Introduction

The interaction between spherical designs, Euclidean lattices, and modular forms provides a particularly rigid setting in which high-dimensional cubature problems can be reduced to finite-dimensional linear algebra.

Let $\mathbb{S}^{d-1}\subset\mathbb{R}^d$ be the unit sphere equipped with normalized rotation-invariant measure $\sigma$.

By harmonic decomposition, this is equivalent to the vanishing of all nonconstant harmonic moments through degree $t$: $$\begin{equation} \sum_{x\in X}W(x)P_j(x)=0, \qquad P_j\in\mathop{\mathrm{Harm}}_j(\mathbb{R}^d), \qquad 1\leq j\leq t. \label{eq:harmonic_vanishing} \end{equation}$$

For the antipodal shell-supported constructions considered here, the odd-degree harmonic conditions vanish automatically. Thus the nontrivial conditions arise from the even degrees.

The Leech lattice

Let $\Lambda_{24}\subset\mathbb{R}^{24}$ denote the Leech lattice. For $m\geq2$, define the shell $$\begin{equation} S_m= \{x\in\Lambda_{24}:\|x\|^2=2m\}. \end{equation}$$

The minimal shell is $S_2$, consisting of $196560$ vectors. After radial projection to $\mathbb{S}^{23}$, this shell forms a spherical $11$-design.

The theta series of the Leech lattice is $$\begin{equation} \Theta_{\Lambda_{24}}(\tau) = \sum_{x\in\Lambda_{24}}q^{\|x\|^2/2} = 1+\sum_{m=2}^{\infty}N_mq^m, \qquad q=e^{2\pi i\tau}. \end{equation}$$

Since $\Theta_{\Lambda_{24}}$ is a modular form of weight $12$ and has no $q$-term, one has $$\begin{equation} \Theta_{\Lambda_{24}} = E_{12}-\frac{65520}{691}\Delta, \end{equation}$$ and therefore $$\begin{equation} N_m = \frac{65520}{691} \left(\sigma_{11}(m)-\tau(m)\right). \label{eq:shell_cardinality} \end{equation}$$

Harmonic Theta Series and Modular Reduction

Let $P_j\in\mathop{\mathrm{Harm}}_j(\mathbb{R}^{24})$ be homogeneous harmonic of even degree $j$. Its harmonic theta series is $$\begin{equation} \Theta_{\Lambda_{24},P_j}(\tau) = \sum_{x\in\Lambda_{24}} P_j(x)q^{\|x\|^2/2}. \label{eq:harmonic_theta} \end{equation}$$

For $j>0$, this is a cusp form of weight $j+12$.

Because the Leech lattice has no vectors of norm $2$, the harmonic theta series has no constant or $q^1$ term. Consequently, $$\begin{equation} \Theta_{\Lambda_{24},P_j} \in S_{j+12}^{(2)}(\mathop{\mathrm{SL}}_2(\mathbb{Z})), \end{equation}$$ where $$\begin{equation} S_w^{(2)}(\mathop{\mathrm{SL}}_2(\mathbb{Z})) = \{f\in S_w(\mathop{\mathrm{SL}}_2(\mathbb{Z})):\mathop{\mathrm{ord}}_q(f)\geq2\}. \end{equation}$$

The automorphism group $\mathrm{Co}_0$ acts on harmonic polynomials. For $P_j$ orthogonal to the invariant subspace, its shell sums vanish.

For the invariant subspace, Venkov’s harmonic theta construction gives the relevant modular-form model.

Since every element of $S_w^{(2)}$ is divisible by $\Delta^2$, division by $\Delta^2$ gives an isomorphism $$\begin{equation} S_{j+12}^{(2)}(\mathop{\mathrm{SL}}_2(\mathbb{Z})) \xrightarrow{\;\sim\;} M_{j-12}(\mathop{\mathrm{SL}}_2(\mathbb{Z})), \qquad f\longmapsto\frac{f}{\Delta^2}. \end{equation}$$

We therefore define the single-degree modular dimension $$\begin{equation} d_j = \dim M_{j-12}(\mathop{\mathrm{SL}}_2(\mathbb{Z})). \label{eq:single_degree_dimension} \end{equation}$$

For the two target degrees considered below, $$\begin{equation} d_{114} = \dim M_{102}(\mathop{\mathrm{SL}}_2(\mathbb{Z})) = 9, \qquad d_{120} = \dim M_{108}(\mathop{\mathrm{SL}}_2(\mathbb{Z})) = 10. \end{equation}$$

These numbers describe the modular-form dimension at the individual top degree. They are not the dimensions of the cumulative evaluation spaces used when all intermediate even harmonic conditions are imposed.

