The H=40 Equation: Descent, Rank-Equality Obstructions, and Galois Descent of Rosati-Symmetric Classes

We give a rigorous status report on the primitive integer equation which is the equation customarily designated in the relevant open-problem list. The first descent reduces every primitive solution to a rational point on one of six explicit genus-two curves. Four of these curves are resolved by fake -descent and classical Chabauty, leaving two quadratic twists where is an explicit sextic. Their Jacobians have Mordell–Weil rank equal to the genus, so ordinary Chabauty is at its boundary. A singleton fake-Selmer computation yields a common quartic cover over , but the associated elliptic-Chabauty problem is likewise at its rank boundary. We then describe exact elliptic quotient maps with -invariant , correct the geometric automorphism group from a dihedral subgroup to the exceptional order- group, and compute the genuine Galois action on the Rosati-fixed subspace. The field of one lifted involution has degree and no quadratic subfield; its normal closure has degree and Galois group . The six involution lines span the full three-dimensional non-scalar Rosati-fixed subspace. Over they remain a single signed Galois orbit whose Reynolds average vanishes. Consequently so the only descended Rosati-symmetric class is the polarization. This rules out the specific quadratic-Chabauty strategy over that depends on an additional Néron–Severi class. It does not solve and does not prove the equation inaccessible to other methods. We conclude with the remaining viable approaches and a reproducibility standard.