I'm impressed with what is being reported, and I'm looking forward to reading their paper whenever it is published. But, I've also seen the Pythagorean theorem proved with geometry.
In short. Take a right triangle with sides a,b, and c. Make four copies, rotating each 90 degrees, and placing it so that the points are touching, and sides a of the copy and b of the previous triangle are in line with each other. This creates a square with sides of length a+b. Inside of this square is a smaller square with sides of length c (rotated a bit). The area of the large square is (a+b)^2. The area of each triangle is 1/2(ab). The area of the smaller square is c^2. The area of the large square is also equal to the sum of the areas of the four triangles and the smaller square, 4[ 1/2(ab)] + c^2 (which reduces to 2ab + c^2). This leaves us with (a+b)^2 = 2ab + c^2. Multiply out the left side, and we get a^2 + 2ab + b^2 = 2ab + c^2. Subtract 2ab from both sides, and you are left with a^2 + b^2 = c^2.