Field-Free Transverse Aharonov--Bohm Phase Gate for an Orbital $l$-Qubit

Free-space Aharonov--Bohm (AB) Bessel modes are known to carry a flux-dependent azimuthal Schr\"odinger probability current and kinetic orbital angular momentum. We extend this response to the spin-resolved conserved Dirac current of a core-excluded finite-wall annular guide and show that confinement converts its surviving orbital component into a transverse Aharonov--Bohm (TAB) propagation phase for a straight traveling mode whose centroid path has zero projection onto $\mathbf A$. The radial-gradient current reverses under spin reversal; retaining the complete evanescent tail closes it as a boundary contribution, leaving a spin-independent orbital phase $\Delta\phi_{ln}\propto l\Phi L_{\rm int}\langle\rho^{-2}\rangle_{ln}/v_z$. Opposite-winding modes $|\pm l\rangle$ form a same-path qubit implementing $R_z(2\delta_l)$ with differential readout and common-mode phase rejection, while direct first-order crosstalk requires the angular harmonic $m=\pm2l$. For a $100\,\mu\mathrm{m}$ section with $a=20\,\mathrm{nm}$, $R=30\,\mathrm{nm}$, $E_z=10\,\mathrm{meV}$, and $|l|=10$, the sensitivity is $0.1885\,\mathrm{mrad/mG}$ and $R_z(\pi)$ occurs at $16.67\,\mathrm{G}$. Finite-wall confinement thus turns intrinsic azimuthal Dirac current into a guided field-free phase operation.