OAI.SquareDifference.interval_low_pair: intermediate lemma from OpenAI, A power saving for square-difference-free sets

Proved

OAI.SquareDifference.interval_low_pair

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.SquareDifference.power_saving, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, A power saving for square-difference-free sets, OpenAI Math Release, September 24, 2026, result family 182, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.SquareDifference.interval_low_pair of lean/OAI/Combinatorics/SquareDifference/IntervalPair.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SquareDifference, with open Finset, open scoped BigOperators ComplexConjugate, open LiftTheory.SquareDifference PairBridge):

lemma interval_low_pair (hinj : Function.Injective (extendedPrime ps p)) (hp2 : ∀j,p j≠2)
    (B : Finset J) (a b L N Q H : ℕ) (hN : 1≤N) (hH : 1≤H) (hab : a+L≤b)
    (hHQ : H*(∏j∈B,p j)≤Q)
    (hD : (smallModulus ps*(∏j∈B,p j):ℝ)≤(N:ℝ)^((1:ℝ)/1000))
    (hHn : (H:ℝ)≤(N:ℝ)^((1:ℝ)/1000)) (hlarge : 8≤(N:ℝ)^((1:ℝ)/16))
    (hcover : ∀r : ℕ,r.Prime → r≤N → ∃i,extendedPrime ps p i=r)
    (A : Finset ℕ) (hA : A⊆range N) (hfree : NatSquareFree A)
    (s t : ZMod (smallModulus ps)) (hst : smallStrictPair ps s t)
    (z w : ResidueSpace p) (hzw : ∀j∈B,w j-z j≠0 ∧ IsSquare (w j-z j)) :
    ‖∑ξ∈univ.filter (fun ξ => (supportDenominator (OutsideModulus p B) ξ:ℝ)≤H),
      residueFourier (OutsideModulus p B)
        (fun x => (intervalLift p a L Q (smallResidueInput (smallModulus ps) A s) (assembleOutside p B z x):ℂ)) (-ξ)*
      residueFourier (OutsideModulus p B)
        (fun x => (intervalLift p b L Q (smallResidueInput (smallModulus ps) A t) (assembleOutside p B w x):ℂ)) ξ*
      squareMultiplier (OutsideModulus p B) ξ‖≤
      ((N:ℝ)/L)^2*(smallModulus ps)*(∏j∈B,p j:ℝ)*
        (kernelAbsoluteConstant*(H:ℝ)^(-(1:ℝ)/3))

Definitions. The statement is written with OAI.SquareDifference.NatSquareFree, OAI.SquareDifference.extendedPrime, OAI.SquareDifference.intervalLift, OAI.SquareDifference.kernelAbsoluteConstant, OAI.SquareDifference.smallModulus, OAI.SquareDifference.smallResidueInput, OAI.SquareDifference.smallStrictPair, OAI.LiftTheory.SquareDifference.OutsideModulus, OAI.LiftTheory.SquareDifference.ResidueSpace, OAI.LiftTheory.SquareDifference.residueFourier, OAI.LiftTheory.SquareDifference.squareMultiplier, OAI.LiftTheory.SquareDifference.supportDenominator and OAI.SquareDifference.PairBridge.assembleOutside (definitions bundle OAIB3Sq000 of this split).

Used by. The proof of OAI.SquareDifference.source_ordered_contraction of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIB3Sq000

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

open Finset

open scoped BigOperators ComplexConjugate

namespace SquareDifference

open LiftTheory.SquareDifference PairBridge

section LowPair

variable {S J : Type*} [Fintype S] [DecidableEq S] [Fintype J] [DecidableEq J]
  (ps : S → ℕ) (p : J → ℕ) [∀i,Fact (ps i).Prime] [∀j,Fact (p j).Prime]

lemma interval_low_pair (hinj : Function.Injective (extendedPrime ps p)) (hp2 : ∀j,p j≠2)
    (B : Finset J) (a b L N Q H : ℕ) (hN : 1≤N) (hH : 1≤H) (hab : a+L≤b)
    (hHQ : H*(∏j∈B,p j)≤Q)
    (hD : (smallModulus ps*(∏j∈B,p j):ℝ)≤(N:ℝ)^((1:ℝ)/1000))
    (hHn : (H:ℝ)≤(N:ℝ)^((1:ℝ)/1000)) (hlarge : 8≤(N:ℝ)^((1:ℝ)/16))
    (hcover : ∀r : ℕ,r.Prime → r≤N → ∃i,extendedPrime ps p i=r)
    (A : Finset ℕ) (hA : A⊆range N) (hfree : NatSquareFree A)
    (s t : ZMod (smallModulus ps)) (hst : smallStrictPair ps s t)
    (z w : ResidueSpace p) (hzw : ∀j∈B,w j-z j≠0 ∧ IsSquare (w j-z j)) :
    ‖∑ξ∈univ.filter (fun ξ => (supportDenominator (OutsideModulus p B) ξ:ℝ)≤H),
      residueFourier (OutsideModulus p B)
        (fun x => (intervalLift p a L Q (smallResidueInput (smallModulus ps) A s) (assembleOutside p B z x):ℂ)) (-ξ)*
      residueFourier (OutsideModulus p B)
        (fun x => (intervalLift p b L Q (smallResidueInput (smallModulus ps) A t) (assembleOutside p B w x):ℂ)) ξ*
      squareMultiplier (OutsideModulus p B) ξ‖≤
      ((N:ℝ)/L)^2*(smallModulus ps)*(∏j∈B,p j:ℝ)*
        (kernelAbsoluteConstant*(H:ℝ)^(-(1:ℝ)/3)) := by
  sorry

end LowPair

end SquareDifference

end

end OAI
end

Source

OpenAI, A power saving for square-difference-free sets, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 182; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/SquareDifference/IntervalPair.lean (declaration OAI.SquareDifference.interval_low_pair)

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