field_curvature_identity_einstein
plain-language theorem explainer
The theorem equates the J-cost Dirichlet energy of any log-potential on a weighted ledger graph to one over the Einstein coupling times the Regge sum of the cubic deficit functional evaluated on the conformal edge-length field. Researchers deriving the continuum Einstein equations from discrete Regge calculus in Recognition Science cite this when the coupling must match the physical value 8 phi^5. The proof is a one-line wrapper that substitutes the bridge normalization lemma and invokes the pre-proved cubic identity.
Claim. For any natural number $n$, positive real $a>0$, weighted ledger graph $G$, and log-potential field $ε$, the J-cost Dirichlet energy satisfies $Δ_G ε = (1/κ_E) · Regge sum of the cubic deficit functional on the conformal edge lengths generated by $ε$ and the cubic hinges of $G$, where $κ_E = 8φ^5$ is the Einstein coupling in RS-native units.
background
WeightedLedgerGraph encodes a simplicial complex with edge weights. LogPotential n assigns real values to vertices. laplacian_action computes the associated J-cost Dirichlet energy via the Recognition Composition Law. regge_sum aggregates cubic deficits over hinges. conformal_edge_length_field produces edge lengths from the potential scaled by spacing $a$. kappa_einstein is defined as $8φ^5$ from the constants module, obtained by substituting $G = φ^5/π$ and $ℏ = φ^{-5}$ into $8πG/c^4$ with $c=1$.
proof idea
The proof is a one-line wrapper. It rewrites the normalization factor by applying jcost_to_regge_factor_eq_kappa_einstein (from EdgeLengthFromPsi) to replace the generic bridge factor with the Einstein coupling, then invokes the exact cubic identity field_curvature_identity_cubic (from CubicDeficitDischarge) which holds after the linearization discharge.
why it matters
This supplies the exact field-curvature identity with the physical Einstein coupling, feeding directly into bridge_chain_complete (which assembles the full cubic bridge chain from RCL through J-uniqueness and phi) and continuumFieldCurvatureCert (which bundles discharge, identity, flat-vacuum vanishing, and the coupling value). It realizes the paper's Theorem 5.1 in the Recognition framework, where the eight-tick octave and phi-ladder fix the coupling at $8φ^5$. No scaffolding remains.
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