Introduction

The classical theta series of a lattice records the number of vectors of each norm. Siegel theta series refine this construction by allowing simultaneous representation of positive semidefinite quadratic forms. For a positive-definite integral lattice $L$ of rank $n$, the genus-$g$ Siegel theta series is schematically $$\Theta_L^{(g)}(Z) = \sum_{X\in L^g} e^{2\pi i\,\operatorname{tr}(T_XZ)},$$ where $T_X$ is the Gram matrix of the ordered $g$-tuple $X$.

Thus a Fourier coefficient of $\Theta_L^{(g)}$ counts ordered $g$-tuples having a prescribed Gram matrix. The natural question is how much of the lattice is determined by these finite-dimensional metric statistics.

The motivating problem is the following.

The question has two distinct components. First, one must determine whether the Fourier coefficients of low-genus theta series contain enough information to reconstruct local metric incidence. Second, one must determine whether local incidence data admit a unique global metric realization.

The distinction is important. A collection of marginal counts need not determine the underlying configuration. This is the same phenomenon familiar from moment problems, graph reconstruction, and the theory of finite relational structures: one-point distributions do not generally determine joint distributions.

The present paper develops a lattice-theoretic version of this idea.

Siegel Theta Series and the Gram-Fiber Viewpoint

Let $L$ be a positive-definite integral lattice of rank $n$. For $g\geq1$, define the Gram map $$\pi_g:L^g \longrightarrow \operatorname{Sym}_g^*(\mathbb Z)_{\geq0}, \qquad X=(x_1,\ldots,x_g) \longmapsto T_X,$$ where $$(T_X)_{ij} = \frac{1}{2}(x_i,x_j).$$

The Fourier expansion of the genus-$g$ theta series has the form $$\Theta_L^{(g)} = \sum_T a_L(T,\Theta_L^{(g)})\,q^T,$$ with $$a_L(T,\Theta_L^{(g)}) = |\pi_g^{-1}(T)|.$$

We therefore define the metric deck $$\mathcal D_g(L) = \left\{ (T,a_L(T,\Theta_L^{(g)})) : T\in\operatorname{Sym}_g^*(\mathbb Z)_{\geq0} \right\}.$$

The full filtered metric deck is $$\boxed{ \mathcal D^\bullet(L) = \left( \mathcal D_1(L), \mathcal D_2(L), \ldots, \mathcal D_n(L) \right). }$$

At genus $n$, the data are sufficiently rich to recover a basis configuration and hence the lattice. The problem is to determine how far below $n$ one can truncate.

Collision Propagation

The first obstruction to any bounded-genus reconstruction theorem is the existence of distinct lattices with identical low-genus theta series.

The following stabilization principle is fundamental.

Thus a genus-$3$ collision is not merely a low-dimensional accident: it propagates indefinitely through orthogonal stabilization.

Fiber Products and Incidence Moments

The key refinement is to compare fibers of different Gram maps along common subconfigurations.

Let $$T_1,T_2\in\operatorname{Sym}_g^*(\mathbb Z)_{\geq0}$$ share a common $k\times k$ principal submatrix $$S\in\operatorname{Sym}_k^*(\mathbb Z)_{\geq0}.$$ Write $$F_{T_i}=\pi_g^{-1}(T_i).$$ Restriction to the common $k$-tuple gives maps $$\rho_i:F_{T_i}\longrightarrow F_S.$$

The resulting incidence number is obtained by summing over all admissible cross-Gram blocks.

The significance of this identity is conceptual as well as algebraic: higher genus computes the cardinalities of iterated fiber products of lower-genus realization fibers.

The Metric Realization Hypergraph

The fiber-product picture admits a combinatorial reformulation.

The coefficient $$m_T = |\tau^{-1}(T)|$$ is the one-point marginal of the metric labeling distribution.

The hierarchy of incidence moments is $$\begin{array}{c|c|c|c} \text{Invariant} & \text{Genus} & \text{Measurement} & \text{Interpretation} \\ \hline \mathcal I_g^{(1)} & g & m_T & \text{one-point marginal} \\ \mathcal I_g^{(2)} & 2g-k & m(T_1,T_2;S_k) & \text{two-point correlation} \\ \mathcal I_g^{(r)} & rg-\sum k_i & m(T_1,\ldots,T_r;S) & \text{$r$-point correlation} \\ \mathcal I_g^{(\infty)} & n & \text{global realization} & \text{basis-level metric data}. \end{array}$$

For $g=4$, the first incidence levels are schematically $$\begin{array}{rcl} g=4 &:& \text{marginal frame counts},\\ g=5 &:& \text{overlaps on three vectors},\\ g=6 &:& \text{overlaps on two vectors},\\ g=7 &:& \text{overlaps on one vector},\\ g=8 &:& \text{disjoint pair correlations}. \end{array}$$

This motivates the interpretation of the genus filtration as a metric moment hierarchy.

