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

The zero-momentum quadratic form

In the framework's study of gravity as a discrete geometry, the action is built from the areas of triangles in a four-dimensional Regge calculus. The declaration trueWeightZeroMomQuadratic assembles the second variation of this action for a constant perturbation of edge lengths. The key result, trueWeightZeroMomQuadratic_eq_bilinear, proves that this quadratic expression is exactly equal to a symmetric bilinear form, meaning it behaves like a product of two linear factors. This is the polarization identity: for any two perturbations, the quadratic form can be recovered from the bilinear form, and vice versa.

The theorem also proves that this form evaluates to zero on three specific test directions: a pure gauge direction, a trace direction, and a homothety direction. In plain terms, these are perturbations that do not change the physical geometry, only its description or its overall scale. The framework's choice of weights, called true weights, makes these spurious directions cost nothing. The earlier provisional weight-1 aggregate gave a nonzero value of 32 on the same gauge decoy, so the true weights are what kill these unphysical modes.

The scope of this result is deliberately narrow. It applies only to the zero-momentum case, where the perturbation is constant across the lattice. The finite-momentum case, where the perturbation varies with a Bloch phase, remains open. The theorem does not prove that the discrete Regge action converges to the Einstein-Hilbert action in four dimensions, and it does not establish the recovery of the gap action. The weights themselves are not derived from Einstein-Hilbert; they come from the committed geometry modules that define the orbit counts, area gradients, and deficit kernels.

What this means for the reader is that the framework has a machine-checked handle on a specific piece of the discrete gravity action: the second variation at zero momentum is a well-defined quadratic form, and the true weights correctly eliminate pure gauge and scale artifacts. This is a necessary step, but it is only one step. The path from this local quadratic form to a full quantum gravity theory remains long and mostly open.

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 theorem does not prove convergence of the discrete Regge action to the Einstein-Hilbert action in four dimensions. The theorem does not establish the recovery of the gap action. The weights are not reverse-engineered 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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