We extend the conductor rigidity framework to prime quadruplets (P, P+2, P+6, P+8) by constructing a genus-4 hyperelliptic curve C_P : y² = x(x²−P²)(x²−(P+2)²)(x²−(P+6)²)(x²−(P+8)²). We prove four unconditional results: (1) the odd involution x → −x induces an order-4 automorphism over K = Q(√−1) whose eigenspace decomposition yields, by the Kani-Rosen theorem, an isogeny Jac(C_P) ⊗ K ~ A × A^σ for a 2-dimensional abelian variety A over K; (2) the discriminant contains five multiplicative conduits at P+1, P+3, P+4, P+5, P+7, determined by the combinatorics of the admissible pattern (0,2,6,8), with a double conduit of toric rank 2 at P+4 where two independent root pairs collide; (3) the 8-dimensional ℓ-adic Galois representation factors through the Weil restriction Res_{K/Q}(A); (4) the Sato-Tate group is confined to (USp(4) × USp(4)) ⋊ Z/2 ⊂ USp(8), a proper subgroup of codimension 16. We conjecture that the five-fold descent obstruction, constrained by the Asai transfer structure, provides a representation-theoretic explanation for the rarity of prime quadruplets as predicted by the Hardy-Littlewood singular series. This is the sixth and final paper in a series. See also:[1] R. Chen, "Conductor Incompressibility for Frey Curves Associated to Prime Gaps," Zenodo, 2026. https://zenodo.org/records/18682375[2] R. Chen, "Density Thresholds for Equidistribution in Prime-Indexed Geometric Families," Zenodo, 2026. https://zenodo.org/records/18682721[3] R. Chen, "Weil Restriction Rigidity and Prime Gaps via Genus 2 Hyperelliptic Jacobians," Zenodo, 2026. https://zenodo.org/records/18683194[4] R. Chen, "On Landau's Fourth Problem: Conductor Rigidity and Sato-Tate Equidistribution for the n²+1 Family," Zenodo, 2026. https://zenodo.org/records/18683712[5] R. Chen, "The 2-2 Coincidence: Conductor Rigidity for Primes in Arithmetic Progressions and the Bombieri-Vinogradov Barrier," Zenodo, 2026. https://zenodo.org/records/18684151