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000064 | \section*{Target A --- the $(E8)_1$ boundary-net identification}\label{sec:A}
000065 | % =====================================================================
000066 | \noindent\textbf{Minimal claim.} The ambient metric/projective measure is reduced to identifying the
000067 | seam--Calder\'on boundary measure with the $(E8)_1$ lattice net, carrier subnet
000068 | $(D5)_1\times(A3)_1$.\veri{redteam/rt\_A\_e8net.py}
000069 | 
000070 | \smallskip\noindent\textbf{Logical chain (established).} The Sugawara level-1 central charges are
000071 | $c(E8)_1=\tfrac{248}{31}=8$, $c(D5)_1=5=\gcar$, $c(A3)_1=3=\Nfam$, so the conformal-embedding
000072 | criterion $c_{\mathrm{coset}}=c(E8)-c(D5)-c(A3)=0$ holds \mE. This is \emph{compatibility}.
000073 | 
000074 | \begin{gapbox}[Where it breaks: central charge underdetermines the net]
000075 | Counterexample: $(D8)_1=SO(16)_1$ also has $c=\tfrac{120}{15}=8$, but is \emph{not} holomorphic
000076 | (four primaries $1,v,s,c$) --- a distinct $c=8$ net. Hence equal central charge is \emph{necessary,
000077 | not sufficient}: $(E8)_1$ is the unique \emph{holomorphic} $c=8$ net (even unimodular rank-8
000078 | lattice $=E8$), so \textbf{holomorphy (single vacuum sector, $\mu$-index $1$) is the load-bearing
000079 | extra assumption}. Dropping chirality is worse: the $c=8$ Narain family is positive-dimensional.
000080 | The constructive map seam--Calder\'on kernel $\to$ $(E8)_1$ and the uniqueness of bulk
000081 | reconstruction are not established; the gap $\Delta_{\mathrm{eff}}>0$ \emph{supports} tightness
000082 | (clustering) but does not prove it.
000083 | \end{gapbox}
000084 | 
000085 | \noindent\textbf{Verdict.} Keep ``reduced to a rigorous boundary-net identification problem''; do
000086 | \emph{not} write ``metric sector closed''. The red team confirms the conservative wording. Status
000087 | \mC. Residual: holomorphy proof $+$ constructive boundary map $+$ bulk-reconstruction uniqueness.
000088 | 
000089 | % =====================================================================