IndisputableMonolith.Geometry.Schlaefli
The Schlaefli module supplies abstract edge-length data structures for finite simplicial complexes together with total deficit and Schlaefli identity definitions. It bridges Cayley-Menger volume formulas and dihedral angle computations to deficit linearization. Workers on discrete gravity and Regge calculus cite it when discharging curvature identities in simplicial settings. The module is purely definitional, introducing types and functions without theorems or proofs.
claimLet $n_E$ be a natural number. An abstract simplicial edge-length datum is a map $ell : Fin n_E to R_{>0}$. The module defines total deficit $delta$ and certifies the Schlaefli identity relating edge-length variations to vanishing first-order curvature terms.
background
This module sits in the geometry layer after the Cayley-Menger module (Phase C1), which encodes simplex volumes via the determinant of squared edge lengths, and the DihedralAngle module (Phase C2), which extracts dihedral angles at edges and 2-faces from the same data. It introduces SimplicialEdgeData as the core type for edge lengths indexed by Fin nE, SimplicialHingeData for the 2-faces, and auxiliary functions totalTheta, deficit, and totalDeficit that aggregate angles and curvature measures around hinges. The local setting prepares the algebraic substrate for linearizing Regge deficits around flat backgrounds.
proof idea
This is a definition module, no proofs. It declares the core types SimplicialEdgeData and SimplicialHingeData, then defines totalTheta, deficit, totalDeficit_flat, SchlaefliIdentity, and the certificate schlaefliCert that the identity annihilates first-order dtheta contributions.
why it matters in Recognition Science
The module supplies the Schlaefli identity and deficit structures that feed directly into SimplicialDeficitDischarge (Phase C5, discharging the ReggeDeficitLinearizationHypothesis for paper Theorem 5.1) and DeficitLinearization (Phase C4). It also supports WeakFieldConformalRegge by furnishing the edge-length data layer required for the conformal ansatz and second-order expansion of the Regge action.
scope and limits
- Does not embed the complex into Euclidean space or compute explicit volumes.
- Does not assume a concrete triangulation or numerical edge values.
- Does not address higher-dimensional curvature beyond the imported simplex setup.
- Does not derive the Schlaefli identity from variational principles.