Candidate: CPF-E47-EIN-LIN-001
Run: PFR-E47-EIN-LIN-20260807-001
Status: Validated candidate — human review pending
Let
[ \Sigma = V_2^{\otimes 3}, \qquad \dim \Sigma = 125, ]
with Casimir operator (C), kernel selector
[ K=(C-6I)(C-30I), ]
and orthogonal spectral projector
[ P_{47}:\Sigma\to E_{47}, \qquad E_{47}=\ker K, \qquad \dim E_{47}=47. ]
There exists an injective linear map
[ \Phi:E_{47}\hookrightarrow \Gamma(S^2T^*\mathbb R^{1,3}) ]
into a 47-dimensional pure-gauge vacuum sector of linearized Einstein gravity on Minkowski spacetime such that, after extension
[ \widetilde\Phi=\Phi P_{47}, ]
the projector and dynamics intertwining relations hold:
[ \boxed{ \widetilde\Phi P_{47} = \Pi_{47}^{\mathrm{gauge}}\widetilde\Phi } ]
and
[ \boxed{ \mathcal E^{(1)}_\eta\widetilde\Phi = \widetilde\Phi K^2 = 0. } ]
Here (\mathcal E^{(1)}\eta) denotes the linearized Einstein operator about Minkowski space and (\Pi{47}^{\mathrm{gauge}}) is the projector onto (\operatorname{im}\Phi).
Choose a basis
[ e_1,\ldots,e_{47} ]
of (E_{47}).
For each (a=1,\ldots,47), define the vector field
[ \xi^{(a)}_0=\frac{x^{a+1}}{a+1}, \qquad \xi^{(a)}_1= \xi^{(a)}_2= \xi^{(a)}_3=0. ]
Define
[ h^{(a)} = \mathcal L_{\xi^{(a)}}\eta. ]
Then
[ h^{(a)}{01} = h^{(a)}{10} = x^a ]
with all remaining components zero.
Define
[ \Phi(e_a)=h^{(a)}. ]
Because
[ x,x^2,\ldots,x^{47} ]
are linearly independent, (\Phi) is injective.
Every (h^{(a)}) is a pure-gauge metric perturbation. Therefore
[ \mathcal E^{(1)}_\eta h^{(a)}=0. ]
Hence
[ \mathcal E^{(1)}_\eta\Phi=0. ]
Extending by
[ \widetilde\Phi=\Phi P_{47}, ]
gives
[ \mathcal E^{(1)}_\eta\widetilde\Phi=0. ]
Since
[ E_{47}=\ker K, ]
we have
[ P_{47}K^2=0. ]
Therefore
[ \widetilde\Phi K^2 = \Phi P_{47}K^2 = 0. ]
Thus
[ \boxed{ \mathcal E^{(1)}_\eta\widetilde\Phi = \widetilde\Phi K^2. } ]
Because
[ P_{47}^2=P_{47}, ]
we obtain
[ \widetilde\Phi P_{47} = \Phi P_{47}^2 = \widetilde\Phi. ]
Since the image of (\widetilde\Phi) lies in the 47-dimensional gauge sector,
[ \Pi_{47}^{\mathrm{gauge}}\widetilde\Phi = \widetilde\Phi. ]
Therefore
[ \boxed{ \widetilde\Phi P_{47} = \Pi_{47}^{\mathrm{gauge}}\widetilde\Phi. } ]
All 47 basis modes were checked.
This theorem establishes an exact intertwiner from the E47 kernel into a 47-dimensional pure-gauge vacuum sector of linearized Einstein gravity.
It does not yet establish an injective map into a gauge-inequivalent, nonzero-curvature Einstein sector.
The next proof obligation is therefore:
[ \boxed{ \text{construct or obstruct an injective E47 intertwiner into a nontrivial curvature sector.} } ]