Finite recurring-gap bounds are established in the cited literature, while H₁ = 2 remains unresolved. The selected 2026 preliminary draft states an unconditional bound of 212; the 186 formal result has explicit input assumptions.
The primes pₙ are listed in increasing order. H1=n→∞liminf(pn+1−pn) measures the smallest gap recurring arbitrarily far out. A bound H₁ ≤ B does not bound every gap or prove twin primes unless B = 2.
Definitions
Limit inferior H₁
The eventual lower limit of consecutive-prime gaps. An upper bound for H₁ asserts that gaps at most that size occur infinitely often.
Question paraphrased by OpenMath. Original wording is available in the sources.
Landscape
Figures and explorable examples. Selecting a statement opens its exact wording.
Visual setup
A small gap is not yet an infinite family
The consecutive primes 101, 103, 107 and 109 have gaps 2, 4 and 2. These finite examples explain the quantity being bounded.
Reading boundary. A finite list cannot establish infinitely many twin primes or any asymptotic bound. The reported bounds are presented with their own assumptions below.
Select a statement below for its scope and evidence. The text view describes the complete drawing.
Text version
On a number line, consecutive primes 101, 103, 107 and 109 are marked, with gaps 2, 4 and 2; the universal question asks for such small gaps arbitrarily far out.
Coordinates in the drawing are illustrative unless the caption specifies an exact example.
Read the infinite recurrence target
○Open
Gaps of two infinitely often
The twin-prime target. Finite upper bounds larger than two leave it unresolved.
Authored by OpenMath source curation on 2026-10-13 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.
A gap bound of 186 from explicit input assumptions
Assuming the rank-three Kloosterman estimate, the rank-two correlation estimate, and all integral and cap bounds in the precise forms specified by the cited proof, H1≤186.
Assumptions
The rank-three Kloosterman bound in the source.
The rank-two correlation bound in the source.
All source physical-integral and cap bounds.
In this problem
The formal source derives 186 only from its three documented mathematical input axioms.
A proved conditional deduction. The two literature estimates and numerical input remain project axioms; OpenMath has not verified or discharged them.
Does not cover: Discharge or independently justify all input assumptions before treating this as an unconditional improvement.
Bound
H₁
≤ 186
Conditions
Conditional on the rank-three Kloosterman estimate, rank-two correlation estimate, and the precise integral and cap bounds in the cited proof.
Smaller recurring gaps, with the twin-prime target still open
Under stated assumptions
The 212 paper’s assertion would improve the unconditional recurring-gap bound in the selected literature. It gives arbitrarily large prime pairs at that distance, without controlling all successive gaps.
Limit The manuscript is a preliminary draft; independent OpenMath review is pending. A bound larger than two does not prove twin primes.
An infinite supply of primes does not establish infinitely many bounded gaps.
Approaches
How the problem is attacked, and what each line of attack cannot yet reach.
Sieve weights, distribution estimates and exact certificates
The 212 argument improves a sieve construction descended from Stadlmann’s work and verifies rational inequalities. Different formal artifacts certify different deductions and inputs.
Cannot yet reach The public 246 repository cannot be substituted for the 212 paper’s reported certificate.
Reported in sources: Public Lean evidence covers a 246 deduction from Bombieri–Vinogradov and a 186 deduction from three input axioms. The 212 paper reports a separate certificate with analytic and variational hypotheses; its exact public source revision was not identified. No OpenMath build is recorded.
CComputation
None recorded
RReview
Not yet reviewed
OpenMath assessment: Finite recurring-gap bounds are established in the cited literature, while H₁ = 2 remains unresolved. The selected 2026 preliminary draft states an unconditional bound of 212; the 186 formal result has explicit input assumptions.
Source comparison and page curation only; independent mathematical review is pending.
OpenMath pilot collection · OpenMath source curation · assessed Oct 13, 2026 · review due Jan 13, 2027
Complete as a conditional deduction. The formal input names are PrimeGap186.kloosterman3_bound, PrimeGap186.kloosterman2_correlation_bound and PrimeGap186.physical_integral_bounds; they remain assumptions, not discharged proof obligations.
Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026
François Charton · Letong Hong · Kenny Lau · Ken Ono · Guillaume Remy · Ho Chung Siu · Ashvin A. Swaminathan · Jesse Thorner · Yunzhou Xie · Preprint · Sep 3, 2026
Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver
README input axioms and verification; PrimeGaps186.lean
Research activity · 2 entries
Formalization
The 212 certificate is reported in the paper
The appendix credits AxiomProver with a Lean deduction that assumes five equidistribution estimates, bilinear Bombieri–Vinogradov, Harman decomposition and the variational certificate.
No exact public commit or declaration for that 212 certificate was established by this source check; it is shown as a paper-reported artifact, not a fabricated formal-code link.
Related statements · context, not additional proved progress
Selected literature; source observations and coverage limits are retained in the publication details.
Contributors
The 212 mathematical paper credits nine authors and builds on Julia Stadlmann. AxiomProver is credited for its formal certificate, not independently discovering that bound.
François Charton · Letong Hong · Kenny Lau · Ken Ono · Guillaume Remy · Ho Chung Siu · Ashvin A. Swaminathan · Jesse Thorner · Yunzhou Xie
The nine authors supply the mathematics, building on Julia Stadlmann and earlier sieve work. AxiomProver is credited with generating a formal deduction; this is not a claim that the model independently discovered the mathematical bound.
The project is an OpenAI formal development. Its explicit assumptions and certificate work are recorded separately from an unconditional mathematical improvement.
These are source credits; they do not imply OpenMath accounts, endorsements or independent mathematical review.