Note: $\zeta^{1,1}=\Bbb M^{1,1}$ (under a $\log-\log$ change of coordinates). In other words, $\zeta^{1,1}$ with coordinates $(x,\phi)\in \zeta^{1,1}$ equals $\Bbb M^{1,1}$ with coordinates $(u,v)\in \Bbb M^{1,1},$ where $x=e^u$ and $\phi=e^v.$
Q: How does one lift the class structures $K_S$ and $K_T$ into $\Bbb R^3$ s.t. the resulting pseudo-riemannian space $\zeta^{1,2},$ under a $\log-\log-\log$ change of coordinates, corresponds to $\Bbb R^{1,2}:=\Bbb M^{1,2}$ (2-Minkowski space)? Another equivalent way of phrasing it is: How does one project the lines of constant time and space under an orthographic projection from $\zeta^{1,2}$ to $\zeta^{1,1}?$
The main idea is that I need to be able to map $\zeta^{1,2}$ and $\Bbb M^{1,2}$ back and forth under a $\log-\log-\log$ change in coordinates.
Context and Background:
Define a class (space class) $K_S:=\{\phi_S(x)\}$ and a class (time class) $K_T:=\{\phi_T(x)\}$ where $S,$ and $T$ generate the respective classes: $$\phi_S(x):=e^{\frac{\ln^2(s)}{\ln(x)}}.$$ $$\phi_T(x):=e^{\frac{\ln^2(t)}{\ln(1-x)}}.$$
To understand how $S$ and $T$ generate the space see: Example of an equation that generates solutions?.
With the (static, i.e. fixed background) classes generated, we can define $K_{S\cup T}:=\{\phi_S(x)\}\cup\{\phi_T(x)\}.$ We can also define $K_{S\cap T}:=\{\phi_S(x)\}\cap\{\phi_T(x)\}.$ Consider $K_{S\cap T}$.
Unpacking this explicitly, implies we can "stamp" our spacetime discretely with parametrised polynomials, $p_{s,t}(x)=x^s-(1-x)^t-$ which follows directly from equating every possible curve in each class. One can think of the "phase space" of a dynamical system if it helps.
In this view, generating our universe, How does this Lie algebra react to this change of coordinates?. $\zeta$ really means defining two "phase spaces" of curves collected into two classes.
Define a new coordinate system, based on the powers of the polynomial(s). Use $(s,t)$ to represent elements in our universe $\zeta.$ Find sequential orthographic projections, linking three different manifolds of dimension $n=1,2,3$ This puts elements in $\zeta$ in bijection with the natural numbers. This shows that initialising $\zeta$ with this coordinate system allows us to work in the countable natural number setting, and define such things as successors after defining a partial order lattice.
Using $(s,t)$ as the coordinate system with $p_{s,t}(x),$ codes $\zeta$. $(s,t)$ is a coordinate system which is more global. Penetrating down to the local coordinate system, we can look at $(x,\phi)$ which acts to code local objects in $\zeta.$ In other words the local code is embedded and parametrised by the global code.
We can also transfer this model back to $\zeta:=\Bbb M^{1,1}$ with a change of coordinates. Since $\zeta$ is defined as Minkowski spacetime in $\exp-\exp$ space. So in order to revert back to the pseduo-riemannian manifold landscape representing the backdrop of special relativity, one does $u=\ln(x)$ and $v=\ln(y).$ One can easily define lorentz boosts and the like. If this isn't clear, $uv=\ln^2(s)$ is a rectangular hyperbola with parameter $s.$ And it's implicitly assumed that the lorentz metric is being used.
