Gravity D2 Scalar Dirichlet Partial Scalar Dirichlet Limit Nonempty Iff Tendsto

The limit criterion

The Dirichlet energy measures how much a field, like a gravitational potential, bends or varies across space. On a discrete grid, you approximate it by summing squared differences between neighboring points. The question is whether these approximations settle down to a single number as the grid gets finer.

In the Recognition Science framework, this settling-down is captured by a formal object called a scalar Dirichlet energy limit, a record that names the limiting value and witnesses that the approximations approach it. The theorem, proved in the framework's machine-checked library of formal theorems, states a simple equivalence: such a limit exists if and only if the sequence of approximate energy values converges in the usual sense of calculus. The proof is direct, building the limit from the convergent sequence and vice versa. A second theorem restates this in terms of the quadrature integral, the specific discrete sum used in the framework's gravity models.

This is a structural result, not a physical one. It does not say that any particular approximation scheme converges, nor that the limit equals the classical continuum Dirichlet energy. It only says that the formal existence of the limit is exactly the same condition as ordinary convergence of the numbers. The theorem holds for any refinement family and any filter, so it applies broadly wherever the framework defines such approximations.

In Recognition Science, this criterion is a bridge. It lets the framework's discrete constructions inherit the language of limits from standard analysis, without assuming those limits exist. The theorem does not prove convergence for any specific physical case; it proves that the two notions, formal limit and numerical convergence, are interchangeable. That interchangeability is what makes the framework's discrete gravity models amenable to ordinary mathematical scrutiny.

theorem scalar_dirichlet_limit_nonempty_iff_tendsto
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (refinementFilter : Filter ρ) (continuumIntegral : ℝ)
    (g : ρ → ℝ)
    (hg : ∀ r : ρ, (F.slice r).quadratureIntegral = g r) :
    Nonempty (ScalarDirichletEnergyLimit F refinementFilter continuumIntegral) ↔
    Filter.Tendsto g refinementFilter (nhds continuumIntegral) := by
  constructor
  · intro h
    obtain ⟨H⟩ := h
    have heq : g = H.scalarEnergy := by
      funext r
      exact (hg r).symm.trans (H.proxy_eq r)
    rw [heq]
    exact H.tendsto
  · intro h
    refine ⟨?_⟩
    exact { scalarEnergy := g, proxy_eq := hg, tendsto := h }

-- §2. The constructive direction

theorem scalar_dirichlet_limit_nonempty_iff_tendsto
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (refinementFilter : Filter ρ) (continuumIntegral : ℝ)
    (g : ρ → ℝ)
    (hg : ∀ r : ρ, (F.slice r).quadratureIntegral = g r) :
    Nonempty (ScalarDirichletEnergyLimit F refinementFilter continuumIntegral) ↔
    Filter.Tendsto g refinementFilter (nhds continuumIntegral) := by
  constructor
  · intro h
    obtain ⟨H⟩ := h
    have heq : g = H.scalarEnergy := by
      funext r
      exact (hg r).symm.trans (H.proxy_eq r)
    rw [heq]
    exact H.tendsto
  · intro h
    refine ⟨?_⟩
    exact { scalarEnergy := g, proxy_eq := hg, tendsto := h }

-- §2. The constructive direction

What this page does not claim

The theorem does not prove that any particular approximation scheme converges. The theorem does not state that the limit equals the classical continuum Dirichlet energy. The theorem does not establish any physical law of gravity.

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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