The Cumulative Modular Evaluation Space

A spherical design of strength $t$ must satisfy harmonic conditions at every degree $j\leq t$. For the antipodal shell-supported constructions considered here, the relevant degrees are the even degrees beginning at $12$.

For an even target degree $k\geq12$, define the cumulative modular space $$\begin{equation} \mathcal M_{\leq k} = \bigoplus_{\substack{12\leq j\leq k\\j\ {\rm even}}} S_{j+12}^{(2)}(\mathop{\mathrm{SL}}_2(\mathbb{Z})). \label{eq:cumulative_modular_space} \end{equation}$$

Its dimension is $$\begin{equation} D_k = \dim\mathcal M_{\leq k} = \sum_{\substack{12\leq j\leq k\\j\ {\rm even}}} d_j. \label{eq:cumulative_dimension} \end{equation}$$

For the two target degrees, $$\begin{equation} D_{114}=243, \qquad D_{120}=271. \label{eq:cumulative_dimensions} \end{equation}$$

Thus the values $9$ and $10$ refer to the top-degree spaces, whereas $243$ and $271$ refer to the cumulative direct-sum spaces.

This distinction is essential for interpreting the computational matrices.

Choose, for each even $j$ with $12\leq j\leq k$, a basis $$F_{j,1},\ldots,F_{j,d_j}$$ of $S_{j+12}^{(2)}(\mathop{\mathrm{SL}}_2(\mathbb{Z}))$.

For a shell index $m$, define the degree-$j$ evaluation vector $$\begin{equation} v_j(m) = (2m)^{-j/2} \left( a_m(F_{j,1}), \ldots, a_m(F_{j,d_j}) \right)^T. \label{eq:degree_evaluation_vector} \end{equation}$$

The cumulative evaluation vector is then $$\begin{equation} v_{\leq k}(m) = \bigoplus_{\substack{12\leq j\leq k\\j\ {\rm even}}} v_j(m) \in\mathbb{R}^{D_k}. \label{eq:cumulative_evaluation_vector} \end{equation}$$

For a finite shell set $\mathcal S$, the cumulative evaluation matrix is $$\begin{equation} A_{\leq k,\mathcal S} = \left( v_{\leq k}(m) \right)_{m\in\mathcal S} \in \mathbb{R}^{D_k\times|\mathcal S|}. \label{eq:cumulative_evaluation_matrix} \end{equation}$$

A positive shell-weight vector $$W=(W_m)_{m\in\mathcal S} \in\mathbb{R}_{>0}^{|\mathcal S|}$$ annihilates all relevant harmonic conditions through degree $k$ precisely when $$\begin{equation} A_{\leq k,\mathcal S}W=0. \label{eq:cumulative_annihilation} \end{equation}$$

The Modular Evaluation Cone

Fix an even degree $k\geq12$. The modular evaluation cone generated by a finite shell set $\mathcal S$ is $$\begin{equation} \mathcal C_{\leq k}(\mathcal S) = \mathop{\mathrm{cone}} \{v_{\leq k}(m):m\in\mathcal S\}. \label{eq:modular_cone} \end{equation}$$

The cubature problem is therefore a finite-dimensional positive dependence problem in this cone.

A positive cubature supported on $\mathcal S$ corresponds to $$W\in\mathbb{R}_{>0}^{|\mathcal S|}$$ such that $$A_{\leq k,\mathcal S}W=0.$$

Equivalently, the origin lies in the relative interior of the convex cone generated by the selected evaluation vectors, subject to the chosen normalization.

Gordan’s alternative

The following convex-duality statement is the finite-dimensional tool needed for the sign-variation discussion.

Sparse Non-Negative Least Squares

Let $$\mathcal D=\{s_{\min},\ldots,s_{\max}\}$$ be a candidate shell pool, and choose an anchor shell $s_0\in\mathcal D$. Normalize its weight to $1$ and solve $$\begin{equation} \min_{W_{\rm rest}\geq0} \left\| A_{\leq k,\mathcal D\setminus\{s_0\}} W_{\rm rest} + v_{\leq k}(s_0) \right\|_2^2. \label{eq:nnls} \end{equation}$$

This is a convex non-negative least-squares problem.

If the minimum is exactly zero, the resulting vector gives an exact non-negative annihilation certificate. If the computed minimum is merely small, the result is instead a numerical approximate annihilator.

