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module module moderate

IndisputableMonolith.Relativity.Geometry

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This module assembles the geometric primitives for pseudo-Riemannian spacetime in the Recognition Science relativity setting. It supplies manifold, metric, connection, curvature, and discrete-bridge structures to downstream GR derivations. Researchers working on lattice-to-continuum limits from J-cost models would cite it. The module consists of nine imports and placeholder definitions rather than a single unified proof.

claimThe module defines a pseudo-Riemannian manifold $(M, g)$ equipped with the unique torsion-free metric-compatible connection whose Christoffel symbols are derived from $g$, together with curvature tensors and the discrete-to-continuum bridge mapping lattice J-cost to the Einstein tensor.

background

The module imports scaffold placeholders for manifolds and connections in which all Christoffel symbols default to zero and covariant derivatives collapse to zero. It includes the Levi-Civita theorem establishing the unique torsion-free metric-compatible connection on any pseudo-Riemannian manifold, with coefficients defined in the Curvature submodule. The DiscreteBridge submodule maps the RS discrete lattice theory to continuum GR via the chain J-cost lattice to quadratic defect to lattice Laplacian to Ricci scalar to Einstein tensor.

proof idea

This is a definition module, no proofs. It organizes nine imported submodules (Manifold, Tensor, Metric, Connection, Curvature, MetricUnification, LeviCivitaTheorem, ParallelTransport, DiscreteBridge) to supply the geometric setting for relativity applications.

why it matters in Recognition Science

This module feeds the geometry into parent theorems in Compact.StaticSpherical, Cosmology.FRWMetric, Geodesics.NullGeodesic, GW.ActionExpansion, and Fields.Integration. It supplies the continuum infrastructure that links the discrete Recognition Science lattice to general relativity, as described in the DiscreteBridge architecture.

scope and limits

used by (9)

From the project-wide theorem graph. These declarations reference this one in their body.

depends on (9)

Lean names referenced from this declaration's body.