Gravity Analysis Regge Flat4 Dhessian Assembly True Weight Zero Mom Quadratic Ax

A gauge test for a gravity action

In the search for a quantum theory of gravity, one standard strategy is to discretize spacetime into a grid of flat pieces, an approach known as Regge calculus. The action, the quantity whose variation gives the equations of motion, is built from the areas of the triangular faces and the angles between them. A fundamental requirement is that the action must not change when you merely relabel the coordinates, a symmetry called diffeomorphism invariance. In the discrete setting, this becomes a test: a pure coordinate change, a gauge direction, should produce zero change in the action.

The framework's machine-checked library of formal theorems has now verified this test for a specific, carefully constructed version of the discrete action. The declaration trueWeightZeroMomQuadratic_axisTTPlus establishes that the second variation of the action, the quadratic form that governs small perturbations, evaluates to zero along a particular gauge direction called axisTTPlus. This direction corresponds to a pure coordinate transformation at zero momentum, meaning a uniform, constant change across the whole grid. The same zero result holds for two other gauge directions, decoyGauge and decoyTrace, and for a uniform scaling of the grid, the homothety direction.

The result is not a numerical approximation. It is a theorem, checked by the Lean proof assistant, with no unproven axioms beyond the standard logical ones. The calculation assembles the full second-variation quadratic from the area gradients of four distinct triangle shapes, weighted by their symmetry orbit counts (72, 48, 48, 24, 24, 24). The key improvement over a previous, provisional version is that the new weights correctly kill all pure gauge directions, where the old version gave a nonzero value of 32 on the same test. This is a necessary condition for the action to be a physically viable candidate for quantum gravity.

In Recognition Science, this is a step toward a full theory of quantum gravity, but it is a narrow one. The theorem applies only to the zero-momentum, per-cell Hessian. It does not prove that the full action converges to the Einstein-Hilbert action of general relativity, nor does it address finite-momentum effects where the gauge symmetry is more subtle. The weights themselves are not derived from Einstein-Hilbert; they come from the committed geometry modules, and the result is a consistency check on that construction, not a derivation of it.

What this means for the reader is a concrete, verified fact: a specific discrete gravity action passes a fundamental gauge test. The framework has shown that its proposed action is not immediately ruled out by a failure of coordinate invariance. This is a small but essential brick in the wall of a quantum gravity theory, a result that can be built upon with confidence because it is machine-checked.

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

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

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

What this page does not claim

This does not prove the full action converges to the Einstein-Hilbert action of general relativity. This does not address finite-momentum effects where the gauge symmetry is more subtle. The weights are not derived from Einstein-Hilbert; they come from the committed geometry modules.

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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