A Four-Sheeted Orbifold Foam as a Planck-Cell Carrier

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 One possible way to think about Planck-scale structure is not to begin with a smooth spacetime manifold, but with a finite geometric carrier on which quantum data can be placed. In this note I describe a speculative mathematical object of this kind: a four-sheeted football-orbifold foam suspended over the face poset of a cube. The proposal is not that ordinary spacetime literally contains small Euclidean cubes. Rather, the cube poset should be understood as a finite incidence scaffold: it records which zero, one, two, and three-dimensional strata are allowed to meet, while the actual geometric realization is an orbifold/Klein foam built over that combinatorial skeleton.  Let $P_{\Box}$ denote the face poset of a cube. The eight vertices of the cube serve as incidence markers for the eight cone points of the foam. Since a football orbifold has two cone points, one can naturally attach four football sheets by pairing the eight vertices into four antipodal pairs. Thus we begin w...
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A description of $\zeta-$space

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I would like to attempt to define a mathematical space I call $\zeta-$space.  $\zeta-$space has a description as the transversal intersection of two isometric pseudo-Riemannian manifolds (Lorentzian submanifolds) both equipped with the Lorentz metric. It is somewhat laborious to describe and derive $\zeta$ because it is not a standard approach and $\zeta$ does not have a clean representation so to speak. However it is a fundamentally important mathematical and physical object in its own right and should be studied.

 I define $\zeta$ in the following way:

 $\zeta:=\{\varphi_S \} \cap \{ \varphi_T \}$

 Under this first level description, $\zeta$ is the intersection between two class structures denoted as $\varphi_S$ and $\varphi_T.$ These class structures can be thought of as a family of functions and also as lines of constant time and constant space respectively:

 $$ \varphi_S(x) = e^{\frac{S}{\log(x)}} $$

 $$ \varphi_T(x) = e^{\frac{T}{\log(1-x)}}$$

 Here I view $S,T$ as fundamental generators. That is, they completely generate the space. Here is how I derived them and began to think of them as generators:

 I set $\varphi_S(x)=x$ and solved for $S.$ I got $S=\{\log^2(x) : x \in \Bbb R \cap (0,1) \}.$ However I thought it better to replace $x$ with some parameter $s.$ So, $S=\{\log^2(s) : s \in \Bbb R \cap (0,1) \}.$ Likewise I set $\varphi_T(x)=1-x$ and solved for $T.$

 Here we have a visualization of $\zeta$ showing the overlapping class structures $\{ \varphi_S\}$ and $\{\varphi_T\}:$

$\zeta-$space as the intersection of two Lorentzian submanifolds


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