TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--04_qft_source/sections/002-section-closure-theorems-from-the-admissibility-complex.tex
Original file SHA-256: 61ecee7727bdec47936bbffc028d3dbac5260a025b6dbcce30124a60d635fc9c
Inclusive original lines: 167-198
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000167 | \begin{theorem}[Calder\'on upgrade of the APS boundary domain]
000168 | \label{thm:apsdom-upgrade}
000169 | Let $\APSdomprov$ be the provisional APS realization of the geometric anchor used in
000170 | \TFPTxref{thm:well-posed-primitive-dynamics}, and let $D_b$ be the canonical doubling of
000171 | $D_{\mathrm{rel}}$ across $\Seam$. Let $C(D_b)$ denote the Calder\'on projector of $D_b$ on the
000172 | seam collar. Then there is a canonical boundary-elliptic upgrade
000173 | \[
000174 | \APSdomprov
000175 | \;\xrightarrow{\;C(D_b)\;}\;
000176 | \APSdom
000177 | \]
000178 | such that:
000179 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000180 |     \item $\APSdom$ lies in the same admissible elliptic class as $\APSdomprov$ and preserves the
000181 |     self-adjoint Fredholm realization of the geometric anchor;
000182 |     \item the relative graph domain of $D_{\mathrm{rel}}$ is unchanged, so all primitive results
000183 |     that depended only on $\APSdomprov$ continue to hold with $\APSdom$;
000184 |     \item the family-pairing data $\APSfam$ are induced canonically on the doubled admissible
000185 |     branch and are not a separate primitive datum.
000186 | \end{enumerate}
000187 | \end{theorem}
000188 | 
000189 | \begin{proof}[Proof sketch]
000190 | The Calder\'on projector $C(D_b)$ is canonically associated with the doubled operator on the
000191 | seam collar and produces a boundary-elliptic projection compatible with the provisional APS
000192 | choice. Both $\APSdomprov$ and $\APSdom$ are admissible elliptic boundary conditions for the
000193 | same Dirac-type operator and therefore define equivalent self-adjoint Fredholm realizations of
000194 | the geometric anchor (boundary-elliptic equivalence preserves the Sobolev graph domain).
000195 | Consequently every statement made on the primitive scaffold under $\APSdomprov$ persists on the
000196 | closed branch under $\APSdom$. The induction of $\APSfam$ from canonical doubling is a standard
000197 | consequence of the cobordism structure of the doubled APS pair.
000198 | \end{proof}