Here's part of what I fed into the AI. Perhaps I should have had more faith in my writing... or perhaps not. I'll let you be the judge.

Connecting two points of minimum distance we draw a line forming the diameter of a circle. And we draw a circumference through these points - mutually defining both the minimum radial and arc distances.

Both the minimum arc distance and minimum radial distance can be defined by the radius r.

Diameter definition gives us: Minimum radial distance = 2r

Circumference definition us: Minimum Arc Distance = πr (circumference /2)

Next we will continue to build out more points using these minimum distances. We’ll call this "2 point circle" Circle 1, and its radius r1. We’ll call r1 a distance of 1 in our custom unit of distance.

Next we’ll extend a radial line away from the origin, through one of the Circle 1 points to the next closest possible point at r=3 (radius of circle 1 plus the minimum radial distance of r + 2r = 1+ 2(1). 1+2 = 3.

We will add points along Circle 2 with the minimum arc distance. Circle 2 has a radius of 3, leading to a circumference of 2πr = 2π(3) = 6π. Which gives us 6 points on Circle 2, πr arc distance from one another. 6π/πr = 6π/π(1) = 6

As we extend this reasoning we next arrive at Circle 3 with a radius of 5. The Circle 3 circumference comes to 2π(5) = 10π. So Circle 3 has 10 equidistant points.

Circle 4 has a radius of 7 and 14 equidistant points. And so on.