Gravity Regge Convergence Cubic Shape Optimal

The cubic lattice's shape bound

The declaration cubic_shape_optimal is a theorem in the framework's machine-checked library of formal theorems. It establishes that a certain number, called the cubic shape bound, is positive. That number is defined to be 1. The proof is a direct computation: it unfolds the definition and verifies that 1 is greater than 0. In plain terms, the theorem says that a cube, as a lattice cell, has a well-defined, positive shape quality, and in this framework that quality is set to the optimal value of 1.

The context matters. In numerical relativity, a lattice is a discrete grid used to approximate continuous spacetime. The Regge action is a way of writing gravity on such a lattice, and a key question is whether it approaches the smooth Einstein-Hilbert action as the grid spacing shrinks to zero. For that convergence to hold at a good rate, the lattice cells must not be too distorted. A shape bound is a number that controls how skewed a cell can be. For a cubic lattice, all cells are identical cubes, so the shape bound is exactly 1, the best possible value. The theorem cubic_shape_optimal records that this best value is positive, which is a necessary condition for the convergence statements that use it.

In Recognition Science, the framework models the cost of recognition events, and from that cost it derives a lattice structure. This theorem is a small but load-bearing piece of the framework's gravity program. It is used in the definition of a conditional convergence statement for the full nonlinear regime of general relativity. That statement, rs_regge_convergence, assumes a bounded curvature and a positive shape bound, and then asserts that the Regge action converges to the Einstein-Hilbert action at order a squared. The theorem cubic_shape_optimal supplies the positive shape bound for the cubic case. Without it, the conditional statement would not even be well-formed.

What the theorem does not claim is equally important. It does not prove that the Regge action converges to the Einstein-Hilbert action. That convergence is a separate, much harder statement, and in the nonlinear regime it is conditional on additional assumptions. The theorem also does not claim that cubes are the only optimal shapes, or that the shape bound of 1 is derived from first principles; it is a definitional choice. Finally, the theorem says nothing about the physical correctness of the cubic lattice as a model of spacetime. It is a formal fact about a defined quantity, not an empirical statement.

theorem cubic_shape_optimal : 0 < cubic_shape_bound := by
  unfold cubic_shape_bound; norm_num

theorem cubic_shape_optimal : 0 < cubic_shape_bound := by
  unfold cubic_shape_bound; norm_num

/-- For a cubic lattice (the RS case), the shape bound is 1 (all cubes
    have the same shape, optimal aspect ratio). -/
def cubic_shape_bound : ℝ := 1

/-- The RS-specific special quadratic convergence statement: on the cubic
    lattice Z^3, with metric g having bounded curvature, the RS Regge action
    (= J-cost sum) is assumed to converge to the EH action at `O(a^2)`.
    This is stronger than the general CMS Theorem 5.1 measure bound. -/
def rs_regge_convergence (K : ℝ) (hK : 0 < K) : Prop :=
  let cond : CMSConditions := ⟨K, hK, cubic_shape_bound, cubic_shape_optimal, 1, one_pos⟩
  nonlinear_convergence_with_conditions cond

What this page does not claim

The theorem does not prove that the Regge action converges to the Einstein-Hilbert action in the nonlinear regime. The theorem does not claim that the shape bound of 1 is derived from physical principles rather than chosen by definition. The theorem says nothing about whether the cubic lattice is the correct model of physical spacetime.

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

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