IndisputableMonolith.Geometry.CayleyMenger
Module Geometry.CayleyMenger supplies edge-length data structures and Cayley-Menger identities for tetrahedra. It is cited by the four downstream geometry modules that discharge the Regge deficit linearization hypothesis. The module consists entirely of definitions and basic identities with no proofs.
claimTetEdges records the six edge lengths of a tetrahedron; TetCMData packages the associated Cayley-Menger 5-by-5 matrix; TetVolumeIdentity equates the squared volume to the determinant of that matrix.
background
The module sits in the geometry layer that prepares piecewise-flat simplicial complexes for Regge calculus. It introduces TetEdges as the assignment of six edge lengths to the complete graph K4 and TetCMData as the matrix whose determinant yields the squared volume. These objects are the geometric primitives required by the Piran-Williams linearization and Schläfli identity.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module feeds SimplicialDeficitDischarge (Phase C5, proving the paper's Theorem 5.1 field-curvature identity) together with DeficitLinearization (Phase C4), DihedralAngle (Phase C2), and Schlaefli (Phase C3). All four downstream modules import it to discharge the Regge deficit linearization hypothesis on general simplicial complexes.
scope and limits
- Does not compute numerical volumes or angles.
- Does not treat higher-dimensional simplices.
- Does not encode curvature or deficit quantities.
- Does not assume any embedding into Euclidean space.