## JSH: Using tautological spaces

Maybe the best way to move things forward is to show how to use these things I call tautological spaces, so others can see their value, so I'll explain things in this post where the simplest thing to point out is that what I call a tautological space is a mathematical identity.

So what I do is get identities, subtract an equation to be analyzed from that identity, and probe the residue, where to maneuver with the identities I use "mod" from modular arithmetic, for instance:

x+y+vz = x+y+vz is the identity but I present it as

x+y+vz = 0(mod x+y+vz)

and then you can manipulate that identity easily with much of the complexity of what you're doing hidden away.

For instance, moving forward with the above:

x+y = -vz(mod x+y+vz), and I can square both sides to get

x^2 + 2xy + y^2 = v^2 z^2 (mod x+y+vz)

and now if I were analyzing x^2 + y^2 = z^2, I can just subtract it away to get

2xy = (v^2 - 1)z^2 mod (x+y+vz), which is

(v^2 - 1)z^2 - 2xy = 0 mod (x+y+vz)

and that is the residue. Notice that v is a free variable, so you can make it whatever you want, while x, y and z are constrained by the equation to be analyzed which I call the conditional.

I use x+y+vz because I invented this technique while tackling Fermat's Last Theorem, so I was always subtracting x^p + y^p = z^p, but eventually found I actually needed

x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)

which goes to the issue of the exponents of the variables in the identity and the answer is, they can be almost whatever you want (non- zero and positive). Since I invented this field I've thought about terminology, and I call the set of exponents the hyperdimensional set, and you can describe the tautological space completely by that set, so that last is:

{2,2,1,2}

so it's a 4 dimensional tautological space, and I call it a tautological space as I'm using identities--which are tautologies as well--and manipulations with them gives you a space where the identity holds.

When I introduced tautological spaces on the sci.math newsgroup back in December 1999, I got a lot of various skeptical reactions, where the most important was the belief that identities can't be useful for analysis.

It's pointless to lay claim to my disputed proof of Fermat's Last Theorem here as an example of the power of analyzing the residue from these identities though I will point out that I *could* tackle x^p + y^p = z^p by using a rather complicated identity, though I ended up subtracting out x^{2p} + 2x^p y^p + y^{2p} = z^{2p}.

But more recently, and more easily considered than FLT, I subtracted out:

c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy

from an identity in a tautological space with the hyperdimensional set {1,1,1,1}.

That is, I used x+y+vz=0(mod x+y+vz), got a more complex identity with a very few simple algebraic manipulations so that I could subtract out the conditional, and got a result.

With z=1, I found I had a way to simplify checking that expression and solving it as a Diophantine equation and though I've given it many times before in recent threads as I've tried to publicize this result, I'll give it now:

(2A(x+y) - B)^2 - 4AS^2 = B^2 - 4AC

where

A =(c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3)

B = 2(c_2 - 2c_1)*(c_6 - c_5) + 4c_5*(c_2 - c_1 - c_3)

and

C = (c_6 - c_5)^2 - 4c_4*(c_2 - c_1 - c_3).

So you now have to find some integer S and x+y, but then you can easily enough get x and y, so it's a vast simplification, where using classical methods, with

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

you'd do different things depending on the coefficients but would do some completion of the square and other manipulations to get to something of the form u^2 - Dy^2 = N, so using tautological space I found a way to do that all in one sweep.

That is more to demonstrate that using identities does not just lead in circles, though it can, which is one of the warnings for those who try to use this analysis technique.

Picking values for your free variable is a critical step which can requires some creativity and fumbling as you figure out how to probe your residue.

On a sidenote, one way of looking at what I've done is that I've simply extended from Gauss who pioneered use of "mod" by having the modulus be an equation, to use an identity, to get complicated identities with less effort and subtract some equation to be analyzed.

Why I got to invent this technique, I don't know.

Note that I use x, y and z with v the free variable for historical reasons but you can use as many variables as you like, and of course, call them what you want, while one requirement is that the free variable be multiplied times at least one of your other variables, so no, it's not an accident that I have x+y+vz, but a necessity learned when I was exploring this approach, as I tried several different forms at first before I settled on that one.

And that is a quick go over of tautological spaces.

Any equation you have from wherever you can put into a tautological space with an additional variable, so you can force a degree of freedom, which shows up in the residue.

It can take some thinking to figure out how to manipulate from the basic start, so for instance with a hyperdimensional set of {3,7,2,5}, you'd start with:

a^3 + b^7+v^2 c^5 = 0(mod a^3 + b^7+v^2 c^5)

where I like v for the free variable but changed other variable letters just to be a little creative.

I always use positive signs but don't see why that's a necessity. It's just been the way I do it.

Some may wonder, what good is that technique with fields? Is it only good with integers?

Answer is, you're using identities, so the result holds in a field, though you may have trouble finding a meaning then for the "mod".

I've used tautological spaces with Diophantine equations just because that's where the motivation for the invention has been, but once trying to get creative I found some funky yet useless trigonometric identities using them, like v^2-1 + 2vcos(θ) + 2cos(θ)^2 ≡ 0(mod sin(θ) +cos(θ)+v)

For more see my math blog: http://mymath.blogspot.com/2005/07/trigonometry-relations.html

But I really don't know what the full power of this technique is, though I'd suspect it'd be up there as I've done a few things with it, and, well, you can pull in ANY mathematical equation, so you can pull in all previously known mathematical equations into a tautological space to subtract them out and look at the residue, so, I like to say, tautological spaces encompass all prior human mathematical knowledge.

Hopefully I helped my case a bit with this post and maybe the physics community will be a better audience than the mathematical community was back in December 1999 and the months thereafter when I talked about these ideas relentlessly, until I stopped and focused more on my results from using them.

The naming conventions are of course my own as the inventor. But I think I did ok.

For more on tautological spaces as well as the research I've done with them, see my math blog.