Introduction and the Problem of the Continuum Limit

The study of discrete approximations of continuous spherical integrals via lattice configurations constitutes a central theme in geometric analysis, sphere-packing theory, and automorphic forms. In Parts I and II of this series , we developed constructions of spherical $t$-designs on the unit sphere $S^{23}\subset\mathbb{R}^{24}$ using concentric shells $$S_m=\{x\in\Lambda_{24}:\|x\|^2=2m\}.$$ After averaging under the Conway group $\mathrm{Co}_0=\mathop{\mathrm{Aut}}(\Lambda_{24})$, the corresponding cubature conditions reduce to a finite-dimensional linear feasibility problem.

Specifically, for each even degree $k\ge12$, the discrete cubature condition is $$\begin{equation} \label{eq:finite_system} \sum_{m\in\mathcal{S}_k} W_m^{(k)}v_{\le k}(m) = \mathbf{0}\in\mathbb{R}^{D_k}, \qquad W_m^{(k)}>0, \end{equation}$$ where $$D_k = \sum_{j=6}^{k/2} \dim M_{2j-12}(\mathop{\mathrm{SL}}_2(\mathbb{Z})) \approx \frac{k^2}{48},$$ and $v_{\le k}(m)$ is the cumulative evaluation vector across the direct sum of cusp spaces $$\begin{equation} S_{2j+12}^{(2)}(\mathop{\mathrm{SL}}_2(\mathbb{Z})) \coloneqq \left\{ f\in S_{2j+12}(\mathop{\mathrm{SL}}_2(\mathbb{Z})): \mathop{\mathrm{ord}}_{q=0}(f)\ge2 \right\} \cong \Delta^2M_{2j-12}(\mathop{\mathrm{SL}}_2(\mathbb{Z})). \end{equation}$$

The Stability Gates for Infinite-Dimensional Limits

In Part II , we identified five independent functional-analytic stability criteria required before finite-dimensional polyhedral certificates can be interpreted as continuous or infinite-dimensional positivity statements:

1. Uniform Total Variation Boundedness: Establish $$\sup_k \sum_{m\in\mathcal{S}_k} W_m^{(k)}\mathcal{K}(m)<\infty$$ under an explicit damping kernel $\mathcal{K}(m)$.

2. Well-Defined Ambient Embedding: Specify a fixed topological vector space $E$ and dual $E^*$ into which all discrete measures $\mu_k$ are embedded.

3. Weak-$*$ Compactness and Accumulation: Extract a weak-$*$ limit $\mu_\infty\in E^*$ via the Banach–Alaoglu theorem in $\sigma(E^*,E)$.

4. Test-Function Compatibility: Verify that the target cusp-form evaluation functionals belong to the predual test space $E$.

5. Non-Triviality / Mass Preservation: Control the asymptotic mass allocation sufficiently to prevent collapse to the trivial zero functional: $$\liminf_{k\to\infty} \langle\mu_k,\mathbf{1}\rangle>0.$$

The Growth Obstacle in Raw Cubatures

In Part II, numerical evaluation of unweighted cubature solutions across degrees $12\le k\le32$ revealed rapid growth in the unnormalized $\ell^1$ norm: $$\begin{equation} \|W^{(12)}\|_1\approx1.24,\qquad \|W^{(20)}\|_1\approx5.29,\qquad \|W^{(28)}\|_1\approx24.88,\qquad \|W^{(32)}\|_1\approx199.73. \end{equation}$$ The growth ratio accelerates to $$\frac{\|W^{(32)}\|_1}{\|W^{(28)}\|_1} \approx8.03.$$

The mathematical origin of this growth is structural: the radial projection factors $(2m)^{-k/2}$ decay rapidly as $m$ increases, requiring outer shells to carry large weights $W_m$ in order to balance the low-order anchor shell $S_{12}$.

In this paper, we analyze the conditions under which an Archimedean damping kernel regularizes this growth, formulate the saddle-point scaling behind the observed desert horizon, and establish the functional-analytic framework for subsequential limits of the resulting measures on $\ell^1(\mathbb{N}_{\ge2})$.

