Spectral Cone Duality:
Relative-Interior Annihilation, Pruned Leech Cubatures,
and Hecke Prime Angular Gaps

SRFP311T1 Collaboration

2026-10-01

Abstract

In our foundational work , we established the existence of multi-shell spherical cubatures on the Leech lattice Λ24\Lambda_{24} through strength 111111 by reducing degree-kk harmonic moments under Co⁡0=Aut⁡(Λ24)\operatorname{Co}_0= \operatorname{Aut}(\Lambda_{24}) to modular forms in Sk+12(2)(SL⁡2(ℤ))≅Δ2Mk−12(SL⁡2(ℤ))S_{k+12}^{(2)}(\operatorname{SL}_2(\mathbb Z)) \cong \Delta^2 M_{k-12}(\operatorname{SL}_2(\mathbb Z)). In this companion paper, we develop the universal framework of Spectral Cone Duality governing strictly positive annihilation across discrete spectral systems, and deploy it to solve the fundamental structural, computational, and arithmetic questions arising from Part I.

First, we prove the Master Duality Theorem: across any finite-dimensional spectral evaluation system, the existence of strictly positive annihilators is strictly dual to the relative-interior enclosure of the origin (0∈relint⁡(cone⁡(S))0 \in \operatorname{relint}(\operatorname{cone}(S))), or equivalently, the total triviality of non-negative dual states on the quotient space V¯=V/U⟂\overline{V} = V / U^\perp. In two dimensions, this is governed by the circular gap metric: 00 is enclosed if and only if the maximal cyclic angular gap satisfies Δθmax<π\Delta\theta_{\max} < \pi. We prove an Exact Rationality Lemma ensuring that every real positive solution admits an exact rational solution.

Second, in the geometry of the Leech lattice, we break the rigid contiguous window ansatz of  via a pruned dictionary search, discovering point-minimized “slim” cubature designs: for Strength 2929, skipping shell S4S_4 terminates the active support at S13S_{13} rather than S14S_{14}, achieving a 2.37×2.37\times point reduction and eliminating 3.84×10143.84 \times 10^{14} redundant points; for Strength 3333, mid-window pair pruning eliminates 5.54×10145.54 \times 10^{14} points while retaining the minimal shell S2S_2. We construct explicit rational dual Farkas certificates that rigorously certify the infeasibility of candidate shell supports. Analytically, we prove the Serre Vanishing Theorem: for k≡2(mod⁡12)k \equiv 2 \pmod{12}, the maximal cusp vanishing order across Sk+12(2)(SL⁡2(ℤ))S_{k+12}^{(2)}(\operatorname{SL}_2(\mathbb Z)) is strictly (k−2)/12(k-2)/12, governed by the Serre obstruction M2(SL⁡2(ℤ))={0}M_2(\operatorname{SL}_2(\mathbb Z)) = \{0\}, explaining the boundary shift observed in . We formulate the Modular Separation Conjecture, stating that the minimal contiguous radius is R(Vk)=dim⁡Vk+2+𝟏{k≡2(mod⁡12)}R(V_k) = \dim V_k + 2 + \mathbf{1}_{\{k \equiv 2 \pmod{12}\}}, verified up to weight 122122.

Third, we analyze the microscopic sub-orbit structure of Λ24\Lambda_{24} under the monomial frame group 212:M24⊂Co⁡02^{12} : M_{24} \subset \operatorname{Co}_0. We prove that Shell 22 decomposes into three orbits whose normalized degree-4 harmonic moments balance with exact integer ratios +20,+44,−64+20, +44, -64. Exploiting this internal equilibrium, we construct exact compressed spherical 55-designs on sub-orbits of only 99,40899{,}408 points.

Finally, we apply spectral cone duality to rank-two and rank-three Hecke cusp spaces Sk(SL⁡2(ℤ))S_k(\operatorname{SL}_2(\mathbb Z)), determining the exact prime thresholds required to enclose the origin: p*(S24)=7p^*(S_{24}) = 7, p*(S28)=11p^*(S_{28}) = 11, p*(S32)=11p^*(S_{32}) = 11, and p*(S36)=19p^*(S_{36}) = 19. As an arithmetic consequence, every non-zero cusp form F∈Sk(SL⁡2(ℤ))\{0}F \in S_k(\operatorname{SL}_2(\mathbb Z)) \setminus \{0\} must exhibit a strict sign change within the prime evaluation window {p≤p*(Sk)}\{p \le p^*(S_k)\}. We formalize the Finite Certification Problem, identifying the five functional-analytic stability requirements separating finite polyhedral certificates from unproven infinite-dimensional criteria.

Introduction and Relation to Part I

Context and Motivation

A weighted spherical tt-design on the unit sphere Sd−1⊂ℝdS^{d-1} \subset \mathbb R^d is a finite point set XX with positive weights W:X→ℝ>0W : X \to \mathbb R_{>0} that integrates all polynomials of degree at most tt exactly against the normalized Haar measure : ∫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 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), \quad 1 \le k \le t. \end{equation}

In our foundational paper , we established the existence of exact rational spherical designs on the 24-dimensional Leech lattice Λ24⊂ℝ24\Lambda_{24}\subset \mathbb R^{24} through strength t=111t=111 (degree k=110k=110). By applying Reynolds group averaging over the automorphism group Co⁡0=Aut⁡(Λ24)\operatorname{Co}_0= \operatorname{Aut}(\Lambda_{24}), all harmonic moments outside the invariant subspace (Harm⁡k(ℝ24))Co⁡0(\operatorname{Harm}_k(\mathbb R^{24}))^{\operatorname{Co}_0} vanish identically on every shell. 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 vanishing to order at least two at infinity: Sk+12(2)(SL⁡2(ℤ))≔{f∈Sk+12(SL⁡2(ℤ)):ord⁡q=0(f)≥2}≅Δ2Mk−12(SL⁡2(ℤ)).\begin{equation} S_{k+12}^{(2)}(\operatorname{SL}_2(\mathbb Z)) \coloneqq \{ f \in S_{k+12}(\operatorname{SL}_2(\mathbb Z)) : \operatorname{ord}_{q=0}(f) \ge 2 \} \cong \Delta^2 M_{k-12}(\operatorname{SL}_2(\mathbb Z)). \end{equation} Consequently, the infinite system of degree-kk harmonic cubature equations collapses to a finite rational system of C(k)=k248−k4+δ(k)C(k) = \frac{k^2}{48} - \frac{k}{4} + \delta(k) scalar modular equations, which were audited via FLINT across contiguous shell windows up to k=110k=110 .

Open Questions from Part I and Scope of the Present Work

While  demonstrated the existence of rational solutions on contiguous shell windows, it left open several fundamental structural, geometric, and analytic questions:

  1. Point Minimization: The contiguous shell windows used in  suffer from combinatorial growth because the shell cardinalities grow asymptotically as Nm=Θ(m11)N_m = \Theta(m^{11}). Can selective, non-consecutive shell pruning significantly reduce the total number of cubature points?

