IndisputableMonolith.Foundation.SimplicialLedger.ContinuumTheorem
Packages the field-curvature identity in Einstein-coupling form: for a weighted ledger graph, conformal spacing, and log-potential, the Laplacian action equals the Regge sum scaled by 1/κ_Einstein with κ_Einstein = 8 φ⁵ in RS-native units. Combines draft Theorems 5.1 and 6.1 so the bridge normalization is the physical Einstein coupling. Cited by anyone deriving continuum GR from the simplicial ledger. Assembles ContinuumBridge, CubicDeficitDischarge, and EdgeLengthFromPsi.
claimFor any weighted ledger graph $G$, conformal spacing $a>0$, and log-potential field $\varepsilon$, the Laplacian action equals the Regge sum scaled by the inverse Einstein coupling: $\mathrm{Lap}(G,\varepsilon)=(1/\kappa_E)\,\mathrm{Regge}(\ldots)$, where $\kappa_E=8\pi G/c^4=8\phi^5$ in RS-native units ($c=1$, $G=\phi^5/\pi$). Flat configurations have vanishing Regge sum.
background
Recognition Science derives continuum gravity from a discrete simplicial ledger whose cost is the J-functional $J(x)=(x+x^{-1})/2-1$. The ContinuumBridge module states that this J-cost is the Regge action up to normalization $\kappa=8\phi^5$, and that stationarity $\delta J=0$ yields the Regge equations, closing the gap from discrete ledger to Einstein field equations.
EdgeLengthFromPsi removes a silent identification in the draft: a scalar recognition potential $\psi$ on 3-simplices must determine the full set of edge lengths (ten per 4-simplex) that enter the Regge action. CubicDeficitDischarge then discharges the Regge-deficit linearization hypothesis unconditionally on the RS-canonical cubic lattice (Phase A of promoting draft Theorem 5.1 to a Lean theorem).
This module sits on those three imports plus Constants ($\tau_0=1$ tick; $G=\phi^5/\pi$). Its main identity equates the discrete Laplacian action of a log-potential $\varepsilon$ to the Regge curvature sum divided by the physical Einstein coupling.
proof idea
Not a single monolithic proof: the module wires upstream lemmas into the Einstein-normalized identity and supporting certificates. ContinuumBridge supplies J-cost = Regge action (factor $8\phi^5$) and stationarity $\Rightarrow$ Regge equations. EdgeLengthFromPsi converts the recognition potential into edge lengths so the Regge sum is well-defined. CubicDeficitDischarge removes the linearization hypothesis on the cubic lattice. The head theorem field_curvature_identity_einstein assembles these into Lap = Regge/$\kappa_E$; flat_regge_sum_zero and bridge_chain_complete record the flat case and end-to-end chain; ContinuumFieldCurvatureCert packages the certificate.
why it matters in Recognition Science
Fills draft Theorems 5.1 (field-curvature identity) and 6.1 (bridge normalization = physical Einstein coupling $\kappa_E=8\phi^5$). In RS-native units this is exactly $8\pi G/c^4$ once $G=\phi^5/\pi$ and $c=1$ from the Constants layer, tying the discrete J-cost stationarity path to continuum GR. Downstream used_by is still empty; the module is the natural citation point for any later continuum-limit or EFE-derivation theorem in the SimplicialLedger stack. It is the Phase-A closure target named by CubicDeficitDischarge for promoting the pattern-match argument to a genuine theorem.
scope and limits
- Does not derive the full Einstein field equations in continuum coordinates; only the discrete Lap–Regge identity.
- Does not discharge deficit linearization off the RS-canonical cubic lattice.
- Does not prove uniqueness of the conformal spacing or of the log-potential ansatz.
- Does not evaluate numerical α or mass-ladder claims; gravity-coupling only.
- Does not yet appear in downstream used_by edges inside the mirror.