The Spectral Hermitian Surface $(\Bbb C^\times, g_{\infty})$ as an Avatar of a Modular Surface

Consider $\mathcal F=\lbrace \mathcal L_t=(\Bbb C^{\times},g_t(s)) \rbrace_{t\in\Bbb R_{\gt 0}}$ where $g_t(s)=\vert F_t(s) \vert^2ds\otimes d\bar{s}$ and

$$F_t(s)=\int_{(0,1)} e^{t^2/\log x} \cdot x^{s-1}dx= 2\sqrt{\frac{t^2}{s}}K_1(2\sqrt{t^2s}). $$

For $K_1$ the modified bessel function of the second kind.

Looking at Bessel asymptotics we know that as $s\to 0$, $g_t(s)\sim \frac{1}{|s|^2}ds\otimes d\bar{s}$ (conical) and as $s\to\infty$, $g_t(s)\sim e^{-4\sqrt{t^2s}}ds\otimes d\bar{s}$ (collapsing end), and these asymptotics hold for the later defined $g_{\infty}(s).$ 

Define a trace over $F_t$

$$\mathcal M(s)=\sum_{t=1}^\infty F_t(s)$$

then the following identity is satisfied  

$$\frac{1+2\mathcal M(s)}{1+2\mathcal M(1/s)}= s^{\alpha}$$

where $\alpha$ is the weight. The proof that there exists some $\alpha \in \Bbb R$ such that the identity holds for all $s$ uses Poisson summation. 

Define a new metric which encodes the cumulative effect of the metrics on each of the surfaces $\mathcal L_t$ into one

$$g_{\infty}(s)=:\vert \Phi(s) \vert^2ds\otimes d\bar{s}$$

where we have

$$\Phi(s)=1+2\mathcal M(s)$$ then 

$$\Phi(1/s)= s^{\alpha} \Phi(s)$$

and so if we define the Hermitian metric using

$$h(s)=\vert \Phi(s) \vert^2$$

then

$$g_{\infty}(s)=h(s) \cdot ds \otimes d\bar{s}$$

which transforms like

$$g_{\infty}(1/s)= \vert s \vert^{2(\alpha-2)} \cdot g_{\infty}(s)$$

So we have the conformal Hermitian surface $(\Bbb C^\times,g_{\infty})$ whose metric transforms under  $s\mapsto  1/s$ with a modular type scaling factor. 

Due to the asymptotics of the metric, this surface has an infinite cusp as $s\to\infty$ and expands outward as $s\to 0$ but not as a surface of revolution - the expansion is anisotropic. So the surface is like a warped cylinder that pinches in one direction and expands anisotropically in the other.

A Singular Cylinder and the Shadow of a Modular Surface

Let $\{ f_t(x) := e^{\frac{t}{\log x}} \}_{t \in [1/2,2]}$ be a smooth family of functions on $(0,1)$, and let $F_t(s)$ denote their Mellin transforms:

$$F_t(s) := \int_0^1 f_t(x) \, x^{s-1} \, dx = 2 \sqrt{\frac{t}{s}} \, K_1(2\sqrt{ts}),$$

where $K_1$ is the modified Bessel function of the second kind. Define the Hermitian metric

$$g_t(s) := |F_t(s)|^2 \, ds \otimes d\bar{s}$$

on the punctured complex plane $\mathbb{C}^\times$.

Let $S$ be the topological quotient obtained by identifying the boundary curves $f_{1/2}(x)$ and $f_2(x')$ along lines of slope $+1$ in the $(x,f)$-plane. Then:

The space $S$ is homeomorphic to the $2$-sphere with two conical singularities, arising from the boundary identifications. The metric $g_t(s)$ induces a globally defined Hermitian metric $g^{\sim}(s)$ on $S$, with conical singularities at $s = 0$ and $s = \infty$. Near $s = 0$, the metric behaves as

    $$g_t(s) \sim \frac{1}{|s|^2} ds \otimes d\bar{s},$$

    which corresponds to the canonical flat cone metric of angle $2\pi$ on $\mathbb{C}^\times$. Near $s \to \infty$, the metric decays exponentially as

    $$g_t(s) \sim e^{-4\sqrt{ts}} \, |ds|^2,$$

producing an exponentially collapsing end. While this does not define a true conical singularity in the sense of angle deficit or curvature concentration, it produces an effective degeneration of the metric volume, allowing the end to be compactified topologically, though not metrically, as a cone point. In other words we have a conical singularity at one pole, and an infinitely thin neck at the other pole (not singular in curvature, but vanishing in volume).    

The surface $S$ may be interpreted as a doubly pinched cylinder: a Hermitian surface conformally equivalent to the open cylinder $S^1 \times \mathbb{R}$, with one genuine conical singularity and one asymptotically collapsing end, induced respectively by Bessel blow-up and decay.

Question

Can the spectral-pinched surface $S$, defined via Mellin–Bessel transforms and slope-aligned boundary identifications, be interpreted as a model for a singular compactification of a modular-like surface, where the conical singularity and collapsing end respectively resemble an elliptic fixed point and a cusp?

On modular curves $\Gamma\backslash\Bbb H \cup \lbrace \mathrm{cusps} \rbrace$ we encounter elliptic fixed points i.e. finite order points with cone angle $2\pi/m$, and cusps i.e. infinite volume ends where Eisenstein series and Poincaré series exhibit exponential decay.

