Emergence of a Discrete Terahertz Frequency from a Closed Quartic Variational Functional

We derive the emergence of a discrete terahertz spectral line from a closed variational structure without the introduction of external parameters or fitting procedures. Starting from a globally defined quartic functional and its stationary condition, we construct the solution space and impose internal constraints of finiteness, normalization, closure, and stability. The resulting second variation defines a structured Hessian with a non-trivial spectral decomposition characterized by a hierarchical scaling factor L = 1/4. A modulated instability analysis leads to the emergence of a fundamental length scale and a corresponding discrete spatial structure. This induces a spectral hierarchy of the form f_{n,k} = n f* 4^{-k}. Imposing admissibility and spectral stability yields a finite truncation of the hierarchy and selects a lowest admissible frequency in the terahertz range. We obtain a concrete prediction: f ≈ 2.35 × 10^12 Hz This frequency does not correspond to known atomic, vibrational, or rotational mechanisms and defines a direct experimental test. The result follows entirely from the internal variational structure and provides a minimal example of how discrete physical observables can emerge from a closed mathematical system.