Kohn-Sham Spectral Embedding on Sparse Graphs at the Nishimori Temperature for Image Classification

Abstract

We introduce Kohn--Sham Spectral Embedding (KSSE), a physics-inspired energy-based model replacing dense CNN classifiers with a sparse-graph spectral embedding evaluated at the Nishimori temperature of an associated Random-Bond Ising Model. By mapping pre-trained features onto quasi-cyclic low-density parity-check graphs and constructing a regularized Laplacian acting as a Kohn--Sham Hamiltonian, we solve $D$ independent channel spectral problems in $\mathcal{O}(N\log N + k^2_{\text{mode}} N)$ time via FFT on circulant blocks (leveraging Pontryagin self-duality of $\mathbb{Z}/p\mathbb{Z}$) and low-order Rayleigh refinement. Graph topology is optimized using \emph{star-domain surgery}: rather than destroying information-carrying codewords by removing frustrated cycles, we construct edge shifts creating local convexity around codewords while bounding residual frustration to $ρ(B_γ)\leq 1+δ$. Multi-scale fractal analysis ($D_2$ spectrum) and fractal learning-rate landscape certifies a landscape transition from rough regimes ($D_2>3$) to star-domain basins ($D_2<1$), enabling Rayleigh refinement with $k_{\text{mode}}=5$ modes. We prove six theoretical results: a generalized Ihara--Bass identity linking belief propagation to the Laplacian; trapping-set eigenvalue correspondence; additive channel separability with an explicit exchange-correlation bound; a surgery theorem bounding frustration with attractor width $Ω(1/\sqrt{d_{\min}})$; a quasi-stationarity perturbation bound; and a fixed-point convergence theorem. In a transductive protocol on ImageNet-1000 with frozen EfficientNet-B4 features ($D=1792$), KSSE achieves \textbf{88.93\%} Top-1 accuracy using $\approx 21.24$M parameters, outperforming Swin-L (197M, 86.4--87.3\%) and matching ViT-H/14 (632M, 88.0--89.5\%) under standard inductive setups, while reducing model footprint by $10\times$ and $30\times$, respectively.