Gradient bound for the comparison difference, with constants independent of $\delta$

Proved

OAI.SmoothLocal.Perturbation.exists_actual_uniform_comparison_gradient

by wurtle · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

differential-geometryisometric-immersion

Notation. $S=[-3,3]^2$ is the model square; a metric is a smooth positive matrix field $g$ ; $z$ is a scalar height; $H=\nabla_g^2z$ is its covariant Hessian; $K_g$ is the Gaussian curvature; $E=\det g\,(1-|dz|_g^2)$ ; $q=H_{xy}/H_{yy}$ is the Hessian quotient; $\kappa(x^2-h(y))$ is the model curvature; the patch box and the central box are the squares $[-1/10,1/10]^2$ and $[-1/5,1/5]^2$ ; the sheared coordinates are $\xi=x,\ \theta=y+q_0x$ , with sheared height $z_s$ ; $L$ and $v_*$ are the class width and speed of (4.2); $\tilde g_\tau$ is the test metric (4.12) built from $g_*$ ; a bounded admissible height of bound $M$ is an admissible height with (4.1).

Let $g_*$ be smooth and positive on an open $V\supseteq S$ with $C^2$ bound $G$ and $\det g_*\ge d$ on $S$ , and fix $\kappa,q_0,M$ . There is $r\in(0,1/2)$ with $Lr\le1/20$ and $r$ small such that for every $N>2$ there are $C_{\mathrm{res}},C_{\mathrm{source}},C_{\mathrm{gradient}}$ such that for $0<\delta\le1/2$ and all large $\tau$ : for every patch perturbation $\eta$ , flow data and bounded admissible height $z$ of bound $M$ with the usual bounds and $g$ within $\tau^{-\tau}$ of the test metric, the Taylor approximation $P$ of the sheared height has residual of size $\le C_{\mathrm{res}}\tau^{-N}$ with weighted moment at most $2C_{\mathrm{res}}\big(\int\chi\big)\delta\,\tau^{-(N+1)}$ , the source satisfies $|F|\le C_{\mathrm{source}}\tau^{2-N}$ , and for $|\theta|\le\delta/\tau$ , $i\in\{1,2\}$ ,

\Big(\int_{|x|\le Lr/16}\big(\partial_i(P-z_s)\big)^2(x,\theta)\,dx\Big)^{1/2}\le C_{\mathrm{gradient}}\,\delta\,\tau^{1-N}.

Formalization Note. This is the upstream Lean declaration OAI.SmoothLocal.Perturbation.exists_actual_uniform_comparison_gradient in lean/OAI/Geometry/IsometricImmersion/Comparison/ActualUniformComparisonGradient.lean, stated verbatim; the Lean statement is authoritative for the precise hypotheses. It is an intermediate result of OpenAI's formalization.

Preamble

import Mathlib
import Definitions.Def_IsometricImmersion
import Definitions.Def_SmoothLocalIsometricSupport

namespace OAI

noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise

namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow
open SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Hyperbolic SmoothLocal.Taylor

