Paramodular Conjecture for the Goldbach–Frey Jacobian: From GSp(4) to Bianchi Modularity via Weil Restriction

The Brumer-Kramer paramodular conjecture predicts that every abelian surface over Q with End_Q(A) = Z and conductor N corresponds to a weight-2 Siegel paramodular newform of level N. We apply this conjecture to the Goldbach-Frey Jacobian Jac(CN,p) from our companion paper, where CN,p: y² = x(x²−p²)(x²−(2N−p)²). We prove three unconditional results: (1) the explicit conductor of Jac(CN,p) from the discriminant Δ = 2¹² p⁶(2N−p)⁶(N−p)⁴N⁴, with f_r = 2 at both static and dynamic conduit primes (two nodes on an irreducible curve give b₁ = 2 and local Euler factor (1−r⁻ˢ)⁻²); (2) the Weil restriction structure Jac ~ ResK/Q(E_p) reduces the paramodular conjecture to the modularity of E_p over K = Q(√−1) via the Asai transfer, with symplectic descent guaranteed by the exterior square L-function criterion of Jacquet-Shalika; (3) the residual representation ρ̄E_p,2 is universally reducible (the 2-2 coincidence: all 2-torsion is Q-rational), while ρ̄E_p,3 is generically absolutely irreducible, identifying ℓ = 3 as the first prime where the ten-author modularity lifting theorem may apply. For fixed N, the paramodular conjecture holds for Jac(CN,p) for all but finitely many Goldbach primes p (Corollary 5.2). The paper does not prove the Goldbach conjecture; it provides a bridge from GSp(4) to GL(2)/K where existing tools apply.