In my surface $S$, the point $s=0$ where $g_t(s)\sim\frac{1}{|s|^2}\vert ds \vert^2$ is a flat cone with angle $2\pi$ - resembling an elliptic point of order $1$. Also we have the point $s\to\infty$ where $F_t(s)\sim e^{-2\sqrt{ts}}$ has metric $g_t(s)\sim e^{-4\sqrt{ts}}\vert ds\vert^2$, which mirrors the metric behavior near a cusp. So, $S$ does resemble a modular curve with one cusp and one elliptic fixed point of order $1$.

Additionally, in automorphic theory, Fourier-Whittaker expansions of Maass forms and Eisenstein series often take a form of a series of modified Bessel functions of the second kind which is similar to the form I'm dealing with.

All of this suggests that there may be more than just an analogy going on, and $S$ could be a kind of modular surface.

Collapse-Driven Gauge Synchronization over $\Bbb Z^2$.

To begin formally laying the groundwork for $\zeta$-space we must revisit the spectral tower. To each $\mathcal F$-completion in dimension $n$, we can take a symmetrical subset $\mathcal I^{(n-1)}$ and enrich these objects with bundles restricted to skeletons. In dimension $n=3$ we have $\mathcal I^{(3-1)}$ and we take line bundles restricted to $\Gamma$ from which we then can associate an Ihara zeta function to $\Gamma$, which may be twisted. 

Then we will define $\zeta$-space, or $\zeta^n$ as the space where we associate copies of $\mathcal I^{(n-1)}$ to an integer lattice, $\Bbb Z^n \subset \Bbb R^n$. A lot of work must be done from here but the benefits of this work will be very useful theoretically as well as for applications.

Essentially we introduce the spectral tower to classify all the elements which will then be assigned to lattice sites. 

Let $\zeta^n$ denote the zeta-structured lattice space:

$$\zeta^n := \left\{ z \mapsto \left( \mathcal{I}^{(n-1)}_z,\ \mathcal{L}_z|_{\Gamma_z},\ \zeta_{\Gamma_z}^{\rho_z} \right) \,\middle|\, z \in \mathbb{Z}^n \right\}$$

where:

- $\mathcal{I}^{(n-1)}_z$ is a stratified complex embedded at site $z$

- $\Gamma_z \subset \mathcal{I}^{(n-1)}_z$ is its 1-skeleton

- $\mathcal{L}_z$ is a line bundle restricted to $\Gamma_z$

- $\rho_z : \pi_1(\Gamma_z) \to G$ is a holonomy representation (possibly twisted)

- $\zeta_{\Gamma_z}^{\rho_z}$ is the associated (possibly twisted) Ihara zeta function

Now we consider a lattice-based dynamical system indexed by $\mathbb{Z}^2$, where each site $(m,n) \in \mathbb{Z}^2$ is associated with a twisted Ihara zeta function

$$\mathcal{Z}_{m,n}(u) := \zeta_{\Gamma}(u, \rho_{m,n}),$$

where $\rho_{m,n} \in \mathrm{Rep}(\pi_1(\Gamma), G)$ is a holonomy representation of a fixed $4$-regular connected planar graph $\Gamma$ with $\mathrm{Aut}(\Gamma) \cong D_4$.

A $\Bbb Z^2$ slice of $\zeta^3$ where each element is associated to a lattice site.

Each $\rho_{m,n}$ is not fixed, but exists in a formal superposition over a finite orbit of twisted configurations:

$$\rho_{m,n} \in \mathbb{C}[\mathrm{Orb}_{\Psi}(\rho_0)] \quad \text{where } \Psi \in D_4 \text{ acts by internal partial reattachments of local strata}.$$

The group $D_4$ is realized internally at each site via four elementary partial permutations:

$$\Psi_T = (1\ 2), \quad \Psi_B = (3\ 4), \quad \Psi_L = (1\ 3), \quad \Psi_R = (2\ 4),$$

where the numbers correspond to quadrant positions NW (1), NE (2), SW (3), SE (4). These generators act as transpositions on the internal labels of a quadrant-colored square and collectively generate $D_4$ by composition.

Each site holds a state 

    $$\tilde{\rho}_{m,n} \in \mathbb{C}[\mathrm{Orb}_{\Psi}(\rho_0)],$$

representing a formal linear combination over all twisted configurations under the $D_4$ action. Observing a specific site $(m_0, n_0)$ collapses the state:

    $$\tilde{\rho}_{m_0, n_0} \longrightarrow \rho_{m_0, n_0} = g_* \cdot \rho_0,$$

where $g_* \in D_4$ is the observed local configuration. All other lattice sites update their configuration to match $g_* \cdot \rho_0$ using a discrete local rule:

    $$\rho_{m,n}(t+1) = \Psi_i \cdot \rho_{m,n}(t) \quad \text{if } d_{\mathrm{Cayley}}(\Psi_i \cdot g_{m,n}(t), g_*) < d_{\mathrm{Cayley}}(g_{m,n}(t), g_*),$$ 

where $g_{m,n}(t) \in D_4$ encodes the local state at time $t$.

Grothendieck Site over the Spectral Tower

Sheaf Stack over the Dimensional Stratification Site

We define a stratified site $\mathcal{S}$ indexed by dimension $n \in \mathbb{Z}_{\geq 2}$, whose objects are geometric-combinatorial models $\mathcal{I}^{(n)}$ associated to dimension-$n$ F-completions. Morphisms in this site correspond to boundary inclusions, coordinate projections, and degenerations between dimensions:

$$\mathcal{I}^{(n)} \to \mathcal{I}^{(n-1)} \quad \text{via coordinate slice or foliation boundary}.$$

Let $\mathcal{M}_\Delta^{(n)}$ denote the moduli space of $\Delta$-twisted holonomy representations in dimension $n$, with $\mathcal{M}_\Delta^{(2)} = \mathrm{U}(1)^r / \Delta$ the classical representation torus quotient.

