The Hecke–Lerch Correspondence VIII: Matrix Architectures and the Resolution of the Non-Abelian Regulator Gap

This paper resolves the non-abelian obstruction in the Hecke–Lerch Correspondence by introducing a matrix-valued kernel architecture. It establishes three foundational theoretical advances:Derivation of the explicit Meijer G-representation of Archimedean kernels via inverse Mellin transform.Proof of a novel matrix-valued Hecke recurrence for Oakes numbers.Definition of the spectral impedance Z(K)=RK/Δ as a measure of arithmetic complexity.The work transforms Stark unit extraction from a problem of analytic continuation to one of matrix inversion and trace residuals, providing a stable linear-algebraic framework for explicit non-abelian class field theory.