Spectral Localization, Probabilistic Gap Mechanisms, and Infrared Stabilization in Yang–Mills Theory - Zsa-G Framework

Zsa-G Framework — Versions V10 (1–3) This repository contains Versions V10 (1–3) of the ongoing Zsa-G research program on spectral structures, infrared localization, and probabilistic stabilization mechanisms in Yang–Mills theory. The primary objective of the program is to isolate mathematically tractable structural components related to: spectral coercivity, localization of unstable modes, operator-theoretic stability, infrared stabilization mechanisms, and conditional probabilistic gap formation. The present collection focuses on three interconnected directions: Functional-analytic spectral decomposition of the Yang–Mills Hessian, Infrared localization of negative modes via the Faddeev–Popov operator, Conditional probabilistic mechanisms for spectral stabilization under curvature suppression assumptions. The framework is formulated explicitly as a conditional and structural program. In particular: ultraviolet coercivity results are established in finite volume under boundedness assumptions, infrared localization is proven via spectral projector methods, KLMN-based self-adjointness and relative compactness arguments are developed rigorously, probabilistic gap mechanisms are formulated conditionally on suitable tail bounds, Osterwalder–Schrader positivity is reduced to a contour-analytic positivity condition, the full continuum and thermodynamic limits remain open. The work does not claim a solution of the Yang–Mills mass gap problem. Rather, it isolates functional-analytic and probabilistic substructures which may contribute to a future constructive framework in non-abelian gauge theory. Included Manuscripts V10-1 Infrared Spectral Localization and Ultraviolet Coercivity in Yang–Mills Theory Main themes Infrared spectral localization Ultraviolet coercivity Spectral projector decomposition Localization of negative modes Quantitative coercivity estimates Functional-analytic Hessian analysis Landau-gauge cancellation mechanism Finite-volume spectral structure Main structural contribution Establishes that negative modes of the Yang–Mills Hessian are confined to a spectrally localized infrared sector under explicit coercivity assumptions. V10-2 Probabilistic Spectral Gap Mechanisms in Yang–Mills Theory Main themes Probabilistic spectral-gap mechanisms Good/bad configuration decomposition Variance decomposition methods Infrared stabilization Conditional tail suppression Effective coercivity mechanisms Finite-volume probabilistic structure Conditional thermodynamic-limit framework Main structural contribution Combines deterministic coercivity with conditional probabilistic suppression mechanisms to formulate a possible route toward effective spectral stabilization. V10-3 Functional-Analytic and Operator-Theoretic Foundations for Infrared Stabilization Main themes KLMN theorem and form-bounded perturbations Relative compactness Weyl stability Self-adjointness of gauge operators Spectral stability under bounded gauge fields Scale-separated Hessian decomposition Operator comparison methods Contour-based positivity framework Main structural contribution Develops the functional-analytic backbone underlying the Zsa-G framework and formulates the perturbative operator structure in finite volume. Archive Overview (V1–V9) V1–V5 Early developmental versions of the foundational manuscript introducing: infrared stabilization concepts, bounded-gauge-field spectral analysis, preliminary coercivity mechanisms, initial contour-positivity ideas, first probabilistic formulations. V6 First major structural expansion introducing: scale-separated Hessian decomposition, spectral projector methods, ultraviolet/infrared decomposition, functional-analytic operator comparisons. V7 Development of: relative compactness arguments, KLMN self-adjointness methods, finite-volume spectral stability, thermodynamic-limit roadmaps. V8 Expansion toward: probabilistic spectral-gap mechanisms, variance decomposition methods, large-curvature suppression heuristics, infrared localization structures. V9 Consolidation and restructuring phase introducing: refined theorem architecture, explicit conditional formulations, improved localization proofs, finite-volume probabilistic structures, contour-based positivity reductions. Repository Links V1: https://zenodo.org/records/19499695 V2: https://zenodo.org/records/19678010 V3: https://zenodo.org/records/19738398 V4: https://zenodo.org/records/19793932 V5: https://zenodo.org/records/20028457 V6: https://zenodo.org/records/20055965 V7: https://zenodo.org/records/20089499 V8: https://zenodo.org/records/20209024 V9: https://zenodo.org/records/20355677 Current Status of the Program Direction Status Functional-analytic finite-volume structure Partially established Relative compactness framework Established in finite volume Ultraviolet coercivity mechanism Structurally established Infrared localization mechanism Established under stated assumptions Probabilistic spectral-gap mechanism Conditional Reflection positivity Reduced to contour positivity problem Continuum thermodynamic limit Open Full Yang–Mills mass gap problem Open The manuscripts are intended as part of an ongoing exploratory research program in mathematical physics focused on spectral structures, operator theory, and infrared stabilization mechanisms in non-abelian gauge theory.