All-order Cauchy data bounds at the upper pulse edge

Proved

OAI.SmoothLocal.Perturbation.exists_actual_upper_off_pulse_cauchy_bounds_at_radius

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).

With $g_*$ , $G,d,\kappa,q_0,r,M$ as above and an order $K$ , there is $B\ge0$ such that for all $N$ and $\delta>0$ and all sufficiently large $\tau$ : for every patch perturbation $\eta$ and bounded admissible height $z$ of bound $M$ for $g=g_0+\eta$ , where $g_0$ has the model curvature on an open $U\supseteq S$ , $g$ has $C^8$ bound $G$ , $|\det g|\ge d$ , $|q(0)-q_0|\le1/(100L)$ and $g$ is within $\tau^{-\tau}$ in $C^\tau(S)$ of the test metric, one has for $|x|\le Lr/2$ , $n\le K$ ,

\big\|D^n z_s(\cdot,\delta/\tau)(x)\big\|\le B,\qquad\big\|D^n\partial_\theta z_s(\cdot,\delta/\tau)(x)\big\|\le B.

This is the upper-edge assertion of Lemma 4.1, obtained from the upper cap.

Formalization Note. This is the upstream Lean declaration OAI.SmoothLocal.Perturbation.exists_actual_upper_off_pulse_cauchy_bounds_at_radius in lean/OAI/Geometry/IsometricImmersion/Caps/ActualUpperOffPulseCauchy.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
open scoped ContDiff Topology Matrix

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

Formal statement

theorem exists_actual_upper_off_pulse_cauchy_bounds_at_radius
    {gStar : MetricField} {V : Set Coord}
    (hgStar : SmoothPositiveOn gStar V) (hV : IsOpen V) (hSV : modelSquare ⊆ V)
    {G d kappa q0 r : ℝ} (M : ℕ) (hG : 0 ≤ G) (hd : 0 < d)
    (hkappa : 0 < kappa) (hM : 0 < M) (hq0 : |q0| ≤ 1/20)
    (hr : 0 < r) (hrhalf : r < 1/2) (hLr : boundedClassWidth kappa M*r ≤ 1/20)
    (hrsmall : heightQuotientJetBound G (M : ℝ) d (1/(M : ℝ))*
      (r+107*(boundedClassWidth kappa M*r)/100) ≤ 9/(100*boundedClassWidth kappa M))
    (K : ℕ) :
    ∃ B : ℝ, 0 ≤ B ∧ ∀ N : ℕ, ∀ delta : ℝ, 0 < delta →
      ∀ᶠ tau : ℕ in atTop,
        ∀ (g0 : MetricField) (eta : metricPatchSet g0 kappa) (U : Set Coord) (z : Coord → ℝ),
          SmoothPositiveOn g0 U → SmoothPositiveOn (perturbedMetric g0 eta.val) U →
          IsOpen U → modelSquare ⊆ U →
          (∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p) →
          CapInductionHeight (perturbedMetric g0 eta.val) U (M : ℝ) (1/(M : ℝ)) (1/(M : ℝ)) z →
          BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z →
          (∀ i j k, k ≤ 8 → ∀ p ∈ modelSquare,
            ‖iteratedFDeriv ℝ k (fun a => perturbedMetric g0 eta.val a 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) →
          ∀ x : ℝ, |x| ≤ boundedClassWidth kappa M*r/2 → ∀ n ≤ K,
            ‖iteratedFDeriv ℝ n (heightCauchyValue (heightInShearCoordinates z q0)
              (delta/(tau : ℝ))) x‖ ≤ B ∧
            ‖iteratedFDeriv ℝ n (heightCauchyVelocity (heightInShearCoordinates z q0)
              (delta/(tau : ℝ))) x‖ ≤ B := 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/Caps/ActualUpperOffPulseCauchy.lean; Section 4.2, Lemma 4.1

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