9-A Closure Theorem for Stable Vacuum Residues

Divergent spectral sums arise ubiquitously in theoretical physics, yet physically meaningful quantities consistently emerge as finite and reproducible residues once universal divergences are removed.While a wide variety of regularization and subtraction procedures are commonly employed, the structural origin and uniqueness of such residues remain only partially understood.In this work, we adopt a deliberately abstract and model-independent perspective to investigate the minimal structural requirements for the existence of finite, invariant, and reproducible residues extracted from divergent spectral sums. We formulate a set of minimal axioms encoding universal asymptotic growth, admissible perturbations, and reproducibility, without reference to any specific physical realization or regularization scheme.Within this framework, we prove that residue extraction schemes based on a single spectral contribution are intrinsically unstable or trivial. We then introduce the notion of conjugated spectral contributions and show that differences between such contributions provide a natural and intrinsic mechanism for canceling universal divergences. Our main result is a closure and uniqueness theorem: any admissible residue extraction scheme satisfying the axioms is structurally equivalent, up to normalization, to a two-sector scheme based on a conjugated pair of spectral contributions.We further demonstrate that multi-sector constructions factorize into sums of conjugated pairs and that the two-sector structure constitutes a minimal normal form for admissible residue extraction schemes. Apparent alternatives based on cutoffs, stochastic filtering, smoothing, or averaging are shown either to reduce to this structure or to violate the axioms.These results identify two-sector conjugation as a structural necessity rather than a model choice.By shifting the focus from regularization techniques to the organization of spectral contributions, the present work provides a rigorous foundation for the stability of vacuum residues and establishes universal constraints that any consistent physical realization must satisfy.