From Zeta Seeds to Spectral Towers

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The basic object in the zeta space framework is a seed $$\mathcal S=(\mathcal I,\Gamma,\Pi),$$ where $\mathcal I$ is a singular stratified surface, $\Gamma\subset \mathcal I$ is a distinguished skeletal graph, and $\Pi$ is a finite symmetry group acting on the seed. The decorated version is $$\mathcal S_{\mathrm{dec}}=(\mathcal I,\Gamma,\Pi,\chi,\rho,\mathcal L),$$ where $\chi$ is coloring data, $\rho$ is holonomy or representation data, and $\mathcal L$ is a line bundle or local system. The guiding idea is that the seed itself is not yet a zeta function. Rather, the seed is a geometric object from which several different spectral realizations may be extracted. For example, one may extract a graph-theoretic realization from $\Gamma$, leading naturally to an Ihara-type zeta function. One may also extract a cone-local analytic realization from neighborhoods of the singular points of $\mathcal I$, leading to theta functions, Mellin transforms, and Bessel kernels. These are not separate ob...
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A Real World Application of $\mathcal F$-completions

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The following diagram is a high explosive lens mold from the Manhattan project, drawn in secret by David Greenglass, to pass on to the Soviets:


Below is a 3D symmetrical extension of this high explosive lens mold:


This is precisely $\mathcal I$ which is a special subset of $CX_V$ in dimension $n=3$. You can see the outer colored loops that wrap around $\mathcal I$. This is a subset of $\Gamma$. This also gives a clean example of the coloring function $\phi: E \to \mathcal C$. 

The flat local system $\mathcal L$ lives on these colored loops and it is inherited directly from the geometry of $\mathcal I$.

Thus, we can form the enrichment $\mathcal X=(\phi,\mathcal L, \rho).$ For more details on this see the previous post: https://jzdynamics.blogspot.com/2025/04/the-geometry-of-delta-groupoid.html. 

Without going into too much detail on the physics - the right thing to do is to take geodesic flows along the white strands and add (i.e. direct sum) the force vectors restricted to the collision interfaces. This encodes the force vectors into a flat bundle or local system giving the net force bundle. We can now interpret $\rho$ (the holonomy representation) in terms of a flat connection (i.e. no curvature is detected). This is because any parallel transport along any colored loop path comprising a subset of $\Gamma$ yields no angle defect.


 





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