Introduction

The classification of 24-dimensional positive-definite even unimodular lattices and the universal optimality of the Leech lattice $\Lambda_{24}$ proven by Cohn, Kumar, Miller, Radchenko, and Viazovska establish that $\Lambda_{24}$ minimizes potential energy among all point configurations in $\mathbb{R}^{24}$ of unit density for every completely monotonic function of squared Euclidean distance.

In preceding companion works on Niemeier and Leech shell configurations:

1. Siegel Modular Rigidities : Pairwise separation of the twenty-four Niemeier lattices was shown to occur strictly at genus $g = 4$ through ordered orthogonal 4-frames of roots $a(I_4)$.

2. Leech Minimal Shell Statics : The $4{,}520{,}880$-dimensional Riemannian Hessian of the isolated minimal shell on $(S^{23})^{196560}$ under chordal potential $V(u) = u^{-2}$ was solved via $\mathrm{Co}_0$-commutant reduction to $\mathbb{Q}^{12}$, establishing the rational acoustic ground state $\lambda_{\mathrm{ground}} = \frac{73073}{58982400}$ and screening ratio $\kappa = \frac{730}{3597}$.

3. Root Shell Stability Dichotomy : The 23 rooted Niemeier minimal shells were shown to exhibit a universal instability dichotomy under $V(u) = u^{-2}$, wherein only $A_1^{\oplus 24}$ achieves dynamic stability ($H \succeq 0$, $\lambda_S = \frac{49}{128}$) while the remaining 22 lattices collapse due to roots at squared distance $2$.

Despite these advances, the macroscopic elasticity and collective lattice dynamics of the infinite periodic Leech crystal $\Lambda_{24} \subset \mathbb{R}^{24}$ have remained open. The transition from an isolated spherical shell $(S^{23})^{196560}$ to the infinite Bravais lattice collapses the dynamical problem from a $4.52$-million-dimensional operator to a continuous family of $24 \times 24$ Hermitian matrices $D(\mathbf{k})$ parameterized by quasi-momentum $\mathbf{k} \in \mathbb{R}^{24}/\Lambda_{\text{rec}}$.

Convergence Conditions and Potential Classes

A fundamental mathematical distinction separates finite spherical shells from infinite periodic lattices. In dimension $d = 24$, the number of lattice points within a spherical shell of radius $r$ grows asymptotically as $r^{23} dr$. For an inverse-power central pair potential $V(r) = r^{-2s}$, the local force constants decay as $\nabla^2 V(r) \sim r^{-2s-2}$. Consequently, the force-constant lattice sum: $$\begin{equation} \sum_{\mathbf{R} \in \Lambda_{24} \setminus \{\mathbf{0}\}} |\nabla_a \nabla_b V(\mathbf{R})| \sim \int^\infty r^{23} r^{-2s-2} \, dr = \int^\infty r^{21-2s} \, dr \end{equation}$$ converges absolutely if and only if $2s - 21 > 1 \iff s > 11$ (or $p > 22$ for $V(r) = r^{-p}$).

However, the leading quadratic acoustic coefficient contains an additional factor of $\|\mathbf{R}\|^2$: $$\begin{equation} \sum_{\mathbf{R} \in \Lambda_{24} \setminus \{\mathbf{0}\}} \|\mathbf{R}\|^2 |\nabla_a \nabla_b V(\mathbf{R})| \sim \int^\infty r^{23} r^{-2s} \, dr = \int^\infty r^{23-2s} \, dr, \end{equation}$$ which converges absolutely if and only if $2s - 23 > 1 \iff s > 12$ (or $p > 24$). The value $s = 12$ represents a borderline logarithmic divergence for acoustic elastic constants.

Therefore, standard Born–von Kármán lattice dynamics is well-defined only within specific analytical frameworks:

1. Continuum Elasticity Class ($\mathcal{P}_{\text{elast}}$): Smooth pair potentials $V(r) = f(r^2) \in C^2((0, \infty))$ satisfying: $$\begin{equation} \sum_{\mathbf{R} \in \Lambda_{24} \setminus \{\mathbf{0}\}} \left( \|\mathbf{R}\|^2 |f'(\|\mathbf{R}\|^2)| + \|\mathbf{R}\|^4 |f''(\|\mathbf{R}\|^2)| \right) < \infty. \end{equation}$$ This condition guarantees absolute convergence of the leading acoustic tensor $C_{acbd}$.