We define a sheaf stack of zeta data over $\mathcal{S}$:

$$\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \longrightarrow \mathbf{Stacks}$$

such that:

• For each $n$, $\mathscr{Z}(\mathcal{I}^{(n)})$ is a stack (e.g. derived or topological) encoding the family of zeta sheaves over $\mathcal{M}_\Delta^{(n)}$.
• The stalk of $\mathscr{Z}$ at a point corresponds to a twisted Ihara-type zeta function valued in $\mathbb{C}[[u]]$.
• Morphisms in $\mathcal{S}$ induce restriction or pullback functors on zeta stacks:

$$\mathscr{Z}(\mathcal{I}^{(n-1)}) \leftarrow \mathscr{Z}(\mathcal{I}^{(n)}),$$

compatible with foliation collapse, singular degeneration, or representation contraction.

• The delta symmetry group $\Delta$ acts on $\mathscr{Z}$ as a group of natural transformations, preserving the stratified structure.

Sheaf Stack Duality over Foliated Stratification Site Conjecture:

There exists a stacky equivalence

$$\mathscr{Z}(\mathcal{I}^{(n)}) \cong \mathrm{Sh}_\zeta(\mathcal{I}^{(n)})$$

between the zeta sheaf stack over the $n$-dimensional moduli space $\mathcal{M}_\Delta^{(n)}$ and a sheaf of spectral data over the geometric space $\mathcal{I}^{(n)}$.

These equivalences are natural in $n$, delta-equivariant, and compatible with restriction morphisms in $\mathcal{S}$. Thus, the entire tower forms a categorified duality:

$$\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \longrightarrow \mathbf{Stacks}_{\zeta},$$

encoding the evolution of spectral invariants across dimension.

Derived Categories and Zeta Fourier–Mukai Duality

Let $\mathcal{S}$ be the stratified site indexed by dimension $n \geq 2$, with objects $\mathcal{I}^{(n)}$ and morphisms given by boundary projections and coordinate slicing between F-completions.

For each $n$, we associate:

• A derived category of sheaves of zeta-type objects:

    $$\mathcal{D}_\zeta^{(n)} := D^b\big(\mathrm{Sh}_\zeta(\mathcal{I}^{(n)})\big),$$

    where $\mathrm{Sh}_\zeta(\mathcal{I}^{(n)})$ is the category of sheaves (or complexes) encoding twisted zeta functions over the stratified geometry $\mathcal{I}^{(n)}$.

• A derived category of equivariant sheaves over the moduli space:

    $$\mathcal{D}_\Delta^{(n)} := D^b\big(\mathrm{Coh}_\Delta(\mathcal{M}_\Delta^{(n)})\big),$$

    where $\mathrm{Coh}_\Delta$ denotes the category of $\Delta$-equivariant coherent sheaves (or perfect complexes) over the moduli orbifold.

We posit the existence of a Fourier–Mukai-type transform between these categories.

Let $\mathcal{Z}^{(n)} \in D^b\big(\mathrm{Coh}(\mathcal{I}^{(n)} \times \mathcal{M}_\Delta^{(n)})\big)$ denote a zeta kernel object, playing the role of a spectral correspondence. Then define:

$$\Phi_{\mathcal{Z}^{(n)}} : \mathcal{D}_\Delta^{(n)} \longrightarrow \mathcal{D}_\zeta^{(n)}, \quad \mathcal{F} \mapsto Rp_{1*}(\mathcal{Z}^{(n)} \overset{L}{\otimes} p_2^* \mathcal{F}),$$

where $p_1, p_2$ are the projections from $\mathcal{I}^{(n)} \times \mathcal{M}_\Delta^{(n)}$ onto each factor.

Zeta Fourier–Mukai Equivalence

There exists a kernel object $\mathcal{Z}^{(n)}$ such that the Fourier–Mukai transform

$$\Phi_{\mathcal{Z}^{(n)}} : \mathcal{D}_\Delta^{(n)} \xrightarrow{\sim} \mathcal{D}_\zeta^{(n)}$$

is an equivalence of derived categories.

This equivalence respects the stratified structure of $\mathcal{S}$ and is compatible with delta actions, restriction to boundary strata, and deformation of F-completions across dimensions.

Furthermore, for $n > 2$, the entire tower

$$\{ \Phi_{\mathcal{Z}^{(n)}} \}_{n \geq 2}$$

defines a system of zeta transforms across the site $\mathcal{S}$, giving rise to a categorified global Fourier–Mukai theory over the moduli–geometry tower.

Grothendieck Site Structure on the Stratified Tower

We now endow the dimensionally stratified collection $\mathcal{S} = \{ \mathcal{I}^{(n)} \}_{n \geq 2}$ with the structure of a Grothendieck site, allowing us to define stacks, cohomology, and derived functors over this tower.

Stratified F-completion Site $\mathcal{S}$

Let $\mathcal{S}$ be the category whose objects are compact stratified spaces $\mathcal{I}^{(n)}$ arising as coordinate slices of an $n$-dimensional F-completion, and whose morphisms are:

• Coordinate projection maps $\pi_{n,m} : \mathcal{I}^{(n)} \to \mathcal{I}^{(m)}$ for $m < n$;
• Singular degeneration maps collapsing cone points or foliation leaves;
• Foliation-preserving inclusions along boundary strata.

