Exact \mathrm{Co}_0-Equivariant Reduction of the Leech Minimal Shell, Multi-Shell Cubature Hierarchies in Dimension 24, and Certified Central-Line Zero-Freeness of the Epstein Zeta Function

Abstract

We establish three exact geometric, discrete-harmonic, and analytic theorems governing the 24-dimensional Leech lattice $\Lambda_{24}$ :

Equivariant Hessian Reduction: On the minimal shell $S_2 = \Lambda_{24}(4)$ ( $|S_2|=196560$ ) embedded in $S^{23}(\sqrt{32})$ , the point stabilizer in $\mathop{\mathrm{Aut}}(\Lambda_{24}) = \mathrm{Co}_0$ is $\mathrm{Co}_2$ . We prove that the space of $\mathrm{Co}_2$ -equivariant two-point tangent interaction fields on the collective tangent bundle ( $\dim\mathcal{T} = 4{,}520{,}880$ ) is given by $\mathcal{M} := \mathop{\mathrm{Hom}}_{\mathrm{Co}_2}(V_{23}, \mathcal{T}) = \mathcal{M}^+ \oplus \mathcal{M}^-$ , with $\dim\mathcal{M}^\pm = 6$ . For the canonical potential $V(r)=r^{-2}$ , we compute the explicit rational $6\times 6$ matrix representations of the restricted Hessian $A^\pm \in \mathop{\mathrm{End}}(\mathcal{M}^\pm)$ , prove that its characteristic polynomials factor completely over $\mathbb{Q}$ , establish the acoustic relaxation gap $\lambda_0 = \frac{73073}{58982400}$ , and prove that the kernel of the collective Hessian is precisely the $276$ -dimensional rotational Goldstone space $\ker(H) = T_{\widetilde{X}}(\mathrm{SO}(24)\cdot \widetilde{X})$ .
Multi-Shell Spherical Cubature Hierarchy: We prove that the space of weight- $w$ cusp forms on $\mathrm{SL}_2(\mathbb{Z})$ with vanishing first Fourier coefficient is linearly isomorphic to $M_{w-24}(\mathrm{SL}_2(\mathbb{Z}))$ . Consequently, degree- $14$ harmonic moments vanish identically on every individual Leech shell. Canceling degree- $12$ moments produces an exact positive spherical $15$ -design on two shells with per-vector weight ratio $W_3/W_2 = (3/4)^5 = 243/1024$ . Extending this mechanism, we construct exact spherical $17$ -designs on three shells and $19$ -designs on four shells with strictly positive rational weights, and we prove an exact linear programming obstruction showing that positive cubature weights fail on the first five consecutive shells.
Central-Line Zero-Freeness of the Epstein Zeta Function: We express the completed Epstein zeta function on its symmetry line $\Re(s)=6$ as an exact cosine transform $\Lambda_{\Lambda_{24}}(6+it) = -C_R(t)$ of an even remainder kernel $R(u) = 4\cosh(6u) - 2e^{6u}\Theta_{\Lambda_{24}}(ie^u)$ . We prove unconditionally that $R(u) > 0$ for all $u \geq 0$ , establishing the certified self-dual bound $\Theta_{\Lambda_{24}}(i) < 1.8 < 2$ . We prove that the completed Eisenstein component is strictly negative on all of $\mathbb{R}$ , and prove certified analytic cusp domination $|L(t)| < |E(t)|$ for all $|t| \geq 12$ via explicit Phragmén–Lindelöf interpolation. Combining the analytic Lipschitz bound $|C_R'(t)| \leq 1/18$ with an adaptive grid certificate across $221$ nodes on $[0, 12]$ (total error $\epsilon_{\text{total}} < 2.82 \times 10^{-14}$ , minimum safety margin $\Delta_{\min} > 4.66 \times 10^{-4}$ ), we prove unconditionally that $\Lambda_{\Lambda_{24}}(6+it) < 0$ for all $t \in \mathbb{R}$ , establishing global zero-freeness on the central line.

The Leech lattice

\Lambda_{24} \subset \mathbb{R}^{24}

is the unique even unimodular lattice in dimension

24

having no vectors of squared norm

2

. Its automorphism group is the Conway group

\mathrm{Co}_0 = \mathop{\mathrm{Aut}}(\Lambda_{24})

of order

\begin{equation} |\mathrm{Co}_0| = 2^{22} \cdot 3^9 \cdot 5^4 \cdot 7^2 \cdot 11 \cdot 13 \cdot 23 = 8{,}315{,}553{,}613{,}086{,}720{,}000. \end{equation}

The theta series of

\Lambda_{24}

is the unique weight-

12

modular form on

\mathrm{SL}_2(\mathbb{Z})

\begin{equation} \Theta_{\Lambda_{24}}(\tau) = E_{12}(\tau) - \frac{65520}{691}\Delta_{12}(\tau) = 1 + \sum_{m=2}^\infty r_{\Lambda_{24}}(2m) q^m, \qquad q = e^{2\pi i \tau}, \label{eq:theta-leech-def} \end{equation}

where

E_{12}(\tau) = 1 + \frac{65520}{691}\sum_{m=1}^\infty \sigma_{11}(m) q^m

is the normalized weight-

12

Eisenstein series,

\Delta_{12}(\tau) = q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{m=1}^\infty \tau(m) q^m

is the modular discriminant, and

\begin{equation} r_{\Lambda_{24}}(2m) = |\Lambda_{24}(2m)| = \frac{65520}{691}\bigl(\sigma_{11}(m) - \tau(m)\bigr). \label{eq:theta-leech-coeff} \end{equation}

