For example the simplest tautological space: x+y+vz = x+y+vz

Where usually show the modular algebra form, but will demonstrate like did long ago, where will show explicit. And use a demonstrative case where here is reference from 2014 for the modular algebra form.

With a bit of algebraic manipulation, have:

x+y = -vz + (x+y+vz), and squaring:

x

^{2}+ 2xy + y

^{2}= v

^{2}z

^{2}- 2vz(x+y+v) + (x+y+vz)

^{2}

Where can see all the growing complexity. Now that is STILL an identity of course, so x, y, z and v are completely free variables. But let's put some conditions on them.

Let: x

^{2}+ y

^{2}= z

^{2}

Then can subtract to get: 2xy= (v

^{2}-1)z

^{2}- 2vz(x+y+vz) + (x+y+vz)

^{2}

Where now can get to:

(v

^{2}-1)z

^{2}= 2xy + 2vz(x+y+vz) - (x+y+vz)

^{2}

With some simple algebraic manipulation. And have the result that (v

^{2}- 1)z

^{2}- 2xy has x+y+vz as a factor, when x

^{2}+ y

^{2}= z

^{2}, and can also easily realize that v is still free.

From the full explicit form can also just look at expressions that result from setting v, like v = 1, gives:

0 = 2xy + 2z(x+y+z) - (x+y+z)

^{2}

And of course then: -2xy = (x+y+z)(z-x-y)

Notice also are not of course limited to integers, as is a trigonometric result if z = 1.

That is: -2cos(x)*sine(x) = (cos(x)+ sine(x) + 1)(1 - cos(x) - sine(x))

And get a better feel for how is a true modular algebra when get to modular form, as does not care about Diophantine or not. Learned then that with an actual modular algebra is not just with integers. Should make sure to emphasize which is why am so doing here.

The modular algebra form of course is: (v

^{2}-1)z

^{2}- 2xy = 0 mod (x+y+vz)

So what the modular algebra form gives is compactness, and is MUCH easier to manipulate.

Where can also notice a sense of how you get algebraic manipulations by how you set v, which can be a HUGE benefit. With explicit is just given, but with modular form you can do equivalent, as MAY find in pieces that way. Just then stitch together. But can also get an explicit result too.

The freedom to change v in ways that help analysis, was the point of the discovery for me, wanted something I could control. And that result of also giving algebraic manipulation of the conditional expression was really cool to finally realize. Was a big surprise for me when came across.

Here demonstrated the full complex identity where notice raised to a form where could meaningfully subtract the conditional expression which leaves a residue, which can be analyzed. With the full modular algebra form I call that the conditional residue.

That actually is an important limitation on this approach. Like is meaningless to subtract that conditional from x+y+vz as you do not eliminate any part of the conditional expression. So had to square to get to something useful here. And depends on the conditions being set on the variables.

In my experience, tautological spaces are effective when you maintain roughly the same number of terms or less than the conditional expression, and are ineffective if you end up with more.

The tautological space had to be an asymmetric form I learned in order for it to work. And have generalized methodology based on my analysis of what seemed to be necessary.

And you can do fewer variables by simply setting for example y or z, where can consider by setting z = 1, with full case of three variable quadratics, which recently posted on, as yeah for years that's what I did. Which probably helped me figure things out with a bit less complexity.

So highlighting a bit I learned just on my own. And do wonder how much more others could figure out, as am certain only scratched the surface.

James Harris