A Unified Multi-Band Secular Framework for the Hierarchical Triple PSR J0435+3233 System

The millisecond pulsar PSR J0435+3233, discovered by FAST, exhibits an unusually large apparent spin-down rate, potentially reflecting line-of-sight acceleration from a distant companion. Recent investigations reveal a hierarchical triple consisting of an inner pulsar-helium white dwarf binary with an orbital period of 8 days, orbited by a G-type subgiant tertiary on an eccentric 74 yr orbit. The current timing description admits two distinct geometrical solutions. We develop a singly averaged (SA) formulation that averages over the inner orbit while retaining the instantaneous phase of the eccentric outer orbit, thereby capturing the time-dependent quadrupolar perturbations induced by the tertiary. We show how this physical three-body description can be incorporated naturally into pulsar-timing analyses and use it to forward-propagate the two currently viable configurations, Tv1 and Tv2, to forecast when their timing signatures become distinguishable with FAST. The SA evolution predicts that the two configurations will become distinguishable in the coming years, with the effective projected semi-major axis, x 1, eff, providing the earliest discriminating observable. We further extend the framework to predict the tertiary's spectroscopic radial-velocity and astrometric evolution, linking FAST timing with LAMOST time-domain spectroscopy and Gaia astrometry. This multi-band framework has the potential to establish PSR J0435+3233 as an exceptionally well-constrained laboratory for three-body dynamics, compact-binary and stellar evolution, and gravitational physics.