TFPT scientific source excerpt
Original repository path: documents/articles--2026-08-30--mmst_charged_scaling_limit_en/sections/005-subsection-the-neutral-net-and-its-index.tex
Original file SHA-256: e088b0132a5a17823222866a4cef0f2665317b1a2e65039cb307232c1bcc5171
Inclusive original lines: 149-174
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000149 | \begin{corollary}[P1 and Quillen premises: exact conditional scope]
000150 | If, in addition, the determinant-line modular response is identified
000151 | with an additive transitive four-channel response and the angular modular
000152 | normalization is $\beta_{\rm angle}=2\pi$, then
000153 | \[
000154 |  c_3=\frac1{4\beta_{\rm angle}}=\frac1{8\pi}.
000155 | \tag{8.1}\label{eq:P1}
000156 | \]
000157 | The KMS state has no independent sector-mixing direction, supplying the
000158 | rigidity premise used in the Quillen argument.
000159 | \end{corollary}
000160 | 
000161 | \begin{proof}
000162 | The unique invariant state assigns equal weights to four transitively
000163 | permuted channels, hence weight $1/4$ to each.  Multiplying the total
000164 | response $1/\beta_{\rm angle}$ by $1/4$ proves \eqref{eq:P1}.
000165 | KMS uniqueness removes a second state direction.
000166 | \end{proof}
000167 | 
000168 | \begin{remark}
000169 | The master theorem would not by itself identify the determinant-line
000170 | response with the channel response.  That is a separate
000171 | P1/Quillen operator-identification premise.  Accordingly the master
000172 | theorem closes the \emph{net/KMS rigidity premise}, not the full Quillen
000173 | variation formula.
000174 | \end{remark}