TFPT scientific source excerpt
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000047 | \iC:=2P_C-1.
000048 | \]
000049 | From that boundary polarization and the exact double one reconstructs the deck involution
000050 | $\taudbl$, the physical quotient $M=\widetilde M/\taudbl$, the seam
000051 | $\Seam=\mathrm{Fix}(\taudbl)$, the reference/relative operator split
000052 | $(D_{\mathrm{ref}},D_{\mathrm{rel}})$, the primitive seam response
000053 | $\chiseed=\lambda_1^+(|B_\Sigma|)^{-1}$, and the doubled closed datum
000054 | $\Tmin^{\mathrm{cl}}$. The seam is therefore no longer primitive ontology but the fixed locus of
000055 | the deck involution extracted from the one-sided boundary problem.
000056 | \end{remark}
000057 | 
000058 | \subsection{Primitive relative spectral dynamics}
000059 | 
000060 | The carrier theorem is not intended to float above an unspecified microscopic dynamics. The
000061 | starting point is therefore the one-sided boundary datum from which the primitive layer, the
000062 | derived closed datum, and the canonical admissibility projector are reconstructed before the later
000063 | closure theorems are invoked.
000064 | 
000065 | \begin{definition}[One-sided boundary admissibility datum]
000066 | \label{def:reduced-theorem-datum}
000067 | The one-sided boundary admissibility datum is
000068 | \[
000069 | \Tbound:=(\mathcal A_+,\mathcal H_+,D_+,J,\Gamma,B_\Sigma)
000070 | \]
000071 | with the following built-in admissibility conditions:
000072 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000073 |     \item $Z(\mathcal A_+)$ is spectrally regular and its Gelfand spectrum is a smooth compact oriented manifold with boundary $M_+$;
000074 |     \item $B_\Sigma$ is a self-adjoint tangential boundary operator on $\Sigma:=\partial M_+$ compatible with the boundary restrictions of $J$ and $\Gamma$;
000075 |     \item in a collar neighborhood of $\Sigma$, the Dirac operator has canonical product form
000076 |     \[
000077 |     D_+=\gamma_n(\partial_n+B_\Sigma);
000078 |     \]
000079 |     \item the boundary collar is reflection-regular so that canonical doubling, the Calder\'on projector, and the admissibility complex are well posed on the same datum.
000080 | \end{enumerate}
000081 | \end{definition}
000082 |