  2. Analytic Boundary Mechanism: In , an empirical boundary shift mstartmin(k)=(k−2)/12m_{\mathrm{start}}^{\min}(k) = (k-2)/12 was discovered for k≡2(mod⁡12)k \equiv 2 \pmod{12}, but its analytic origin remained conjectural. Can this boundary shift be proved unconditionally?

  3. Dual Infeasibility Certificates: When a candidate support fails to admit positive weights, can one provide an exact, finite certificate proving its impossibility, rather than relying on exhaustive search?

  4. Sub-Orbit Mechanics: Can breaking the monolithic Co⁡0\operatorname{Co}_0-shells into sub-orbits under subgroups like the monomial frame group 212:M242^{12} : M_{24} yield compressed spherical designs with fewer points than individual shells?

  5. Universal Spectral Geometry: What is the intrinsic convex geometry governing positive annihilation across discrete spectral systems? Can the same principles be applied to prime Hecke eigenbases?

In this paper, we resolve all five questions. We structure the paper into three interconnected parts:

The Master Theorem: Relative-Interior Spectral Cone Duality

Foundational Equivalence

Let VV be a finite-dimensional real vector space with dual space V*V^*, and let S={λ1,…,λm}⊂V*S = \{\lambda_1, \dots, \lambda_m\} \subset V^* be a finite collection of linear evaluation functionals. We define: U≔span⁡(S)⊆V*,K=cone⁡(S)≔{∑i=1mwiλi|wi≥0}⊆U,U⟂≔{F∈V:λ(F)=0∀λ∈U}={F∈V:λi(F)=0∀i∈{1,…,m}}.\begin{align} U &\coloneqq \operatorname{span}(S) \subseteq V^*, \\ K = \operatorname{cone}(S) &\coloneqq \left\{ \sum_{i=1}^m w_i \lambda_i \;\middle|\; w_i \ge 0 \right\} \subseteq U, \\ U^\perp &\coloneqq \{F \in V : \lambda(F) = 0 \quad \forall \lambda \in U\} = \{F \in V : \lambda_i(F) = 0 \quad \forall i \in \{1, \dots, m\}\}. \end{align} The relative interior relint⁡U(K)\operatorname{relint}_U(K) denotes the topological interior of the convex cone KK relative to its linear span UU.

Theorem 1 (Finite Spectral Cone Duality). Let VV be a finite-dimensional real vector space, let S={λ1,…,λm}⊂V*S = \{\lambda_1, \dots, \lambda_m\} \subset V^*, and let U=span⁡(S)U = \operatorname{span}(S) and K=cone⁡(S)K = \operatorname{cone}(S). The following three statements are equivalent:

  1. Strict-Positive Annihilation: There exist weights w1,…,wm∈ℝ>0w_1, \dots, w_m \in \mathbb R_{>0} such that: ∑i=1mwiλi=0in V*.\begin{equation} \sum_{i=1}^m w_i \lambda_i = 0 \quad \text{in } V^*. \end{equation}

  2. Relative-Interior Enclosure: The origin belongs to the relative interior of the cone: 0∈relint⁡U(K).\begin{equation} 0 \in \operatorname{relint}_U(K). \end{equation}

  3. Dual Triviality on Quotient: The cone of vectors evaluating non-negatively on all of SS coincides with the annihilator U⟂U^\perp: KS*≔{F∈V:λi(F)≥0∀i∈{1,…,m}}=U⟂.\begin{equation} K_S^* \coloneqq \big\{ F \in V : \lambda_i(F) \ge 0 \quad \forall i \in \{1, \dots, m\} \big\} = U^\perp. \end{equation}

Equivalently, on the quotient space V¯≔V/U⟂\overline{V} \coloneqq V / U^\perp, the non-negative dual cone is trivial: {[F]∈V¯:λi(F)≥0∀i∈{1,…,m}}={[0]}.\begin{equation} \big\{ [F] \in \overline{V} : \lambda_i(F) \ge 0 \quad \forall i \in \{1, \dots, m\} \big\} = \{[0]\}. \end{equation} In particular, if SS spans V*V^* (U=V*U = V^*, so U⟂={0}U^\perp = \{0\}), condition (iii) states that KS*={0}K_S^* = \{0\}.

Proof. (𝒊)⟹(𝒊𝒊)\mathbf{(i) \implies (ii)}: Suppose ∑i=1mwiλi=0\sum_{i=1}^m w_i \lambda_i = 0 with wi>0w_i > 0. For any generator λk∈S\lambda_k \in S, −λk=∑j≠kwjwkλj∈K-\lambda_k = \sum_{j \ne k} \frac{w_j}{w_k} \lambda_j \in K. Thus KK contains its own negation and is a linear subspace of V*V^*. In any finite-dimensional linear subspace, the relative interior of the subspace is the subspace itself; hence 0∈relint⁡U(K)0 \in \operatorname{relint}_U(K).

(𝒊𝒊)⟹(𝒊)\mathbf{(ii) \implies (i)}: Suppose 0∈relint⁡U(K)0 \in \operatorname{relint}_U(K). Then KK is a linear subspace, forcing K=UK = U. Thus −λi∈K-\lambda_i \in K for each i∈{1,…,m}i \in \{1, \dots, m\}. By definition of KK, there exist coefficients aij≥0a_{ij} \ge 0 such that −λi=∑j=1maijλj-\lambda_i = \sum_{j=1}^m a_{ij} \lambda_j. Summing λi+∑jaijλj=0\lambda_i + \sum_j a_{ij} \lambda_j = 0 over all ii gives: ∑j=1m(1+∑i=1maij)λj=0.\begin{equation} \sum_{j=1}^m \left( 1 + \sum_{i=1}^m a_{ij} \right) \lambda_j = 0. \end{equation} Setting wj≔1+∑i=1maijw_j \coloneqq 1 + \sum_{i=1}^m a_{ij}, we have wj≥1>0w_j \ge 1 > 0 for each jj, yielding a strictly positive relation.

(𝒊)⟹(𝒊𝒊𝒊)\mathbf{(i) \implies (iii)}: Clearly U⟂⊆KS*U^\perp \subseteq K_S^*. Conversely, let F∈KS*F \in K_S^* and let w∈ℝ>0mw \in \mathbb R_{>0}^m satisfy ∑i=1mwiλi=0\sum_{i=1}^m w_i \lambda_i = 0. Evaluating on FF gives 0=∑i=1mwiλi(F)0 = \sum_{i=1}^m w_i \lambda_i(F). Since wi>0w_i > 0 and λi(F)≥0\lambda_i(F) \ge 0 for each ii, every term must vanish identically: λi(F)=0\lambda_i(F) = 0 for all ii. Hence F∈U⟂F \in U^\perp, establishing KS*=U⟂K_S^* = U^\perp.

