This work derives the periodic structure of the chemical elements from first principles within the Quarkbase framework. Starting from a single variational functional of a vacuum pressure field, the theory produces a discrete ladder of admissible stable configurations whose finiteness, ordering, and repetition follow from geometric closure and phase-coherence constraints, without invoking electronic shells, orbitals, or empirical fitting. Chemical periods emerge as closure intervals of this ladder, while chemical groups arise from the underlying geometric and spectral structure. The observable atomic number is linked to a physical compactation charge through a single global calibration, yielding a finite periodic table with seven complete periods and a predicted subsequent closure beyond the currently known elements.