N^2+1 Chamber Lift: Quadratic Residue Sieve for Landau's Problem

This note gives an expository chamber-coordinate presentation of Landau's problem on primes of the form $n^2+1$. The square value $n^2$ is viewed as a diagonal base point and $n^2+1$ as its unit-square lift. The arithmetic obstruction is classical: for an odd prime $q$, divisibility $q\mid n^2+1$ is equivalent to $n^2\equiv -1\pmod q$, which is solvable precisely for primes $q\equiv 1\pmod 4$. Thus each such prime removes two residue classes of $n$ modulo $q$, while primes $q\equiv3\pmod4$ remove none. The Chamber Lift vocabulary used here is not a proof mechanism. It is an atlas for local obstruction, finite sieve survival, Gaussian-integer intuition, and the conjectural Bunyakovsky/Bateman-Horn picture surrounding Landau's open problem.