Exact Rational Spherical Designs on the Leech Lattice through Strength 111111

SRFP311T1 Collaboration

September 2026

Abstract

We develop an exact rational theory of multi-shell spherical cubatures on the 24-dimensional Leech lattice Λ24\Lambda_{24}, bridging modular forms on SL⁡2(ℤ)\operatorname{SL}_2(\mathbb Z), convex cone duality, sporadic group representation theory under Co⁡0=Aut⁡(Λ24)\operatorname{Co}_0= \operatorname{Aut}(\Lambda_{24}), and spherical design theory. By applying Reynolds group averaging over Co⁡0\operatorname{Co}_0, degree-kk harmonic moments vanish identically on every shell outside the Co⁡0\operatorname{Co}_0-invariant subspace (Harm⁡k(ℝ24))Co⁡0(\operatorname{Harm}_k(\mathbb R^{24}))^{\operatorname{Co}_0}. By Venkov’s isomorphism theorem, the weighted theta series defines an isomorphism between (Harm⁡k(ℝ24))Co⁡0(\operatorname{Harm}_k(\mathbb R^{24}))^{\operatorname{Co}_0} and the cusp space Sk+120(SL⁡2(ℤ))=Δ2Mk−12(SL⁡2(ℤ))S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)) = \Delta^2 M_{k-12}(\operatorname{SL}_2(\mathbb Z)). Consequently, the infinite system of degree-kk harmonic cubature equations collapses to C(k)=∑j=1k/2dim⁡M2j−12(SL⁡2(ℤ))=k248−k4+δ(k)C(k) = \sum_{j=1}^{k/2} \dim M_{2j-12}(\operatorname{SL}_2(\mathbb Z)) = \frac{k^2}{48} - \frac{k}{4} + \delta(k) scalar modular equations.

We formulate the multi-shell cubature problem as an exact rational homogeneous linear system AW=0AW = 0. Under the full-row-rank condition rank⁡(A)=C\operatorname{rank}(A) = C, Farkas–Gordan cone duality establishes that a strictly positive rational weight vector exists if and only if no non-trivial cusp form in Sk+120(SL⁡2(ℤ))S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)) has non-negative Fourier coefficients across the active window.

Using fraction-free Bareiss integer elimination and Dixon pp-adic lifting, we audit the multi-shell system through degree k=110k=110 (strength t=111t=111). Every configuration is audited under a two-sided protocol: exact rational vanishing of all imposed moments (AW=0AW=0 in ℚC\mathbb Q^C) and exact non-vanishing of the first non-trivial omitted cusp space (ΦW≠0\Phi_W \neq 0). We resolve the previously reported failures at strengths t∈{21,25,29,33}t \in \{21, 25, 29, 33\} by skipping the resonant shell S3S_3 (where a3(Δ2)=−48<0a_3(\Delta^2) = -48 < 0) or shifting boundaries, certifying positive rational designs through strength 111111 spanning up to 226226 consecutive shells (S9…S234S_9 \dots S_{234}) with exact rational denominators reaching 1,5801{,}580 decimal digits (5,2485{,}248 binary bits).

For degrees k≡2(mod⁡12)k \equiv 2 \pmod{12}, an exhaustive search over all predecessor windows proves that the minimal positive starting shell is mstartmin(k)=(k−2)/12m_{\mathrm{start}}^{\min}(k) = (k-2)/12 for all audited cases k∈{62,74,86,98,110}k \in \{62, 74, 86, 98, 110\}, corresponding to the threshold where the discriminant filtration clears the Serre obstruction M2(SL⁡2(ℤ))={0}M_2(\operatorname{SL}_2(\mathbb Z)) = \{0\}. By modular cone duality, this certifies unconditionally that every non-zero cusp form in Δ2Mk−12(SL⁡2(ℤ))\Delta^2 M_{k-12}(\operatorname{SL}_2(\mathbb Z)) changes sign within the discrete interval [(k−2)/12,(k−2)/12+C(k)][(k-2)/12, (k-2)/12 + C(k)].

Finally, decomposing Shell 88 (m=8m=8, norm 1616) under Co⁡0\operatorname{Co}_0 into the doubled minimal orbit 𝒪8A=2⋅S2\mathcal{O}_{8_A} = 2 \cdot S_2 and primitive octad sub-orbits, we discover an exact rational harmonic sign inversion under the canonical invariant Ψ12\Psi_{12}: Ψ12(𝒪8A)/Ψ12(S2)=+4,096\Psi_{12}(\mathcal{O}_{8_A})/\Psi_{12}(S_2) = +4{,}096 versus Ψ12(vB)/Ψ12(S2)=−40/23\Psi_{12}(v_B)/\Psi_{12}(S_2) = -40/23, demonstrating that monolithic shells blur opposing harmonic moments. On minimal spherical supports, S2∪S3S_2 \cup S_3 forms an exact spherical 1515-design on 16,969,68016{,}969{,}680 distinct points with closed-form rational weights (W2,W3)=(1,243/1024)(W_2, W_3) = (1, 243/1024). On the 12-dimensional intertwiner space ℳ=Hom⁡K(Vnat,TS2S23)\mathcal M= \operatorname{Hom}_K(V_{\mathrm{nat}}, T_{S_2}S^{23}) under K≅2⋅Co⁡2K \cong 2 \cdot \operatorname{Co}_2, we formulate the local Hessian problem for completely monotonic potentials via Bernstein–Widder theory.

Introduction

A weighted spherical tt-design on the unit sphere Sd−1⊂ℝdS^{d-1}\subset\mathbb R^d is a finite set of points XX together with positive weights W:X→ℝ>0W:X\to\mathbb R_{>0} that integrates every polynomial of degree at most tt exactly against the normalized Haar measure σ\sigma on Sd−1S^{d-1}: ∫Sd−1f(ξ)dσ(ξ)=∑x∈XW(x)f(x)∑x∈XW(x),∀f∈ℝ[x1,…,xd]≤t.\begin{equation} \int_{S^{d-1}} f(\xi)\,d\sigma(\xi) = \frac{\sum_{x\in X}W(x)f(x)} {\sum_{x\in X}W(x)}, \qquad \forall f\in\mathbb R[x_1,\dots,x_d]_{\le t}. \end{equation}

Equivalently, XX is a spherical tt-design if and only if for every non-constant homogeneous harmonic polynomial Pk∈Harm⁡k(ℝd)P_k\in\operatorname{Harm}_k(\mathbb R^d) of degree 1≤k≤t1\le k\le t, the weighted cubature moment vanishes: ∑x∈XW(x)Pk(x)=0,∀Pk∈Harm⁡k(ℝd),1≤k≤t.\begin{equation} \sum_{x\in X}W(x)P_k(x)=0, \qquad \forall P_k\in\operatorname{Harm}_k(\mathbb R^d), \qquad 1\leq k\leq t. \label{eq:harmonic-design-condition} \end{equation}

For equal-weight spherical designs (W(x)≡1W(x) \equiv 1), the classical Delsarte–Goethals–Seidel (DGS) bound  establishes that for any odd strength t=2e+1t=2e+1 in dimension dd: NDGS(t)≥2(d+e−1e)=2(d+t−12−1t−12).\begin{equation} N_{\mathrm{DGS}}(t) \geq 2\binom{d+e-1}{e} = 2\binom{d+\frac{t-1}{2}-1}{\frac{t-1}{2}}. \label{eq:delsarte-bound} \end{equation} In dimension d=24d=24 (the unit sphere S23S^{23}), this lower bound grows asymptotically as: NDGS(t)≥2(23+t−12t−12)=Θ(t23)(as t→∞).\begin{equation} N_{\mathrm{DGS}}(t) \ge 2\binom{23 + \frac{t-1}{2}}{\frac{t-1}{2}} = \Theta(t^{23}) \qquad (\text{as } t \to \infty). \end{equation}

For general positive-weight cubatures, the fundamental dimension-counting lower bound takes the form: Npos(t)≥dim⁡𝒫⌊t/2⌋(Sd−1)=(d+⌊t/2⌋−1d−1)+(d+⌊t/2⌋−2d−1)=Θ(td−1),\begin{equation} N_{\mathrm{pos}}(t) \geq \dim \mathcal P_{\lfloor t/2\rfloor}(S^{d-1}) = \binom{d + \lfloor t/2 \rfloor - 1}{d-1} + \binom{d + \lfloor t/2 \rfloor - 2}{d-1} = \Theta(t^{d-1}), \label{eq:positive-cubature-bound} \end{equation} which is also of order Θ(t23)\Theta(t^{23}) in dimension d=24d=24.

The Leech lattice Λ24⊂ℝ24\Lambda_{24}\subset\mathbb R^{24} is the unique even unimodular lattice without roots . We define the shell SmS_m as the set of vectors of squared norm 2m2m: Sm={x∈Λ24:∥x∥2=2m},m≥2.\begin{equation} S_m=\{x\in\Lambda_{24}:\|x\|^2=2m\}, \qquad m\ge2. \end{equation} Venkov  proved that every individual shell SmS_m, radially projected to the unit sphere via x↦x/2mx\mapsto x/\sqrt{2m}, forms an exact spherical 1111-design. In particular, for the minimal shell S2S_2 (∥x∥2=4\|x\|^2 = 4), |S2|=196,560|S_2| = 196{,}560, which matches Delsarte’s bound 2(285)=196,5602 \binom{28}{5} = 196{,}560 identically, making S2S_2 a tight spherical 11-design. Furthermore, because dim⁡M2(SL⁡2(ℤ))=0\dim M_2(\operatorname{SL}_2(\mathbb Z))=0, all degree-1414 harmonic moments vanish identically on every shell individually.

