TFPT scientific source excerpt
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000108 | \subsection*{2026-07-02 (\texttt{v469}/\texttt{v470} --- the closure-route round on the two named targets: the \texttt{SEAM.EQUIV.01} extension leg re-founded on 1995--2001 peer-reviewed crossed-product theorems $+$ the realisation axiom reduced to invariant level; the $\alpha^3$ level $=$ the \emph{computed} bulk Chern invariant $+$ the seam $F$-normalisation $=$ the embedding index $k_Y{=}5/3$; both targets stay \mO)}
000109 | \begin{itemize}
000110 | \item \textbf{\texttt{v469\_seam\_crossedproduct\_route.py} (\texttt{SEAM.EQUIV.CROSSEDPRODUCT.01}).}
000111 | The $128$-spinor extension leg of \texttt{SEAM.EQUIV.01} no longer rests on 2024/2025 preprints:
000112 | \mE\ the $SO(16)_1$ spinor has $h_s{=}N/16{=}16/16{=}1{\in}\Z$, so its statistics phase is $+1$ ---
000113 | \emph{exactly} the Longo--Rehren locality criterion --- and the $\Z_2$ simple-current crossed product
000114 | $SO(16)_1{\rtimes}\{1,s\}$ is a \emph{local} index-$2$ extension by the peer-reviewed subfactor package
000115 | (Longo--Rehren, Rev.\ Math.\ Phys.\ 7 (1995); the level-$1$ $so(N)$ net with its DHR sectors from
000116 | even-CAR: B\"ockenhauer, Rev.\ Math.\ Phys.\ 8 (1996); simple-current extensions of nets:
000117 | B\"ockenhauer--Evans, Comm.\ Math.\ Phys.\ 197 (1998)); KLM (Comm.\ Math.\ Phys.\ 219 (2001)) gives
000118 | $\mu(B){=}4/2^2{=}1$ $\Rightarrow$ holomorphic $\Rightarrow$ $(E_8)_1$ by the \texttt{v463} pin.
000119 | The AGT/AMT lattice-VOA route (\texttt{v459}) is \emph{demoted to an independent second witness}.
000120 | \mE\ the index-$4$ $\mu_4$ glue runs on the same integer: $h(J^k){=}\{1,1,1\}$ for $k{=}1,2,3$
000121 | (the \texttt{v125} isotropy $q(k(1,1)){=}k^2$ at net level), $\mu{=}16/4^2{=}1$, same endpoint.
000122 | \mE\ the realisation input is reduced from model fiat to \emph{invariants} (R1$'$): quasi-free
000123 | \mC\ (\texttt{v155}/\texttt{v160}) $+$ gap \mE\ (\texttt{v302}) $+$ class D $+$ $c_-{=}8$ \mE\
000124 | (\texttt{v456}, from P1), with the invariants \emph{computed} --- per-copy FHS Chern $|C|{=}1$ vs $0$
000125 | (control), $\nu{=}16{\times}|C|{=}16$, $c_-{=}\nu/2{=}8$ --- and $\nu\equiv 0\ (\mathrm{mod}\ 16)$ is
000126 | exactly Kitaev's class-D phase whose edge admits a purely bosonic description $=$ the $(E_8)_1$
000127 | level-$1$ state (Ann.\ Phys.\ 321 (2006), sixteen-fold way); the phase-invariance of $c_-$ is cited
000128 | (Kapustin--Spodyneiko, PRB 101 (2020); Kim et al., PRL 129 (2022)). \mO\ \emph{not closed}: the cited
000129 | theorems stay cited; \texttt{SEAM.EQUIV.01} stays \mO.