Spectral Vacuum Mechanism - Part XV Unification of the mass formula in SVM particles of the Standard Model

Abstract This paper formulates and tests the Spectral Vacuum Mechanism (SVM) as a specific mechanism within the Vacuum Spectral Model (VSM), in which particle masses emerge from the second-order action spectrum against the background of a universal vacuum configuration. The elementary masses are related to the eigenvalues of the spectral operator L̂ via m²(c,n)=λ_n^(c), where c∈{ξ,φ,χ} is the vacuum channel and n is the excitation number. The key result is the derivation of a universal mass formula, equally applicable to all families and sectors: leptons, quarks and hadrons. For composite states (mesons, baryons) the mass is expressed as the sum of the contributions of the Σ_k √λ_k modes and the calculated interacting corrections E_bind[ψ] and E_kin[ψ], M(composite) = Σₖ₌₁ᴺ √λₖ + E_bind + E_kin which makes it possible to obtain both ground states and excited spectra. Numerical calculations show that for a fixed set of physical parameters of the vacuum profile and minimal channel deformation, an accuracy of about 0.002–5% is achieved for a wide range of particles without individual fitting by family. Thus, for the first time within the VSM/SVM framework, a single (universal) formula for particle masses has been obtained, covering all families in a single computational and conceptual framework, with an explicit comparison with the Standard Model and alternative scenarios. Next, we propose to develop a series of SVMs along the lines of theoretical (constructive) justification: we formulate the model in a strictly gauge-invariant setting, control the restriction and removal of the cutoff (j_max → ∞), and link spectral estimates with non-local diagnostic criteria (for example, via Wilson loops). Keywords Spectral Vacuum Mechanism (SVM); Vacuum Spectral Model (VSM); mass generation; universal mass formula; second variation spectrum; domain wall; eigenvalues; hadronic spectroscopy; reggae patterns; minimum of free parameters. Other related works by the author: [1] M. Nemirovsky,Physical Vacuum – Part I Spectral vacuum mechanism (SVM) (lepton masses), https://zenodo.org/records/17931024 (2025) [2] M. Nemirovsky,Physical Vacuum – Part II Spectral vacuum mechanism (SVM) (quark sector), https://zenodo.org/records/17923554 (2025) [3] M. Nemirovsky,Physical Vacuum – Part III Composite (hadronic) excitations in the Spectral Vacuum Mechanism (SVM), https://zenodo.org/records/17923690 (2025) [4] M. Nemirovsky,Physical Vacuum – Part IV Spectral vacuum mechanism (SVM) Hadronic sector in the Spectral Vacuum Mechanism: baryons and mesons from the universal gate phase, https://zenodo.org/records/17942901 (2025)