SPECTRAL VACUUM MECHANISM — PART XXXVI Spectral Derivation of String Tension and Confinement from Stable Vacuum Structure

Abstract We derive the emergence of confinement-like dynamics from stable spectral vacuum structure within the Spectral Vacuum Mechanism (SVM). The theory is defined by the quartic spectral action S[H] = −α Tr(H²) + β Tr(H⁴), α > 0, β > 0, whose nontrivial stationary configuration generates a unique intrinsic vacuum eigenvalue scale λ₀² = α / (2β). This scale is the structural origin of all subsequent physical quantities. From vacuum stability and positivity of the second variation, a spectral rigidity emerges. When restricted to correlated collective fluctuations, this rigidity produces an effective one-dimensional elastic sector with explicit string tension T_eff = α² / β. Thus the sequence vacuum scale → spectral rigidity → string tension is derived directly from the stable spectral action, without introducing fundamental strings or gauge-field functional integration. In the semiclassical long-wavelength regime, this elastic sector yields Wilson area-law behavior. The corresponding bound-state scale satisfies m_G² ∼ c_g² T_eff, establishing the structural relation string tension → mass gap scale. Finally, we demonstrate a controlled scaling correspondence between spectral tension and curvature scale, T_eff = C_geom α Λ², showing that vacuum scale → confinement → mass generation → geometric scaling arise from a single intrinsic spectral eigenvalue structure. The derivation is semiclassical but mathematically controlled, relying solely on vacuum stability, positivity of the quadratic form, and large-N collective organization. Keywords: Spectral vacuum; String tension; Confinement; Wilson loop; Glueball mass; Large-N dynamics; Emergent locality; Spectral geometry; Vacuum rigidity; Nonperturbative dynamics. Other works by the author on a related topic: [1] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXIX: Spectral Yang–Mills Dynamics: Field Equations, Running Coupling, and Confinement from Vacuum Geometry, Zenodo. DOI: 10.5281/zenodo.18624234 (2026). [2] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXX: Emergent Geometry from Spectral Flow, Zenodo. DOI: 10.5281/zenodo.18636847 (2026). [3] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXIV: Quantum Spectral Flow and Vacuum Stability, DOI: 10.5281/zenodo.18704453 (2026). [4] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXV: Spectral Strings and Emergent Worldsheet Dynamics, DOI: 10.5281/zenodo.18726006 (2026).