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

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 O(N\log N + k^2_{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 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_{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 88.93% Top-1 accuracy using \approx 21.24M 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.