Gravity D2 Scalar Dirichlet Partial Uniform Probe Scalar Dirichlet Limit Iff Qua

The limit equivalence

In the Recognition Science framework, a scalar Dirichlet energy limit is a way of assigning a single number to a sequence of finer and finer approximations of a region. The question is when this limit is well-defined. The theorem uniform_probe_scalar_dirichlet_limit_iff_quadrature_tendsto answers this with a precise equivalence: the limit exists if and only if the sequence of quadrature integrals converges to that same number. In plain terms, the approximating sums must settle down to a definite value for the limit to be meaningful.

The condition for this equivalence is that the probe functions used in the approximation are uniform. This means that at every refinement step, the same probe function is used across all the tetrahedral cells of the discretization. The theorem then states that under this uniformity, the existence of the scalar Dirichlet energy limit is exactly equivalent to the convergence of the quadrature integrals. The proof is short and direct, following from a more general lemma about limits and filters.

This result is a formal theorem in the framework's machine-checked library of formal theorems. It is not a physical claim about gravity or about any specific force law. It is a statement about the mathematical consistency of a particular discretization scheme. The theorem does not say that the limit always exists; it only says that the limit exists exactly when the quadrature sums converge. It also does not claim that the quadrature sums always converge, nor does it identify what the limit value must be.

What this means for the framework is that the scalar Dirichlet energy limit is well-behaved under a natural condition. If one can show that the quadrature integrals converge, then the limit is guaranteed to exist. This is a useful stepping stone for further results about the discretized gravity models in the framework, but it is not itself a statement about the physical content of those models.

theorem uniform_probe_scalar_dirichlet_limit_iff_quadrature_tendsto
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (refinementFilter : Filter ρ) (continuumIntegral : ℝ)
    (huniform : ∀ r : ρ,
      letI : NeZero (F.slice r).Nx := (F.slice r).instNx
      letI : NeZero (F.slice r).Ny := (F.slice r).instNy
      letI : NeZero (F.slice r).Nz := (F.slice r).instNz
      ∃ ξ : VertexPotential
          (canonicalEncodedPeriodicFreudenthalTorus (F.slice r).Nx (F.slice r).Ny (F.slice r).Nz
            (F.slice r).hx (F.slice r).hy (F.slice r).hz).K,
        ∀ τ : Fin (Fintype.card (PeriodicTet (F.slice r).Nx (F.slice r).Ny (F.slice r).Nz)),
          (F.slice r).data.tetProbe (tetFinEquiv (F.slice r).Nx (F.slice r).Ny (F.slice r).Nz τ) = ξ) :
    Nonempty (ScalarDirichletEnergyLimit F refinementFilter continuumIntegral) ↔
    Filter.Tendsto (fun r => (F.slice r).quadratureIntegral) refinementFilter (nhds continuumIntegral) := by
  -- For uniform-probe families, quadratureIntegral_of_uniform_probe (via
  -- uniform_probe_quadratureIntegral_eq_scaled_dirichlet) rewrites the proxy
  -- as the scaled Dirichlet energy. The equivalence then follows from
  -- scalar_dirichlet_limit_nonempty_iff_tendsto.
  exact scalar_dirichlet_limit_nonempty_iff_tendsto F refinementFilter continuumIntegral
    (fun r => (F.slice r).quadratureIntegral) (fun r => rfl)

theorem uniform_probe_scalar_dirichlet_limit_iff_quadrature_tendsto
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (refinementFilter : Filter ρ) (continuumIntegral : ℝ)
    (huniform : ∀ r : ρ,
      letI : NeZero (F.slice r).Nx := (F.slice r).instNx
      letI : NeZero (F.slice r).Ny := (F.slice r).instNy
      letI : NeZero (F.slice r).Nz := (F.slice r).instNz
      ∃ ξ : VertexPotential
          (canonicalEncodedPeriodicFreudenthalTorus (F.slice r).Nx (F.slice r).Ny (F.slice r).Nz
            (F.slice r).hx (F.slice r).hy (F.slice r).hz).K,
        ∀ τ : Fin (Fintype.card (PeriodicTet (F.slice r).Nx (F.slice r).Ny (F.slice r).Nz)),
          (F.slice r).data.tetProbe (tetFinEquiv (F.slice r).Nx (F.slice r).Ny (F.slice r).Nz τ) = ξ) :
    Nonempty (ScalarDirichletEnergyLimit F refinementFilter continuumIntegral) ↔
    Filter.Tendsto (fun r => (F.slice r).quadratureIntegral) refinementFilter (nhds continuumIntegral) := by
  -- For uniform-probe families, quadratureIntegral_of_uniform_probe (via
  -- uniform_probe_quadratureIntegral_eq_scaled_dirichlet) rewrites the proxy
  -- as the scaled Dirichlet energy. The equivalence then follows from
  -- scalar_dirichlet_limit_nonempty_iff_tendsto.
  exact scalar_dirichlet_limit_nonempty_iff_tendsto F refinementFilter continuumIntegral
    (fun r => (F.slice r).quadratureIntegral) (fun r => rfl)

What this page does not claim

The theorem does not prove that the scalar Dirichlet energy limit always exists. The theorem does not assert that the quadrature sums converge for any particular refinement family. The theorem makes no claim about the physical value of the limit or about any specific gravitational interaction.

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/D2ScalarDirichletPartial.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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