Constructive Programme Phase 2 Osterwalder

This document proves the Osterwalder–Schrader Reflection Positivity axiom (OS3) for the Yang–Mills theory projected by the Reynolds Projector . The central difficulty is that Yang–Mills theory with Faddeev–Popov ghosts lives naturally in a Krein space K (a Hilbert space with indefinite inner product) where reflection positivity fails for the full space. The T-DFT argument resolves this in two steps: (1) Elimination of Ghosts (Section 2): We prove that is the strictly positive part of the Krein space. The key ingredient is that Faddeev–Popov ghost fields carry colour in the adjoint representation and are therefore not colour singlets; thus, the Reynolds Projector annihilates them exactly. Formally: which is the positive-definite subspace of K. (2) Proof of OS3 (Section 4): We prove Reflection Positivity on using: (a) the modular involution JG of Tomita–Takesaki established in Companion C2; (b) the reality and positivity of the Euclidean Yang–Mills action restricted to the first Gribov horizon; and (c) the categorical annihilation of negative-norm states by the topological projection. By the Osterwalder–Schrader Reconstruction Theorem, the T-DFT Yang–Mills theory in the sector admits a Wightman QFT on Minkowski space. The complete OS axioms (OS1–OS5) are verified, fulfilling all constraints (R1 through R3) of the Jaffe–Witten Millennium Problem statement. Main result: The projected theory satisfies all five OS axioms, guaranteeing the existence of a Wightman QFT with a deterministic mass gap: