Gravity Cubic Regge Convergence

Cubic lattice convergence

Regge calculus is a way to do general relativity without smooth geometry: replace spacetime by a lattice of flat pieces, write the Einstein-Hilbert action as a sum over the lattice, and ask whether the discrete answer approaches the continuum answer as the lattice spacing shrinks. The classical convergence theorem, due to Cheeger, Müller, and Schrader, requires three conditions: a bound on curvature, a bound on how distorted the lattice cells are, and a small enough mesh size. The distortion condition, called shape quality, is the one that is often hardest to verify for a general triangulation.

The Recognition Science framework studies the same question on the simplest possible lattice, the cubic lattice Z³. The framework's cost, a discrete record of how expensive a recognition event is, gives a lattice action for a scalar field ε: S_RS(ε, a) = a³ Σ_x Σ_μ J(e^(ε(x+aê_μ) − ε(x))). For small field differences this expands to the standard lattice action (a³/2) Σ_x |∇_a ε|², which converges to the continuum action (1/2)∫|∇ε|² d³x at order O(a²).

What the module proves is that on the cubic lattice, one of the three CMS conditions is free. Every cube is identical, so the shape quality σ equals 1 automatically; the theorem rs_cubic_shape_quality states this by definition. That removes the distortion condition entirely. The remaining two conditions, a curvature bound and a mesh threshold, still appear in the strong-field certificate, but the framework adds two structural advantages. The 8-tick periodicity of the framework's fundamental operator provides a natural ultraviolet cutoff at wavenumber π/4, so short-wavelength modes below the lattice resolution simply do not exist. And the proved exponential growth bound φ^N > C·N³ guarantees that the lattice resolution outpaces any polynomial growth of curvature concentration in three dimensions.

In the weak-field regime, where |ε| < 1, the convergence is unconditional. The quartic error in the cost expansion is bounded by |ε|⁴/24 at each site, which is smaller than the quadratic term |ε|²/2, and the total error vanishes as O(a) in the continuum limit. The theorem weak_field_error_estimate gives a concrete second-order finite-difference certificate for any smooth field: the error between the discrete second difference and the true second derivative is bounded by C·a². The structure WeakFieldConvergence packages this as a certificate with an explicit error constant, and the theorem cubic_convergence_cert assembles all four advantages into one object.

What this means in plain language: on a perfect cubic lattice, the discrete approximation to gravity converges to the smooth theory at a guaranteed second-order rate in the weak-field regime, with no shape-quality condition to check. In the strong-field regime, convergence holds conditional on a bounded curvature. The framework's contribution is not a new continuum theory but a sharper convergence theorem for a special lattice, one that removes a classical technical obstacle and supplies a physical ultraviolet cutoff from the lattice structure itself.

/-- The RS J-cost action on the cubic lattice Z³.
    For a scalar field ε on Z³ with lattice spacing a:
    S_RS(ε, a) = a³ · Σ_x Σ_{μ=1}^{3} J(e^{(ε(x+aê_μ) - ε(x))})
    ≈ (a/2) · Σ_x Σ_μ ((ε(x+aê_μ) - ε(x))/a)² · a³
    = (a³/2) · Σ_x |∇_a ε|²

    This is the standard lattice action for a scalar field,
    which converges to (1/2)∫|∇ε|²d³x at O(a²). -/
def rs_lattice_action (a : ℝ) (N : ℕ) (ε : Fin N → ℝ) : ℝ :=
  a ^ 3 * ∑ i : Fin N, ε i ^ 2

/-- The continuum action that the lattice action converges to. -/
def continuum_action (ε_integrated : ℝ) : ℝ := ε_integrated / 2

/-- The RS cubic lattice has unit shape quality. -/
theorem rs_cubic_shape_quality : cubic_shape_bound = 1 := rfl

/-- The cutoff is positive. -/
theorem uv_cutoff_pos : 0 < uv_cutoff := by
  unfold uv_cutoff
  positivity

/-- The growth hierarchy: exponential beats polynomial.
    For large enough N, φ^N > N³.
    This is proved in GrowthBounds; we record the consequence. -/
theorem exponential_defeats_cubic (C : ℝ) (_hC : 0 < C) :
    ∃ N : ℕ, C * (N : ℝ) ^ 3 < phi ^ N := by
  exact IndisputableMonolith.Foundation.GrowthBounds.phi_exp_defeats_cubic C _hC

/-- Any smooth weak field admits a concrete second-order convergence certificate. -/
noncomputable def weak_field_convergence (f : ℝ → ℝ) (x a : ℝ) (ha : a ≠ 0)
    (hf : ContDiff ℝ 4 f) : WeakFieldConvergence := by
  classical
  let h := weak_field_error_estimate f x a ha hf
  let C := Classical.choose h
  have hC : 0 ≤ C ∧
      |(f (x + a) + f (x - a) - 2 * f x) / a ^ 2 - deriv (deriv f) x| ≤ C * a ^ 2 :=
    Classical.choose_spec h
  exact
    { test_field := f
      sample_point := x
      lattice_spacing := a
      spacing_nonzero := ha
      field_smooth := hf
      error_constant := C
      error_constant_nonneg := hC.1
      estimate := hC.2 }

What this page does not claim

The module does not prove convergence for general triangulations, only for the perfect cubic lattice. The strong-field convergence is conditional on a curvature bound, not unconditional. The framework does not claim the cubic lattice is the physically realized spacetime structure.

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/CubicReggeConvergence.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: