The microsecond you made that post, it was slurped by a crawler or a clanker somewhere anyway, be it Bing, Yahoo, Google, etc...., and Fediverse can't do shit about that. Deleting it does not really help you any if your threat profile is "I might get ICE'd". Better to own (literally and metaphorically) what you speak with your own name.
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Always assume public stuff is archived somewhere. It sucks, but it is what it is.
When you delete a post, your instance sends a "this post got deleted"-message to all servers that got the original post. Most fediverse services implement it properly and delete it, but the message may get lost, admins may change the behavior etc. No guarantees.
In my experience (non technical, my delelting post experience) when you create a post it propagates immediatly to any instance that is properly federate to your instance. This usually means same software and people from there following content from your instance. Then, with a lot of conditions to be fullfield, it propaged to partially federated instances. For example instance of another software that makes full federation more difficult, or instance that does a search on your specific post...
When you delete your post, it deletes automatically to any properly federated instance but that could be with a delay (a few minutes, or an hour) but the delete order has issues reaching all the other instances (the ones that were federeate when you create the post but not anymore, the ones from another fediware, ...)
You also have no guaranty that someone doesn't have your post in a backup they may use to restore a fallen instance, or leak through another way. As with every content on the internet, better consider anything once posted as know publically forever.
There will be a trace. First of all, the way the Fediverse it set up, the instances all cooperate from distributing posts and comments, to deleting them. There's no guarantee every instance does it (correctly). And as the Fediverse is made up of different software, it also depends on the specific implementation.
And then we also have AI scrapers, the Wayback Machine and other internet archives. It'll end up there as well.
So better treat everything as easily traceable which you post in public. And it's notoriously difficult to really remove stuff from the internet anyway.
Plus the US has some absurdly large datacenters for surveillance. Idk if it's clever to lie to them about your social media history. They certainly have the capacity to scrape posts and store them forever.
If the idea is to connect with family and friends, there are parts of the Fediverse where you can post stuff that's only visible to followers. Probably not possible on lemmy or piefed though, I think.
If your use case is post to a small known group, PieFed supports creating local private communities where content is t federated at all. This of course only works if your whole group can decide on a single specific instance.
If I delete this post, will it be completely removed across all instances that synchronized it?
It will send out a message to relevant servers that it should be deleted. There is no guarantee that they will comply with that message. If your post has been copied to hundreds or thousands of other servers, there is no guarantee that they will all receive or understand that message. Some may even be actively malicious, for example because they are controlled by exactly the people you want to hide from!
I remember once deleting a comment (on this account) a few seconds after posting it. After that, I kept getting upvotes for it! I found out that that was happening because one very popular instance had for some reason not deleted the comment, so its users had no idea that it was supposed to be gone.
Is a deleted post traceable in any way?
Everything on the public Internet is. Anyone can set up a bot that just scrapes and archives everything on the Internet that it can find; and governments certainly have the resources to do so!
Is it kept in a log or a database on ferdiverse instances?
Potentially.
With governments across the globe increasingly surveiling us online and scrutinizing everything we say, I’m starting to think I should plainly delete any account that has personally identifiable information like my real name and photo. I initially thought it would be easier to connect with family and friends, but now I’m growing increasingly worried about how this can be used against me.
Posting things on the public Internet, especially under one's real name, inherently comes with that risk. Always has.