Gravity D2 Damped Schedule Closure Damping Factor Mul Residual Coefficient Le On

The damping bound

In numerical analysis, when a computation approximates a continuous quantity on a discrete grid, the error between the two usually shrinks as the grid gets finer. The Recognition Science framework's library of formal theorems studies this for a specific case: a discrete version of the Einstein-Hilbert action of general relativity, built on a periodic lattice of tetrahedra. The declaration dampingFactor_mul_residualCoefficient_le_one is a proved inequality about that setting. It says that a certain damping factor, a number used to scale down the grid spacing, multiplied by a residual coefficient, a number that bounds how much the discrete approximation can deviate from the continuum, is always at most 1.

The statement is a theorem in the library's D2DampedScheduleClosure module. It is proved from the definitions of the damping factor and the residual coefficient, using the fact that both are nonnegative. The damping factor is defined as the smaller of two fractions: one involving the local radius of validity of a cubic Taylor bound, and one involving the residual coefficient itself. The theorem then follows from the algebraic property that for any nonnegative numbers a and b, the product of b and the minimum of a and 1/(1+b) is at most 1. This is a purely formal result, verified in the machine-checked library with no unproved assumptions.

The practical consequence is that the framework can construct a modified family of lattice approximations, called a damped family, where the residual error is guaranteed to vanish as the grid is refined. The inequality is the key step: it ensures that the damping factor, chosen based on the local geometry, is small enough to keep the error under control. This is what allows the framework to discharge one of its open analytic targets, the vanishing of the residual, without supplying it as an extra hypothesis.

The theorem does not claim that the residual error itself is zero for any finite grid. It only bounds the product of the damping factor and the residual coefficient by 1. It also does not prove that the discrete approximation converges to the continuum; that requires a separate condition on the quadrature sums, which remains an open target in the framework. The inequality is a precise, local statement about how to choose a damping factor, not a global statement about convergence.

theorem dampingFactor_mul_residualCoefficient_le_one
    {α : Type*} {l : Filter α}
    (S : CanonicalPeriodicTetSixTetVolumeQuadratureSlice l) :
    dampingFactor S * residualCoefficient S ≤ 1 := by
  have hK := residualCoefficient_nonneg S
  have h1K := one_add_residualCoefficient_pos S
  have hd : dampingFactor S ≤ 1 / (1 + residualCoefficient S) := min_le_right _ _
  calc dampingFactor S * residualCoefficient S
      ≤ (1 / (1 + residualCoefficient S)) * residualCoefficient S := by
        exact mul_le_mul_of_nonneg_right hd hK
    _ ≤ 1 := by
        rw [div_mul_eq_mul_div, one_mul, div_le_one h1K]
        linarith

/-- The per-slice damping factor.  The first component keeps every damped
probe inside the local-correspondence radius; the second shrinks the slice
residual coefficient below one. -/
noncomputable def dampingFactor
    {α : Type*} {l : Filter α}
    (S : CanonicalPeriodicTetSixTetVolumeQuadratureSlice l) : ℝ :=
  min (localRadius S / (1 + probeNormSum S)) (1 / (1 + residualCoefficient S))

/-- The slice residual coefficient: limiting cell-volume weight times the
local cubic constant times the cubed probe-norm sum. -/
noncomputable def residualCoefficient
    {α : Type*} {l : Filter α}
    (S : CanonicalPeriodicTetSixTetVolumeQuadratureSlice l) : ℝ :=
  letI : NeZero S.Nx := S.instNx
  letI : NeZero S.Ny := S.instNy
  letI : NeZero S.Nz := S.instNz
  |S.data.limitCellVolume| / 6 * localConstant S * probeCubeSum S

theorem dampingFactor_pos
    {α : Type*} {l : Filter α}
    (S : CanonicalPeriodicTetSixTetVolumeQuadratureSlice l) :
    0 < dampingFactor S := by
  unfold dampingFactor
  exact lt_min
    (div_pos (localRadius_pos S) (one_add_probeNormSum_pos S))
    (div_pos one_pos (one_add_residualCoefficient_pos S))

theorem residualCoefficient_nonneg
    {α : Type*} {l : Filter α}
    (S : CanonicalPeriodicTetSixTetVolumeQuadratureSlice l) :
    0 ≤ residualCoefficient S :=
  mul_nonneg
    (mul_nonneg (div_nonneg (abs_nonneg _) (by norm_num)) (localConstant_nonneg S))
    (probeCubeSum_nonneg S)

What this page does not claim

The theorem does not prove that the residual error is zero for any finite grid. It does not establish convergence of the discrete approximation to the continuum. It does not apply to non-product or non-flat triangulations, which remain an open item.

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.

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