What I like to call tautological spaces are complex identities that can be used to do algebraic manipulation. The benefit of allowing the math to do such manipulations is easier perfection, and insights which seem demonstrably to elude human beings.
My basic demonstration for simplicity and familiarity talks analyzing: x2 + y2 = z2
That can be pulled from a complex identity:
x2 + 2xy + y2 = v2z2 - 2vz(x+y+v) + (x+y+vz)2
Leaving a residue:
(v2-1)z2 = 2xy + 2vz(x+y+vz) - (x+y+vz)2
And since v is still a free variable, you can do equivalent of symbol manipulation by choice of v, for instance, with v = 1 find easily:
2xy = (x+y+z)(x+y-z)
Where notice with our original expression which I like to call the conditional, we can add 2xy to both sides. That simple move is to get a key perfect square, allowing a factorization:
x2 + 2xy + y2 - z2 = 2xy, and then easily: (x+y-z)(x+y+z) = 2xy
Easy example for demonstration to help with grasping the basic concepts.
You can shift through every possible way to manipulate algebraically with tautological spaces using v as your tool to probe an expression. As you shift v, the algebra moves everything around for you. And again used a simple quadratic for simplicity and familiarity, but this approach does not care.
Importantly, you can use modular to greatly simplify complexity that can build rapidly with the complex identities--where is to its proper purpose as you do not need all of that information.
So the actual tautological space for analysis: x+y+vz = 0(mod x+y+vz)
Where the result then is: (v2-1)z2 = 2xy (mod x+y+vz)
Which I call the conditional residue, which is true when: x2 + y2 = z2
You can rapidly zip through EVERY way an expression can be algebraically manipulated, just looking for things, and asking questions.
The greater ease with which algebraic manipulation can be done then is astonishing, and is interesting to me that it took me finding this approach. And found it December 1999 when looking to get a better handle.
That's strange, huh? And have wondered as what do I care really, just glad to have the tools now! Yet yeah, must be the original discoverer as once you have these ways, how do you go back?
Fumbling with equations and trying a few things here or there which you can do indefinitely, when the math can give you huge answers in minutes? There's no way to go back for me, at least. Would be kind of like choosing to walk a thousand miles when you can just fly there in an airplane.
So I talk to the math now, and ask the math questions, where when tautological spaces are available as an option can just get answers so much more quickly than any other way.