We extend the conductor rigidity framework to prime quadruplets (P, P+2, P+6, P+8) by constructing a genus-4 hyperelliptic curve CP: 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 (CP) ⊗ 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₊/ₐ (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/186823752 R. Chen, "Density Thresholds for Equidistribution in Prime-Indexed Geometric Families, " Zenodo, 2026. https: //zenodo. org/records/186827213 R. Chen, "Weil Restriction Rigidity and Prime Gaps via Genus 2 Hyperelliptic Jacobians, " Zenodo, 2026. https: //zenodo. org/records/186831944 R. Chen, "On Landau's Fourth Problem: Conductor Rigidity and Sato-Tate Equidistribution for the n²+1 Family, " Zenodo, 2026. https: //zenodo. org/records/186837125 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
Ruqing Chen (Wed,) studied this question.