Gravity Regge Cubic Lattice Limit Regge Action Second Order Cubic Lattice Limit

The cubic-lattice limit

In numerical relativity, the Regge action is a standard way to approximate the smooth spacetime of general relativity by a discrete network of flat tetrahedral pieces. The second-order Regge action is its refinement that includes curvature-squared corrections. The question of whether such a discrete action has a well-defined continuum limit, as the lattice spacing shrinks to zero, is a central mathematical concern. A new theorem in the Recognition Science framework's machine-checked library of formal theorems addresses this for the special case of a regular cubic lattice.

The declaration reggeActionSecondOrder_cubic_lattice_limit establishes a precise conditional statement. It says: if one is given a regular cubic lattice model, with a positive lattice spacing, a chosen continuum action, and a certified error constant, such that the absolute difference between the discrete second-order Regge action and the continuum action is bounded by that constant times the square of the lattice spacing, then the continuum limit exists. The theorem itself is a formal restatement of this condition. It does not, by itself, prove that any particular physical lattice model satisfies that condition; it proves that the condition is sufficient.

The framework's library also contains a companion theorem showing that a family of such models converges pointwise to the continuum action whenever the certified error envelope tends to zero along a refinement parameter. This is the analytic core that makes the limit statement useful. The library further provides a one-dimensional finite-difference estimate, which bounds the error of the standard second derivative approximation by a constant times the square of the step size, for smooth functions. This is the analytic ingredient that a concrete cubic-lattice model would use to satisfy the main theorem's hypothesis.

What the declaration does not claim is equally important. It does not assert that the canonical second-order Regge action on a physical six-tetrahedra-per-cube triangulation is the finite-difference Dirichlet action; that identification remains a stated target, not a proved fact. The theorem also does not provide the specific error constant or the specific continuum action for any particular physical model; those are supplied by the model structure, not by the theorem. The theorem is a conditional bridge: given a certified estimate, it guarantees the limit. The physical instance that would cross that bridge is still open.

/-- Phase-G cubic-lattice `O(a^2)` continuum-limit theorem for the canonical
second-order Regge action. -/
theorem reggeActionSecondOrder_cubic_lattice_limit
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_limit : ReggeCubicLatticeLimitInput K hK) :
    ReggeSecondOrderCubicLatticeLimit K hK h_limit.model :=
  h_limit.limit_estimate

/-- A family of cubic-lattice comparison models converges pointwise whenever
its certified `O(a^2)` error envelope tends to zero along the refinement
parameter. -/
theorem reggeSecondOrderCubicLatticeLimit_error_vanishes_along_models
    {α : Type*} {l : Filter α}
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (M : α → RegularCubicLatticeModel K hK)
    (hLimit : ∀ t : α, ReggeSecondOrderCubicLatticeLimit K hK (M t))
    (hEnvelope :
      Filter.Tendsto
        (fun t : α => (M t).errorConstant * (M t).latticeSpacing ^ (2 : ℕ))
        l (nhds 0))
    (ξ : VertexPotential K) :
    Filter.Tendsto
      (fun t : α =>
        |reggeActionSecondOrder K hK (canonicalReggeHessian K hK) ξ -
          (M t).continuumAction ξ|)
      l (nhds 0) := by
  apply squeeze_zero
  · intro t
    exact abs_nonneg _
  · intro t
    exact hLimit t ξ
  · exact hEnvelope

/-- The existing one-dimensional finite-difference theorem is the analytic
ingredient used in the cubic-lattice model. -/
theorem finite_difference_second_order_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 :=
  CubicReggeConvergence.weak_field_error_estimate f x a ha hf

What this page does not claim

The theorem does not prove that any particular physical lattice model satisfies the error-bound condition. The theorem does not assert the six-tetrahedra-per-cube action equals the finite-difference Dirichlet action. The theorem does not supply the specific error constant or continuum action for any physical model.

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/ReggeCubicLatticeLimit.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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