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000001 | \subsection{Primitive relative spectral dynamics}
000002 | 
000003 | The carrier theorem is not intended to float above an unspecified microscopic dynamics. The
000004 | starting point is therefore the one-sided boundary datum from which the primitive layer, the
000005 | derived closed datum, and the canonical admissibility projector are reconstructed before the later
000006 | closure theorems are invoked.
000007 | 
000008 | \begin{definition}[One-sided boundary admissibility datum]
000009 | \label{def:reduced-theorem-datum}
000010 | The one-sided boundary admissibility datum is
000011 | \[
000012 | \Tbound:=(\mathcal A_+,\mathcal H_+,D_+,J,\Gamma,B_\Sigma)
000013 | \]
000014 | with the following built-in admissibility conditions:
000015 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000016 |     \item $Z(\mathcal A_+)$ is spectrally regular and its Gelfand spectrum is a smooth compact oriented manifold with boundary $M_+$;
000017 |     \item $B_\Sigma$ is a self-adjoint tangential boundary operator on $\Sigma:=\partial M_+$ compatible with the boundary restrictions of $J$ and $\Gamma$;
000018 |     \item in a collar neighborhood of $\Sigma$, the Dirac operator has canonical product form
000019 |     \[
000020 |     D_+=\gamma_n(\partial_n+B_\Sigma);
000021 |     \]
000022 |     \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.
000023 | \end{enumerate}
000024 | \end{definition}
000025 | 
000026 | \begin{remark}[Almost-commutative realizations]
000027 | \label{ass:almost-commutative-factorization}
000028 | When a factorized realization is useful, one may choose
000029 | \[
000030 | \mathcal H\cong L^2(\widetilde M,S)\otimes \Hint,
000031 | \]
000032 | with $S\to\widetilde M$ the spinor bundle on the oriented double cover and $\Hint$ a finite
000033 | internal Hilbert space. This is an analytic realization choice rather than part of the boundary
000034 | datum.
000035 | \end{remark}