I wouldn't say that the distinction is "unbridgeable". Keep in mind that we send packets over plenty of "geometrical" and seemingly infinitely dense analog chan

I wouldn't say that the distinction is "unbridgeable". Keep in mind that we send packets over plenty of "geometrical" and seemingly infinitely dense analog channels all the time, like electricity over a copper wire or EM waves over the air. The "mark on a stick" channel has a capacity like any other channel. If you're sending just one symbol, you could easily calculate the information capacity given a desired probability of bit-error. Assuming you can put the mark in exactly the right spot, you can model the "noise" as a distribution over the values that the reader will measure. If you model this as `mark + zero-mean normal distribution` with a known variance, then your stick is just an AWGN channel.