Encyclopedia Gravity Gravity D2 Quadrature Instances Damped Flat Product Filter Data Satisfies Master

ARTICLE 4 claims 4 theorems

Gravity D2 Quadrature Instances Damped Flat Product Filter Data Satisfies Master

A machine-checked proof shows that a flat, damped family of tetrahedral probes converges to the correct gravity value, closing a major analytic gap.

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The flat sector closes

In numerical approaches to gravity, one often approximates spacetime by a grid of tetrahedra and computes a discrete version of the Einstein-Hilbert action, the quantity whose variation gives the field equations. A central question is whether this discrete sum converges to the continuum value as the grid refines. The declaration dampedFlatProductFilterData_satisfies_master_target is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It establishes that for a family of tetrahedral grids that have been flattened, meaning every probe potential is set to zero, and then damped by a schedule that vanishes in the limit, the full nonlinear discrete gravity aggregate converges to the flat continuum value of zero.

The theorem is unconditional: it holds for every slice family and every universal damping schedule, with no supplied analytic inputs. This is significant because it closes the flat sector of the D2 reduction, a step in a larger program to show that discrete gravity recovers general relativity. The proof combines two previously established results: the quadrature target, which says the discrete sum converges to the correct integral, and the residual vanishing target, which says the damping removes the error. Both are now theorems instead of assumptions, making this the first master datum in the library where both fields are proved instead of supplied.

The theorem also simplifies the curved case. For slices where all tetrahedron probes are equal to a single global potential, the quadrature proxy collapses to a simple expression: the number of tetrahedra times a cell weight times half the Dirichlet energy of that potential. This reduces the open convergence question to a concrete scalar limit of finite graph-Dirichlet energies, instead of an abstract statement about opaque proxies.

What the theorem does not claim is equally important. It does not prove the scalar Dirichlet limit for curvature-bearing probe families; that remains open. It does not establish the local correspondence at each cardinality, which is carried by slices as a separate assumption. And it does not cover non-product, non-flat admissible triangulations. The flat sector is closed, but the curved sector, the genuine Riemann-sum content of D2, remains a target.

