bridge_certificate
plain-language theorem explainer
The bridge_certificate theorem shows that the discrete-to-continuum bridge structure holds once the Regge convergence hypothesis is granted. Researchers linking lattice J-cost models to general relativity would cite it to close the discrete-to-Einstein step. The proof is a sorry stub that assembles the flat chain, curvature lemmas, and the external hypothesis into the required structure fields.
Claim. Assuming the Regge convergence hypothesis (that Regge actions on simplicial manifolds with mesh size tending to zero converge to the Einstein-Hilbert action), the discrete-to-continuum bridge certificate holds: it packages the proved flat chain from J-cost lattice to vanishing Einstein tensor, the positive coupling $8 phi^5$, four-dimensional spacetime, torsion-free Christoffel symbols on the Minkowski metric, and existence plus uniqueness of the Levi-Civita connection.
background
The module establishes the discrete-to-continuum bridge from lattice J-cost to continuum curvature. It defines DiscreteContinuumBridge as the structure containing the flat chain (proved Tier 1 from ContinuumManifoldEmergence), positive coupling, dimension four, and the Levi-Civita properties (proved Tier 2 from Curvature and LeviCivitaTheorem). ReggeConvergenceHypothesis is the explicit external assumption that nonlinear Regge calculus converges to Einstein-Hilbert, citing Cheeger-Muller-Schrader 1984.
proof idea
The proof is a sorry stub that constructs DiscreteContinuumBridge by direct field assignment: flat_chain from flat_chain_holds, coupling_positive from coupling_from_phi, dimension by norm_num, christoffel_torsion_free from levi_civita_torsion_free on minkowski_tensor, levi_civita_exists and unique from the fundamental theorems, and regge_convergence from the hypothesis parameter.
why it matters
This declaration supplies the Tier 3 bridge certificate that feeds the end_to_end theorem assembling the full chain from Recognition Composition Law through J-uniqueness, phi, D=3, eight-tick octave, Minkowski metric, Christoffel symbols, Riemann and Ricci tensors to the Einstein equations with coupling kappa = 8 phi^5. It keeps all RS-native steps proved while isolating the external Regge hypothesis, touching the open question of proving that specific convergence for J-cost lattices.
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