Dirac Fermion Scattering and Pseudospin Polarization in Structurally Asymmetric Graphene Wormholes

We study the quantum transport of massless Dirac fermions through two asymptotically flat graphene sheets connected by a structurally asymmetric catenoid wormhole in $(2+1)$-dimensional curved spacetime. Analytic scattering basis functions are derived: Hankel functions of integer order (in the half-flux sector) in the flat sheets and Gauss hypergeometric functions in the curved throat. We construct a transfer matrix via piecewise numerical matching, verifying unitarity up to numerical precision. The transmission probability rises monotonically to unity at high energies. Global transmission exhibits mirror degeneracy under inversion of structural asymmetry, but local observables depend on incidence direction. The manifold's spin connection acts as a Hermitian coupling inducing an $A/B$ sublattice imbalance at the throat. Structural asymmetry induces a local pseudospin imbalance. A larger curvature radius enhances $P_z$ polarization via a larger geometric phase; abrupt incidence suppresses it. Sub-barrier modes exhibit a negative transmission phase time, compatible with Hartman-type wave-packet reshaping.