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000179 | \subsection{Tempered admissible Schwinger families}
000180 | 
000181 | \begin{lemma}[Temperedness and moment bounds for admissible Schwinger functions]
000182 | \label{lem:tempered-schwinger-bounds}
000183 | On the Euclidean continuation of $\mathcal U_{\mathrm{scat}}$, each truncated admissible Schwinger
000184 | distribution $S_n^T$ defines a tempered distribution. Moreover, for every compact $K$ and every
000185 | source monomial of total degree $n$, its moments obey polynomial bounds determined by the same
000186 | uniform heat-kernel estimates and finite local moments of the admissible constructive measure.
000187 | \end{lemma}
000188 | 
000189 | \begin{proof}
000190 | By \TFPTxref{thm:well-posed-primitive-dynamics}, the admissible Euclidean generator is realized by a
000191 | self-adjoint elliptic operator on every blocked volume. Together with the constructive measure of
000192 | \TFPTxref{thm:constructive-geometric-measure}, this yields uniform heat-kernel bounds and finite moments
000193 | for local source insertions. Differentiating $W_{\mathrm{rel}}$ with respect to the local sources
000194 | therefore produces distributions whose derivatives are controlled by polynomial combinations of the
000195 | same bounds. Passing to the admissible projective limit preserves these polynomial estimates, so the
000196 | resulting Schwinger family is tempered.
000197 | \end{proof}
000198 | 
000199 | \begin{lemma}[Graded Euclidean covariance and symmetry on \texorpdfstring{$\Padm$}{Padm}]
000200 | \label{lem:graded-euclidean-covariance-padm}
000201 | On the Euclidean continuation of $\mathcal U_{\mathrm{scat}}$, the admissible Schwinger family is
000202 | graded Euclidean covariant, graded symmetric under exchange of external legs, and continuous in the
000203 | Schwartz topology.
000204 | \end{lemma}
000205 | 
000206 | \begin{proof}
000207 | The Dirac-type principal symbol of the geometric anchor and the low-curvature reduction imply local
000208 | Euclidean covariance of the admissible source functional on $\mathcal U_{\mathrm{scat}}$. Bosonic
000209 | source derivatives commute, while fermionic derivatives anticommute, so the mixed Schwinger family
000210 | is graded symmetric with the usual CAR signs. Continuity follows from the tempered moment bounds of
000211 | \Cref{lem:tempered-schwinger-bounds}, which control the action of $S_n^T$ on Schwartz test
000212 | functions.
000213 | \end{proof}
000214 |