Gravity D2 Scalar Dirichlet Quadrature Limit Scalar Limit And Damped Implies Ful

A conditional bridge

In numerical analysis, a quadrature rule approximates an integral by a weighted sum of function values at discrete points. The framework's gravity program studies a family of such rules built from tetrahedral meshes, where the weights come from volumes and the function values from a scalar field on the mesh. The question is whether, as the mesh refines, these discrete approximations converge to the true continuum integral. For a flat geometry the answer is trivial: the scalar field is zero, so every approximation is zero and convergence is immediate.

For curved geometries the question is harder. The framework's library proves a conditional statement: if a certain scalar sequence of graph energies converges to the continuum integral, then the full quadrature convergence target holds. The proof combines this scalar limit hypothesis with a previously established result about damped schedules, which handles the residual terms. The combined theorem, scalar_limit_and_damped_implies_full_convergence, states that the scalar limit alone suffices for the full product-filter convergence of the nonlinear Regge aggregate to the continuum Einstein-Hilbert/Dirichlet integral.

The key point is what the theorem does not prove. The scalar limit itself, the concrete numerical limit of finite graph-Dirichlet energies for curvature-bearing probe families, remains an open analytic input. The theorem is a bridge: it shows that if that numerical limit is established, everything else follows. This is the sharpest honest reduction of the D2 target to a single analytic question, and the framework's library names that question explicitly as the remaining open content.

/-- **The scalar Dirichlet energy limit plus the damped schedule implies full
D2 product-filter convergence.**  Since the damped-schedule closure
discharges the uniform residual target unconditionally for damped schedules
(via `dampedFamily_fullReggeProduct_tendsto_continuum`), the scalar
Dirichlet energy limit alone suffices for the full nonlinear Regge aggregate
to converge to the continuum Einstein-Hilbert/Dirichlet integral on the
product filter.

This is the sharpest reduction of D2 to a single analytic input: the scalar
graph-Dirichlet energy limit for curvature-bearing probe families. -/
theorem scalar_limit_and_damped_implies_full_convergence
    {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) (continuumIntegral : ℝ)
    (H : ScalarDirichletEnergyLimit F refinementFilter continuumIntegral) :
    Filter.Tendsto
      (CanonicalPeriodicTetSixTetVolumeQuadratureProductFullReggeAggregate
        (α := α) (ρ := ρ) (dampedFamily F σ hσ0 hσne))
      (refinementFilter ×ˢ l : Filter (ρ × α))
      (nhds continuumIntegral) :=
  dampedFamily_fullReggeProduct_tendsto_continuum F σ hσ0 hσne
    refinementFilter continuumIntegral
    (scalar_dirichlet_limit_implies_d2_quadrature_target F refinementFilter
      continuumIntegral H)

What this page does not claim

The scalar Dirichlet energy limit for curved geometries is not proved here. The theorem does not establish convergence for undamped schedules. The framework's library does not claim the D2 quadrature target holds unconditionally for curvature-bearing probes.

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/D2ScalarDirichletQuadratureLimit.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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