3. On the l^2-Summability of the Möbius Core Sequence via Abel-Convergence and Infinite Operators

This paper establishes the structural inversion of an infinite-dimensional matrix operator associated with periodic Dirichlet series. By exploiting a rigid abelian cutoff condition anchored at the conformal boundary z=1, we introduce a unified regularizing inverse operator $U_g^{-1}$ that circumvents the non-associativity issues typical of infinite matrix algebra outside the domain of absolute convergence. We demonstrate that the explicit evaluation of this unified framework generates the sequence of partial sums of the Möbius function scaled by their indices. Leveraging the Abel-convergence of the modified Dirichlet series at the specific affix $s=-3/2$, we prove that the auxiliary generating function belongs to the $C^1$ class on the unit circle, thereby ensuring that the core coefficient sequence is strictly square-summable $\ell^2$. As a direct global consequence of this bounded inversion process, the analytic regularity of the inverse system is uniquely extended over the entire half-plane $Re(s) > 1/2$.