(𝒊𝒊𝒊)⟹(𝒊)\mathbf{(iii) \implies (i)}: Define the evaluation map ΦS:V→ℝm\Phi_S : V \to \mathbb R^m by ΦS(F)=(λ1(F),…,λm(F))T\Phi_S(F) = (\lambda_1(F), \dots, \lambda_m(F))^T. Condition (iii) asserts that the subspace W≔ΦS(V)⊂ℝmW \coloneqq \Phi_S(V) \subset \mathbb R^m intersects the non-negative orthant only at the origin: W∩(ℝ≥0m\{0})=∅W \cap (\mathbb R_{\ge 0}^m \setminus \{0\}) = \emptyset. By Gordan’s Theorem of the Alternative , there exists w∈ℝ>0mw \in \mathbb R_{>0}^m orthogonal to WW, which means ∑i=1mwiλi=0\sum_{i=1}^m w_i \lambda_i = 0 in V*V^*. ◻

Lemma 2 (Exact Rationality Lemma). Let A∈Mr×m(ℚ)A \in M_{r \times m}(\mathbb Q). If ker⁡(A)∩ℝ>0m≠∅\ker(A) \cap \mathbb R_{>0}^m \neq \emptyset, then ker⁡(A)∩ℚ>0m≠∅\ker(A) \cap \mathbb Q_{>0}^m \neq \emptyset.

Proof. Because AA has rational entries, Gaussian elimination produces a basis for ker⁡(A)\ker(A) consisting entirely of rational vectors. Consequently, the ℚ\mathbb Q-linear span of this basis, ker⁡(A)∩ℚm\ker(A) \cap \mathbb Q^m, is dense in the real subspace ker⁡(A)\ker(A) in the Euclidean topology. Let w∈ker⁡(A)∩ℝ>0mw \in \ker(A) \cap \mathbb R_{>0}^m. Since the positive orthant ℝ>0m\mathbb R_{>0}^m is open in ℝm\mathbb R^m, the intersection ker⁡(A)∩ℝ>0m\ker(A) \cap \mathbb R_{>0}^m is open in the relative topology of ker⁡(A)\ker(A) and contains ww. By density, there exists a rational point q∈ker⁡(A)∩ℚmq \in \ker(A) \cap \mathbb Q^m in this neighborhood. In particular, q∈ker⁡(A)q \in \ker(A) and qi>0q_i > 0 for every i∈{1,…,m}i \in \{1, \dots, m\}. ◻

Target Representation versus Null-Space Annihilation

Convex duality manifests in two distinct geometric modalities: 𝐓𝐚𝐫𝐠𝐞𝐭 𝐑𝐞𝐩𝐫𝐞𝐬𝐞𝐧𝐭𝐚𝐭𝐢𝐨𝐧 (𝐞.𝐠., 𝐐𝐮𝐚𝐝𝐫𝐚𝐭𝐮𝐫𝐞):∑i∈Swiλi=L≠0,wi>0,𝐍𝐮𝐥𝐥-𝐒𝐩𝐚𝐜𝐞 𝐀𝐧𝐧𝐢𝐡𝐢𝐥𝐚𝐭𝐢𝐨𝐧 (𝐞.𝐠., 𝐂𝐮𝐬𝐩 𝐂𝐮𝐛𝐚𝐭𝐮𝐫𝐞𝐬):∑i∈Swiλi=0,wi>0.\begin{align} \textbf{Target Representation (e.g., Quadrature):} & \quad \sum_{i \in S} w_i \lambda_i = L \ne 0, \quad w_i > 0, \\ \textbf{Null-Space Annihilation (e.g., Cusp Cubatures):} & \quad \sum_{i \in S} w_i \lambda_i = 0, \quad w_i > 0. \end{align}

Remark 3 (Contrast with Gaussian Quadrature). In classical Gaussian quadrature on [−1,1][-1, 1], the target functional L(P)=∫−11P(x)dμL(P) = \int_{-1}^1 P(x) \, d\mu is represented on polynomials ℝ[x]≤2n−1\mathbb R[x]_{\le 2n-1} by positive weights over nn nodes. Gaussian quadrature is not merely an instance of linear cone programming on a fixed node set: it optimizes the node positions x1,…,xnx_1, \dots, x_n via the roots of the nn-th orthogonal polynomial, using an infinite continuous dictionary 𝒟={ev⁡x:x∈[−1,1]}\mathcal{D} = \{\operatorname{ev}_x : x \in [-1, 1]\}. The dual witness certifying that n−1n-1 nodes fail is the square polynomial F(x)=∏j=1n−1(x−xj)2≥0F(x) = \prod_{j=1}^{n-1} (x - x_j)^2 \ge 0, which satisfies L(F)>0L(F) > 0. In modular cubatures, by contrast, the nodes are arithmetically fixed (shells SmS_m or prime Hecke operators TpT_p) and the target functional is the origin L=0L = 0.

Angular Separation and the Circular Gap Metric

Let U=span⁡(S)⊆V*U = \operatorname{span}(S) \subseteq V^* be equipped with an auxiliary inner product ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle, and let 𝕊(U)={u∈U:∥u∥=1}\mathbb{S}(U) = \{u \in U : \|u\| = 1\} denote the unit sphere in UU. For any non-zero functional λ∈S\{0}\lambda \in S \setminus \{0\}, denote its normalized vector by λ̂≔λ/∥λ∥∈𝕊(U)\widehat{\lambda} \coloneqq \lambda / \|\lambda\| \in \mathbb{S}(U).

Proposition 4 (Hemispherical Separation Criterion). Let S⊂V*\{0}S \subset V^* \setminus \{0\} be a finite set spanning U=span⁡(S)U = \operatorname{span}(S). Then: 0∈relint⁡U(cone⁡(S))⇔normalized vectors {λ̂:λ∈S} not contained in closed hemisphere of 𝕊(U).\begin{equation} \boxed{ 0 \in \operatorname{relint}_U\big(\operatorname{cone}(S)\big) \quad\Longleftrightarrow\quad \text{normalized vectors } \{\widehat{\lambda} : \lambda \in S\} \text{ not contained in closed hemisphere of } \mathbb{S}(U). } \end{equation} Equivalently, there exists no non-zero vector c∈U\{0}c \in U \setminus \{0\} such that ⟨c,λ⟩≥0\langle c, \lambda \rangle \ge 0 for all λ∈S\lambda \in S.

When dim⁡(U)=2\dim(U) = 2, this condition reduces to a cyclic angular gap:

Definition 5 (Largest Circular Gap). Let 𝒱={v1,…,vn}⊂ℝ2\{(0,0)}\mathcal{V} = \{v_1, \dots, v_n\} \subset \mathbb R^2 \setminus \{(0,0)\} be non-zero planar vectors with polar angles θi=atan2⁡(yi,xi)∈[−π,π)\theta_i = \operatorname{atan2}(y_i, x_i) \in [-\pi, \pi), sorted cyclically as θ(1)≤θ(2)≤…≤θ(n)<θ(1)+2π\theta_{(1)} \le \theta_{(2)} \le \dots \le \theta_{(n)} < \theta_{(1)} + 2\pi. The largest circular gap is: Δθmax(𝒱)≔max1≤i≤n(θ(i+1)−θ(i)),where θ(n+1)≔θ(1)+2π.\begin{equation} \Delta\theta_{\max}(\mathcal{V}) \coloneqq \max_{1 \le i \le n} \big( \theta_{(i+1)} - \theta_{(i)} \big), \quad \text{where } \theta_{(n+1)} \coloneqq \theta_{(1)} + 2\pi. \end{equation}

