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IndisputableMonolith.Foundation.SimplicialLedger.ContinuumTheorem

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The ContinuumTheorem module states the field-curvature identity that equates the Laplacian action on any weighted ledger graph to the normalized Regge sum with Einstein coupling 8φ^5. Discrete-gravity researchers would cite it as the explicit bridge from J-cost stationarity to the Einstein equations. The module assembles the stationarity result, the edge-length identification, and the cubic-lattice discharge into a single unconditional statement.

claimFor any weighted ledger graph $G$, conformal spacing $a>0$, and log-potential field $\varepsilon$, the identity reads $\Delta_G\varepsilon = \frac{1}{8\phi^5}\cdot\operatorname{ReggeSum}(G,a,\varepsilon)$, where $\kappa_E=8\phi^5$ is the Einstein coupling in RS-native units.

background

The module belongs to the SimplicialLedger hierarchy and imports four upstream modules. Constants supplies the RS time quantum $\tau_0=1$ tick. ContinuumBridge proves that the J-cost functional on the simplicial ledger coincides with the Regge action (normalized by $\kappa=8\phi^5$) and that J-cost stationarity yields the Regge equations. EdgeLengthFromPsi identifies the scalar recognition-potential field $\psi$ on 3-simplices with the full set of edge lengths needed for a Regge action. CubicDeficitDischarge removes the ReggeDeficitLinearizationHypothesis for the RS-canonical cubic lattice, completing Phase A of the program to convert the draft paper's Theorem 5.1 into a Lean theorem.

proof idea

The module has no internal proof body. It structures the argument by importing the three supporting modules and declaring the combined field-curvature identity once the cubic-lattice case is discharged. The overall structure therefore consists of the stationarity theorem from ContinuumBridge, the geometric identification from EdgeLengthFromPsi, and the unconditional discharge from CubicDeficitDischarge.

why it matters in Recognition Science

This module supplies the paper's Theorem 5.1 (field-curvature identity) together with Theorem 6.1 by discharging the remaining linearization hypothesis. It feeds the derivation of the Einstein field equations from the Recognition Composition Law and J-uniqueness (T5). In the RS framework the result fixes the gravitational coupling at $\kappa_E=8\phi^5$, which is consistent with the native value $G=\phi^5/\pi$.

scope and limits

depends on (4)

Lean names referenced from this declaration's body.

declarations in this module (6)