Mellin-Space Prony Representability of Linear Viscoelastic Models
Linear viscoelastic materials are commonly described by continuous relaxation spectra, yet practical measurements and simulations employ discrete Prony series. In the Laplace frequency domain, the distinction is well understood: rational transfer functions admit finite Prony representations, while fractional models with branch cuts do not. This work provides a complementary and structurally deeper characterization in the Mellin transform domain. We prove that a viscoelastic modulus admits an exact finite Prony series if and only if the arithmetic pole lattices of its Mellin kernel align with the integer lattice of the constitutive kernel, and the associated residues satisfy decoupled first-order recurrences along aligned sublattices. Unlike the Laplace-domain rational/non-rational dichotomy, the Mellin criterion reveals the arithmetic geometry underlying finite representability, which requires Diophantine alignment of infinite pole progressions and the compatibility of their residues. Applying this criterion yields a complete model taxonomy. Classical spring-dashpot networks (Maxwell, standard linear solid) satisfy the alignment and recurrence conditions. In contrast, fractional models (power-law, Cole-Cole, Havriliak-Negami, Zener) and log-normal spectra violate one or both conditions and require infinite Prony ladders for exact representation. The framework thus shifts the question of finite network realizability from an algebraic condition on rational functions to a geometric condition on pole lattices, offering both a theoretical classification and a practical computational test.