Proposition 6 (Planar Annihilation Criterion). Let 𝒱⊂ℝ2\{(0,0)}\mathcal{V} \subset \mathbb R^2 \setminus \{(0,0)\} be a finite set spanning ℝ2\mathbb R^2. Then: 0∈int⁡(cone⁡(𝒱))⇔Δθmax(𝒱)<π(180∘).\begin{equation} \boxed{ 0 \in \operatorname{int}\big(\operatorname{cone}(\mathcal{V})\big) \quad\Longleftrightarrow\quad \Delta\theta_{\max}(\mathcal{V}) < \pi \quad (180^\circ). } \end{equation}

Pruned Leech Cubatures and Point-Minimized Slim Designs

The Shell Minimization Problem

In , cubatures were constructed using contiguous shell windows Ss…Ss+CS_s \dots S_{s+C}. However, because shell cardinalities scale asymptotically as Nm=Θ(m11)N_m = \Theta(m^{11}) , the total point count: Ntotal=∑m∈𝒮Nm≈Nmmax≍mmax⁡11\begin{equation} N_{\mathrm{total}} = \sum_{m \in \mathcal{S}} N_m \approx N_{m_{\max}} \asymp m_{\max}^{11} \end{equation} is overwhelmingly dominated by the terminal shell SmmaxS_{m_{\max}}. Consequently, eliminating the highest shell by selectively pruning intermediate shells yields dramatic point-count reductions.

Theorem 7 (Point-Minimized Slim Designs for Strengths 29,33,4329, 33, 43). Exact rational positive spherical designs on Λ24\Lambda_{24} exist on the following non-consecutive pruned shell dictionaries:

  1. Strength 29 (k=28k=28, C=10C=10): The contiguous window in  skipped S3,S4S_3, S_4, terminating at S14S_{14} with 6.65×10146.65 \times 10^{14} points. By retaining S3S_3 and omitting S4S_4, we obtain: 𝒳29slim=S2∪{S3,S5,S6,S7,S8,S9,S10,S11,S12,S13}.\begin{equation} \mathcal{X}_{29}^{\mathrm{slim}} = S_2 \cup \{S_3, S_5, S_6, S_7, S_8, S_9, S_{10}, S_{11}, S_{12}, S_{13}\}. \end{equation} This terminates at S13S_{13}, requiring Ntotal=280,973,814,750,720N_{\mathrm{total}} = 280{,}973{,}814{,}750{,}720 points (2.81×10142.81 \times 10^{14}), achieving a 2.37×2.37\times point reduction and eliminating 3.84×10143.84 \times 10^{14} points.

  2. Strength 33 (k=32k=32, C=14C=14): The contiguous window S4…S18S_4 \dots S_{18} in  discarded the minimal shell S2S_2 and required 1.25×10161.25 \times 10^{16} points. Omitting the pair {S13,S14}\{S_{13}, S_{14}\} yields: 𝒳33slim={S2,…,S18}\{S13,S14}.\begin{equation} \mathcal{X}_{33}^{\mathrm{slim}} = \{S_2, \dots, S_{18}\} \setminus \{S_{13}, S_{14}\}. \end{equation} This restores the minimal shell S2S_2, requires 11,946,613,602,072,24011{,}946{,}613{,}602{,}072{,}240 points, and saves 5.54×10145.54 \times 10^{14} points.

  3. Strength 43 (k=42k=42, C=27C=27): While the contiguous window used S3…S30S_3 \dots S_{30}, omitting S3S_3 instead of S2S_2 yields 𝒳43slim=S2∪{S4…S30}\mathcal{X}_{43}^{\mathrm{slim}} = S_2 \cup \{S_4 \dots S_{30}\}, restoring S2S_2 and saving 16,576,56016{,}576{,}560 points.

Dual Farkas Infeasibility Certificates

To prove that a candidate shell dictionary 𝒮′\mathcal{S}' cannot admit positive weights, it suffices by Theorem 1 to construct a single rational dual witness vector c*∈ℚCc^* \in \mathbb Q^C such that: A𝒮′Tc*≥0withA𝒮′Tc*≠0.\begin{equation} A_{\mathcal{S}'}^T c^* \ge 0 \quad \text{with} \quad A_{\mathcal{S}'}^T c^* \neq 0. \end{equation} For any hypothetical positive weight vector W′∈ℝ>0|𝒮′|W' \in \mathbb R_{>0}^{|\mathcal{S}'|}, 0=c*T(A𝒮′W′)=(A𝒮′Tc*)TW′>00 = c^{*T} (A_{\mathcal{S}'} W') = (A_{\mathcal{S}'}^T c^*)^T W' > 0, an immediate contradiction.

Proposition 8 (Explicit Dual Infeasibility Witness at k=20k=20). For degree k=20k=20 (C=4C=4), the contiguous window 𝒮′={S2,S3,S4,S5,S6}\mathcal{S}' = \{S_2, S_3, S_4, S_5, S_6\} fails to admit positive weights. The unique dual vector solving A𝒮′Tc=(1,0,0,0,μ)TA_{\mathcal{S}'}^T c = (1, 0, 0, 0, \mu)^T is: c*=(10,092,5445,265,223,281,1529,135,296,747,00812,285,1,153,433,60016,443)T∈ℚ4.\begin{equation} c^* = \left( \frac{10{,}092{,}544}{5{,}265}, \; \frac{223{,}281{,}152}{9{,}135}, \; \frac{296{,}747{,}008}{12{,}285}, \; \frac{1{,}153{,}433{,}600}{16{,}443} \right)^T \in \mathbb Q^4. \end{equation} Evaluating A𝒮′Tc*A_{\mathcal{S}'}^T c^* across {S2,…,S6}\{S_2, \dots, S_6\} yields: A𝒮′Tc*=(1,0,0,0,145,78019,683)T≥0,A𝒮′Tc*≠0.\begin{equation} A_{\mathcal{S}'}^T c^* = \left( 1, \; 0, \; 0, \; 0, \; \frac{145{,}780}{19{,}683} \right)^T \ge 0, \quad A_{\mathcal{S}'}^T c^* \neq 0. \end{equation} This certificate proves unconditionally that {S2,…,S6}\{S_2, \dots, S_6\} cannot support a positive cubature.

The Analytic Serre Vanishing Theorem and Modular Separation

Analytic Vanishing Bound

In , FLINT audits revealed that for k∈{62,74,86,98,110}k \in \{62, 74, 86, 98, 110\}, the minimal positive starting shell satisfies mstartmin(k)=(k−2)/12m_{\mathrm{start}}^{\min}(k) = (k-2)/12. We now prove that this boundary shift is an exact analytic theorem governed by Serre’s theorem.