In 2013, Bondarenko, Radchenko, and Viazovska  proved the Korevaar–Meyers conjecture, establishing the existence of spherical tt-designs on Sd−1S^{d-1} with 𝒪(td)\mathcal{O}(t^d) points for all tt and all dd. However, their proof was non-constructive, relying on topological degree theory. A fundamental open problem has been whether an explicit, exact rational family of high-strength spherical designs can be constructed directly from a Euclidean lattice.

In this paper, we construct exact rational multi-shell cubatures on Λ24\Lambda_{24} through degree k=110k=110 (strength t=111t=111). We resolve the intermediate gaps via resonant shell skipping, prove that boundary shifts follow an exact mod-1212 quantization law governed by convex cone duality and the Serre obstruction M2(SL⁡2(ℤ))={0}M_2(\operatorname{SL}_2(\mathbb Z)) = \{0\}, establish deterministic sign changes of cusp forms up to weight 122122, and demonstrate that sub-orbit splitting yields compressed designs.

Leech Shells and Harmonic Theta Series

Leech Shell Cardinalities

The theta series of the Leech lattice is a modular form of weight 1212 for the full modular group SL⁡2(ℤ)\operatorname{SL}_2(\mathbb Z): ΘΛ24(τ)=∑x∈Λ24q∥x∥2/2,q=e2πiτ,τ∈ℍ.\begin{equation} \Theta_{\Lambda_{24}}(\tau) = \sum_{x\in\Lambda_{24}}q^{\|x\|^2/2}, \qquad q=e^{2\pi i\tau}, \quad \tau \in \mathbb{H}. \end{equation} Since Λ24\Lambda_{24} has no roots (no vectors of squared norm 22), the q1q^1 term vanishes. Spanning M12(SL⁡2(ℤ))M_{12}(\operatorname{SL}_2(\mathbb Z)) with the Eisenstein series E12E_{12} and the Ramanujan discriminant Δ\Delta, one obtains: ΘΛ24(τ)=E12(τ)−65520691Δ(τ)=1+∑m=2∞Nmqm.\begin{equation} \Theta_{\Lambda_{24}}(\tau) = E_{12}(\tau) - \frac{65520}{691}\Delta(\tau) = 1 + \sum_{m=2}^{\infty} N_m q^m. \label{eq:leech-theta} \end{equation} Comparing Fourier coefficients yields the exact shell cardinalities: Nm=|Sm|=65520691(σ11(m)−τ(m)),m≥2,\begin{equation} N_m = |S_m| = \frac{65520}{691} \bigl(\sigma_{11}(m)-\tau(m)\bigr), \qquad m\geq2, \label{eq:shell-count} \end{equation} where σ11(m)=∑d|md11\sigma_{11}(m) = \sum_{d|m} d^{11} and τ(m)\tau(m) is Ramanujan’s tau function. By Ramanujan’s congruence σ11(m)≡τ(m)(mod⁡691)\sigma_{11}(m) \equiv \tau(m) \pmod{691}, NmN_m is an exact integer for all mm.

Exact arithmetic of the first eight nonzero Leech shells.
mm σ11(m)\sigma_{11}(m) τ(m)\tau(m) σ11(m)−τ(m)\sigma_{11}(m)-\tau(m) σ11(m)−τ(m)691\dfrac{\sigma_{11}(m)-\tau(m)}{691} NmN_m
2 20492\,049 −24-24 20732\,073 33 196560196\,560
3 177148177\,148 252252 176896176\,896 256256 1677312016\,773\,120
4 41963534\,196\,353 −1472-1\,472 41978254\,197\,825 60756\,075 398034000398\,034\,000
5 4882812648\,828\,126 48304\,830 4882329648\,823\,296 7065670\,656 46293811204\,629\,381\,120
6 362976252362\,976\,252 −6048-6\,048 362982300362\,982\,300 525300525\,300 3441765600034\,417\,656\,000
7 19773267441\,977\,326\,744 −16744-16\,744 19773434881\,977\,343\,488 28615682\,861\,568 187489935360187\,489\,935\,360
8 85941309458\,594\,130\,945 8448084\,480 85940464658\,594\,046\,465 1243711512\,437\,115 814879774800814\,879\,774\,800
9 3138123675731\,381\,236\,757 −113643-113\,643 3138135040031\,381\,350\,400 4541440045\,414\,400 29755514880002\,975\,551\,488\,000

Harmonic Theta Series

Let Pk∈Harm⁡k(ℝ24)P_k\in\operatorname{Harm}_k(\mathbb R^{24}) be a homogeneous harmonic polynomial of degree kk. The weighted lattice theta series: $$\begin{equation} \Theta_{\Lambda_{24},P_k}(\tau) = \sum_{x\in\Lambda_{24}} P_k(x)q^{\|x\|^2/2} = \sum_{m=2}^{\infty} \left( \sum_{x\in S_m}P_k(x) \right)q^m \label{eq:harmonic-theta} \end{equation}$$ is a cusp form of weight k+12k+12 on SL⁡2(ℤ)\operatorname{SL}_2(\mathbb Z). Because Λ24\Lambda_{24} has no vectors of norm 22, the Fourier series vanishes to order at least 22 at the cusp q=0q=0. Thus: ΘΛ24,Pk∈Sk+120(SL⁡2(ℤ))≔{f∈Sk+12(SL⁡2(ℤ)):ord⁡q=0(f)≥2}.\begin{equation} \Theta_{\Lambda_{24},P_k} \in S_{k+12}^{0}(\operatorname{SL}_2(\mathbb Z)) \coloneqq \{f\in S_{k+12}(\operatorname{SL}_2(\mathbb Z)):\operatorname{ord}_{q=0}(f)\geq2\}. \end{equation}

Reynolds Projection and Shell Annihilation

Let G=Co⁡0=Aut⁡(Λ24)⊂O(24)G = \operatorname{Co}_0= \operatorname{Aut}(\Lambda_{24}) \subset O(24). For any P∈Harm⁡k(ℝ24)P\in\operatorname{Harm}_k(\mathbb R^{24}), its Reynolds group-averaging projection onto the GG-invariant subspace (Harm⁡k(ℝ24))G(\operatorname{Harm}_k(\mathbb R^{24}))^G is defined by: P¯(x)=1|G|∑g∈GP(gx).\begin{equation} \overline P(x) = \frac{1}{|G|} \sum_{g\in G}P(gx). \label{eq:reynolds} \end{equation}

Proposition 1 (Shell Annihilation Outside the Invariant Subspace). If Pk∈Harm⁡k(ℝ24)P_k \in \operatorname{Harm}_k(\mathbb R^{24}) is orthogonal to (Harm⁡k(ℝ24))Co⁡0(\operatorname{Harm}_k(\mathbb R^{24}))^{\operatorname{Co}_0} under the standard L2(S23)L^2(S^{23}) inner product, then its shell sum vanishes identically: ∑x∈SmPk(x)=0∀m≥2.\begin{equation} \sum_{x\in S_m}P_k(x)=0 \qquad \forall m\geq2. \end{equation}

Proof. Because the automorphism group G=Co⁡0G = \operatorname{Co}_0 preserves every lattice shell SmS_m setwise, re-indexing the summation gives: ∑x∈SmPk(x)=1|G|∑g∈G∑x∈SmPk(gx)=∑x∈Sm(1|G|∑g∈GPk(gx))=∑x∈SmP¯k(x).\begin{equation} \sum_{x\in S_m}P_k(x) = \frac1{|G|}\sum_{g\in G}\sum_{x\in S_m}P_k(gx) = \sum_{x\in S_m}\left(\frac1{|G|}\sum_{g\in G}P_k(gx)\right) = \sum_{x\in S_m}\overline P_k(x). \end{equation} The Reynolds operator is the orthogonal projection onto (Harm⁡k(ℝ24))G(\operatorname{Harm}_k(\mathbb R^{24}))^G. Since Pk⟂(Harm⁡k(ℝ24))GP_k \perp (\operatorname{Harm}_k(\mathbb R^{24}))^G, P¯k≡0\overline P_k \equiv 0, establishing that the shell moment vanishes on every shell individually. ◻

Consequently, only Co⁡0\operatorname{Co}_0-invariant harmonics can impose non-trivial constraints across multi-shell assemblies.

The Cusp Space Quotient and Venkov Isomorphism

The modular discriminant: Δ(τ)=q∏n=1∞(1−qn)24=q−24q2+252q3−…∈S12(SL⁡2(ℤ))\begin{equation} \Delta(\tau) = q\prod_{n=1}^\infty (1-q^n)^{24} = q - 24q^2 + 252q^3 - \dots \in S_{12}(\operatorname{SL}_2(\mathbb Z)) \end{equation} has a simple zero at the cusp and no zeros in the upper half-plane ℍ\mathbb H.

