. (IV) DAM dynamics with a bounded-belowenergy functional give dE/dt ≤ 0 and unconditional convergence to a topological attractor.Version 2 replaces the arithmetic rationale for the projector rank with the correct and strongerprinciple: the rank must sit at a spectral gap, because eigenvalue separationand nothing elsecontrols perturbative mode mixing. We prove the Perturbative Firewall Theorem, which bounds the leakage of the projected dynamics under perturbation V by ∥(I − P)δx∥ ≤ ∥V ∥/∆ with ∆ the gap at the cut, converting the previously sampled stability into a theorem with a computable constant; and we exhibit the exact gap structure of the 861-node triangular Laplacian, whose protected cuts in the working region are M = 62, 110, 126, 143, 159, 215, 237, 256, with M = 215 (∆ = 0.094) the natural rewall for a quarter-spectrum truncation. Hardwarerealizations photonic bandgap, microwave-ltered superconducting, Rydberg-dressed neutral atomimplement the projector as a physical mode cuto placed at the gap.

DOI: 10.5281/zenodo.21328160

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