The Geometric–Arithmetic Balance Lemma: Unconditional Formalization of the Topological Cancellation $c_2 + p_1 = 0 \implies A_1 = 0$ and the Analytic Normalization of the $\zeta$-Determinant

This manuscript presents a fully formalized and unconditional proof of the core analytical lemma required for the absolute spectral equivalence within the TEBAC Program (Topological Enforcement of the Balance Lemma in Analytic Continuum). Core Contribution: Geometric–Arithmetic Balance Lemma: We rigorously demonstrate that the Topological Cancellation Rule $c_2 + p_1 = 0$ is the necessary and sufficient precondition for the vanishing of the global first Seeley–DeWitt coefficient, $A_1 = 0$. Analytic Normalization: The vanishing of $A_1$ ensures the analytic normalization ($\alpha=\beta=0$) of the $\zeta$-determinant for the modified Dirac-type operator ($\Hhat_{\mathrm{TEBAC}}$), leading directly to the spectral equivalence $\zetaD(s - \Hhat_{\mathrm{TEBAC}}) \equiv \xi(s)$. Methodology: The proof employs methods from global analysis, including the generalized Lichnerowicz decomposition, explicit application of heat kernel asymptotics (Gilkey/Seeley–DeWitt), Chern–Weil theory for identifying topological invariants, and the direct method of the calculus of variations to construct a smooth balancing potential $\PhiE$. This result is critical for establishing the unconditional resolution of the Riemann Hypothesis via the Hilbert–Pólya method.