On the irrationality of ζ₃(5)

We prove that the Kubota–Leopoldt 3-adic zeta value \(ζ_3(5)\) is irrational. In the published literature the only individual 3-adic zeta value known to be irrational was ζ₃(3), and results for larger s only show that many values in a range are irrational. This preprint isolates the next value and proves ζ₃(5) itself irrational. The proof constructs Hankel determinants from moments of an explicit Volkenborn integral whose values lie in \(Q+Qζ_3(5)\). A quadratic 3-adic valuation bound, an archimedean logarithmic-energy estimate, and explicit control of the denominators through an exact rational certificate yield the required contradiction, while nonvanishing follows from strict \(p\)-adic dominance at a suitable auxiliary prime.