Mass as a Spectral Level of a Closed Quartic Variational Functional

Mass is derived as a purely structural quantity emerging from the spectral hi-erarchy of a closed quartic variational functional S[Ψ]. No particles, elds, or con-tinuous parameters are introduced. The derivation follows a constrained emergencesequence in which the domain operator ΓD generates a discrete spectrum {ωk},entirely determined by the invariant set extracted from the Hessian.Each admissible conguration corresponds to a stable spectral mode satisfyingthe structural conditions of the functional. The only scalar quantity associated withsuch congurations is the spectral value ωk, which is therefore identied as mass.This leads to the relationmk = E⋆Lkwhere L = 1/4 is the contraction scale and E⋆ the global spectral scale.The resulting structure implies that mass is not a continuous parameter anddoes not arise from composition or interaction, but is instead a discrete label cor-responding to the position of a conguration within the spectral hierarchy. Theframework excludes the existence of arbitrary mass values and constrains the ad-missible spectrum to a nite, structured set.The result demonstrates that mass is not a fundamental property of physicalobjects, but an emergent consequence of the admissibility structure dened by thevariational functional.