IndisputableMonolith.Geometry.ReggeHessian3D
Defines vertex conformal potentials and the quadratic Hessian form of the Regge action on a finite 3D triangulation. Supplies the data package that equates the second variation of the Regge action to a sum of local Hessian contributions. Downstream gravity and action modules import this interface to connect genuine 3D geometry to the weak-field conformal bridge. The content is definitional scaffolding plus algebraic identities for the quadratic form.
claimOn a finite 3D triangulation, assign a conformal potential $\phi_v$ to each vertex $v$. The zero potential is the constant assignment. The Hessian quadratic form is $\sum_{\sigma} H_\sigma(\phi)$, summed over top simplices, and the second variation of the Regge action equals this quadratic form under the conformal ansatz.
background
Regge calculus discretizes gravity on a triangulation by assigning edge lengths and deficit angles. In three dimensions the Schläfli identity relates variations of dihedral angles to variations of edge lengths; the upstream module SchlaefliTriangulation3D packages the global cancellation as the finite sum of local tetrahedral identities.
This module works in the conformal sector: each vertex carries a real potential, and edge lengths (or metric scale factors) are deformed by differences of those potentials. The zero potential is the flat reference assignment. The Hessian of the Regge action at that reference is a quadratic form on the space of vertex potentials.
The local theoretical setting is finite combinatorial 3-manifolds (or regions) triangulated by tetrahedra, with no continuum limit taken here. Notation follows the sibling definitions: vertex potentials, the zero section, the Hessian quadratic form, and a data bundle packaging those objects for downstream use.
proof idea
Primarily a definition module. It introduces the type of vertex conformal potentials, the zero potential, the Hessian quadratic form as a sum over simplices, and a ReggeHessianData package bundling those pieces. Algebraic lemmas establish that the double sum defining the quadratic form may be reordered (sum-commute) and that the second variation of the Regge action equals the Hessian quadratic form under the conformal ansatz. No deep analytic argument lives here; the equalities are rearrangements and interface wiring onto the Schläfli triangulation substrate.
why it matters in Recognition Science
This module is the geometric Hessian interface for 3D Regge calculus in the Recognition stack. ReggeActionConcrete imports it to isolate the final analytic Hessian step: a concrete action package supplies the Regge action under the conformal ansatz and proves its second variation, then turns that into the existing Hessian data interface defined here.
ReggeComponentTheorem3D further connects the genuine 3D Regge Hessian package to the weak-field conformal Regge bridge, producing a component comparison that feeds the Dirichlet-form reduction already proved in the weak-field layer. In the broader framework this sits under the D = 3 forcing (T8) and supplies the discrete second-variation object needed before continuum or continuum-limit gravity statements.
Without this package, downstream modules would re-encode vertex potentials and Hessian sums ad hoc; the shared interface keeps the geometric computation separate from the gravity-bridge wiring.
scope and limits
- Does not prove continuum Einstein equations or any smooth limit of the triangulation.
- Does not treat dimensions other than three or non-conformal metric variations.
- Does not establish positivity or signature of the Hessian spectrum.
- Does not define the full Regge action; only the Hessian interface and second-variation identity.
- Does not address curvature sourcing, matter coupling, or boundary terms beyond the finite sum.