A chunk of iron put into a 1μT magnetic field will behave differently depending on what other magnetic fields it has been in recently.
How exactly does this address "whether it was recently thrown upward or dropped from above"?
A ball in a given location will do something unpredictable if you have no information about its past. Is that hysteresis?
So its past is kind of irrelevant. At some point if nothing is supporting it, it will fall towards the centre of the Earth.
This is just high school physics.
I'd be surprised if that were the case: seems like it'd violate some sort of "locality", where the future should depend entirely on the present state.
Is it just that there's unidentified present state that's perhaps more easily characterized by knowing the history? Or is it somehow unidentifiable in principle?
The hysteresis in the material only means that if you apply 1T to it to magnetise (supposing the ball starts with zero magnetisation) it and then apply 0T to demagnetise it, the remanent field in the ball would be like 0.4T due to hystetesis, you need to apply something like -0.4T to leave it as it started.