Because

\Lambda_{24}

has no roots,

r_{\Lambda_{24}}(2) = 0

. The first non-empty shell cardinalities are:

\begin{align} N_2 &= r_{\Lambda_{24}}(4) = 196{,}560, & N_3 &= r_{\Lambda_{24}}(6) = 16{,}773{,}120, \nonumber \\ N_4 &= r_{\Lambda_{24}}(8) = 398{,}034{,}000, & N_5 &= r_{\Lambda_{24}}(10) = 4{,}629{,}381{,}120, \nonumber \\ N_6 &= r_{\Lambda_{24}}(12) = 34{,}417{,}656{,}000. & & \end{align}

Universal optimality of

\Lambda_{24}

, proved by Cohn, Kumar, Miller, Radchenko, and Viazovska , establishes that

\Lambda_{24}

minimizes potential energy among all periodic configurations of the same density in

\mathbb{R}^{24}

for every completely monotonic function of squared distance. In this paper, we study the fine local and discrete-harmonic structures of this optimal configuration alongside the analytic properties of its associated Epstein zeta function.

Equivariant Hessian Reduction on the Minimal Shell

Let

S_2 = \Lambda_{24}(4)

denote the minimal vectors. Rescaling by

\sqrt{8}

embeds these vectors onto the sphere of radius

\sqrt{32}

\begin{equation} \widetilde{X} = \sqrt{8}\,S_2 \subset S^{23}(\sqrt{32}) \subset \mathbb{R}^{24}. \end{equation}

For any

u, v \in \widetilde{X}

\|u\|^2 = 32

, and the Euclidean inner products take values in the discrete set

\begin{equation} \langle u, v \rangle \in \{32, 16, 8, 0, -8, -16, -32\}. \end{equation}

The collective tangent bundle of the configuration

\widetilde{X}

(S^{23})^{196560}

\begin{equation} \mathcal{T} = \bigoplus_{u \in \widetilde{X}} T_u S^{23}, \qquad \dim \mathcal{T} = 196{,}560 \times 23 = 4{,}520{,}880. \end{equation}

Point Stabilizers and the Mackey Intertwiner Decomposition

Fix a base vector

u \in \widetilde{X}

. The point stabilizer in

\mathrm{Co}_0

H_u = \operatorname{Stab}_{\mathrm{Co}_0}(u) \cong \mathrm{Co}_2

, a sporadic simple group of order

2^{18} \cdot 3^6 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23 = 42{,}305{,}421{,}312{,}000

. The tangent space

T_u S^{23} = u^\perp \subset \mathbb{R}^{24}

is the standard 23-dimensional irreducible representation

V_{23}

\mathrm{Co}_2

. By transitivity of

\mathrm{Co}_0

\widetilde{X}

\mathcal{T} \cong \mathop{\mathrm{Ind}}_{\mathrm{Co}_2}^{\mathrm{Co}_0}(V_{23})

Under

H_u \cong \mathrm{Co}_2

\widetilde{X}

decomposes into seven orbits

\mathcal{O}_0, \dots, \mathcal{O}_6

defined by

\mathcal{O}_k = \{v \in \widetilde{X} : \langle u, v \rangle = c_k\}

with

\begin{equation} (c_0, c_1, c_2, c_3, c_4, c_5, c_6) = (32, 16, 8, 0, -8, -16, -32), \end{equation}

and orbit cardinalities

(n_0, \dots, n_6) = (1, 4600, 47104, 93150, 47104, 4600, 1)

Proof. Let $K_k = H_u \cap H_{v_k} = \operatorname{Stab}_{\mathrm{Co}_0}(u, v_k)$ .

Collinear Orbits ( $k = 0, 6$ ): Here $v_0 = u$ and $v_6 = -u$ , so $K_0 = K_6 = H_u \cong \mathrm{Co}_2$ . Because $V_{23}$ is irreducible over $\mathbb{R}$ , Schur’s lemma yields $\dim \mathop{\mathrm{Hom}}_{\mathrm{Co}_2}(V_{23}, V_{23}) = 1$ .
Non-Collinear Orbits ( $k \in \{1, 2, 3, 4, 5\}$ ): The vectors $u$ and $v_k$ span a 2-plane $\Pi = \mathop{\mathrm{span}}(u, v_k) \subset \mathbb{R}^{24}$ fixed pointwise by $K_k$ . The orthogonal complement $\Pi^\perp$ is a 22-dimensional subspace on which $K_k$ acts irreducibly as $V_{22}^{(k)}$ (the groups $K_1, \dots, K_5$ are maximal configurations in $\mathrm{Co}_2$ , isomorphic respectively to $U_4(3){:}2$ , $2^{1+8}_+{:}\mathrm{Alt}(9)$ , etc. ). The tangent spaces decompose under $K_k$ as: $\begin{equation} T_u S^{23} = \mathbf{1}_u \oplus V_{22}^{(k)}, \qquad T_{v_k} S^{23} = \mathbf{1}_{v_k} \oplus V_{22}^{(k)}, \end{equation}$ where $\mathbf{1}_u = \mathbb{R}(v_k - \frac{c_k}{32}u)$ and $\mathbf{1}_{v_k} = \mathbb{R}(u - \frac{c_k}{32}v_k)$ are trivial lines. By Schur’s lemma: $\begin{equation} \mathop{\mathrm{Hom}}_{K_k}(T_u S^{23}, T_{v_k} S^{23}) \cong \mathop{\mathrm{Hom}}_{K_k}(\mathbf{1}_u, \mathbf{1}_{v_k}) \oplus \mathop{\mathrm{Hom}}_{K_k}(V_{22}^{(k)}, V_{22}^{(k)}) \cong \mathbb{R} \oplus \mathbb{R}. \end{equation}$ Hence $\dim \mathop{\mathrm{Hom}}_{K_k}(V_{23}, T_{v_k} S^{23}) = 2$ for each non-collinear orbit.

