TFPT scientific source excerpt
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000026 |     d\nu(t)\ge 0;
000027 |     \end{equation}
000028 |     \item the seam collar is reflection-regular so that the doubled operator and admissibility complex defined later are well posed on the same datum.
000029 | \end{enumerate}
000030 | \end{definition}
000031 | 
000032 | \begin{assumption}[Almost commutative factorization of the Hilbert space]
000033 | \label{ass:almost-commutative-factorization}
000034 | The reduced datum admits an almost commutative factorization
000035 | \[
000036 | \mathcal H\cong L^2(\widetilde M,S)\otimes \Hint,
000037 | \]
000038 | with $S\to\widetilde M$ the spinor bundle on the oriented double cover and $\Hint$ a finite
000039 | internal Hilbert space.
000040 | \end{assumption}
000041 | 
000042 | \begin{theorem}[Primitive kinematic scaffold reconstructed from the admissible reduced datum]
000043 | \label{thm:primitive-datum-from-reduced-datum}
000044 | \label{thm:primitive-scaffold-from-reduced-datum}
000045 | Let $\Tmin^{\mathrm{adm}}$ be as above. Then the primitive kinematic scaffold is reconstructed
000046 | canonically as
000047 | \[
000048 | \widetilde M=\mathrm{Spec}\,Z(\mathcal A),\qquad
000049 | M=\widetilde M/\inv,\qquad
000050 | \Seam=\mathrm{Fix}(\inv),\qquad
000051 | X=\widetilde M\setminus \Seam,
000052 | \]
000053 | \[
000054 | D_{\mathrm{ref}}:=\frac12(D+\inv D\inv),\qquad
000055 | D_{\mathrm{rel}}:=\frac12(D-\inv D\inv),\qquad
000056 | \chiseed:=\lambda_1^+(|B_\Sigma|)^{-1}.
000057 | \]
000058 | Hence the primitive scaffold
000059 | \[
000060 | \Tprim^{\mathrm{kin}}
000061 | :=
000062 | (X,\widetilde M,M,\Seam,\inv,D_{\mathrm{ref}},D_{\mathrm{rel}},f,\chiseed)
000063 | \]
000064 | is derived, not assumed.
000065 | If one further chooses an almost-commutative factorization
000066 | \[
000067 | \mathcal H_{\mathrm{prim}}\cong L^2(X,S)\otimes \Hint,
000068 | \]
000069 | then $S$ and $\Hint$ refer to that chosen factorization and are not claimed as
000070 | additional reconstruction outputs of the present theorem.