Hmmm, what's the algorithm to extract 101 and not 10101 or 1010103 as the next prime? I suppose you can remember how many digits were in the last prime and only allow the same or 1 more, which should be unambiguous. Though it relies on the unproven conjecture that there are no very long runs (an entire order of magnitude!) with no primes.

The fact that any order of magnitude (base 2) contains at least one prime was proven in 1852: https://en.wikipedia.org/wiki/Bertrand%27s_postulate

Of course, the aliens listening might not have proven it yet.

The point is, with a constructed transcendent number, you can represent any series at all, so long as you have "enough precision."

That last part means you can simply adjust your precision to your desired constraints. If you want to unambiguously represent the primes up through all the four-digit primes, simply change your padding accordingly: 0.000200030005....

Obviously the Buenos Aires Constant in the article is a neater representation, and relies on neater math. But GP's point was that you can stuff anything at all into a transcendent number.

Yes, that was my non-mathematician's intuition.

Also, if I want to transmit a large set of primes (the complete calculated constant), I'll have to /retransmit/. How do I do that unambiguously? Well, "SOS". Transmit the set of primes. Pause long enough. Retransmit them. Ad infinitum. Ensure the pause is very very precise too.

It would be fun to pause for 2.<precision> seconds between retransmits.

Things are observable in the universe because we can tell when they are there and /not/ there. Any homogenous continuum ought to be undetectable (e.g. relativistic frame of reference, so my non-physicist brain says).

Edit: singular/plural bugfix

Even if that ambiguity did exist, it would just complicate the issue of extracting primes from the sequence. It wouldn't complicate recognizing the sequence.