TFPT scientific source excerpt
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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 | 
000085 | 
000086 | % ---- Operational seed, induced boundary data, and primitive setup only: source lines 8907-9157 ----
000087 | \section{Intrinsic closed branch and canonical rigid object}
000088 | \label{sec:rg-map-compiler-outputs}
000089 | 
000090 | The closed branch is not selected by a comparison functional. At the top of the main-text theorem
000091 | stack now stands a thinner operational seed layer: quasilocal Euclidean dynamics, reflection
000092 | positivity, finite relative action, a nontrivial seam class, and one elliptic collar generator.
000093 | That seed induces the one-sided admissible boundary datum. Once the induced datum is fixed, the
000094 | remaining closure problem splits into one joint discrete admissibility theorem and one continuous
000095 | retained-sector variational problem. The discrete theorem fixes carrier, family, determinant, and
000096 | strong-CP data simultaneously; the continuous sector is the unique admissible critical point of one
000097 | projected modular free energy on the retained branch. Their compatibility determines one intrinsic
000098 | branch and therefore one canonical rigid object inside the theorem class.
000099 | 
000100 | \subsection{Operational seed and boundary generation}
000101 | \label{sec:operational-seed-generation}
000102 | 
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}.