5-On the Necessity of Conjugated Spectral Contributions in Vacuum Energies

Recent structural analyses of Casimir-type vacuum energies have shown that finite residues emerge as robust invariants once universal quadratic divergences are removed. These results raise a fundamental question: what is the minimal structural mechanism capable of producing such stable spectral residues without reliance on arbitrary regularization schemes?In this work, we address this question from a deliberately model-independent perspective.We analyze the general structure of vacuum energy subtractions and show that constructionsbased on a single spectral contribution, whether through cutoffs or internal filtering, fail toproduce invariant and scheme-independent residues. Such approaches inevitably lead toambiguities, instability under perturbations, or dependence on microscopic details.We demonstrate that the existence of a stable and reproducible vacuum residue requiresthe presence of at least two conjugated spectral contributions sharing the same dominantasymptotic growth. In this minimal setting, the leading divergences cancel structurally, whilea finite residual term remains invariant under admissible perturbations.Our results establish a necessity theorem for conjugated spectral structures in vacuumenergy constructions. While no specific physical interpretation is imposed at this stage, thisframework provides a natural foundation for more detailed models of vacuum structure, tobe developed elsewhere.