Spectral Dynamics of Sieves in Cyclotomic Extensions: 𝑳-functions, Catalan's Constant, and Asymptotic Saturation

In this work, we extend the analytical principles of asymptotic modular filtering from the real axis to the complex plane by rigorously developing deterministic sieves over rings of algebraic integers O𝐾 . We investigate the spectral dynamics of residue classes in cyclotomic extensions, constructively demonstrating that the topology of search subspaces in the Gaussian ring Z𝑖 is intrinsically governed by the ramification properties and the underlying manifold structure of the Dedekind Zeta Function. It is analytically established that the critical coupling constant of the stochastic ensemble in Z𝑖 exhibits a geometric renormalization dependent on Catalan’s constant 𝐺, a phenomenon directly emerging from the evaluation of the spectral entropy density at 𝑠 = 2. We introduce the concept of the asymptotic parsimony threshold for extensions of degree 𝑑, formally proving that the Gaussian ideal primorial Π2 = ⟨3(1 + 𝑖)⟩ operates as an optimal dynamical attractor that bounds and minimizes the combinatorial divergence of the search lattice. Finally, we provide analytical evidence and rigorous bounds for the variance saturation in the exponent distribution of Mersenne primes, demonstrating the stabilization of asymptotic states towards regimes of low effective dimensionality. This spectral confinement phenomenon challenges the ergodicity assumptions (ETH) underlying the maximum entropy models commonly employed in algebraic lattices. Mathematics Subject Classification 2020: 11R42.Keywords: Analytic number theory, 𝐿-functions, Catalan’s constant, Cyclotomic extensions, Asymptoticdistribution, Arithmetic quantum chaos.