I see.

The calculations he offered WRT the "round" earth were from an on-line calculator. The page with the calculator describes how the calculations are done. The resulting prediction was correct.

The other calculations were done using one particular flat earth model and some simple grade 9 math. As he pointed out, there seems to be some issue with which flat earth model to choose and what the elevation of the sun is as flat earther's themselves cannot agree on those important aspects of their "theory".

However, you mention "physical aspects of the celestial sphere". Perhaps they would be a way to make a reliable prediction. So, which physical aspects are we talking about and what is the mathematical model for their use?