The Goldbach Mirror: Conductor Rigidity and the Static Conduit in GSp(4)

We construct the Goldbach-Frey curve C_{N,p} : y² = x(x²−p²)(x²−(2N−p)²) for a fixed even integer 2N and prime candidate p. This genus-2 curve is the additive mirror of the twin prime curve: the sum p + (2N−p) = 2N is fixed, producing a static conduit at divisors of N. We prove three unconditional results. (1) The discriminant factors as Δ = 2¹² · p⁶ · (2N−p)⁶ · (N−p)⁴ · N⁴, with N⁴ appearing as a parameter-independent factor shared by every curve in the family {C_{N,p}}. (2) The Kani-Rosen splitting yields Jac(C_{N,p}) ⊗ K ~ E_p × E_p^σ for an elliptic curve E_p over K = Q(√−1), so the Jacobian is a Weil restriction Res_{K/Q}(E_p). (3) At any odd prime r dividing N with r ∤ p(2N−p)(N−p), the curve has multiplicative reduction with ord_r(Δ) = 4·ord_r(N), and this local conductor data is uniform across the entire family — independent of p. The static conduit factor ∏_{r|N, r>2} (r−1)/(r−2) corresponds precisely to the Hardy-Littlewood singular series for Goldbach representations, providing a geometric explanation for the banded structure of the Goldbach comet. The passage from this framework to a proof of the Goldbach conjecture remains open. Code repository: https://github.com/Ruqing1963/goldbach-mirror-conductor-rigidity This is the seventh paper in a series. See also:[1] R. Chen, "Conductor Incompressibility for Frey Curves Associated to Prime Gaps," Zenodo, 2026. https://zenodo.org/records/18682375[2] R. Chen, "Density Thresholds for Equidistribution in Prime-Indexed Geometric Families," Zenodo, 2026. https://zenodo.org/records/18682721[3] R. Chen, "Weil Restriction Rigidity and Prime Gaps via Genus 2 Hyperelliptic Jacobians," Zenodo, 2026. https://zenodo.org/records/18683194[4] R. Chen, "On Landau's Fourth Problem: Conductor Rigidity and Sato-Tate Equidistribution for the n²+1 Family," Zenodo, 2026. https://zenodo.org/records/18683712[5] R. Chen, "The 2-2 Coincidence: Conductor Rigidity for Primes in Arithmetic Progressions and the Bombieri-Vinogradov Barrier," Zenodo, 2026. https://zenodo.org/records/18684151[6] R. Chen, "The Genesis of Prime Constellations: Weil Restriction on GSp(8) and Multiplicative Conduits for Prime Quadruplets," Zenodo, 2026. https://zenodo.org/records/18684352