Closed-form solution of the exact propagator recursion

Derive a closed-form expression for the recursive relation governing the exact neuron-neuron propagator, including all leading-order O(1) loop corrections in the lattice gauge-theory description of fully connected deep neural networks.

Background

The paper derives a recursive formula for the exact propagator X_l by summing infinitely many cactus and related loop diagrams at leading order O(1) in the 1/N expansion. This recursion incorporates both bare propagators and recursively dressed propagators from preceding layers.

Although the authors reorganize the infinite series and prove its convergence, they do not obtain a closed-form solution. Such a solution would make the statistical fluctuation corrections to neuron variance more directly analyzable, particularly at strong coupling and across deep networks.

References

Unfortunately, despite some effort, we have not managed to obtain a closed-form expression for this relation.

— Deep neural networks as lattice gauge theories  (2608.19331 - Jefferson et al., 19 Aug 2026) in Section 2, subsection “Perturbative corrections to the neuron propagator”

Thus, it is unclear if truncating this sum to any fixed $k$ has a sensible physical interpretation.

— Deep neural networks as lattice gauge theories  (2608.19331 - Jefferson et al., 19 Aug 2026) in Appendix B, subsection “Not-quite-full computation”