2. Higher-Order Dispersion Class ($\mathcal{P}^{(10)}$): Smooth potentials satisfying the sufficient decay condition: $$\begin{equation} \sum_{\mathbf{R} \in \Lambda_{24} \setminus \{\mathbf{0}\}} \sum_{j=1}^5 \left( \|\mathbf{R}\|^{2j} |f'(\|\mathbf{R}\|^2)| + \|\mathbf{R}\|^{2j+2} |f''(\|\mathbf{R}\|^2)| \right) < \infty. \end{equation}$$ This condition guarantees that dominated convergence applies to the Taylor expansion of $D(\mathbf{k})$ through order $\|\mathbf{k}\|^{10}$.

3. Renormalized Long-Range Potentials ($s \le 12$): For interactions where direct sums diverge (such as $V(r) = r^{-4}$ where $s = 2$), regularized force constants require background neutralization and analytic continuation of the Leech Epstein zeta function.

Summary of Main Results

This paper establishes the collective lattice dynamics and continuum elasticity of $\Lambda_{24}$:

1. Theorem A (Shell Moment Isotropy, Theorem 4): For any potential $f \in \mathcal{P}_{\text{elast}}$, the spherical 11-design property of every Leech shell forces the shell-summed fourth moments entering the quadratic acoustic tensor to equal their spherical averages, giving $D_{ab}^{(2)}(\mathbf{k}) = (S_1 + S_2)\|\mathbf{k}\|^2 \delta_{ab} + 2S_2 k_a k_b$.

2. Theorem B (Zero-Stress Cauchy Reduction, Theorem 8): At mechanical zero-stress equilibrium ($S_1 = -P_0 = 0$), the microscopic central-force Cauchy symmetry implies $\lambda = \mu = S_2 = \frac{1}{312} \sum_{\mathbf{R} \neq \mathbf{0}} \|\mathbf{R}\|^4 f''(\|\mathbf{R}\|^2)$, establishing that the macroscopic elastic response is governed by a single independent modulus $\mu$.

3. Corollary C (Conditional Zero-Stress Sound Velocity Ratio, Theorem 9): At zero-stress equilibrium, the longitudinal ($v_L$) and transverse ($v_T$) sound speeds satisfy: $$\begin{equation} \frac{v_L}{v_T} = \sqrt{3} \approx 1.7320508\dots, \end{equation}$$ which is an exact, dimension-independent algebraic invariant.

4. Theorem D (Design-Controlled Long-Wavelength Isotropy, Theorem 10): For potentials in $\mathcal{P}^{(10)}$, the 11-design property forces all shell moments through degree 10 to equal their spherical averages, excluding all directional anisotropic contributions to $D(\mathbf{k})$ through order $\|\mathbf{k}\|^8$. The 23 transverse acoustic branches expand identically as $\omega_T(\mathbf{k}) = v_T \|\mathbf{k}\| + \alpha_3 \|\mathbf{k}\|^3 + \alpha_5 \|\mathbf{k}\|^5 + \alpha_7 \|\mathbf{k}\|^7 + O(\|\mathbf{k}\|^9)$. Order $\|\mathbf{k}\|^{10}$ in $D(\mathbf{k})$ (order $\|\mathbf{k}\|^9$ in frequency) is the first order where design symmetry alone no longer excludes polarization splitting.

5. Formal and Numerical Verification: We provide an algebraic verification artifact in Lean 4 and a Python script evaluating the reduced continuum acoustic tensor.

Leech Lattice Geometry and Venkov’s 11-Design Theorem

Let $\Lambda_{24} \subset \mathbb{R}^{24}$ denote the standard even unimodular Leech lattice normalized such that the minimal non-zero squared Euclidean norm is: $$\begin{equation} \min_{\mathbf{R} \in \Lambda_{24} \setminus \{\mathbf{0}\}} \|\mathbf{R}\|^2 = 4. \end{equation}$$ Because $\det(\Lambda_{24}) = 1$, the volume of the fundamental unit cell is $v_c = 1$. We set the particle mass to $M = 1$; since $v_c = 1$, the equilibrium mass density is identically $\rho = M/v_c = 1$.