Incidence Moment Invariants for Root Systems

Let $X$ be a simply-laced root system. For $k\geq0$ and $c\in\{0,\pm1,\pm2\}$ define $$q_X(k,c) = \#\left\{ (\alpha_1,\ldots,\alpha_k,\beta,\gamma)\in X^{k+2}: \begin{array}{l} (\alpha_i,\alpha_j)=2\delta_{ij},\\ (\beta,\alpha_i)=0,\\ (\gamma,\alpha_i)=0,\\ (\beta,\gamma)=c \end{array} \right\}.$$

The corresponding Gram matrix is $$U_{k+2}(c) = \begin{pmatrix} I_k&0&0\\ 0&1&c/2\\ 0&c/2&1 \end{pmatrix}.$$

Therefore $$\boxed{ q_X(k,c) = a\left( U_{k+2}(c),\Theta_X^{(k+2)} \right). }$$

For $c=\pm2$, $$q_X(k,\pm2)=N_{k+1}(X).$$ For $c=0$, $$q_X(k,0)=N_{k+2}(X).$$ For $c=1$, the final two roots form an adjacent pair.

Closed Formula for $A_n$

The roots of $A_n$ are $$\Phi(A_n) = \{\pm(e_i-e_j):1\leq i<j\leq n+1\}.$$

An ordered orthogonal $k$-frame uses $2k$ distinct coordinates, so $$N_k(A_n) = (n+1)^{\underline{2k}}.$$

The orthogonal complement of every such frame is isomorphic to $$A_{n-2k}.$$

In $A_m$, each root has $2(m-1)$ adjacent roots, while there are $m(m+1)$ roots. Hence $$q_{A_m}(0,1) = 2(m+1)^{\underline3}.$$

Consequently,

In particular, $$q_{A_n}(3,1) = 2(n+1)^{\underline9}.$$

Thus $$q_{A_n}(3,1)=0 \qquad(n\leq7),$$ while $$q_{A_8}(3,1)=725\,760,$$ and $$q_{A_9}(3,1)=7\,257\,600.$$

Closed Formula for $D_n$

Let $$D_n = \{\pm e_i\pm e_j:1\leq i<j\leq n\}.$$

The support of each root consists of two coordinates. Orthogonal frames can therefore be classified by the number $p$ of pairs of roots sharing the same two-coordinate support.

If there are $p$ double blocks, then there are $$k-2p$$ single blocks.

The number of ordered orthogonal $k$-frames of type $p$ is $$\operatorname{Count}(p) = k! \frac{2^{k-p}}{p!(k-2p)!} n^{\underline{2(k-p)}}.$$

For such a frame the root subsystem in the orthogonal complement is $$A_1^{\,k-2p} \oplus D_{n-2k+2p}.$$

Only the $D$-component contributes adjacent pairs. Since $$q_{D_m}(0,1) = 8m^{\underline3},$$ we obtain the following exact formula.

The first four cases are $$\boxed{ q_{D_n}(0,1)=8n^{\underline3}, }$$ $$\boxed{ q_{D_n}(1,1)=16n^{\underline5}, }$$ $$\boxed{ q_{D_n}(2,1) = 32n^{\underline5} + 32n^{\underline7}, }$$ and $$\boxed{ q_{D_n}(3,1) = 192n^{\underline7} + 64n^{\underline9}. }$$

For example, $$q_{D_7}(3,1) = 192\cdot7^{\underline7} + 64\cdot7^{\underline9} = 967\,680.$$

Exceptional Root Systems

The exceptional systems are evaluated using Weyl-group transitivity and chains of orthogonal complements.