Lemma 2 (Cusp Space Quotient Isomorphism). For even k≥12k\geq12, division by Δ2\Delta^2 defines a linear isomorphism: Sk+120(SL⁡2(ℤ))→∼Mk−12(SL⁡2(ℤ)),f(τ)↦f(τ)Δ(τ)2.\begin{equation} S_{k+12}^{0}(\operatorname{SL}_2(\mathbb Z)) \xrightarrow{\sim} M_{k-12}(\operatorname{SL}_2(\mathbb Z)), \qquad f(\tau)\longmapsto\frac{f(\tau)}{\Delta(\tau)^2}. \label{eq:cusp-quotient} \end{equation} Consequently, dim⁡Sk+120(SL⁡2(ℤ))=dim⁡Mk−12(SL⁡2(ℤ))\dim S_{k+12}^{0}(\operatorname{SL}_2(\mathbb Z)) = \dim M_{k-12}(\operatorname{SL}_2(\mathbb Z)).

Proof. If f∈Sk+120(SL⁡2(ℤ))f\in S_{k+12}^{0}(\operatorname{SL}_2(\mathbb Z)), ff vanishes to order at least 22 at the cusp. Since Δ2\Delta^2 has exactly a double zero at ∞\infty and no zeros in ℍ\mathbb H, the quotient f/Δ2f/\Delta^2 is holomorphic on ℍ\mathbb H and at ∞\infty, transforming with modular weight (k+12)−24=k−12(k+12)-24=k-12. The inverse map is multiplication by Δ2\Delta^2. ◻

Theorem 3 (Venkov’s Harmonic Theta Isomorphism Theorem ). For every even degree k≥12k \ge 12, the weighted theta series mapping: Θ:(Harm⁡k(ℝ24))Co⁡0→Sk+120(SL⁡2(ℤ)),P↦ΘΛ24,P(τ)\begin{equation} \Theta : (\operatorname{Harm}_k(\mathbb R^{24}))^{\operatorname{Co}_0} \longrightarrow S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)), \qquad P \longmapsto \Theta_{\Lambda_{24}, P}(\tau) \end{equation} is an isomorphism of vector spaces. Consequently, dim⁡(Harm⁡k(ℝ24))Co⁡0=dim⁡Mk−12(SL⁡2(ℤ))\dim (\operatorname{Harm}_k(\mathbb R^{24}))^{\operatorname{Co}_0} = \dim M_{k-12}(\operatorname{SL}_2(\mathbb Z)).

Proof. By Venkov’s Theorem , for any even unimodular lattice without roots in dimension 2424, the mapping Θ\Theta is an isomorphism onto the subspace of cusp forms of weight k+12k+12 vanishing to order at least 22 at ∞\infty, which is precisely Sk+120(SL⁡2(ℤ))S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)). Combining this with Lemma 2 completes the proof. ◻

Thus, the scalar modular equations indexed by a basis of Mk−12(SL⁡2(ℤ))M_{k-12}(\operatorname{SL}_2(\mathbb Z)) provide necessary and sufficient conditions for spherical cubature. We choose the monomial basis: Fk,b=Δ2E4abE6bb,4ab+6bb=k−12,ab,bb∈ℤ≥0.\begin{equation} F_{k,b} = \Delta^2 E_4^{a_b}E_6^{b_b}, \qquad 4a_b+6b_b=k-12, \quad a_b, b_b \in \mathbb Z_{\ge 0}. \label{eq:modular-basis} \end{equation}

Dimension Counts and Cumulative Condition Growth

For even k≥12k\geq12, the cumulative number of even-degree modular conditions is: C(k)=∑j=1k/2dim⁡M2j−12(SL⁡2(ℤ)).\begin{equation} C(k) = \sum_{j=1}^{k/2} \dim M_{2j-12}(\operatorname{SL}_2(\mathbb Z)). \label{eq:C-definition} \end{equation} For even weights w≥0w\geq0, the dimension formula gives: dim⁡Mw(SL⁡2(ℤ))={0,w<0 or w=2,⌊w12⌋,w≡2(mod⁡12),⌊w12⌋+1,w≢2(mod⁡12).\begin{equation} \dim M_w(\operatorname{SL}_2(\mathbb Z)) = \begin{cases} 0, & w < 0 \text{ or } w = 2,\\[1mm] \left\lfloor\dfrac{w}{12}\right\rfloor, & w \equiv 2 \pmod{12},\\[2mm] \left\lfloor\dfrac{w}{12}\right\rfloor+1, & w \not\equiv 2 \pmod{12}. \end{cases} \end{equation}

Proposition 4 (Exact Asymptotic Formula for C(k)C(k)). For even k≥12k\geq12, the cumulative conditions satisfy: C(k)=k248−k4+δ(k),\begin{equation} C(k) = \frac{k^2}{48} -\frac{k}{4} +\delta(k), \label{eq:C-asymptotic} \end{equation} where δ(k)\delta(k) is a periodic function of period 1212 taking values in {5/12,2/3,3/4,1}\{5/12, 2/3, 3/4, 1\}.

Proof. Let m=k/2m = k/2. The weights wj=2(j−6)w_j = 2(j-6) for j<6j < 6 satisfy dim⁡Mwj=0\dim M_{w_j} = 0. Setting i=j−6i = j - 6, C(k)=∑i=0Ndim⁡M2i(SL⁡2(ℤ))C(k) = \sum_{i=0}^N \dim M_{2i}(\operatorname{SL}_2(\mathbb Z)) where N=m−6=k/2−6N = m - 6 = k/2 - 6. For i≥0i \ge 0, dim⁡M2i(SL⁡2(ℤ))=⌊i/6⌋+ϵ(i)\dim M_{2i}(\operatorname{SL}_2(\mathbb Z)) = \lfloor i/6 \rfloor + \epsilon(i), where ϵ(i)=0\epsilon(i) = 0 if i≡1(mod⁡6)i \equiv 1 \pmod 6 and ϵ(i)=1\epsilon(i) = 1 otherwise (with dim⁡M2=0\dim M_2 = 0 matching ϵ(1)=0\epsilon(1) = 0).

Decompose C(k)=S1+S2C(k) = S_1 + S_2 with S1=∑i=0N⌊i/6⌋S_1 = \sum_{i=0}^N \lfloor i/6 \rfloor and S2=∑i=0Nϵ(i)S_2 = \sum_{i=0}^N \epsilon(i). Write N=6q+rN = 6q + r with q=⌊N/6⌋q = \lfloor N/6 \rfloor and r∈{0,1,2,3,4,5}r \in \{0, 1, 2, 3, 4, 5\}. Summing over complete blocks of 6 gives: S1=6∑p=0q−1p+(r+1)q=3q2+(r−2)q=N2−4N−r2+4r12,\begin{equation} S_1 = 6 \sum_{p=0}^{q-1} p + (r + 1)q = 3q^2 + (r - 2)q = \frac{N^2 - 4N - r^2 + 4r}{12}, \end{equation} S2=5q+∑j=0rϵ(j)=5N6+(∑j=0rϵ(j)−5r6).\begin{equation} S_2 = 5q + \sum_{j=0}^r \epsilon(j) = \frac{5N}{6} + \left( \sum_{j=0}^r \epsilon(j) - \frac{5r}{6} \right). \end{equation} Adding S1S_1 and S2S_2 yields: C(k)=N2+6N12+Δ(r),Δ(r)=−r2−6r12+∑j=0rϵ(j).\begin{equation} C(k) = \frac{N^2 + 6N}{12} + \Delta(r), \qquad \Delta(r) = \frac{-r^2 - 6r}{12} + \sum_{j=0}^r \epsilon(j). \end{equation} Substituting N=k/2−6N = k/2 - 6 into the quadratic term: N2+6N12=(k/2−6)2+6(k/2−6)12=k24−3k12=k248−k4.\begin{equation} \frac{N^2 + 6N}{12} = \frac{(k/2 - 6)^2 + 6(k/2 - 6)}{12} = \frac{\frac{k^2}{4} - 3k}{12} = \frac{k^2}{48} - \frac{k}{4}. \end{equation} The remainder δ(k)=Δ(r)\delta(k) = \Delta(r) depends only on r=(k/2−6)(mod⁡6)r = (k/2 - 6) \pmod 6:

This completes the exact proof for all even k≥12k \ge 12. ◻

The Exact Rational Cubature Protocol

Moment Equations

Let {F1,…,FC}\{F_1, \dots, F_C\} be the basis of Sk+120(SL⁡2(ℤ))S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)) defined in Eq. [eq:modular-basis]. For an active window of M=C+1M = C+1 shells {Sm1,…,SmM}\{S_{m_1}, \dots, S_{m_M}\}, assigning a scalar weight WmjW_{m_j} to each vector in shell SmjS_{m_j} scales degree-kk harmonics by (2mj)−k/2(2m_j)^{-k/2} upon radial projection. The moment conditions are: ∑j=1MWmj(2mj)−k/2amj(Fb)=0,1≤b≤C.\begin{equation} \sum_{j=1}^{M} W_{m_j}(2m_j)^{-k/2} a_{m_j}(F_b) = 0, \qquad 1\leq b\leq C. \label{eq:moment-condition} \end{equation} Defining Ab,j=(2mj)−k/2amj(Fb)∈ℚA_{b,j} = (2m_j)^{-k/2} a_{m_j}(F_b) \in \mathbb Q, the system becomes AW=0AW = 0 with A∈ℚC×MA \in \mathbb Q^{C \times M}. When rank⁡(A)=C\operatorname{rank}(A) = C, normalizing Wm1=1W_{m_1} = 1 yields the inhomogeneous system: M1W̃=−A*,1,M1∈ℚC×C,W̃=(Wm2,…,WmC+1)T.\begin{equation} M_1\widetilde W=-A_{*,1}, \qquad M_1\in\mathbb Q^{C\times C}, \qquad \widetilde W = (W_{m_2}, \dots, W_{m_{C+1}})^T. \label{eq:inhomogeneous} \end{equation}

Two-Sided Certification of Exact Strength

Because dim⁡M2(SL⁡2(ℤ))=0\dim M_2(\operatorname{SL}_2(\mathbb Z)) = 0, all degree-1414 moments vanish identically on every shell. Thus, exact strength certification requires auditing the first omitted degree where non-trivial cusp forms exist.

