Programmable k-local Ising interactions and shallow optical Kolmogorov--Arnold networks through repeated data encounters

Photonic processors are naturally suited to linear transformations, but independently programmable higher-order interactions usually require nonlinear media or a reduction to pairwise models. We introduce a repeated-encounter architecture that combines linear optical propagation with square-law detection to evaluate sparse and structured k-local Ising objectives. Each hyperedge is routed to a resolved channel, returned through the spin-dependent mask, and reconstructed from calibrated encounter-order signals. Within the stated architecture class, a k-spin Walsh term requires at least lceil k/2rceil data encounters. A reciprocal rank-one recollection attains this bound for every finite order, allowing hyperedge identity, interaction order, and signed coupling to be programmed independently without quadratization ancillas or material optical nonlinearities. We test the R=2, k=4 member in a finite discrete-Fourier model of an ideal folded 4f relay. Reciprocal recollection produces the four-body response, whereas a fixed-patch control does not; configuration-level calibration measures the leakage caused by finite windows. Under signed-amplitude encoding, the same encounter hierarchy spans polynomial edge functions for shallow optical Kolmogorov--Arnold networks. The arbitrary-k result is analytic; the finite Fourier calculation tests its two-encounter member and does not replace experimental validation.