The author could instead have used only abstract mathematical language (like "a configuration space with a basis of cardinality 4" and "parameter s defined by s = ..."), but this would be accessible to fewer readers. If you prefer dryer language, read the second half of the article, or the other resources it links to.

Using a configuration space that is a good fit for a specific system is not unusual in math or physics. It's the same thing that astrophysicists do when they work in orbital elements, or that engineers and programmers do when they make calculations in Fourier space. It's worth understanding these concepts even if you're not a physicist.

Configuration spaces are so fundamental and useful that they are introduced in the very first lecture on classical mechanics in Leornard Susskind's excellent series of Stanford physics courses: http://theoreticalminimum.com/courses

(Neal Stephenson's novel Anathem also has a simple introduction to configuration spaces in an appendix, because that's the sort of thing that ends up in a Neal Stephenson novel.)

So it's horseshit because it "seems like horseshit" to you?

In fact, coordinate transformations are at the heart of the modern conception of classical mechanics: http://en.wikipedia.org/wiki/Lagrangian_mechanics

Your "invisible frogs" would not be a reformulation of Newtonian Mechanics (NM). A reformulation needs to make the exact same predictions as the original theory. There are many reformulations of both NM and GR, and they are quite useful (and interesting) in various contexts. This is a solid, interesting work that does not deserve being dubbed "horse shit" by a person who does not even have a formal training in the field.

As stated in the top portion of the thread, the original paper is completely not horsesh!t. It is a reformulation of a 3 dimensional problem in 4 dimensions that significantly simplifies the specific calculations/relationships of interest.