Gravity Nonlinear Convergence Quadratic Error Vanishes

The vanishing error

In numerical relativity, the Regge calculus approximates smooth spacetime by a lattice of flat simplices, much like approximating a curved surface by a mesh of flat triangles. The approximation depends on the mesh size, often written as a. A central question is whether the error in the computed action or curvature vanishes as the mesh is refined. The declaration quadratic_error_vanishes addresses a specific piece of this question.

The declaration is a theorem in the framework's machine-checked library of formal theorems. It states that for any positive constant C, the expression C times a squared tends to zero as a tends to zero. In symbols, it proves that the limit of C * a^2 as a approaches 0 is 0. This is a simple mathematical fact: a positive constant multiplied by a quantity that shrinks to zero also shrinks to zero. The proof uses the continuity of the function f(a) = C * a^2 at zero.

The context matters. In the framework's files on nonlinear convergence, this theorem appears alongside hypotheses about how Regge calculus converges to Einstein-Hilbert geometry. The general theorem from Cheeger, Müller, and Schrader (1984) gives a bound with a square-root term in the mesh size, not a squared term. The stronger quadratic bound, O(a^2), is a separate special-purpose hypothesis used in some weak-field or numerical settings. The theorem quadratic_error_vanishes is a formal statement about the limit of that quadratic error term, not a proof that the stronger hypothesis holds.

In Recognition Science, the declaration is part of a larger structure that links the discrete lattice of the framework to the full nonlinear Einstein field equations. The structure assumes convergence axioms, including the quadratic one, and then derives the field equations conditionally. The theorem quadratic_error_vanishes is a small but necessary piece: it shows that if the quadratic error bound holds, then the error indeed disappears in the continuum limit. It is a building block, not the whole bridge.

What the declaration does not claim is equally important. It does not prove that Regge calculus converges to general relativity. That convergence is an input, axiomatized from the literature, not a result derived within the framework. The theorem also does not establish the rate of convergence; it only shows that a specific quadratic expression vanishes. The general CMS bound, with its square-root term, is a different statement. Finally, the declaration says nothing about the physical validity of the framework's assumptions; it is a formal result about a limit.

/-- Second-order convergence implies the special quadratic error vanishes as
`a -> 0`. -/
theorem quadratic_error_vanishes (C : ℝ) (_hC : 0 < C) :
    Filter.Tendsto (fun a => C * a ^ 2) (nhds 0) (nhds 0) := by
  have h : Continuous (fun a : ℝ => C * a ^ 2) := by continuity
  have := h.tendsto (0 : ℝ)
  simp at this
  exact this

/-- **CMS Theorem 5.1 shape, scalar real abstraction.**

For a smooth Riemannian manifold `M`, a sufficiently fine `Θ`-fat
triangulation with mesh `η`, and a submanifold `U` with smooth boundary, CMS
prove a curvature-measure estimate of the form

`|R_i(U) - R_{i,η}(U)| ≤ c · (Vol(U) · sqrt η + Vol(B_{sqrt η}(∂U)))`.

The real variables here are the scalar placeholders for those geometric
quantities:

* `RiU`: smooth Lipschitz-Killing curvature measure on `U`;
* `RiEtaU`: piecewise-flat / Regge curvature measure on `U`;
* `VolU`: volume of `U`;
* `boundaryTubeVol`: volume of the `sqrt η`-tubular neighborhood of `∂U`;
* `η`: mesh size;
* `c`: the CMS constant depending on curvature bounds and fatness.

This is deliberately **not** an `O(η^2)` statement. -/
def cms_theorem_5_1_measure_bound : Prop :=
  ∀ (RiU RiEtaU VolU boundaryTubeVol η c : ℝ),
    0 ≤ VolU → 0 ≤ boundaryTubeVol → 0 < η → η < 1 → 0 < c →
      |RiU - RiEtaU| ≤ c * (VolU * Real.sqrt η + boundaryTubeVol)

/-- **Special stronger hypothesis (not CMS Theorem 5.1 in general).**

Some weak-field cubic-lattice or numerical Regge settings can carry
second-order truncation/convergence estimates.  That is a separate input from
the general CMS curvature-measure theorem above.

This is the old `regge_to_eh_convergence_axiom` statement, retained under a
more honest name. -/
def special_quadratic_regge_to_eh_convergence_hypothesis : Prop :=
  ∀ (S_EH : ℝ) (a : ℝ), 0 < a → a < 1 →
    ∃ (S_Regge : ℝ) (C : ℝ), 0 < C ∧
      |S_Regge - S_EH| ≤ C * a ^ 2

/-- In the RS framework, the Regge action convergence gives:
    S_Regge(J-cost lattice, a) -> (1/2*kappa_RS) * integral R sqrt(g)

    Combined with:
    - J-cost minimization implies delta S_Regge = 0 (variational dynamics)
    - delta S_EH = 0 implies EFE (Hilbert variation)
    - kappa_RS = 8*phi^5 (derived coupling)

    This gives the FULL (nonlinear) Einstein field equations
    from the RS discrete ledger, conditional on the convergence axiom. -/
structure RSReggeConvergence where
  /-- General CMS Theorem 5.1-style curvature-measure convergence. -/
  cms_measure_convergence : cms_theorem_5_1_measure_bound
  /-- Stronger special-purpose action convergence, if a module needs `O(a^2)`. -/
  action_convergence : regge_to_eh_convergence_axiom
  ricci_convergence : regge_ricci_convergence_axiom
  kappa_derived : rs_kappa = 8 * phi ^ 5
  kappa_positive : 0 < rs_kappa

What this page does not claim

The declaration does not prove that Regge calculus converges to general relativity. The declaration does not establish the rate of convergence for the general CMS bound. The declaration does not assert the physical validity of the framework's convergence assumptions.

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/NonlinearConvergence.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.

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