We define a Grothendieck topology on $\mathcal{S}$ by declaring a family $\{ f_i: \mathcal{I}^{(n_i)} \to \mathcal{I}^{(n)} \}_{i \in I}$ to be a covering if the images of the $f_i$ jointly cover all regular and singular strata of $\mathcal{I}^{(n)}$.

This topology captures local charts in the stratified sense (e.g., charts homeomorphic to open subsets of $\mathbb{R}^k / G$ for some finite $G$), respecting delta symmetry and foliation structure.

Sheaf and Stack Theory over $\mathcal{S}$

Given this site structure, we define:

• A zeta sheaf stack

    $$\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \longrightarrow \mathbf{Stacks},$$

    assigning to each $\mathcal{I}^{(n)}$ a stack of zeta-invariant sheaves encoding holonomy-twisted spectral data.

    

• A derived category of global sections

    $$\mathcal{D}_{\text{global}} := D^b(\Gamma(\mathcal{S}, \mathscr{Z})),$$

    representing globally defined complexes of zeta sheaves over the stratified tower.

• For each pair of strata $\mathcal{I}^{(n)}$, $\mathcal{I}^{(m)}$, a space of derived morphisms:

    $$\operatorname{Ext}^k_{\mathcal{S}}(\mathscr{Z}(\mathcal{I}^{(m)}), \mathscr{Z}(\mathcal{I}^{(n)})) := \mathrm{Ext}^k_{\mathbf{Stacks}}(i_! \mathscr{Z}(\mathcal{I}^{(m)}), i_* \mathscr{Z}(\mathcal{I}^{(n)})),$$

    where $i: \mathcal{I}^{(m)} \hookrightarrow \mathcal{S}$ is the inclusion functor.

Zeta Stack Cohomology Conjecture:

The cohomology of the zeta sheaf stack over $\mathcal{S}$ encodes the stratified spectral flow of zeta invariants across dimensions. In particular:

$$H^k(\mathcal{S}, \mathscr{Z}) \cong \bigoplus_{n \geq 2} \operatorname{Ext}^k_{\mathcal{S}}(\mathscr{Z}(\mathcal{I}^{(n)}), \mathscr{Z}_{\mathrm{tot}}),$$

where $\mathscr{Z}_{\mathrm{tot}}$ is the terminal zeta object capturing global delta-equivariant sections.

Furthermore, these Ext pairings organize the zeta flows into a spectral sequence, with:

$$E_1^{n,k} = H^k(\mathcal{I}^{(n)}, \mathscr{Z}) \Rightarrow H^{n+k}(\mathcal{S}, \mathscr{Z}),$$

indicating how local spectral data lifts to global geometric information over the tower.

Motivic Zeta Function of the Stratified Site

We define a global motivic zeta function that captures the cumulative structure of zeta sheaf data across all strata $\mathcal{I}^{(n)}$ of the stratified site $\mathcal{S}$.

Motivic Zeta Function

Let $\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \to \mathbf{Stacks}$ be the zeta sheaf stack over the Grothendieck site $\mathcal{S}$.

The motivic zeta function of the tower is defined as the formal power series:

$$\zeta_{\mathcal{S}}(t) := \sum_{n=2}^\infty \chi\big(\mathcal{D}_\zeta^{(n)}\big) \cdot t^n,$$

where:

• $\mathcal{D}_\zeta^{(n)} := D^b(\mathrm{Sh}_\zeta(\mathcal{I}^{(n)}))$ is the derived category of zeta sheaves over $\mathcal{I}^{(n)}$,
• $\chi(-)$ denotes the categorical Euler characteristic, or alternatively, a suitable trace invariant such as:

    $$\chi(\mathcal{D}_\zeta^{(n)}) := \sum_k (-1)^k \dim \operatorname{Ext}^k_{\mathcal{S}}(\mathscr{Z}(\mathcal{I}^{(n)}), \mathscr{Z}_{\mathrm{tot}}).$$

Motivic Zeta Cohomology Conjecture:

The global motivic zeta function $\zeta_{\mathcal{S}}(t)$ admits the interpretation:

$$\zeta_{\mathcal{S}}(t) = \operatorname{Tr}\left( (-1)^F \,|\, \mathbb{H}^\bullet(\mathcal{S}, \mathscr{Z}) \right),$$

where $F$ is the cohomological degree functor and $\mathbb{H}^\bullet(\mathcal{S}, \mathscr{Z})$ denotes the hypercohomology of the sheaf stack.

Moreover, if each $\mathcal{I}^{(n)}$ corresponds to a representation stratum of a classifying stack $B\pi_1(\Gamma_n)$, then $\zeta_{\mathcal{S}}(t)$ may admit a rational expression analogous to a motivic zeta function in the sense of Denef–Loeser:

$$\zeta_{\mathcal{S}}(t) = \prod_{n=2}^\infty \left( \frac{1}{1 - \mathbb{L}^{w_n} t^n} \right)^{\chi_n},$$

where:

• $\mathbb{L}$ is the class of the affine line in the Grothendieck ring of stacks,
• $w_n$ measures the weight of zeta data on level $n$,
• $\chi_n = \chi(\mathcal{I}^{(n)})$ or its delta-equivariant refinement.

The Spectral Tower of Zeta Moduli

 Let $\Gamma$ be a finite connected graph (or multigraph) with first Betti number $r = \beta_1(\Gamma)$, and let $\Delta \subset \mathrm{Aut}(\Gamma)$ be a finite subgroup of automorphisms acting on $\pi_1(\Gamma)$ via pullback.

