The 2-2 Coincidence: Conductor Rigidity for Primes in Arithmetic Progressions and the Bombieri-Vinogradov Barrier

We construct the EH-Frey curve E_{p,a} : y² = x(x−p)(x−a) to encode the condition p ≡ a (mod q) and prove a 2-2 coincidence theorem: for the generic case ord_q(p−a) = 1, the minimal discriminant satisfies ord_q(Δ_min) = 2, the curve has multiplicative reduction of Kodaira type I_2, and level lowering is possible only for ℓ = 2 — where the mod-2 representation is reducible due to full rational 2-torsion. This is the same conductor incompressibility mechanism identified in our companion papers on twin primes and Landau's problem, establishing it as a recurring geometric signature of the parity barrier across additive prime problems.

We then prove the Bombieri-Vinogradov theorem over the function field F_r(t) with the sharp exponent θ = 1/2, arising directly from the Riemann Hypothesis for curves over finite fields (Weil's theorem). The exponent 1/2 is not an artifact of sieve methods but the exact position of L-function zeros on the critical line.

Together, these results identify a double dead end: the 1/2 barrier is a GL(1) phenomenon (weight-1 purity of Dirichlet L-functions), while the 2-2 coincidence is a GL(2) phenomenon (conductor incompressibility of Frey curves). These are logically independent obstructions at different automorphic ranks that together insulate the Elliott-Halberstam conjecture from all classical methods.

This is the fifth 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