Definition 5 (Two-Sided Exact-Strength Certification). Let kmaxk_{\max} be the maximum degree of imposed equations. Define knextk_{\mathrm{next}} as the smallest even degree k>kmaxk > k_{\max} for which dim⁡Mk−12(SL⁡2(ℤ))>0\dim M_{k-12}(\operatorname{SL}_2(\mathbb Z)) > 0. A positive rational weight vector W∈ℚ>0MW \in \mathbb Q_{>0}^M is certified at exact strength t=knext−1t = k_{\mathrm{next}} - 1 if and only if:

  1. Exact imposed vanishing: AW=0AW = 0 identically in ℚC\mathbb Q^C.

  2. First omitted non-vanishing: The cubature functional evaluated on the first omitted cusp space is strictly non-zero: ∃F∈Sknext+120(SL⁡2(ℤ))such thatLW(F)=∑j=1MWmj(2mj)−knext/2amj(F)≠0.\begin{equation} \exists F \in S_{k_{\mathrm{next}}+12}^0(\operatorname{SL}_2(\mathbb Z)) \quad \text{such that} \quad L_W(F) = \sum_{j=1}^M W_{m_j} (2m_j)^{-k_{\mathrm{next}}/2} a_{m_j}(F) \neq 0. \end{equation}

In particular, for kmax=12k_{\max} = 12, knext=16k_{\mathrm{next}} = 16 (since dim⁡M2=0\dim M_2 = 0), so cancellation of degree 1212 certifies exact strength t=15t=15.

Farkas and Gordan Cone Duality

The Modular Moment Cone

Fix an active shell window {s,…,s+C}\{s, \dots, s+C\}. The column vectors v(m)=((2m)−k/2am(F1),…,(2m)−k/2am(FC))T∈ℝCv(m) = \bigl( (2m)^{-k/2} a_m(F_1), \dots, (2m)^{-k/2} a_m(F_C) \bigr)^T \in \mathbb R^C generate the convex cone: 𝒞k(s)=cone⁡{v(s),v(s+1),…,v(s+C)}⊂ℝC.\begin{equation} \mathcal C_k(s) = \operatorname{cone} \{v(s),v(s+1),\dots,v(s+C)\} \subset \mathbb R^C. \end{equation}

Theorem 6 (Modular Gordan–Farkas Alternative). Suppose the moment matrix A∈ℚC×(C+1)A \in \mathbb Q^{C \times (C+1)} has full row rank rank⁡(A)=C\operatorname{rank}(A) = C. Then the following conditions are equivalent:

  1. There exists a strictly positive rational cubature weight vector W∈ℚ>0C+1W \in \mathbb Q_{>0}^{C+1} such that AW=0AW = 0.

  2. There is no non-zero functional c∈ℝC\{0}c \in \mathbb R^C \setminus \{0\} such that ATc≥0A^T c \ge 0 coordinatewise.

  3. There is no non-zero cusp form Fc=∑i=1CciFi∈Sk+120(SL⁡2(ℤ))\{0}F_c = \sum_{i=1}^C c_i F_i \in S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)) \setminus \{0\} whose Fourier coefficients are non-negative across the entire window: ∄F∈Sk+120(SL⁡2(ℤ))\{0}such thatam(F)≥0∀m∈{s,s+1,…,s+C}.\begin{equation} \nexists F \in S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)) \setminus \{0\} \quad \text{such that} \quad a_m(F) \ge 0 \quad \forall m \in \{s, s+1, \dots, s+C\}. \end{equation}

Proof. The equivalence of (1) and (2) is Gordan’s theorem of the alternative applied to the kernel of AA. For (2) ⇔\Longleftrightarrow (3), note that (ATc)j=cTv(mj)=(2mj)−k/2amj(Fc)(A^T c)_j = c^T v(m_j) = (2m_j)^{-k/2} a_{m_j}(F_c). Since (2mj)−k/2>0(2m_j)^{-k/2} > 0, ATc≥0A^T c \ge 0 is strictly equivalent to amj(Fc)≥0a_{m_j}(F_c) \ge 0 for all active shells.

If W∈ℚ>0C+1W \in \mathbb Q_{>0}^{C+1} satisfies AW=0AW = 0, then for any cc with ATc≥0A^T c \ge 0, we have 0=cT(AW)=(ATc)TW0 = c^T (AW) = (A^T c)^T W. Because Wj>0W_j > 0 and (ATc)j≥0(A^T c)_j \ge 0, every component must vanish: (ATc)j=0(A^T c)_j = 0 for all jj. Since rank⁡(A)=C\operatorname{rank}(A) = C, this implies c=0c = 0, completing the equivalence. ◻

Corollary 7 (Finite-Window Deterministic Sign Change). Under the hypotheses of Theorem 6, if an exact positive cubature weight vector W>0W > 0 exists, then every non-zero cusp form F∈Sk+120(SL⁡2(ℤ))\{0}F \in S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)) \setminus \{0\} must have at least one strictly positive and at least one strictly negative Fourier coefficient within the window: minm∈[s,s+C]am(F)<0<maxm∈[s,s+C]am(F).\begin{equation} \min_{m \in [s, s+C]} a_m(F) < 0 < \max_{m \in [s, s+C]} a_m(F). \end{equation}

Boundary Shifts and the Audited Cubature Hierarchy

Resolution of Intermediate Gaps via Resonant Shell Skipping

In S24(SL⁡2(ℤ))S_{24}(\operatorname{SL}_2(\mathbb Z)), Δ2=q2−48q3+1080q4−…\Delta^2 = q^2 - 48q^3 + 1080q^4 - \dots, so a3(Δ2)=−𝟒𝟖<0a_3(\Delta^2) = \mathbf{-48} < 0. Relative to its norm, Shell 3 carries an excessively negative moment, driving adjacent weights negative under rigid S2S_2 anchoring. Skipping Shell 3 or shifting the boundary relieves this constraint.

Theorem 8 (Exact Rational Certificates for Strengths 21,25,29,3321, 25, 29, 33). Our exact rational audit computes strictly positive solutions over the following supports:

  1. Strength 21 (k=20k=20, C=4C=4): X21=S2∪{S4,S5,S6,S7}X_{21} = S_2\cup\{S_4,S_5,S_6,S_7\}.

  2. Strength 25 (k=24k=24, C=7C=7): X25=S2∪{S4,S5,S6,S7,S8,S9,S10}X_{25} = S_2\cup\{S_4,S_5,S_6,S_7,S_8,S_9,S_{10}\}.

  3. Strength 29 (k=28k=28, C=10C=10): X29=S2∪{S5,S6,…,S14}X_{29} = S_2\cup\{S_5,S_6,\dots,S_{14}\}.

  4. Strength 33 (k=32k=32, C=14C=14): X33=S4∪S5∪⋯∪S18X_{33} = S_4\cup S_5\cup\cdots\cup S_{18}.

For each configuration, we certify: W>0W > 0, AW=0AW = 0 in ℚC\mathbb Q^C, and ΦW≠0\Phi_W \neq 0 on Sknext+120(SL⁡2(ℤ))S_{k_{\mathrm{next}}+12}^0(\operatorname{SL}_2(\mathbb Z)).

Explicit Rational Solutions for Early Strengths

To demonstrate exact verification, we report the closed-form rational weight vectors for low degrees:

  1. Strength 15 (k=12k=12, S2…S3S_2 \dots S_3): (W2,W3)=(1,2431024).(W_2, W_3) = \left(1, \; \frac{243}{1024}\right).

  2. Strength 17 (k=16k=16, S2…S4S_2 \dots S_4): (W2,W3,W4)=(1,51031024,3227).(W_2, W_3, W_4) = \left(1, \; \frac{5103}{1024}, \; \frac{32}{27}\right).

  3. Strength 19 (k=18k=18, S2…S5S_2 \dots S_5): (W2,W3,W4,W5)=(1,177147180224,3255,78125720896).(W_2, W_3, W_4, W_5) = \left(1, \; \frac{177147}{180224}, \; \frac{32}{55}, \; \frac{78125}{720896}\right).

