Refined Constraints on the Hard X-ray Polarization of the Crab Pulsar and Nebula Derived from an Extended XL-Calibur Dataset

We present updated hard X-ray polarization measurements of the Crab pulsar and nebula obtained with the balloon-borne polarimeter XL-Calibur in the ~19-64 keV energy range. During the flight, intermittent GPS-failure resulted in poorly constrained timing for ~38% of the Crab dataset. By implementing a new phase-recovery method that reconstructs timing during extended GPS-off intervals, phase tag data is recovered for ~95% of the GPS-off dataset, increasing the precision of the phase-resolved analysis. Phase-information for the data is recovered by using the Crab pulsar, with its 33 ms period, as an external timing source. Using a Markov-Chain Monte-Carlo framework to jointly fit phase offsets and frequency derivatives, sufficient phase accuracy is achieved, across multiple periods without GPS for a phase-resolved analysis. This enables inclusion of nearly the full dataset in the polarization study. The polarization degree of the nebular emission is found to be (27.7${\pm}$4.9)% at a polarization angle of 127.2{\deg}${\pm}$5.1{\deg} confirming previous XL-Calibur results and remaining aligned with the Crab's spin axis, consistent with synchrotron emission from the inner nebula. Phase-resolved measurements show that the off-pulse and bridge intervals exhibit a strong polarization, while the pulsar peaks, although weakly constrained, remain in agreement with the softer-energy trends of IXPE. These findings reinforce a scenario in which hard X-ray emission arises primarily in the nebular torus and wind regions. The successful recovery of precise phase tagging from GPS-off data demonstrates the capacity to use the pulsar as an external clock even in the case of sparsely populated data.