Gravity Analysis Regge Edge Stencil4 D Finite Ttquadratic Not Gauge Invariant On

A provisional gravity probe

In numerical gravity, researchers often approximate continuous spacetime with a discrete grid, and they need a way to measure the energy of the gravitational field on that grid. The declaration under discussion defines one such probe, called finiteTTQuadratic, which is a sum of squared contributions from fifteen distinct edge directions in a four-dimensional hypercube. The probe is designed to isolate transverse-traceless (TT) modes, the radiative part of the gravitational field. The central result is that this probe is not gauge invariant: its value changes under a pure gauge transformation, a coordinate change that should not affect physical predictions.

The machine-checked library of formal theorems proves that the probe returns 32 on a pure gauge input, not zero as exact gauge invariance would require. This is not an approximation or a numerical glitch; it is an exact identity. The theorem also shows the probe is nonzero on a specific TT seed, axisTTPlus, and that it evaluates to 80 on a pure trace input. These values serve as decoys, distinguishing the probe's behavior on different types of perturbations. The result is a concrete, if provisional, diagnostic: the probe is not a reliable physical observable because it does not respect the symmetry that separates real effects from coordinate artifacts.

The declaration does not claim to have found the correct gravity energy probe. The weights used in the sum are provisional, set to one for every edge class, and the true weights from the full theory remain an open problem. The result does not prove that the framework's action converges to the Einstein-Hilbert action in four dimensions, nor does it complete the decomposition of the edge TT operator. It is a single, sharp negative result: a specific candidate fails a specific test, and the failure is recorded exactly.

This matters because it shows the framework's method of building physical quantities from discrete geometry is testable. A candidate that fails gauge invariance is not discarded vaguely; it is shown to fail by a precise amount. The next step is to find weights that make the probe gauge invariant, or to derive the true weights from the underlying theory. The declaration is a checkpoint, not a destination.

theorem finiteTTQuadratic_gaugePart_axisWave :
    finiteTTQuadratic (gaugePart axisWave axisGaugeVector) = 32 := by
  unfold finiteTTQuadratic
  have hterm :
      ∀ d : Fin 15,
        classCoeff (gaugePart axisWave axisGaugeVector) d ^ 2 =
          (4 : ℝ) * (hasBit0 d : ℝ) := by
    intro d
    rw [classCoeff_gaugePart_axis]
    unfold hasBit0
    cases classBit d 0 <;> norm_num
  simp_rw [hterm, ← Finset.mul_sum, ← Nat.cast_sum, sum_hasBit0]
  norm_num

theorem finiteTTQuadratic_axisTTPlus_ne_zero :
    finiteTTQuadratic axisTTPlus ≠ 0 := by
  rw [finiteTTQuadratic_axisTTPlus]
  norm_num

theorem sum_weightSqNat : (∑ d : Fin 15, classWeightNat d ^ 2) = 80 := by
  unfold classWeightNat classBit maskOf
  decide

What this page does not claim

This does not prove the framework's action converges to the Einstein-Hilbert action in four dimensions. This does not provide the true Regge weights for the 4D Hessian. This does not establish a gauge-invariant energy observable for discrete gravity.

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/ReggeEdgeStencil4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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