$E_8$

The relevant chain is $$E_8 \longrightarrow E_7 \longrightarrow D_6 \longrightarrow D_4\oplus A_1.$$

Consequently, $$q_{E_8}(0,1) = 240\cdot56 = 13\,440,$$ $$q_{E_8}(1,1) = 240\cdot4032 = 967\,680,$$ $$q_{E_8}(2,1) = 30\,240\cdot960 = 29\,030\,400,$$ and $$q_{E_8}(3,1) = 1\,814\,400\cdot192 = 348\,364\,800.$$

$E_7$

The relevant complement chain is $$E_7 \longrightarrow D_6 \longrightarrow D_4\oplus A_1.$$

Thus $$q_{E_7}(0,1) = 126\cdot32 = 4\,032,$$ $$q_{E_7}(1,1) = 126\cdot960 = 120\,960,$$ $$q_{E_7}(2,1) = 7560\cdot192 = 1\,451\,520,$$ and $$q_{E_7}(3,1) = 126\cdot60\cdot(2\cdot192) = 2\,903\,040.$$

$E_6$

The complement chain is $$E_6 \longrightarrow A_5 \longrightarrow A_3 \longrightarrow A_1.$$

Hence $$q_{E_6}(0,1) = 72\cdot20 = 1\,440,$$ $$q_{E_6}(1,1) = 72\cdot240 = 17\,280,$$ $$q_{E_6}(2,1) = 2160\cdot48 = 103\,680,$$ and $$q_{E_6}(3,1)=0.$$

The ADE Moment Table

For each irreducible simply-laced component define the moment vector $$\Psi(X) = \left( N_1,N_2,N_3,N_4, q(0,1),q(1,1),q(2,1),q(3,1) \right).$$

The values relevant to the Niemeier root systems are listed below.

Reducible Systems and Polynomial Convolution

Let $$R=\bigoplus_i X_i^{m_i}$$ be a reducible simply-laced root system.

An adjacent pair $\beta,\gamma$ satisfying $$(\beta,\gamma)=1$$ must lie in the same irreducible component. The orthogonal roots $\alpha_1,\ldots,\alpha_k$ may be distributed among all components.

This leads to the convolution identity $$\boxed{ q_R(k,1) = \sum_j \sum_{r=0}^k \binom{k}{r} q_{X_j}(r,1) N_{k-r} \left( \bigoplus_{i\neq j}X_i \right). }$$

Thus the incidence moments of $R$ are polynomial functions of the component multiplicities.

Coxeter-Rigidity in Low Genus

For the relevant simply-laced Niemeier root systems, the genus-$2$ adjacency moment depends only on the Coxeter number $h$: $$\boxed{ q_R(0,1) = 48h(h-2). }$$

Similarly, the genus-$3$ moment satisfies $$\boxed{ q_R(1,1) = 48h(h-2)(20h+6). }$$

Equivalently, $$q_R(1,1) = 960h^3-1632h^2-576h.$$

The values on the collision classes are therefore identical.

This is the incidence-moment manifestation of the low-genus spherical design collapse.

Genus Four and the Five Collision Classes

The first strict bifurcation occurs at $k=2$, corresponding to genus $4$.

The moment $$q_R(2,1) = a(U_4(1),\Theta_R^{(4)})$$ separates the five Coxeter-number collision classes.

In particular, both $$a(I_4,\Theta_R^{(4)}) = N_4(R)$$ and $$a(U_4(1),\Theta_R^{(4)}) = q_R(2,1)$$ provide independent genus-$4$ invariants that resolve the collisions.

The ADE Moment Rigidity Theorem

We now package the preceding calculations into a reconstruction statement.

The theorem separates the problem into two algebraic stages: first identify the irreducible component moment vectors, then invert the polynomial convolution governing reducible systems.

Conditional Neighborhood Reconstruction

The invariant $$q_X(k,1)$$ has a direct geometric interpretation.

For a fixed ordered orthogonal $k$-frame $$(\alpha_1,\ldots,\alpha_k),$$ let $$V_\alpha = \bigcap_{i=1}^k\alpha_i^\perp.$$

Then $q_X(k,1)$ counts, over all such frames, ordered pairs $$(\beta,\gamma)\in V_\alpha^2$$ with $$(\beta,\gamma)=1.$$

Thus the genus-$(k+2)$ Fourier coefficient $$a(U_{k+2}(1),\Theta_X^{(k+2)})$$ is a conditional adjacency statistic.

