Prime Quintuplets and the Sextuplet Boundary for Prime Quintuplets and the Sextuplet Boundary for Q (n) = n⁴7 - (n-1) ⁴7: Extreme Prime Clusters in a High-Degree Polynomial: Extreme Prime Clusters in a High-Degree Polynomial

Among the 742 prime quadruplets for Q(n) = n⁴⁷ − (n−1)⁴⁷ over 1 ≤ n ≤ 2×10¹¹, exactly 7 extend to prime quintuplets (five consecutive integers generating probable primes of 487–519 digits), while no sextuplet was found. We prove that Q(n) has a bifurcated root structure: ω₁(p) = 0 for primes p ≢ 1 (mod 47), but ω₁(p) = 46 for all resonant primes p ≡ 1 (mod 47). This periodic sieve obstruction produces dramatic "cliffs" in the Bateman–Horn Euler product, the first at p = 283. Computing the corrected singular series to sieve bound B = 10,000 yields 𝔖₅ ≈ 57,100 and 𝔖₆ ≈ 520,000. The predictions EC₅ = 5.83 and EC₆ = 0.047 agree with observation to within Poisson noise. The sextuplet boundary is N* ≈ 1.05 × 10¹³, requiring a 52-fold search extension. Part III of the Titan Project.