No external drive was connected to the Mac during these 48hrs for TM backup. at time t=1hr, TM tried to make the fist snapshots, and found that the size would be 500GB. Together with the original 500GB data, this snapshot would have caused the total storage to exceed the 1TB SSD capacity (or the 80% ceiling). Thus, TM did not make the first ever snapshot. As a result, at t=48 hrs, there was no snapshot on the SSD. Yes?
I am not sure what happens when there is no backup disk connected, since Time Machine can't be enabled.
You have misunderstood the nature of snapshots. A snapshot is not a copy of your data - there is no question of needing 500+500GB. It is a view of your data as it was at particular points in time. This is due to the magic of the APFS file system. It does not duplicate any data. The file system and its snapshots will consume (in your example) 500GB + 24 MB. This is sufficient to give you 24 views of the file system as it was in the past 24 hours.
And this answers q2. Yes.
As it is intended to be used, TM will also be backing up to external hard disk and after 24 hours it starts a) removing old snapshots on the system disk and b) thinning the quantity of backups on the hard disk from hourly to daily to weekly. The thinning is what Apple is referring to when talking about the last 24 hours. When I look on my TM disk I can see backups for each of the last 24 hours, the first backup yesterday and all the preceding 30 days, and one backup every week before that.
My understanding of last successful TM backup is the last backup to hard disk.
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Arq is not really what you want for the purposes you stated...
Just to add you your excellent reply. Two key parameters in disaster recovery are "time to recover" and what "points in time" are available.
Assuming a system disk hardware failure:
CCC, without doubt, has the fastest time to recover (a reboot and away you go), but will be more limited in the points in time available. The simple quick recovery is to the time when you last ran CCC.
TM is slower to recovery (a few hours), but has many points time available.
Arq, like TM, has lots of points in time but has a long time to recover.
For the need to recover a few files (e.g. you deleted a folder by mistake, or you want an old version of a document), the time to recover is similar for all there methods (slowest for Arq from the cloud) and easy access to multiple points in time is more important. This is where TM and Arq shine.
Personally I use TM and Arq. The only time I had trouble with TM was when the external hard dive was about to die. Like you I use different products according to the nature of the disaster I want to recover from. If my income depended on fast time to recover, I would use CCC too.
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