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

The zero-momentum gauge check

In numerical relativity, the Regge action is a discrete version of Einstein's theory built from flat four-dimensional simplices. The declaration trueWeightZeroMomQuadratic is a quadratic form, a second-order measure of how the action responds to small changes in edge lengths. The specific object decoyGauge is a test deformation, a direction in the space of edge-length perturbations designed to mimic a coordinate change rather than a physical distortion.

The key result is that the quadratic form evaluates to zero on this test direction. In the framework's library, a machine-checked collection of formal theorems, this is stated as the theorem trueWeightZeroMomQuadratic_decoyGauge, which proves the value is 0. This is a gauge check: a pure coordinate deformation should not change the physical content of the action, and the zero confirms the construction passes this necessary test at zero momentum, meaning for constant edge-class perturbations across the lattice.

The calculation is not a free parameter fit. The weights come from committed geometry modules: orbit counts (72, 48, 48, 24, 24, 24) and Heron area gradients for four triangle types. The result replaces a provisional weight-1 aggregate that gave a nonzero value of 32 on the same decoy, so the new assembly behaves better under the gauge test. The theorem is kernel-checked with no admitted axioms, within the framework's standard axiom footprint.

What this does not claim is broader physical significance. It does not prove that the full Regge action converges to the Einstein-Hilbert action in four dimensions. It does not establish that the gap action recovery holds. It does not reverse-engineer weights from Einstein-Hilbert; all inputs come from the committed geometry modules. The scope is strictly zero-momentum per-cell Hessian; finite-momentum Bloch phase folding remains an open target.

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

The declaration does not prove S_RS_converges_EH_4d. The declaration does not flip gap_action_recovery. The declaration does not reverse-engineer weights from Einstein-Hilbert.

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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