ζ(5) is irrational

We prove that ζ(5) is irrational. The proof constructs integer polynomials Qₙ of degree 37n with 0 < Qₙ(ζ(5)) < exp(−139n²/5) for all sufficiently large n. These polynomials are rationally normalized Hankel determinants. A positive moment representation gives nonvanishing and, through a logarithmic-energy estimate, bounds their real values. Local estimates for the entire rational functional control the coefficients and show that the normalization preserves the required decay. A perturbation estimate also gives |ζ(5) − a/b| > b⁻²⁶⁰ for every integer a and all sufficiently large positive integers b. Version note: This Zenodo record preserves the initial public preprint version. Subsequent revisions will be maintained on arXiv once the paper is announced there.