I will present in detail a new twist in the subject of arithmetic algebraization theorems. It comes out of a joint work in progress with Frank Calegari and Yunqing Tang on irrational periods, and bears also a relation to a variation by Zudilin around the classical Polya-Bertrandias determinantal criterion for the rationality of a formal function on the projective line. Time permitting, I will sketch an application to an irrationality proof of the 2-adic avatar of $\zeta(5)$.