For reproducibility, define the active support using the explicit numerical threshold $$\varepsilon_{\rm act}=10^{-12}.$$ Thus $$\begin{equation} \mathcal S_{\rm active} = \left\{ m\in\mathcal D: W_m>\varepsilon_{\rm act} \right\}. \label{eq:active_threshold} \end{equation}$$

The threshold is a computational convention, not a mathematical invariant. Consequently, support counts should be accompanied by stability tests under changes in precision, solver tolerances, and activation threshold.

Numerical Exclusion Gaps

The computations reported below were performed using the cumulative modular evaluation matrices described in Sections 3 and 4. The reported residuals are floating-point quantities.

Degree $k=114$

For $$k=114,$$ we used the candidate pool $$\mathcal D=\{12,\ldots,265\}.$$

The numerical solution contains $54$ active shells under the threshold $\varepsilon_{\rm act}=10^{-12}$. The first active shell is $m=12$, while the second active shell is $m=129$.

Thus the observed first internal gap has index width $$129-12=117,$$ corresponding to the empty shell indices $$13,14,\ldots,128.$$

Representative weights from the numerical solution are $$\begin{align*} W_{12} &\approx1.000000\times10^0,\\ W_{129} &\approx1.170522\times10^{-7},\\ W_{153} &\approx2.860106\times10^{-8},\\ W_{265} &\approx9.316376\times10^{-9}. \end{align*}$$

These values should be regarded as floating-point numerical data rather than exact weights.

Degree $k=120$

For $$k=120,$$ we used $$\mathcal D=\{12,\ldots,295\}.$$

The reported numerical solution contains $53$ active shells. The beginning of the support is $$12,\quad89,\quad150,\quad168,\ldots.$$

Consequently, the first exclusion gap has index width $$89-12=77,$$ corresponding to the empty shell indices $$13,\ldots,88.$$

This differs from the first gap observed at $k=114$ and demonstrates that the support pattern is degree-dependent.

A Conjectural Horizon for the Exclusion Gap

The numerical experiments motivate a possible relationship between the degree $k$ and the location of an exclusion boundary.

A heuristic scale considered in the underlying computational study is $$\begin{equation} m_*(k)\asymp\frac{k^2}{96}. \label{eq:horizon} \end{equation}$$

We stress that the present manuscript does not prove this asymptotic law.

For $k=114$, $$\frac{114^2}{96}=135.375,$$ while the observed second active shell is $129$.

For $k=120$, $$\frac{120^2}{96}=150,$$ while the observed second active shell is $89$.

The two reported computations therefore do not by themselves constitute evidence for a monotone realization of the proposed quadratic scale. They are better viewed as motivating data for a broader computational test.

Additional computations over a substantially wider range of degrees, candidate pools, precisions, and support-selection procedures would be required before drawing an asymptotic conclusion.

Rank and Sparse Support

For $k=120$, the active shell index set is $$\begin{equation} \begin{aligned} \mathcal{S}_{120} = \bigl\{ &12, 89, 150, 168, 177, 178, 180, 189, 196, 200, 208, 212, 213, 214, \\ &215, 218, 220, 222, 227, 230, 235, 236, 242, 243, 246, 248, 250, \\ &252, 253, 256, 257, 258, 262, 267, 269, 271, 272, 273, 274, 276, \\ &277, 281, 282, 283, 285, 286, 287, 288, 289, 290, 291, 292, 295 \bigr\}. \end{aligned} \end{equation}$$

This set contains $53$ shell indices.

Choose the normalized integral basis $$\begin{equation} F_{a,b} = \Delta^2E_4^aE_6^b, \qquad 4a+6b=j-12, \end{equation}$$ for each relevant even degree $j$, with the standard normalized Eisenstein-series conventions $$E_4(q)=1+240\sum_{n=1}^\infty \sigma_3(n)q^n,$$ $$E_6(q)=1-504\sum_{n=1}^\infty \sigma_5(n)q^n,$$ and $$\Delta(q)=q\prod_{n=1}^\infty(1-q^n)^{24}.$$

Under these conventions, every Fourier coefficient of $F_{a,b}$ is an integer.

For the cumulative degree-$120$ space, assemble all such basis elements for $$j=12,14,\ldots,120.$$

Define the exact integer coefficient matrix $$\begin{equation} B_{\rm active} = \left( a_m(F) \right)_{ \substack{ F\in\mathcal B_{\leq120}\\ m\in\mathcal S_{120} }} \in \mathbb{Z}^{271\times53}, \label{eq:integer_matrix} \end{equation}$$ where $\mathcal B_{\leq120}$ is the chosen integral basis of the cumulative modular space $$\mathcal M_{\leq120}.$$

This matrix is deliberately distinguished from the real evaluation matrix $A_{\leq120,\mathcal S_{120}}$, whose columns contain the additional radial factors $(2m)^{-j/2}$ degree by degree.

