On Landau's Fourth Problem: Conductor Rigidity and Sato-Tate Equidistribution for the n²+1 Family

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 E_n : y² = x³ − 2nx² + (n²+1)x.

We establish three unconditional results. (1) The discriminant Δ(E_n) = −64(n²+1)² rigidly encodes the target value: if P = n²+1 is an odd prime, then E_n has multiplicative reduction at P with ord_P(Δ_min) = 2 (Kodaira type I_2). (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 F_q(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/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