IndisputableMonolith.Geometry.Schlaefli
The Schlaefli module supplies abstract edge-length data for finite simplicial complexes indexed by Fin nE. It is cited by researchers discharging the Regge deficit linearization hypothesis in phases C3 onward. The module consists of data definitions and identity statements that organize edge and hinge structures for later deficit calculations.
claimLet $n_E$ be a natural number indexing a finite set of edges. An abstract edge-length datum is a map $\ell: \mathrm{Fin}(n_E) \to \mathbb{R}^+$ assigning positive lengths to each edge of a simplicial complex. The module introduces associated hinge data, total angle sums $\Theta$, deficit functions $\delta$, and the Schläfli identity relating infinitesimal edge-length variations to dihedral-angle changes.
background
The module belongs to the Geometry domain and imports the Cayley-Menger module (Phase C1), which encodes simplex volumes via the determinant on edge lengths, and the DihedralAngle module (Phase C2), which extracts dihedral angles from the same data. These upstream modules establish the algebraic foundation for volume and angle extraction on simplices. The local setting is the incremental program to discharge ReggeDeficitLinearizationHypothesis on general simplicial complexes, with this module supplying the shared data layer for deficit and linearization steps.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds SimplicialDeficitDischarge (Phase C5), which composes prior phases into a general simplicial discharge of the Regge deficit linearization hypothesis for the paper's Theorem 5.1. It also supports DeficitLinearization (Phase C4) and the algebraic core of WeakFieldConformalRegge, which reduces the Regge action $S = (1/\kappa) \sum_h A_h \delta_h$ under conformal edge ansatz and weak-field expansion.
scope and limits
- Does not compute explicit numerical volumes or angles from edge lengths.
- Does not discharge the full ReggeDeficitLinearizationHypothesis.
- Does not extend to infinite or non-simplicial complexes.
- Does not incorporate background curvature or non-flat metrics.