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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Gluing leaves in zeta space and automorphisms on the resulting quotient spaces

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Define an analytic planar homotopy $h_t: (0,1) \to (0,1)$ for $t\in[1/3,3]$ where $h_t(x)=e^{\frac{t}{\log x}}$ s.t. $h_{1/3}=e^{\frac{1/3}{\log x}}$ and $h_3=e^{\frac{3}{\log x}}$ for $x\in(0,1)$. I want a rotational homotopy that projects onto the planar homotopy. So, define the equivalence relation on the boundary curves of the closed subset of $(0,1)^2$, that is $h_{1/3}\sim h_3$, where points are identified by straight lines with slope $1$, connecting $h_{1/3}$ and $h_3$. This yields a spherical topology $\mathcal S$ embedded in $(0,1)^3$ with the endpoints of the sphere being singularities at $p=(0,1,1)$ and $q=(1,0,0)$. 

Conceptually it's clear that a projection $\pi: (0,1)^3 \to (0,1)^2$ takes the leaves on $\mathcal S$ to $h_t(x)$ as a double cover. But my main objective of study is to put a metric on $\mathcal S$. The $1$-parameter metric I derived on the planar space is $g_t(r)=\int_{(0,1)} x^{r-1}h_t(x)~dx = 2\sqrt{\frac{t}{r}}K_1(2\sqrt{tr})$ and it measures the distance between the $h_t(x)$ leaves. For example the distance between leaf $t=1/3$ and leaf $t=3$ is $d(1/3,3)= \bigg| g_{1/3}(r)-g_{3}(r) \bigg |.$

If one repeats this gluing process by taking $t\in[1/k,k]$ for every real $k>1$ then one obtains shells, at least mutually diffeomorphic to $\mathcal S$ embedded in $[0,1]^3$ which can be made into a smooth codimension one foliation of $(0,1)^3$ with $k=1$ corresponding to the only codimension two leaf. Call the codimension one set of shells, $\mathcal G$. 

Since the analytic homotopy $h_t(x)$ satisfies

$$t \frac{\partial^2}{\partial t^2} h_t(x) = - x \frac{\partial}{\partial x} h_t(x)$$

I would expect a compatible differential equation for $\mathcal G$ s.t. each leaf in $\mathcal G$ corresponds to an instant in time, as was such for $h_t(x)$. This is desirable as it would give a geometric flow that shrinks the surfaces asymptotically to the codimension 2 leaf. 

Some evidence for this already is that the metric $g_t(r)$ satisfies a linear third order PDE and geometric flow

$$t^2 \frac{\partial ^3}{\partial t^3}g_t(r) =r^2 \frac{\partial}{\partial r}g_t(r)$$

where we took the Mellin transform of the leaves of $h_t(x)$ to obtain the desired PDE.

Evidently, the leaves in $\mathcal G$ are compactly supported with sufficient decay properties giving access to the Paley-Wiener theorem, allowing one to conclude analyticity of the Fourier transform of these leaves. 

Now if we act on $\mathcal G$ by rigid automorphisms in $\Bbb R^3$ then the only allowable action is a $360$ degree rotation which leaves the set of leaves in $\mathcal G$ invariant. We can layer on additional symmetry by letting $\mathcal G$ be just one element and taking $\mathcal G_{\alpha}$ for $\alpha=1,2,3,4$ gaining octahedral symmetry. It can be shown that the Euler characteristic vanishes i.e. $\chi(M)=0$ where $M$ is just the union of the $\mathcal G_{\alpha}$ but that is another story. 

How do the automorphisms of $\mathcal G $ interact with the complex space? We know by the Paley-Weiner theorem that analyticity is inherited, but does an automorphism of $\mathcal G$ necessarily correspond to some invariance of the Fourier transform of a generic leave in $\mathcal G$? What I am looking for is some symmetric functional equation in $\Bbb C$ that is unchanged under an automorphism of a generic leaf in $\mathcal G$.

For example clearly $h_t(x)$ is involutive. It also satisfies $h(x^y)=h(x)^{1/y}$ which can be viewed as a eigenfunction problem. I am also reminded of the Mellin transform of the Jacobi third theta function completing the Riemann zeta function and through Poisson summation giving rise to a functional equation on the complex space. But I don't think Poisson summation applies here because $h_t(x)$ is not a series of functions.

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