Constructive Programme Phase 1 the Reeh–schlieder Property in the Physical Hilbert Space Hᵍ of the T-DFT Framework via Reynolds Projection

This document proves that the physical vacuum vector is cyclic and separating for the local von Neumann algebra for every open region , within the physical Hilbert space of the T-DFT framework. The proof follows the classical Reeh–Schlieder argument adapted to the colour-singlet sector projected by the Reynolds Projector . The necessary conditions—Poincaré covariance of the vacuum and the strict spectral condition with mass gap —have already been established in previous companion documents. The new contributions are: (1) Formalisation of as a Haag–Kastler von Neumann algebra network; (2) Proof that the Reynolds Projection preserves the analyticity of the projected Green functions (Lemma 3.3); (3) Closure of the analytic-continuation argument for cyclicity and separation (Theorem 4.1). Main result: The original Reeh–Schlieder theorem applies without modification to the projected sector , because the Reynolds Projector preserves all hypotheses of the theorem. This result opens access to the Tomita–Takesaki modular theory on the physical sector—the technical tool required for the Osterwalder–Schrader reconstruction of the constructive programme.