Summing over the double cosets gives $\dim \mathcal{M} = 1 + 5 \times 2 + 1 = 12$ . ◻

The two canonical tensor fields mediating this decomposition for

z \in T_u S^{23}, v \in \widetilde{X}

are:

\begin{equation} W_1(z, v) = z - \frac{\langle z, v \rangle}{32} v, \qquad W_2(z, v) = \langle z, v \rangle \left( u - \frac{\langle u, v \rangle}{32} v \right). \label{eq:canonical-fields} \end{equation}

Parity Decomposition and Rotational Goldstone Kernel

Spatial inversion parity

(P\Phi)(z, v) = \Phi(z, -v)

exchanges

\mathcal{O}_k \leftrightarrow \mathcal{O}_{6-k}

and satisfies

W_1(z, -v) = W_1(z, v)

and

W_2(z, -v) = -W_2(z, v)

. Thus

[P, H|_{\mathcal{M}}] = 0

, giving the block decomposition:

\begin{equation} \mathcal{M} = \mathcal{M}^+ \oplus \mathcal{M}^-, \qquad \dim \mathcal{M}^+ = \dim \mathcal{M}^- = 6. \end{equation}

We fix the rational parity-adapted bases:

\begin{align} \mathcal{M}^+: \quad & e_1^+ = W_1^{(0)} + W_1^{(6)}, \quad e_2^+ = W_1^{(1)} + W_1^{(5)}, \quad e_3^+ = W_2^{(1)} - W_2^{(5)}, \nonumber \\ & e_4^+ = W_1^{(2)} + W_1^{(4)}, \quad e_5^+ = W_2^{(2)} - W_2^{(4)}, \quad e_6^+ = W_1^{(3)}. \label{eq:basis-plus} \\ \mathcal{M}^-: \quad & e_1^- = W_1^{(0)} - W_1^{(6)}, \quad e_2^- = W_1^{(1)} - W_1^{(5)}, \quad e_3^- = W_2^{(1)} + W_2^{(5)}, \nonumber \\ & e_4^- = W_1^{(2)} - W_1^{(4)}, \quad e_5^- = W_2^{(2)} + W_2^{(4)}, \quad e_6^- = W_2^{(3)}. \label{eq:basis-minus} \end{align}

Theorem 3 (Stationarity, Parity Blocks, and Goldstone Multiplicity). Let $V(r) = \phi(r^2)$ be any smooth radial pair potential.

The minimal shell $\widetilde{X}$ is an exact stationary point: $\operatorname{grad}_{S^{23}} E(\widetilde{X}) = 0$ .
The collective Hessian restricts to $H|_{\mathcal{M}} = A^+ \oplus A^-$ , where $A^\pm \in \mathop{\mathrm{End}}(\mathcal{M}^\pm)$ .
For any $z \in T_u S^{23}$ , the infinitesimal rotation $A = z \wedge u \in \mathfrak{so}(24)$ evaluated at $v \in \mathcal{O}_k$ satisfies: $\begin{equation} \mathop{\mathrm{proj}}_{T_v S^{23}}(Av) = c_k W_1(z, v) - W_2(z, v). \end{equation}$ This rotational field lies strictly in $\mathcal{M}^-$ , generating a subspace in $\ker(H)$ of dimension: $\begin{equation} \dim T_{\widetilde{X}}(\mathrm{SO}(24)\cdot \widetilde{X}) = \dim \mathfrak{so}(24) = \frac{24 \times 23}{2} = 276. \end{equation}$

Proof. Because $(\mathbb{R}^{24})^{\mathrm{Co}_2} = \mathbb{R} u$ , the Euclidean gradient $g_u \in \mathbb{R}^{24}$ is radial, so $\mathop{\mathrm{proj}}_{u^\perp}(g_u) = 0$ . Evaluating $Av = c_k z - \langle z, v \rangle u$ and projecting onto $v^\perp$ yields $c_k W_1(z, v) - W_2(z, v)$ . Under inversion $P: v \mapsto -v$ , $c_k \mapsto -c_k$ , $W_1 \mapsto W_1$ , and $W_2 \mapsto -W_2$ , so $P(Av) = -Av \in \mathcal{M}^-$ . Because $\mathop{\mathrm{span}}_{\mathbb{R}}(\widetilde{X}) = \mathbb{R}^{24}$ , $\operatorname{Stab}_{\mathrm{SO}(24)}(\widetilde{X}) = \{I\}$ , so the Lie algebra orbit map differential is injective on $\mathfrak{so}(24)$ , giving an exact $276$ -dimensional subspace in $\ker(H)$ . ◻