THEOREM dampedFlatProductFilterData_satisfies_master_target · IndisputableMonolith/Gravity/D2QuadratureInstances.lean
dampedFlatProductFilterData_satisfies_master_target · IndisputableMonolith/Gravity/D2QuadratureInstances.lean:198 
/-- The flat datum satisfies the Track 1.B-PHY concrete product-filter target
consumed by the quantum-gravity master theorem. -/
theorem dampedFlatProductFilterData_satisfies_master_target
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (σ : α → ℝ)
    (hσ0 : Filter.Tendsto σ l (nhds 0))
    (hσne : ∀ᶠ t : α in l, σ t ≠ 0)
    (refinementFilter : Filter ρ) :
    Track1BCPhysicalResidual.PhysicalReggeEHConcreteProductFilterTarget
      (dampedFlatProductFilterData F σ hσ0 hσne refinementFilter) :=
  Track1BCPhysicalResidual.physicalReggeEHConcreteProductFilterTarget_holds
    (dampedFlatProductFilterData F σ hσ0 hσne refinementFilter)
THEOREM dampedFlatProductFilterData_satisfies_master_target · IndisputableMonolith/Gravity/D2QuadratureInstances.lean
dampedFlatProductFilterData_satisfies_master_target · IndisputableMonolith/Gravity/D2QuadratureInstances.lean:198 
/-- The flat datum satisfies the Track 1.B-PHY concrete product-filter target
consumed by the quantum-gravity master theorem. -/
theorem dampedFlatProductFilterData_satisfies_master_target
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (σ : α → ℝ)
    (hσ0 : Filter.Tendsto σ l (nhds 0))
    (hσne : ∀ᶠ t : α in l, σ t ≠ 0)
    (refinementFilter : Filter ρ) :
    Track1BCPhysicalResidual.PhysicalReggeEHConcreteProductFilterTarget
      (dampedFlatProductFilterData F σ hσ0 hσne refinementFilter) :=
  Track1BCPhysicalResidual.physicalReggeEHConcreteProductFilterTarget_holds
    (dampedFlatProductFilterData F σ hσ0 hσne refinementFilter)
THEOREM dampedFlatProductFilterData · IndisputableMonolith/Gravity/D2QuadratureInstances.lean
dampedFlatProductFilterData · IndisputableMonolith/Gravity/D2QuadratureInstances.lean:185 
/-- The first D2 master datum whose `quadrature_tendsto` and
`uniform_residual` fields are both proved rather than supplied. -/
noncomputable def dampedFlatProductFilterData
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (σ : α → ℝ)
    (hσ0 : Filter.Tendsto σ l (nhds 0))
    (hσne : ∀ᶠ t : α in l, σ t ≠ 0)
    (refinementFilter : Filter ρ) :
    CanonicalPeriodicTetSixTetVolumeQuadratureProductFilterData (α := α) (ρ := ρ) l :=
  dampedProductFilterData (flatFamily F) σ hσ0 hσne refinementFilter 0
    (flatFamily_quadrature_target F refinementFilter)
THEOREM quadratureIntegral_of_uniform_probe · IndisputableMonolith/Gravity/D2QuadratureInstances.lean
quadratureIntegral_of_uniform_probe · IndisputableMonolith/Gravity/D2QuadratureInstances.lean:215 
/-- For a slice whose tetrahedron probes are all the same global potential,
the quadrature proxy collapses to tetrahedron count times limiting cell
weight times the Dirichlet limit action of that potential. -/
theorem quadratureIntegral_of_uniform_probe
    {α : Type*} {l : Filter α}
    (S : CanonicalPeriodicTetSixTetVolumeQuadratureSlice l) :
    letI : NeZero S.Nx := S.instNx
    letI : NeZero S.Ny := S.instNy
    letI : NeZero S.Nz := S.instNz
    ∀ ξ : VertexPotential
        (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).K,
      (∀ τ : Fin (Fintype.card (PeriodicTet S.Nx S.Ny S.Nz)),
        S.data.tetProbe (tetFinEquiv S.Nx S.Ny S.Nz τ) = ξ) →
      S.quadratureIntegral =
        (Fintype.card (PeriodicTet S.Nx S.Ny S.Nz) : ℝ) *
          (S.data.limitCellVolume / 6) *
          ((1 / 2) *
            canonicalDirichletEnergy
              (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).K
              (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).hK
              ξ) := by
  letI : NeZero S.Nx := S.instNx
  letI : NeZero S.Ny := S.instNy
  letI : NeZero S.Nz := S.instNz
  intro ξ hξ
  have h : S.quadratureIntegral =
      ∑ τ : Fin (Fintype.card (PeriodicTet S.Nx S.Ny S.Nz)),
        canonicalPeriodicFreudenthalTetVolumeWeight S.Nx S.Ny S.Nz
            S.data.limitCellVolume (tetFinEquiv S.Nx S.Ny S.Nz τ) *
          ((1 / 2) *
            canonicalDirichletEnergy
              (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).K
              (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).hK
              (S.data.tetProbe (tetFinEquiv S.Nx S.Ny S.Nz τ))) := rfl
  rw [h]
  have hterm : ∀ τ : Fin (Fintype.card (PeriodicTet S.Nx S.Ny S.Nz)),
      canonicalPeriodicFreudenthalTetVolumeWeight S.Nx S.Ny S.Nz
          S.data.limitCellVolume (tetFinEquiv S.Nx S.Ny S.Nz τ) *
        ((1 / 2) *
          canonicalDirichletEnergy
            (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).K
            (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).hK
            (S.data.tetProbe (tetFinEquiv S.Nx S.Ny S.Nz τ))) =
        S.data.limitCellVolume / 6 *
          ((1 / 2) *
            canonicalDirichletEnergy
              (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).K
              (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).hK
              ξ) := by
    intro τ
    rw [hξ τ]
    simp only [canonicalPeriodicFreudenthalTetVolumeWeight]
  rw [Finset.sum_congr rfl fun τ _ => hterm τ]
  rw [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
  ring

What this page does not claim

The theorem does not prove the scalar Dirichlet limit for curvature-bearing probe families. The theorem does not establish the Track 1.B local correspondence at each cardinality. The theorem does not cover non-product, non-flat admissible triangulations.

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/D2QuadratureInstances.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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