Pulse Graphs: Prime-Activated Boolean Dynamics on Directed Graphs
arXiv:2607.10453
2026
Architecture
1 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper gives an explicit construction of sparse Boolean recurrent systems whose state evolution contains a maximal-length affine feedback register, producing periods exponential in the number of state bits. The transferable asset is not primality itself but the compilation of prime-count tests into copy, XNOR, XOR, NAND, and NOT gates using a few permanently active control vertices. This yields a fixed recurrent feature map with bounded in-degree, linear edge count, and deterministic long-range state diversity. The most credible ML use is a cheap long-memory reservoir or binary state-space layer, trained only through an input projection and readout and compared against random Boolean reservoirs, ESNs, and small GRUs.
Ideas from this paper
Unverified
2026
Replace a learned recurrent transition with a sparse binary reservoir compiled from prime-count gates. Three permanently active control vertices implement copy and XNOR transitions, allowing the signal state to realize an affine feedback register over \(\mathbb F_2\) with period \(2^m-1\) using only \(O(m)\) edges and bounded in-degree. Train only an input projection and readout, or use straight-through estimators if end-to-end adaptation is required.
Useful5/10
Difficulty5/10
Novelty6/10