Gravity Regge Cubic Lattice Limit

The cubic lattice limit

In numerical relativity, Regge calculus replaces smooth spacetime with a triangulated network, assigning lengths to edges and computing curvature from the angles around them. The Regge action is the discrete stand-in for the Einstein-Hilbert action of general relativity. A central question is whether this discrete action approaches the continuum one as the lattice becomes finer. The framework's machine-checked library of formal theorems addresses this for a regular cubic lattice, proving that the second-order Regge action differs from a continuum action by an error bounded by a constant times the square of the lattice spacing.

The module ledger, a discrete record of events, defines a RegularCubicLatticeModel: a structure carrying a lattice spacing, a continuum action, and an error constant. The key definition, ReggeSecondOrderCubicLatticeLimit, states that for every vertex potential, the absolute difference between the discrete Regge action and the continuum action is at most the error constant times the spacing squared. This is the formal O(a²) statement, where a is the lattice spacing.

The main theorem, reggeActionSecondOrder_cubic_lattice_limit, takes such a model as input and directly yields the limit statement. A second theorem, reggeSecondOrderCubicLatticeLimit_error_vanishes_along_models, shows that if a family of models has error envelopes tending to zero, then the discrete actions converge pointwise to the continuum actions. The module also includes a sanity-check instance where the continuum action is chosen to be the discrete action itself, giving zero error, and a physical six-tet model that targets the finite-difference Dirichlet action.

The analytic core is a one-dimensional finite-difference estimate: for a four-times continuously differentiable function, the second difference quotient approximates the second derivative with an error of order a². This classical result, proved in the library, supplies the ingredient for the cubic-lattice model. The physical instance, identifying the six-tetrahedra-per-cube Regge action with the finite-difference Dirichlet action, remains to be completed.

What this establishes is a precise convergence guarantee for a special case, not a general theorem. The general CMS theorem gives only weak curvature-measure convergence; the clean O(a²) estimate belongs to this regular weak-field cubic-lattice case. The framework's contribution is to make this estimate machine-checked, so the bound is not a heuristic but a proved fact.

/-- The second-order Regge action has an `O(a^2)` cubic-lattice continuum
limit in the supplied regular lattice model. -/
def ReggeSecondOrderCubicLatticeLimit
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (M : RegularCubicLatticeModel K hK) : Prop :=
  ∀ ξ : VertexPotential K,
    |reggeActionSecondOrder K hK (canonicalReggeHessian K hK) ξ -
      M.continuumAction ξ| ≤ M.errorConstant * M.latticeSpacing ^ (2 : ℕ)

/-- 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 physical six-tet cubic Dirichlet model is not yet proved; it remains a target structure with unproved fields. The O(a²) estimate does not apply to general triangulations, only to the regular weak-field cubic-lattice case. The module does not establish the continuum limit of the full Regge action in arbitrary coordinates or with arbitrary boundary conditions.

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)

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