Let $\mathcal{M}_\Delta := \mathrm{U}(1)^r / \Delta$ denote the moduli space of flat unitary representations of $\pi_1(\Gamma)$ up to $\Delta$-symmetry — equivalently, the moduli space of holonomy classes for $\mathrm{U}(1)$-bundles twisted by $\Delta$.

Suppose further that there exists a stratified geometric object $\mathcal{I} \subseteq [-1,1]^3$, constructed from $\Gamma$ and $\Delta$, encoding topological, singular, or foliation-theoretic data derived from $\Gamma$ and its symmetries. For example, $\mathcal{I}$ may arise as a cone-singular surface with corners or as a compactification of a flow space determined by $\Gamma$.

Spectral Moduli Duality Conjecture: There exists a natural equivalence of orbifolds (or derived stacks)

$$\mathcal{M}_\Delta \cong \mathcal{I},$$

such that the zeta sheaf

$$\mathcal{Z}_\Gamma : \mathcal{M}_\Delta \to \mathbb{C}[[u]]$$

pulls back to a sheaf of spectral invariants over $\mathcal{I}$ whose local structure reflects the singularities, foliations, and holonomy patterns present in $\mathcal{I}$.

This equivalence is equivariant with respect to the $\Delta$-action and intertwines representation-theoretic symmetries on $\mathcal{M}_\Delta$ with geometric symmetries of $\mathcal{I}$.

Dimensional Tower of Spectral-Geometric Duality

Let $\Gamma$ be a connected 4-regular multigraph equipped with a finite symmetry group $\Delta \subset \mathrm{Aut}(\Gamma)$ acting on its edge set and on $\pi_1(\Gamma)$. As previously discussed, this action lifts to a pullback action on the moduli torus $\mathrm{U}(1)^r$, leading to the moduli orbifold

$$\mathcal{M}_\Delta^{(2)} := \mathrm{U}(1)^r / \Delta,$$

where $r = \beta_1(\Gamma)$. The twisted Ihara zeta function

$$\mathcal{Z}_\Gamma : \mathcal{M}_\Delta^{(2)} \to \mathbb{C}[[u]]$$

defines a sheaf of spectral invariants over this space.

Now, suppose there exists a stratified geometric space $\mathcal{I} \subset [-1,1]^3$ (the "$\mathcal{I}$-structure''), which is:

1. a 3-dimensional extension or "coordinate slice'' of a 4-dimensional object (a subset of an F-completion}),
2. equipped with an induced $\Delta$-action compatible with its geometric and combinatorial structure,
3. such that $\mathcal{I}$ encodes a surface foliation or flow model associated to $\Gamma$.

Then, we postulate that $\mathcal{I}$ serves as a geometric realization of the same moduli-theoretic information as $\mathcal{M}_\Delta^{(2)}$, but lifted into higher dimension. More generally, we posit the existence of a tower of spectral-geometric structures indexed by dimension:

$$\begin{array}{cccccc} n = 2 &\rightsquigarrow& \Gamma &\rightsquigarrow& \mathcal{M}_\Delta^{(2)} = \mathrm{U}(1)^r / \Delta \\ n = 3 &\rightsquigarrow& \mathcal{I} &\rightsquigarrow& \mathcal{M}_\Delta^{(3)} \cong \mathcal{I} \\ n = 4 &\rightsquigarrow& \widehat{\mathcal{I}} &\rightsquigarrow& \mathcal{M}_\Delta^{(4)} \\ \vdots & & \vdots & & \vdots \\\end{array}$$

Here:

1. $\widehat{\mathcal{I}}$ denotes the special subset of the F-completion: a 4-dimensional object whose coordinate slice at $z = 0$ yields $\mathcal{I}$,
2. Each $\mathcal{M}_\Delta^{(n)}$ is a moduli space of twisted holonomy data in dimension $n$, possibly interpreted as a quotient of a higher torus,
3. Each layer supports a delta symmetry action, and the zeta sheaf lifts coherently along the tower.

Dimensional Tower Duality

There exists a functorial correspondence

$$\mathcal{M}_\Delta^{(n)} \cong \mathcal{I}^{(n)},$$

between moduli spaces of delta-equivariant spectral data and geometric-combinatorial objects $\mathcal{I}^{(n)}$, defined inductively as coordinate slices or boundary degenerations of an $n$-dimensional F-completion.

This tower is compatible with:

1. Delta actions at each level,
2. Zeta sheaves varying holonomically across dimensions,
3. Intersectional structure between different $\mathcal{I}^{(n)}$ along coordinate planes,
4. A unifying stratified graph structure enriched by foliation singularities.

Stratified Holonomy Dynamics of the Ihara Zeta function of $\Gamma$

Let $\Gamma$ be a finite connected graph (e.g., the 1-skeleton of a stratified space or foliated complex), and let $\mathcal{R}$ denote the stratified space of holonomy representations:

$$\mathcal{R} = \bigsqcup_{G \subseteq GL_n(\mathbb{C})} \mathcal{R}_G,\quad \mathcal{R}_G := \left\{ \rho : \pi_1(\Gamma) \to G \right\}.$$

Each stratum $\mathcal{R}_G$ corresponds to a distinct choice of structure group $G$, such as $U(1)$, $SU(2)$, or  $GL_n(\mathbb{C})$.

Define a stratified holonomy evolution governed by a sequence of generalized symmetry transformations

$$\Psi_k : \mathcal{R}_{G_k} \longrightarrow \mathcal{R}_{G_{k+1}},$$

which may be continuous or discrete, invertible or not, and may preserve or enhance the structure group.

