IndisputableMonolith.Geometry.DeficitLinearization
The module defines a flat-background simplicial complex as a finite collection of hinges and edges obeying the flat-sum condition on dihedral angles. Researchers linearizing Regge deficits in discrete gravity cite it as the Phase C4 object. It assembles volume, angle, and identity results from three imported modules into one setting without new derivations.
claimA flat-background simplicial complex consists of finite index sets for hinges and edges together with edge lengths and dihedral angles such that the sum of dihedral angles around each hinge equals $2π$.
background
Recognition Science models spacetime via piecewise-flat simplicial complexes. This module introduces the flat-background variant as the setting for deficit linearization, relying on the Cayley-Menger module for simplex volumes from edge lengths, the DihedralAngle module for computing angles at edges, and the Schlaefli module for the identity relating angle variations to volume changes. The central object is a collection of hinges indexed by a finite set and edges by another finite set, each hinge satisfying the flat-sum condition.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
This module supplies the structural foundation for the SimplicialDeficitDischarge module, which completes Phase C5 by discharging the Regge deficit linearization hypothesis and thereby proving the field-curvature identity stated as Theorem 5.1 in the paper.
scope and limits
- Does not treat curved or non-simplicial geometries.
- Does not derive the explicit form of the linearized deficit.
- Does not include numerical verification for example complexes.
- Does not address higher-dimensional extensions.