TFPT scientific source excerpt
Original repository path: documents/archive--paper-latex--tfpt-31/sections/014-section-downstream-cosmology-module-on-the-observation-c.tex
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Inclusive original lines: 4-24
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000004 | \begin{lemma}[Automatic seam-transfer convergence]
000005 | \label{lem:automatic-seam-transfer-convergence}
000006 | On the admissible branch the seam transfer operator
000007 | \[
000008 | \mathcal U_\Seam:=\Padm e^{-\ell_\Seam|B_\Seam|}\Padm
000009 | \]
000010 | is positive and trace class. Moreover,
000011 | \[
000012 | 0<\mathcal U_\Seam\le e^{-S_\Seam}\Padm,
000013 | \qquad
000014 | S_\Seam:=\ell_\Seam\lambda_1^+(B_\Seam)>0.
000015 | \]
000016 | Hence
000017 | \[
000018 | \mathrm{Spec}(\mathcal U_\Seam)\subset (0,e^{-S_\Seam}] \subset (0,1),
000019 | \]
000020 | so the Fredholm determinant
000021 | \[
000022 | -\log\det_{\mathrm{adm}}(1-\mathcal U_\Seam)
000023 | \]
000024 | converges absolutely and defines a positive quantity.