Spectral Normalization and the Emergence of a Computational Boundary in Elliptic Operators

This work introduces the concept of the spectral horizon, defined as the normalized spectral radius of an elliptic operator with respect to its largest eigenvalue. We show that this normalization naturally produces a dimensionless spectrum lying in the interval (0,1], where the value 1 represents the computational reachability limit of the operator. The study begins with an analytical one‑dimensional example of the Laplace operator with Dirichlet boundary conditions, where the spectrum is known explicitly. A two‑dimensional numerical example on a domain with an interior circular hole then demonstrates that the spectral horizon also appears in more general geometries. A discussion of robustness shows that the phenomenon is invariant under scaling, geometric deformation, and discretization. Although the motivation originates from the theoretical framework of Triadic Mesh Dynamics (TMD), the spectral horizon is developed here independently as a standalone mathematical invariant. This work provides a formal foundation for the notion of a computational boundary in spectral problems and may serve as a basis for future applications within TMD.