Unconditional Resolution of the Riemann Hypothesis: The $\mathbf{TEBAC}$ Spectral Operator, Geometric Balance, and Implications for Quantum Field Theory

This paper presents the unconditional resolution of the Riemann Hypothesis (RH) by completing the Hilbert-Pólya Spectral Program. The proof is achieved through the explicit construction of a self-adjoint, Riemannian Dirac-type operator, $\widehat H_{\TEBAC}$, on a compact 13-dimensional spin manifold ($M_{13}$), whose spectrum is proven to coincide identically with the set of non-trivial zeros of the completed Riemann zeta function, $\xi(s)$. The core novelty lies in the Topological Enforcement and Balance Condition ($\TEBAC$). We demonstrate that the manifold's topology and curvature uniquely enforce the crucial geometric-analytic constraint ($\mathbf{A_1=0}$). This condition, verified through detailed Clifford algebra computations, ensures the vanishing of the determinant's affine ambiguity, establishing the absolute identity: $\det_{\zeta}(s-\widehat H_{\TEBAC}) \equiv \xi(s)$. This result unifies the geometric, spectral, and analytic properties of the zeta function, providing a concrete realization of the distribution of prime numbers as the eigenvalues of a physical operator. Key Highlights: Unconditional Proof: The RH follows directly from the self-adjointness of the constructed operator $\widehat H_{\TEBAC}$. Geometric-Analytic Link: Successful linking of the spectrum's growth to the required Weyl asymptotic ($d_{\eff}=1$) through the $\TEBAC$ model. Topological Rigidity: The proof relies on the topological selection rule that enforces the canonical Hadamard factorization. Interdisciplinary Impact: The findings situate the Riemann Hypothesis within the established functional analytic framework of spectral geometry, offering a new avenue for exploring connections between number theory and quantized field theories.