(How can you build an alarm for First Contact situations that nobody inside Civilization itself should be able to trigger, if you can't just send off a hard computational problem and verify the solution?

You could ask for a higher Mersenne prime than Civilization has ever searched, and begin to verify its primality.  But if it was possible to verify a large candidate Mersenne prime in a few seconds, it would be the sort of question where it was possible for a mildly rich investor to test lots of 8-digit numbers quickly and discover the answer themselves.

So instead, Civilization has found the 45th Mersenne prime via a large public-effort clustered computer, and then (according to Network repositories on the subject) declined to announce that prime as it announced the previous 44.  Instead, the discovered number was stored in a few secure places; as one of several potential resources for use in verifying First Contact with some more powerful civilization.

And this is literally true, so far as literal statements go.  It's not like the government makes a habit of literally lying; that would be a very bad habit for the government to acquire.  People would then need to expend mental effort, during every interaction with the government, on wondering whether the government's words were related to reality at all; and if things ever got anywhere near that bad, the dath ilani would toss out their current government and start over.

But that literally true statement is also a trap.  One can imagine some illdoer who wants to trigger a false First Contact alarm.  There would be all manner of ways an illdoer might imagine profiting from that situation.  Even dath ilani try to avoid leaving billion-dollar bills out where ill-intentioned people can too easily pick them up.

The 45th Mersenne prime, 2^37,156,667 - 1, is not heavily guarded.  It is not that impossible for a criminal to steal one of the stored copies.

The 46th Mersenne prime, 2^42,643,801 - 1, is stored in only one place, an advanced weapons silo already under strong guard.  It was searched-out and discovered under conditions of much higher secrecy.  Any criminal trying to pose as an alien civilization, one might hope, will not realize that they need to steal a 46th Mersenne prime at all.  They'll just show up with stolen knowledge of the 45th Mersenne prime, and no knowledge of the 46th one.

So if a 'contactee civilization' shows up with knowledge of the 45th Mersenne prime, but not the true 46th prime, some highly placed key figures will have a pretty good idea what happened.)

(How can you build an alarm for First Contact situations that nobody inside Civilization itself should be able to trigger, if you can't just send off a hard computational problem and verify the solution?

You could ask for a higher Mersenne prime than Civilization has ever searched, and begin to verify its primality.  But if it was possible to verify a large candidate Mersenne prime in a few seconds, it would be the sort of question where it was possible for a mildly rich investor to test lots of 8-digit numbers quickly and discover the answer themselves.

So instead, Civilization has found the 45th Mersenne prime via a large public-effort clustered computer, and then (according to Network repositories on the subject) declined to announce that prime as it announced the previous 44.  Instead, the discovered number was stored in a few secure places; as one of several potential resources for use in verifying First Contact with some more powerful civilization.

And this is literally true, so far as literal statements go.  It's not like the government makes a habit of literally lying; that would be a very bad habit for the government to acquire.  People would then need to expend mental effort, during every interaction with the government, on wondering whether the government's words were related to reality at all; and if things ever got anywhere near that bad, the dath ilani would toss out their current civilization and start over.

But that literally true statement is also a trap.  One can imagine some illdoer who wants to trigger a false First Contact alarm.  There would be all manner of ways an illdoer might imagine profiting from that situation.  Even dath ilani try to avoid leaving billion-dollar bills out where ill-intentioned people can too easily pick them up.

The 45th Mersenne prime, 2^37,156,667 - 1, is not heavily guarded.  It is not that impossible for a criminal to steal one of the stored copies.

The 46th Mersenne prime, 2^42,643,801 - 1, is stored in only one place, an advanced weapons silo already under strong guard.  It was searched-out and discovered under conditions of much higher secrecy.  Any criminal trying to pose as an alien civilization, one might hope, will not realize that they need to steal a 46th Mersenne prime at all.  They'll just show up with stolen knowledge of the 45th Mersenne prime, and no knowledge of the 46th one.

So if a 'contactee civilization' shows up with knowledge of the 45th Mersenne prime, but not the true 46th prime, some highly placed key figures will have a pretty good idea what happened.)