continuum_bridge_cert
plain-language theorem explainer
The continuum bridge certificate establishes that J-cost stationarity on the simplicial ledger produces the Einstein field equations in the continuum limit with coupling κ = 8φ^5. A physicist deriving general relativity from discrete structures would cite this result. The proof constructs the certificate in term mode by direct reference to the J-cost to EFE chain lemma together with the flat vacuum property and Regge factor identities.
Claim. Stationarity of the J-cost functional on a simplicial ledger implies the Einstein field equations in the continuum limit, with gravitational coupling constant κ = 8 φ^5.
background
A simplicial ledger is a collection of 3-simplices forming a manifold covering. The J-cost functional on ledger edges expands in log coordinates ε = ln ψ as J(e^ε) = cosh(ε) − 1 = ε²/2 + O(ε⁴), so the coupling term between neighbors is (ε₁ − ε₂)²/2 + O((ε₁ − ε₂)⁴). This quadratic form is identical to the discrete Laplacian action and to the Regge action S_Regge = Σ_h δ_h · A_h. The module states that the field-curvature identity is a theorem following from this quadratic structure, not an extra hypothesis. Upstream, the flat vacuum theorem shows that zero potential is a stationary point of J-cost by direct computation that the discrete Laplacian vanishes on constant fields.
proof idea
The proof is a term-mode construction of the certificate structure. It sets the main chain field to the lemma establishing the J-cost to EFE correspondence, assigns the flat vacuum property to the theorem that the discrete Laplacian vanishes for zero potential, derives the equality of the Regge factor to 8 φ^5, confirms positivity of that factor, verifies zero J-cost for the flat configuration, and records the exact quadratic correspondence for deficit angles.
why it matters
This declaration closes the critical gap between the discrete RS ledger and Einstein's field equations, completing the derivation chain from J_global through log-coordinate expansion and discrete Laplacian identification to Regge action identification and the continuum limit. It realizes the module's key insight that J-cost stationarity yields the same equations as Regge action minimization. In the Recognition Science framework it connects the forcing chain to general relativity via the eight-tick octave and D = 3 spatial dimensions. It touches the open question of full nonlinear convergence proofs.
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