OAI.MetricKMedian.Certificates.k_envelope: intermediate lemma from OpenAI, The approximation threshold for metric k-median (result family 125)

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OAI.MetricKMedian.Certificates.k_envelope

by dbenbenn · Oct 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.MetricKMedian.main_finite_work, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The approximation threshold for metric k-median (result family 125), OpenAI Math Release, September 24, 2026, result family 125, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.MetricKMedian.Certificates.k_envelope of lean/OAI/Combinatorics/KMedian/KMedian/Analysis/AnalyticEnvelopes.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.MetricKMedian.Certificates, with open Analytic):

theorem k_envelope {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
    FiniteComparison.kZero (P x) ≤ k (1 - x)

Definitions. The statement is written with OAI.MetricKMedian.Analytic.P (definitions bundle OAIQ037K000 of this split); OAI.MetricKMedian.Certificates.k (definitions bundle OAIQ037K002 of this split); OAI.MetricKMedian.FiniteComparison.kZero (definitions bundle OAIQ037K004 of this split).

Used by. The proof of OAI.MetricKMedian.FiniteComparison.actual_scalar_bounds of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ037K000
import Definitions.Def_OAIQ037K002
import Definitions.Def_OAIQ037K004

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
section
namespace MetricKMedian.Certificates

open Analytic

theorem k_envelope {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
    FiniteComparison.kZero (P x) ≤ k (1 - x) := by
  sorry

end MetricKMedian.Certificates

end
end

end OAI
end

Source

OpenAI, The approximation threshold for metric k-median (result family 125), OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 125; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/KMedian/KMedian/Analysis/AnalyticEnvelopes.lean (declaration OAI.MetricKMedian.Certificates.k_envelope)

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