The three-loop banana crystal: rank one, the window theorem, and depth one

The companion paper "Hasse–Witt obstructions for the Aₙ root polytopes" (DOI 10.5281/zenodo.22135651) shows that the Hasse–Witt conditions fail for those polytopes, and the sufficient criterion of Beukers and Vlasenko therefore does not apply to the banana family and their level-two Frobenius lift does not reach the constants. This paper builds the replacement for three loops. I determine the p-adic structure of the full Cartier matrix beyond its sharp layer, prove a rank-one structure theorem modulo p, and develop a single-term window calculus in which each pairing collapses to one term. The window annihilation theorem is then proved unconditionally at every prime p ≥ 5, by reducing a chain of exact statements to one identity over ℚ that is proved in full. Finally I prove that the layered depth-one system T(1, L) is solvable for every layer at p = 5, p = 7, p = 11 and p = 13, by a finite certified fixed-point computation whose exactness rests on the double-annihilator duality of finite abelian p-groups and whose stabilization is certified by module equality. The depth-one half of the main conjecture is therefore a theorem at the target prime. Falsified intermediate hypotheses are reported as negative results throughout because the proofs are shaped by their failure.