Infrared Spectral Localization: Zsa-G Framework (Version 11)

This release (Version 11) extends the Zsa-G framework through three interconnected investigations addressing spectral stability, localization mechanisms, and thermodynamic aspects of infrared-regularized Yang–Mills-type models. The present version contains: 11.1 Relative Compactness and Essential Spectrum Stability for Gauge-Covariant Elliptic Operators This paper establishes deterministic and probabilistic criteria ensuring stability of the essential spectrum under gauge-covariant perturbations on unbounded domains. Relative compactness of resolvent differences is analyzed using KLMN theory, Rellich-Kondrachov compactness, Weyl's theorem, Sobolev embeddings, and Borel-Cantelli arguments. 11.2 Localization of Spectral Projectors via Auxiliary Fields This work develops a conditional localization framework for non-local spectral projectors arising in infrared-regularized gauge theories. The construction combines Hubbard-Stratonovich transformations, Dunford-Taylor contour representations, Bochner-Minlos theory, weak convergence methods, conditional Osterwalder-Schrader positivity mechanisms, BRST structures, and ultraviolet decoupling considerations. 11.3 Toward Non-Perturbative Thermodynamic Stability in Infrared-Regularized Yang–Mills Theory This paper investigates a conditional framework for thermodynamic stability in the infinite-volume limit. The analysis includes cluster-expansion methods, polymer representations, asymptotic decay bootstrap mechanisms via Riesz potentials, and a conditional reconstruction framework identifying future directions toward continuum measure reconstruction. These articles should be interpreted as contributions to a broader functional-analytic and constructive program. Statements concerning continuum Yang-Mills measures, reflection positivity, BRST cohomology, thermodynamic limits, and mass-gap generation remain explicitly identified as conditional whenever rigorous proofs are not presently available. Contents of previous versions: V1–V5:• Initial formulation of the Zsa-G framework and infrared spectral regularization program. V6:• Functional-analytic foundations.• Spectral localization concepts.• Early infrared stabilization mechanisms. V7:• Development of auxiliary-field localization strategies.• Spectral projector representations. V8:• Measure-theoretic and operator-theoretic refinements.• Preliminary continuum considerations. V9:• Spectral stability analysis.• Bounded perturbation framework.• Initial discussions of localization and gap formation. V10:• Functional-analytic spectral decomposition via the Faddeev-Popov operator.• Infrared localization of negative modes.• Conditional probabilistic gap framework.• Relative compactness, spectral localization, and continuum extensions.