Landau's fourth problem asks whether there are infinitely many primes of the form n²+1. We transpose the problem into arithmetic geometry by constructing the Landau-Frey elliptic curve Eₙ: y² = x³ − 2nx² + (n²+1) x. We establish three unconditional results. (1) The discriminant Δ (Eₙ) = −64 (n²+1) ² rigidly encodes the target value: if P = n²+1 is an odd prime, then Eₙ has multiplicative reduction at P with ordP (Δₘin) = 2 (Kodaira type I₂). (2) The conductor exponent at P is incompressible: level lowering modulo an auxiliary prime ℓ can remove P only when ℓ = 2, but the mod-2 representation is reducible (due to Q (√−1) -rational 2-torsion), blocking all level-lowering attempts. (3) Over the function field Fq (t), the associated elliptic surface has SL (2) geometric monodromy, and Deligne's equidistribution theorem yields unconditional Sato-Tate distribution with error O (q^−1/2). The "2-2 coincidence" (valuation 2 forces ℓ = 2, which is neutralized by reducibility) is structurally identical to the twin prime case, suggesting a universal geometric signature of the parity barrier for polynomial prime problems. This is the fourth 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/18683194
Ruqing Chen (Wed,) studied this question.