TFPT scientific source excerpt
Original repository path: documents/articles--2026-08-30--mmst_charged_scaling_limit_en/sections/001-section-verdict-and-logical-form.tex
Original file SHA-256: e75e10cc33137187e6c494e8486ffdec91e43d6b636d434ce0735d97e56a9c7f
Inclusive original lines: 45-99
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.

000045 | \section{Verdict and logical form}
000046 | 
000047 | \begin{theorem}[Charged MMST master theorem: exact proved form]
000048 | \label{thm:master}
000049 | Let the finite collars, scaling maps, neutral current limit and charged
000050 | operators be as in Section~\ref{sec:setup}.  Suppose the scaling algebra
000051 | is exhausted by the neutral currents and
000052 | the four charged grades as specified in \textup{(ALG-EXH)} there.  Then:
000053 | \begin{enumerate}[label=\textup{(L\arabic*)}]
000054 | \item For every $f\in C^\infty(S^1)$ there are $p<\infty$ and
000055 |   $C_f<\infty$, independent of $N$, such that on the finite-energy core
000056 |   \[
000057 |     \norm{\Psi_{\lambda,N}(f)\xi}
000058 |     \le C_f\norm{(1+L_{0,N})^p\xi}.
000059 |   \tag{1.1}\label{eq:L1}
000060 |   \]
000061 | \item After the canonical scaling identifications,
000062 |   $\Psi_{\lambda,N}(f)$ converges strongly on the common finite-energy
000063 |   core to a nonzero closable field $\Psi_\lambda(f)$.  The generated local
000064 |   algebras converge to a local, conformal extension net
000065 |   $I\mapsto\B(I)$ of the neutral net $I\mapsto\A(I)$.
000066 | \item The finite $\Z_4$ crossed products, the canonical coefficient
000067 |   expectations, and the fixed-point expectations commute with the
000068 |   inductive/scaling limit.  The limiting Pimsner--Popa bound is still
000069 |   $E(x)\ge\frac14x$ for $x\ge0$.
000070 | \item The resulting extension is the strongly local lattice net
000071 |   $\A_{V_{E_8}}$, equivalently the level-one $E_8$ current net.  It is
000072 |   holomorphic, has $\mu=1$, central charge $8$, vacuum character
000073 |   $E_4/\eta^8$, and local algebras equal to the hyperfinite
000074 |   type-$\mathrm{III}_1$ factor.
000075 | \end{enumerate}
000076 | \end{theorem}
000077 | 
000078 | \begin{proof}
000079 | L1 follows from Theorem~\ref{thm:L1reduction}.  L2 follows from
000080 | Theorem~\ref{thm:L2reduction}.  L3 is the unconditional
000081 | Theorem~\ref{thm:crossedlimit}.  L4 follows from
000082 | Theorem~\ref{thm:identification}.  The Green estimate used by L1 is the
000083 | all-volume Theorem~\ref{thm:UGF}; the graph estimate used by L2 is the
000084 | all-volume dyadic Theorem~\ref{thm:TELa}.  The condition still used by L2
000085 | is precisely \textup{(ALG-EXH)} in the final gap box; it is not silently
000086 | replaced by a finite-volume fit.
000087 | \end{proof}
000088 | 
000089 | \begin{remark}[What is and is not proved]
000090 | The implication in Theorem~\ref{thm:master} is proved.  Uniform Green
000091 | domination and local Sobolev convergence are proved in
000092 | Theorem~\ref{thm:UGF}, and edge-mode graph convergence is proved in
000093 | Theorem~\ref{thm:TELa}.  The antecedent \textup{(ALG-EXH)} is not proved
000094 | for the frozen microscopic collar.  Therefore
000095 | this memorandum does not close
000096 | \texttt{SEAM.MMST.TYPEIII.CHARGED.01}.  It proves L3, completes the
000097 | classical reduction of L4, closes L1 and the edge-projected TEL-a part
000098 | of L2, and isolates the remaining microscopic local-algebra statement.
000099 | \end{remark}