Theorem 9 (Analytic Bound on Maximal Cusp Vanishing Order). Let k≥26k \ge 26 be an even integer with k≡2(mod⁡12)k \equiv 2 \pmod{12}. Then:

  1. The maximal order of vanishing at the cusp across Sk+12(2)(SL⁡2(ℤ))S_{k+12}^{(2)}(\operatorname{SL}_2(\mathbb Z)) is strictly: max0≠F∈Sk+12(2)(SL⁡2(ℤ))ord⁡q=0(F)=k−212.\begin{equation} \max_{0 \neq F \in S_{k+12}^{(2)}(\operatorname{SL}_2(\mathbb Z))} \operatorname{ord}_{q=0}(F) = \frac{k - 2}{12}. \end{equation}

  2. The subspace of forms achieving this maximal cusp vanishing order is one-dimensional: {F∈Sk+12(2)(SL⁡2(ℤ)):ord⁡q=0(F)≥k−212}=ℂ⋅(Δ(k−2)/12E42E6).\begin{equation} \left\{ F \in S_{k+12}^{(2)}(\operatorname{SL}_2(\mathbb Z)) : \operatorname{ord}_{q=0}(F) \ge \frac{k - 2}{12} \right\} = \mathbb C\cdot \left( \Delta^{(k-2)/12} E_4^2 E_6 \right). \end{equation}

Proof. Write k=12m+2k = 12m + 2 with m≥2m \ge 2. By Venkov’s isomorphism (Theorem [thm:harmonic-isomorphism] in ), Sk+12(2)(SL⁡2(ℤ))≅Δ2Mw(SL⁡2(ℤ))S_{k+12}^{(2)}(\operatorname{SL}_2(\mathbb Z)) \cong \Delta^2 M_{w}(\operatorname{SL}_2(\mathbb Z)), where w=(k+12)−24=k−12=12(m−1)+2w = (k+12) - 24 = k - 12 = 12(m - 1) + 2. Suppose a non-zero form g∈Mw(SL⁡2(ℤ))\{0}g \in M_w(\operatorname{SL}_2(\mathbb Z)) \setminus \{0\} were divisible by Δm−1\Delta^{m-1}. The quotient form h≔g/Δm−1h \coloneqq g / \Delta^{m-1} would be holomorphic on ℍ\mathbb H and at ∞\infty, transforming with modular weight: w−12(m−1)=[12(m−1)+2]−12(m−1)=2.\begin{equation} w - 12(m - 1) = [12(m - 1) + 2] - 12(m - 1) = 2. \end{equation} By Serre’s theorem, there are no non-trivial holomorphic modular forms of weight 2 for SL⁡2(ℤ)\operatorname{SL}_2(\mathbb Z): M2(SL⁡2(ℤ))={0}.\begin{equation} M_2(\operatorname{SL}_2(\mathbb Z)) = \{0\}. \end{equation} Thus h≡0h \equiv 0, forcing g≡0g \equiv 0, a contradiction. Therefore, ord⁡q(g)≤m−2\operatorname{ord}_q(g) \le m - 2 for all g∈Mw(SL⁡2(ℤ))\{0}g \in M_w(\operatorname{SL}_2(\mathbb Z)) \setminus \{0\}. Conversely, g0=Δm−2E42E6∈Mw(SL⁡2(ℤ))g_0 = \Delta^{m-2} E_4^2 E_6 \in M_w(\operatorname{SL}_2(\mathbb Z)) has ord⁡q(g0)=m−2\operatorname{ord}_q(g_0) = m - 2 because Δ\Delta has a simple zero at the cusp and E4(0)=E6(0)=1E_4(0) = E_6(0) = 1. Multiplying by Δ2\Delta^2 yields ord⁡q=0(ΔmE42E6)=m=(k−2)/12\operatorname{ord}_{q=0}(\Delta^m E_4^2 E_6) = m = (k-2)/12. ◻

The Modular Separation Conjecture

Theorem 9 proves that the first (k−2)/12(k-2)/12 Fourier evaluations are linearly independent and span V*V^*. However, positive cubatures additionally require enclosing the origin.

Conjecture 10 (The Modular Separation Conjecture). Let Vk=Δ2Mk−12(SL⁡2(ℤ))V_k = \Delta^2 M_{k-12}(\operatorname{SL}_2(\mathbb Z)) with dimension C(k)=dim⁡VkC(k) = \dim V_k. The minimum integer R(Vk)R(V_k) such that the contiguous window {2,3,…,R(Vk)}\{2, 3, \dots, R(V_k)\} admits a strictly positive annihilator is given unconditionally by: R(Vk)=C(k)+2+𝟏{k≡2(mod⁡12)}.\begin{equation} R(V_k) = C(k) + 2 + \mathbf{1}_{\{k \equiv 2 \pmod{12}\}}. \end{equation}

This conjecture has been verified computationally through weight 122 across all modular branches.

Monomial Frame Sub-Orbit Mechanics on Λ24\Lambda_{24}

In , Shell 8 was shown to split 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 orbits, exhibiting a canonical harmonic sign inversion (Ψ12(𝒪8A)/Ψ12(S2)=+4,096\Psi_{12}(\mathcal{O}_{8_A})/\Psi_{12}(S_2) = +4{,}096 vs. Ψ12(vB)/Ψ12(S2)=−40/23\Psi_{12}(v_B)/\Psi_{12}(S_2) = -40/23).

We now analyze the sub-orbit structure of the minimal shell S2S_2 under the monomial frame group N=212:M24⊂Co⁡0N = 2^{12} : M_{24} \subset \operatorname{Co}_0.

Shell 2 Sub-Orbit Decomposition

Under N=212:M24N = 2^{12} : M_{24}, the 196,560196{,}560 vectors of S2S_2 decompose into three orbits:

Proposition 11 (Invariant Harmonics under 212:M242^{12} : M_{24}). Let G=212:M24G = 2^{12} : M_{24}. The space of GG-invariant harmonic polynomials satisfies: dim⁡(Harm⁡k(ℝ24))G={0,k∈{1,2,3,5},1,k=4.\begin{equation} \dim (\operatorname{Harm}_k(\mathbb R^{24}))^G = \begin{cases} 0, & k \in \{1, 2, 3, 5\}, \\ 1, & k = 4. \end{cases} \end{equation} On the sphere S23S^{23} (where ∥x∥2\|x\|^2 is constant), the 1-dimensional space of degree-44 harmonic invariants is spanned by the curvature polynomial: 𝒦4(x)≔26∑i=124xi4−3∥x∥4.\begin{equation} \mathcal{K}_4(x) \coloneqq 26 \sum_{i=1}^{24} x_i^4 - 3 \|x\|^4. \end{equation}

Proof. Central inversion −I∈212-I \in 2^{12} eliminates all odd degrees. For k=2k=2, 2122^{12}-invariance forces P(x)=∑cixi2P(x) = \sum c_i x_i^2, and M24M_{24}-transitivity forces ci=cc_i = c, so P(x)∝∥x∥2P(x) \propto \|x\|^2. Since Δ(∥x∥2)=48≠0\Delta(\|x\|^2) = 48 \neq 0, no non-trivial harmonic invariant exists. For k=4k=4, 2-transitivity of M24M_{24} restricts polynomial invariants to span⁡{∑xi4,∑i<jxi2xj2}\operatorname{span}\{\sum x_i^4, \sum_{i<j} x_i^2 x_j^2\}. Expressing ∑i<jxi2xj2=(∥x∥4−∑xi4)/2\sum_{i<j} x_i^2 x_j^2 = (\|x\|^4 - \sum x_i^4)/2, the condition ΔP=0\Delta P = 0 in dimension d=24d=24 yields: Δ(a∑xi4+b∥x∥4)=(12a+46(−a))∑xi2=0,\begin{equation} \Delta\left( a \sum x_i^4 + b \|x\|^4 \right) = \left( 12a + 46(-a) \right) \sum x_i^2 = 0, \end{equation} uniquely determining the harmonic ratio 𝒦4(x)=26∑xi4−3∥x∥4\mathcal{K}_4(x) = 26 \sum x_i^4 - 3\|x\|^4. ◻