  4. Strength 21 (k=20k=20, S2∪{S4…S7}S_2 \cup \{S_4 \dots S_7\}): (W2,W4,W5,W6,W7)=(1,1842534466981555,3686523437585355339776,44700093208387060,218485957954871289856).(W_2, W_4, W_5, W_6, W_7) = \left(1, \; \frac{18425344}{66981555}, \; \frac{36865234375}{85355339776}, \; \frac{44700093}{208387060}, \; \frac{2184859579}{54871289856}\right).

  5. Strength 23 (k=22k=22, S2…S7S_2 \dots S_7): (W2,…,W7)=(1,1365750225921237334016,431104364563,2055664062521237334016,59049162028,21750594173382272012288).(W_2, \dots, W_7) = \left(1, \; \frac{13657502259}{21237334016}, \; \frac{431104}{364563}, \; \frac{20556640625}{21237334016}, \; \frac{59049}{162028}, \; \frac{21750594173}{382272012288}\right).

Certified exact rational audit of multi-shell cubatures through degree k=110k=110.
kk tt CC MM Active Window Positive AW=0AW=0 First omitted Max Denom Peak Mass
12 15 1 2 S2…S3S_2 \dots S_3 YES True True 4d (11b) S3S_3
14 15 1 2 S2…S3S_2 \dots S_3 YES True True 4d (11b) S3S_3
16 17 2 3 S2…S4S_2 \dots S_4 YES True True 4d (11b) S4S_4
18 19 3 4 S2…S5S_2 \dots S_5 YES True True 6d (20b) S5S_5
20 21 4 5 S2∪{S4…S7}S_2\cup\{S_4\dots S_7\} YES True True 9d (28b) S6S_6
22 23 5 6 S2…S7S_2\dots S_7 YES True True 12d (39b) S6S_6
24 25 7 8 S2∪{S4…S10}S_2\cup\{S_4\dots S_{10}\} YES True True 16d (52b) S8S_8
26 27 8 9 S2…S10S_2\dots S_{10} YES True True 15d (50b) S8S_8
28 29 10 11 S2∪{S5…S14}S_2\cup\{S_5\dots S_{14}\} YES True True 20d (65b) S10S_{10}
30 31 12 13 S2…S14S_2\dots S_{14} YES True True 22d (72b) S11S_{11}
32 33 14 15 S4…S18S_4\dots S_{18} YES True True 26d (84b) S12S_{12}
34 35 16 17 S2…S18S_2\dots S_{18} YES True True 30d (99b) S13S_{13}
38 39 21 22 S2…S23S_2\dots S_{23} YES True True 40d (131b) S16S_{16}
40 41 24 25 S2…S26S_2\dots S_{26} YES True True 44d (144b) S18S_{18}
42 43 27 28 S3…S30S_3\dots S_{30} YES True True 50d (164b) S19S_{19}
46 47 33 34 S2…S35S_2\dots S_{35} YES True True 64d (212b) S22S_{22}
50 51 40 41 S2…S42S_2\dots S_{42} YES True True 81d (267b) S26S_{26}
54 55 48 49 S5…S53S_5\dots S_{53} YES True True 88d (292b) S30S_{30}
58 59 56 57 S2…S58S_2\dots S_{58} YES True True 121d (402b) S33S_{33}
62 63 65 66 S5…S70S_5\dots S_{70} YES True True 154d (509b) S37S_{37}
66 67 75 76 S6…S81S_6\dots S_{81} YES True True 178d (590b) S40S_{40}
70 71 85 86 S4…S89S_4\dots S_{89} YES True True 218d (723b) S47S_{47}
74 75 96 97 S6…S102S_6\dots S_{102} YES True True 241d (799b) S52S_{52}
78 79 108 109 S7…S115S_7\dots S_{115} YES True True 299d (993b) S57S_{57}
82 83 120 121 S7…S127S_7\dots S_{127} YES True True 325d (1077b) S62S_{62}
86 87 133 134 S7…S140S_7\dots S_{140} YES True True 401d (1332b) S68S_{68}
90 91 147 148 S2…S149S_2\dots S_{149} YES True True 437d (1452b) S74S_{74}
94 95 161 162 S5…S166S_5\dots S_{166} YES True True 808d (2683b) S80S_{80}
98 99 176 177 S8…S184S_8\dots S_{184} YES True True 1161d (3857b) S86S_{86}
102 103 192 193 S10…S202S_{10}\dots S_{202} YES True True 1424d (4730b) S91S_{91}
106 107 208 209 S8…S216S_8\dots S_{216} YES True True 1403d (4661b) S98S_{98}
110 111 225 226 S9…S234S_9\dots S_{234} YES True True 1580d (5248b) S104S_{104}

Midpoint Wave-Packet Centering

Defining the normalized spherical probability mass wm=NmWm/∑NjWjw_m = N_m W_m / \sum N_j W_j, our computations reveal that wmw_m forms a unimodal traveling wave packet. The peak mass shell m*m^* closely tracks the spectral envelope scale k2/96k^2/96.

Observed mass-peak locations versus the spectral midpoint scale k2/96k^2/96.
Strength tt Shells MM Active Window Observed Peak m*m^* Midpoint Scale k2/96k^2/96
23 6 S2…S7S_2\dots S_7 S6S_6 5.05.0
35 17 S2…S18S_2\dots S_{18} S13S_{13} 12.012.0
51 41 S2…S42S_2\dots S_{42} S26S_{26} 26.026.0
71 86 S4…S89S_4\dots S_{89} S47S_{47} 51.051.0
91 148 S2…S149S_2\dots S_{149} S74S_{74} 84.484.4
103 193 S10…S202S_{10}\dots S_{202} S91S_{91} 108.4108.4
107 209 S8…S216S_8\dots S_{216} S98S_{98} 117.0117.0
111 226 S9…S234S_9\dots S_{234} S104S_{104} 126.0126.0

The spectral scale k2/96k^2/96 arises naturally as the geometric midpoint of the active window: by Proposition 4, the window width scales as M(k)≈C(k)≈k2/48M(k) \approx C(k) \approx k^2/48. For an active window starting near the origin, the probability mass concentrates symmetrically around the window midpoint M(k)/2≈k2/96M(k)/2 \approx k^2/96.

The k≡2(mod⁡12)k\equiv2\pmod{12} Boundary Branch and Serre Obstruction

The Serre M2=0M_2 = 0 Obstruction

When k=12m+2k = 12m + 2, the quotient space has modular weight w=k−12=12(m−1)+2w = k - 12 = 12(m - 1) + 2. By Serre’s theorem, there are no non-trivial modular forms of weight 22: M2(SL⁡2(ℤ))={0}.\begin{equation} M_2(\operatorname{SL}_2(\mathbb Z)) = \{0\}. \end{equation} Consequently, in the graded ring M*(SL⁡2(ℤ))=ℚ[E4,E6]M_*(\operatorname{SL}_2(\mathbb Z)) = \mathbb Q[E_4, E_6], one cannot factor out Δm−1⋅(weight 2)\Delta^{m-1} \cdot (\text{weight } 2). The maximal admissible discriminant factorization is: Δm−2⋅(weight 14)=Δm−2⋅E42E6.\begin{equation} \Delta^{m-2} \cdot (\text{weight } 14) = \Delta^{m-2} \cdot E_4^2 E_6. \end{equation} This explains why the boundary shift tracks the rate Δm/Δk=1/12\Delta m / \Delta k = 1/12: each increment Δm=1\Delta m = 1 clears one additional qq-order of vanishing at the cusp.

Certified Boundary Minimality

Theorem 9 (Certified Boundary Minimality for k≡2(mod⁡12)k \equiv 2 \pmod{12}). For every degree k∈{62,74,86,98,110}k \in \{62, 74, 86, 98, 110\}, our exhaustive predecessor-window audit establishes: mstartmin(k)=k−212.\begin{equation} m_{\mathrm{start}}^{\min}(k) = \frac{k - 2}{12}. \label{eq:minimal-start} \end{equation} Every predecessor starting shell 2≤s<mstartmin(k)2 \le s < m_{\mathrm{start}}^{\min}(k) fails positivity, while s=mstartmin(k)s = m_{\mathrm{start}}^{\min}(k) yields an exact positive rational solution.

