A Poisson Limit Model for Landau's First Problem via Bounded-Variance Multi-Residue Sieves

We establish a rigorous Poisson limit model for the distribution of primes of the form $n^2+1$ (Landau's First Problem). Adapting a bounded-variance multi-residue sieve framework, we model the surviving residue classes that generate OEIS sequence A005574. We overcome the challenge of a fluctuating sieve dimension—where only primes $p \equiv 1 \pmod 4$ contribute active exclusions—by embedding the Legendre symbol directly into the single-event density. Furthermore, we bound the local sieve variance by demonstrating that overlaps between shifted quadratic roots are strictly finite and governed by elementary Diophantine constraints.MSC (2020): 11N05 (primary), 11N36 (secondary)