Axiomatic Verification: Spectral Reconstruction of the Yang-Mills Hamiltonian (CGTP Vol. III)

Update in Version 2.0 (Rigorous Formalization): This version expands the theoretical framework to include an explicit functional verification of the Osterwalder-Schrader Axiom 3 (Reflection Positivity), reinforcing the mathematical foundations of the program. Explicit Verification: We provide a rigorous functional proof based on the Hubbard-Stratonovich transformation, formalizing the arguments presented in previous drafts. Ferromagnetic Measure: The derivation explicitly maps gauge-invariant observables to a dual scalar theory, certifying that the local Boltzmann weights correspond to ferromagnetic interactions (Modified Bessel functions). Unitarity Certification: By invoking the generalized FKG inequalities, this update provides a definitive mathematical certification of the positivity of the physical Hilbert space metric. This update aligns Volume III with the detailed analytical derivations provided in the "CGTP Companion v2.0". Abstract This document constitutes the mathematical culmination of the Constructive Gauge Theory Program, dedicated to the rigorous axiomatic verification and spectral reconstruction of pure Yang-Mills theory. We verify that the correlation functions constructed via the hybrid Renormalization Group flow in Volume II satisfy the Osterwalder-Schrader axioms (Regularity, Invariance, Reflection Positivity, and Clustering). Consequently, we reconstruct the physical Hilbert space and the Hamiltonian operator, proving the existence of a strictly positive mass gap in the energy spectrum on ℝ⁴. Series Context: This work constitutes Volume III of the Constructive Gauge Theory Program (CGTP). It provides the mathematical certification of the theory, transforming the constructive stability proofs of Volume II into a rigorous definition of a Relativistic Quantum Field Theory.