On the Spectral Decomposition and Arithmetic Degeneracies of Isolated Multiplicity Dirichlet Series

We formalize a universal spectral decomposition framework for infinite series de-fined over discrete parameter spaces embedded in algebraic number rings. By isolating thearithmetic baseline capacity from the topological geometric sum, we construct the isolatedmultiplicity series MΓ(s). For geometrically massive thin arithmetic groups (δ > 1/2), werigorously establish that MΓ(s) absorbs the strict exponential explosion of trace collisions, pos-sessing an abscissa of absolute convergence exactly equal to the Hausdorff dimension δ. Webypass heuristic integral bounds by deriving this phase transition directly via non-negativearithmetic excess coefficients and Patterson-Sullivan spectral bounds. Furthermore, utilizingFredholm determinants of nuclear transfer operators and shifted Hadamard factorizations, weestablish the meromorphic continuation of MΓ(s) into the critical strip.