The 5-Integrality of the Four-Loop Equal-Mass Banana Integral, Part II: The Criterion and the Letters of the Excellent Lift

The conjecture called no-5 says that the ε-factorized differential equation of the four-loop equal-mass banana integral is free of the prime 5 at every order. Part I is the measurement record. Here I prove theorems. The Wronskian of the underlying Hulek–Verrill operator is determined modulo 5 at all orders by an explicit Mahler product with alphabet P = 1 + 3t + 3t³ + 4t⁴, the determinant digit of the rank-two Hasse–Witt matrix. The criterion, the congruence u[5m] ≡ 0 mod 5 on which the first 5-adic level turned, is proved in the closed form u ≡ x₅(t⁵)(3t + t³ + t⁴) mod 5, and with it the Transfer theorem W[25n] ≡ W[n] + 5n u[5n] mod 25, its mod-125 form with every correction term explicit, and the Operator form become theorems about one another. The level-125 splitting fails because the third Hasse–Witt invariant is a non-unit at 5. The excellent Frobenius lift of Beukers and Vlasenko is determined digit by digit, its first four letters are explicit rational functions, and the Cartier calculus built on them proves the first three faces of the tower that carries the integrality obligation of the record. That obligation is the integrality of the horizontal Frobenius of one explicit extension of ∧²M by a Tate-twisted trivial crystal, verified to order 650, and the two remaining open statements are functionals of the same six series. An impossibility theorem shows that no finite-digit invariant closes any of them. The prime is 5 because of the cyclic symmetry of the five propagators.