Gravity D2 Scoping Audit D2 Scope Status Damped

The damped scope record

In the Recognition Science framework's machine-checked library of formal theorems, D2ScopeStatusDamped is a bookkeeping structure. It records, as a set of true-or-false flags, the current state of a specific derivation: showing that a discrete model of gravity, built from tetrahedra on a periodic grid, converges to the continuous Einstein-Hilbert action as the grid refines. The structure itself is a definitional choice, a way to make the scope of a proof explicit rather than hidden in a long argument.

The flags tell a precise story. The reduction theorem is proved: if two analytic conditions hold, then the full nonlinear Regge aggregate converges to the continuum integral. One of those conditions, the vanishing of a residual error term, has been discharged for a class of damped schedules, meaning it is derived from a per-slice cubic bound and is no longer an independent input. The other condition, quadrature convergence, remains open as a target. The residual target for arbitrary undamped schedules also remains open, as does the generalization to non-product, non-flat triangulations.

This is a reduction, not a closure. The declaration does not claim that gravity has been derived from first principles. It claims that a specific convergence statement has been reduced to two named analytic targets, one of which has been closed for a particular schedule class. The open frontier is named, not asserted away. This is the honest status of the D2 master witness, and the structure makes that status auditable at a glance.

/-- **D2 REDUCTION THEOREM (what is actually proved).**  On the canonical
periodic six-tet cubic torus, the two named analytic targets — quadrature
convergence to the continuum integral and a vanishing Regge-residual — imply
that the full nonlinear Regge aggregate converges to the continuum
Einstein-Hilbert/Dirichlet integral on the product filter.

This is the honest content of the D2 master witness: convergence is reduced
to the two analytic inputs, which remain the open targets (§2).  It makes no
claim about non-product or non-flat triangulations. -/
theorem d2_reduction
    {α ρ : Type*} {l : Filter α}
    (family : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ)
    (refinementFilter : Filter ρ) (continuumIntegral : ℝ)
    (hquad : D2QuadratureConvergenceTarget l family refinementFilter continuumIntegral)
    (hres : D2ResidualVanishingTarget l family refinementFilter) :
    Filter.Tendsto
      (CanonicalPeriodicTetSixTetVolumeQuadratureProductFullReggeAggregate
        (α := α) (ρ := ρ) family)
      (refinementFilter ×ˢ l : Filter (ρ × α))
      (nhds continuumIntegral) :=
  (CanonicalPeriodicTetSixTetVolumeQuadratureProductFilterData.fullReggeProduct_tendsto_continuum
    (α := α) (ρ := ρ) (l := l)
    { family := family
      refinementFilter := refinementFilter
      continuumIntegral := continuumIntegral
      quadrature_tendsto := hquad
      uniform_residual := hres })

/-- D2 scope after `Gravity.D2DampedScheduleClosure`: one analytic input
(the quadrature limit) instead of two. -/
def d2ScopeStatusDamped : D2ScopeStatusDamped where
  reduction_proved := true
  quadrature_target_open := true
  residual_target_discharged_for_damped_schedules := true
  residual_target_open_for_arbitrary_schedules := true
  general_triangulation_open := true

What this page does not claim

The declaration does not prove that the Einstein-Hilbert action emerges from Regge calculus without any analytic hypotheses. It does not claim the quadrature convergence target is closed. It does not address physically admissible non-product, 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/D2ScopingAudit.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

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