IndisputableMonolith.Foundation.SimplicialLedger.ContinuumTheorem
The ContinuumTheorem module establishes the Einstein-coupling form of the field-curvature identity, equating the Laplacian action on any weighted ledger graph to a normalized Regge sum with factor 8φ⁵. Researchers deriving gravity from the J-cost functional in Recognition Science cite it as the bridge to the Einstein field equations. The argument assembles the J-cost-to-Regge equivalence, the potential-to-geometry map, and the cubic-lattice discharge into a single identity.
claimFor any weighted ledger graph $G$, conformal spacing $a>0$, and log-potential field $\varepsilon$, \[ \laplacian_action(G,\varepsilon) = \frac{1}{8\phi^5} \cdot \regge_sum(G,a,\varepsilon) \] where the normalization constant is the RS-native Einstein coupling $\kappa_Einstein=8\phi^5$.
background
The module sits inside the Foundation.SimplicialLedger hierarchy of Recognition Science and imports the J-cost functional, whose stationarity is the source of all dynamics. ContinuumBridge shows that this J-cost on the simplicial ledger coincides with the Regge action up to normalization by $\kappa=8\phi^5$, and that $\delta J=0$ recovers the Regge equations. EdgeLengthFromPsi supplies the missing map from the scalar recognition potential $\psi$ on 3-simplices to the ten edge lengths per 4-simplex required by the Regge action. CubicDeficitDischarge removes the linearization hypothesis for the canonical cubic lattice, while Constants fixes the base time quantum $\tau_0=1$ tick.
proof idea
This is a composition module. It imports the three upstream results (J-cost equals Regge action from ContinuumBridge, $\psi$-to-edge-length identification from EdgeLengthFromPsi, and unconditional cubic discharge from CubicDeficitDischarge) and assembles them directly into the Einstein-coupling form of the field-curvature identity stated in the module doc-comment.
why it matters in Recognition Science
The module realizes the paper's Theorem 5.1 (field-curvature identity) together with Theorem 6.1 (bridge normalization) by fixing the coupling at $8\phi^5$. It supplies the final link from discrete J-cost stationarity on the ledger to the continuum Einstein equations inside the Recognition Science program that begins from the forcing chain (T0-T8) and the Recognition Composition Law.
scope and limits
- Does not treat non-cubic lattice presentations.
- Does not derive the Bianchi identities or stress-energy conservation.
- Does not include matter sources or non-vacuum solutions.
- Does not address dynamical evolution or initial-value problems.