Prime Gaps at Most 186

Researchers have formalized a mathematical proof in Lean 4 demonstrating that the gap between consecutive prime numbers is at most 186, building on the Deligne-type Kloosterman sum estimates. The formalization derives the result DHL[40,2]—stating that every admissible set of forty integer shifts contains infinitely many translates with at least two primes—and applies it to a specific tuple of diameter 186 to establish the prime gap bound. The proof remains conditional on three explicit mathematical axioms related to Kloosterman sum bounds that have been validated numerically but not yet fully formalized.