Prime Constellations of nᵏ − (n−1)ᵏ: Algebraic Obstructions, Bateman–Horn Verification, and the Non-Monotonicity of Maximum Constellation Lengths

We investigate the polynomial family Qₖ(n) = nᵏ − (n − 1)ᵏ for prime exponents k = 3, 5, 7, 11, 13, enumerating all prime values for n ≤ 10⁸ and searching for consecutive prime constellations up to n ≤ 2 × 10⁹. Main results: (1) The maximum constellation length L(k) is determined by the longest gap in the forbidden residue set modulo the smallest splitting prime p₀(k) ≡ 1 (mod k), yielding L(3) = 3, L(5) = 6, L(7) = 17, L(11) = 4, L(13) = 8. All five bounds are verified computationally with zero exceptions. (2) The Bateman–Horn conjecture is confirmed to 0.2% accuracy for polynomial degrees 4, 6, 10, and 12, with the Euler product C(k) converged to ≤ 0.03% precision over all primes p ≤ 10⁷. (3) L(k) is non-monotonic in k, controlled by the forbidden density ρ(k) = (k − 1)/p₀(k). (4) Residue locking: all 239 quadruplets for k = 11 are confined to exactly 2 of 23 residue classes modulo p₀ = 23. (5) A unique septuplet is found at n = 77,664,241 for k = 7, matching the Bateman–Horn prediction of 1.1 expected occurrences. Repository contents: LaTeX source and compiled PDF (14 pages, 6 figures, 20 references); computational scripts (segmented sieve, Bateman–Horn constant computation, algebraic verification, figure generation); summary data tables. MSC 2020: 11N32, 11N05, 11C08, 11Y35