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

Proved

OAI.SquareDifference.source_ordered_contraction

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.source_ordered_contraction of lean/OAI/Combinatorics/SquareDifference/IntervalContraction.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

lemma source_ordered_contraction {S : Type} [Fintype S] [DecidableEq S]
    (ps : S → ℕ) [∀s,Fact (ps s).Prime] :
    ∃C : ℝ,0<C ∧ ∀ᶠ N : ℕ in atTop,
      ∀{J : Type} [Fintype J] [DecidableEq J] (p : J → ℕ) [∀j,Fact (p j).Prime],
      Function.Injective (extendedPrime ps p) → (∀j,64≤p j) →
      (∀j,max tupleMassThreshold tupleConditionalThreshold≤(p j:ℝ)) →
      (∀r : ℕ,r.Prime → r≤N → ∃i,extendedPrime ps p i=r) →
      ∀(A : Finset ℕ),A⊆range N → NatSquareFree A →
      ∀s : TupleVertex → ZMod (smallModulus ps),
      (∀k<24,smallStrictPair ps (s (cycleVertex k)) (s (cycleVertex (k+1)))) →
      |multilinearIntegral (tensorLaw (fun j => tupleGoodLaw (p:=p j)))
        (fun v => actualTruncatedLift p N (powerCutoff N sourceBeta)
          (smallResidueInput (smallModulus ps) A (s v)))|≤C*(N:ℝ)^(-sourceSigma (Fintype.card TupleVertex))

Definitions. The statement is written with OAI.SquareDifference.NatSquareFree, OAI.SquareDifference.TupleVertex, OAI.SquareDifference.actualTruncatedLift, OAI.SquareDifference.cycleVertex, OAI.SquareDifference.extendedPrime, OAI.SquareDifference.multilinearIntegral, OAI.SquareDifference.powerCutoff, OAI.SquareDifference.smallModulus, OAI.SquareDifference.smallResidueInput, OAI.SquareDifference.smallStrictPair, OAI.SquareDifference.sourceBeta, OAI.SquareDifference.sourceSigma, OAI.SquareDifference.tensorLaw, OAI.SquareDifference.tupleBlocks, OAI.SquareDifference.tupleConditionalThreshold, OAI.SquareDifference.tupleGoodLaw, OAI.SquareDifference.tupleH and OAI.SquareDifference.tupleMassThreshold (definitions bundle OAIB3Sq000 of this split).

Used by. The proof of OAI.SquareDifference.power_saving 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 Filter

open scoped BigOperators Topology

namespace SquareDifference

open LiftTheory.SquareDifference

lemma source_ordered_contraction {S : Type} [Fintype S] [DecidableEq S]
    (ps : S → ℕ) [∀s,Fact (ps s).Prime] :
    ∃C : ℝ,0<C ∧ ∀ᶠ N : ℕ in atTop,
      ∀{J : Type} [Fintype J] [DecidableEq J] (p : J → ℕ) [∀j,Fact (p j).Prime],
      Function.Injective (extendedPrime ps p) → (∀j,64≤p j) →
      (∀j,max tupleMassThreshold tupleConditionalThreshold≤(p j:ℝ)) →
      (∀r : ℕ,r.Prime → r≤N → ∃i,extendedPrime ps p i=r) →
      ∀(A : Finset ℕ),A⊆range N → NatSquareFree A →
      ∀s : TupleVertex → ZMod (smallModulus ps),
      (∀k<24,smallStrictPair ps (s (cycleVertex k)) (s (cycleVertex (k+1)))) →
      |multilinearIntegral (tensorLaw (fun j => tupleGoodLaw (p:=p j)))
        (fun v => actualTruncatedLift p N (powerCutoff N sourceBeta)
          (smallResidueInput (smallModulus ps) A (s v)))|≤C*(N:ℝ)^(-sourceSigma (Fintype.card TupleVertex)) := by
  sorry

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/IntervalContraction.lean (declaration OAI.SquareDifference.source_ordered_contraction)

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