Gravity Analysis Regge Flat4 Dhessian Assembly

The flat 4D gravity Hessian

Regge calculus is a discrete approach to general relativity in which spacetime is approximated by flat triangular pieces, called simplices, joined along their edges. The action is a sum over hinge edges of the deficit angle times the area of the hinge. In four dimensions the hinges are triangles, and the second variation of this action, the Hessian, controls how small perturbations of the edge lengths behave. The module ReggeFlat4DHessianAssembly builds that Hessian explicitly for a flat four-dimensional lattice and proves several of its properties in a machine-checked library of formal theorems.

The construction starts with the four flat triangle shapes that occur in the lattice, with squared side lengths (1,1,2), (1,2,3), (1,3,4), and (2,2,4). For each, the module derives the gradient of the triangle area with respect to its side lengths using Heron's formula. The area squared is A² = (2ab+2bc+2ca−a²−b²−c²)/16, and the gradients are explicit rational or square-root values at the flat configuration. These area gradients feed into the Hessian through a sum over the six orbit types of the permutation group S₄, weighted by counts 72, 48, 48, 24, 24, 24.

The central result is that this true-weight Hessian, called trueWeightZeroMomQuadratic, evaluates to zero on three test directions: the axis translation, a decoy gauge direction, and a decoy trace direction. The provisional weight-1 version gave 32 on the same decoy gauge, so the true weights kill pure gauge motions at zero momentum. A homothety direction, a uniform scaling of all edges, also evaluates to zero. The associated bilinear form is proved to be symmetric and bilinear, and the complement transport identity shows that edge-class indices are invariant under a vertex complement operation.

In Recognition Science, this is a step toward connecting the discrete gravity action to the Einstein–Hilbert action of continuum general relativity. The module does not prove that convergence, and it does not reverse-engineer weights from Einstein–Hilbert; all orbit counts, area gradients, and deficit kernels come from committed geometry modules. The scope is zero momentum, meaning constant edge-class perturbations only. Finite-momentum Bloch phase folding across hinge translates remains open.

What this establishes in plain language: on a flat four-dimensional lattice, the correctly weighted second variation of the discrete gravity action has the gauge-killing property one expects from a physical action. Pure coordinate deformations cost no energy, and uniform scaling costs none either. That is a necessary consistency check for any candidate discrete gravity action, and it now holds with machine-checked proofs.

def trueWeightZeroMomQuadratic (H : Mat4) : ℝ :=
  ∑ ty : HingeOrbitType, orbitZeroMomQuadratic ty H

def trueWeightZeroMomQuadratic (H : Mat4) : ℝ :=
  ∑ ty : HingeOrbitType, orbitZeroMomQuadratic ty H

def homothetyClassCoeff : Mat4 := decoyTrace

def trueWeightZeroMomBilinear (A B : Mat4) : ℝ :=
  ∑ ty : HingeOrbitType, orbitZeroMomBilinear ty A B

What this page does not claim

This module does not prove convergence of the discrete Regge action to the Einstein–Hilbert action. This module does not derive the weights from Einstein–Hilbert; they come from committed geometry modules. This module does not address finite-momentum perturbations or Bloch phase folding.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/Analysis/ReggeFlat4DHessianAssembly.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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