I'm unable to see how you can choose a stateless value which is guaranteed to not change for a finite period of time.

Since the number is stateless, it must depend on some external reference (eg. a timet).  However, you want the value to change, there's no way to avoid getting the value at some arbitrary number of seconds N prior to the number changing, for some N less than the duration you're interested in (here 600 seconds).  So unless I've misunderstood your requirements, I don't see how you what you're looking for is possible.

Now, if you're willing to accept numbers which are the same for a 10-minute period, just use:

my $value = int(time / 600);
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But, of course, you still have the problem of having the value change very soon after you choose it.