Finite-field rank

Let $$p=2^{31}-1=2\,147\,483\,647.$$

The exact integer matrix [eq:integer_matrix] can be reduced entrywise modulo $p$. The reported computation gives $$\begin{equation} \mathop{\mathrm{rank}}_{\mathbb{F}_p}(B_{\rm active})=53. \label{eq:finite_rank} \end{equation}$$

Numerical Cubature Data

The following table summarizes the computations discussed above.

Here $D_k$ denotes the dimension of the cumulative modular evaluation space defined in Section 3. It should not be confused with the top-degree dimension $d_k$.

The residual values in Table 1 are floating-point numerical residuals. They do not constitute exact proofs that the corresponding modular equations vanish.

Point counts

The total number of lattice points represented by a shell-supported cubature is $$\begin{equation} N_{\rm total} = \sum_{m\in\mathcal S_{\rm active}}N_m. \end{equation}$$

Using [eq:shell_cardinality], this quantity can be evaluated exactly once the support is fixed.

Since $$N_m\asymp m^{11},$$ the shell cardinality has polynomial order $m^{11}$. Consequently, a comparison of total point counts should use the complete shell list rather than only the largest shell index.

For the $k=120$ support, the relevant exact quantity is $$N_{\rm total}(120) = \sum_{m\in\mathcal S_{120}} \frac{65520}{691} \left(\sigma_{11}(m)-\tau(m)\right).$$ An explicit integer evaluation should be included once the computational data are independently verified.

Sign Variation: What Remains to Be Proved

The convex-duality argument gives a clean route from an exact positive cubature to sign variation, but the required hypotheses must be stated carefully.

Let $$A_{\rm active} \in \mathbb{R}^{D_k\times r}$$ be an exact cumulative evaluation matrix for a support of $r$ shells.

Suppose there exists $$W\in\mathbb{R}_{>0}^{r}$$ such that $$A_{\rm active}W=0.$$

Then Gordan’s theorem rules out a nonzero vector $c$ satisfying $$A_{\rm active}^Tc\geq0.$$

Thus every nonzero modular coordinate vector has at least one negative evaluation on the selected support. Applying the same argument to $-c$ gives a positive evaluation as well, provided the zero-evaluation kernel has been eliminated.

Exact Certification Versus Floating-Point Evidence

The distinction between numerical evidence and mathematical certification is central to the present work.

The residual $$5.92\times10^{-16}$$ is evidence that a double-precision numerical solver found a vector close to the nullspace of the evaluation matrix. It does not prove that an exact null vector exists.

Several routes are available for upgrading the computation to an exact certificate.

1. Exact rational reconstruction. Given sufficiently accurate numerical weights, one may attempt to recover rational values and verify $$AW=0$$ using exact arithmetic.

2. Interval arithmetic. One may compute rigorous enclosures for the residual and establish a certified bound.

3. Exact nullspace computation. For sufficiently small support, rational linear algebra can determine whether the matrix has a positive null vector exactly.

4. Certified cone separation. A rational dual vector can establish the relevant Gordan or Farkas alternative exactly.

5. Higher-precision stability testing. Repeating the NNLS computation at substantially higher precision and comparing support and weights can test whether the observed support is numerically stable, although stability alone is not an exact proof.

Until such a procedure is supplied, the appropriate terminology is “numerical candidate,” “approximate annihilator,” or “computational evidence,” rather than “unconditional proof.”

Finite-Field Verification

Finite-field arithmetic is useful for detecting exact rank without the coefficient growth associated with fraction-free elimination over $\mathbb{Q}$.

Let $$p=2^{31}-1=2\,147\,483\,647.$$

The matrix used for the rank computation is the exact integer matrix $B_{\rm active}$ defined in [eq:integer_matrix], not the radially scaled real matrix $A_{\rm active}$.

The reported computation gives $$\mathop{\mathrm{rank}}_{\mathbb{F}_p}(B_{\rm active})=53.$$

For a reproducible certificate, the accompanying computational data should include at least one of the following:

1. the complete finite-field row-reduction record;

2. the $53$ pivot rows and columns; or

3. a single $53\times53$ minor whose determinant is explicitly shown to be nonzero modulo $p$.

The third option gives a particularly compact certificate: if $$\det M\not\equiv0\pmod p,$$ then the corresponding integer determinant is nonzero.

