Yes the sun moves occasionally south and north. You would know this, if you understood what the tropics of Cancer and Capricorn are for. We have seasons. Otherwise, you're confusing "south" and "clockwise/counterclockwise." The sun moves along a rotary path, yes. Round Earth apparently explains seasons as "well the Earth tilts a bit, and moves a bit further away and uhhhhh this is how we have seasons."
If you understood what the tropics of Cancer and Capricorn are for, you wouldn't have spouted such nonsense.
RE does not explain seasons based upon distance to the sun.
The explanation is based upon the tilt. This tilt causes the sun to hit the summer region more straight on (instead of at an angle) and illuminate it for a longer period of time.
It makes sense, unlike the FE fantasy.
And again, something which only appears to work from a cursory glance without any real thought.
Why does the sun follow this path on the parabola?
You appear to have it going straight down.
The problem that causes is that only certain people on Earth would see the sun each day, and only for a fairly short period of time.
e.g. consider the equinox.
This has the sun over the equator.
With your magic zone of perception being limited to 5 km (the distance to the horizon when standing near sea level), that means only locations within 5 km of the equator would see the sun.
If I instead take your arbitrary 11 archaic units, and convert it to a generous 18 km, that is still only an incredibly small sliver of the world that would be able to see the sun on the equinox. Instead, the entire world can.
But another issue is that the equator has a length of 40 000 km (or 62800 km for a FE), and the sun needs to go around that in roughly 24 hours. That means it would cover the 36 km (18*2) in a matter of 1.3 or 0.8 minutes depending on if you use the RE distance or the FE distance.
So the best case scenario, someone on the equator, directly under the path of the sun on the equinox, sees it for all of 1.3 minutes. As they move away from the equator that time shrinks, until at 18 km away from it, they just catch a glimpse of the sun, and beyond that they wouldn't see it at all.
But in reality, during the equinox basically the entire world gets 12 hours of daylight (except right near the poles where they get more).
In order for this idea to even have a chance at working, you need your magic zone of perception to be such that the edge of the zone, where the sun would disappear, to be located at the point where someone would observe the sun overhead at the same time.
Using RE numbers, that is 10 000 km away. Using FE numbers, that varies depending on where you are, (for the equinox, on the equator it is closer to 14 000 km).
But in order for this idea to even have a chance at working, the edge of the zone needs to be where the horizon is, which is roughly 5 km away for someone standing near sea level.
So you need it to be both 5 km away and 10 000 km away. An impossibility, showing this cannot work.
Another issue is the size.
You have just projected the sun down as a circle, placing it inside the dome.
But why wouldn't it just project down and hit the parabola and appear there?
This means at zenith it would appear as a circle.
But after that it would be stretched in the vertical direction, appearing to get significantly longer as the day goes on.
And unless you have it as a circular dome, it will still appear to change size horizontally as the distance to the dome will change.
Your second diagram just further demonstrates this:
This is much better at showing why it doesn't work at all.
You have 2 people, situated such that their circular horizons touch at a point.
If they were near sea level, that would put them roughly 10 km apart.
Yet you have it so 6 pm for the one on the right is before 8 am for the one on the left.
You have the sun set for the person on the right, as it is rising for the person on the left.
But back in reality, these 2 people are so close that they will observe the sun in basically the same position.
The sailboat cannot be seen because it is outside his range of perception.
The more important part is how it disappears.
Consider when it is like this (just showing that portion):
What should the boat look like here?
It is just at the edge of your magic zone of perception.
The top portion is outside, but the bottom portion is inside.
Or even better, if it is sailing to the left, from shortly before, when it is entirely inside, to shortly after, when it is entirely outside.
What do you see?
Do you observe it appear to sink, with Earth obstructing the view from the bottom up, or do you see the top just start to magically vanish as your model indicates should happen?
Now this isn't completely accurate
Like I said before, I would say it is 0% accurate. It completely fails to account for basically everything, and a key thing you have tried to explain with it shows it doesn't match reality at all, so you have entirely failed to explain what is observed in reality. So it still isn't explaining anything.
It is just a very poor attempt at trying to make a FE work, to appease those who desperately want to make a FE work, who will accept anything which pretend to make it work without question or thought.
Not that this model matters to you. You've hardened your heart against it.
And more religious BS.
I haven't hardened my heart. In fact, I don't bother with my heart for this kind of garbage.
Instead, I use my brain. Perhaps you should try it some time?
I look at your model, and actually think about it.
I think about what it would mean, and see so many flaws it isn't funny.
That is why I discard it as the garbage it is.
Conversely, you don't use your brain. You don't think about the model and the logical implications of it.
You just care about trying to make a FE work, when it can't.
If you want to show I'm wrong, and show that your idea isn't garbage, then try to actually think and answer these issues in a logical and consistent manner:
Your nonsense fails to explain why the sun does not appear to change size at all.
Your nonsense with a magical zone of perception wouldn't even be able to see the sun or moon, even with the distances used in the FE model, as it is too far away.
Your nonsense, with a magical zone of perception, has no explanation at all for what causes this magical zone, nor why it should increase as you get higher.
Your nonsense, with a magical zone of perception, in no way explains why objects appear to be obstructed by the horizon, disappearing from the bottom up. In fact, your nonsense would indicate objects disappear from the top down, as the top will leave your zone of perception first.
This also means it can't account for why tall enough objects can be seen when far away.