UNCONDITIONAL GEOMETRIC-SPECTRAL RESOLUTION OF THE RIEMANN HYPOTHESIS: Topological Enforcement of the $A_1=0$ Balance Condition (TEBAC).

This work presents the unconditional resolution of the Riemann Hypothesis (RH), achieved within the rigorous framework of the Hilbert–Pólya Spectral Program. We construct a self-adjoint, Riemannian Dirac-type operator ($\widehat{H}_{\text{TEBAC}}$) on a specific 13-dimensional compact spin manifold ($M^{13}$) equipped with a balanced geometry. Through a stringent geometric and analytic process, we prove the absolute identity of its zeta-regularized spectral determinant and the completed Riemann zeta function, $\xi(s)$: $$\det_{\zeta}(s-\widehat{H}_{\text{TEBAC}})\equiv\xi(s)$$ The Unconditional Breakthrough: This spectral identity holds true only if the geometric-analytic Balance Condition ($A_1=0$) is satisfied, which is essential for achieving a canonical Hadamard factorization (order 1, genus 1) free of any affine ambiguity. We demonstrate that this condition is unconditionally enforced by the topology and curvature of the manifold via the Topological Enforcement and Balance Condition (TEBAC) selection rule. TEBAC compels the global disappearance of the critical Seeley–DeWitt coefficient $A_1$ through the cancellation of characteristic forms: $$\int_{M} A_1(x)\,\mathrm{d}V_g \;\propto\; \int_{M}\!\big(\Tr(F\wedge F)-\Tr(R\wedge R)\big)=0$$ This rigorous, geometrically-enforced balance guarantees that the spectrum of $\widehat{H}_{\text{TEBAC}}$ coincides precisely with the non-trivial zeros of $\xi(s)$, thus providing the unconditional resolution of the Riemann Hypothesis.