Primes of the Form n^2+1 on the 6N Skeleton: the Wing Decomposition and the Character Spectrum of the Landau–Shanks Constant

Every constellation this series has measured so far — twin pairs, Goldbach sums, arithmetic progressions — is a linear translate of the prime set. We turn to the first nonlinear sequence: primes of the form n^2+1, the subject of Landau's fourth problem. Using a square-root sieve on n (marking n ≡ ±r mod q where r^2 ≡ −1), we enumerate all n ≤ 3×10^7 with n^2+1 prime — 1,275,229 of them, with n^2+1 as large as 9×10^14 — and read the result through the 6N lattice. Three findings. (i) Wing decomposition: n must be even (odd n annihilates n^2+1 into the even residue class), and among even n, 6 | n sends n^2+1 to the right break-zone 6N+1 while n ≡ 2,4 (mod 6) sends it to the left zone 6N−1. We measure the split as right:left = 0.50015, exactly the 1:2 of pure counting: the n^2+1-prime density per channel is identical on the two wings, so the quadratic deformation induces no Chebyshev-type bias. (ii) Character spectrum: the local factor g(q) = (1 − ω(q)/q)/(1 − 1/q) splits the odd primes by the quadratic character of −1. Primes q ≡ 3 (mod 4) never divide n^2+1 and enhance it by q/(q−1) > 1; primes q ≡ 1 (mod 4) divide it on two residues and suppress it by (q−2)/(q−1) < 1. (iii) The product of these factors is the Landau–Shanks constant C = 1.37281, and it controls the count: the measured Q(N) / ∫ dt/log(t^2+1) equals 1.37253, matching C to 0.02%, with the empirical constant converging onto C as N grows. We verify the nonlinear (degree-two) Bateman–Horn heuristic to better than a part in 10^3 at n^2+1 ~ 10^15. None of this is new as theory — ω(q), the constant C, and the asymptotic are due to Hardy–Littlewood, Bateman–Horn, and Shanks. What this paper adds is the 6N wing decomposition, the character-spectrum reading of the local factors, and a high-precision verification via the square-root sieve. We make no claim about the infinitude of such primes: Landau's problem is the backdrop, not a target. As throughout this series, this is a measurement, not a theorem; the Hardy–Littlewood heuristic is taken as input. Part XXXI of "Arithmetic Geodynamics on the 6N Skeleton." Code and measured data: https://github.com/Ruqing1963/6N-n2plus1