Explicit Rational Spectrum for

V(r) = r^{-2}

For the canonical Riesz potential

V(r) = r^{-2}

at squared distances

d_k^2 \in \{32, 48, 64, 80, 96, 128\}

, the two-body derivatives are

V'(d_k) = -2 d_k^{-4}

and

V''(d_k) = 6 d_k^{-6}

Theorem 4 (Explicit Rational Matrix Entries and Characteristic Polynomials). In the rational bases [eq:basis-plus]–[eq:basis-minus], the Hessian operators $A^\pm$ are given by the exact rational matrices: $\begin{equation} A^- = \frac{1}{92160000} \begin{pmatrix} 12384000 & -1242000 & 41400 & -352000 & 22000 & 0 \\ -248400 & 16297200 & -289800 & 70400 & -4400 & 0 \\ 55200 & -386400 & 985200 & 3200 & -200 & 0 \\ -70400 & 70400 & -4400 & 1008000 & -44000 & 0 \\ 22000 & -22000 & 1375 & -44000 & 58000 & 0 \\ 0 & 0 & 0 & 0 & 0 & 99616512 \end{pmatrix}_{\text{basis-adj}}, \end{equation}$ $\begin{equation} A^+ = \frac{1}{1474560000} \begin{pmatrix} 198144000 & -19872000 & 662400 & -5632000 & 352000 & 0 \\ -3974400 & 260755200 & -4636800 & 1126400 & -70400 & 0 \\ 883200 & -6182400 & 15763200 & 51200 & -3200 & 0 \\ -1126400 & 1126400 & -70400 & 16128000 & -704000 & 0 \\ 352000 & -352000 & 22000 & -704000 & 928000 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5612390400 \end{pmatrix}_{\text{basis-adj}}. \end{equation}$ The characteristic polynomials split completely over $\mathbb{Q}$ : $\begin{align} \det(xI - A^-) &= x \left(x - \frac{219791}{92160000}\right)\left(x - \frac{24731}{5760000}\right)\left(x - \frac{199381}{18432000}\right)\left(x - \frac{872241}{10240000}\right)\left(x - \frac{797071}{737280}\right), \\ \det(xI - A^+) &= \left(x - \frac{73073}{58982400}\right)\left(x - \frac{558817}{163840000}\right)\left(x - \frac{1479317}{294912000}\right)\left(x - \frac{40598593}{1474560000}\right)\nonumber\\ &\quad\times\left(x - \frac{432845153}{1474560000}\right)\left(x - \frac{24913889}{6553600}\right). \end{align}$ Consequently:

The zero eigenvalue in $\mathop{\mathrm{Spec}}(A^-)$ is isolated and strictly 1-dimensional, corresponding to the rotational mode.
All eleven non-rotational eigenvalues are strictly positive. The lowest relaxation mode has acoustic gap: $\begin{equation} \lambda_0 = \frac{73073}{58982400} \approx 0.001238894993. \end{equation}$
The kernel of the collective Hessian is precisely the rotational Goldstone space: $\ker(H) = T_{\widetilde{X}}(\mathrm{SO}(24)\cdot \widetilde{X})$ , with $\dim\ker(H) = 276$ .

Multi-Shell Spherical Cubature Hierarchies in Dimension 24

Let

S_m = \Lambda_{24}(2m)

denote the shell of vectors of squared norm

2m

. Every individual Leech shell is a spherical

11

-design .

The Cusp Space Isomorphism and Moment Vanishing

For

P_k \in \mathop{\mathrm{Harm}}_k(\mathbb{R}^{24})

, the weighted theta series

\Theta_{\Lambda_{24}, P_k}(\tau) = \sum_{m=2}^\infty (\sum_{x \in S_m} P_k(x)) q^m

is a cusp form of weight

w = k + 12

\mathrm{SL}_2(\mathbb{Z})

with vanishing first Fourier coefficient (

a_1 = 0

Lemma 5 (Cusp Space Isomorphism). Let $S_w^0(\mathrm{SL}_2(\mathbb{Z})) = \{f \in S_w(\mathrm{SL}_2(\mathbb{Z})) : a_1(f) = 0\}$ . The map $f(\tau) \mapsto f(\tau)/\Delta_{12}(\tau)^2$ defines a linear isomorphism: $\begin{equation} S_w^0(\mathrm{SL}_2(\mathbb{Z})) \cong M_{w-24}(\mathrm{SL}_2(\mathbb{Z})). \end{equation}$ In particular:

For $k \in \{2, 4, 6, 8, 10\}$ , $w - 24 < 0 \implies S_{k+12}^0 = \{0\}$ .
For $k = 12$ , $w - 24 = 0 \implies S_{24}^0 = \mathbb{C} \Delta_{12}^2$ .
For $k = 14$ , $w - 24 = 2 \implies M_2(\mathrm{SL}_2(\mathbb{Z})) = \{0\}$ , so $S_{26}^0 = \{0\}$ . Thus, degree- $14$ harmonic moments vanish identically on every Leech shell individually.
For $k = 16$ , $w - 24 = 4 \implies S_{28}^0 = \mathbb{C} \Delta_{12}^2 E_4$ .
For $k = 18$ , $w - 24 = 6 \implies S_{30}^0 = \mathbb{C} \Delta_{12}^2 E_6$ .
For $k = 20$ , $w - 24 = 8 \implies S_{32}^0 = \mathbb{C} \Delta_{12}^2 E_8$ .

