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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 | 
000215 | \subsection{OS and CAR reconstruction on \texorpdfstring{$\Padm$}{Padm}}
000216 | 
000217 | \begin{theorem}[Tailored OS/CAR reconstruction on \texorpdfstring{$\Padm$}{Padm}]
000218 | \label{thm:tailored-os-car-reconstruction}
000219 | Let $\{S_n^T\}_{n\ge 1}$ be the admissible Schwinger family on the Euclidean continuation of
000220 | $\mathcal U_{\mathrm{scat}}$. If reflection positivity, temperedness, graded Euclidean covariance,
000221 | graded symmetry, and clustering hold on $\Padm$, then there exist:
000222 | \begin{enumerate}[label=(\roman*), leftmargin=1.8em, topsep=3pt, itemsep=2pt]
000223 |     \item a Hilbert space $\mathcal H_{\mathrm{OS}}$ with vacuum vector $\Omega$,
000224 |     \item a strongly continuous positive-energy semigroup $T_t=e^{-tH_{\mathrm{adm}}}$ with
000225 |     self-adjoint generator $H_{\mathrm{adm}}\ge 0$,
000226 |     \item a reconstructed graded local field net on $\Padm$ whose Euclidean boundary values are the
000227 |     original admissible Schwinger functions.
000228 | \end{enumerate}
000229 | Equivalently, the admissible Euclidean family reconstructs a Wightman/CAR theory on the admissible
000230 | sector.
000231 | \end{theorem}
000232 | 
000233 | \begin{proof}
000234 | Define the OS sesquilinear form by
000235 | \[
000236 | (F,G)_{\mathrm{OS}}:=\langle \Theta F,G\rangle_{\mathrm{rel}}
000237 | \]
000238 | for positive-time admissible observables. Reflection positivity from
000239 | \TFPTxref{thm:reflection-positivity-admissible} and the fermionic quotient-form analysis of
000240 | \Cref{app:os-fermions} show that this form is positive semidefinite on the full graded admissible
000241 | algebra. Quotienting by the null space and completing gives $\mathcal H_{\mathrm{OS}}$.
000242 | 
000243 | Positive Euclidean time translation preserves the null space and acts by contractions, hence defines
000244 | a strongly continuous contraction semigroup $T_t$ on $\mathcal H_{\mathrm{OS}}$. The dense domain of
000245 | positive-time polynomial observables gives the symmetric generator domain. By Hille--Yosida, the
000246 | closure of this generator is self-adjoint and nonnegative, so $T_t=e^{-tH_{\mathrm{adm}}}$ for a
000247 | self-adjoint $H_{\mathrm{adm}}\ge 0$.
000248 | 
000249 | The temperedness, covariance, graded symmetry, and clustering established in
000250 | \Cref{lem:tempered-schwinger-bounds,lem:graded-euclidean-covariance-padm} are precisely the
000251 | remaining OS/CAR reconstruction hypotheses. Therefore analytic continuation yields the reconstructed
000252 | Wightman/CAR family with vacuum $\Omega$. Because $\Theta$, $P_\Theta$, and $\Padm$ commute, the
000253 | entire reconstruction stays on the admissible sector $\Padm$.
000254 | \end{proof}