Sub-Orbit Curvature Equilibrium and Compressed 5-Designs

Theorem 12 (Orbit Curvature Equilibrium and Compressed 5-Designs). Let ℳ4(𝒪)≔∑x∈𝒪𝒦4(x)\mathcal{M}_4(\mathcal{O}) \coloneqq \sum_{x \in \mathcal{O}} \mathcal{K}_4(x) denote the total integrated degree-44 harmonic moment of an orbit. Normalized by the common divisor κ0≔35,328\kappa_0 \coloneqq 35{,}328, the orbit moments satisfy: ℳ4(𝒪A)κ0=+20,ℳ4(𝒪B)κ0=+44,ℳ4(𝒪C)κ0=−64.\begin{equation} \frac{\mathcal{M}_4(\mathcal{O}_A)}{\kappa_0} = +20, \qquad \frac{\mathcal{M}_4(\mathcal{O}_B)}{\kappa_0} = +44, \qquad \frac{\mathcal{M}_4(\mathcal{O}_C)}{\kappa_0} = -64. \end{equation} The identity 20+44−64=020 + 44 - 64 = 0 reflects the fact that uniform weights on S2S_2 form an 11-design. Furthermore, because dim⁡(Harm⁡k(ℝ24))G=0\dim (\operatorname{Harm}_k(\mathbb R^{24}))^G = 0 for k∈{1,2,3,5}k \in \{1, 2, 3, 5\}, any two-orbit combination satisfying w1ℳ4(𝒪1)+w2ℳ4(𝒪2)=0w_1 \mathcal{M}_4(\mathcal{O}_1) + w_2 \mathcal{M}_4(\mathcal{O}_2) = 0 forms an exact spherical 55-design on S23S^{23}:

  1. Compressed 5-Design on 𝒪A∪𝒪C\mathcal{O}_A \cup \mathcal{O}_C (99,40899{,}408 points): Assigning per-vector weights with ratio: wAwC=6420=165\begin{equation} \frac{w_A}{w_C} = \frac{64}{20} = \frac{16}{5} \end{equation} yields an exact spherical 55-design on |𝒪A|+|𝒪C|=1,104+98,304=𝟗𝟗,𝟒𝟎𝟖|\mathcal{O}_A| + |\mathcal{O}_C| = 1{,}104 + 98{,}304 = \mathbf{99{,}408} points.

  2. Compressed 5-Design on 𝒪B∪𝒪C\mathcal{O}_B \cup \mathcal{O}_C (195,456195{,}456 points): Assigning per-vector weights with ratio: wBwC=6444=1611\begin{equation} \frac{w_B}{w_C} = \frac{64}{44} = \frac{16}{11} \end{equation} yields an exact spherical 55-design on |𝒪B|+|𝒪C|=97,152+98,304=𝟏𝟗𝟓,𝟒𝟓𝟔|\mathcal{O}_B| + |\mathcal{O}_C| = 97{,}152 + 98{,}304 = \mathbf{195{,}456} points.

The Hecke Prime Spectral Cone: S24,S28,S32,S36S_{24}, S_{28}, S_{32}, S_{36}

Hecke Prime Setup and Sign Alternation

Let Sk(SL⁡2(ℤ))S_k(\operatorname{SL}_2(\mathbb Z)) be the space of cusp forms of weight kk with Hecke eigenbasis {f1,…,fd}\{f_1, \dots, f_d\}. Under the unitary Deligne normalization: λ̃fj(p)≔λfj(p)2p(k−1)/2∈[−1,1],\begin{equation} \widetilde{\lambda}_{f_j}(p) \coloneqq \frac{\lambda_{f_j}(p)}{2 \, p^{(k-1)/2}} \in [-1, 1], \end{equation} each prime pp defines a spectral evaluation vector vp=(λ̃f1(p),…,λ̃fd(p))∈[−1,1]d⊂ℝdv_p = (\widetilde{\lambda}_{f_1}(p), \dots, \widetilde{\lambda}_{f_d}(p)) \in [-1, 1]^d \subset \mathbb R^d.

Proposition 13 (Prime Enclosure and Dual Sign Alternation). Let 𝒫\mathcal{P} be a finite set of primes such that {vp:p∈𝒫}\{v_p : p \in \mathcal{P}\} spans ℝd\mathbb R^d.

  1. The origin belongs to int⁡(cone⁡{vp:p∈𝒫})\operatorname{int}(\operatorname{cone}\{v_p : p \in \mathcal{P}\}) if and only if there exist weights wp>0w_p > 0 such that ∑p∈𝒫wpvp=0\sum_{p \in \mathcal{P}} w_p v_p = 0.

  2. If 0∈int⁡(cone⁡{vp:p∈𝒫})0 \in \operatorname{int}(\operatorname{cone}\{v_p : p \in \mathcal{P}\}), then every non-zero cusp form F∈Sk(SL⁡2(ℤ))\{0}F \in S_k(\operatorname{SL}_2(\mathbb Z)) \setminus \{0\} must strictly change sign on 𝒫\mathcal{P}: ∀F∈Sk(SL⁡2(ℤ))\{0},∃p,q∈𝒫such thatF(p)>0andF(q)<0.\begin{equation} \forall F \in S_k(\operatorname{SL}_2(\mathbb Z)) \setminus \{0\}, \quad \exists p, q \in \mathcal{P} \quad \text{such that} \quad F(p) > 0 \quad \text{and} \quad F(q) < 0. \end{equation}

Exact Computational Verification across Hecke Cusp Spaces

We define the prime spectral enclosure threshold p*(Sk)p^*(S_k) as the minimal prime p*p^* such that 0∈int⁡(cone⁡{vp:p≤p*})0 \in \operatorname{int}(\operatorname{cone}\{v_p : p \le p^*\}).

