Responding to both the OP and Ravenwood since they both raise points that need to be addressed.
Does this flat Earth still have a solid, Iron Based core?
How come we can triangulate, to within a few Micrometers, the position of any vibration in the Earth using a Round Earth mathematical modeling equation?
Um, micrometers? Really? Vibrational sources are rarely even defined to that level of precision (explosions and earthquakes are much larger). Simple earthquake triangulation generally works within a few kilometers, not micrometers. There are way too many unknowns for that level of precision in general.
Secondary, this mathematical system can be derived from first principles (therefore self proving) and performed by a persons mental capability, no computers involved.
Stronger than that. The math can be done by a highschool student. In fact, when I've tutored kids in geometry before I've given this as an example of something they can do (although simplified to only using three stations of known approximate distance from each other and over a small enough area that the Earth can be assumed to be flat. If you have more than 3 stations or if you try to do this over a larger area and use actual data you will get nonsense). The technology to measure points accurately enough to detect vibrations roughly isn't easy but it is reasonably doable as a hobby object by someone with a decent engineering background and time on their hands.
How does the FE theory explain the accurate discovery and proven theories of ''Pi'' using a RE model?
Can you explain what you mean here? Whether or not the Earth is flat doesn't alter the value of Pi at all.
Now, is there anything in your first question that cannot be done in a FE model? I haven't actually looked at the math for it, so this is an honest question. Would that same math find the vibrations in a FE?
More or less the answer to this is no. They would not agree.
Ok, let's back this up a bit:
Vibration sources create a variety of different types of vibrations. The most easy to understand and to study are P waves, S waves and Surface Waves. P waves are the standard wave you think about when visualizing vibrations through an elastic medium (think a series of shock waves). P waves have vibrations which are parallel to the direction of travel. S-waves move perpendicular to the direction of travel and they are a bit harder to intuit. Finally, surface waves are waves that travel along the surface (or really along any point where there is a transition between substances).
Assume we are in a reasonably not that large area so that we don't need to worry too much about whether the Earth is flat or not (say North America for our purposes). You can then triangulate the location of a vibration source based on any one of the three waves by timing the delay. We can calculate and confirm quite easily the times it takes for all three types of waves to propagate. Now,
if we have many different locations that are checking for vibrations we can get very precise data. Here's where this becomes a problem for FE. The numbers will be substantially different for FE and RE. This is before we get to looking at things on a larger scale.
Moreover, for large vibration sources (say decent size earthquakes and nuclear explosions) we can map pretty precisely where the sources are compared to each other without regard to looking at surface waves (using just S and P). This gives us an approximate location in 3 dimensions (you need 4 observation points to triangulate in 3 dimensions but in practice there are many world wide). This builds a decent 3-picture of sources. If the earth were flat you would get all the points lying on the same plane. But you don't, they form a sphere roughly. There's no way to place points in this fashion. You do need to know roughly the relative location of some of the stations but you can do this easily with some of the stations being only a few hundred miles apart. Moreover, if your initial assumptions in that regard are wrong you would quickly end up with nonsense and contradictory numbers.
There are other wave types you can use to confirm this such as looking for Love waves and Rayleigh waves. This gives completely consistent data with the RE seismology. The numbers for these just won't work on a flat earth.
I'm not even dealing with the use of these techniques to map inside the Earth because the FE model fails even before we get to that point. The fact that the RE model gives us further consistent predictions in that regard is simply icing on the cake.
Now, an FEer could claim that the geologists at every major university and at all the testing stations are in on the conspiracy and are altering the data. As I've remarked before, if the world is flat the conspiracy is much much larger than just the space programs.
Edit for formatting