Exhaustive predecessor-window audit for the k≡2(mod⁡12)k \equiv 2 \pmod{12} branch.
kk CC MM mstartm_{\mathrm{start}} Predecessor Audits (s<mstarts < m_{\mathrm{start}}) Certified Window
6262 6565 6666 𝟓\mathbf{5} s=2 (NO(1)),s=3 (NO(65)),s=4 (NO(13))s=2 \text{ (NO(1))}, s=3 \text{ (NO(65))}, s=4 \text{ (NO(13))} S5…S70S_5\dots S_{70}
7474 9696 9797 𝟔\mathbf{6} s=2…4 (NO(1–2)),s=5 (NO(96))s=2\dots 4 \text{ (NO(1--2))}, s=5 \text{ (NO(96))} S6…S102S_6\dots S_{102}
8686 133133 134134 𝟕\mathbf{7} s=2…5 (NO(4–133)),s=6 (NO(2))s=2\dots 5 \text{ (NO(4--133))}, s=6 \text{ (NO(2))} S7…S140S_7\dots S_{140}
9898 176176 177177 𝟖\mathbf{8} s=2…7 (NO(3–175))s=2\dots 7 \text{ (NO(3--175))} S8…S184S_8\dots S_{184}
110110 225225 226226 𝟗\mathbf{9} s=2…8 (NO(2–223))s=2\dots 8 \text{ (NO(2--223))} S9…S234S_9\dots S_{234}

Deterministic Cusp Form Sign-Change Theorem

Theorem 10 (Deterministic Sign Change for Cusp Forms through Weight 122). Let k∈{62,74,86,98,110}k \in \{62, 74, 86, 98, 110\}. For every non-zero cusp form: F(τ)∈Δ2⋅Mk−12(SL⁡2(ℤ))\{0},\begin{equation} F(\tau) \in \Delta^2 \cdot M_{k-12}(\operatorname{SL}_2(\mathbb Z)) \setminus \{0\}, \end{equation} the Fourier coefficients {am(F)}\{a_m(F)\} cannot have constant sign on the discrete interval: m∈[k−212,k−212+C(k)].\begin{equation} m \in \left[ \frac{k - 2}{12}, \; \frac{k - 2}{12} + C(k) \right]. \label{eq:sign-window} \end{equation} That is, every non-zero cusp form takes both strictly positive and strictly negative values within this window.

Proof. For each listed degree kk, Table 4 provides an exact rational weight vector W∈ℚ>0C+1W \in \mathbb Q_{>0}^{C+1} satisfying AW=0AW = 0 with rank⁡(A)=C\operatorname{rank}(A) = C. By Theorem 6, no non-zero cusp form in Sk+120(SL⁡2(ℤ))S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)) can have am(F)≥0a_m(F) \ge 0 for all m∈[s,s+C]m \in [s, s+C]. Applying the same result to −F-F proves that FF must take both strictly positive and strictly negative values. ◻

Orbit Splitting and Internal Shell Cancellation

Evaluation of the Canonical Invariant Harmonic Ψ12\Psi_{12}

Projecting the degree-1212 Gegenbauer polynomial C12(11)(t)C_{12}^{(11)}(t) onto S2S_2 defines the unique Co⁡0\operatorname{Co}_0-invariant harmonic polynomial: Ψ12(x)=∥x∥12∑y∈S2C12(11)(x⋅y∥x∥∥y∥).\begin{equation} \Psi_{12}(x) = \|x\|^{12} \sum_{y\in S_2} C_{12}^{(11)} \left( \frac{x\cdot y}{\|x\|\|y\|} \right). \label{eq:Psi12} \end{equation} The degree-1212 Gegenbauer polynomial with α=11\alpha = 11 is: C12(11)(t)=2,648,662,016t12−3,972,993,024t10+2,128,389,120t8−496,624,128t6+49,008,960t4−1,633,632t2+8,008.\begin{equation} \begin{aligned} C_{12}^{(11)}(t) ={}& 2{,}648{,}662{,}016\,t^{12} - 3{,}972{,}993{,}024\,t^{10} + 2{,}128{,}389{,}120\,t^8 \\ &- 496{,}624{,}128\,t^6 + 49{,}008{,}960\,t^4 - 1{,}633{,}632\,t^2 + 8{,}008. \end{aligned} \end{equation} Because the argument t=x⋅y∥x∥∥y∥t = \frac{x \cdot y}{\|x\| \|y\|} is a scale-invariant cosine, evaluating on the sphere S23S^{23} yields integer sums. For minimal vectors u∈S2u \in S_2, integer contraction over S2S_2 yields: Ψ12(S2)=1,771,619,850,Ψ12(S3)Ψ12(S2)=−916.\begin{equation} \Psi_{12}(S_2) = 1{,}771{,}619{,}850, \qquad \frac{\Psi_{12}(S_3)}{\Psi_{12}(S_2)} = -\frac{9}{16}. \label{eq:Psi-S3} \end{equation}

Shell 8 Orbit Splitting and Harmonic Sign Inversion

In the intrinsic lattice metric, Shell 88 consists of vectors with squared norm ∥x∥2=2m=16\|x\|^2 = 2m = 16. It decomposes under Co⁡0\operatorname{Co}_0 into the doubled minimal orbit 𝒪8A\mathcal{O}_{8_A} and multiple primitive orbits.

Theorem 11 (Conway Orbit Decomposition and Harmonic Sign Inversion). The shell S8S_8 contains:

  1. The doubled minimal orbit 𝒪8A=2⋅S2\mathcal{O}_{8_A} = 2 \cdot S_2 of cardinality |𝒪8A|=|S2|=196,560|\mathcal{O}_{8_A}| = |S_2| = 196{,}560. By linearity of group actions (g(2u)=2g(u)g(2u) = 2g(u)), Stab⁡Co⁡0(2u)=Stab⁡Co⁡0(u)≅2⋅Co⁡2\operatorname{Stab}_{\operatorname{Co}_0}(2u) = \operatorname{Stab}_{\operatorname{Co}_0}(u) \cong 2 \cdot \operatorname{Co}_2 .

  2. The complement S8\𝒪8AS_8 \setminus \mathcal{O}_{8_A} consisting of N8−196,560=814,879,578,240N_8 - 196{,}560 = 814{,}879{,}578{,}240 primitive vectors.

The values of Ψ12\Psi_{12} on 𝒪8A\mathcal{O}_{8_A} and on the representative primitive vector vB=18(10,27,016)∈S8v_B = \frac{1}{\sqrt{8}}(10, 2^7, 0^{16}) \in S_8 satisfy: Ψ12(𝒪8A)Ψ12(S2)=+4,096=212,Ψ12(vB)Ψ12(S2)=−4023.\begin{equation} \frac{\Psi_{12}(\mathcal{O}_{8_A})}{\Psi_{12}(S_2)} = +4{,}096 = 2^{12}, \qquad \frac{\Psi_{12}(v_B)}{\Psi_{12}(S_2)} = -\frac{40}{23}. \label{eq:orbit-ratios} \end{equation} Consequently, Ψ12(𝒪8A)/Ψ12(vB)=−11,776/5=−2,355.2\Psi_{12}(\mathcal{O}_{8_A}) / \Psi_{12}(v_B) = -11{,}776 / 5 = -2{,}355.2.

Proof. Homogeneity gives Ψ12(2u)=212Ψ12(u)=4,096Ψ12(S2)\Psi_{12}(2u) = 2^{12} \Psi_{12}(u) = 4{,}096 \Psi_{12}(S_2).

In the standard octad frame scaled by 8\sqrt{8}, the primitive vector vB=18(10,27,016)v_B = \frac{1}{\sqrt{8}}(10, 2^7, 0^{16}) has squared norm ∥vB∥2=100+288=16\|v_B\|^2 = \frac{100+28}{8} = 16. Its inner products against the 196,560196{,}560 vectors of S2S_2 take values vB⋅y∈{±6,±5,±4,±3,±2,±1,0}v_B \cdot y \in \{\pm 6, \pm 5, \pm 4, \pm 3, \pm 2, \pm 1, 0\} with multiplicities {8,256,2,268,9,472,23,608,39,424,46,488}\{8, 256, 2{,}268, 9{,}472, 23{,}608, 39{,}424, 46{,}488\}.

Evaluating ∑y∈S2C12(11)(vB⋅y8)\sum_{y \in S_2} C_{12}^{(11)}\left( \frac{v_B \cdot y}{8} \right) against these multiplicities yields −192,567,375256-\frac{192{,}567{,}375}{256}. Multiplying by the norm dilation factor (∥vB∥/∥u∥)12=(16/4)6=4,096(\|v_B\|/\|u\|)^{12} = (16/4)^6 = 4{,}096 yields: Ψ12(vB)=4,096×(−192,567,375256)=−3,081,078,000.\begin{equation} \Psi_{12}(v_B) = 4{,}096 \times \left( -\frac{192{,}567{,}375}{256} \right) = -3{,}081{,}078{,}000. \end{equation} Dividing by Ψ12(S2)=1,771,619,850\Psi_{12}(S_2) = 1{,}771{,}619{,}850 yields −3,081,078,0001,771,619,850=−4023-\frac{3{,}081{,}078{,}000}{1{,}771{,}619{,}850} = -\frac{40}{23}. ◻

Remark 12 (Multi-Orbit Structure of Shell 8). Because the full Fourier coefficient a8(Δ2)=+5,145,888>0a_8(\Delta^2) = +5{,}145{,}888 > 0 is strictly positive, the total shell sum satisfies ∑x∈S8Ψ12(x)=1,011,475,745,280×Ψ12(S2)>0\sum_{x \in S_8} \Psi_{12}(x) = 1{,}011{,}475{,}745{,}280 \times \Psi_{12}(S_2) > 0. Since Ψ12(vB)<0\Psi_{12}(v_B) < 0, the primitive vectors cannot form a single transitive orbit under Co⁡0\operatorname{Co}_0, proving that Shell 8 decomposes into multiple orbits of opposing harmonic signs.

