Regularization Independence for the Arithmetic Potential $\Phi=\sum_p f_p\log p$

This paper provides the crucial technical validation for the regularization scheme applied to the arithmetic potential $\Phi$ in the Spectral-Topological Program toward the Riemann Hypothesis (RH) \cite{KaradzhovRH}. The core result is a theorem on the independence of the renormalized quadratic functional $I_{\mathrm{ren}}=\int_M\Phi^2$, where $\Phi$ is defined as the sum over primes $\Phi=\sum_p f_p\log p$ on a compact Riemannian manifold $(M,g)$. We prove that the finite (renormalized) part of $I_{\mathrm{ren}}$ is equivalent across the three standard regularization methods: $\zeta$-Functional Regularization ($\mathrm{FP}_{s=0}\zeta_\Phi(s)$), Heat-Kernel Cutoff ($\mathrm{FP}_{t\downarrow0} I_{\mathrm{heat}}(t)$), Exponential Spectral Cutoff ($\mathrm{FP}_{\varepsilon\downarrow0} I_{\mathrm{exp}}(\varepsilon)$). The proof relies on establishing that the Dirichlet-type conditions (D1–D3) required for the analytic control of the arithmetic coefficients $f_p$ are sufficient to satisfy the necessary analytic hypotheses (H1–H3) for the classical Mellin transform and Seeley expansion theorems. This rigorous equivalence guarantees the unambiguous physical definition of the arithmetic energy quantity in the main program. Furthermore, we prove a triviality theorem showing that if the renormalized finite part $I_{\mathrm{ren}}$ vanishes, the potential must be identically zero ($\Phi \equiv 0$), establishing the fundamental non-negativity of the arithmetic energy. Keywords: Spectral Geometry, Zeta Regularization, Heat Kernel Asymptotics, Renormalization, Riemann Hypothesis, Arithmetic Potential, Mathematical Physics.