I think there are three main issues (though I might be overlooking something):

1. AI taking over peoples work is as much an issue for mathematicians here, as it is for programers, artists, writers, etc. elsewhere.
2. We still need to check if these 700 papers are correct (AI makes mistakes at least as often as humans, usually more often). This involves a huge amount of effort. And if it turns out that they contain a bunch of mistakes, OpenAI is unlikely to care. To them, these papers have already served their purpose (generating headlines).
3. Solving a problem in mathematics means being able to (formally) convince other people that your opinion is correct (massively simplified, but imo that is what mathematics boils down to), by getting them to understand why your opinion is correct. Having a pile of linear algebra spit out 700 texts about certain math problems removes the whole “human understanding” part of that.

However, it’s not entirely unprecedented for a computer to provide a proof that we don’t completely understand. I forgot the details (and I’ll edit them in if I can find it), but there was at least one problem that was solved using a computer and brute force trying millions of cases. That produces a proof much longer than any human could read in a lifetime, and iirc there where quite a few mathematicians unhappy about it at the time too.

Debian is the OS.

Proxmox builds on Debian to turn it into a hypervisor. That’s an OS for managing VMs and containers. Proxmox (like most hypervisors) gives you a handy webinterface.

There are easier hypervisors to use, but Proxmox is great for learning, and allows you to expand well later.

tbh, except for gaming and AI, you don’t need nearly as much processing power as you might think. I run most of my stuff on a (fairly new, can’t remember which one exactly) i5, and it sits at an average 2-3% CPU usage, with the maximum its recorded being 10.9% (there’s two VMs and 8 containers running on it). 11th gen i5 will probably be more than enough to start with.

In my non-professional opinion, what you really want to look at is how much power your system is going to use when idle, since that’s what (small) servers are doing most of the time. This is not something specified by manufacturers, and it depends on all the parts going into the computer (CPU, mainboard, PSU, fans, how the BIOS is set up, …).

I would also recommend getting at least 16 GB of RAM (honestly I would say 32, if not for the current prices). Containers don’t use too much, but if you want to set up VMs for services, each VM will want at least 2-4 GB for itself.

LXC-Containers (https://en.wikipedia.org/wiki/LXC) are like extremely lightweight and slightly less secure Linux VMs. They are similar to Docker, but while Docker is containerizing applications, LXC containerizes Linux distros.

Running Docker inside a VM is the recommended way to run Docker on Proxmox.

Personally, I just run Docker inside LXC containers. Aparently, this is supposed to be not entirely stable, but its never given me any problems. (I commit an entirely different sin against efficiency, by keeping each service in its own LXC container, each with its own Docker host.)

I also run Immich (in Docker inside a container), and haven’t noticed any performance issues. But I’m the only user, and I don’t use it heavily, so ymmv.

I’ve never used CasaOS, so I can’t help you with that, but I run NetBird for remote access to my homelab. What you’re describing sounds to me like an issue I used to have with Tailscale.

I think NetBird might be trying to route your traffic through a STUN/TURN server to get around a NAT or firewall. A NAT (in case you haven’t heard of it) is a sort of “mapping” from one IPv4 address to many IPv4 addresses. Its used to "make more "addresses, otherwise we would have run out decades ago. (this is a simplification, its actually much more complicated)

The first thing to check here is: Are you behind a CG-NAT? (there’s probably some websites to check this)

If yes, then there is little you can do without setting up a VPS to get around it.

If that’s not it, then you’ll have to figure out how CasaOS handles networking internally. Your services might be behind a NAT inside.

As for ressources/information, I mostly stuck to the official docs.