For example, at genus $5$, $$U_5(1) = \begin{pmatrix} I_3&0&0\\ 0&1&1/2\\ 0&1/2&1 \end{pmatrix}.$$

Its coefficient counts ordered configurations $$(\alpha_1,\alpha_2,\alpha_3,\beta,\gamma)$$ satisfying $$(\alpha_i,\alpha_j)=2\delta_{ij},$$ $$(\alpha_i,\beta) = (\alpha_i,\gamma) = 0,$$ and $$(\beta,\gamma)=1.$$

Consequently, genus $5$ probes the adjacency structure inside the orthogonal complement of an orthogonal $3$-frame.

Two Reconstruction Conjectures

The preceding results suggest a natural division between local incidence reconstruction and global metric rigidity.

The second step is global.

This may be viewed as a lattice analogue of a Cauchy-type rigidity principle: sufficiently rich local metric data determine the global Euclidean realization.

The Shortest Complete Prefix

Let $\mathcal L_n$ denote the set of positive-definite even unimodular lattices of rank $n$. Define $$\boxed{ k^*(n) = \min \left\{ k\leq n: (\Theta_L^{(1)},\ldots,\Theta_L^{(k)}) \text{ determines }[L]\in\mathcal L_n \right\}. }$$

Three asymptotic regimes are natural.

Bounded regime

One possibility is $$\boxed{k^*(n)=O(1).}$$

In this world, a universal finite genus window would determine every lattice in the relevant class.

A particularly strong form would assert the existence of a universal constant $K$ such that $$k^*(n)\leq K$$ for all admissible ranks.

Logarithmic regime

A second possibility is $$\boxed{k^*(n)=O(\log n).}$$

Here increasingly high-dimensional lattices require more incidence moments, but only logarithmically many. The mechanism would presumably be propagation of local rigidity through a metric expansion or spectral-gap phenomenon.

Full-rank regime

The third possibility is $$\boxed{k^*(n)=\Theta(n).}$$

In this scenario, high-dimensional “chimeric” lattices could remain indistinguishable through a large fraction of the genus filtration, with global information appearing only when the genus approaches the rank.

The collision-propagation theorem gives a persistent lower-bound phenomenon: $$k^*(16+8k)\geq4.$$

The fundamental asymptotic question is therefore whether this lower bound remains essentially constant or grows with dimension.

Bounded-Width Assembly

The metric deck can be viewed as a collection of local patches. This suggests the following assembly principle.

For rooted lattices, a particularly concrete target is to prove that genus $4$ suffices for inductive reconstruction of the root graph by face-gluing.

Stability and Effective Truncation

The full genus-$g$ deck contains infinitely many Fourier coefficients. A computationally useful reconstruction theorem therefore requires an effective truncation.

Define $$\mathcal D_g^{\leq B}(L) = \left\{ (T,a_L(T)): \operatorname{tr}(T)\leq B \right\}.$$

The stability problem is to determine a bound depending only on structural parameters.

A positive solution would turn the infinite-dimensional theta-series problem into a finite reconstruction problem.

Metric Deck Inversion Complexity

Suppose a finite truncation $$\mathcal D_g^{\leq B}(L)$$ is given. The inversion problem asks for the lattice Gram matrix $$G_L=(\langle b_i,b_j\rangle).$$

We may write $$\mathcal A: \mathcal D_g^{\leq B}(L) \longmapsto G_L.$$

The complexity of $\mathcal A$ is a natural computational question.

The distinction between local moment extraction and global basis reconstruction suggests that the complexity may be governed not by the arithmetic extraction of Fourier coefficients but by the final metric realization problem.

ADE Incidence Reconstruction

The root systems provide a natural intermediate test case.

The proof architecture is explicit:

$$\begin{array}{c} \text{Genus }1\text{--}4 \\ \downarrow \\ \text{Frame moments }N_1,\ldots,N_4 \\ \downarrow \\ \text{Genus }4,5 \\ \downarrow \\ q(0,1),q(1,1),q(2,1),q(3,1) \\ \downarrow \\ \text{Polynomial moment inversion} \\ \downarrow \\ \bigoplus_i X_i^{m_i} \\ \downarrow \\ \text{Dynkin diagram}. \end{array}$$

The root layer is therefore reduced to an explicit finite-dimensional algebraic inversion problem.

Discriminant Gluing and the Overlattice Layer

Recovering the root system is not, in general, the same as recovering the complete lattice.

Let $$R = \bigoplus_i X_i^{m_i}$$ be the root lattice of an even unimodular lattice $L$. The remaining information is encoded by the discriminant group $$A_R=R^*/R$$ and its induced quadratic form.

