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 functional without introducing external physical structures, phenomenological fitting, numerical calibration, spectroscopic input, renormalization procedures, or experimentally imposed scales. Starting exclusively from the exact quartic functional, the complete derivational chain is developed 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 a recursively constrained cluster structure satisfying \[\{f,4f,f/4\}.\] The framework predicts that the observable hierarchy is non-atomic, non-rotational, non-vibrational, recursively organized, finitely admissible, and directly falsifiable through terahertz spectroscopy. All quantities emerge exclusively from the internal variational structure of the framework and do not rely on fitting procedures or externally introduced assumptions.