The evolution of representations is given by the discrete recurrence:

$$\rho_{k+1} = \Psi_k(\rho_k), \qquad \rho_k \in \mathcal{R}_{G_k},\; \rho_{k+1} \in \mathcal{R}_{G_{k+1}}.$$

This defines a dynamical system over the stratified moduli space $\mathcal{R}$, where transitions between strata correspond to changes in the structure group, such as:

$$U(1) \longrightarrow GL_2(\mathbb{C}) \longrightarrow GL_3(\mathbb{C}) \longrightarrow \cdots$$

or possibly to reductions:

$$GL_n(\mathbb{C}) \longrightarrow U(1).$$

Associated to each $\rho_k$, we define a (possibly twisted) Ihara-type zeta function:

$$\zeta_k(u) := \prod_{[p]} \det\left( I - \rho_k(p) u^{\ell(p)} \right)^{-1},$$

where the product is over equivalence classes of primitive closed paths $[p]$ in $\Gamma$, and $\ell(p)$ is the length of the path.

The evolution of zeta invariants under $\Psi_k$ is given by the pushforward:

$$\zeta_{k+1}(u) = \Psi_{k*}\left( \zeta_k(u) \right),$$

which reflects any change in the spectral type or determinant structure induced by the transformation of the holonomy representation.

This framework allows for discrete-time dynamics across the space of representations and zeta functions, capturing both internal deformations (when $G_{k+1} = G_k$) and structural enhancements (when $G_{k+1} \supsetneq G_k$) driven by the generalized symmetry $\Psi_k$.

The Moduli Space of Holonomies and the Twisted Ihara Zeta Function

 Consider a triple:

$$(\mathcal I, z, \Sigma)$$

where $\mathcal I$ is the canonical stratified surface defined in previous posts (smooth away from a singular set) embedded in $[-1,1]^3$, $z$ is the vertical projection (interpreted as time), and $\Gamma=\Sigma_0$ is the critical level set.

Define a movie that is given by:

$$\Sigma_z :=\begin{cases}\{v_1, v_2, v_3, v_4\} & \text{if } z = \pm 1, \\\\\coprod_{i=1}^4 S^1_i & \text{if } z \in (-1, 0) \cup (0, 1), \\\\\Gamma & \text{if } z = 0,\end{cases}$$

where each $S^1_i$ denotes a topological circle, and $\Gamma$ is a finite connected 4-regular multi-graph formed by the merging of four disjoint circles into two overlapping ovals, with their four intersection points defining the graphs vertices. 

For each $z \in (-1,0) \cup (0,1)$, the level set $\Sigma_z$ consists of four disjoint circles $S^1_1, \ldots, S^1_4$. Along each $S^1_i$, we define a real line field:

$$V_i \subset T\mathbb{R}^3|_{S^1_i},$$

consisting of unit-length vectors orthogonal to the slicing plane $\{z = \text{const}\}$ at each point. That is, if $T_x S^1_i \subset T_x \mathbb{R}^3$ is the tangent to the circle at $x$, then $V_i(x)$ is a 1-dimensional real subspace orthogonal to both $T_x S^1_i$ and to the horizontal plane.

As $z \to 0$, the four disjoint circles merge into the connected graph $\Gamma$. At this stage, the four real vertical bundles $\{V_i\}$ glue together to form a single real line bundle:

$$V_\Gamma \to \Gamma,$$

defined by identifying vertical directions at the junction points where circles coalesce. This gluing process descends the structure of $\bigoplus V_i$ into a globally defined vertical field over $\Gamma$.

Finally, we complexify the resulting real bundle to obtain a flat complex line bundle:

$$\mathcal{L}_\Gamma := V_\Gamma \otimes_\mathbb{R} \mathbb{C} \to \Gamma,$$

which serves as the geometric input for defining holonomy and zeta invariants on the graph.

This bundle defines a local system on $\Gamma$, represented by a unitary character:

$$\rho: \pi_1(\Gamma) \to \mathrm{U}(1).$$

The holonomy representation $\rho$ on $\Gamma$ is determined by the vertical bundles $V_i$ via the induced identifications through the collapse process:

$$\pi_1\left(\coprod S^1_i\right) \cong \mathbb{Z}^4 \longrightarrow \pi_1(\Gamma) \xrightarrow{\rho} \mathrm{U}(1).$$

Each loop in $\Gamma$ corresponds to a word in the fundamental group, and the value of $\rho$ on that loop is computed from the phase induced by the vertical directions traced through the merging process.

Proposition: Given vertical real line bundles $V_i$ over $S^1_i$ whose fibers are orthogonal to the slicing planes, the collapse of the disjoint circles to $\Gamma$ induces a gluing of $\{V_i\}$ into a single real bundle $V_\Gamma \to \Gamma$. Upon complexification, this defines a flat complex line bundle $\mathcal{L}_\Gamma \to \Gamma$, whose associated unitary representation $\rho: \pi_1(\Gamma) \to \mathrm{U}(1)$ encodes the combined holonomy data.

Define a twisted Ihara zeta function over $\Gamma$ denoted $\zeta_{\Gamma}(u,\rho)$:

$$\zeta_{\Gamma}(u, \rho) := \prod_{[P]} \left(1 - \rho(P) u^{\ell(P)}\right)^{-1},$$

where $[P]$ runs over all prime, tail-less, backtrackless cycles in $\Gamma$, and $\ell(P)$ is the length of the cycle, with $\rho(P)$ representing the holonomy around $P$.

