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 u[5m] ≡ 0 mod 5 is proved in closed form, and with it the Transfer theorem, its mod-125 form, 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 letters rational at every depth by a theorem of theirs read at the source, and the Cartier calculus built on them proves the first three faces of the tower that carries the integrality obligation of the record, with the theorem that the Cartier operator kills the form those faces have, so each level follows from the one before. 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 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 the cyclic symmetry of the five slots collapses the period.