The Analytical Structure of the Arithmetic Potential $\Phi(x)$ and Unconditional Self-Adjointness: Convergence, Regularity, and Topological Enforcement of $A_1=0$

This technical preprint establishes and proves the fundamental analytical properties of the geometric-arithmetic potential $\Phi(x)=\sum_{p\in\mathbb P} f_p(x)\log p$, which defines the core spectral operator $\widehat{H}=\Dop+\Phi$ within the Hilbert–Pólya Program. We first prove the absolute convergence of $\Phi$ in the Sobolev space $H^s(M_{13})$ for $s>13/2$. This high regularity guarantees that $\Phi$ is a bounded, symmetric operator. By the Kato–Rellich Theorem, this establishes the unconditional self-adjointness ($\widehat{H}^*=\widehat{H}$) and the discrete real spectrum of $\widehat{H}$—a key requirement for the spectral resolution of the Riemann Hypothesis. The second and most crucial part integrates this analytical structure with the Topological Enforcement Theorem (TET). We prove that the Geometric-Arithmetic Balance condition ($A_1(\widehat{H}^2)=0$) is necessary and sufficient to satisfy the Topological Selection Rule ($c_2+p_1=0$). This result is critical, as $A_1=0$ is the requirement for the spectral equivalence $\det_{\zeta}(s-\widehat{H})\equiv\xi(s)$. The paper further relies on the key elliptic regularity of the potential ($\Phi \in C^\infty$) enforced by the topological constraint, as fully demonstrated in the companion work \cite{KaradzhovTET}.