Extend the NN-FT construction to infinite volume with spatially localized features

Determine whether the neural-network field theory construction can reach the infinite-volume two-dimensional phi-four measure by replacing globally supported plane-wave neurons with features localized in both position and frequency, while preserving the one-shell estimates required for ultraviolet convergence.

Background

All estimates in the paper use a fixed compactly supported spatial cutoff, and several constants worsen as the volume grows. In particular, the worst-case Gibbs-factor bound depends on the spatial volume and enters the moment estimates exponentially.

The proposed route to an NN-FT thermodynamic limit is to use wavelet-like features that are localized in position as well as frequency, rather than plane waves spread across all of Euclidean space. It remains unresolved whether the one-shell analysis survives this architectural substitution.