ContinuumBridgeCert
plain-language theorem explainer
The ContinuumBridgeCert structure certifies that J-cost stationarity on a weighted simplicial ledger recovers the Einstein field equations in the continuum limit, with the coupling fixed at κ = 8φ⁵. Researchers deriving general relativity from discrete or emergent models would cite it to anchor the Recognition Science ledger in classical gravity. The declaration is a structure that packages the full JCostToEFE chain together with vacuum flatness, the explicit κ value and its positivity, zero flat cost, and the quadratic deficit identity.
Claim. A certificate structure asserting the complete J-cost to Einstein field equations derivation chain, the vanishing of the discrete Laplacian on the zero perturbation for any weighted simplicial graph, the equality of the Regge normalization factor to $8φ^5$, positivity of that factor, vanishing of J-cost at the flat configuration, and the identity $J^{(2)}$-log quadratic deficit equals $δε^2/2$.
background
The module closes the gap between the discrete RS ledger and Einstein's equations by showing that J-cost stationarity on simplicial graphs is equivalent to Regge action minimization. WeightedLedgerGraph is a structure carrying nonnegative symmetric edge weights that encode ledger connectivity. The discrete Laplacian at vertex i is defined as the weighted sum ∑_j w_ij (ε_i − ε_j), which is precisely the quadratic form of the J-cost action in log coordinates ε = ln ψ. J_log_quadratic supplies the leading term ε²/2 of the expansion J(e^ε) = cosh(ε) − 1, with the quartic remainder controlled by the standard Taylor bound.
proof idea
The declaration is a structure definition with an empty proof body. It directly assembles five independent properties: the JCostToEFE chain object, the flat-vacuum statement that the discrete Laplacian vanishes on the zero field, the explicit equality jcost_to_regge_factor = 8φ^5, its positivity, the flat cost identity Jcost 1 = 0, and the quadratic deficit correspondence. No tactics or lemmas are invoked inside the declaration itself.
why it matters
This certificate supplies the missing link that lets J-cost stationarity on the simplicial ledger produce the Einstein field equations via the Regge action. It is instantiated by the theorem continuum_bridge_cert and supplies the normalization used in the corollary kappa_calibration_positive that equates the derived κ to the Einstein coupling. The value κ = 8φ^5 matches the RS-native gravitational constant G = φ^5/π (in units c = ħ = 1) and realizes the T8 forcing of three spatial dimensions through the simplicial structure. It touches the open question of rigorous convergence of the discrete Regge equations to the smooth EFE.
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