The Spectral Agora: Schrödinger Diagnostics for Internal Deliberation in Large Reasoning Models

Abstract We propose a formal bridge between two seemingly unrelated programmes: (1) the spectral analysis of Schrödinger operators encoding the Riemann zeta zeros (Blum, 2026a), and (2) the Neural Agora framework for internal deliberation in Large Reasoning Models (Blum, 2026b). The key insight is that propagation of the residual stream through the transformer's middle layers ("the Agora," L8–16) can be modelled as a discrete scattering problem, where the layer-wise transformations play the role of a potential and the SAE feature decomposition provides the mode structure. We adapt the eight diagnostic tests developed for the Riemann problem to classify deliberation quality, and demonstrate on a minimal synthetic sample (6 scenarios × 9 layers × 6 SAE modes) that the adapted diagnostics correctly differentiate healthy deliberation from five reasoning pathologies. We identify four confirmed constraint-correspondences, one structural correspondence, and three testable predictions for real LLM traces. Keywords: Schrödinger operator, transfer matrix, spectral diagnostics, mechanistic interpretability, reasoning models, deliberation, coupling pincer