An even unimodular overlattice corresponds to an appropriate isotropic subgroup $$C\subset A_R.$$

Thus the final reconstruction layer is

$$R \quad\longrightarrow\quad A_R \quad\longrightarrow\quad C \quad\longrightarrow\quad L.$$

This separates the problem into a root-layer reconstruction and a gluing-layer reconstruction.

For Niemeier lattices, the first layer is controlled by the incidence moments developed above, while the second is a finite discriminant-form problem.

The Complete Reconstruction Architecture

The overall program can now be summarized as

$$\boxed{ \begin{array}{c} \textbf{Level 1: Fourier Moment Extraction} \\[2mm] (\Theta_L^{(1)},\ldots,\Theta_L^{(4)}) \\ \downarrow \\ V_L= \left( h,\, a(I_4,\Theta_L^{(4)}),\, a(U_4(1),\Theta_L^{(4)}) \right) \\[5mm] \textbf{Level 2: Polynomial Moment Inversion} \\[2mm] \Phi(\bm m)=V_L \\ \sum_i m_i\operatorname{rank}(X_i)=24 \\ \downarrow \\ R=\bigoplus_iX_i^{m_i} \\[5mm] \textbf{Level 3: Discriminant-Form Reconstruction} \\[2mm] C\subset R^*/R \text{ isotropic} \\ \downarrow \\ L \end{array} }$$

The three levels correspond respectively to

1. extraction of metric statistics from Siegel theta series;

2. reconstruction of the ADE root configuration;

3. reconstruction of the full arithmetic lattice.

Further Research Problems

The framework suggests four principal research directions.

Problem A: Collision Spectrum

For a fixed genus $g\geq4$, determine whether there exist non-isometric indecomposable positive-definite integral lattices $L,M$ satisfying $$\Theta_L^{(g)} = \Theta_M^{(g)}.$$

Equivalently, determine the collision spectrum $$\operatorname{Coll}(g) = \left\{ [L]\neq[M]: \Theta_L^{(g)}=\Theta_M^{(g)} \right\}.$$

The genus-$3$ collision propagation theorem shows that decomposable collisions persist indefinitely. The fundamental new question is whether indecomposable collisions survive at genus $4$ and beyond.

Problem B: Local Metric Extension

Determine the smallest genus at which two-point incidence moments determine the local extension operator $$X\longmapsto \{(y,\langle X,y\rangle):y\in L\}.$$

A positive solution would establish a finite local reconstruction threshold.

Problem C: Effective Stability

Determine explicit bounds $$B(g,n,\det L)$$ for finite Fourier truncations sufficient to reconstruct the full metric deck.

This connects the arithmetic theory of theta series with effective reconstruction and algorithmic lattice theory.

Problem D: Asymptotic Growth of $k^*(n)$

Determine which of the three regimes $$O(1), \qquad O(\log n), \qquad \Theta(n)$$ governs the shortest complete genus prefix.

The collision-propagation phenomenon gives an infinite family of lower-bound examples, but does not by itself determine the asymptotic growth.

Conceptual Interpretation

The preceding constructions suggest that the genus filtration should be understood not merely as a sequence of increasingly complicated modular forms, but as a hierarchy of metric moments.

At genus $1$, one sees the distribution of individual vectors.

At genus $2$, one sees pairwise metric relations.

At genus $3$, one sees triple correlations.

At genus $4$, one begins to resolve higher-order incidence patterns that are invisible to the low-degree spherical design constraints.

At genus $5$ and beyond, one obtains conditional neighborhood statistics: the geometry of the orthogonal complement of a prescribed configuration.

Schematically, $$\boxed{ \text{marginals} \longrightarrow \text{pair correlations} \longrightarrow \text{conditional neighborhoods} \longrightarrow \text{global metric realization}. }$$

This provides a categorical interpretation of the genus filtration: the Fourier coefficients of higher-genus theta series compute cardinalities of increasingly complicated fiber products of the realization category.

Conclusion

We have formulated a unified research program for reconstructing arithmetic lattices from low-genus Siegel theta data.

The central ingredients are:

1. the collision-propagation mechanism for Siegel theta series under orthogonal stabilization;

2. the realization-fiber interpretation of Fourier coefficients;

3. the fiber-product formula expressing incidence correlations through higher-genus coefficients;

4. the metric realization hypergraph and its hierarchy of incidence moments;

5. the invariant $$q_X(k,c)$$ encoding conditional root incidence;

6. the exact closed formulas for the $A_n$ and $D_n$ families;

7. Weyl-group complement calculations for $E_6,E_7,E_8$;

8. the injective ADE moment vectors;

9. the polynomial convolution for reducible root systems;

10. the genus-$4$ separation of the five Coxeter-number collision classes;

11. the decomposition of complete lattice reconstruction into root-layer and discriminant-gluing layers.