Stability Questions and the Infinite-Dimensional Limit

The finite-dimensional cone problem raises a natural question: whether families of finite positive certificates can converge to a meaningful positive functional in an infinite-dimensional setting.

Such a statement requires substantially more structure than the finite-dimensional Gordan alternative.

A future infinite-dimensional formulation would need to specify:

1. a precise topological vector space $E$ of test functions;

2. a dual space $E^*$ in which the limiting functionals live;

3. a topology in which bounded sequences of certificates possess convergent subsequences;

4. compatibility between finite-dimensional projections and the limiting functional;

5. closedness of the relevant positivity cone; and

6. a normalization preventing convergence to the zero functional.

These conditions constitute a framework for future analysis rather than a theorem in the absence of a fully specified functional-analytic model.

Observed growth of numerical weights

For illustration, the numerical calculations reported in the underlying computational study gave the following unnormalized $\ell^1$ norms: $$\begin{align*} k=12&:\quad1.24\times10^0,\\ k=16&:\quad7.17\times10^0,\\ k=20&:\quad5.29\times10^0,\\ k=24&:\quad1.35\times10^1,\\ k=28&:\quad2.49\times10^1,\\ k=32&:\quad1.99\times10^2. \end{align*}$$

These values suggest rapid growth in the normalization used by the computation. They do not, by themselves, establish exponential growth, nor do they prove failure of every alternative normalization or limiting procedure.

Reproducibility Requirements

To make the numerical claims independently verifiable, the following information should accompany the manuscript:

1. the exact definitions and normalizations of the modular bases;

2. the complete candidate pools $\mathcal D$;

3. the precise construction of the cumulative evaluation matrices;

4. the NNLS solver and numerical precision;

5. stopping tolerances and conditioning diagnostics;

6. the activation threshold $\varepsilon_{\rm act}$;

7. the complete numerical weight vectors;

8. the exact integer Fourier-coefficient matrices;

9. the finite-field pivot data or a nonzero maximal minor;

10. independent residual calculations at higher precision;

11. support-stability tables under threshold variation; and

12. scripts sufficient to reproduce every table in the manuscript.

In particular, the number of active shells should be regarded as a reproducible computational statistic rather than a mathematical invariant until its stability under precision, tolerance, and threshold changes has been established.

Discussion

The principal computational observation of this work is that sparse non-negative optimization produces shell supports very different from the contiguous supports used in earlier computations.

At $k=114$, the reported support has $54$ shells and begins with $$12,\ 129,\ldots,$$ while at $k=120$ the reported support has $53$ shells and begins with $$12,\ 89,\ 150,\ldots.$$

The appearance of large internal gaps is therefore a notable numerical feature of the two computations. Its dependence on degree, candidate pool, solver tolerance, precision, and regularization remains to be studied.

The finite-field rank calculation provides independent algebraic information about the selected $k=120$ support: its selected Fourier-evaluation columns are linearly independent, assuming the stated exact integer matrix and modular computation are correct.

Several important questions remain open:

1. Can the numerical positive weights be replaced by exact positive weights?

2. Is the observed support stable under increased arithmetic precision and changes in NNLS implementation?

3. Does the selected support generate a face of the full modular cone?

4. Is there a genuine asymptotic law governing the first exclusion gap?

5. What is the smallest support size capable of separating the relevant cumulative modular space?

6. Can an exact positive certificate be combined with a sufficiently large separating support to prove sign variation for an entire modular space?

These questions distinguish the experimentally observed sparse geometry from the theorems that follow from exact algebraic certificates.

Conclusion

Sparse non-negative optimization provides a computational framework for studying high-strength spherical cubatures supported on Leech-lattice shells.

The calculations reported here produce numerical candidates at strengths $115$ and $121$ with substantially non-contiguous supports. In both cases the supports exhibit pronounced internal exclusion gaps. For $k=120$, an independent finite-field calculation verifies linear independence of the selected Fourier-evaluation columns, assuming the stated exact integer matrix and modular computation.

The modular-cone formulation also identifies precisely what would be needed for a rigorous sign-variation theorem: an exact strictly positive annihilating vector together with an appropriate injectivity or separation statement for the evaluation map. A floating-point residual alone is insufficient.

Likewise, the proposed quadratic horizon $$m_*(k)\asymp\frac{k^2}{96}$$ is best viewed as a conjectural asymptotic scale suggested by a computational heuristic, rather than as an established law.

The main contribution is therefore a computationally testable framework for sparse modular cubature geometry, together with explicit statements of the additional certificates required to convert numerical observations into rigorous mathematical theorems.