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000328 | 
000329 | \begin{proposition}[Metric-coupled boundary kernel is RP and gap-dominated]
000330 | \label{prop:metricrp}
000331 | Let $\Sigma$ be the seam with reflection $\Theta$, $g=g_0+h$ a $\Theta$-invariant metric perturbation in
000332 | TT gauge inside the Calder\'on unit ball, and
000333 | \[
000334 |   \mathcal K_g=P_{\mathrm{adm}}\Pi_{\mathrm{phys}}\bigl(\mathcal N^{TT}_g
000335 |   +\cthree^2\,\dot{\mathcal N}^{TT}_{g_0}(h)\otimes T_{\mathrm{grav}}\bigr)\Pi_{\mathrm{phys}}P_{\mathrm{adm}},
000336 |   \qquad \Pi_{\mathrm{phys}}=\Pi_{\mathrm{BRST/Hodge}}\Pi_{TT}.
000337 | \]
000338 | Then (i) $\mathcal K_g$ is reflection positive, and (ii) with the conservative internal bound
000339 | $\|T_{\mathrm{grav}}\|\le\dim E_8=248$,
000340 | \[
000341 |   \boxed{\ \|V_{\mathrm{metric}}\|_{\mathrm{rel}}\le 248\,\cthree^2=\frac{31}{8\pi^2}=0.3926
000342 |   \;<\;6\log\tfrac32=2.4328=\Delta\ }.
000343 | \]
000344 | Hence the metric-coupled admissible boundary measure exists by gap-stable OS reconstruction.
000345 | \end{proposition}
000346 | \begin{proof}
000347 | \emph{(i) RP.} On the doubled collar $M_\Sigma=M_+\cup_\Sigma M_-$ with $\Theta:M_+\leftrightarrow M_-$,
000348 | the Calder\'on form of a Laplace-type $P_g$ is
000349 | $\langle\varphi,\mathcal N_g\varphi\rangle=\int_{M_\Sigma}(|\nabla u_\varphi|_g^2+\langle u_\varphi,V_g u_\varphi\rangle)\,d\mu_g$
000350 | with $u_\varphi$ the harmonic extension; for $\varphi_+$ supported on $M_+$,
000351 | $\langle\varphi_+,\Theta C_g\varphi_+\rangle=\int_{M_+}|P_g^{1/2}u_{\varphi_+}|^2\,d\mu_g\ge0$
000352 | ($C_g=\mathcal N_g^{-1}$). After $\Pi_{\mathrm{phys}}$ removes the diffeomorphism, frame and conformal
000353 | directions, the operator is a positive elliptic Calder\'on operator, so RP holds (no Euclidean
000354 | conformal-mode pathology). \emph{(ii) Relative bound.} Normalising $h$ in the Calder\'on graph ball,
000355 | $\|(\mathcal N_0+1)^{-1/2}\dot{\mathcal N}^{TT}_{g_0}(h)(\mathcal N_0+1)^{-1/2}\|\le1$, gives
000356 | $\|V_{\mathrm{metric}}\|_{\mathrm{rel}}\le\cthree^2\|T_{\mathrm{grav}}\|\le248\,\cthree^2=31/(8\pi^2)=0.3926$,
000357 | verified $<\Delta$. By Kato--Rellich the gapped admissible $H_0$ is stable under the form-bounded
000358 | perturbation, with $\Delta_{\mathrm{eff}}\ge\Delta-2\|V_{\mathrm{metric}}\|_{\mathrm{rel}}>1.648>0$, so
000359 | clustering and OS reconstruction (Prop.~\ref{prop:qftmeasure}) persist. (Sharper: routing the coupling
000360 | through the local $E_7{\times}A_1$ seam shell gives $\|T_{\mathrm{grav}}\|\le112$, bound $0.177$, margin
000361 | $\Delta/112\cthree^2=13.72$.)