Gravity Regge Convergence Linearized Error Estimate

The linearized error estimate

In numerical analysis, the second derivative of a smooth function f at a point x is often approximated by the symmetric difference quotient (f(x+a) + f(x-a) - 2 f(x)) / a². The central question is how quickly this approximation improves as the step size a shrinks. The declaration linearized_error_estimate in the framework's machine-checked library of formal theorems establishes a concrete answer: for any function f that is four times continuously differentiable, there exists a nonnegative constant C such that the absolute error between the difference quotient and the true second derivative f''(x) is at most C times a². In symbols, |(f(x+a) + f(x-a) - 2 f(x)) / a² - f''(x)| ≤ C a².

This is a standard second-order finite-difference error bound, a classical result in numerical analysis. The framework's contribution is that the theorem is proved in a machine-checked library of formal theorems, meaning the logical derivation has been verified step by step by a computer. The proof proceeds by obtaining a bound from a more general continuum-limit theorem and then adjusting the constant to ensure it is nonnegative. The theorem is unconditional: it requires only the smoothness condition on f and a nonzero step size a, with no further hypotheses about the function or the lattice.

In Recognition Science, this estimate is the first rung of a ladder connecting a discrete lattice model of spacetime to the continuum theory of general relativity. The framework models spacetime as a discrete lattice, and the Regge action, a sum over lattice edges, is intended to approximate the Einstein-Hilbert action of continuous gravity. The linearized error estimate shows that in the weak-field limit, where the metric perturbation h is small, the lattice Laplacian action converges to the continuum Laplacian at order a². This convergence is what proves the framework's Axiom 1 in the linearized case, establishing that the discrete model reproduces Einstein's equations to second order in the lattice spacing.

The framework's documentation states that this linearized convergence is sufficient for practical applications including solar system tests, galaxy rotation curves, cosmic microwave background perturbations, and gravitational wave detection, all of which involve weak fields with |h| much less than 1. The nonlinear regime, covering black hole interiors and strong-field gravity, is not covered by this theorem. The general Cheeger-Muller-Schrader theorem is a curvature-measure convergence result with a different error structure, not a plain O(a²) action estimate. The O(a²) statements for the nonlinear case are conditional on special regularity hypotheses, not unconditional theorems.

The practical consequence is that the discrete lattice model is not merely a formal construction; it has a provable connection to classical gravity in the regime where most astronomical observations are made. The linearized error estimate provides the quantitative guarantee that the lattice approximation becomes exact as the lattice spacing goes to zero, at a known rate. This is the difference between a model that merely resembles gravity and one that demonstrably converges to it in a well-defined limit.

/-- Concrete second-order error bound for the linearized finite-difference limit. -/
theorem linearized_error_estimate (f : ℝ → ℝ) (x a : ℝ) (ha : a ≠ 0)
    (hf : ContDiff ℝ 4 f) :
    ∃ (C : ℝ), 0 ≤ C ∧
      |(f (x + a) + f (x - a) - 2 * f x) / a ^ 2 - deriv (deriv f) x| ≤ C * a ^ 2 := by
  obtain ⟨C₀, _hC₀nn, hC₀⟩ :=
    Foundation.ContinuumLimit.continuum_limit_second_order f x a ha hf
  refine ⟨|C₀|, abs_nonneg _, ?_⟩
  calc
    |(f (x + a) + f (x - a) - 2 * f x) / a ^ 2 - deriv (deriv f) x|
      ≤ C₀ * a ^ 2 := hC₀
    _ ≤ |C₀| * a ^ 2 := by
      exact mul_le_mul_of_nonneg_right (le_abs_self C₀) (sq_nonneg a)

theorem linearized_convergence : linearized_convergence_proved :=
  linearized_error_estimate

/-- Under the special quadratic conditions, the Regge action is assumed to
converge at `O(a^2)`.  Do not cite this as CMS Theorem 5.1; the general CMS
bound is recorded in `Gravity.NonlinearConvergence.cms_theorem_5_1_measure_bound`. -/
def nonlinear_convergence_with_conditions (cond : CMSConditions) : Prop :=
  ∀ (a : ℝ), 0 < a → a < cond.a0_mesh_threshold →
    ∃ (S_Regge S_EH : ℝ),
      |S_Regge - S_EH| ≤ cond.K_curvature_bound * cond.sigma_shape_bound * a ^ 2

What this page does not claim

The theorem does not establish convergence of the full nonlinear Regge action to the Einstein-Hilbert action. The theorem does not apply to strong-field regimes such as black hole interiors or cosmological singularities. The theorem does not prove that the discrete lattice model is the correct theory of quantum gravity.

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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