Size of the comparison source for the metric class

Proved

OAI.SmoothLocal.Perturbation.exists_actual_class_comparison_source_bound_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, $g_*$ of $C^2$ bound $G$ with $\det g_*\ge d$ on $S$ , $N>2$ and $C_{\mathrm{res}}\ge0$ , there is $C_{\mathrm{source}}>0$ such that for $0<\delta\le1/2$ and all large $\tau$ (with $\tau\ge1$ ): for every patch perturbation $\eta$ , flow data and bounded admissible height $z$ of bound $M$ with the usual bounds, $g$ within $\tau^{-\tau}$ of the test metric, and every function $P$ with $|Q$ -residual $_{g_*}(P)|\le C_{\mathrm{res}}\tau^{-N}$ on the strip of half-width $Lr/2$ , the comparison source satisfies on that strip

|F(p)|\le C_{\mathrm{source}}\,\tau^{2-N}.

Formalization Note. This is the upstream Lean declaration OAI.SmoothLocal.Perturbation.exists_actual_class_comparison_source_bound_at_radius in lean/OAI/Geometry/IsometricImmersion/Comparison/ActualClassComparisonSourceBound.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_class_comparison_source_bound_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)
    (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)
    (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))
    (N : ℕ) (hN : 2 < N) (Cres : ℝ) (hCres : 0 ≤ Cres) :
    ∃ Csource : ℝ, 0 < Csource ∧
      ∀ delta : ℝ, 0 < delta → delta ≤ 1/2 →
      ∀ᶠ tau : ℕ in atTop,
        1 ≤ tau ∧
        ∀ (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) →
          ∀ P : Coord → ℝ,
            (∀ p ∈ pulseStrip (boundedClassWidth kappa M*r/2) delta (tau : ℝ),
              |qResidual (metricInShearCoordinates gStar q0) P p| ≤ Cres/(tau : ℝ)^N) →
            ∀ 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 := 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/ActualClassComparisonSourceBound.lean; Section 4.4, (4.18)

View graph