Gravity D2 Quadrature Instances Flat Family Quadrature Target

The flat sector closes

In numerical approaches to gravity, one often approximates a smooth spacetime by a discrete mesh of tetrahedra. A key question is whether the discrete approximation converges to the continuum result as the mesh is refined. In the Recognition Science framework, this convergence is formalized through a quadrature target, a precise statement that a certain discrete sum approaches a continuum integral as the mesh gets finer.

The declaration flatFamily_quadrature_target proves a special case of this convergence. It considers a 'flat family' of meshes, where every tetrahedron probe, the quantity used to measure the geometry, is replaced by the zero potential. Since the canonical Dirichlet energy of the zero potential is zero, the discrete sum is identically zero at every refinement level. The theorem then states that this zero sum converges to the flat continuum Einstein-Hilbert value, which is also zero. This holds for any refinement schedule, with no additional hypotheses.

This result is part of a larger effort to close the 'D2' sector, a set of analytic inputs needed for a master theorem in the framework's quantum gravity program. The flat case is the simplest possible scenario, and its closure is a necessary step. The theorem, tagged THEOREM, is machine-checked with no unproved axioms. It does not apply to curved geometries, where the probes are non-zero and the convergence remains an open problem.

In Recognition Science, this proof is a building block. It shows that the framework's machinery works in the trivial case, providing a solid foundation for the more difficult curved case. The flat sector's closure is a concrete, verified result that advances the program's overall goal of deriving gravity from discrete recognition events.

/-- **The flat-sector quadrature target holds with no hypothesis.**  The
flattened family's quadrature proxies are identically zero, so they converge
to the flat continuum Einstein-Hilbert value `0` along every refinement
filter. -/
theorem flatFamily_quadrature_target
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (refinementFilter : Filter ρ) :
    CanonicalPeriodicTetSixTetVolumeQuadratureCrossCardinalityTarget
      (flatFamily F) refinementFilter 0 := by
  unfold CanonicalPeriodicTetSixTetVolumeQuadratureCrossCardinalityTarget
  have h : (fun r : ρ => ((flatFamily F).slice r).quadratureIntegral) =
      fun _ : ρ => (0 : ℝ) := by
    funext r
    exact flattenSlice_quadratureIntegral (F.slice r)
  rw [h]
  exact tendsto_const_nhds

/-- **The flat-sector quadrature target holds with no hypothesis.**  The
flattened family's quadrature proxies are identically zero, so they converge
to the flat continuum Einstein-Hilbert value `0` along every refinement
filter. -/
theorem flatFamily_quadrature_target
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (refinementFilter : Filter ρ) :
    CanonicalPeriodicTetSixTetVolumeQuadratureCrossCardinalityTarget
      (flatFamily F) refinementFilter 0 := by
  unfold CanonicalPeriodicTetSixTetVolumeQuadratureCrossCardinalityTarget
  have h : (fun r : ρ => ((flatFamily F).slice r).quadratureIntegral) =
      fun _ : ρ => (0 : ℝ) := by
    funext r
    exact flattenSlice_quadratureIntegral (F.slice r)
  rw [h]
  exact tendsto_const_nhds

What this page does not claim

This result does not apply to curved geometries, where the convergence remains open. This theorem does not prove the convergence for non-flat, non-product admissible triangulations. This does not establish the scalar Dirichlet limit for curvature-bearing probe families.

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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Derived articles

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