IndisputableMonolith.Gravity.ReggeComponentTheorem3D
Packages the genuine geometric component of the 3D Regge action under the conformal weak-field ansatz: the Cayley–Menger, dihedral, and Hessian data that feed the component comparison. Gravity workers cite it when separating the true geometric quadratic form from dual-weight scaffolding. The module is mostly definitional packaging plus reduction lemmas that hand a clean comparison object to the proof module.
claimOn a finite 3D triangulation, under the conformal edge ansatz $\ell_{ij}=\ell_0\exp((\xi_i+\xi_j)/2)$ and weak-field expansion of the Regge action $S=(1/\kappa)\sum_h A_h\delta_h$ to second order in $\xi$, the module assembles a genuine geometric component package (Hessian/quadratic Taylor data from the Cayley–Menger and dihedral geometry) and records the comparison and Dirichlet-reduction maps that turn that package into a component-comparison statement.
background
Recognition Science gravity work reduces continuum GR-like dynamics to discrete Regge calculus on a triangulation, then specializes to three spatial dimensions (the T8 landmark $D=3$). The Regge action is $S=(1/\kappa)\sum_h A_h\delta_h$ over hinges. The conformal edge ansatz sets every edge length from vertex fields $\xi$ via $\ell_{ij}=\ell_0\exp((\xi_i+\xi_j)/2)$; the weak-field step expands $S$ to quadratic order in $\xi$.
Upstream, ReggeHessian3D supplies the Hessian interface for the genuine Regge action on a finite 3D triangulation: an explicit action, its Hessian matrix, and the theorem that the quadratic Taylor coefficient is that matrix. WeakFieldConformalRegge proves the algebraic core of that reduction (action, ansatz, second-order expansion). This module sits between those geometric inputs and the final comparison target: it names the genuine component package produced by the Cayley–Menger / dihedral / Hessian computation, rather than a dual-weight surrogate.
proof idea
Definition-and-packaging module, not a single deep proof. It introduces the genuine component package as the geometric object coming out of the Cayley–Menger, dihedral, and Hessian pipeline under the 3D conformal weak-field ansatz. Sibling declarations then (i) build a component-comparison witness from a genuine package and (ii) record the Dirichlet-style reduction that relates the genuine quadratic form to the comparison matrix. Heavy analytic work is imported from the Hessian and weak-field modules; this file organizes those outputs into the comparison-facing API consumed by the proof module.
why it matters in Recognition Science
Feeds ReggeComponentTheorem3DProof, which "separates the independent dual-weight construction from the weak-field coefficient matrix and records the theorem that turns that geometric computation into ReggeComponentComparison." Without a clean genuine-package interface, the final comparison would mix dual-weight scaffolding with true 3D Regge geometry.
In the broader RS gravity chain this is the discrete, $D=3$ step that makes the weak-field conformal Regge quadratic form auditable: Cayley–Menger volumes and dihedral defects determine the Hessian, the conformal ansatz reduces edge variations to vertex fields, and the package is what gets compared to continuum or dual-weight targets. It is infrastructure for the component theorem, not the continuum limit itself.
scope and limits
- Does not prove the final ReggeComponentComparison theorem; that lives in the Proof module.
- Does not derive continuum Einstein equations or fix Newton's constant.
- Does not treat Lorentzian signature, curvature singularities, or non-conformal edge variations.
- Does not replace dual-weight constructions; it only packages the genuine geometric side.
- Does not claim numerical values for masses, alpha, or other RS constants.