ContinuumBridgeCert
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.
claimA 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 in Recognition Science
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.
scope and limits
- Does not prove convergence of the discrete Regge equations to the continuum Einstein equations.
- Does not derive the explicit stress-energy tensor from ledger degrees of freedom.
- Does not control higher-order curvature corrections beyond the quadratic J approximation.
- Does not establish uniqueness of the flat vacuum solution on arbitrary graphs.
Lean usage
theorem continuum_bridge_cert : ContinuumBridgeCert where chain := jcost_to_efe_chain flat_vacuum := fun G => flat_is_vacuum G kappa_derived := jcost_to_regge_factor_eq kappa_pos := jcost_to_regge_factor_pos flat_cost_zero := flat_zero_cost deficit_correspondence := fun δε => J_log_quadratic_def δεformal statement (Lean)
304structure ContinuumBridgeCert where
305 chain : JCostToEFE
306 flat_vacuum : ∀ {n : ℕ} (G : WeightedLedgerGraph n),
307 ∀ i, discrete_laplacian G (fun _ => (0 : ℝ)) i = 0
308 kappa_derived : jcost_to_regge_factor = 8 * phi ^ 5
309 kappa_pos : 0 < jcost_to_regge_factor
310 flat_cost_zero : Jcost 1 = 0
311 deficit_correspondence : ∀ δε : ℝ, J_log_quadratic δε = δε ^ 2 / 2
312