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

This work derives a discrete terahertz spectral hierarchy directly from a closed quartic variational structure without introducing external physical postulates, phenomenological fitting, renormalization procedures, experimentally calibrated scales, spectroscopic input, or auxiliary theoretical sectors. Starting exclusively from the exact quartic variational functional, the complete derivational chain is constructed through explicit first variation, exact second variation, Hessian construction, spectral decomposition, recursive Z3 closure, Fourier linearization, finite hierarchical admissibility, and global decisional selection. The exact Hessian spectrum generates the coherent structural scale \(E^\star\) together with the universal recursive suppression ratio \[ L=\frac14. \] The recursive hierarchy generated by the normalized spectrum produces a finite admissible closure level \[ k^\star=5, \] selected globally through the decisional functional \[ \Psi^\star = \mathrm{Sel}_{\Psi\in A} C(\Psi\mid\Omega). \] The resulting oscillatory hierarchy generates the exact terahertz prediction \[ f_{\mathrm{THz}} = 2.353515625\times10^{12}\ \mathrm{Hz}, \] together with the recursively constrained cluster structure \[ \{f,4f,f/4\}. \] The framework predicts that the observable hierarchy is non-atomic, non-rotational, non-vibrational, recursively organized, finitely admissible, and directly falsifiable through terahertz spectroscopy. The work further establishes the non-local nature of the decisional functional, the exact separation between the global admissibility structure \(\Omega\) and the dynamical oscillatory frequencies \(\varpi\), the non-fundamental nature of SI units, the absence of fitting procedures, and the structural rigidity of the complete variational chain. All quantities emerge exclusively from the internal variational structure of the framework and do not rely on fitting procedures or externally introduced assumptions.