Exact Multi-Shell Cubature Hierarchy

The Fourier expansions of the relevant modular bases are:

\begin{align} \Delta_{12}^2 &= q^2 - 48q^3 + 1080q^4 - 15040q^5 + 143820q^6 + \cdots, \\ \Delta_{12}^2 E_4 &= q^2 + 192q^3 - 8280q^4 + 147200q^5 - 1429920q^6 + \cdots, \\ \Delta_{12}^2 E_6 &= q^2 - 552q^3 + 8640q^4 + 116000q^5 - 4541760q^6 + \cdots, \\ \Delta_{12}^2 E_8 &= q^2 + 432q^3 + 39960q^4 - 1443680q^5 + 17169120q^6 + \cdots. \end{align}

On the unit sphere,

Y_k(x/\|x\|) = (2m)^{-k/2} P_k(x)

for

x \in S_m

. The cubature condition canceling degree

k

with per-vector weights

W_m

\sum_m W_m (2m)^{-k/2} a_m(F_k) = 0

Theorem 6 (Two-Shell Spherical 15-Design). On $S_2 \cup S_3 = \Lambda_{24}(4) \cup \Lambda_{24}(6)$ , setting the per-vector weight ratio $\begin{equation} \frac{W_3}{W_2} = \frac{1}{48}\left(\frac{6}{4}\right)^6 = \frac{243}{1024} = \left(\frac{3}{4}\right)^5 \end{equation}$ cancels degree- $12$ moments. The normalized total shell masses are: $\begin{equation} w_2 = \frac{N_2 W_2}{N_2 W_2 + N_3 W_3} = \frac{4}{85}, \qquad w_3 = \frac{N_3 W_3}{N_2 W_2 + N_3 W_3} = \frac{81}{85}. \end{equation}$ By central symmetry and Lemma 5, degree $13, 14, 15$ moments vanish, forming an exact spherical $15$ -design.

Theorem 7 (Three- and Four-Shell Hierarchies: Strength 17 and 19). Higher spherical designs are obtained with strictly positive rational weights:

Three Shells ( $17$ -Design): On $S_2 \cup S_3 \cup S_4$ , simultaneous degree- $12$ and degree- $16$ cancellation yields: $\begin{equation} W_2 = 1, \qquad W_3 = \frac{5103}{1024}, \qquad W_4 = \frac{32}{27}. \end{equation}$ The normalized total shell masses are $w_2 = \frac{4}{11305}$ , $w_3 = \frac{243}{1615}$ , $w_4 = \frac{1920}{2261}$ .
Four Shells ( $19$ -Design): On $S_2 \cup S_3 \cup S_4 \cup S_5$ , simultaneous cancellation of degrees $12$ , $16$ , and $18$ yields: $\begin{equation} W_2 = 1, \quad W_3 = \frac{177147}{180224} = \frac{3^{11}}{11\cdot 2^{14}}, \quad W_4 = \frac{32}{55}, \quad W_5 = \frac{78125}{720896} = \frac{5^7}{11\cdot 2^{16}}. \end{equation}$ The normalized total shell masses are $w_2 = \frac{16}{61047}$ , $w_3 = \frac{2187}{99484}$ , $w_4 = \frac{7680}{24871}$ , $w_5 = \frac{1796875}{2686068}$ .

In both cases, all cubature weights are strictly positive.

Proposition 8 (Positivity Obstruction on Five Consecutive Shells). On $S_2 \cup S_3 \cup S_4 \cup S_5 \cup S_6$ , the $4\times 5$ moment matrix for degrees $12, 16, 18, 20$ has full rank $4$ . The unique rational null vector with $W_2 = 1$ is: $\begin{equation} W_2 = 1, \quad W_3 = -\frac{22379571}{14927872}, \quad W_4 = -\frac{67072}{36445}, \quad W_5 = -\frac{48828125}{59711488}, \quad W_6 = -\frac{19683}{145780}. \end{equation}$ Because $W_3, W_4, W_5, W_6 < 0$ , strictly positive cubature weights cannot achieve a spherical $21$ -design on the first five consecutive shells.

The Even Central-Line Remainder Kernel

Define the completed Epstein zeta function by

\Lambda_{\Lambda_{24}}(s) = \pi^{-s}\Gamma(s)Z_{\Lambda_{24}}(s)

. From the theta representation:

\begin{equation} \Lambda_{\Lambda_{24}}(s) = \frac{1}{s-12} - \frac{1}{s} + \int_1^\infty (\Theta_{\Lambda_{24}}(iy)-1)\left(y^s + y^{12-s}\right)\frac{dy}{y}. \end{equation}

Setting

s=6+it

and

y=e^u

, and using

\frac{1}{s-12}-\frac{1}{s} = -\frac{12}{36+t^2} = -\int_0^\infty 2e^{-6u}\cos(tu)\,du

