The Odlyzko-Poonen conjecture states that a random monic polynomial with constant coefficient 1 and other coefficients uniform in {0,1} is irreducible over Q with probability tending to one as the degree grows. Earlier proofs were conditional on the Generalized Riemann Hypothesis or covered other coefficient distributions.

Any factorization P=AB yields a twin polynomial Q=AB* with QQ*=PP* and Q also a 0-1 polynomial. The core new step bounds the probability that a nontrivial twin satisfies PP*≡QQ* mod 4. It does this by conditioning on sums of opposite coefficients and revealing coefficients sequentially. Reciprocal non-cyclotomic factors are handled separately by combining a separation argument of Breuillard-Varjú with the same conditioning. The full proof is formalized in Lean.

The conjecture is proved unconditionally, with P(reducible) = sqrt(2/(πn)) + O(n^{-1}). With probability 1-Ce^{-c√n}, the polynomial factors as a cyclotomic part times an irreducible noncyclotomic part, which strengthens a previously GRH-conditional result.