From Discrete Spectra to Spectral Manifolds: A Geometric Reformulation

In this paper, we introduce a mathematically explicit bridge between discrete spectral descriptions and continuous geometric representations. The core construction is a continuous spectral embedding in which the discrete index becomes a coordinate on a spectral manifold. We establish its basic structural properties (including sampling consistency, order preservation, and a controlled discrete-to-continuous limit) and illustrate the framework on Bohr-type levels and the spectral distribution underlying Planck’s law. This construction provides a geometric reformulation of spectral structures without introducing additional physical assumptions, thereby providing a geometric framework open to possible physical interpretations.