Semigroup Spectral Extraction from Euclidean Exponential Clustering

This companion note isolates and makes fully self-contained the “spectral extraction” step used in macroscopic mass-gap constructions for four-dimensional Yang–Mills (and similar theories). It shows that Osterwalder–Schrader (OS) positivity together with exponential decay of Euclidean time-separated correlations on a bounded local net of gauge-invariant observables at fixed macroscopic scale ℓ > 0 is sufficient to obtain a strictly positive spectral gap for the reconstructed Hamiltonian. All statements are formulated on the bounded macroscopic observable net O_ℓ (no sharp local operator-valued distributions or t → 0 limits are constructed). The logical chain is: OS reconstruction on the net yields a Hilbert space H with vacuum Ω and a strongly continuous positivity-preserving contraction semigroup T_t = e^{-tH} generated by Euclidean time translations. Exponential time clustering of the Schwinger functional (a direct consequence of Dobrushin contractivity or finite-depth entrance at the blocked scale) produces exponential upper bounds on the semigroup matrix elements ⟨[F̂], e^{-tH}[F̂]⟩ for centered observables F̂ = F − ⟨F⟩1. A standard spectral-theorem argument on the dense subspace of centered vectors (orthogonal to Ω) then excludes spectrum in the interval (0, m), yielding Spec(H) ⊂ {0} ∪ [m, ∞). The note provides: • Complete definitions of the bounded macroscopic net, OS-positive functional, and OS reconstruction package (Lemmas 3.2–3.4). • An elementary spectral exclusion lemma (Lemma 4.1) in the precise quantifier form produced by clustering bounds. • The main “entrance-to-gap” theorem (Theorem 5.5) with full proof, converting Euclidean exponential time clustering directly into the Hamiltonian spectral gap. • A corollary establishing uniqueness of the vacuum (one-dimensional ground-state sector) under the same hypotheses. • A short appendix lemma on the spectral theorem for completeness. No renormalization-group flow analysis, stable-manifold geometry, or model-dependent estimates appear. The only inputs are OS positivity (from the Euclidean theory) and exponential clustering (obtained upstream, e.g., via the finite-depth entrance mechanism of the companion Route 1 paper). The discussion clarifies that the relevant “time” parameter is the OS semigroup time, not an RG scale. This note can be cited independently as the precise bridge from Euclidean clustering to the continuum mass-gap statement on any bounded positive-scale observable algebra.