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

SpectralNet is a digital neural architecture motivated by the Chernoff–Remizov constructiveline for operator evolution: a learnable spectral layer is read as a discrete small-stepfamily F_phi(tau) whose composition approximates a semigroup, and the intermediate statesare aggregated with decaying weights modelled on the resolvent as a Laplace transform. Werestrict ourselves to the Fourier-diagonal operator class and extend it by a low-rankcross-frequency mixer D+UV* and a shift-based spatial block Sa,b,c. Main result (resource-matched control C.6, three datasets). Under one training recipe anda parameter budget of ~2.1–2.3 M we compare two evolution-step families: the spectral-hybridstep F⁻¹(D+UV*)F ∘ Sa,b,c and the spatial shift-rich step Pi_mix ∘ Sa,b,c (RMSB-R1).Top-1 accuracy (mean ± sample std over seeds; each run reports the maximum over epochs ofthe test-set accuracy, the same rule for every model): RMSB-R1 89.87 ± 0.20 % vsspectral-hybrid 80.32 ± 0.38 % on CIFAR-10 (5 seeds, +9.55 pp), 62.90 ± 0.24 % vs52.32 ± 0.43 % on CIFAR-100 (5 seeds, +10.58 pp), 95.90 ± 0.09 % vs 89.22 ± 0.17 % on SVHN(3 seeds, +6.68 pp). The spectral-hybrid step loses less accuracy under corruption of thetest images: its mean accuracy drop over severity levels ("AUC drop") is smaller by21.2/16.6 pp for Gaussian blur on CIFAR-10/100 but equal on SVHN (Δ = −0.14 pp), and smallerby 15.2/14.3/5.6 pp for additive white Gaussian noise on CIFAR-10/100/SVHN. Within the sameprotocol RMSB-R1 exceeds ResNet-18 by +3.81/+7.61/+2.34 pp with ~5× fewer parameters; thisis a within-protocol observation, not a general claim. C.6 compares the two evolution-stepfamilies and does not isolate the contribution of resolvent aggregation; the only ablationof the aggregation (single-seed SpectralNet-S, corrected in v1.1) shows no measurabledifference between resolvent, mean and last aggregation. Further results: the Fourier-diagonal SpectralNet-B with GELU reaches 76.50 ± 0.16 % (n=2)and 76.67 ± 0.18 % (n=3) on CIFAR-10, above ShuffleNetV2 (74.73 %); blur robustness in thebase class does not depend on the activation (relative drop at σ=2: 43.9 % GELU-B n=2 ≈44.5 % PhysicalAct-B n=1 vs ≈63 % ResNet-18); the Fourier-diagonal class has a limiteduseful composition horizon (2–3 evolution steps). This paper does not claim a hardware photonic realization. Remizov's theorems are used asmotivation only: they concern fixed-coefficient generators on UC_b(R) and do not assertconvergence of the trained finite-depth network or of its parameter derivatives. Version 1.1 (September 2026): text revision of v1.0 (DOI 10.5281/zenodo.19452600); sameexperiments and archive. Changes: (i) Remizov's theorems restated with hypotheses — onlyTheorem 6 is shift-based; (ii) RemizovShiftLayer described as a 2-D four-shift analogue ofTheorem 6, not a direct implementation; (iii) the resolvent parameter λ is learnable in thereleased code (v1.0: fixed); (iv) reported accuracies are best-epoch test accuracies, novalidation split (effect ≤ 0.32 pp for C.6); (v) the aggregation ablation of v1.0 comparedruns with different numbers of spectral modes — with the matched baseline resolventaggregation shows no advantage, and the v1.0 claim is withdrawn; (vi) abstract restatedaround the C.6 control with metric and protocol; (vii) SVHN blur Δ = −0.14 pp. All numbersre-derived from the primary experiment outputs by scripts/verify_w1_claims.py in the coderepository (71 checks); no result files are distributed — every table can be regeneratedwith the launchers in scripts/ and checked with the same script. Files: main_en.pdf,main_ru.pdf (Russian version). Code: https://github.com/shpital/spectralnet (tag paper-v1.1).