Gravity D2 Damped Schedule Closure Damped Family Full Regge Product Tendsto Cont

The damped convergence theorem

In numerical gravity, physicists often approximate a smooth spacetime by a grid of tetrahedra, then refine the grid to approach the continuum. The question is whether the discrete approximation genuinely converges to the continuous result. The Recognition Science library, a machine-checked collection of formal theorems, addresses this for a specific family of periodic tetrahedral grids.

The central theorem, dampedFamily_fullReggeProduct_tendsto_continuum, establishes that a particular construction, the damped family (a refinement schedule where the grid spacing shrinks at a controlled rate), produces a sequence of discrete gravity values that converges to a single continuum integral. This convergence holds for the full nonlinear Regge action, the discrete version of the Einstein-Hilbert action, across the product of refinement and cardinality filters. The theorem requires two inputs: a family of grids and a damping schedule that tends to zero, plus a separate assumption that the quadrature approximations converge.

The key achievement is that the theorem derives the vanishing of the residual, the difference between the discrete and continuous actions, from a local bound on the curvature, rather than assuming it. The damping factor, computed from each grid slice's own local properties, ensures the residual shrinks below a universal envelope. This means the convergence to the continuum is not an extra hypothesis but a consequence of the local correspondence already present in each slice.

What the theorem does not claim is equally important. It does not show that the quadrature sums themselves converge; that remains a supplied input, family-specific geometric data. It also does not establish the local correspondence bound from first principles; that is a field carried by every slice, a pre-existing assumption. The theorem's scope is limited to product filters and flat periodic triangulations, not arbitrary non-product or non-flat grids.

In plain terms, the result shows that if you slow down the refinement appropriately, the discrete gravity calculation reliably homes in on the continuum value, provided the quadrature sums behave. The residual error, the part that could spoil convergence, is forced to vanish by the damping schedule alone. This turns a previously supplied analytic field into a proved consequence, a meaningful step in the framework's program of deriving physics from recognition events.

/-- **D2 DAMPED-SCHEDULE CLOSURE.**  For every varying-cardinality slice
family and every universal schedule `σ → 0`, the full nonlinear Regge
aggregate of the damped family converges to the continuum integral on the
product filter, given only the cross-cardinality quadrature limit.  The
two-scale uniform residual is derived, not assumed. -/
theorem dampedFamily_fullReggeProduct_tendsto_continuum
    {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)
    (hquad :
      CanonicalPeriodicTetSixTetVolumeQuadratureCrossCardinalityTarget
        F refinementFilter continuumIntegral) :
    Filter.Tendsto
      (CanonicalPeriodicTetSixTetVolumeQuadratureProductFullReggeAggregate
        (α := α) (ρ := ρ) (dampedFamily F σ hσ0 hσne))
      (refinementFilter ×ˢ l : Filter (ρ × α))
      (nhds continuumIntegral) :=
  (dampedProductFilterData F σ hσ0 hσne refinementFilter continuumIntegral
    hquad).fullReggeProduct_tendsto_continuum

/-- **DERIVED UNIFORM RESIDUAL (D2 open item 2 discharged).**  The damped
family satisfies the product uniform-residual target for every refinement
filter.  No analytic residual field is supplied: the bound comes from each
slice's own local cubic Taylor correspondence, the flat-action normalization,
the edge-stencil Dirichlet identification, and the constructed damping. -/
theorem dampedFamily_uniformResidual
    {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) :
    CanonicalPeriodicTetSixTetVolumeQuadratureProductUniformResidualTarget
      (dampedFamily F σ hσ0 hσne) refinementFilter := by
  refine canonicalPeriodicTetSixTetVolumeQuadratureProductUniformResidualTarget_of_snd_abs_bound
    (dampedFamily F σ hσ0 hσne) refinementFilter (fun t => |σ t|) ?_ ?_
  · simpa using hσ0.abs
  · have h1 : ∀ᶠ t : α in l, |σ t| ≤ 1 := by
      have hball : Metric.closedBall (0 : ℝ) 1 ∈ nhds (0 : ℝ) :=
        Metric.closedBall_mem_nhds 0 one_pos
      have := hσ0.eventually_mem hball
      simpa [Metric.mem_closedBall, Real.dist_eq] using this
    refine ((hσne.and h1).prod_inr refinementFilter).mono ?_
    rintro ⟨r, t⟩ ⟨ht_ne, ht_le⟩
    exact dampedSlice_residual_abs_le (F.slice r) σ hσ0 hσne t ht_ne ht_le

What this page does not claim

The theorem does not prove the quadrature convergence target, which remains a supplied input. It does not establish the local correspondence bound from first principles; that is a pre-existing assumption on each slice. It does not cover non-product or 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/D2DampedScheduleClosure.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: