The pure-gauge intertwiner has been closed exactly.
The next question is no longer whether an E47-to-Einstein map can exist.
It can.
The next question is stronger:
[ \boxed{ \text{Can }E_{47}\text{ embed injectively into a gauge-inequivalent Einstein sector?} } ]
We seek
[ \Phi_{\mathrm{phys}}: E_{47} \hookrightarrow \mathcal H_{\mathrm{Einstein}} ]
such that
[ \boxed{ \Phi_{\mathrm{phys}}P_{47} = \Pi_{\mathrm{phys}}\Phi_{\mathrm{phys}} } ]
and, for an explicitly declared Einstein-side operator (\mathcal L_E),
[ \boxed{ \mathcal L_E\Phi_{\mathrm{phys}} = \Phi_{\mathrm{phys}}K^2. } ]
The image must contain gauge-inequivalent modes satisfying at least one nontriviality condition such as
[ R^{(1)}_{\mu\nu\rho\sigma}[h]\neq0, ]
or an equivalent gauge-invariant curvature observable.
Exactly three outcomes are admissible.
An injective nontrivial intertwiner exists.
No such injective map exists under the declared operator, gauge, regularity, or dimensional assumptions.
Existence depends on an explicit additional structure such as background curvature, boundary conditions, matter coupling, or mode restriction.
Failure is evidence.
A failed intertwiner must be retained with its residual and obstruction condition.