## JSH: Basic tautological spaces example

Ten years ago this month I discovered what I call tautological spaces. Partly as a commemoration of the anniversary this month here is a demonstration post of a basic tautological spaces example.

What are tautological spaces? Most simply they are identities.

A simple tautological space: x+y+vz = x+y+vz

Identities are so dismissed that I also have the bonus of being someone who found a surprisingly simple use for them—probing equations for analysis. I invented the term "tautological space" as well.

The 'v' variable is free, so you always have one extra degree of freedom.

My innovation moving forward from Gauss was to use "mod" with identities:

x+y+vz = 0(mod x+y+vz), which is the equivalent of x+y+vz = x+y+vz

You use tautological spaces with non-identities which I call conditionals.

Here's using tautological spaces with x^2 + y^2 = z^2.

x+y+vz = 0(mod x+y+vz), so x+y = -vz(mod x+y+vz), squaring both sides:

x^2 + 2xy + y^2 = v^2 z^2 (mod x + y + vz), now subtract out

x^2 + y^2 = z^2, giving

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

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

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

and I can get rid of x, with x = -y-vz (mod x+y+vz), so I have:

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

which just says that for ANY v you choose, you have that x+y+vz, will be a factor of what's on the left hand side which is the residue.

And that is not really ring specific, as if you don't use "mod" you can do it all explicitly. But it's most meaningful in rings where "factor" is meaningful.

So like with v=1, I have 2yz + 2y^2 = 0 (mod x+y+z), and trivially with x=3, y=4, z=5, you have:

2(4)(5) + 2(16) = 72 has 12 as a factor.

So you get the result that if x^2 + y^2 = z^2, then 2y(z+y) has x+y+z as a factor.

Notice that is true over infinity. Tautological spaces give answers over infinity in general. So their use always encapsulates an infinite set.

They add on to existing knowledge, as notice you can analyze x^2 + y^2 = z^2 separately, or throw anything you want at its residue using the variable 'v' which is your control variable.

Now this month ten years of tautological spaces! A fascinatingly simple idea using identities, which just so happens to capture infinity.