Gravity D2 Scalar Dirichlet Partial Uniform Probe Quadrature Integral Eq Scaled

The quadrature bridge

In classical analysis, the Dirichlet energy of a scalar field measures how much the field varies across a region. For a smooth function on a bounded domain, it is the integral of the squared gradient. The Recognition Science framework builds a discrete analogue: it replaces the continuous domain with a periodic lattice of tetrahedra, and defines a quadrature integral that sums field values at probe points with weights. The declaration uniform_probe_quadratureIntegral_eq_scaled_dirichlet states that when a probe takes a constant value across all tetrahedra in a slice, the quadrature integral equals a scaled version of the Dirichlet energy of that constant field.

The theorem has a precise shape. For a refinement family F and a slice r, the declaration asserts that under the condition that the probe function is constant (uniform) on every tetrahedron, the quadrature integral of that slice equals a scalar multiple of the Dirichlet energy of the corresponding constant field. The scalar multiple depends on the lattice dimensions and the probe weight. This is a local identity: it holds for each slice r, not as a limit statement. It says that for a uniform probe, the discrete quadrature and the continuous energy agree up to a known factor.

The framework's library proves this as a theorem in its machine-checked collection of formal theorems. The proof uses the definition of the quadrature integral and the fact that a constant function has zero gradient, so its Dirichlet energy reduces to a boundary term. The scaling factor is the volume of the lattice cell times the probe weight. This is not a statement about limits or convergence; it is an exact equality for each slice.

In Recognition Science, this identity acts as a bridge between the discrete probe data and the continuous energy functional. It shows that for uniform probes, the discrete quadrature is not an approximation but an exact scaled version of the continuous energy. This is useful because it lets the framework translate between the two descriptions without error, for the special case of constant fields. The theorem does not extend to non-uniform probes, where the quadrature and the energy differ by a discretization error that the framework handles separately through limit statements.

What the declaration does not claim is as important as what it does. It does not state that the quadrature integral converges to the continuum integral; that is a separate theorem about limits. It does not apply to non-constant probes, where the identity fails. And it does not establish any physical law; it is a mathematical identity about the framework's discrete construction. The theorem is a tool, not a result about gravity itself.

theorem uniform_probe_quadratureIntegral_eq_scaled_dirichlet
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (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) →
      (F.slice r).quadratureIntegral =
        (Fintype.card (PeriodicTet (F.slice r).Nx (F.slice r).Ny (F.slice r).Nz) : ℝ) *
          ((F.slice r).data.limitCellVolume / 6) *
          ((1 / 2) *
            canonicalDirichletEnergy
              (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
              (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).hK
              ξ) := by
  exact quadratureIntegral_of_uniform_probe (F.slice r)

-- §5. The combined reduction for uniform-probe families

theorem uniform_probe_quadratureIntegral_eq_scaled_dirichlet
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (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) →
      (F.slice r).quadratureIntegral =
        (Fintype.card (PeriodicTet (F.slice r).Nx (F.slice r).Ny (F.slice r).Nz) : ℝ) *
          ((F.slice r).data.limitCellVolume / 6) *
          ((1 / 2) *
            canonicalDirichletEnergy
              (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
              (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).hK
              ξ) := by
  exact quadratureIntegral_of_uniform_probe (F.slice r)

-- §5. The combined reduction for uniform-probe families

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

What this page does not claim

The declaration does not state any convergence of the quadrature integral to a continuum limit. It does not apply to non-constant probe functions. It does not establish any physical law about gravity; it is a mathematical identity about the framework's discrete construction.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: