Exponential Euclidean Decay and Spectral Support Separation in Osterwalder-Schrader Reconstruction

We establish exponential Euclidean decay and spectral support separation conditions sufficient for Osterwalder–Schrader reconstruction of relativistic quantum field theories. The analysis isolates a structural separation between the spectral support of excitations and Euclidean decay scales, providing a framework in which reflection positivity and spectral positivity jointly enforce reconstruction. The results clarify how Euclidean correlation decay encodes the spectral structure of physical excitations and provide a pathway toward spectral-gap formulations in constructive quantum field theory.