Gravity D2 Scalar Dirichlet Partial Scalar Dirichlet Limit Iff Quadrature Tendst

The limit condition

In the Recognition Science framework, the question of whether a physical quantity has a well-defined limiting value as a computational grid is refined comes down to a simple test. The theorem scalar_dirichlet_limit_iff_quadrature_tendsto states that a scalar Dirichlet energy limit exists if and only if the sequence of quadrature integrals, one for each refinement step, converges to the expected continuum value. This is a formal equivalence, proved in the machine-checked library of formal theorems, not a heuristic approximation.

The statement is precise. Given a family of periodic tetrahedral volume quadrature refinements, the existence of a limit object is logically equivalent to the condition that the function mapping each refinement to its quadrature integral tends to the continuum integral. In plain language: if the computed sums approach a single number as the grid gets finer, then the energy limit exists, and conversely. The proof is short because it follows directly from a more general theorem about nonempty limit sets and filter convergence.

This result does not claim that the quadrature integrals always converge. It does not assert that any particular physical system satisfies the convergence condition. It only establishes the equivalence: the limit exists if and only if the sums tend to the right value. The theorem is a bridge between a topological statement about existence and a concrete numerical condition that can be checked in practice.

The practical consequence is that checking convergence of the quadrature sums is sufficient to guarantee the existence of the energy limit. This turns a potentially abstract existence question into a computational one. It also means that if the sums fail to converge, the limit does not exist, which is a useful diagnostic for whether a refinement scheme is well-behaved.

In Recognition Science, this fits into a broader program where physical structure is derived from a ledger of recognition events. The theorem itself is a technical tool within that program, not a claim about the physical world. It is a statement about the relationship between two mathematical objects: a limit structure and a sequence of sums.

theorem scalar_dirichlet_limit_iff_quadrature_tendsto
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (refinementFilter : Filter ρ) (continuumIntegral : ℝ) :
    Nonempty (ScalarDirichletEnergyLimit F refinementFilter continuumIntegral) ↔
    Filter.Tendsto (fun r => (F.slice r).quadratureIntegral) refinementFilter (nhds continuumIntegral) := by
  exact scalar_dirichlet_limit_nonempty_iff_tendsto F refinementFilter continuumIntegral
    (fun r => (F.slice r).quadratureIntegral) (fun r => rfl)

-- §4. The uniform-probe identification

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 assert that quadrature integrals always converge for any refinement family. The theorem does not claim that any particular physical system satisfies the convergence condition. The theorem does not establish the value of the continuum integral; it only relates its existence to convergence.

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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