The 5-Integrality of the Four-Loop Equal-Mass Banana Integral

The ε-factorized differential equation of the four-loop equal-mass banana integral is conjectured to be free of the prime 5 at every order. I study the first leg as a tower of power series under the operator U₅ and prove a reduction theorem, by which the full tower of congruences follows from one growth statement about an explicitly computed series. I exhibit an exact rational fixed point of the tower's transport map and measure where rational fits fail. On a build twenty-five times deeper than the first version's, with per-index precision certificates, P(5) = 17. The first version's plateau was a precision artifact, and the depth profile is the exact line P(k) = 3k + 2 through depth seven on 78,126 certified rows. The second leg's census reads Q(k) = 2k exactly through depth eight, and the divided de Rham Frobenius modulo 5 is an explicit polynomial matrix. The operator is globally nilpotent with bad primes 2, 3 and 5, and its p-curvature is nilpotent at every prime checked from 7 to 47, so the conjecture is the single unsettled prime of an otherwise closed question. One measurement outside the conjecture is recorded, the modulo p²ˢ upgrade of Beukers and Vlasenko, holding in every case tested.