The nature of number

[Categories+Sheaves]

I still have a problem, Looking at the results I am getting. Is there any work about gaps in the series, how little or much of a gap can there be before the series separates, or do you only need the parameters to create the full series?

There aren't any gaps in the real numbers, and the 'gaps' in the rational numbers (if they can be called that ) are infinitely small. The "size" of the gap between 0 and 1 (if we're talking about the natural numbers) doesn't matter, as long as we can tell the difference between one of them, two of them, etc.

Is there a mathematical theory about how much of the Argos you need to have an Argos, or how small a hole makes it sink?

You would still have to define what it means for the Argo to be the Argo.
That being said, the image of those "x0" maps should not be considered equivalent to the preimage. You seem to realize that doing so would be problematic, but the way you're considering numbers (It's not clear what the necessary/sufficient properties of a "coherent series" are) aren't giving you any leverage on the problem. I'm not going to shove some mathematical definitions down your throat, but you have to nom on something if you want to get ahead on these sorts of questions.

Or, from another angle: if you're this curious about this stuff, you're doing yourself a disservice by not sitting through some online lectures or reading through a relevant textbook. (Also, this would force you to phrase your questions in a way that I can make better sense of )