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

We construct the Goldbach-Frey curve C₍, ₏: 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₍, ₏. (2) The Kani-Rosen splitting yields Jac (C₍, ₏) ⊗ K ~ Eₚ × Eₚ^σ for an elliptic curve Eₚ over K = Q (√−1), so the Jacobian is a Weil restriction Res₊/ₐ (Eₚ). (3) At any odd prime r dividing N with r ∤ p (2N−p) (N−p), the curve has multiplicative reduction with ordᵣ (Δ) = 4·ordᵣ (N), and this local conductor data is uniform across the entire family — independent of p. The static conduit factor ∏ₑ|₍, ₑ>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/186823752 R. Chen, "Density Thresholds for Equidistribution in Prime-Indexed Geometric Families, " Zenodo, 2026. https: //zenodo. org/records/186827213 R. Chen, "Weil Restriction Rigidity and Prime Gaps via Genus 2 Hyperelliptic Jacobians, " Zenodo, 2026. https: //zenodo. org/records/186831944 R. Chen, "On Landau's Fourth Problem: Conductor Rigidity and Sato-Tate Equidistribution for the n²+1 Family, " Zenodo, 2026. https: //zenodo. org/records/186837125 R. Chen, "The 2-2 Coincidence: Conductor Rigidity for Primes in Arithmetic Progressions and the Bombieri-Vinogradov Barrier, " Zenodo, 2026. https: //zenodo. org/records/186841516 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