All-order Cauchy data bound on a cut, from all-order bounds on the lower cut

Proved

OAI.SmoothLocal.Perturbation.exists_bounded_patch_cauchy_bound

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).

Fix $G,d,\kappa,q_0,r,C\ge0$ as above, with $M>0$ , $r<1/2$ , $|q_0|\le1/20$ , $Lr\le1/20$ and $r$ small, and an order $K$ . There is $B\ge0$ such that for every patch perturbation $\eta$ , every bounded admissible height $z$ with bound $M$ , and every $b\in[-r,0]$ : if $g=g_0+\eta$ has $C^4$ bound $G$ and $|\det g|\ge d$ on $S$ , $|q(0)-q_0|\le1/(100L)$ , the sheared metric has derivatives bounded by $C$ through order $\max(9,K+3)$ on the slab between $\theta=-r$ and $\theta=b$ , and the sheared height has derivatives bounded by $A$ through order $\max(8,K+2)$ on the cut $\theta=-r$ , then for $|x|\le Lr/2$ and $n\le K$ ,

\big\|D^n\big[z_s(\cdot,b)\big](x)\big\|\le B,\qquad\big\|D^n\big[\partial_\theta z_s(\cdot,b)\big](x)\big\|\le B,

where $z_s$ is the sheared height. It is the step from the cut $\theta=-r$ to any later cut below the pulse.

Formalization Note. This is the upstream Lean declaration OAI.SmoothLocal.Perturbation.exists_bounded_patch_cauchy_bound in lean/OAI/Geometry/IsometricImmersion/Caps/BoundedPatchCauchy.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 NNReal

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_bounded_patch_cauchy_bound
    (G d kappa q0 r C : ℝ) (M : ℕ) (A : ℝ≥0) (hG : 0 ≤ G) (hd : 0 < d)
    (hkappa : 0 < kappa) (hM : 0 < M) (hr : 0 < r) (hrhalf : r < 1/2)
    (hC : 0 ≤ C) (hq0 : |q0| ≤ 1/20)
    (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 ∧ ∀ (g0 : MetricField) (eta : metricPatchSet g0 kappa)
      (z : Coord → ℝ) (b : ℝ),
      BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z → -r ≤ b → b ≤ 0 →
      (∀ i j k, k ≤ 4 → ∀ 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, CoordinateBound (fun p => metricInShearCoordinates (perturbedMetric g0 eta.val) q0 p i j)
        (shrinkingSlab (-(boundedClassWidth kappa M*r)) (boundedClassWidth kappa M*r)
          (-r) b (boundedClassWidth kappa M/4)) (max 9 (K+3)) C) →
      (∀ ds : List (Fin 2), ds.length ≤ max 8 (K+2) → ∀ x : ℝ, |x| ≤ boundedClassWidth kappa M*r →
        |iteratedCoordPartial ds (heightInShearCoordinates z q0) (coordinatePoint x (-r))| ≤ (A : ℝ)) →
      ∀ x : ℝ, |x| ≤ boundedClassWidth kappa M*r/2 → ∀ n ≤ K,
        ‖iteratedFDeriv ℝ n (heightCauchyValue (heightInShearCoordinates z q0) b) x‖ ≤ B ∧
        ‖iteratedFDeriv ℝ n (heightCauchyVelocity (heightInShearCoordinates z q0) b) 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/BoundedPatchCauchy.lean; Section 4.2, proof of Lemma 4.1

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