Parity families and a kernel-averaged L-function for near-Ramanujan signings

What the math gives to ML

The paper gives a constructive way to choose signs on graph edges so that parity constraints on short even cycles induce cancellation among closed walks, while preserving a controlled subset of walk classes. The transferable asset is the finite-field character representation: graph propagation spectra can be searched over an affine family of signings using cheap trace estimates rather than optimizing an unconstrained discrete matrix. This suggests a signed graph-neural propagation layer whose adjacency has a deliberately reduced spectral radius, improving stability in deep message-passing networks. The most practical first test is to generate constrained signings, select one by Hutchinson estimates of even matrix traces, and compare deep signed GNNs against unsigned propagation and unconstrained random signs.