Necessity of a Constitutive Dirac Fermion for BPS Closure in a U(1) Higgs–Einstein–Cartan Model on Hyperbolic Space H4

We consider a U(1) Higgs gauge theory minimally coupled to Einstein–Cartan gravity on the hyperbolic space H4 (sectional curvature K = −1, Ricci scalar R = −12). Imposing the BPS vortex condition together with a one-loop renormalization group fixed point for the quartic coupling λ, we prove that no real solution exists in the purely bosonic sector: the one-loop β-function contribution from the torsion vertex is positive-definite, making simultaneous BPS saturation and RG fixed-point incompatible. The addition of a single four-component Dirac fermion (constitutive fermion) with Yukawa coupling y and U(1) charge resolves this incompatibility uniquely, yielding closed-form constitutive relations among all couplings. The unique solution is κ2v2 = 1, λ = g2/10, κ2 = g2/5, v2 = 5/g2, y4 = 9/4 + 147 g4/800. We further show that a Weyl spinor generates a U(1) gauge anomaly and a Majorana spinor is incompatible with the required U(1) charge, establishing that the Dirac spinor is the unique consistent fermionic extension at one loop.