Modern multi-class image classification uses high-dimensional CNN features that incur large memory and computational costs and obscure the data manifold's geometry. Existing graph-based spectral classifiers work on synthetic or binary tasks but degrade on natural images with many classes because feature manifolds have non-trivial topology. We introduce a physics-inspired pipeline where frozen MobileNetV2 features are interpreted as Ising spins on a sparse multi-edge type quasi-cyclic LDPC graph, defining a Random-Bond Ising Model (RBIM). The model is operated at its Nishimori temperature -- where the smallest eigenvalue of the Bethe-Hessian matrix vanishes. A spectral-topological correspondence links trapping sets in the Tanner graph to topological invariants via poles of the Ihara-Bass zeta function, enabling systematic suppression of harmful substructures that otherwise reduce top-1 accuracy by more than a factor of four. A fast quadratic-Newton estimator finds the Nishimori temperature in $\sim 9$ Arnoldi iterations, a sixfold speed-up over bisection. The resulting ensembles compress the original $1280$-dimensional MobileNetV2 representation to $32$ dimensions (ImageNet-10) or $64$ dimensions (ImageNet-100). We achieve $98.7\%$ top-1 accuracy on ImageNet-10 and $84.92\%$ on ImageNet-100 using a three-graph soft ensemble. Relative to MobileNetV2, our hard ensemble increases accuracy by $0.10\%$ while reducing FLOPs by a factor of $2.67$. Against ResNet-50, the soft ensemble drops only 1.09% accuracy yet cuts FLOPs by $29\times$. The novelty lies in (a) establishing a rigorous link between graph trapping sets and algebraic-topological defects, (b) an efficient Nishimori-temperature estimator, and (c) demonstrating topology-guided LDPC graph embedding for highly compressed classifiers.