4-Casimir Energies as Structural Residues of Normalized Mode Sums

Casimir-type vacuum energies arise from differences between formally divergent mode sums associated with distinct physical configurations. While a variety of regularization procedures are commonly employed to extract finite results, the structural origin of these finite residues remains only partially understood. In this work, we adopt a deliberately minimal and model-independent perspective basedon normalized mode sums. We show that a broad class of Casimir-type constructions can bereformulated as comparisons between quadratic spectral sums whose dominant divergencesare universal and cancel structurally. Once this leading behavior is factored out, the remaining finite contribution is governed by coarse properties of the mode organization, chiefly theasymptotic density of admissible modes.Building on recent results concerning deterministic subsequences of linearly growing spectra, we establish a rigidity property for normalized sums: whenever an asymptotic densityexists, the normalized sum converges toward a universal limit independent of microscopicdetails. Finite-size convergence profiles, however, retain sensitivity to the geometric organization of the spectrum.Within this framework, Casimir energies emerge as structural spectral residues associatedwith admissible mode partitions imposed by boundary geometry and physical constraints,rather than as artifacts of specific regularization schemes. Numerical simulations confirmthe analytical results and demonstrate the failure of stochastic mode selection to producestable residues.These findings suggest a reinterpretation of vacuum energies in terms of spectral organization and admissibility, providing a unified and physically transparent framework forCasimir-type phenomena.