Seeing as how I haven't checked in in awhile, hello everything. Special welcome back to Teach. So I've been working on some patterns, and have reduced my work into one, fairly underwhelming formula.

N = (D-A)^2+E / (2*A).

I'll be extra salty if this was stated somewhere else, other than the known "X^2+E = 2NA". I'm using the form D-A instead of X, because this is the fruit of working on the X+N square, and my observation that it seems like an overly complex way of iterating over A.

Here in this formula, if (2A) evenly divides (D-A)^2+E, A is valid, thus X, and you get N. Which isn't useful except, finding the intercept of these two equations, (where (D-A)^2+E < 2A) gives us the upper bound for A, (i.e, the lower bound for X). It may also be possible to push X into the negative (where B and A become swapped) and give us the upper bound for B.

There's some more patterns with this I'm exploring, and I could do a visualization, but what's happening here is fairly complex to show without coloring it manually.

So a visual of what I've been playing with, every single cell comes from the origin cells in N0. So N0, F0, F1, F4, F9, F16, etc etc. The square root of these original cell's F, is X. The color coding guide for this is green is A=1. Red is A=2. Magenta A=3. Orange A=4. Again, all these blocks share the same X, with the origin at F(X^2), N0.

So there are an infinite amount of these, and their spacing is what controls if a cell exists at a certain E,N or not. This is the key to unlocking T as well. What we understand as an X-chain, (where the cell's B becomes the next cell's A), is a deep relationship to where these objects overlap, and multiple X chains are where multiples of THAT pattern overlap.

Going deeper, this first picture is the X=1 object in Red. In the second picture, we add the X=3 object in Green. The third, the X=5 object in Blue. Lastly, the X=7 object in Yellow.

X=1 in Red, X = -1 in light blue, VS. X=12 in Red, X = -11 in light blue. This is part of my tangent that the original VQC code only generates a part of the grid, which obscures some patterns like this one.