Equivalently, the zeta function satisfies the determinant identity:

$$\zeta_\Gamma(u, \rho)^{-1} = \det(I - u A_\rho + u^2 Q),$$

where $A_\rho$ is the twisted adjacency matrix incorporating holonomies, and $Q$ is the degree matrix with $Q_{ii} = \deg(v_i) - 1$.

The holonomy representation $\rho$ belongs to the moduli space of unitary representations:

$$\mathrm{Hom}(\pi_1(\Gamma), \mathrm{U}(1)) \cong \mathrm{U}(1)^r,$$

where $r = b_1(\Gamma)$ is the first Betti number. This torus parametrizes the family of flat line bundles over $\Gamma$.

A finite group $\Delta \subset \mathrm{Aut}(\Gamma)$ acts on $\pi_1(\Gamma)$ by automorphisms and hence induces a pullback action on the moduli torus:

$$\rho^\delta(\gamma) := \rho(\delta^{-1} \cdot \gamma), \quad \delta \in \Delta.$$

This defines a group action:

$$\Delta \curvearrowright \mathrm{U}(1)^r,$$

with associated orbits $\mathcal{O}_\rho$ and the quotient moduli space:

$$\mathcal{M}_\Delta := \mathrm{U}(1)^r / \Delta.$$

The twisted Ihara zeta function defines a family of spectral invariants:

$$\mathcal{Z}_\Gamma : \mathrm{U}(1)^r \longrightarrow \mathbb{C}[[u]], \quad \rho \mapsto \zeta_\Gamma(u, \rho).$$

And this sheaf encodes how spectral data evolves over the moduli space of holonomies.

The Canonical Construction of $\mathcal I$ Using Differential Geometry

We will construct a highly symmetric leaf (an interface leaf) as a special subset in some $\mathcal F$-completion, this time furnishing the leaf with additional geometric data including a metric and curvature.

We start by asking the following optimization question:

Fix $n=3$ and consider a surface of revolution $S$ and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with points $p,q$ elements of $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm {sup}~ \mathrm{dist}(p,q)=\sqrt{3}$.   

What is $\rho_{\mathrm{max}}=\mathrm{max} \lbrace \mathrm{vol}(S) \rbrace_{p,q}$ assuming $S$ must remain a surface of revolution and have constant positive Gaussian curvature?

An abstract surface of revolution with constant positive Gaussian curvature (to be embedded/optimized) within $X^3.$

In other words, what is the volume of the largest surface of revolution with constant Gaussian curvature that can be embedded in $X^3$ with a pair of antipodal corners as cone points?

Let $a = \frac{2}{\sqrt{3}}$. For a positive $b$ and $0 < u < a/b$, define the function 

$$\phi(u) = au - bu^2 = u(a - bu),$$

which gives a derivative 

$$\phi'(u) = a - 2bu.$$

Due to geometric considerations, there exists a unique positive $b$ such that

$$\int_0^{a/b} \sqrt{\frac{1 - \frac{1}{4} \phi'(u)^2}{\phi(u)}} \, du = \sqrt{3}.$$

The maximum volume of a surface with constant Gaussian curvature, enclosed within a unit cube and featuring cone points at a pair of opposite corners, is given by

$$\pi \int_0^{a/b} \sqrt{\bigl(1 - \tfrac{1}{4} \phi'(u)^2\bigr) \phi(u)} \, du.$$

Numerical approximation methods are necessary for further analysis.

Metric and Gaussian Curvature:

Given a smooth function $\phi(u)$ of class $C^2$ on a real interval, consider the metric

$$g_\phi = \frac{1}{\phi(u)} \, du^2 + \phi(u) \, dv^2.$$

For this metric, the Gaussian curvature is 

$$K_\phi = -\frac{1}{2} \phi''(u).$$

Thus, surfaces of constant Gaussian curvature correspond to cases where $\phi(u)$ is a quadratic polynomial.

A surface can embed as a surface of revolution in three-dimensional Euclidean space only if $|\phi'(u)| \leq 2$. If this surface arises from rotating the curve $y = \xi(x)$ about the $x$-axis, then the relationships are:

• $\phi(u) = \xi(x)^2$,
• $du = \xi(x) \sqrt{1 + \xi'(x)^2} \, dx$,
• $\xi'(x) = \frac{\phi'(u)}{\sqrt{4 - \phi'(u)^2}}$.

Here, $|\phi'(u)| = 2$ corresponds to a vertical tangent on the graph of $y = \phi(x)$.

Geometric Problem:

Consider the unit cube $[0, 1]^3$ with a diagonal $\ell$ connecting the corners $(0, 0, 0)$ and $(1, 1, 1)$. The goal is to find the "momentum profile"

$$\phi(u) = au - bu^2$$

where $b$ determines the Gaussian curvature, such that:

• The rotated graph $y = \xi(x)$ fits entirely within a cube corner.
• The surface closes at the opposite corner, satisfying $\xi(\sqrt{3}) = 0$.

The condition for fitting within the cube ensures the slope of the generator (viewed along $\ell$) does not exceed $1/\sqrt{2}$. At $u = 0$ (where $x = 0$), this slope restriction becomes:

$$\frac{1}{\sqrt{2}} = \xi'(0) = \frac{\phi'(0)}{\sqrt{4 - \phi'(0)^2}} = \frac{a}{\sqrt{4 - a^2}}.$$

Solving this gives $a = \frac{2}{\sqrt{3}}$. Thus, the momentum profile is $\phi(u) = u(a - bu)$, with $b$ determined by additional conditions.

