We construct a genus-2 hyperelliptic curve Cₚ: y² = x (x²−p²) (x²− (p+2) ²) whose Jacobian yields a 4-dimensional Galois representation in GSp (4). The odd involution x → −x forces Jac (Cₚ) to split over Q (√−1) into conjugate elliptic curves E₁ × E₁^σ, making Jac (Cₚ) the Weil restriction Res₊/ₐ (E₁) — absolutely simple over Q with indecomposable L-function L (s, E₁/K). In the Langlands framework, the associated GSp (4) automorphic form is the automorphic induction of a Bianchi modular form over GL (2, OK), fusing the parameters p and p+2 into a single automorphic object. We prove four unconditional results: (1) discriminant entanglement with explicit factorization, (2) the additive/multiplicative reduction dichotomy at p, p+1, p+2, (3) Jacobian splitting via the Kani-Rosen theorem, and (4) absolute simplicity with Weil restriction structure. We formulate the Weil Rigidity Conjecture: this indecomposable structure, combined with the multiplicative conduit at p+1, imposes strict constraints on admissible prime gaps. Numerical evidence is provided via two large prime pairs discovered in the algebraic family Q₄₇ (n) = n⁴⁷− (n−1) ⁴⁷: a 484-digit sexy prime pair (gap 6) and a 490-digit twin prime pair (gap 2), confirming that gaps are bounded, not forbidden. Code repository: https: //github. com/Ruqing1963/weil-restriction-prime-gaps This is the third 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/18682721
Ruqing Chen (Wed,) studied this question.