Formal statement

theorem exists_actual_uniform_comparison_gradient
    {gStar : MetricField} {V : Set Coord}
    (hgStar : SmoothPositiveOn gStar V) (hV : IsOpen V) (hSV : modelSquare ⊆ V)
    {G d kappa q0 : ℝ} (M : ℕ) (hG : 0 ≤ G) (hd : 0 < d)
    (hgStarB : ∀ i j k, k ≤ 2 → ∀ p ∈ modelSquare,
      ‖iteratedFDeriv ℝ k (fun q => gStar q i j) p‖ ≤ G)
    (hdStar : ∀ p ∈ modelSquare, d ≤ (gStar p).det)
    (hkappa : 0 < kappa) (hM : 0 < M) (hq0 : |q0| ≤ 1/20) :
    ∃ r : ℝ, 0 < r ∧ r < 1/2 ∧ boundedClassWidth kappa M*r ≤ 1/20 ∧
      heightQuotientJetBound G (M : ℝ) d (1/(M : ℝ))*
        (r+107*(boundedClassWidth kappa M*r)/100) ≤ 9/(100*boundedClassWidth kappa M) ∧
      ∀ N : ℕ, 2 < N → ∃ Cres Csource Cgradient : ℝ,
        0 ≤ Cres ∧ 0 < Csource ∧ 0 ≤ Cgradient ∧
        ∀ delta : ℝ, 0 < delta → delta ≤ 1/2 →
        ∀ᶠ tau : ℕ in atTop,
          ∀ (g0 : MetricField) (eta : metricPatchSet g0 kappa) (U W : Set Coord)
            (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ),
            SmoothPositiveOn (perturbedMetric g0 eta.val) U → IsOpen U →
            CapInductionHeight (perturbedMetric g0 eta.val) U (M : ℝ) (1/(M : ℝ)) (1/(M : ℝ)) z →
            CapInductionFlow (perturbedMetric g0 eta.val) U G (M : ℝ) d (1/(M : ℝ)) (1/(M : ℝ)) kappa z Y W →
            BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z →
            (∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare,
              ‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j) p‖ ≤ G) →
            (∀ p ∈ modelSquare, d ≤ |(perturbedMetric g0 eta.val p).det|) →
            |hessianQuotient (perturbedMetric g0 eta.val) z 0-q0| ≤ 1/(100*boundedClassWidth kappa M) →
            (∀ i j : Fin 2, ∀ k ≤ tau, ∀ p ∈ modelSquare,
              ‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j-
                testMetric gStar q0 (boundedClassWidth kappa M*r/16) N delta (tau : ℝ) q i j) p‖ ≤
                  metricApproximationAccuracy tau) →
            let zs := heightInShearCoordinates z q0
            let gs := metricInShearCoordinates gStar q0
            let P := taylorApproximation gs (-delta/(tau : ℝ))
              (heightCauchyValue zs (-delta/(tau : ℝ)))
              (heightCauchyVelocity zs (-delta/(tau : ℝ))) N
            ContDiffOn ℝ ∞ P (spatialStrip (Ioo (-(boundedClassWidth kappa M*r)) (boundedClassWidth kappa M*r))) ∧
            (∀ x ∈ Ioo (-(boundedClassWidth kappa M*r)) (boundedClassWidth kappa M*r),
              P ![x,-delta/(tau : ℝ)] = zs ![x,-delta/(tau : ℝ)] ∧
              coordPartial 1 P ![x,-delta/(tau : ℝ)] = coordPartial 1 zs ![x,-delta/(tau : ℝ)]) ∧
            (∀ x, ∃ Q : Polynomial ℝ, Q.natDegree ≤ N+1 ∧ ∀ t, Q.eval t = P ![x,t]) ∧
            (∀ x : ℝ, |x| ≤ boundedClassWidth kappa M*r/2 →
              ∀ t ∈ Icc (-delta/(tau : ℝ)) (delta/(tau : ℝ)),
                (boundedClassSpeed kappa M)^2/(8*(M : ℝ)) ≤ |covHessian gs P ![x,t] 0 0| ∧
                |qResidual gs P ![x,t]| ≤ Cres/(tau : ℝ)^N) ∧
            ContinuousOn (qResidual gs P)
              (pulseStrip (boundedClassWidth kappa M*r/16) delta (tau : ℝ)) ∧
            |pulseWeightedMoment (boundedClassWidth kappa M*r/16) delta (tau : ℝ) (qResidual gs P)| ≤
              (2*Cres*(∫ x : ℝ, axisBump (boundedClassWidth kappa M*r/16) x))*delta/(tau : ℝ)^(N+1) ∧
            (∀ p ∈ pulseStrip (boundedClassWidth kappa M*r/2) delta (tau : ℝ),
              |actualComparisonSource gStar (perturbedMetric g0 eta.val) z q0 P p| ≤
                Csource*(tau : ℝ)^2/(tau : ℝ)^N) ∧
            (∀ theta ∈ Icc (-(delta/(tau : ℝ))) (delta/(tau : ℝ)), ∀ i : Fin 2,
              Real.sqrt (∫ x in Icc (-(boundedClassWidth kappa M*r/16)) (boundedClassWidth kappa M*r/16),
                (coordPartial i (comparisonDifference P zs) (boxPoint x theta))^2) ≤
                Cgradient*delta*(tau : ℝ)/(tau : ℝ)^N) := by
  sorry

end SmoothLocal.Perturbation
end
end OAI

Source

OpenAI, A Smooth Metric with No Local Isometric Immersion into Three-Space, September 24, 2026, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf; Lean: openai/math at adc7f1241b42e322a6451854ab7e4b4c146bf78a, lean/OAI/Geometry/IsometricImmersion/ (Apache License 2.0), file lean/OAI/Geometry/IsometricImmersion/Comparison/ActualUniformComparisonGradient.lean; Section 4.4, (4.19)

View graph

Gradient bound for the comparison difference, with constants independent of δ\deltaδ

Gradient bound for the comparison difference, with constants independent of $\delta$