Date: September 6, 2026
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Bottom line. E47 has a closed finite-dimensional theorem and reproducible computational certificates. The earlier bare-projector record did not independently generate spacetime. Successor certificate MC-E47-INTRINSIC-SPACETIME-UNMARKED/1.0 now constructs an intrinsic Lorentzian four-plane and an injective 47-real-dimensional image in the completely unmarked vacuum-Einstein quotient from the full locked tensor-cube datum. Within Einstein gravity, $T_{\mu\nu}^{(E)}=-\rho_Eg_{\mu\nu}$ still implies $\Lambda_E=8\pi G\rho_E$. The further substitution $\rho_E=\rho_0(47/125)$ remains conditional on a vacuum-density ansatz whose E47 derivation is 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
A first covariant finite-to-field bridge is now certified as citizen E47–GR–QFT Covariant Intertwiner · Exact Proof and Machine Certificate: $\mathcal E=M\times\mathbb C^{125}$, $\Pi=I\otimes P_{47}$, and $\mathfrak A_g=((\Box_g-m^2)\otimes I)-\kappa(I\otimes K^2)$ satisfy $[\mathfrak A_g,\Pi]=0$ and $\mathfrak A_g\circ\iota_v=\iota_v\circ(\Box_g-m^2)$ for $v\in E_{47}$. The intrinsic Lorentzian geometry and unmarked vacuum-Einstein injection are now supplied by successor page E47 Intrinsic Lorentzian Spacetime and Unmarked Einstein-Moduli Embedding using the full locked tensor-cube datum. Remaining constructive tasks are the covariant vacuum-sector constitutive derivation, dimensionful scale setting, Einstein–matter closure beyond the vacuum plane-wave branch, and 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; the covariant internal-fiber/product-kernel stage is closed; the marked nonlinear Einstein sector is closed; and successor certificate MC-E47-INTRINSIC-SPACETIME-UNMARKED/1.0 closes intrinsic Lorentzian 3+1 emergence and injectivity onto a 47-real-dimensional image in the completely unmarked vacuum-Einstein quotient for the full locked tensor-cube datum. Vacuum-density constitutive derivation, dimensionful scale setting, broader Einstein–matter closure, interacting/renormalized QFT completion, and empirical quantum-gravity discrimination remain open.