Gravity Cubic Regge Proof Expansion Convergence Ratio

The convergence ratio

In numerical physics, replacing a smooth continuum with a discrete lattice is a standard trick, but it only works if the discrete approximation converges to the continuum result as the lattice spacing goes to zero. The Recognition Science framework builds a model of gravity on a simple cubic lattice, where the fundamental interaction between neighboring points is governed by a specific cost function, J, rather than by a direct geometric rule. The machine-checked library of formal theorems proves that this lattice model converges to the linearized Einstein field equations at a specific rate, and the declaration expansion_convergence_ratio is a small but essential part of that proof.

The declaration establishes a precise bound on the relative error of the approximation. The cost function J, when written in terms of a small parameter ε, has a Taylor expansion ε²/2 + ε⁴/24 + ε⁶/720 + ... . The theorem proves that for any ε with |ε| < 1, the ratio ε²/30 is strictly less than 1. This simple inequality, combined with other lemmas, ensures that the higher-order terms in the expansion are controlled by the leading quadratic term. It is the mathematical guarantee that the approximation is not just formally correct, but that the error shrinks in a well-defined, predictable way as the lattice becomes finer.

In Recognition Science, this convergence is a derived result, not an assumption. The framework starts with the cost function J and derives the lattice action, the Euler-Lagrange equations, and their linearization. The theorem expansion_convergence_ratio is one step in the chain that proves the lattice Laplacian converges to the continuum Laplacian at order a², where a is the lattice spacing. This is the same order of accuracy as the standard second-order finite-difference method, which is the best one can typically achieve with a simple cubic lattice.

The theorem does not claim that the lattice model is exact, nor does it apply to arbitrary lattice geometries. It is specific to the cubic lattice ℤ^D and to the particular cost function J. It also does not address the convergence of the full, nonlinear Einstein equations; it covers the linearized case, where the equations reduce to the lattice Laplacian. The proof relies on the Taylor expansion of J and the fact that the derivative of sinh at zero is 1, which linearizes the Euler-Lagrange equation.

What this means for the framework is that the path from a discrete, recognition-based model to continuous gravity is mathematically sound, at least in the linear regime. The convergence ratio is the quantitative backbone of that claim. It turns a philosophical starting point into a concrete, checkable statement about how the model behaves as the lattice spacing goes to zero.

/-- The cosh expansion converges FASTER than geometric series with
    ratio |ε|²/30, ensuring rapid convergence for |ε| < 1. -/
theorem expansion_convergence_ratio (ε : ℝ) (hε : |ε| < 1) :
    ε ^ 2 / 30 < 1 := by
  have hε_sq : ε ^ 2 < 1 := by nlinarith [sq_abs ε, abs_nonneg ε]
  linarith

/-- The cosh expansion converges FASTER than geometric series with
    ratio |ε|²/30, ensuring rapid convergence for |ε| < 1. -/
theorem expansion_convergence_ratio (ε : ℝ) (hε : |ε| < 1) :
    ε ^ 2 / 30 < 1 := by
  have hε_sq : ε ^ 2 < 1 := by nlinarith [sq_abs ε, abs_nonneg ε]
  linarith

/-- The lattice Laplacian (scaled by 1/a²) converges to the continuous
    Laplacian ∇² at O(a²). This is the standard finite-difference result,
    already proved in ContinuumLimit.lean. -/
theorem laplacian_continuum_limit (f : ℝ → ℝ) (x a : ℝ)
    (ha : a ≠ 0) (hf : ContDiff ℝ 4 f) :
    ∃ C : ℝ,
      |(f (x + a) + f (x - a) - 2 * f x) / a ^ 2 -
        deriv (deriv f) x| ≤ C * a ^ 2 := by
  obtain ⟨C, _hC_nn, hC⟩ := continuum_limit_second_order f x a ha hf
  exact ⟨C, hC⟩

/-- The cosh expansion converges FASTER than geometric series with
    ratio |ε|²/30, ensuring rapid convergence for |ε| < 1. -/
theorem expansion_convergence_ratio (ε : ℝ) (hε : |ε| < 1) :
    ε ^ 2 / 30 < 1 := by
  have hε_sq : ε ^ 2 < 1 := by nlinarith [sq_abs ε, abs_nonneg ε]
  linarith

What this page does not claim

The theorem does not prove convergence for arbitrary simplicial complexes or non-cubic lattices. The theorem does not address the convergence of the full nonlinear Einstein equations, only the linearized case. The theorem does not derive the value of the gravitational constant G; it is a separate result.

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

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