NSk--Wightman Correlator and Osterwalder--Schrader Reconstruction Core for Fields in the NSk/ψ Program

NSk--Wightman establishes a minimal bosonic correlator and Osterwalder-Schrader reconstruction core for fields in the NSk/ψ program. The aim of the module is to isolate and close the part of the formalism that allows one to pass from Euclidean Schwinger correlator families to a Hilbert space, a positive-time semigroup, and an energy generator, without yet developing the full Minkowski layer. In its pure layer, the module organizes Schwinger correlator families, time reflection, Osterwalder-Schrader positivity, and the quotient construction of the pre-Hilbert and Hilbert spaces. It then defines the vacuum, the positive-time translation semigroup, its generator, and the minimal compatibility package required for downstream YM/Wightman applications. The main core result states that a positive spectral gap of the generator implies exponential decay of the reduced two-point correlator. In this way, the module provides a precise bridge between OS structure, the notion of vacuum, semigroup evolution, and correlation decay rates. Version 1 is intentionally restricted to the ungraded bosonic OS reconstruction core. It does not yet develop the full Minkowski reconstruction, full field operators on the reconstructed Hilbert space, or the fermionic version of the formalism. Its role is to provide a compact, self-contained, and citable base module for further field-theoretic work within the NSk/ψ architecture.