TFPT scientific source excerpt
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000033 | \begin{remark}[APS roles on the admissible branch]
000034 | \label{rem:aps-roles}
000035 | At the primitive level $\APSdomprov$ is the provisional elliptic boundary choice for the
000036 | geometric anchor: an analytic starting representative chosen so that
000037 | \TFPTxref{thm:well-posed-primitive-dynamics} applies. After the carrier theorem and the master
000038 | closure roadmap of \TFPTxref{sec:closure-theorems} the Calder\'on projector of the doubled operator
000039 | $D_b$ canonically promotes $\APSdomprov$ to the closed-branch boundary domain $\APSdom$ in the
000040 | same admissible class. The family-pairing data $\APSfam$ are then induced canonically and remain
000041 | a derived consequence rather than a second primitive input package. The explicit upgrade
000042 | statement is recorded as \TFPTxref{thm:apsdom-upgrade} in \TFPTxref{sec:closure-theorems}.
000043 | \end{remark}
000044 | 
000045 | \begin{figure}[t]
000046 | \centering
000047 | \scriptsize
000048 | \begin{tikzpicture}[
000049 |     root/.style={draw=black!65, fill=gray!10, rounded corners, text width=3.3cm, minimum height=0.9cm, align=center},
000050 |     prov/.style={draw=blue!60!black, fill=blue!6, rounded corners, text width=3.15cm, minimum height=1.45cm, align=center},
000051 |     dom/.style={draw=blue!70!black, fill=blue!10, rounded corners, very thick, text width=3.15cm, minimum height=1.45cm, align=center},
000052 |     fam/.style={draw=orange!75!black, fill=orange!8, rounded corners, text width=3.15cm, minimum height=1.45cm, align=center},
000053 |     note/.style={font=\footnotesize\itshape, text=black!70, align=center}
000054 | ]
000055 | \node[root] (aps) at (0,0.45) {APS roles in the present manuscript};
000056 | \node[prov] (prov) at (-4.05,-1.95) {\textbf{$\APSdomprov$}\\provisional APS realization\\(analytic starting choice)};
000057 | \node[dom]  (dom)  at (0,-1.95)    {\textbf{$\APSdom$}\\closed-branch APS domain\\via Calder\'on upgrade};
000058 | \node[fam]  (fam)  at (4.05,-1.95)  {\textbf{$\APSfam$}\\family pairing induced\\by canonical doubling};
000059 | \draw[->, thick] (aps.south west) -- ($(prov.north)+(0,0.05)$);
000060 | \draw[->, thick] (aps.south) -- ($(dom.north)+(0,0.05)$);
000061 | \draw[->, thick] (aps.south east) -- ($(fam.north)+(0,0.05)$);
000062 | \draw[->, thick, blue!70!black] ($(prov.north east)+(0.08,0.16)$)
000063 |     .. controls +(0.7,0.75) and +(-0.7,0.75) ..
000064 |     node[above, fill=white, inner sep=1.5pt, font=\scriptsize\itshape, text=blue!70!black]
000065 |     {Calder\'on upgrade}
000066 |     ($(dom.north west)+(-0.08,0.16)$);
000067 | \draw[->, dashed, black!45, line width=0.8pt] ($(dom.north east)+(0.08,0.14)$)
000068 |     .. controls +(0.7,0.65) and +(-0.7,0.65) ..
000069 |     node[above, fill=white, inner sep=1.5pt, font=\scriptsize\itshape, text=black!65]
000070 |     {induced}
000071 |     ($(fam.north west)+(-0.08,0.14)$);
000072 | \node[note] at (0,-3.2) {Provisional choice $\to$ Calder\'on upgrade $\to$ derived family pairing.};
000073 | \end{tikzpicture}
000074 | \caption{APS organization with Calder\'on upgrade and induced family pairing.}
000075 | \label{fig:aps-split}
000076 | \end{figure}
000077 | 
000078 | \begin{corollary}[Carrier-compatible Lorentzian dynamics after closure]
000079 | Once the later carrier-compatible fluctuation package and the admissibility closure hypotheses of \TFPTxref{sec:closure-theorems} are imposed, the resulting admissible Euclidean theory reconstructs a self-adjoint Lorentzian Hamiltonian $H_{\mathrm{adm}}$ and a one-parameter automorphism group
000080 | \[
000081 | \alpha_t(O)=e^{itH_{\mathrm{adm}}}Oe^{-itH_{\mathrm{adm}}}.
000082 | \]
000083 | \end{corollary}
000084 |