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Fast variability of the X-ray corona in black hole binaries can produce asoft lag by reverberation, where the reprocessed thermalized disc photons lagbehind the illuminating hard X-rays. This lag is small, and systematicallydecreases with increasing mass accretion rate towards the hard-soft transition,consistent with a decreasing truncation radius between the thin disc and X-rayhot inner flow. However, the soft lag suddenly increases dramatically justbefore the spectrum becomes disc-dominated (hard-intermediate state).Interpreting this as reverberation requires that the X-ray source distance fromthe disc increases dramatically, potentially consistent with switching toX-rays produced in the radio jet. However, this change in lag behaviour occurswithout any clear change in hard X-ray spectrum, and before the plasmoidejection event which might produce such a source (soft-intermediate state).Instead, we show how the soft lag can be interpreted in the context ofpropagation lags from mass accretion rate fluctuations. These normally producehard lags, as the model has radial stratification, with fluctuations fromlarger radii modulating the harder spectra produced at smaller radii. However,all that is required to switch the sign is that the hottest Comptonizedemission has seed photons which allow it to extend down in energy below thesofter emission from the slower variable turbulent region from the inner edgeof the disc. Our model connects the timing change to the spectral change, andgives a smooth transition of the X-ray source properties from the bright hardstate to the disc-dominated states.

The origin of long soft lags and the nature of the hard-intermediate state in black hole binaries