IndisputableMonolith.Geometry.DeficitLinearization
The module defines structures for flat-background simplicial complexes with finite hinges and edges obeying the flat-sum condition on dihedral angles. Researchers discharging the Regge deficit linearization hypothesis in the Recognition Science program cite it as the Phase C4 assembly of geometric primitives. It supplies the perturbation and certification objects that close the linearization step before downstream composition.
claimA flat-background simplicial complex consists of finitely many hinges indexed by $n_H$ and edges indexed by $n_E$, each hinge obeying the flat-sum condition on its dihedral angles derived from edge lengths.
background
The module sits inside the program to formalize Regge calculus for piecewise-flat complexes. It imports the Cayley-Menger determinant that encodes simplex volumes from edge lengths, the dihedral-angle construction from those lengths, and Schläfli's identity relating angle and volume variations. The supplied doc-comment states the central object: a flat-background simplicial complex with finitely many hinges and edges satisfying the flat-sum condition.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the linearization definitions that the SimplicialDeficitDischarge module composes to prove the paper's Theorem 5.1 on the field-curvature identity. It completes Phase C4 of the discharge sequence for the ReggeDeficitLinearizationHypothesis on general simplicial complexes.
scope and limits
- Does not treat non-flat background geometries.
- Does not compute explicit numerical deficit values.
- Does not address infinite or continuous complexes.
- Does not discharge the full hypothesis without the downstream module.