Secular Evolution of PSR J2021+4026: Long-Term γ-Ray Flux and Spin-Down Variability Beyond State Transitions

Abstract: PSR J2021+4026 is a remarkable $γ$-ray pulsar exhibiting repeated transitions between high $γ$-ray flux (HGF) and low $γ$-ray flux (LGF) states. With 17-yr Fermi-LAT monitoring, we reveal persistent secular evolution and enhanced spin-down rate variability within individual emission states -- beneath the quasi-periodic state transitions. After removing discrete jumps, the jump-corrected flux $δF_γ$ shows a three-phase evolution: rise ($+2.02^{+0.17}_{-0.15}\%~\mathrm{yr}^{-1}$), decline ($-3.72^{+0.34}_{-0.47}\%~\mathrm{yr}^{-1}$), and rapid rise ($+14.9^{+6.4}_{-4.4}\%~\mathrm{yr}^{-1}$), with all rates quoted relative to the long-term mean flux $\langle F_γ\rangle=7.8\times 10^{-10}\,\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. Moreover, the flux of the LGF state is gradually approaching the stable HGF level at a rate of $+0.72 \pm 0.11\%~\mathrm{yr}^{-1}$. These results demonstrate that secular flux evolution in PSR J2021+4026 operates largely independently of discrete state transitions, yet jointly with them drives the system toward a stable high-flux equilibrium.