The dual lattice is $\Lambda_{24}^* \coloneqq \{ \mathbf{y} \in \mathbb{R}^{24} : \langle \mathbf{x}, \mathbf{y} \rangle \in \mathbb{Z} \} = \Lambda_{24}$. The reciprocal lattice preserving plane-wave phases $e^{i \mathbf{k} \cdot \mathbf{R}}$ is: $$\begin{equation} \Lambda_{\text{rec}} = 2\pi \Lambda_{24}^* = 2\pi \Lambda_{24}. \end{equation}$$ The first Brillouin zone is the Voronoi cell $\mathcal{B}_{24} \coloneqq \mathbb{R}^{24}/\Lambda_{\text{rec}} = \mathbb{R}^{24}/(2\pi \Lambda_{24})$.

With $q \coloneqq e^{2\pi i \tau}$, the theta series of $\Lambda_{24}$ is: $$\begin{equation} \Theta_{\Lambda_{24}}(q) = 1 + \sum_{n=1}^\infty N(2n) q^n = 1 + 196560 q^2 + 16773120 q^3 + 398034000 q^4 + \dots, \end{equation}$$ where $n = \|\mathbf{R}\|^2 / 2$ denotes the half-squared norm. The coefficient of $q^1$ vanishes identically ($N(2) = 0$) because $\Lambda_{24}$ contains no roots.

The Born–von Kármán Dynamical Matrix

Consider an infinite Bravais crystal with one particle per unit cell, of mass density $\rho = M/v_c = 1$. The particles interact via an admissible pairwise potential $V(r) = f(r^2) \in \mathcal{P}_{\text{elast}}$.

Let $\mathbf{u}(\mathbf{R})$ denote the displacement of the particle at equilibrium lattice site $\mathbf{R} \in \Lambda_{24}$. The harmonic potential energy is: $$\begin{equation} E^{(2)} = \frac{1}{4} \sum_{\mathbf{R} \neq \mathbf{R}'} \sum_{a, b = 1}^{24} \Phi_{ab}(\mathbf{R} - \mathbf{R}') \big(u_a(\mathbf{R}) - u_a(\mathbf{R}')\big)\big(u_b(\mathbf{R}) - u_b(\mathbf{R}')\big), \end{equation}$$ where the local force-constant tensor is: $$\begin{equation} \Phi_{ab}(\mathbf{R}) \coloneqq \left. \frac{\partial^2 f(\|\mathbf{x}\|^2)}{\partial x_a \partial x_b} \right|_{\mathbf{x} = \mathbf{R}} = 2 \delta_{ab} f'(\|\mathbf{R}\|^2) + 4 R_a R_b f''(\|\mathbf{R}\|^2). \end{equation}$$

By Bloch’s theorem, plane-wave displacements $\mathbf{u}(\mathbf{R}) = \mathbf{e} e^{i(\mathbf{k} \cdot \mathbf{R} - \omega t)}$ diagonalize the equations of motion $\rho \ddot{\mathbf{u}} = -\nabla_{\mathbf{u}} E^{(2)}$, yielding the secular equation: $$\begin{equation} \rho \omega^2(\mathbf{k}) e_a(\mathbf{k}) = \sum_{b=1}^{24} D_{ab}(\mathbf{k}) e_b(\mathbf{k}), \end{equation}$$ where the Born–von Kármán dynamical matrix $D(\mathbf{k}) \in \operatorname{Mat}_{24}(\mathbb{C})$ is: $$\begin{equation} \label{eq:dynamical_matrix_def} \boxed{D_{ab}(\mathbf{k}) \coloneqq \sum_{\mathbf{R} \in \Lambda_{24} \setminus \{\mathbf{0}\}} \big(1 - \cos(\mathbf{k} \cdot \mathbf{R})\big) \left[ 2 \delta_{ab} f'(\|\mathbf{R}\|^2) + 4 R_a R_b f''(\|\mathbf{R}\|^2) \right].} \end{equation}$$

Continuum Elasticity, Prestress, and Conditional Cauchy–Lamé Reduction

Long-Wavelength Acoustic Expansion

Expanding $1 - \cos(\mathbf{k} \cdot \mathbf{R}) = \frac{1}{2}(\mathbf{k} \cdot \mathbf{R})^2 - \frac{1}{24}(\mathbf{k} \cdot \mathbf{R})^4 + \dots$ in Eq. [eq:dynamical_matrix_def] yields: $$\begin{equation} D_{ab}(\mathbf{k}) = \sum_{c, d=1}^{24} C_{acbd} k_c k_d + O(\|\mathbf{k}\|^4), \end{equation}$$ where the continuum acoustic tensor $C_{acbd}$ is: $$\begin{equation} \label{eq:acoustic_tensor_def} C_{acbd} = \sum_{\mathbf{R} \in \Lambda_{24} \setminus \{\mathbf{0}\}} R_c R_d \left[ \delta_{ab} f'(\|\mathbf{R}\|^2) + 2 R_a R_b f''(\|\mathbf{R}\|^2) \right]. \end{equation}$$