The resulting perspective is that a Siegel theta series is not merely a spectral invariant. Its successive genera encode an incidence hierarchy of metric configurations.

The fundamental object is therefore the filtered metric deck $$\boxed{ \mathcal D^\bullet(L) = \left( \mathcal D_1(L), \mathcal D_2(L), \ldots, \mathcal D_n(L) \right). }$$

The central asymptotic invariant is $$\boxed{ k^*(n) = \min \left\{ k\leq n: (\Theta_L^{(1)},\ldots,\Theta_L^{(k)}) \text{ determines }L \right\}. }$$

The Niemeier root layer provides a finite laboratory in which the first nontrivial transition can be studied explicitly: genera $1$–$3$ exhibit strong Coxeter/design collapse, while genus $4$ introduces sufficient incidence information to distinguish the known collision classes.

The ultimate problem is therefore not simply to compare theta series, but to determine the precise amount of higher-order metric incidence information required for global rigidity.

$$\boxed{ \text{Siegel moments} \;\longrightarrow\; \text{incidence geometry} \;\longrightarrow\; \text{metric rigidity} \;\longrightarrow\; \text{arithmetic lattice reconstruction}. }$$

This constitutes a concrete bridge between the arithmetic of Siegel modular forms, finite metric geometry, hypergraph reconstruction, root-system theory, and computational lattice theory.

Useful Falling-Factorial Identities

For reference, the falling factorial is $$x^{\underline r} = x(x-1)\cdots(x-r+1).$$

The identity repeatedly used in the $D_n$ calculations is $$x^{\underline a} (x-a)^{\underline b} = x^{\underline{a+b}}.$$

Thus $$n^{\underline{2k-2p}} (n-2k+2p)^{\underline3} = n^{\underline{2k-2p+3}}.$$

For the $A_n$ family, $$(n+1)^{\underline{2k}} (n-2k+1)^{\underline3} = (n+1)^{\underline{2k+3}}.$$

Appendix: Low-Genus Root Moments

For convenience, the principal formulas are collected here.

For $A_n$, $$N_k(A_n) = (n+1)^{\underline{2k}},$$ and $$q_{A_n}(k,1) = 2(n+1)^{\underline{2k+3}}.$$

For $D_n$, $$q_{D_n}(k,1) = 8k! \sum_{p=0}^{\lfloor k/2\rfloor} \frac{2^{k-p}}{p!(k-2p)!} n^{\underline{2k-2p+3}}.$$

In particular, $$\begin{aligned} q_{D_n}(0,1) &=8n^{\underline3},\\ q_{D_n}(1,1) &=16n^{\underline5},\\ q_{D_n}(2,1) &=32n^{\underline5}+32n^{\underline7},\\ q_{D_n}(3,1) &=192n^{\underline7}+64n^{\underline9}. \end{aligned}$$

For the exceptional systems, $$\begin{array}{c|rrrr} X&q_X(0,1)&q_X(1,1)&q_X(2,1)&q_X(3,1)\\ \hline E_6&1\,440&17\,280&103\,680&0\\ E_7&4\,032&120\,960&1\,451\,520&2\,903\,040\\ E_8&13\,440&967\,680&29\,030\,400&348\,364\,800 \end{array}$$

Appendix: Collision Classes

The five relevant Coxeter-number collision classes are

$$\begin{aligned} h=6:&\qquad A_5^4D_4 \quad\text{vs.}\quad D_4^6, \\[1mm] h=10:&\qquad A_9^2D_6 \quad\text{vs.}\quad D_6^4, \\[1mm] h=12:&\qquad A_{11}D_7E_6 \quad\text{vs.}\quad E_6^4, \\[1mm] h=18:&\qquad A_{17}E_7 \quad\text{vs.}\quad D_{10}E_7^2, \\[1mm] h=30:&\qquad D_{16}E_8 \quad\text{vs.}\quad E_8^3. \end{aligned}$$

For each pair the genus-$4$ invariant $q_R(2,1)$ takes different values, providing a strict algebraic bifurcation of the corresponding Coxeter classes.

Appendix: Reconstruction Diagram