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