Osterwalder-Schrader Reflection Positivity in the Colour-Singlet Sector Hᶢ

This paper establishes the Osterwalder–Schrader Reflection Positivity axiom (OS3) for Yang–Mills theory projected by the Reynolds Projector P̂G, within the Holographic Vacuum Elasticity (HVE) framework governed by the Vacuum Suppression Law (VSL): Oobs = Oideal · exp(−χ · σ0G · W(x) · Ω3 · fG). 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 HVE resolution proceeds in two steps. First (Section 2), we prove that HG = P̂GH is the strictly positive part of K: Lemma 2.1 (Gribov copies are non-singlets) shows that large gauge fluctuations generating zeros of the Faddeev–Popov determinant det(Mghost) reside entirely in non-trivial adjoint-representation colour sectors, so P̂G annihilates them exactly and the measure dμHVE supported on the first Gribov horizon Ω1 carries no copy contamination; Lemma 2.4 (BRST filter) shows that ghost fields likewise carry adjoint colour, so HG ↪ HBRST (BRST cohomology), which is the positive-definite subspace of K. Second (Section 4), OS3 on HG is proved using the modular involution JG of Tomita–Takesaki theory (Companion C2), the reality and positivity of the Euclidean Yang–Mills action restricted to Ω1, and the annihilation of negative-norm states by P̂G. The complete OS axioms OS1–OS5 are verified (Section 5), the Osterwalder–Schrader Reconstruction Theorem yields a Wight