Corollary 13 (Internal Shell 8 Annihilation). Let 𝒪8B=Co⁡0⋅vB\mathcal{O}_{8_B} = \operatorname{Co}_0\cdot v_B. Assigning positive scalar weights wA,wBw_A, w_B cancels degree 1212 internally across 𝒪8A∪𝒪8B\mathcal{O}_{8_A} \cup \mathcal{O}_{8_B}: wA|𝒪8A|Ψ12(𝒪8A)+wB|𝒪8B|Ψ12(𝒪8B)=0.\begin{equation} w_A |\mathcal{O}_{8_A}| \Psi_{12}(\mathcal{O}_{8_A}) + w_B |\mathcal{O}_{8_B}| \Psi_{12}(\mathcal{O}_{8_B}) = 0. \end{equation} Taking |𝒪8B|=814,879,578,240|\mathcal{O}_{8_B}| = 814{,}879{,}578{,}240 yields the exact rational ratio: wBwA=|𝒪8A|⋅4,096|𝒪8B|⋅(40/23)=196,560⋅4,096⋅23814,879,578,240⋅40=64112,655≈11,760.23.\begin{equation} \frac{w_B}{w_A} = \frac{|\mathcal{O}_{8_A}| \cdot 4{,}096}{|\mathcal{O}_{8_B}| \cdot (40/23)} = \frac{196{,}560 \cdot 4{,}096 \cdot 23}{814{,}879{,}578{,}240 \cdot 40} = \frac{64}{112{,}655} \approx \frac{1}{1{,}760.23}. \label{eq:orbit-weight-ratio} \end{equation}

The Explicit Strength-15 Construction

For a shell or orbit 𝒪\mathcal O of squared norm 2m2m, its normalized degree-1212 moment contribution is: M(𝒪)=|𝒪|(2m)−6Ψ12(𝒪)Ψ12(S2).\begin{equation} M(\mathcal O) = |\mathcal O|(2m)^{-6} \frac{\Psi_{12}(\mathcal O)}{\Psi_{12}(S_2)}. \label{eq:moment-M} \end{equation} Evaluating on the minimal shells: M2=|S2|⋅(4)−6=196,5604,096=12,285256,M3=|S3|⋅(6)−6(−916)=16,773,12046,656(−916)=29,12081(−916)=−1,8209.\begin{align} M_2 &= |S_2| \cdot (4)^{-6} = \frac{196{,}560}{4{,}096} = \frac{12{,}285}{256}, \label{eq:M2}\\ M_3 &= |S_3| \cdot (6)^{-6} \left(-\frac{9}{16}\right) = \frac{16{,}773{,}120}{46{,}656} \left(-\frac{9}{16}\right) = \frac{29{,}120}{81} \left(-\frac{9}{16}\right) = -\frac{1{,}820}{9}. \label{eq:M3} \end{align}

Theorem 14 (Closed-Form Spherical 15-Design on S23S^{23}). The two-shell configuration S2∪S3S_2 \cup S_3 with per-vector shell weights: W(S2)=1,W(S3)=2431024=35210\begin{equation} W(S_2) = 1, \qquad W(S_3) = \frac{243}{1024} = \frac{3^5}{2^{10}} \label{eq:strength15-weights} \end{equation} forms an exact spherical 1515-design on Ntotal=196,560+16,773,120=16,969,680N_{\mathrm{total}} = 196{,}560 + 16{,}773{,}120 = 16{,}969{,}680 distinct spherical points.

Proof. Solving W(S2)M2+W(S3)M3=0W(S_2) M_2 + W(S_3) M_3 = 0 with W(S2)=1W(S_2) = 1 yields: W(S3)=M2−M3=12,285/2561,820/9=12,285×9256×1,820=110,565465,920=2431024.\begin{equation} W(S_3) = \frac{M_2}{-M_3} = \frac{12{,}285 / 256}{1{,}820 / 9} = \frac{12{,}285 \times 9}{256 \times 1{,}820} = \frac{110{,}565}{465{,}920} = \frac{243}{1024}. \end{equation} Equivalently, in terms of modular Fourier coefficients: W(S2)(4)−6a2(Δ2)+W(S3)(6)−6a3(Δ2)=0⇔W214096−W34846656=0⇔W3W2=9724096=2431024.\begin{equation} W(S_2)(4)^{-6} a_2(\Delta^2) + W(S_3)(6)^{-6} a_3(\Delta^2) = 0 \iff W_2 \frac{1}{4096} - W_3 \frac{48}{46656} = 0 \iff \frac{W_3}{W_2} = \frac{972}{4096} = \frac{243}{1024}. \end{equation} All odd moments vanish by central symmetry, degrees 2≤k≤102 \le k \le 10 vanish by Venkov’s 11-design theorem, and degree 1414 vanishes because M2(SL⁡2(ℤ))={0}M_2(\operatorname{SL}_2(\mathbb Z)) = \{0\}. ◻

Three-Layer Euclidean Harmonic Cancellation

In ℝ24\mathbb R^{24}, regarding S2S_2, S3S_3, and 𝒪8A\mathcal{O}_{8_A} as three distinct radial layers (∥x∥2∈{4,6,16}\|x\|^2 \in \{4, 6, 16\}), radial scaling gives M8A=|𝒪8A|(16)−6(4096)=196,560⋅2−24⋅212=12,285256=M2M_{8_A} = |\mathcal{O}_{8_A}| (16)^{-6} (4096) = 196{,}560 \cdot 2^{-24} \cdot 2^{12} = \frac{12{,}285}{256} = M_2. Setting unit weights W2=W8A=1W_2 = W_{8_A} = 1: 2M2+W3M3=0⟹W3=2M2−M3=243512=3529.\begin{equation} 2 M_2 + W_3 M_3 = 0 \implies W_3 = \frac{2 M_2}{-M_3} = \frac{243}{512} = \frac{3^5}{2^9}. \label{eq:three-layer-weight} \end{equation} This provides exact degree-12 harmonic cancellation across three concentric radii in ℝ24\mathbb R^{24}.

Intertwiner Elasticity

Let u∈S2u \in S_2 be a minimal vector. In scaled coordinates where ∥u∥2=32\|u\|^2 = 32, the stabilizer subgroup in Co⁡0\operatorname{Co}_0 is: K=Stab⁡Co⁡0(u)≅2⋅Co⁡2.\begin{equation} K = \operatorname{Stab}_{\operatorname{Co}_0}(u) \cong 2 \cdot \operatorname{Co}_2. \end{equation} Under KK, the 196,560196{,}560 vectors of S2S_2 decompose into exactly 7 orbits characterized by u⋅v∈{±32,±16,±8,0}u \cdot v \in \{\pm 32, \pm 16, \pm 8, 0\}: $$\begin{equation} \begin{tabular}{rccccccc} \toprule $u \cdot v$ : & $\pm 32$ & $\pm 16$ & $\pm 8$ & $0$ \\ \text{Orbit Size} : & $1$ each & $4{,}600$ each & $47{,}104$ each & $93{,}150$ \\ \bottomrule \end{tabular} \end{equation}$$

The space of KK-equivariant tangent vector fields on S2S_2 is: ℳ=Hom⁡K(Vnat,TS2S23).\begin{equation} \mathcal M= \operatorname{Hom}_K(V_{\mathrm{nat}}, T_{S_2}S^{23}). \end{equation} On each of the 5 non-polar orbits (s∈{±16,±8,0}s \in \{\pm 16, \pm 8, 0\}), the space of equivariant fields has dimension 2, spanned by:

On the 2 polar orbits (s=±32s = \pm 32), v=±u⟹W2≡0v = \pm u \implies W_2 \equiv 0, leaving 1 field each. Thus: dim⁡ℳ=5×2+2×1=12.\begin{equation} \dim \mathcal M= 5 \times 2 + 2 \times 1 = 12. \end{equation}

Proposition 15 (Rotational Goldstone Mode and Parity Splitting). For any smooth radial energy functional E(X)=∑x≠yF(∥x−y∥2)E(X) = \sum_{x \ne y} F(\|x-y\|^2):

  1. The rotational field Φrot,z(v)=(z⋅v)u−(u⋅v)z\Phi_{\mathrm{rot}, z}(v) = (z \cdot v)u - (u \cdot v)z satisfies HΦrot=0H \Phi_{\mathrm{rot}} = 0, spanning the kernel ker⁡(H)=span⁡{crot}\ker(H) = \operatorname{span}\{c_{\mathrm{rot}}\}.

  2. The antipodal pullback (PΦ)(z,v)=Φ(z,−v)(P\Phi)(z, v) = \Phi(z, -v) satisfies P2=IP^2 = I and [P,H]=0[P, H] = 0, decomposing ℳ\mathcal M into 66 even (+1+1) and 66 odd (−1-1) modes, with P(crot)=−crotP(c_{\mathrm{rot}}) = -c_{\mathrm{rot}}.

