Hasse–Witt obstructions for the Aₙ root polytopes

The Beukers–Vlasenko theory of Dwork crystals derives Frobenius structures and p-adic zeta values for Calabi–Yau families whose Newton polytopes have simplicial faces of volume one. The Aₙ root polytopes of the equal-mass banana amplitudes violate that hypothesis because their two-dimensional faces include squares, and I prove that the violation is essential. For every prime p ≥ 5 and every n ≥ 3 the second Hasse–Witt determinant of the full polytope has p-adic valuation exactly L(2) + C(n+1, 2)·C(n−1, 2), one power of p above the Beukers–Vlasenko baseline for each square face, so the k-th Hasse–Witt condition fails for 2 ≤ k < p, and the interior crystal fails at level three by the same count. The excess follows a closed form through level four except at p = 5, and depends on the prime thereafter. I locate the failure in the square-center forms, show that they are exact p³-eigenvectors of the local Cartier operator, with eigenvalue C(3p−3, 2p−2)·C(2p−2, p−1)², prove a corrected decomposition theorem that survives the failure modulo a quotient killed by p, and show that the strong decomposition holds when the Cartier image of the crystal equals that of level k modulo pᵏ, verified at depth one at three primes. For A₃ this yields a theorem conditional on the depth-∞ membership of the corrected center forms, proved at depth one at four primes in the companion paper "The three-loop banana crystal: rank one, the window theorem, and depth one" (DOI 10.5281/zenodo.22135653), on an independence hypothesis, and on irreducibility of the Picard–Fuchs operator. What is obstructed is a sufficient criterion, not its conclusion.