Gravity Analysis Regge Flat4 Dhessian Assembly True Weight Zero Mom Bilinear Add

The bilinear form

In the Recognition Science framework's study of gravity, the starting point is a discrete geometry built from flat four-dimensional pieces, in the spirit of Regge calculus. The relevant object is a Hessian, a table of second derivatives that describes how a quantity such as an action changes when the geometry is perturbed. The framework's library has constructed a specific Hessian for a flat four-dimensional Regge geometry, using a particular weighting of the contributions from different types of hinges, the edges around which the flat pieces meet.

The declaration trueWeightZeroMomBilinear_add_left concerns the associated bilinear form, a function that takes two perturbations and returns a number, built from the Hessian by a standard polarization procedure. The theorem states that this form is additive in its first argument: the value for the sum of two perturbations equals the sum of the values for each perturbation separately. This is a linearity property, a basic structural feature that makes the form a well-behaved object for further analysis. The same property holds for the second argument by symmetry.

The proof is part of a larger, machine-checked effort to assemble the correct quadratic approximation to the Regge action in four dimensions. The library has also proved that this weighted Hessian gives zero when evaluated on certain pure gauge deformations, which are changes that do not alter the physical geometry. This is a desirable property: a physically meaningful action should be insensitive to such redundancies. The additivity theorem is a technical but necessary step in establishing that the Hessian is a genuine quadratic form, suitable for studying the stability and dynamics of the discrete geometry.

What the theorem does not claim is broader. It does not prove that the discrete Regge action converges to the Einstein-Hilbert action of continuous general relativity in any limit. It does not establish that the Hessian is positive definite, which would indicate a stable minimum. It does not address perturbations with finite momentum, only those that are constant across the geometry. These remain open targets within the framework, and the additivity result is a foundation stone for them, not a shortcut past them.

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

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

What this page does not claim

This theorem does not prove convergence of the discrete action to the Einstein-Hilbert action. This theorem does not establish positivity or stability of the geometry. This theorem does not cover finite-momentum perturbations.

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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