We report on a spectral-timing analysis of the neutron star low-mass X-ray binary Aql~X-1 with the Neutron Star Interior Composition Explorer (NICER) on the International Space Station. Aql~X-1 was observed with NICER during a dim outburst in 2017 July, collecting approximately $50$ ks of good exposure. The spectral and timing properties of the source correspond to that of an (hard) extreme island state in the atoll classification. We find that the fractional amplitude of the low frequency ($<0.3$ Hz) band-limited noise shows a dramatic turnover as a function of energy: it peaks at 0.5 keV with nearly 5% rms, drops to $12%$ rms at 2 keV, and rises to $15%$ rms at 10 keV. Through the analysis of covariance spectra, we demonstrate that band-limited noise exists in both the soft thermal emission and the power-law emission. Additionally, we measure hard time lags, indicating the thermal emission at $0.5$ keV leads the power-law emission at 10 keV on a timescale of $sim100$ ms at $0.3$ Hz to $sim10$ ms at $3$ Hz. Our results demonstrate that the thermal emission in the hard state is intrinsically variable, and driving the modulation of the higher energy power-law. Interpreting the thermal spectrum as disk emission, we find our results are consistent with the disk propagation model proposed for accretion onto black holes.
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