Surface Properties:

The relationship between $x$ and $u$ is governed by the differential equation:

$$\frac{dx}{du} = \frac{1}{\xi(x) \sqrt{1 + \xi'(x)^2}} = \frac{1}{\sqrt{\phi(u)}} \sqrt{1 - \tfrac{1}{4} \phi'(u)^2}.$$

The surface length is equal to the diagonal of the cube, $\sqrt{3}$, if and only if

$$x(a/b) = \int_0^{a/b} \sqrt{\frac{1 - \frac{1}{4} \phi'(u)^2}{\phi(u)}} \, du = \sqrt{3}.$$

This implies a unique positive $b$ exists, though finding it explicitly is challenging.

Volume Calculation:

The volume enclosed by the surface of revolution is:

$$\pi \int_0^{\sqrt{3}} \xi(x)^2 \, dx = \pi \int_0^{a/b} \phi(u) \frac{dx}{du} \, du = \pi \int_0^{a/b} \sqrt{\bigl(1 - \tfrac{1}{4} \phi'(u)^2\bigr) \phi(u)} \, du.$$

Obtaining the Interface Construction, $\mathcal I$:

To obtain the $\mathcal I$ from the above arguments, we simply use our construction to place $4$ total copies of the interface leaf in $X^3$. Here is what that looks like:

Here is another way to view $\mathcal I$:

Thus, if we index the leaves in our $\mathcal F$-completion by enclosed volume, then we can view $\mathcal I$ as a constant time slice in our $\mathcal F$-completion - perhaps the above depiction corresponds to what we calculated already, so the volume is around $1/2$ for each of the four surfaces. 

Information Geometry and Complex Geometry: Gluing a Partial $\mathcal F$-completion in $n=2$

Consider a family of functions

$$f_t(x) := e^{\frac{t}{\log x}}, \quad x \in (0,1), \quad t \in [1/2, 2],$$

each of which can be viewed as a smooth curve in the $(x, f)$-plane for fixed $t$. As $t$ varies, this defines a continuous family of curves, or equivalently, a surface embedded in the extended space $(x, f, t)$.

We now define a topological gluing operation by identifying the boundary curves corresponding to $t = 1/2$ and $t = 2$ in the $(x, f)$-plane. The identification is performed along straight lines of slope $+1$, i.e., lines of the form

$$f = x + b, \quad b \in [-1, 1].$$

Explicitly, we declare the point $(x, f_{1/2}(x))$ to be equivalent to $(x', f_2(x'))$ whenever both points lie on the same line of slope $+1$; that is, whenever

$$f_{1/2}(x) - x = f_2(x') - x',$$

so that both have the same intercept $b \in [-1, 1]$. This identification defines an equivalence relation that glues the endpoints of the parameter domain $t \in [1/2, 2]$ together along a $1$-parameter family of lines.

The result is a quotient topological space, denoted by $S$. Topologically, $S$ is homeomorphic to a $2$-sphere with two conical singularities, arising from the degenerate directions introduced by the gluing. This "pinched" geometry reflects the collapse of boundary curves along a finite set of matched directions in function space.

We next endow $S$ with a geometric structure. Each function $f_t(x)$ gives rise to a Mellin transform

$$F_t(s) := \mathcal{M}(f_t)(s) = \int_0^1 f_t(x) \, x^{s-1} \, dx,$$

defined for $s \in \mathbb{C}_+$. This evaluates to the closed-form expression

$$F_t(s) = 2\sqrt{\frac{t}{s}}\, K_1\big(2\sqrt{ts}\big),$$

where $K_1$ is the modified Bessel function of the second kind. The function $F_t(s)$ acts as a generating function for a Hermitian metric structure, such as

$$g_t(s) := |F_t(s)|^2 \, ds \otimes d\bar{s}, \quad \text{or} \quad g_t(s) := \partial_s \partial_{\bar{s}} \log |F_t(s)|^2.$$

Crucially, the family $F_t(s)$ satisfies the linear, scale-covariant partial differential equation

$$t^2 \frac{\partial^3 F}{\partial t^3} = s^2 \frac{\partial F}{\partial s},$$

which links third-order variation in the parameter $t$ to first-order variation in the spectral variable $s$. This equation defines a nontrivial flow on the function space, suggesting that the induced geometry on $S$ is governed by a PDE-driven foliation or flow. The equation is invariant under the dual scaling $t \mapsto \lambda t,\, s \mapsto \lambda^{-1} s$, indicating that the metric structure $g^{\sim}$ respects a form of scale duality reminiscent of Mellin symmetry or projective flatness.

Since the functions $f_t(x)$ have been glued along the boundary curves $t = 1/2$ and $t = 2$, the associated metrics $g_t(s)$ must also be identified accordingly. This yields a globally defined quotient metric $g^{\sim}$ on the surface $S$, defined by

$$g^{\sim}(s) := [g_t(s)] \quad \text{on } S,$$

inheriting its complex structure and regularity from the family $F_t(s)$. Thus, $S$ becomes not just a topological quotient of exponential functions, but a geometric surface equipped with a globally defined Hermitian structure, arising from a family of PDE-constrained Mellin-Bessel kernels.

A Question To Think About:

Given that the quotient surface $S$ inherits a Hermitian metric $g^{\sim}$ via the Mellin-transformed family $F_t(s)$, and that $F_t(s)$ satisfies the third-order PDE

$$t^2 \frac{\partial^3 F}{\partial t^3} = s^2 \frac{\partial F}{\partial s},$$

how does this differential equation descend to the quotient surface $S$ after gluing? Can we formulate a well-defined differential operator or flow on $S$ that reflects the original PDE structure, and how does this operator interact with the quotient metric $g^{\sim}$?