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

We present a parameter-free derivation of a discrete terahertz spectral line emerging from a closed quartic variational functional. Starting exclusively from a globally defined variational structure and its associated stationary condition, we construct the solution space and impose internal constraints of finiteness, normalization, closure, and stability. The second functional variation yields 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 discrete spatial structure. This induces a quantized spectral hierarchy of the form fn,k = n f* 4⁻ᵏ. Imposing admissibility, spectral stability, and closure constraints results in a finite truncation of the hierarchy, selecting 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 standard atomic, vibrational, or rotational transitions and defines a direct experimental test. The result follows entirely from the internal variational structure without external assumptions or fitting procedures.