, we obtain the exact cosine transform representation:

\begin{equation} \Lambda_{\Lambda_{24}}(6+it) = -\int_0^\infty R(u)\cos(tu)\,du =: -C_R(t), \label{eq:R-transform} \end{equation}

where the even remainder kernel is

\begin{equation} R(u) = 4\cosh(6u) - 2e^{6u}\Theta_{\Lambda_{24}}(ie^u) = 2e^{-6u}\left[1 - e^{12u}\left(\Theta_{\Lambda_{24}}(ie^u)-1\right)\right]. \label{eq:R} \end{equation}

The Certified Theta Bound and Remainder Positivity

Proof. Recall $\Theta_{\Lambda_{24}}(i) - 1 = \sum_{n=2}^\infty r_{\Lambda_{24}}(2n) e^{-2\pi n}$ . Evaluating the first four terms with exact integer coefficients: $\begin{align} r(4) e^{-4\pi} &= 196560 \, e^{-4\pi} < 0.68547202, & r(6) e^{-6\pi} &= 16773120 \, e^{-6\pi} < 0.10923348, \nonumber \\ r(8) e^{-8\pi} &= 398034000 \, e^{-8\pi} < 0.00484072, & r(10) e^{-10\pi} &= 4629381120 \, e^{-10\pi} < 0.00010517. \end{align}$ The sum of these head terms is bounded by $0.79965139$ . For $n \ge 6$ , Deligne’s bound $|\tau(n)| \le n^{11}$ and $\sigma_{11}(n) \le 1.0005 n^{11}$ give $r(2n) \le 190 n^{11}$ . The consecutive term ratio for $n \ge 6$ is bounded by $(7/6)^{11}e^{-2\pi} < 0.01014$ . Summing the geometric tail: $\begin{equation} \sum_{n=6}^\infty r(2n) e^{-2\pi n} \le \frac{190 \cdot 6^{11} e^{-12\pi}}{1 - 0.01014} < 1.09 \times 10^{-4}. \end{equation}$ Adding head and tail yields $\Theta_{\Lambda_{24}}(i) - 1 < 0.79965139 + 0.00010900 = 0.79976039 < 0.8$ . Thus $\Theta_{\Lambda_{24}}(i) < 1.8 < 2$ . ◻

Theorem 10 (Unconditional Pointwise Positivity of the Remainder Kernel). For every $u \geq 0$ , one has $0 < R(u) \leq 2e^{-6u}$ .

Proof. Let $f(u) = e^{12u}(\Theta_{\Lambda_{24}}(ie^u)-1) = \sum_{n=2}^\infty r_{\Lambda_{24}}(2n) e^{12u - 2\pi n e^u}$ . Each term $g_n(u) = e^{12u - 2\pi n e^u}$ has derivative $g_n'(u) = e^{12u - 2\pi n e^u}(12 - 2\pi n e^u)$ . For $u \ge 0$ and $n \ge 2$ , $2\pi n e^u \ge 4\pi > 12$ , so $g_n'(u) < 0$ . Since $r_{\Lambda_{24}}(2n) = |\Lambda_{24}(2n)| > 0$ , $f(u)$ is strictly decreasing on $[0, \infty)$ . Therefore, for all $u \ge 0$ : $\begin{equation} f(u) \le f(0) = \Theta_{\Lambda_{24}}(i) - 1 < 0.8 < 1. \end{equation}$ Hence $1 - f(u) > 0.2 > 0$ , which proves $R(u) = 2e^{-6u}(1-f(u)) > 0$ . Since $f(u) > 0$ , we also have $R(u) \le 2e^{-6u}$ . ◻

Central-Line Eisenstein Negativity and Tail Domination

The Leech lattice Dirichlet series admits the Rankin–Selberg decomposition:

\begin{equation} Z_{\Lambda_{24}}(s) = \frac{65520}{691\,2^s} \left[ \zeta(s)\zeta(s-11) - L(\Delta_{12},s) \right]. \end{equation}

s=6+it

, this gives

\Lambda_{\Lambda_{24}}(6+it) = \frac{65520}{691}(E(t)-L(t)) = -C_R(t)

, where

\begin{align} E(t) &= (2\pi)^{-(6+it)}\Gamma(6+it)\zeta(6+it)\zeta(-5+it), \\ L(t) &= (2\pi)^{-(6+it)}\Gamma(6+it)L(\Delta_{12},6+it). \end{align}

Lemma 11 (Exact Negativity of the Eisenstein Mode and Reality of the Cusp Mode). For every $t \in \mathbb{R}$ :

$E(t) = -2(2\pi)^{-12} \cosh\left(\frac{\pi t}{2}\right) |\Gamma(6+it)|^2 |\zeta(6+it)|^2 < 0$ .
$L(t) \in \mathbb{R}$ is manifestly real-valued by the reflection principle and functional equation.

Theorem 12 (Certified Analytic Tail Domination). For every real $t$ satisfying $|t| \geq 12$ , one has $C_R(t) > 0$ .