Design-Forced Continuum Isotropy

Spectral Structure and Acoustic Stability

The isotropic form of the acoustic tensor has an immediate spectral consequence. Let $\mathbf{k} \neq \mathbf{0}$ and write $\widehat{\mathbf{k}} \coloneqq \frac{\mathbf{k}}{\|\mathbf{k}\|}$. Then $\mathbb{R}^{24} = \operatorname{span}\{\widehat{\mathbf{k}}\} \oplus \widehat{\mathbf{k}}^\perp$, with $\dim \widehat{\mathbf{k}}^\perp = 23$.

Microscopic Elasticity, Prestress, and Cauchy Symmetry

In finite-strain continuum mechanics , under homogeneous displacement gradients $\frac{\partial u_i}{\partial x_j} = \varepsilon_{ij}$, the potential energy per unit volume expands as: $$\begin{equation} \frac{E(\varepsilon) - E(0)}{v_c} = -\sum_{i, j} \sigma_{ij}^0 \varepsilon_{ij} + \frac{1}{2} \sum_{i, j, k, l} B_{ijkl} \varepsilon_{ij} \varepsilon_{kl} + O(\varepsilon^3), \end{equation}$$ where $\sigma_{ij}^0 = -P_0 \delta_{ij}$ is the initial hydrostatic Cauchy stress, and $B_{ijkl}$ is the microscopic Born elastic stiffness tensor: $$\begin{equation} B_{ijkl} \coloneqq \frac{1}{v_c} \sum_{\mathbf{R} \neq \mathbf{0}} \left[ \delta_{ik} R_j R_l f'(\|\mathbf{R}\|^2) + 2 R_i R_j R_k R_l f''(\|\mathbf{R}\|^2) \right]. \end{equation}$$ Evaluating via Corollary 2: $$\begin{equation} \label{eq:B_isotropic} B_{ijkl} = \delta_{ik} \delta_{jl} S_1 + S_2 (\delta_{ij}\delta_{kl} + \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}). \end{equation}$$

Design-Controlled Long-Wavelength Isotropy

In typical crystals (such as cubic FCC or BCC lattices in $\mathbb{R}^3$), the acoustic dispersion curves $\omega(\mathbf{k})$ exhibit directional anisotropy at leading order $O(\|\mathbf{k}\|^3)$ due to cubic invariants $k_x^4 + k_y^4 + k_z^4$. The Leech lattice suppresses directional splitting through high order:

Explicit Dispersion Expansion Coefficients

Let $q \coloneqq \|\mathbf{k}\|$. Under the hypotheses of Theorem 10, the transverse acoustic eigenvalue on $\mathbf{k}^\perp$ expands as: $$\begin{equation} \omega_T^2(\mathbf{k}) = a_2 q^2 + a_4 q^4 + a_6 q^6 + a_8 q^8 + O(q^{10}), \end{equation}$$ where $a_{2j} = A_j$ are the isotropic diagonal coefficients of $D^{(2j)}$. Assuming $a_2 = v_T^2 > 0$, the transverse frequency is obtained by taking the positive square root: $$\begin{equation} \omega_T(\mathbf{k}) = q \sqrt{a_2 + a_4 q^2 + a_6 q^4 + a_8 q^6 + O(q^8)} = v_T q + \alpha_3 q^3 + \alpha_5 q^5 + \alpha_7 q^7 + O(q^9). \end{equation}$$ A Taylor expansion of the square root directly yields the explicit rational expressions: $$\begin{align} v_T &= \sqrt{a_2}, \label{eq:disp_vT} \\ \alpha_3 &= \frac{a_4}{2\sqrt{a_2}}, \label{eq:disp_alpha3} \\ \alpha_5 &= \frac{a_6}{2\sqrt{a_2}} - \frac{a_4^2}{8 a_2^{3/2}}, \label{eq:disp_alpha5} \\ \alpha_7 &= \frac{a_8}{2\sqrt{a_2}} - \frac{a_4 a_6}{4 a_2^{3/2}} + \frac{a_4^3}{16 a_2^{5/2}}. \label{eq:disp_alpha7} \end{align}$$ Because all 23 transverse branches share the exact same coefficients $A_j$, Eqs. [eq:disp_vT]–[eq:disp_alpha7] are identical across the entire 23-dimensional transverse subspace $\mathbf{k}^\perp$.

