Gravity Analysis Regge Flat4 Dhessian Assembly True Weight Kills Gauge At Zero M

The zero-momentum gauge test

In numerical relativity, the Regge action is a way to approximate Einstein's equations by replacing smooth spacetime with a grid of flat simplices, like a geodesic dome made of triangles. The action's second variation, its Hessian, tells you how the grid's energy responds to small distortions of the edge lengths. A long-standing problem is that this Hessian can respond to distortions that are pure gauge: changes that do not alter the physical geometry, only its description, like re-labeling points on a map. A healthy action should be blind to such distortions, giving zero response.

The declaration trueWeight_kills_gauge_at_zero_momentum reports a machine-checked theorem in the framework's library of formal proofs. It concerns a four-dimensional flat grid, the simplest Regge setting, and specifically the zero-momentum case: a uniform distortion applied everywhere at once, rather than a wave with a finite wavelength. The theorem evaluates the newly assembled, orbit-count-weighted Hessian on three specific test distortions: a pure coordinate shear, a pure gauge mode, and a pure trace mode. On all three, the Hessian returns exactly zero. The previous, provisional weighting scheme gave a nonzero value of 32 on the same gauge test, a sign that it was not yet physically sound. The new result shows that the corrected weights remove this spurious response.

The calculation builds on explicit formulas. The area of a flat triangle is given by Heron's formula, and the theorem verifies the derivatives of that area with respect to each edge length at the flat configuration. These area gradients are then combined with the deficit kernels, the geometric objects encoding the curvature at each hinge, using orbit counts 72, 48, 48, 24, 24, 24 that come from the symmetry group of the four-simplex. The final quadratic form is assembled as a sum over these orbit types. The theorem also confirms that the associated bilinear form is symmetric and that a uniform scaling of the whole grid, a homothety, also evaluates to zero.

In Recognition Science, this result is a step toward showing that the discrete Regge action, built from first principles, recovers the continuum Einstein-Hilbert action in the appropriate limit. The theorem is tagged THEOREM, meaning it is kernel-checked with no gaps in its proof. Its scope is deliberately narrow: it applies only to the zero-momentum, per-cell Hessian in four dimensions. The finite-momentum case, where the distortion varies across the grid like a wave, remains an open target. The theorem does not prove that the discrete action converges to the Einstein-Hilbert action, and it does not reverse-engineer the orbit weights from the continuum theory. Those weights are derived from the committed geometry modules, not fitted to match known physics.

What this means for the reader is a concrete, checkable improvement in the health of a discrete gravity action. A gauge-invariant Hessian at zero momentum is a necessary condition for the action to be a trustworthy starting point for further analysis. The result does not, by itself, establish the full recovery of general relativity, but it removes a known pathology and provides a clean test that future constructions can be held against.

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

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

theorem hasDerivAt_heronSq_a (a0 b c : ℝ) :
    HasDerivAt (fun t : ℝ => heronSq t b c) ((b + c - a0) / 8) a0 := by
  have hfun :
      (fun t : ℝ => heronSq t b c) =
        fun t : ℝ =>
          ((2 * b + 2 * c) * t - t ^ 2 + (2 * b * c - b ^ 2 - c ^ 2)) / 16 := by
    funext t; unfold heronSq; ring
  rw [hfun]
  have h :=
    (hasDerivAt_quad_sub_sq (2 * b + 2 * c) (2 * b * c - b ^ 2 - c ^ 2) a0).div_const
      (16 : ℝ)
  convert h using 1
  ring

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

What this page does not claim

This does not prove that the discrete Regge action converges to the Einstein-Hilbert action. This does not establish gauge invariance for finite-momentum distortions. This does not reverse-engineer the orbit weights from the continuum theory.

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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