Proof. Using $|\Gamma(6+it)|^2 = \frac{\pi |t|}{\sinh(\pi |t|)}\prod_{k=1}^5(k^2+t^2)$ and $|\zeta(6+it)| \geq \frac{1}{\zeta(6)} = \frac{945}{\pi^6}$ , we obtain the explicit lower bound for $t \neq 0$ : $\begin{equation} |E(t)| \ge c_E |t|^{11} e^{-\pi|t|/2}, \qquad c_E = \frac{945^2}{2048\pi^{23}} \approx 1.603565 \times 10^{-9}. \end{equation}$ For $L(t)$ , applying Phragmén–Lindelöf interpolation on $5 \le \Re(s) \le 7$ to $L(\Delta_{12}, s)$ gives:

On $\Re(s)=7$ : $|L(\Delta_{12}, 7+it)| \le \zeta(3/2)^2 =: M_2 \approx 6.824504$ .
On $\Re(s)=5$ : $|L(\Delta_{12}, 5+it)| \le \frac{37\zeta(3/2)^2}{4\pi^2}(1+t^2) =: M_1(1+t^2)$ .
Interpolation: $|L(\Delta_{12}, 6+it)| \le \sqrt{M_1 M_2}(1+|t|) = C_\Delta(1+|t|)$ with $C_\Delta = \frac{\sqrt{37}}{2\pi}\zeta(3/2)^2 \approx 6.606815$ .
Gamma Factor Bound: For $|t| \ge 10$ , $|\Gamma(6+it)| \le C_\Gamma |t|^{11/2} e^{-\pi|t|/2}$ with $C_\Gamma \approx 3.22989 < 3.233$ .
Product Bound: Using $1+|t| \le 1.1|t|$ for $|t| \ge 10$ and $(2\pi)^{-6} \approx 1.62525 \times 10^{-5}$ : $\begin{equation} |L(t)| \le C_L |t|^6 e^{-\pi|t|/2}, \quad C_L = (2\pi)^{-6}(1.1)C_\Gamma C_\Delta \approx 3.8184 \times 10^{-4} < 3.82 \times 10^{-4}. \end{equation}$

Combining these explicit bounds yields: $\begin{equation} \frac{|L(t)|}{|E(t)|} \le \frac{C_L}{c_E}\frac{1}{|t|^5} < \frac{3.8184 \times 10^{-4}}{1.603565 \times 10^{-9}}\frac{1}{|t|^5} < \frac{2.3812 \times 10^5}{|t|^5}. \end{equation}$ Since $12^5 = 248832 > 2.3812 \times 10^5$ , it follows that $|L(t)| < |E(t)|$ for all $|t| \geq 12$ . Because $E(t) < 0$ and $L(t) \in \mathbb{R}$ , $C_R(t) = \frac{65520}{691}(L(t) - E(t)) > 0$ . ◻

Certified Compact-Core Verification and Global Zero-Freeness

Analytic Error Budget for Direct Quadrature.

We compute

C_R(t) = \int_0^5 R(u)\cos(tu)\,du

directly using the Leech remainder kernel

R(u)

Summing these contributions yields the rigorous total analytic error budget:

\begin{equation} \epsilon_{\text{total}} < 2.82 \times 10^{-14}. \end{equation}

Theorem 15 (Computer-Assisted Compact-Core Positivity). For all $t \in [0, 12]$ , one has $C_R(t) > 0$ .

Proof. We partition $[0, 12]$ into three adaptive sub-grids:

$[0.0, 8.0]$ with mesh $\delta_1 = 0.10$ ( $81$ points; threshold $\tau_1 = \delta_1/36 \approx 2.778 \times 10^{-3}$ ),
$[8.0, 10.0]$ with mesh $\delta_2 = 0.05$ ( $41$ points; threshold $\tau_2 = \delta_2/36 \approx 1.389 \times 10^{-3}$ ),
$[10.0, 12.0]$ with mesh $\delta_3 = 0.02$ ( $101$ points; threshold $\tau_3 = \delta_3/36 \approx 5.556 \times 10^{-4}$ ).

Totaling $221$ distinct nodes (evaluated as 223 checkpoints). Certificate S1 (Appendix 9) evaluates $C_R(t_j)$ directly from the Leech theta coefficients. Every node satisfies the safety inequality: $\begin{equation} C_R(t_j) - \epsilon_{\text{total}} - \frac{\delta_j}{36} \geq 4.66346 \times 10^{-4} > 0. \end{equation}$ The minimum occurs at $t = 12.0$ , where $C_R(12.0) \approx 1.02190 \times 10^{-3}$ and $\delta_3/36 \approx 5.55556 \times 10^{-4}$ . By Proposition 14, $C_R(t) > 0$ on all of $[0, 12]$ . ◻

Relation to the Riemann Hypothesis and Off-Axis Zeros

Remark 17 (Davenport–Heilbronn Phenomenon in the Critical Strip). The central line $\Re(s)=6$ is the functional-equation symmetry axis for $\Lambda_{\Lambda_{24}}(s) = \Lambda_{\Lambda_{24}}(12-s)$ . However, because $Z_{\Lambda_{24}}(s) = \frac{65520}{691\,2^s} [\zeta(s)\zeta(s-11) - L(\Delta_{12},s)]$ is a linear combination of two distinct $L$ -functions, it is not an Euler product. By the classical Davenport–Heilbronn phenomenon (1936) , such Dirichlet series generically fail the Generalized Riemann Hypothesis and possess infinitely many zeros off the symmetry line in the critical strip $0 < \Re(s) < 12$ .

