GenuineComponentPackage
plain-language theorem explainer
A data package that packages a weak-field Regge configuration on a finite 3D triangulation together with a symmetric nonnegative geometric area matrix whose off-diagonal entries match the bilinear coefficients of that configuration, plus a Schläfli row-sum identity. Gravity workers cite it as the interface between Cayley–Menger/dihedral Hessian geometry and the existing conformal Regge comparison layer. As a structure, it is pure data with no proof body.
Claim. For a finite 3D Regge triangulation $K$, a genuine component package consists of weak-field Regge data $W$ on the vertices of $K$, a symmetric nonnegative geometric area function $A(i,j)$ on vertex pairs, the matching condition that off-diagonal bilinear coefficients of $W$ equal $-A(i,j)$, and a Schläfli row-sum identity for $W$.
background
The module bridges genuine 3D Regge Hessian geometry to the weak-field conformal Regge layer already developed elsewhere. A triangulation $K$ is a finite abstract 3D Regge complex: vertex, edge, and tetrahedron counts with incidence maps and nondegenerate squared-edge data on each tetrahedron.
Weak-field Regge data $W$ on $K.nV$ vertices carry the bilinear coefficients that appear in the second-order expansion of the Regge action. The geometric area matrix is the dual weight coming from Cayley–Menger volumes and dihedral angles; symmetry and nonnegativity are the elementary geometric constraints. The off-diagonal match forces those bilinear coefficients to equal minus the geometric areas, so the discrete Hessian is identified with a graph Laplacian weighted by areas. The Schläfli row-sum condition is the discrete identity that makes the conformal (trace) sector reduce cleanly.
Upstream, the comparison and Dirichlet-form machinery already lives in the weak-field conformal Regge development; this package is the geometric input that package expects.
proof idea
No proof: the declaration is a structure (bundled data and propositions). Inhabitants are built downstream by supplying weak-field data, a geometric area matrix with the stated symmetry and sign, the off-diagonal coefficient match, and a proof of the Schläfli row-sum. Concrete constructors appear in the companion proof module, which lift a finished Cayley–Menger/dihedral computation into this bundle.
why it matters
This is the hinge between the geometric 3D Regge Hessian computation and the already-proved weak-field conformal reduction. Downstream, componentComparison_of_genuine turns any such package into the existing ReggeComponentComparison interface by reading off the area matrix and its symmetry/nonnegativity. From there, genuine_component_dirichlet_reduction applies the weak-field conformal reduction: the second-order Regge action equals half the Dirichlet form of the edge-area weights on a log-potential, using the packaged Schläfli row-sum.
The companion proof module closes the loop the other way: a concrete component comparison (or a final target theorem under incidence consistency) produces a nonempty package for every admissible triangulation. In the Recognition gravity stack this is the 3D spatial step that feeds continuum-limit and weak-field claims once the Hessian geometry is discharged; it sits under the $D=3$ landmark of the forcing chain without itself deriving dimensionality.
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