A Boundary-Consistent Two-Zone Electron Kernel for Distant Pulsar Contributions to Positron Flux and Anisotropy

We present a semi-analytical series solution for electron and positron propagation in a spherical two-zone diffusion model. The solution treats slow diffusion inside a near-source region and standard interstellar diffusion outside it, while synchrotron and Klein--Nishina inverse-Compton cooling are included through energy characteristics. The formulation avoids the oscillatory cancellations of direct two-zone integral evaluations and preserves the sharp radiative cooling boundary seen in finite-volume checks. We apply the kernel to pulsar contributions to the local cosmic-ray lepton flux. Nearby pulsars remain natural candidates near the TeV cutoff, but at tens to hundreds of GeV the larger source volume allows more distant pulsars to contribute collectively: for a disk half-thickness of 0.2, kpc, sources beyond 1, kpc can still provide 37 -- 47% of the 10 -- 100, GeV flux. Comparing with AMS-02 positron data and all-electron anisotropy limits, and imposing an inner 100, pc cavity motivated by the Local Bubble and pulsar proper motions, we find that Geminga-scale slow-diffusion halos remain compatible with current data. The fitted pulsar component is dominated by sources beyond 0.3, kpc, but flux and anisotropy data alone do not uniquely determine the halo size; external information such as TeV halo morphology is still required.