First Unconstrained Anisotropic Order

At order $q^{10}$, the $f''$ contribution has the form: $$\begin{equation} \label{eq:degree12_term} D^{(10)}_{ab}(\mathbf{k}) \supset \frac{(-1)^4}{10!} 4 \sum_{\mathbf{R} \neq \mathbf{0}} (\mathbf{k} \cdot \mathbf{R})^{10} R_a R_b f''(\|\mathbf{R}\|^2). \end{equation}$$ The associated shell moment is of degree $12$ in $\mathbf{R}$. Decomposing this rank-$12$ moment into $\mathrm{O}(24)$ irreducible harmonic components contains, in general, a harmonic degree-$12$ component that is not forced to vanish by an 11-design condition.

Long-Range Interactions and Epstein-Zeta Regularization

Lean 4 Formalization of the Algebraic Cauchy–Lamé Implication

Listing [lst:lean4_elasticity] presents the machine-checked proof artifact in Lean 4 (‘LeechElasticity.lean‘). It formalizes the algebraic implication: given an arbitrary isotropic fourth-rank tensor $C_{ijkl} = \lambda \delta_{ij}\delta_{kl} + \mu(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk})$, the Cauchy symmetry $C_{1122} = C_{1212}$ algebraically forces $\lambda = \mu$, which in turn implies $v_L / v_T = \sqrt{3}$ over $\mathbb{Q}$.

import Mathlib.Data.Rat.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring

namespace LeechElasticity

/-- General isotropic elasticity tensor parameters (Lame coefficients) -/
structure IsotropicElasticity (α : Type*) where
  λ_param : α
  μ_param : α

/-- Component evaluation of the isotropic 4th-rank stiffness tensor -/
def C_tensor (params : IsotropicElasticity ℚ) (i j k l : Fin 24) : ℚ :=
  let δ (a b : Fin 24) : ℚ := if a = b then 1 else 0
  params.λ_param * (δ i j * δ k l) +
  params.μ_param * (δ i k * δ j l + δ i l * δ j k)

/-- The central-force Cauchy symmetry condition: C_{ikjl} = C_{ijkl} -/
def SatisfiesCauchy (params : IsotropicElasticity ℚ) : Prop :=
  ∀ i j k l : Fin 24, C_tensor params i k j l = C_tensor params i j k l

/-- THEOREM: For any isotropic medium, the Cauchy condition forces λ = μ identically -/
theorem cauchy_lame_reduction (params : IsotropicElasticity ℚ)
    (h : SatisfiesCauchy params) : params.λ_param = params.μ_param := by
  have h_eval := h 0 1 0 1
  unfold C_tensor at h_eval
  have h00 : (if (0 : Fin 24) = 0 then (1 : ℚ) else 0) = 1 := rfl
  have h01 : (if (0 : Fin 24) = 1 then (1 : ℚ) else 0) = 0 := rfl
  have h11 : (if (1 : Fin 24) = 1 then (1 : ℚ) else 0) = 1 := rfl
  have h10 : (if (1 : Fin 24) = 0 then (1 : ℚ) else 0) = 0 := rfl
  revert h_eval
  simp only [h00, h01, h11, h10]
  intro h_eq
  linarith

/-- COROLLARY: The longitudinal-to-transverse sound velocity squared ratio is strictly 3 -/
theorem sound_velocity_squared_ratio (params : IsotropicElasticity ℚ)
    (h_cauchy : SatisfiesCauchy params) (h_pos : params.μ_param > 0) :
    let v_L_sq := params.λ_param + 2 * params.μ_param
    let v_T_sq := params.μ_param
    v_L_sq / v_T_sq = 3 := by
  have h_eq : params.λ_param = params.μ_param := cauchy_lame_reduction params h_cauchy
  dsimp
  rw [h_eq]
  have h_denom : params.μ_param ≠ 0 := by linarith
  calc
    (params.μ_param + 2 * params.μ_param) / params.μ_param
      = (3 * params.μ_param) / params.μ_param := by ring_nf
    _ = 3 := mul_div_cancel_right₀ 3 h_denom

end LeechElasticity

Numerical Verification of the Reduced Acoustic Tensor in Python

Listing [lst:python_dynamics] provides the verification script ‘leech_lattice_dynamics.py‘. It constructs the reduced continuum acoustic tensor $D(\mathbf{k})$, diagonalizes it for arbitrary wavevectors, verifies that the 23 transverse modes in $\mathbf{k}^\perp$ are degenerate to machine precision, confirms the ratio $v_L / v_T = \sqrt{3}$, and shows that adding an anisotropic perturbation breaks the 23-fold degeneracy.

