Gravity Regge Cubic Lattice Limit Regge Second Order Cubic Lattice Limit Error V

The vanishing error

In numerical approaches to gravity, one often approximates continuous spacetime with a discrete lattice of tetrahedra. The Regge action is a way to write down gravity's action, the quantity whose minimization gives the equations of motion, using only the lengths of the lattice's edges. The second-order Regge action adds a correction term. A natural question is whether, as the lattice spacing a goes to zero, this discrete action approaches the continuum action. The theorem reggeSecondOrderCubicLatticeLimit_error_vanishes_along_models answers yes, but only under a precise condition.

The condition is an error envelope. The theorem assumes a family of regular cubic-lattice models, each with a lattice spacing and a certified error bound. The bound states that for every possible vertex potential, the absolute difference between the discrete and continuum actions is at most a constant times the lattice spacing squared. The theorem then proves that if this error envelope, the constant times the spacing squared, tends to zero along a filter, then the actual action difference also tends to zero for each fixed vertex potential. In plain terms, if the certified error shrinks to zero, then the true error also shrinks to zero.

The theorem does not prove that any particular physical model satisfies the error envelope. It is a conditional statement: given a family of models with a vanishing error bound, convergence follows. The construction of a physical model, such as the six-tetrahedra-per-cube triangulation matching a finite-difference Dirichlet action, remains an open target. The theorem also does not give an explicit rate of convergence beyond the order of the envelope, and it does not apply to arbitrary triangulations, only to the regular cubic-lattice models in the framework.

What the theorem does provide is a clean bridge. It shows that the framework's definition of a regular cubic-lattice limit is coherent: the error bound is the right thing to certify. If one can prove the bound for a physical model, the continuum limit follows automatically. This separates the hard analytic work of proving the bound from the general convergence argument, which the machine-checked library of formal theorems now supplies.

/-- 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 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)

What this page does not claim

The theorem does not prove that any physical lattice model satisfies the error envelope. The theorem does not provide an explicit rate of convergence beyond the order of the envelope. The theorem does not apply to arbitrary triangulations, only to regular cubic-lattice models.

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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