Toward a Spectral Principle: Extending the TEQ Framework<span class="math-tex">\( \)
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We reformulate the Total Entropic Quantity (TEQ) framework using two axioms, extending the second to include spectral comparison via analytic continuation. This extension formalizes the treatment of renormalization, vacuum energy suppression, and spectral anomalies as structural consequences of entropy geometry. Using the extended Minimal Principle, we derive the exact Casimir energy, explain the chiral anomaly, and reinterpret zeta regularization as a physically grounded method for comparing entropy-curved spectra. Appendices confirm that core quantum corrections—including the Lamb shift and the running of the coupling constant α—remain derivable from the original two axioms. Crucially, these results are obtained without recourse to ad hoc regularization, arbitrary subtractions, or postulated operator structure; instead, regularization and anomaly arise as necessary features of entropy geometry and analytic continuation. These results reinforce TEQ's explanatory economy: a single resolution-based variational principle governs not only quantum dynamics but also spectral comparisons and anomalies. This work preserves axiomatic minimality while extending the empirical and structural reach of the TEQ framework.