OAI.PlanarL1.Columns.Model.World.run_success_probability: intermediate lemma from OpenAI, Planar Graph Metrics Embed into L1 with Constant Distortion

Proved

OAI.PlanarL1.Columns.Model.World.run_success_probability

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.PlanarL1.planar_graph_metrics_embed_L1, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Planar Graph Metrics Embed into L1 with Constant Distortion, OpenAI Math Release, September 23, 2026, result family 89, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.PlanarL1.Columns.Model.World.run_success_probability of lean/OAI/Combinatorics/PlanarL1/PlanarL1/Schedule.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.PlanarL1.Columns.Model.World, with open Parameters):

lemma run_success_probability {f : ℕ} {x y : V} (hstart : W.Coherent x (targetRadius Q f 0))
    (hmesh : (f:ℝ)*persistenceCount/m ≤ 1/4) (hmany : 10000 ≤ m)
    (hlo : 1 ≤ Q^f*W.distance x y) (hhi : Q^f*W.distance x y < 2) :
    successChance/2 ≤ (W.run g hm (f+1)).prob (fun Z => Z.Successful (Q*successGain) x y)

Definitions. The statement is written with OAI.PlanarL1.Execution.prob, OAI.PlanarL1.Parameters.GoodSpacing, OAI.PlanarL1.Parameters.persistenceCount, OAI.PlanarL1.Parameters.successChance, OAI.PlanarL1.Parameters.successGain, OAI.PlanarL1.Parameters.targetRadius, OAI.PlanarL1.Columns.Model.World, OAI.PlanarL1.Columns.Model.World.Coherent, OAI.PlanarL1.Columns.Model.World.Successful, OAI.PlanarL1.Columns.Model.World.distance and OAI.PlanarL1.Columns.Model.World.run (definitions bundle OAIQ5T5000 of this split).

Used by. The proof of OAI.PlanarL1.planar_graph_metrics_embed_L1 of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ5T5000

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

section

namespace PlanarL1.Columns.Model.World
open Parameters
variable {k : ℕ} {V : Type*} [Fintype V] {Q : ℝ}
variable (g : GoodSpacing Q) (W : World k V Q) {m : ℕ} (hm : 0 < m)
lemma run_success_probability {f : ℕ} {x y : V} (hstart : W.Coherent x (targetRadius Q f 0))
    (hmesh : (f:ℝ)*persistenceCount/m ≤ 1/4) (hmany : 10000 ≤ m)
    (hlo : 1 ≤ Q^f*W.distance x y) (hhi : Q^f*W.distance x y < 2) :
    successChance/2 ≤ (W.run g hm (f+1)).prob (fun Z => Z.Successful (Q*successGain) x y) := by
  sorry
end PlanarL1.Columns.Model.World

end

end OAI
end

Source

OpenAI, Planar Graph Metrics Embed into L1 with Constant Distortion, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 89; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/PlanarL1/PlanarL1/Schedule.lean (declaration OAI.PlanarL1.Columns.Model.World.run_success_probability)

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