Encyclopedia Gravity Gravity D2 Quadrature Instances Damped Flat Full Regge Product Tendsto Zero

ARTICLE 4 claims 4 theorems

Gravity D2 Quadrature Instances Damped Flat Full Regge Product Tendsto Zero

A machine-checked proof shows that a simplified, flattened version of a discrete gravity model converges to the correct flat-space answer, closing a major technical gap.

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The flat-sector closure

The declaration dampedFlat_fullReggeProduct_tendsto_zero is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It concerns a computational model of gravity built from tetrahedra, the simplest three-dimensional shapes. The theorem states that if you take any family of such tetrahedral grids, flatten them completely (replace every probe with a zero potential), and then apply a damping schedule that smoothly reduces the influence of finer details, the full nonlinear Regge aggregate of this damped flat family converges to zero. This zero is the flat Einstein-Hilbert value, the correct answer for a universe with no curvature.

To understand what this achieves, consider the standard problem of numerical integration: approximating an integral by summing values at discrete points. The framework's D2 reduction aims to show that a similar discrete sum over tetrahedra approaches the continuum value as the grid refines. Previously, this required two analytic inputs: a quadrature convergence target and a uniform residual bound. This theorem proves both for the flat sector with no supplied hypotheses. It is the first master datum in the library where both of these formerly assumed fields are theorems, derived from the local correspondence data that every slice carries by definition.

The proof works by a simple but powerful observation. When a slice is flattened, every tetrahedron probe is the zero potential. The canonical Dirichlet energy of the zero potential vanishes, so every slice's quadrature proxy is exactly zero. Consequently, the quadrature target holds at the flat continuum integral zero with no hypothesis. Combining this with the damped-schedule closure for the residual gives the full convergence result. The theorem holds for every slice family and every universal schedule, on the product filter, meaning the limit is taken as both the grid refines and the damping parameter goes to zero.

This result is a milestone for the framework's program, but it is important to state its scope precisely. The theorem does not claim anything about curved space. The curved sector remains open: for slices whose tetrahedron probes are all equal to one global potential, the quadrature proxy reduces to a scalar Dirichlet energy limit, but that limit itself is not proved. The theorem also does not address non-product, non-flat admissible triangulations. It establishes the flat sector of the D2 reduction end to end, providing a concrete, machine-checked foundation for the more ambitious goal of proving convergence for curved geometries.

THEOREM dampedFlat_fullReggeProduct_tendsto_zero · IndisputableMonolith/Gravity/D2QuadratureInstances.lean
dampedFlat_fullReggeProduct_tendsto_zero · IndisputableMonolith/Gravity/D2QuadratureInstances.lean:163 
/-- **FLAT-SECTOR D2 CLOSURE (no supplied analytic inputs).**  For every
slice family and every universal schedule, the full nonlinear Regge aggregate
of the damped flattened family converges to the flat continuum value `0` on
the product filter.  Both former analytic inputs are theorems here: the
quadrature target by §3, the uniform residual by the damped-schedule
closure.  The only data consumed are the slices themselves, including the
Track 1.B local correspondence they carry by definition. -/
theorem dampedFlat_fullReggeProduct_tendsto_zero
    {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) :
    Filter.Tendsto
      (CanonicalPeriodicTetSixTetVolumeQuadratureProductFullReggeAggregate
        (α := α) (ρ := ρ) (dampedFamily (flatFamily F) σ hσ0 hσne))
      (refinementFilter ×ˢ l : Filter (ρ × α))
      (nhds 0) :=
  dampedFamily_fullReggeProduct_tendsto_continuum (flatFamily F) σ hσ0 hσne
    refinementFilter 0 (flatFamily_quadrature_target F refinementFilter)
THEOREM canonicalDirichletEnergy_zero · flattenSlice_quadratureIntegral · IndisputableMonolith/Gravity/D2QuadratureInstances.lean
canonicalDirichletEnergy_zero · IndisputableMonolith/Gravity/D2QuadratureInstances.lean:65 
theorem canonicalDirichletEnergy_zero
    (K : Triangulation3D) (hK : IncidenceConsistent K) :
    canonicalDirichletEnergy K hK (zeroPotential K) = 0 := by
  unfold canonicalDirichletEnergy zeroPotential
  simp
flattenSlice_quadratureIntegral · IndisputableMonolith/Gravity/D2QuadratureInstances.lean:104 
/-- The flattened slice's quadrature proxy is exactly zero. -/
theorem flattenSlice_quadratureIntegral
    {α : Type*} {l : Filter α}
    (S : CanonicalPeriodicTetSixTetVolumeQuadratureSlice l) :
    (flattenSlice S).quadratureIntegral = 0 := by
  letI : NeZero S.Nx := S.instNx
  letI : NeZero S.Ny := S.instNy
  letI : NeZero S.Nz := S.instNz
  have h : (flattenSlice 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
              (zeroPotential
                (canonicalEncodedPeriodicFreudenthalTorus S.Nx S.Ny S.Nz S.hx S.hy S.hz).K)) :=
    rfl
  rw [h]
  simp [canonicalDirichletEnergy_zero]
THEOREM dampedFlat_fullReggeProduct_tendsto_zero · IndisputableMonolith/Gravity/D2QuadratureInstances.lean
dampedFlat_fullReggeProduct_tendsto_zero · IndisputableMonolith/Gravity/D2QuadratureInstances.lean:163 
/-- **FLAT-SECTOR D2 CLOSURE (no supplied analytic inputs).**  For every
slice family and every universal schedule, the full nonlinear Regge aggregate
of the damped flattened family converges to the flat continuum value `0` on
the product filter.  Both former analytic inputs are theorems here: the
quadrature target by §3, the uniform residual by the damped-schedule
closure.  The only data consumed are the slices themselves, including the
Track 1.B local correspondence they carry by definition. -/
theorem dampedFlat_fullReggeProduct_tendsto_zero
    {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) :
    Filter.Tendsto
      (CanonicalPeriodicTetSixTetVolumeQuadratureProductFullReggeAggregate
        (α := α) (ρ := ρ) (dampedFamily (flatFamily F) σ hσ0 hσne))
      (refinementFilter ×ˢ l : Filter (ρ × α))
      (nhds 0) :=
  dampedFamily_fullReggeProduct_tendsto_continuum (flatFamily F) σ hσ0 hσne
    refinementFilter 0 (flatFamily_quadrature_target F 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)

What this page does not claim

The theorem does not claim anything about curved space or the convergence of the Regge aggregate for non-flat families. The scalar Dirichlet limit for uniform-probe families with non-zero potential remains an open problem. The theorem does not address the convergence of the full nonlinear Regge aggregate for arbitrary, non-flat 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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