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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Automorphisms and zeta space

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Here I will operate under the assumption that $\phi_s$ is a unique and well-posed solution to the following PDE. This means whatever conditions are needed for that to occur, will be imposed.


$$s\frac{\partial^2}{\partial s^2}\phi_s(x)=\mp x\frac{\partial}{\partial x}\phi_s(x) $$

for which we have our pre-assigned unique solution:

$$\mathcal \phi_{s}:=\bigg\lbrace e^{\frac{\pm s}{\log x}}:s \in \Bbb R \bigg\rbrace$$

We have singularities at $s=0$ and $x=0,1$ and this will be called a "cross singularity" or  $CS$. A natural idea would be to intialize some distribution on this $CS$ and run time forward giving a wellposed analytic solution for all time. So we choose our pre-assigned solution $\phi_s$ and run the solution back in time where we obtain what we wanted, at least qualitatively.

We can throw out our solution for $s<0$ because it doesn't directly apply to the situation, and it does not rapidly decay. Meanwhile, for $s>0$ it is Schwartz i.e. has a rapid decay criterion, and we know its Mellin transform exists and has a closed Bessel form. This will be employed to extract a *1-parameter family* of functional equations for the Riemann zeta function.

$$\Gamma(\phi,r,s)\cdot \zeta(s) = \Gamma(\hat{\phi},r,1-s)\cdot \zeta(1-s)$$

whereby moving $\zeta(s)$ to the RHS gives us a partial differential relation on the factor $\Gamma(\phi_s,r)$

$$r^2 \frac{\partial ^3}{\partial r^3}\Gamma(\phi_s,r) =s^2 \frac{\partial}{\partial s}\Gamma(\phi_s,r) $$

where we recognize that the Mellin transform of our Schwartz family, $\phi_s$ must obey a third order (shallow water wave) PDE. Here we explicitly calculate the factor which we can quickly verify as a solution.

$$\Gamma(\phi,r,s)=\int_{\mathbb R^\times \cap ~(0,1)} |x|^r~\phi_s(x)~{dx\over |x|}=2 \sqrt{\frac{s}{r}}K_1(2\sqrt{r s})$$

My motivation at least, stems from building a space, a so called zeta-space, or $\zeta$-space which somehow encodes all of what I wrote above and goes further, letting one express automorphisms of $\zeta$-space as leaving critical strips invariant under these automorphisms. 

I will now transition to more visual diagrams in part because it's much easier to convey the point and because I am much less sure how to rigorously show the mathematics in this part of the post.

$\phi_s$ is now shown to be decorated on all the panels of the unit cube in $\Bbb R^3$. I left out two panels because it would get cluttered. Here $\phi_s$ are understood to be our $1$-parameter family of Schwartz functions (the solutions we did not discard from the original 2nd order PDE).






We can build sections within the cube by gluing bounded subsets of the $\phi_s$ solutions to the original 2nd order PDE and requiring (by definition of a section) that projections take us onto the leaves, $\phi_s$ on the base space. If we do this correctly, the sections should look like this:






Finally we symmetrize to get symmetrical $\zeta$-space:





This is not rigorously posed yet because I haven't figured out how to say it precisely, but if the 3 rubber band loops around the object allow the object to act like a Rubiks cube group we can get a permutation structure on zeta-space. Any Rubiks cube transformation leaves the boundary invariant i.e. leaves the Schwartz families unchanged corresponding to an invariant quantity of the automorphisms of zeta-space.




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