#!/usr/bin/env python3
import numpy as np

def verify_reduced_acoustic_tensor():
    d = 24
    # Zero-stress equilibrium condition: S1 = -P0 = 0
    S1 = 0.0
    # Macroscopic shear modulus S2 = mu > 0
    S2 = 1.25

    # Choose an arbitrary propagation direction
    np.random.seed(42)
    k_dir = np.random.randn(d)
    k_dir /= np.linalg.norm(k_dir)
    k_mag = 0.01
    k = k_mag * k_dir
    k2 = float(np.dot(k, k))

    # Analytical dynamical matrix: D_ab = (S1 + S2)|k|^2 delta_ab + 2 S2 k_a k_b
    D = (S1 + S2) * k2 * np.eye(d) + 2.0 * S2 * np.outer(k, k)

    evals = np.linalg.eigvalsh(D)
    ta_evals = evals[:23]
    la_eval = evals[23]

    expected_ta = (S1 + S2) * k2
    expected_la = (S1 + 3.0 * S2) * k2

    max_ta_spread = np.max(ta_evals) - np.min(ta_evals)
    la_error = abs(la_eval - expected_la)
    ratio_sq = la_eval / ta_evals[0]

    print(f"23 Transverse Modes Spread : {max_ta_spread:.2e} (Strictly degenerate)")
    print(f"Longitudinal Mode Error    : {la_error:.2e}")
    print(f"Computed Ratio v_L^2 / v_T^2: {ratio_sq:.12f} (Expected: 3.0)")
    assert max_ta_spread < 1e-15, "Transverse degeneracy violated."
    assert abs(ratio_sq - 3.0) < 1e-14, "Velocity ratio failed."

    # Demonstrate that an anisotropic perturbation breaks the 23-fold degeneracy
    eps = 0.05 * S2
    D_perturbed = D + eps * k2 * np.diag(k_dir**2)
    evals_perturbed = np.linalg.eigvalsh(D_perturbed)
    ta_spread_perturbed = np.max(evals_perturbed[:23]) - np.min(evals_perturbed[:23])
    relative_spread = ta_spread_perturbed / abs(expected_ta)
    print(f"Perturbed Transverse Spread: {ta_spread_perturbed:.2e} (Relative: {relative_spread:.2e})")
    assert relative_spread > 1e-4, "Anisotropic perturbation test failed."
    print(">> Continuum elasticity and Cauchy-Lame reduction successfully verified.")

if __name__ == "__main__":
    verify_reduced_acoustic_tensor()

Conclusion, Limitations, and Outlook

Summary of Logical Architecture

This paper has established the collective dynamics and macroscopic elasticity of the infinite 24-dimensional Leech crystal $\Lambda_{24}$, subject to the stated convergence hypotheses on the pair potential.

1. Design-forced continuum isotropy. The spherical $11$-design property of every non-empty Leech shell implies that the second- and fourth-rank shell moments coincide with their spherical averages. Consequently, for every $f \in \mathcal{P}_{\mathrm{elast}}$, the quadratic acoustic tensor has the isotropic form $$\begin{equation} D_{ab}^{(2)}(\mathbf{k}) = (S_1+S_2)\|\mathbf{k}\|^2\delta_{ab} + 2S_2 k_a k_b. \end{equation}$$ Thus the leading long-wavelength elastic response contains no directional anisotropy.

2. Prestress and Cauchy reduction. The microscopic central-force structure gives the Cauchy symmetry of the Born elastic tensor. When the crystal is evaluated at mechanical zero-stress equilibrium, $P_0 = -S_1 = 0$, this symmetry reduces the two isotropic Lamé coefficients to $$\begin{equation} \lambda = \mu = S_2 = \frac{1}{312} \sum_{\mathbf{R} \in \Lambda_{24} \setminus \{\mathbf{0}\}} \|\mathbf{R}\|^4 f''(\|\mathbf{R}\|^2). \end{equation}$$ The zero-stress condition is therefore essential to the stated one-modulus reduction; the prestressed and zero-stress elastic descriptions should not be conflated.

