full_gr_certificate
plain-language theorem explainer
Master certificate packaging the full RS-to-Einstein-field-equations chain: 4D spacetime, coupling κ = 8φ⁵ > 0, Hilbert variation and matter-coupling closures, flat Regge vanishing, discrete Bianchi, and linearized plus second-order Regge convergence. Gravity auditors cite it as the single closure object for the sourced nonlinear EFE derivation. The body is pure structure assembly wiring already-proved sibling certificates.
Claim. The Recognition Science master GR certificate asserts: spacetime dimension is $4$; the Einstein coupling equals $\kappa = 8\phi^5$ with $\kappa > 0$; Hilbert-variation and matter-coupling closures hold; the Regge action vanishes whenever all hinge deficits are zero; the discrete Bianchi identity holds in the flat case; linearized Regge-to-Einstein-Hilbert convergence holds; and the nonlinear convergence error is second order under the stated mesh-regularity hypotheses.
background
Module Gravity.FullEFE derives the complete nonlinear sourced Einstein field equations from the RS discrete ledger, conditional on Regge continuum convergence. The chain runs: J-cost minimization on the ledger yields the Regge action on $\mathbb{Z}^3$; Regge converges to the Einstein-Hilbert action; stationary Regge implies stationary EH; Hilbert variation gives vacuum EFE; adding matter yields $G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}$; Bianchi gives $\nabla^\mu T_{\mu\nu}=0$; and $\kappa=8\phi^5$ is forced from $\phi$, not fitted.
FullGRCertificate is the structure that records every link: a FullEFEData payload (dimension and $\kappa$), positivity and non-vanishing of $\kappa$, HilbertVariationClosure, MatterCouplingClosure, flat Regge vanishing, flat Bianchi, linearized convergence, and a second-order nonlinear error bound. Sibling defs rs_efe_data, rs_efe_dimension, rs_efe_kappa, hilbert_variation_closure, and matter_coupling_closure supply the concrete witnesses.
Curvature infrastructure (Levi-Civita connection, Riemann/Ricci/Einstein tensors, EH action density, stress-energy conservation) is built in-tree; vacuum Minkowski is already a proved EFE solution.
proof idea
Not a tactic proof: a structure-value definition. Each field of FullGRCertificate is filled by a named sibling or import lemma.
- Dimension payload and checks:
rs_efe_data,rs_efe_dimension,rs_efe_kappa,rs_kappa_pos,rs_conservation. - Variational and matter sides:
hilbert_variation_closure,matter_coupling_closure. - Discrete geometry:
regge_action_flat,flat_bianchi. - Continuum limit:
ReggeConvergence.linearized_convergencefor the weak-field regime;NonlinearConvergence.convergence_is_second_order(with triviallinarithside conditions) for the second-order error claim.
No new algebra is performed here; the certificate only wires existing results into one exportable object.
why it matters
This is the gravity-facing master export for the full EFE program in Recognition Science. Module status: steps 1-2 (J-cost quadratic limit, Regge from J) are unconditional; step 3 (nonlinear Regge-to-EH) remains conditional on CMS-type regularity; step 4 has an explicit linearized second-order estimate; steps 5-6 now ride theorem-backed EinsteinHilbertAction and StressEnergyTensor certificates rather than placeholder Props; steps 7-8 (Bianchi conservation, $\kappa=8\phi^5$) are proved.
Framework landmarks: $\phi$ enters the coupling exactly as $\kappa=8\phi^5$ (RS-native units with $G\sim\phi^5/\pi$); spatial $D=3$ from the forcing chain (T8) appears here as spacetime dimension 4; the eight-tick/octave discrete ledger sits upstream of the Regge action. No downstream consumers are wired yet (used_by empty), so this object is the intended citation point for any later global "RS derives GR" theorem. Open gap remains full nonlinear CMS formalization in a proof assistant.
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