Completely Monotonic Potentials and the Local Hessian

A potential F:(0,∞)→ℝF : (0, \infty) \to \mathbb R is completely monotonic if (−1)nF(n)(r)≥0(-1)^n F^{(n)}(r) \ge 0 for all n≥0n \ge 0. By the Bernstein–Widder theorem: F(r)=∫[0,∞)e−trdμ(t),dμ≥0.\begin{equation} F(r) = \int_{[0, \infty)} e^{-tr} \, d\mu(t), \qquad d\mu \ge 0. \end{equation} Constant potentials (t=0t=0) have identically zero Hessian. For non-constant potentials, μ((0,∞))>0\mu((0, \infty)) > 0.

Proposition 16 (Conditional Local Positivity Criterion). On the 12-dimensional intertwiner space ℳ\mathcal M, the Hessian for a Gaussian Gt(r)=e−trG_t(r) = e^{-tr} (t>0t > 0) decomposes as A(Gt)=λS(Gt)I−Ktan(Gt)A(G_t) = \lambda_S(G_t) I - K_{\mathrm{tan}}(G_t), where the diagonal curvature: λS(Gt)=∑s[4t2e−tUscs+2te−tUss32count(s)]\begin{equation} \lambda_S(G_t) = \sum_{s} \left[ 4 t^2 e^{-t U_s} c_s + 2 t e^{-t U_s} \frac{s}{32} \operatorname{count}(s) \right] \end{equation} satisfies the exact hyperbolic-sine identity: ∑s∈{±32,±16,±8,0}2te−tUss32count⁡(s)=4te−64t[sinh(64t)+2,300sinh(32t)+11,776sinh(16t)]>0\begin{equation} \sum_{s \in \{\pm 32, \pm 16, \pm 8, 0\}} 2 t e^{-t U_s} \frac{s}{32} \operatorname{count}(s) = 4 t e^{-64t} \left[ \sinh(64t) + 2{,}300 \sinh(32t) + 11{,}776 \sinh(16t) \right] > 0 \end{equation} for all t>0t > 0.

Condition H: Assume that for all t>0t > 0, the operator norm of the tangential repulsion tensor restricted to the orthogonal complement of the Goldstone mode satisfies: ∥Ktan(Gt)∥ℳ⊖ℝcrot<λS(Gt).\begin{equation} \|K_{\mathrm{tan}}(G_t)\|_{\mathcal M\ominus \mathbb Rc_{\mathrm{rot}}} < \lambda_S(G_t). \label{eq:operator-bound} \end{equation}

Under Condition H, every non-constant completely monotonic potential FF has a strictly positive Hessian on ℳ⊖ℝcrot\mathcal M\ominus \mathbb Rc_{\mathrm{rot}}: vTHFv=∫(0,∞)(vTA(Gt)v)dμ(t)>0∀v∈ℳ\ℝcrot.\begin{equation} v^T H_F v = \int_{(0, \infty)} \left( v^T A(G_t) v \right) d\mu(t) > 0 \qquad \forall v \in \mathcal M\setminus \mathbb Rc_{\mathrm{rot}}. \end{equation}

Point Counts and Asymptotic Delsarte Comparison

Comparison of certified Leech cubatures against the equal-weight DGS bound.
Strength tt DGS Bound NDGSN_{\mathrm{DGS}} Active Leech Configuration Points Used NtotalN_{\mathrm{total}} Ratio NtotalNDGS\frac{N_{\mathrm{total}}}{N_{\mathrm{DGS}}}
𝟏𝟏\mathbf{11} 𝟏𝟗𝟔,𝟓𝟔𝟎\mathbf{196{,}560} S2S_2 𝟏𝟗𝟔,𝟓𝟔𝟎\mathbf{196{,}560} 𝟏.𝟎𝟎𝟎𝟎(𝐓𝐈𝐆𝐇𝐓)\mathbf{1.0000 \quad (\text{TIGHT})}
𝟏𝟓\mathbf{15} 𝟒,𝟎𝟕𝟏,𝟔𝟎𝟎\mathbf{4{,}071{,}600} S2∪S3S_2 \cup S_3 𝟏𝟔,𝟗𝟔𝟗,𝟔𝟖𝟎\mathbf{16{,}969{,}680} 𝟒.𝟏𝟕(𝐍𝐄𝐀𝐑-𝐎𝐏𝐓.)\mathbf{4.17 \quad (\text{NEAR-OPT.})}
1717 15,777,45015{,}777{,}450 S2…S4S_2 \dots S_4 415,003,680415{,}003{,}680 26.326.3
1919 56,097,60056{,}097{,}600 S2…S5S_2 \dots S_5 5,044,384,8005{,}044{,}384{,}800 89.989.9
2121 185,122,080185{,}122{,}080 S2∪{S4…S7}S_2 \cup \{S_4 \dots S_7\} 226,935,203,040226{,}935{,}203{,}040 1,2261{,}226
2323 572,195,520572{,}195{,}520 S2…S7S_2 \dots S_7 226,951,976,160226{,}951{,}976{,}160 397397
9191 1.60×10181.60 \times 10^{18} S2…S149S_2 \dots S_{149} 9.85×10269.85 \times 10^{26} 6.1×1086.1 \times 10^8
𝟏𝟏𝟏\mathbf{111} 𝟔.𝟗𝟎×𝟏𝟎𝟏𝟗\mathbf{6.90 \times 10^{19}} S9…S234S_9 \dots S_{234} 𝟐.𝟏𝟗×𝟏𝟎𝟐𝟗\mathbf{2.19 \times 10^{29}} 𝟑.𝟐×𝟏𝟎𝟗\mathbf{3.2 \times 10^9}

The number of conditions scales as C(t)∼t2/48⟹M(t)=Θ(t2)C(t) \sim t^2/48 \implies M(t) = \Theta(t^2). The boundary shift mstart(t)∼t/12=Θ(t)m_{\mathrm{start}}(t) \sim t/12 = \Theta(t) is subdominant: lim⁡t→∞mstart(t)M(t)=0\lim_{t \to \infty} \frac{m_{\mathrm{start}}(t)}{M(t)} = 0. Since Nm=Θ(m11)N_m = \Theta(m^{11}), the total vector count satisfies: Ntotal(t)=∑m=mstartmstart+M−1Nm≍M(t)12≍(t2)12=Θ(t24).\begin{equation} N_{\mathrm{total}}(t) = \sum_{m = m_{\mathrm{start}}}^{m_{\mathrm{start}}+M-1} N_m \asymp M(t)^{12} \asymp (t^2)^{12} = \Theta(t^{24}). \end{equation} In dimension d=24d=24, the general positive cubature lower bound is Θ(t23)\Theta(t^{23}). The multi-shell construction achieves an exponent of 2424, which is within an additional factor of 𝒪(t)\mathcal{O}(t) of the lower bound, matching the Korevaar–Meyers upper bound scale 𝒪(td)=𝒪(t24)\mathcal{O}(t^d) = \mathcal{O}(t^{24}).

Computational Certification Protocol

All computational certificates in this paper are exact rational statements verified via FLINT . The verification package contains:

  1. The exact monomial basis Fb=Δ2E4abE6bbF_b = \Delta^2 E_4^{a_b} E_6^{b_b} of Sk+120(SL⁡2(ℤ))S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)).

  2. The exact rational moment matrix A∈ℚC×(C+1)A \in \mathbb Q^{C \times (C+1)}.

  3. A certificate proving rank⁡(A)=C\operatorname{rank}(A) = C.

  4. The exact rational solution vector W∈ℚ>0C+1W \in \mathbb Q_{>0}^{C+1} with certified positive coordinates.

  5. Exact rational verification that the residual vector satisfies AW=0AW = 0 identically in ℚC\mathbb Q^C.

  6. An exact witness F∈Sknext+120(SL⁡2(ℤ))F \in S_{k_{\mathrm{next}}+12}^0(\operatorname{SL}_2(\mathbb Z)) proving LW(F)≠0L_W(F) \neq 0.

  7. For boundary minimality, exact negative-weight certificates for every predecessor window s<mstartmin(k)s < m_{\mathrm{start}}^{\min}(k).

  8. Cryptographic SHA-256 hashes of all input matrices and output weight vectors.

Scope of the Results

To maintain strict scientific standards, we delineate our results into three categories:

Proved Structural Results

Certified Computational Results

Conjectures and Open Questions

Conclusion

We have established an exact rational multi-shell cubature framework on the Leech lattice Λ24\Lambda_{24}. By leveraging Reynolds group averaging and the isomorphism Sk+120(SL⁡2(ℤ))≅Mk−12(SL⁡2(ℤ))S_{k+12}^0(\operatorname{SL}_2(\mathbb Z)) \cong M_{k-12}(\operatorname{SL}_2(\mathbb Z)), the infinite system of degree-kk harmonic moment conditions collapses to a finite rational linear system.

Our exact FLINT computations certify positive rational spherical cubatures through strength 111111, proving that intermediate gaps are resolved by resonant shell skipping. Through modular cone duality, our certified positive weights provide an unconditional proof of finite-window sign changes for cusp forms up to weight 122122. Finally, Co⁡0\operatorname{Co}_0-orbit splitting reveals an exact harmonic sign inversion on Shell 8, enabling internal spherical designs and compressed cubature hierarchies. Unified master exact audit file is available at:

https://srfp311t1.com/leech_cubature_111.py

99

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