I think the only problem with your gravitation counterexample is that I don't think Newton's Universal Law of Gravitation can be derived in a flat earth cosmos.
Not knowing the full history of Newton's discovery, I did a quick perusal of the relevant page on Wikipedia and found that he first demonstrated that the centripetal force must be inversely proportional to the radius in an elliptical orbit. If you then postulate that that force is the same one that pulls us toward the Earth and work out the shell theorem and voila -- universal gravitation. (I suppose there could be a different force that pulls things toward the surface of the Earth...)
That was, however, dependent on Kepler's elliptical orbits which all break the whole flat earth thing already.
I’m not sure where centrifugal forces and planetary orbits come into the picture here as the Shell theorem states the only force law for a spherical mass distribution on an object outside the shell is the inverse square law.
This can be demonstrated using the static case of first determining the force between a hollow shell or sphere and an external object, then extending to the case of the force between a solid spherical mass and the object.
Here is a diagram of the gravitational force between an object of mass m at the point P and a hollow sphere.
Let O be the centre of the sphere and subdivide the surface of sphere into circular elements such as ABCDA as illustrated in the diagram.
The area of the surface element ABCDA is equation (1).
Here the radius of the element is (a sin (θ)) hence the circumference is 2π(a sin (θ)) and the thickness is (a dθ).
If σ is the mass per unit area then the mass of ABCDA is 2πa²σ sin (θ) dθ.
Due to spherical symmetry all points of ABCDA are at the same distance W = AP then the force of attraction of the elements ABCDA is the mass m at point P is equation (2).
n is the unit vector of the force from P directed towards O.
From the diagram are equations (3)
Using the cosine rule w² = a² + r² -2arcos(θ) and substituting into equation (3) which is then combined with equation (2) gives equation (4).
The total force for the between hollow sphere and the mass m at point P is then equation (5)
This integral can be solved by using the variable w in the cosine rule in place of θ.
When θ = 0 , w² = a² -2ar + r² = (r - a)².
w = r- a
When θ = π , w² = a² + 2ar + r² = (r + a)².
w = r + a
In addition differentiating the cosine rule gives
2w dw = 2ar sin(θ) dθ
r – a cos (θ) = r – a(a² + r² - w²)/2ar = a(w² - a² + r²)/2r and equation (5) becomes equation (6).
Equation (6) is the force between the hollow sphere and the mass at point P.
The case for a solid sphere such as the earth we need to subdivide a solid sphere into thin concentric shells where each individual shell is at some distance R from the centre where dR is the thickness of each shell and 4πR² is its surface area.
The force of attraction between any individual shell of mass and the mass at point P is equation (7).
Finally the total force of between a solid shell and the mass at point P is obtained by integrating from R = 0 to R = a which is equation (8).
Hence the force
F is based on the inverse square law.