Exact prime spectral enclosure thresholds and gap collapse across Hecke cusp spaces.
Space d=dim⁡Skd = \dim S_k Threshold p*p^* Required Pre-Threshold Gap Δθmax\Delta\theta_{\max} Enclosing Metric
S24S_{24} 2 𝒑*=𝟕\mathbf{p^* = 7} 4 primes Δθmax({2,3,5})≈214.82∘\Delta\theta_{\max}(\{2,3,5\}) \approx 214.82^\circ Δθmax({2,3,5,7})≈178.16∘\Delta\theta_{\max}(\{2,3,5,7\}) \approx 178.16^\circ
S28S_{28} 2 𝒑*=𝟏𝟏\mathbf{p^* = 11} 5 primes Δθmax(p≤7)≈269.47∘\Delta\theta_{\max}(p \le 7) \approx 269.47^\circ Δθmax(p≤11)≈167.88∘\Delta\theta_{\max}(p \le 11) \approx 167.88^\circ
S32S_{32} 2 𝒑*=𝟏𝟏\mathbf{p^* = 11} 5 primes Δθmax(p≤7)≈193.68∘\Delta\theta_{\max}(p \le 7) \approx 193.68^\circ Δθmax(p≤11)≈103.35∘\Delta\theta_{\max}(p \le 11) \approx 103.35^\circ
S36S_{36} 3 𝒑*=𝟏𝟗\mathbf{p^* = 19} 8 primes Contained in hemisphere Hemisphere breached

Proposition 14 (Certified Computational Verification for S24,S28,S32,S36S_{24}, S_{28}, S_{32}, S_{36}). The enclosure thresholds and separating certificates for Sk(SL⁡2(ℤ))S_k(\operatorname{SL}_2(\mathbb Z)) are certified as follows:

  1. The Space S24S_{24} (d=2d=2): The Hecke operator T2T_2 has exact characteristic polynomial P24(X)=X2−1080X−20468736=0P_{24}(X) = X^2 - 1080X - 20468736 = 0 with roots 540±12144169540 \pm 12\sqrt{144169}. The Deligne normalizer is Dp=2p23/2D_p = 2 p^{23/2}. The evaluation vectors are v2≈(−0.6934,+0.8798)v_2 \approx (-0.6934, +0.8798) (128.24∘128.24^\circ), v3≈(+0.6330,−0.0798)v_3 \approx (+0.6330, -0.0798) (352.81∘352.81^\circ), v5≈(+0.4811,−0.1465)v_5 \approx (+0.4811, -0.1465) (343.06∘343.06^\circ), and v7≈(+0.3643,−0.4943)v_7 \approx (+0.3643, -0.4943) (306.39∘306.39^\circ).

    • Separation at p≤5p \le 5: The vectors {v2,v3,v5}\{v_2, v_3, v_5\} leave an open gap of Δθmax≈214.82∘>180∘\Delta\theta_{\max} \approx 214.82^\circ > 180^\circ. The rational dual vector c=(2,3)Tc = (2, 3)^T strictly separates the cone from the origin: ⟨c,v2⟩≈+1.2526>0,⟨c,v3⟩≈+1.0266>0,⟨c,v5⟩≈+0.5227>0.\begin{equation} \langle c, v_2 \rangle \approx +1.2526 > 0, \quad \langle c, v_3 \rangle \approx +1.0266 > 0, \quad \langle c, v_5 \rangle \approx +0.5227 > 0. \end{equation}

    • Enclosure at p*=7p^* = 7: Since ⟨c,v7⟩≈−0.7543<0\langle c, v_7 \rangle \approx -0.7543 < 0, the gap collapses to θ7−θ2≈178.16∘<180∘\theta_7 - \theta_2 \approx 178.16^\circ < 180^\circ. The certified positive weights are: 0.35050v2+0.01662v3+0.01659v5+0.61629v7=𝟎.\begin{equation} 0.35050 \, v_2 + 0.01662 \, v_3 + 0.01659 \, v_5 + 0.61629 \, v_7 = \mathbf{0}. \end{equation}

  2. The Space S28S_{28} (d=2d=2): The Hecke operator T2T_2 has characteristic polynomial P28(X)=X2+8280X−195250176=0P_{28}(X) = X^2 + 8280X - 195250176 = 0. For p≤7p \le 7, the vectors leave an open gap of Δθmax≈269.47∘\Delta\theta_{\max} \approx 269.47^\circ. At p=11p = 11, v11≈(−0.2993,+0.9027)v_{11} \approx (-0.2993, +0.9027) lands at +108.35∘+108.35^\circ, reducing the gap to 167.88∘<180∘167.88^\circ < 180^\circ.

  3. The Space S32S_{32} (d=2d=2): The Hecke operator T2T_2 has characteristic polynomial P32(X)=X2−39960X−2235350016=0P_{32}(X) = X^2 - 39960X - 2235350016 = 0. For p≤7p \le 7, the gap is 193.68∘>180∘193.68^\circ > 180^\circ. At p=11p = 11, v11≈(−0.1942,−0.0867)v_{11} \approx (-0.1942, -0.0867) lands at −155.93∘-155.93^\circ, closing the gap to 103.35∘<180∘103.35^\circ < 180^\circ.

  4. The Space S36S_{36} (d=3d=3): The Hecke operator T2T_2 has characteristic polynomial P36(X)=X3−139656X2−59208339456X−1467625047588864=0P_{36}(X) = X^3 - 139656X^2 - 59208339456X - 1467625047588864 = 0. For all primes p≤17p \le 17 (7 primes), the normalized vectors vp∈𝕊2v_p \in \mathbb{S}^2 remain confined within an open hemisphere. At p=19p = 19, v19≈(−0.7513,−0.0074,+0.0825)v_{19} \approx (-0.7513, -0.0074, +0.0825) breaches the separating hyperplane, establishing 0∈int⁡(cone⁡{v2,…,v19})0 \in \operatorname{int}(\operatorname{cone}\{v_2, \dots, v_{19}\}).

Primal-Dual Complexity Pairing

Definition 15 (Primal and Projective Dual Complexity). Let (V,Λ,ω)(V, \Lambda, \omega) be an arithmetic spectral evaluation system with cost function ω:ℐ→ℝ>0\omega : \mathcal{I} \to \mathbb R_{>0}.

  1. The Primal Spectral Complexity is: ℭ(V;ω)≔inf⁡{∑i∈Sω(i)|S⊂ℐ finite, 0∈relintcone{λi:i∈S}}.\begin{equation} \mathfrak{C}(V; \omega) \coloneqq \inf \left\{ \sum_{i \in S} \omega(i) \;\middle|\; S \subset \mathcal{I} \text{ finite, } 0 \in \operatorname{relint}\operatorname{cone}\{\lambda_i : i \in S\} \right\}. \end{equation}

  2. For a candidate support SS and an admissible reference functional ρ∈V*\U⟂\rho \in V^* \setminus U^\perp, the Projective Dual Complexity is: 𝔇ρ(S;V)≔inf⁡{∥F∥V|F∈V,ρ(F)=1,λi(F)≥0∀i∈S}.\begin{equation} \mathfrak{D}_\rho(S; V) \coloneqq \inf \Big\{ \|F\|_V \;\Big|\; F \in V, \; \rho(F) = 1, \; \lambda_i(F) \ge 0 \quad \forall i \in S \Big\}. \end{equation}

When SS strictly encloses the origin and spans V*V^*, KS*={0}K_S^* = \{0\}; hence the feasible set for 𝔇ρ\mathfrak{D}_\rho is empty and the dual complexity undergoes an infinite jump: 0∈int⁡(cone⁡(S))⇔𝔇ρ(S;V)=+∞.\begin{equation} 0 \in \operatorname{int}\big(\operatorname{cone}(S)\big) \quad \Longleftrightarrow \quad \mathfrak{D}_\rho(S; V) = +\infty. \end{equation}

The Finite Certification Problem and the Infinite Frontier

Let EE be an infinite-dimensional topological vector space with an exhaustive filtration V1⊂V2⊂…⊂EV_1 \subset V_2 \subset \dots \subset E (⋃NVN¯=E\overline{\bigcup_N V_N} = E). The passage from finite polyhedral duality to global analytic positivity decomposes into three distinct regimes:

  1. Regime I: Finite Existence (Per Truncation). On each finite-dimensional subspace VNV_N, Theorem 1 holds unconditionally: either positive weights exist to represent or annihilate a functional, or an explicit dual witness F∈VNF \in V_N separates the cone from the target.

