TFPT scientific source excerpt
Original repository path: documents/archive--paper-latex--tfpt-32/sections/015-subsection-internal-closure-candidates-beyond-the-theore.tex
Original file SHA-256: 8582c396e7a64e91ef798ea9feb978dede36c7fb7468c903f08e84bf15368c30
Inclusive original lines: 54-110
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000054 | \begin{figure}[t]
000055 | \centering
000056 | \scriptsize
000057 | \begin{tikzpicture}[
000058 |     closed/.style={draw=blue!60!black, fill=blue!6, rounded corners, text width=3.6cm, minimum height=1.05cm, align=center},
000059 |     future/.style={draw=orange!75!black, fill=orange!8, rounded corners, dashed, text width=4.2cm, minimum height=1.2cm, align=center},
000060 |     note/.style={font=\footnotesize\itshape, text=black!70, align=center}
000061 | ]
000062 | \node[closed] (now) at (-3.2,0) {\textbf{closed now}\\admissible vacuum theorem\\$\Lambda_{\Seam,\mathrm{ren}}$};
000063 | \node[future] (future) at (3.2,0) {\textbf{future Fredholm route}\\$\mathcal U_\Seam \mapsto -\log\det_{\mathrm{adm}}(1-\mathcal U_\Seam)$\\$\Lambda_{\mathrm{obs}}$};
000064 | \draw[->, thick, dashed, orange!75!black] (now) -- node[above] {\small later observation-level upgrade} (future);
000065 | \node[note] at (0,-1.55) {Current closure stops at the renormalized seam residue; the observed hierarchy belongs to the later determinant program.};
000066 | \end{tikzpicture}
000067 | \caption{Vacuum-energy status split. The current theorem stack closes only the renormalized seam vacuum energy $\Lambda_{\Seam,\mathrm{ren}}$, while the tiny observed hierarchy is deferred to the later Fredholm-determinant route for $\Lambda_{\mathrm{obs}}$.}
000068 | \label{fig:vacuum-energy-status}
000069 | \end{figure}
000070 | 
000071 | \begin{conjecture}[Seam Fredholm-determinant route for the cosmological constant]
000072 | Because TFPT is formulated relatively, bulk zero-point contributions are natural
000073 | candidates to cancel in the reference subtraction, leaving only a genuine seam
000074 | residue. The corresponding completion route is to define the positive trace-class
000075 | seam transfer operator
000076 | \[
000077 | \mathcal U_\Seam
000078 | :=
000079 | \Padm\,e^{-\ell_\Seam |B_\Seam|}\,\Padm,
000080 | \qquad
000081 | \ell_\Seam\,\lambda_1^+(B_\Seam)=S_\Seam,
000082 | \qquad
000083 | S_\Seam:=\frac{2\pi}{3}\alpha^{-1}(0),
000084 | \]
000085 | and then set
000086 | \[
000087 | \Lambda_{\mathrm{obs}}
000088 | =
000089 | \Mpl^4\Bigl[-\log\det\nolimits_{\mathrm{adm}}(1-\mathcal U_\Seam)\Bigr].
000090 | \]
000091 | The Fredholm expansion gives
000092 | \[
000093 | \Lambda_{\mathrm{obs}}
000094 | =
000095 | \Mpl^4\sum_{k\ge1}\frac{1}{k}\,\tr(\mathcal U_\Seam^k),
000096 | \]
000097 | so the old seam transseries is recovered as the determinant expansion of one
000098 | positive operator. At leading order this gives
000099 | \[
000100 | \frac{\Lambda_{\mathrm{obs}}}{\Mpl^4}
000101 | \approx
000102 | \Omega_{\mathrm{adm}}\exp\!\Bigl[-\frac{2\pi}{3}\alpha^{-1}(0)\Bigr]
000103 | =
000104 | 48\,e^{-2\pi\alpha^{-1}(0)/3}
000105 | \approx
000106 | 1.1\times10^{-123}.
000107 | \]
000108 | This remains a conjectural closure route rather than a proved theorem, but it
000109 | turns the tiny cosmological constant into a seam-action effect with controlled
000110 | sign and convergence properties instead of a free transseries ansatz.