From Eigenmodes to Proofs: Integrating Graph Spectral Operators with Symbolic Interpretable Reasoning

We introduce Spectral NSR, a fully spectral neuro-symbolic reasoning framework that embeds logical rules as spectral templates and performs inference directly in the graph spectral domain. By leveraging graph signal processing (GSP) and frequency-selective filters grounded in the Laplacian eigenstructure of knowledge graphs, the architecture unifies the interpretability of symbolic reasoning with the scalability and adaptability of spectral learning. Beyond the core formulation, we incorporate a comprehensive set of extensions, including dynamic graph and basis learning, rational and diffusion filters for sharper spectral selectivity, mixture-of-spectral-experts for modular specialization, proof-guided training with spectral curricula, and uncertainty quantification for calibrated confidence. Additional enhancements such as large language model coupling, co-spectral transfer alignment, adversarial robustness, efficient GPU kernels, generalized Laplacians, and causal interventions further expand the versatility of the framework. Empirical evaluation on state-of-the-art reasoning benchmarks such as ProofWriter and CLUTRR demonstrates that Spectral NSR achieves superior accuracy, faster inference, improved robustness to adversarial perturbations, and higher interpretability compared to leading baselines including transformers, message-passing neural networks, and neuro-symbolic logic programming systems. Spectral attribution and proof-band agreement analyses confirm that model decisions align closely with symbolic proof structures, while transfer experiments validate effective domain adaptation through co-spectral alignment. These results establish Spectral NSR as a scalable and principled foundation for the next generation of reasoning systems, offering transparency, robustness, and generalization beyond conventional approaches.