Archimedean Damping and Total Variation Bounds

Let $$\mathbb{N}_{\ge2}=\{2,3,4,\dots\}$$ denote the discrete index space of Leech shells. We equip $\ell^1(\mathbb{N}_{\ge2})$ with its standard norm $$\|\mu\|_{\mathrm{TV}} = \sum_{m=2}^{\infty}|\mu(m)|.$$

In the linear feasibility problem $$A_{\le k}W^{(k)}=\mathbf{0},$$ the matrix entries are $$A_{b,m} = (2m)^{-k/2}a_m(F_b).$$ This motivates introducing the rescaled cubature variables $$\begin{equation} \label{eq:rescaled_variables} \widetilde{W}_m^{(k)} \coloneqq W_m^{(k)}(2m)^{-k/2}. \end{equation}$$

For each target degree $k$, let $(W^{(k)},\mathcal{S}_k)$ be a non-negative cubature solution. We associate to it the regularized discrete measure $$\begin{equation} \label{eq:regularized_measure} \mu_k^{(s,\beta)} \coloneqq \sum_{m\in\mathcal{S}_k} \widetilde{W}_m^{(k)} \mathcal{K}_{s,\beta}(m)\,\delta_m \in\ell^1(\mathbb{N}_{\ge2}), \end{equation}$$ where $\delta_m$ denotes the standard Dirac point mass at $m\in\mathbb{N}_{\ge2}$.

To establish uniform boundedness in $\ell^1(\mathbb{N}_{\ge2})$, we formalize the necessary growth hypothesis directly on the rescaled variables.

The Spectral Horizon Conjecture

In Part II , numerical experiments revealed an exclusion gap, referred to there as the Desert Phenomenon:

• At $k=114$ ($C=243$), the active support was $S_{12}\cup\{S_{129},\dots,S_{265}\}$, with gap $\Delta m=117$.

• At $k=120$ ($C=271$), the active support was $S_{12}\cup\{S_{89},\dots,S_{295}\}$, with gap $\Delta m=77$.

Saddle-Point Asymptotic Scaling

The proposed scale $$m^*(k)\asymp\frac{k^2}{96}$$ is motivated by the asymptotic behavior of the modular basis $$F_{a,b} = \Delta^2E_4^aE_6^b, \qquad 4a+6b=k-12.$$ The corresponding modular weight is $k+12$.

By Cauchy’s integral formula, the $m$-th Fourier coefficient is $$\begin{equation} \label{eq:cauchy_integral} a_m(F_{a,b}) = \frac{1}{2\pi i} \oint_{|q|=r} \Delta(q)^2E_4(q)^aE_6(q)^b q^{-m-1}\,dq. \end{equation}$$ Writing $q=e^{-2\pi u}$ for $u>0$, the corresponding logarithmic phase along the positive real axis has the form $$\Phi_k(u) = a\log E_4(e^{-2\pi u}) + b\log E_6(e^{-2\pi u}) + 2\log\Delta(e^{-2\pi u}) + 2\pi m u.$$

Under modular inversion $\tau\mapsto-1/\tau$, the large modular weight contributes a logarithmic term of order $$\frac{k}{2}\log(1/u).$$ Thus, schematically, for small $u$ one obtains $$\begin{equation} \label{eq:phase_asymptotic} \Phi_k(u) \approx \frac{k}{2}\log\left(\frac1u\right) + \frac{c_1}{u} + 2\pi m u, \end{equation}$$ where the constant $c_1$ depends on the precise modular factors under consideration.

Differentiating the displayed model phase gives $$\begin{equation} \label{eq:saddle_condition} \Phi_k'(u) = -\frac{k}{2u} - \frac{c_1}{u^2} + 2\pi m. \end{equation}$$ When the first and third terms dominate, the associated radial scale is $$\begin{equation} \label{eq:saddle_point_scaling} u_0 \asymp \frac{k}{m}. \end{equation}$$

This calculation therefore identifies a natural $k/m$ saddle-point scaling. The more specific numerical transition scale $m^*(k)\asymp k^2/96$ is motivated by the observed cone geometry and numerical data; deriving the constant $1/96$ rigorously from the full modular saddle-point analysis remains open.

