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000138 | \section*{Research Contract --- \texttt{SEAM.MMST.TYPEIII.CHARGED.01} (one scaling-limit theorem)}
000139 | \addcontentsline{toc}{section}{Research Contract: SEAM.MMST.TYPEIII.CHARGED.01}
000140 | % =====================================================================
000141 | \noindent\emph{Typing.} A research contract \mO\ (ledger row
000142 | \texttt{SEAM.MMST.TYPEIII.CHARGED.01}), registered 2026-08-29 from review wave~5.
000143 | \textbf{Demand.}  \emph{One} scaling-limit theorem $\Psi_{\lambda,N}\to\Psi_\lambda$ strongly on
000144 | the finite-energy core $+$ crossed-product continuity, from which $B\cong(E_8)_1$, $\mu=1$,
000145 | type III$_1$, unique rotational KMS, unique conditional expectation, deck equidistribution and
000146 | modular invariance all follow as corollaries.  \textbf{Children} (listed here, not listing back):
000147 | \texttt{SEAM.SIMPLECURRENT.GENERATOR.01}, the continuum leg of \texttt{SEAM.STATE.RPMIXING.01},
000148 | the rigidity premise of \texttt{ALPHA.QUILLEN.EXACT.01}, the P1 reading of \texttt{AX.P1.01}.
000149 | \textbf{Cited document.}
000150 | \texttt{articles/2026-08-29/holomorphic\_kms\_extension\_en.tex} ($13$~pp, compiled):
000151 | unconditional rotational-KMS uniqueness via KLM $+$ Longo--Tanimoto \emph{given} the
000152 | identification; P1 $=$ theorem-conditional on seam identification $+$ $\beta_{\mathrm{angle}}=2\pi$.
000153 | Axiom typing of \texttt{AX.P1.01} UNCHANGED.  \textbf{Kill tests.}  The scaling limit fails to
000154 | exist; uniqueness of rotational KMS fails; type III$_1$ fails; a competing conditional
000155 | expectation exists.  Status: \texttt{SEAM.MMST.TYPEIII.CHARGED.01} \mO.
000156 | 
000157 | \par\smallskip\noindent\textbf{Executed update (2026-08-30, review wave~7): five lemmas
000158 | in-house; residual TEL-B-EXTERNAL $+$ ALG-EXH boxed
000159 | (\veri{v1006\_\allowbreak mmst\_\allowbreak lemma\_\allowbreak battery.py}; the contract stays \mO).}
000160 | Cited \texttt{articles/2026-08-30/\allowbreak mmst\_\allowbreak charged\_\allowbreak scaling\_\allowbreak limit\_\allowbreak en.tex} ($21$~pp) and
000161 | \texttt{articles/2026-08-30/\allowbreak externalization\_\allowbreak mmst\_\allowbreak handoff\_\allowbreak v2\_\allowbreak en.html}.
000162 | In-house: L3 crossed-product; UGF $K_G=\pi^2/4$ (measured plateau $\sim1.023$, max measured $K$
000163 | $1.029399$ at $N=16$); TEL-a $N^{-2}$ with exact isometries
000164 | ($A_{\mathrm{TEL}}=5120(1+\pi/2)$, measured envelope $3.420384<8$);
000165 | integer D5+A3 pairings; $\mathbb Z_4$ outerness in the limit $+$ finite-inner correction
000166 | ($256$-dim gauge unitary is inner).  Residual boxed: TEL-B-EXTERNAL (Hankel $C_R$; HS remainder
000167 | sourced from the paper max $2.105079358$ / fit $2.107565497$ / v998 bound $2.11$, not
000168 | recomputed) $+$ ALG-EXH (two Buchholz--Verch estimates).  Display marker stays \mO.
000169 | 
000170 | \par\smallskip\noindent\textbf{Executed update (2026-08-31, Monday-morning harvest;
000171 | \veri{v1016\_\allowbreak state\_\allowbreak gap\_\allowbreak batteries.py};
000172 | the contract stays \mO).}
000173 | The third (certified-tail) TEL-B route is BLOCKED: $N^2 D_N$ grows $530\to19032$,
000174 | remainder UV-supported.  Strictly smaller external target: the scalar dyadic increment
000175 | $\lvert\lVert R_{2N}\rVert-\lVert R_N\rVert\rvert\le A_R/N$ with $A_R<43$ sufficient for
000176 | $C_R<3$ (measured $N$-increments $0.120$--$0.123$, factor $\sim350$ headroom).  $N$-uniform
000177 | CAR nuclearity PROVED ($\nu_N\le e^{16 C_\beta}$; $C_\beta=(67.97,29.73,12.15)$ for
000178 | $\beta=(0.5,1,2)$) as ALG specialist compactness input; ALG-EXH projector-identification
000179 | UNCHANGED.  Display marker stays \mO.
000180 |