An understanding of spin frequency ($ u$) evolution of neutron stars in the low-mass X-ray binary (LMXB) phase is essential to explain the observed $ u$-distribution of millisecond pulsars (MSPs), and to probe the stellar and binary physics, including the possibility of continuous gravitational wave emission. Here, using numerical computations we conclude that $ u$ can evolve in two distinctly different modes, as $ u$ may approach a lower spin equilibrium value ($ u_{rm eq,per}$) for persistent accretion for a long-term average accretion rate ($dot{M}_{rm av}$) greater than a critical limit ($dot{M}_{rm av,crit}$), and may approach a higher effective spin equilibrium value ($ u_{rm eq,eff}$) for transient accretion for $dot{M}_{rm av} < dot{M}_{rm av,crit}$. For example, when $dot{M}_{rm av}$ falls below $dot{M}_{rm av,crit}$ for an initially persistent source, $ u$ increases considerably due to transient accretion, which is counterintuitive. We also find that, contrary to what was suggested, a fast or sudden decrease of $dot{M}_{rm av}$ to zero in the last part of the LMXB phase is not essential for the genesis of spin-powered MSPs, and neutron stars could spin up in this $dot{M}_{rm av}$-decreasing phase. Our findings imply that the traditional way of $ u$-evolution computation is inadequate in most cases, even for initially persistent sources, and may not even correctly estimate whether $ u$ increases or decreases.
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