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

A check against pure scaling

In physics, a symmetry is a transformation that leaves the laws unchanged. One of the oldest and most basic symmetries is scale invariance: the idea that stretching or shrinking an entire system uniformly should not change its fundamental behavior. For a candidate theory of gravity, this means that if you take a valid configuration of space and simply multiply all distances by the same factor, the action, the quantity whose minimization determines the dynamics, should not change. A theory that responds to such a uniform scaling would single out a preferred size for the universe, a feature no known fundamental theory possesses.

The declaration trueWeightZeroMomQuadratic_homothety is a formal, machine-checked proof that a specific candidate action for gravity in four dimensions satisfies this symmetry test. The action in question is built from a discrete approximation of spacetime known as Regge calculus, where continuous space is replaced by a network of flat triangular pieces, and the curvature is concentrated along their shared edges, called hinges. The candidate action is a quadratic form, a kind of energy function, assembled from the areas of these triangles and the angles between them. The theorem evaluates this quadratic form on a special direction, the homothety direction, which represents a uniform scaling of all edge lengths. The result is exactly zero.

This zero is not an accident of a crude approximation. The proof is carried out in a machine-checked library of formal theorems, meaning every step is verified by a computer. The computation uses the exact Heron formula for triangle areas and a specific weighting scheme for the hinges, with orbit counts 72, 48, 48, 24, 24, and 24. The theorem is part of a larger campaign to test whether this discrete action can recover the known Einstein-Hilbert action of general relativity in the appropriate limit. Passing this symmetry check is a necessary condition for that goal, but it is far from sufficient.

The theorem's scope is deliberately narrow. It applies only to a zero-momentum perturbation, meaning the scaling is constant across the entire space, with no spatial variation. The behavior of the action under a scaling that varies from point to point, a finite-momentum perturbation, remains an open problem. The theorem does not prove that this discrete action converges to the Einstein-Hilbert action, nor does it establish that the framework's own gravity action recovers the known gap in the energy spectrum. These are separate, much harder questions. The homothety check is one clean, verified stone in a much larger edifice, confirming that the candidate action does not possess an obvious fatal flaw.

In Recognition Science, this result is a concrete example of how the framework's formal methods are applied to a physical question. The framework does not merely assert that its gravity action is well-behaved; it provides a machine-checked proof of a specific symmetry property. This is a model of how the framework aims to build confidence in its claims: one precise, verifiable statement at a time. The homothety theorem is a small but solid piece of evidence that the candidate action is a serious contender, not a random construction.

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

This theorem does not prove that the discrete action converges to the Einstein-Hilbert action of general relativity. This theorem does not apply to finite-momentum perturbations, where the scaling varies from point to point. This theorem does not establish that the framework's gravity action recovers the known energy gap.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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