Remark. The scalar result and the gauge transfer question [fdip-007R]

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The sufficient criterion here comes from the compact-spin argument of Friedli–Velenik [friedli2017statistical, Lemma 10.8, pp. 442–443], with an explicit Gaussian cutoff passage. It concerns an ordinary scalar field with a chosen reflection. The classical small-spacing expansion of the Wilson action supplies no reflection-positive probability law by itself. Nor does integrating matter into a connection marginal establish such a property.

For gauge fields, the positive-character kernel conditions give a different starting point for an explicit finite-volume transfer construction. One must specify the convolution kernel, gauge projection and time-boundary convention and prove the operator properties. The failure of a quotient shift explains why those steps cannot be replaced by reflection positivity alone.

Osterwalder–Seiler [osterwalder1978gauge] provides historical context for gauge and fermion positivity. It is cited here for that context only; no theorem or proof from it is used to extend the scalar result. General gauge–fermion reflection positivity, an infinite-time vacuum construction and continuum reconstruction require separate hypotheses and arguments. Time here remains a chosen lattice direction, rather than an additional field observable.