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Lean formalization shows conditional proof that prime gaps are infinitely often ≤186
A Lean 4 formalization provides a conditional proof that there are infinitely many prime pairs with gaps no larger than 186.
Such a formalization shows how advanced analytic number theory results can be encoded in a proof assistant, offering a reusable framework for verifying other bounds. It also highlights the current reliance on unproven axioms, clarifying where formal guarantees exist and where assumptions remain for engineers working on cryptographic or algorithmic number-theory code.
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The formalization derives the bound liminf(p_{n+1}-p_n) ≤ 186 from the hypothesis that every admissible 40-tuple has infinitely many translates containing at least two primes.
It depends on three explicit axioms: Deligne-type Kloosterman sum bounds, a Kloosterman correlation bound, and physical-integral numerical estimates that are not proved in Lean.
A Python certificate reproduces the numerical work, while the Lean kernel checks the conditional proofs under those axioms.
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The Lean development encodes the statement that the limit inferior of consecutive prime gaps is at most 186. It does so by first proving DHL[40,2] for every admissible integer tuple from the assumed axioms. Then it applies this result to a specific tuple of diameter 186, yielding infinitely many translates with at least two primes. This chain of reasoning gives the desired bound on prime gaps.
The proof leans on three axioms that are taken from the literature but remain unverified in the Lean environment. Two axioms concern bounds on Kloosterman sums and their correlations, while the third supplies numerical upper bounds for certain physical integrals. A separate Python script recomputes those integrals and produces a certificate, but the script does not discharge the axioms themselves.
To verify the formal part, the project is built with Lean 4 and its Mathlib library; the kernel accepts the three main theorems assuming the axioms. The numerical certificate is run with a standard Python environment, using NumPy and a custom FLINT build, and must pass floating-point and signed-convolution checks. Successful runs produce a receipt indicating passed checks, confirming the numerical side of the work.
Adopting this result requires trusting the unverified axioms; if any of them fails, the conditional proof no longer guarantees the gap bound. The approach does not yield an unconditional improvement over known prime-gap results, nor does it resolve the twin prime conjecture. Engineers who rely on formal guarantees must therefore treat the bound as provisional pending further proof of the underlying assumptions.
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