Spectral Degeneracy Operators: Spectral Theory, Uncertainty Principles, and Well-Posed Neural Architectures

This work develops a mathematical framework for Spectral Degeneracy Operators (SDOs), a class of degenerate elliptic operators with anisotropic diffusion tensors that vanish at a prescribed degeneracy point. We establish the functional-analytic setting in weighted Sobolev spaces and prove a spectral decomposition for the diagonal case on rectangular domains, together with a Weyl-type asymptotic law and a Bessel-type asymptotic expansion for the one-dimensional components. We further prove a strong maximum principle and local H\"older regularity away from the degeneracy locus. A Landau-type uncertainty inequality is obtained with an explicit but not necessarily optimal constant, and the framework is extended to compact Riemannian manifolds (weighted Poincar\'e inequality) and to globally hyperbolic spacetimes with a suitably regular foliation (relativistic Landau inequality for functions with spacelike compact support). For neural applications, we introduce SDO-Nets and prove well-posedness, Lipschitz stability, and continuous dependence on parameters for each layer. A neural-turbulence correspondence theorem guarantees that, under explicit hypotheses, every sequence of minimizers of the calibrated loss admits a subsequence converging to the true turbulence structure, with full convergence under uniqueness. Finally, a H\"older-type stability estimate is obtained for the inverse problem of calibrating degeneracy points from boundary flux measurements. The work bridges degenerate PDE theory, spectral analysis, and physics-informed machine learning, while leaving several questions open, including sharp constants, the case of exponents θᵢ≥ 2, and empirical validation of the neural constructions.