TFPT scientific source excerpt
Original repository path: documents/articles--2026-08-29--holomorphic_kms_extension_en/sections/003-section-what-this-closes-and-does-not-close.tex
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Inclusive original lines: 73-92
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000073 | \section*{Conclusion}
000074 | \addcontentsline{toc}{section}{Conclusion}
000075 | 
000076 | The abstract operator-algebraic theorem is closed at its natural
000077 | strength.  Existence comes from the vacuum-sector rotational Gibbs
000078 | state of the holomorphic extension.  The index-four group average is
000079 | the unique expectation by the finite-index Radon--Nikodym theorem
000080 | and irreducibility.  Takesaki's theorem gives the explicit modular
000081 | restriction and intertwining chain.  Gauge-invariant extension
000082 | uniqueness is an elementary averaging identity, while unconditional
000083 | rotational KMS uniqueness requires both KLM ($\mu=1$ leaves only the
000084 | vacuum sector) and Longo--Tanimoto (rotational KMS extreme points are
000085 | sector Gibbs states).  Araki's theorem identifies the exact frozen
000086 | finite covariance and explains the finite KMS rank collapse.
000087 | 
000088 | The remaining issue is not another uniqueness argument.  It is the
000089 | construction and identification of the concrete TFPT collar
000090 | scaling-limit net, including its non-bilinear extension fields and
000091 | its determinant-line response.  That issue is isolated in the
000092 | single box above and lies outside Theorem~\ref{thm:main}.