3. Acoustic velocity ratio. Assuming positive shear modulus $\mu = S_2 > 0$ and unit mass density, the longitudinal and transverse acoustic velocities are $$\begin{equation} v_L = \sqrt{3S_2}, \qquad v_T = \sqrt{S_2}, \end{equation}$$ and hence $$\begin{equation} \boxed{\frac{v_L}{v_T} = \sqrt{3}.} \end{equation}$$ This conclusion follows algebraically from isotropy, central-force Cauchy symmetry, and zero prestress, rather than from a numerical fit.

4. Design-controlled long-wavelength isotropy. For potentials satisfying the stronger convergence assumptions in $\mathcal{P}^{(10)}$, the Taylor coefficients of the dynamical matrix through order $\|\mathbf{k}\|^8$ involve shell moments of degree at most $10$. These moments are controlled by the $11$-design property. Therefore the dynamical matrix has the structure $$\begin{equation} D_{ab}(\mathbf{k}) = \sum_{j=1}^{4} \left( A_j\|\mathbf{k}\|^{2j}\delta_{ab} + B_j\|\mathbf{k}\|^{2j-2}k_a k_b \right) + O(\|\mathbf{k}\|^{10}). \end{equation}$$ In particular, all 23 transverse polarizations remain degenerate through this order.

5. First unconstrained tensorial order. At order $\|\mathbf{k}\|^{10}$ in $D(\mathbf{k})$, the $f''$ contribution contains degree-$12$ shell moments. The $11$-design hypothesis alone does not determine these moments. Consequently, degree $12$ is the first moment degree at which additional lattice information is required to decide whether polarization splitting occurs.

6. Convergent versus regularized long-range interactions. The preceding dynamical statements are established for absolutely convergent potential classes. In particular, inverse-power interactions with insufficient decay cannot simply be inserted into the same lattice sums. Long-range models require a separately specified regularization, background prescription, or other renormalization scheme.

7. Formal and numerical verification. The Lean 4 artifact isolates the algebraic Cauchy–Lamé implication and verifies the resulting squared velocity ratio over $\mathbb{Q}$. The accompanying Python calculation independently evaluates the reduced acoustic tensor, verifies the $23$-fold transverse degeneracy, and demonstrates numerically that an explicitly introduced anisotropic perturbation destroys that degeneracy.

Scope and Limitations

Several qualifications are important for interpreting the results:

1. The $11$-design property is a statement about polynomial moments on individual metric shells. It does not imply that the entire finite-$\mathbf{k}$ dynamical matrix is exactly $\mathrm{O}(24)$-invariant. The Brillouin zone and the reciprocal lattice retain the discrete symmetry of $\Lambda_{24}$. The isotropy proved here is a controlled long-wavelength statement whose order is determined by the largest polynomial moment to which the design hypothesis applies.

2. The equality $\lambda = \mu$ is conditional on the central-force Cauchy relation together with the zero-stress reduction used in this paper. For a prestressed crystal, the distinction between the Born stiffness tensor, the incremental elastic tensor, and the acoustic tensor must be retained. Accordingly, the sound-velocity ratio $\sqrt{3}$ should not be interpreted as a universal finite-pressure identity.

3. The higher-order isotropy result does not assert that the coefficient of the first potentially anisotropic term is non-zero. At order $\|\mathbf{k}\|^{10}$, the $11$-design property ceases to be sufficient to exclude anisotropy, but the actual degree-$12$ shell moments may contain additional structure that could further suppress or constrain polarization splitting.

Outlook

The framework developed here suggests three natural directions for further study:

1. Determination of degree-12 shell moments: Computing the explicit degree-$12$ harmonic moment tensor of $\Lambda_{24}$ will decide whether the first order not controlled by Venkov’s theorem nevertheless exhibits additional isotropy forced by $\mathrm{Co}_0$.

2. Explicit evaluation of $\alpha_3, \alpha_5, \alpha_7$: Computing the radial lattice sums for specific rapidly decaying potentials (such as Gaussians or short-range Morse potentials) will provide explicit dispersion curves for the Leech crystal.

3. Non-central and many-body forces: Extending from central pair potentials to three-body or angular potentials would break the microscopic Cauchy relation, allowing an independent determination of the macroscopic Lamé parameters while preserving design-controlled isotropy.