I don't think that you actually understand any of the words you use there

Lol.

but please show me the calculations and distances involved.

Calculations are what we contrive AFTER we measure first. If you want to calculate, make some measurements of the same star in the same position (multiple nights, obviously) from different known distances/locations to it and extrapolate from there. You will see that it does exactly as i said; as you recede the star will lower in azimuth until it converges with the horizon and you can’t see it anymore. Do you honestly doubt that?

How far away from the constellation of Orion am I on the Northern Hemisphere and how far on the Southern and what exactly is the cut-off distance beyond which it isn't visible anymore?

I’m sure those answers are available to you if you want to go find them. It’s pretty standard astronomy stuff - go look it up if you’re interested! It doesn’t have much to do with what you asked though - that will just tell you how far you’ll have to travel to “restore” the stars you can’t see anymore.

I am talking about why you can’t see them anymore (they’re too far away), which is a different matter than how far you need to travel to restore them.

I don't think that you actually understand any of the words you use there

Lol.

but please show me the calculations and distances involved.

Calculations are what we contrive AFTER we measure first. If you want to calculate, make some measurements of the same star in the same position (multiple nights, obviously) from different known distances/locations to it and extrapolate from there. You will see that it does exactly as i said; as you recede the star will lower in azimuth until it converges with the horizon and you can’t see it anymore. Do you honestly doubt that?

How far away from the constellation of Orion am I on the Northern Hemisphere and how far on the Southern and what exactly is the cut-off distance beyond which it isn't visible anymore?

I’m sure those answers are available to you if you want to go find them. It’s pretty standard astronomy stuff - go look it up if you’re interested! It doesn’t have much to do with what i’m talking about though - that will just tell you how far you’ll have to travel to “restore” the stars you can’t see anymore. I am talking about why you can’t see them anymore (they’re too far away), which is a different matter than how far you need to travel to restore them.