TFPT scientific source excerpt
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000103 | \begin{definition}[Operational seed category]
000104 | \label{def:operational-seed-category}
000105 | Let $\Seedop$ denote the category whose objects are tuples
000106 | \[
000107 | \mathfrak S=(\mathfrak A_{\mathrm{loc}},\tau_t,\Theta,\omega,[u_\Sigma],\mathcal D_{\mathrm{coll}})
000108 | \]
000109 | satisfying:
000110 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000111 |     \item $\mathfrak A_{\mathrm{loc}}$ is a quasilocal Euclidean observable net with positive-time
000112 |     reflection $\Theta$;
000113 |     \item $\omega$ is reflection positive and clustering on the admissible local algebra, with finite
000114 |     relative action with respect to the declared reference sector;
000115 |     \item $[u_\Sigma]\neq 0$ is a nontrivial seam class;
000116 |     \item $\mathcal D_{\mathrm{coll}}$ is a first-order elliptic collar generator whose positive-side
000117 |     restriction is reflection regular.
000118 | \end{enumerate}
000119 | Morphisms are unitary intertwiners preserving the displayed structures.
000120 | \end{definition}
000121 | 
000122 | \begin{theorem}[Operational collar completion]
000123 | \label{thm:operational-collar-completion}
000124 | There exists a functor
000125 | \[
000126 | \Cseed:\Seedop\longrightarrow \mathbf{SBord}^{\mathrm{adm}}
000127 | \]
000128 | which assigns to every operational seed $\mathfrak S$ a canonically induced one-sided admissible
000129 | spectral bordism
000130 | \[
000131 | \Cseed(\mathfrak S)=\mathcal B(\mathfrak S)
000132 | \]
000133 | with induced boundary datum
000134 | \[
000135 | \mathfrak T_{\partial}(\mathfrak S)=(\mathcal A_+,\mathcal H_+,D_+,J,\Gamma,B_\Sigma).
000136 | \]
000137 | \end{theorem}
000138 | 
000139 | \begin{proof}[Proof architecture]
000140 | Reflection positivity and the positive-time net reconstruct the one-sided OS Hilbert space on the
000141 | collar. The first-order elliptic collar generator determines the boundary operator $B_\Sigma$,
000142 | while the nontrivial seam class gives the induced spectral-flow class. The resulting one-sided
000143 | tuple satisfies exactly the axioms of \Cref{def:admissible-spectral-bordism-category}.