The first problem is to get a clear understanding of what the curve of "Curvity" looks like and draw connections between its many parts. For example:
But wasn't it negative gravity that made the ship fall away from earth and didn't you need "c2" amount of energy in order to "see" the negative side of gravity? It also wasn't clear how a law that worked on inter-galactic scales could also work on the atomic scales. However, galaxy rotation curves could hold the key and the graphs of orbital speeds could literally be the curve of Curvity written across the heavens.
If MOND's "gravity falls off more slowly" could be explained by Curvity's "gravity falls off more quickly and goes negative", then it should be possible to put together a graph which shows just that. Which is what is shown below:
The light blue line, just shows a scalar quantity that can be any one of the "factors", like energy, distance, mass, time, etc and its most important point is c2 (90 billion of a "factor") or as Alan called it "C". From Curvity, we know that at that point, we start to see the negative side of things, for example, if you travel at the speed "C" away from earth, gravity goes to zero (on earth from our perspective) and time stops. Go faster than the speed of light and gravity and time go negative on Earth (again, only from our perspective), hence the negative part of the curve. Of course, it was Alan who suggested the use of a sine wave before, on his visit with Daniel in Oregon, where he said:
"You must always remember that your ordinary physical laws as they are usually expressed, do not hold true when carried to an extent which permits the error to be measured because they do not follow a straight line reaching to infinity but a curve of finite radius. In a timeless universe, this curve would be represented by a circle but since the laws operate through time as well as space, the curve is more readily understood if depicted as a 'sine wave'. In this case the base line of the wave represents zero and the portions above and below the line represent the positive and negative aspects of the law." (Sine wave & Zero Point, Curvity)
The reason the curve above is a big breakthrough, as simple as it looks, is because it can explain why a resonant cavity produces movement, using a positive attraction of gravity. In our daily existence, we reside entirely at the zero starting point on the graph (far left), but inside the cavity because of resonance, the energy can build up to incredible levels, in the case of Shawyer's cavity, some 17 megawatts. As the power builds up, we are moving up the sine curve until the first peak (above the "e" in Positive), at which point the maximum force we can generate is limited by the gravity field of earth, or 9.8m/s2.
The graph is also useful for explaining things at the atomic scale as well, for example, if the Strong force really is gravity in the atom, then as we travel over the first half of the sine wave, we should see the strong force get stronger, with the most stable atoms like those of iron, at the very peak. This may also explain why the Strong force gets stronger as you try to pull a proton or a neutron out of an atom. As we travel over the peak and back down the other side of the sine wave, we get to the zero point again, at C (c2) on the linear line, and here we have the least stable atoms, like those of Uranium 235 and Plutonium, which when an extra neutron is added, slip well past the zero point and into the negative section of the curve, thus exploding apart. The graph also shows why it is necessary to slow the neutron in order for the heavy atoms to absorb it and that is because the strong/gravity force so close to the "zero" point, that it is really weak.
Another great result is that we can use the sine wave nature to test if Curvity can explain galaxy rotation curves that MOND cannot. For example, the reason some galaxies may have an unexpected rotation curves (vs MOND) is because the surrounding galaxies are far enough away that their negative gravity contribution does not line up with the dropping gravity of the galaxy under study. Hence a galaxy rotation curve may not be flat, but have a slope, where if the surrounding galaxy(ies) are too far, the slope will drop, or if the galaxies are nearer, the slope will rise.
The diagram is a bit confusing because it isn't showing a rotation curve, but a measurement of gravity versus distance (and not an accurate one). The point is that it shows that if a negative gravity contribution from a surrounding galaxy is strong, the rotation curve will have a positive slope (bottom example). If the negative gravity contribution is not large, the rotation curve slope will be negative (pointed down in the top example). Remember, the red lines represent negative gravity which push star systems toward the galaxy center, in the same direction that positive gravity, represented by blue lines, pulls. The purple line is the combination of the Newtonian and negative gravity.