The Limiting Measure on the Sequence Space $\ell^1(\mathbb{N}_{\ge2})$

To avoid topological type mismatches, the discrete measures $\mu_k^{(s,\beta)}$ are embedded into a single fixed Banach dual pair rather than into the varying finite-dimensional spaces $\mathbb{R}^{D_k}$ or the continuous modular curve $\mathop{\mathrm{SL}}_2(\mathbb{Z})\backslash\mathbb{H}$.

The Sequence Dual Pair

We take $$E=c_0(\mathbb{N}_{\ge2})$$ and identify its Banach dual canonically with $$E^*=\ell^1(\mathbb{N}_{\ge2}).$$ Explicitly, $$\begin{equation} E^* = \ell^1(\mathbb{N}_{\ge2}) = \left\{ \mu:\mathbb{N}_{\ge2}\to\mathbb{R}: \sum_{m=2}^\infty|\mu(m)|<\infty \right\}, \end{equation}$$ with norm $$\|\mu\|_{\mathrm{TV}} = \sum_{m=2}^\infty|\mu(m)|.$$ The predual is $$\begin{equation} E = c_0(\mathbb{N}_{\ge2}) = \left\{ \phi:\mathbb{N}_{\ge2}\to\mathbb{R}: \lim_{m\to\infty}\phi(m)=0 \right\}, \end{equation}$$ equipped with $$\|\phi\|_\infty = \sup_{m\ge2}|\phi(m)|.$$

We equip $\ell^1(\mathbb{N}_{\ge2})$ with the weak-$*$ topology $$\sigma(\ell^1,c_0).$$

The Escape of Outer Mass

The constant sequence $$\mathbf{1}=(1,1,1,\dots)$$ does not belong to $c_0(\mathbb{N}_{\ge2})$. Consequently, weak-$*$ convergence in $\sigma(\ell^1,c_0)$ does not preserve total mass: $$\mu_k(\mathbf{1}) \not\to \mu_\infty(\mathbf{1})$$ in general.

This phenomenon should be interpreted as escape of mass relative to the chosen topology: mass moving to indices $m\to\infty$ becomes invisible to every test sequence in $c_0$.

Duality Comparison: The Modular Cone vs. Guinand–Weil Positivity

The existence of positive annihilating cones on the Leech lattice raises the question of whether this framework can be transferred to the Guinand–Weil explicit formula to deduce the Riemann Hypothesis (RH).

The relevant structural differences are summarized below.

Conclusion and Research Roadmap

Part III develops a conditional asymptotic framework for the infinite-dimensional limit of Leech lattice spherical cubatures.

1. We introduced the Archimedean Damping Kernel $\mathcal{K}_{s,\beta}(m)$ and showed that Gate 1 follows from the polynomial-exponential rescaled weight growth hypothesis.

2. We formulated the Spectral Horizon Conjecture $$m^*(k)\asymp\frac{k^2}{96}.$$ The saddle-point analysis identifies the underlying $k/m$ radial scaling, while the specific constant $1/96$ remains conjectural and is motivated by the numerical desert phenomenon and associated cone geometry.

3. We established weak-$*$ compactness in the sequence space $\ell^1(\mathbb{N}_{\ge2})$ and proved the Escape of Outer Mass theorem in $\sigma(\ell^1,c_0)$.

4. We identified structural differences between finite-dimensional modular cone duality on $\Lambda_{24}$ and the infinite-dimensional Guinand–Weil positivity problem. In particular, the finite-dimensional annihilation mechanism does not by itself furnish a positivity theorem for the Guinand–Weil functional.

The principal open mathematical challenges are therefore:

1. proving Hypothesis 2 directly from the dual cone geometry or from the optimization/KKT structure;

2. deriving the precise horizon constant $1/96$, if the conjectured scaling is correct, from a rigorous modular saddle-point analysis;

3. establishing an exact normalization protocol for Gate 5 so that a non-trivial limiting functional can be identified; and

4. determining whether any rigorous correspondence exists between the finite-dimensional modular cone and the test-function spaces appearing in the Guinand–Weil explicit formula.

Thus the present work isolates the functional-analytic conditions required for a meaningful continuum limit without asserting that those conditions are automatically satisfied by the finite-dimensional constructions.