Date: September 6, 2026
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Bottom line. E47 has a closed finite-dimensional theorem and a reproducible computational certificate. The Python record confirms the implementation of that theorem; it does not independently generate spacetime. Within Einstein gravity, $T_{\mu\nu}^{(E)}=-\rho_Eg_{\mu\nu}$ implies $\Lambda_E=8\pi G\rho_E$. The further substitution $\rho_E=\rho_0(47/125)$ is conditional on a vacuum-density ansatz whose E47 derivation remains open.
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On $V=V_2^{\otimes3}\cong\mathbb C^{125}$, the Hermitian total-spin Casimir $C$ defines $K=(C-6I)(C-30I)$. Its kernel is the exact sector $E_{47}=E_6\oplus E_{30}$, with $\dim E_{47}=47$, orthogonal projector $P_{47}$, and dimensionless occupancy $\Omega_c=47/125$. The finite filters $e^{-tK^2}$ and $(I-\varepsilon K^2)^n$ converge to $P_{47}$ under the stated spectral conditions. E47 Projector-Valued Vacuum and Einstein–AQSFT Closure
The deterministic Python witnesses reproduce the declared dimensions, spectrum, projector identities, kernel annihilation, convergence rate, and perturbation bounds at floating-point precision. Their certified scope is the finite operator model they execute. E47 Executable Validation Certificate — Exact Spectral Core and Continuum Evidence Boundary and Python and Numerical Validation
The remaining task is constructive: derive the map from the finite E47 operator algebra to a nondegenerate Lorentzian spacetime model, derive its vacuum stress tensor and dimensionful scale, solve the resulting Einstein–matter system, and obtain a discriminating empirical prediction.
| Layer | Established result | Evidence class | Boundary |
|---|---|---|---|
| Finite carrier | $V=V_2^{\otimes3}$; $\dim V=125$ | E0 exact | Finite representation theory only. |
| Casimir spectrum | $\operatorname{spec}(C)=\{0,2,6,12,20,30,42\}$; multiplicities $(1,9,25,28,27,22,13)$ | E0 exact | Determines the eigenspaces on this carrier. |
| Selector and invariant | $K=(C-6I)(C-30I)$; $\ker K=E_6\oplus E_{30}=E_{47}$; rank $47$ | E0 exact | Rank alone supplies no continuum interpretation. |
| Projector | $P_{47}=Q_6+Q_{30}$; $P_{47}^2=P_{47}=P_{47}^{\dagger}$; $KP_{47}=0$ | E0 exact | $47/125$ is an exact dimensionless occupancy. |
| Contraction | $A=K^2$; $\delta=11664$; $L=186624$; $\varepsilon_=1/99144$; $|\Gamma_{\varepsilon_}^n-P_{47}|=(15/17)^n$ | E0 exact | Converges to the finite spectral projector. |
| Product-kernel lift | If $L_g\succeq0$ is self-adjoint, $\ker(L_g\otimes I+I\otimes K^2)=\ker L_g\otimes E_{47}$ | E0 under stated hypotheses | Does not derive $L_g$, $g$, or Einstein dynamics. |
| Executable reconstruction | Seed 470125; generic-basis 125×125 realization; 37 PASS; 10 formal revisions; 8 constitutive requirements; 0 FAIL | E1 machine | Certifies the implemented objects and tests only. |
| Numerical residuals | $|P_{47}^2-P_{47}|2=2.48\times10^{-15}$; $|KP{47}|2=5.07\times10^{-13}$; $|[C,P{47}]|_2=7.69\times10^{-14}$ | E1 machine | Witnesses exact identities; supplies no physical evidence. |
| Perturbation trials | 20/20 Hermitian trials preserved rank 47 and satisfied the projector bounds for $|E|\lt\delta/2$ | E1 machine | Finite tested perturbation regime only. |
| Vacuum equivalence | If $T_{\mu\nu}^{(E)}=-\rho_Eg_{\mu\nu}$, then $\Lambda_E=8\pi G\rho_E$ | Exact conditional theorem | The stress-tensor form is an input, not an E47-derived output. |
| E47-scaled vacuum | If $\rho_E=\rho_0(47/125)$, then $\Lambda_E=8\pi G\rho_0(47/125)$ | Conditional corollary | The multiplier ansatz and $\rho_0$ remain unproved. |
The continuum branch closes when an explicit map
$$ (V,C,K,P_{47})\longrightarrow(M,g,\Phi,S,T_{\mu\nu}) $$
is constructed, the Einstein equations are solved with declared data, finite and continuum residuals both pass, and at least one scale-setting or discriminating prediction is independently confronted with evidence.
Current disposition: finite E47 core exact; executable reconstruction validated; Einstein vacuum equivalence conditional on its stress-tensor input; E47-to-continuum derivation open.