SpectralNet: A Resolvent-Inspired Neural Architecture Based on Chernoff Approximations and Photonic Motivation

We introduce SpectralNet, a digital neural architecture motivated by the Chernoff-Remizov constructive line for operator evolution. The core idea is to interpret a learnable spectral layer as a discrete approximation of a small-step family Fₚhi (tau) whose repeated composition approximates a semigroup Tₚhi (t), while dense aggregation of intermediate states with decaying weights approximates the resolvent as a Laplace transform of the evolution. In this work we restrict ourselves to the Fourier-diagonal operator class, where the learnable spectral mask can be interpreted as the symbol of a discrete evolution step. On this basis we introduce a compact resolvent aggregation mechanism, study the choice of nonlinearity (PhysicalAct, GELU, ReLU), and conduct a systematic investigation of scaling, robustness, and composition depth. Key results: - SpectralNet-B with GELU achieves 76. 50-76. 67% on CIFAR-10, surpassing ShuffleNetV2 (74. 73%). The n=2 and n=3 configurations reach 76. 50% +/- 0. 16% and 76. 67% +/- 0. 18%, respectively. In the GELU-B n=4 configuration the gap to MobileNetV2 on SVHN narrows to 0. 74 pp (91. 44% vs 92. 18%). - For the base Fourier-diagonal SpectralNet, blur robustness is consistent across different activations: GELU-B n=2 (43. 9% drop) ~ PhysicalAct-B n=1 (44. 5%) vs 63. 0% for ResNet-18. In the resource-matched comparison of spectral-hybrid vs spatial shift-rich (C. 6), the blur advantage depends on data structure (CIFAR-10/100: Delta +17-21 pp; SVHN: Delta ~ 0). - We identify a limited useful composition horizon in the Fourier-diagonal class: 2-3 evolution steps form an accuracy plateau; further depth increase leads to representational saturation. - A resource-matched control C. 6 + 3-dataset verification (CIFAR-10, CIFAR-100, SVHN) establishes an empirical dichotomy between two trajectories: spectral-hybrid (80-89%) vs spatial shift-rich RMSB-R1 (90-96%), accuracy gap > +5 pp on all three. RMSB-R1 surpasses ResNet-18 on all three datasets with ~5x fewer parameters. The AWGN dichotomy is the most fundamental (Delta +5. 6-15. 2 pp across all three). This paper does not claim a hardware photonic realization. The photonic 4f-system and physical adjoint propagation are considered as physical motivation and a natural direction for future work.