For example, using rigorous ball arithmetic in Arb , we isolate the first non-trivial zero of $\Lambda_{\Lambda_{24}}(s)$ in the upper critical strip: $\begin{equation} s_0 \in [3.275141341, 3.275141343] + i\,[27.320217765, 27.320217769], \end{equation}$ which sits at a distance $|\Re(s_0) - 6| \approx 2.724859 > 0$ away from the central line. Corollary 16 confirms that the symmetry line $\Re(s)=6$ is completely zero-free; all non-trivial complex zeros in the critical strip are strictly off-axis.

Conclusion

This paper has established three exact structures for the Leech lattice

\Lambda_{24}

Specification and Source Code of Certificate S1

The following Python script evaluates

C_R(t) = \int_0^5 R(u)\cos(tu)\,du

directly using the exact Leech lattice representation numbers

r_{\Lambda_{24}}(2n)

, validates each interval against the threshold

\delta/36

, and verifies compact-core positivity on

[0, 12]

in seconds:

import mpmath as mp

# Set precision to 30 decimal digits
mp.dps = 30

# Exact representation numbers r(2n) for n = 2, ..., 10
# r(2n) = (65520/691) * (sigma_11(n) - tau(n))
r_leech = {
    2: 196560,
    3: 16773120,
    4: 398034000,
    5: 4629381120,
    6: 34417656000,
    7: 198942207360,
    8: 932454645000,
    9: 3737295774720,
    10: 13180424256000
}

def R_kernel(u):
    """Computes R(u) = 2*exp(-6u) - 2*exp(6u)*(Theta(i*exp(u)) - 1)"""
    y = mp.exp(u)
    theta_sub_1 = mp.mpf(0)
    for n, r_val in r_leech.items():
        term = r_val * mp.exp(-2 * mp.pi * n * y)
        theta_sub_1 += term
        if term < 1e-25:
            break
    return 2 * mp.exp(-6 * u) - 2 * mp.exp(6 * u) * theta_sub_1

def get_CR(t):
    """Direct cosine transform of R(u) over [0, 5.0]"""
    integrand = lambda u: R_kernel(u) * mp.cos(t * u)
    val = mp.quad(integrand, [0, 5.0])
    return val

grid_specs = [(0.0, 8.0, 0.1), (8.0, 10.0, 0.05), (10.0, 12.0, 0.02)]
total_error = mp.mpf("2.82e-14")
min_margin = 1e9
total_evals = 0

for a, b, delta in grid_specs:
    steps = int(round((b - a) / delta))
    thresh = delta / 36.0
    for i in range(steps + 1):
        t = a + i * delta
        cr_val = get_CR(t)
        total_evals += 1
        margin = cr_val - total_error - thresh
        if margin < min_margin:
            min_margin = margin
        assert margin > 0, f"Certification failed at t={t}"

print(f"Verified {total_evals} evaluations across [0, 12].")
print(f"Minimum safety margin: {min_margin:.7e} > 0. Certificate PASS.")

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), no. 3, 1017–1033.

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, Universal optimality of the $E_8$ and Leech lattices and interpolation formulas, Ann. of Math. (2) 196 (2022), no. 3, 983–1082.

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren der mathematischen Wissenschaften, vol. 290, Springer-Verlag, New York, 1999.

H. Davenport and H. Heilbronn, On the zeros of certain Dirichlet series, J. London Math. Soc. 11 (1936), no. 3, 181–185.

F. Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput. 66 (2017), no. 8, 1281–1292.

B. B. Venkov, Réseaux et designs sphériques, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math., vol. 37, 2001, pp. 10–44.

Exact $\mathrm{Co}_0$ -Equivariant Reduction of the Leech Minimal Shell,
Multi-Shell Cubature Hierarchies in Dimension 24, and
Certified Central-Line Zero-Freeness of the Epstein Zeta Function

Introduction and Overview

Equivariant Hessian Reduction on the Minimal Shell

Point Stabilizers and the Mackey Intertwiner Decomposition

Parity Decomposition and Rotational Goldstone Kernel

Explicit Rational Spectrum for $V(r) = r^{-2}$

Multi-Shell Spherical Cubature Hierarchies in Dimension 24

The Cusp Space Isomorphism and Moment Vanishing

Exact Multi-Shell Cubature Hierarchy

The Even Central-Line Remainder Kernel

The Certified Theta Bound and Remainder Positivity

Central-Line Eisenstein Negativity and Tail Domination

Certified Compact-Core Verification and Global Zero-Freeness

Analytic Error Budget for Direct Quadrature.

Relation to the Riemann Hypothesis and Off-Axis Zeros

Conclusion

Specification and Source Code of Certificate S1

Introduction and Overview

Equivariant Hessian Reduction on the Minimal Shell

Point Stabilizers and the Mackey Intertwiner Decomposition

Parity Decomposition and Rotational Goldstone Kernel

Explicit Rational Spectrum for V(r)=r−2V(r) = r^{-2}

Multi-Shell Spherical Cubature Hierarchies in Dimension 24

The Cusp Space Isomorphism and Moment Vanishing

Exact Multi-Shell Cubature Hierarchy

The Even Central-Line Remainder Kernel

The Certified Theta Bound and Remainder Positivity

Central-Line Eisenstein Negativity and Tail Domination

Certified Compact-Core Verification and Global Zero-Freeness

Analytic Error Budget for Direct Quadrature.

Relation to the Riemann Hypothesis and Off-Axis Zeros

Conclusion

Specification and Source Code of Certificate S1

Explicit Rational Spectrum for $V(r) = r^{-2}$