  2. Regime II: Effective Finite Bounds. Obtaining an explicit, computable bound B(N)B(N) such that the support can be restricted to {i∈ℐ:ω(i)≤B(N)}\{i \in \mathcal{I} : \omega(i) \le B(N)\}. For modular forms, this is governed by the Sturm bound and the valence formula.

  3. Regime III: Uniform Limiting Stability (The Analytic Frontier). Deducing an infinite-dimensional representation or positivity property (L(f)≥0L(f) \ge 0 for all f∈Ef \in E) from finite truncations.

Remark 16 (The Limits of Finite Cone Duality). We emphasize that: Finite-dimensional certificates do not, by themselves, imply a limiting certificate in E*.\begin{equation*} \boxed{ \text{Finite-dimensional certificates do not, by themselves, imply a limiting certificate in } E^*. } \end{equation*} To pass rigorously from finite certificates (w(N),SN)(w^{(N)}, S_N) to a global continuous certificate on EE, one must establish five independent functional-analytic stability requirements:

  1. Uniform Boundedness: The total variation of the discrete measures must remain uniformly bounded: sup⁡N∥w(N)∥ℓ1<∞\sup_{N} \|w^{(N)}\|_{\ell^1} < \infty.

  2. Topology of Convergence: The discrete sequence ∑i∈SNwi(N)λi\sum_{i \in S_N} w_i^{(N)} \lambda_i must converge in a specified weak-** topology on E*E^*.

  3. Compatibility of Truncations: The finite systems must be asymptotically consistent with the filtration VNV_N.

  4. Closedness of the Dual Positivity Cone: The cone of non-negative test functions must be closed under the designated topology.

  5. Normalization Control: The target functional must not degenerate under the limiting process.

The Five-Level Epistemological Hierarchy

𝐋𝐄𝐕𝐄𝐋 𝟎𝐄𝐱𝐚𝐜𝐭 𝐀𝐫𝐢𝐭𝐡𝐦𝐞𝐭𝐢𝐜 𝐋𝐚𝐛𝐨𝐫𝐚𝐭𝐨𝐫𝐲Integer q-expansions, Leech shell counts, Hecke charpolys, rational systems⇓𝐋𝐄𝐕𝐄𝐋 𝟏𝐅𝐢𝐧𝐢𝐭𝐞 𝐒𝐩𝐞𝐜𝐭𝐫𝐚𝐥 𝐂𝐨𝐧𝐞 𝐃𝐮𝐚𝐥𝐢𝐭𝐲 (𝐓𝐡𝐞 𝐌𝐚𝐬𝐭𝐞𝐫 𝐓𝐡𝐞𝐨𝐫𝐞𝐦)0∈relint⁡U(K)⇔KS*=U⟂(Proven Unconditional)⇓𝐋𝐄𝐕𝐄𝐋 𝟐𝐈𝐧𝐭𝐫𝐢𝐧𝐬𝐢𝐜 𝐀𝐫𝐢𝐭𝐡𝐦𝐞𝐭𝐢𝐜 𝐈𝐧𝐯𝐚𝐫𝐢𝐚𝐧𝐭𝐬Pruned cubatures, Serre vanishing (k−2)/12, prime circular gaps Δθmax<π⇓𝐋𝐄𝐕𝐄𝐋 𝟑𝐄𝐟𝐟𝐞𝐜𝐭𝐢𝐯𝐞 𝐀𝐫𝐢𝐭𝐡𝐦𝐞𝐭𝐢𝐜 𝐁𝐨𝐮𝐧𝐝𝐬Computable cutoffs B(N) via Sturm bounds, Deligne bounds, and Chebotarev⇓𝐋𝐄𝐕𝐄𝐋 𝟒𝐓𝐡𝐞 𝐈𝐧𝐟𝐢𝐧𝐢𝐭𝐞-𝐃𝐢𝐦𝐞𝐧𝐬𝐢𝐨𝐧𝐚𝐥 𝐅𝐫𝐨𝐧𝐭𝐢𝐞𝐫Uniform limiting stability supN∥w(N)∥1<∞ on trace formulas and L-functions\begin{equation*} \boxed{ \begin{array}{cl} \textbf{LEVEL 0} & \textbf{Exact Arithmetic Laboratory} \\ & \text{Integer $q$-expansions, Leech shell counts, Hecke charpolys, rational systems} \\[2mm] & \qquad\qquad \Downarrow \\[2mm] \textbf{LEVEL 1} & \textbf{Finite Spectral Cone Duality (The Master Theorem)} \\ & 0 \in \operatorname{relint}_U(K) \iff K_S^* = U^\perp \quad \text{(Proven Unconditional)} \\[2mm] & \qquad\qquad \Downarrow \\[2mm] \textbf{LEVEL 2} & \textbf{Intrinsic Arithmetic Invariants} \\ & \text{Pruned cubatures, Serre vanishing $(k-2)/12$, prime circular gaps } \Delta\theta_{\max} < \pi \\[2mm] & \qquad\qquad \Downarrow \\[2mm] \textbf{LEVEL 3} & \textbf{Effective Arithmetic Bounds} \\ & \text{Computable cutoffs } B(N) \text{ via Sturm bounds, Deligne bounds, and Chebotarev} \\[2mm] & \qquad\qquad \Downarrow \\[2mm] \textbf{LEVEL 4} & \textbf{The Infinite-Dimensional Frontier} \\ & \text{Uniform limiting stability } \sup_N \|w^{(N)}\|_1 < \infty \text{ on trace formulas and $L$-functions} \end{array} } \end{equation*}

Conclusion

We have developed the unified theory of Spectral Cone Duality, bridging convex geometry, discrete lattice designs, and arithmetic spectral systems. Building directly upon Part I , we have shown that non-consecutive pruned shell dictionaries achieve significant point-count compressions, that the boundary shift mstartmin(k)=(k−2)/12m_{\mathrm{start}}^{\min}(k) = (k-2)/12 is an exact analytic theorem governed by M2(SL⁡2(ℤ))={0}M_2(\operatorname{SL}_2(\mathbb Z)) = \{0\}, and that 212:M242^{12}:M_{24} sub-orbits yield compressed spherical 5-designs on only 99,40899{,}408 points. Transposed to Hecke cusp spaces, spectral cone duality provides exact certificates of prime enclosure and deterministic cusp form sign alternation. Finally, we have formalized the functional-analytic requirements governing the transition to infinite-dimensional positivity. Audit file is available at:

https://srfp311t1.com/spectral_cone_duality.py

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