# Hyperbolic-Modular Neural Operators: Noncommutative Spectral Geometry and Asymptotic Analysis in PDE Learning

Rômulo Damasclin Chaves Dos Santos, Jorge Henrique de Oliveira Sales

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Rômulo Damasclin Chaves Dos Santos, Jorge Henrique de Oliveira Sales. Hyperbolic-Modular Neural Operators: Noncommutative Spectral Geometry and Asymptotic Analysis in PDE Learning. 2025. [hal-05304474]

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# Hyperbolic-Modular Neural Operators: Noncommutative Spectral Geometry and Asymptotic Analysis in PDE Learning

Rômulo Damasclín Chaves dos Santos
Santa Cruz State University
rdcsantos@uesc.br

Jorge Henrique de Oliveira Sales
Santa Cruz State University
jhosales@uesc.br

October 8, 2025

## Abstract

We introduce a unified theoretical framework for *Hyperbolic-Modular Neural Operators* (HMNOs), integrating principles from noncommutative spectral geometry, anisotropic harmonic analysis, and operator learning for partial differential equations (PDEs). The proposed formulation incorporates hyperbolic symmetry and modular spectral damping into neural operator architectures, providing a curvature-aware and spectrally stable approximation mechanism on non-Euclidean domains. Within this setting, we derive a rigorous *Voronovskaya-type asymptotic expansion* that quantifies the bias-variance decomposition of modularly damped operator families in anisotropic Besov and Sobolev scales. The expansion reveals an intrinsic connection between kernel moment asymptotics and spectral-geometric invariants, linking learning dynamics with the Chern character and Dixmier traces of the associated spectral triples. Our results establish minimax-optimal convergence rates and open a path toward geometrically consistent operator learning in curved and arithmetic spaces.

**Keywords:** Neural operators, Anisotropic Besov spaces, Noncommutative Chern character, Hyperbolic symmetry, Ramanujan modularity.

MSC: 46E35. 41A25. 35Q68. 42B35. 68T07. 58J20. 58B34. 65D15. 81T75.

## 1 Introduction

Neural operator learning aims to approximate mappings between infinite-dimensional function spaces, typically those induced by partial differential equations (PDEs). Classical frameworks such as the Fourier Neural Operator (FNO) and DeepONet have achieved remarkable success on Euclidean domains, but they fail to capture anisotropic smoothness, curvature, and modular symmetries essential for describing non-Euclidean PDE dynamics.
The field's trajectory began with foundational architectures like Fourier Neural Operators (FNO) [1], which introduced mesh-independent spectral representations, and DeepONet [2], which established universal approximation theory for nonlinear operators through decoupled branch-trunk networks. While these methods demonstrated remarkable success on Euclidean domains, their geometric limitations soon became apparent, sparking a new wave of research into geometrically adaptive architectures.

The pursuit of geometric expressivity led to significant innovations: CORAL [3] integrated neural fields with coordinate-aware representations for non-Euclidean domains, while Geo-FNO [4] learned spectral-aligned deformations for complex geometries. This evolution continued with Neural Manifold Operators [5], which intrinsically embed Riemannian geometric priors, and probabilistic frameworks like NOGaP [6] that incorporate uncertainty quantification particularly crucial for ill-posed inverse problems.

Concurrent architectural refinements have further expanded the operator learning landscape. Derivative-informed approaches [7] enabled PDE-constrained optimization under uncertainty, while neural inverse operators [8] tackled high-dimensional inverse problems. Fourier architectures found specialized applications in wave propagation [9] and structure-preserving modeling [10], with ensemble methods [11] and rigorous error analysis [12] enhancing robustness and theoretical understanding. Architectural variants like HyperFNO [13], Factorized FNO [14], and Invertible FNO [15] pushed boundaries in expressivity, efficiency, and bidirectional solvability.

Despite these advances, a fundamental challenge persists: existing architectures struggle to capture mixed anisotropic smoothness, modular arithmetic structure, and hyperbolic curvature effects—precisely the features that govern phenomena like spectral asymmetry, directional transport on curved domains, and modular invariance. Classical approximation theory underscores this difficulty: anisotropic Besov spaces [16, 23] present inherent challenges for uniform approximation, while decoupling theory [17] and mixed smoothness analysis [18] reveal fundamental limitations in harmonic analysis [19, 20, 21, 22].

Addressing these limitations requires a fundamentally new approach. We introduce *Hypermodular Neural Operators with Hyperbolic Symmetry (ONHSH)*—a mathematically grounded framework that synthesizes directional hyperbolic activations, modular spectral damping, and curvature-aware density functions. Our architecture draws inspiration from approximation theory on special domains [24] while incorporating deep mathematical insights from noncommutative geometry [25] and index theory [26].

The ONHSH architecture unifies:

(i) *Hyperbolic symmetry*: through Lorentz-invariant kernel formulations;

(ii) *Modular damping*: via arithmetic decay in frequency space, ensuring spectral compactness;

(iii) *Anisotropic regularity*: grounded in Besov and Triebel–Lizorkin scales, guaranteeing minimax convergence.

We provide a detailed analytical foundation for ONHSH, culminating in a strengthened Voronovskaya-type theorem quantifying asymptotic bias and variance. By incorporating statistical estimation tools from nonparametric theory [27], we further quantify approximation uncertainty in highly anisotropic regimes.

## Main Contributions
* A novel modular-symmetric operator architecture unifying hyperbolic activations, arithmetic damping, and kernel curvature for PDE learning on anisotropic and algebraic domains

* Minimax-optimal approximation rates in weighted anisotropic Besov spaces with high-order spectral expansions

* A groundbreaking connection between operator spectral variance and noncommutative Chern geometry, bridging functional approximation with arithmetic topology

This work establishes a new foundation for operator learning one that is mathematically rigorous, geometrically adaptive, and spectrally structured significantly advancing our capacity to solve PDEs on complex, curved, and number-theoretically enriched domains.

## 1.1 Research Scope and Methodological Positioning

This work introduces Hypermodular Neural Operators with Hyperbolic Symmetry (ONHSH), advancing neural operator learning through a mathematically rigorous framework that explicitly incorporates geometric and arithmetic structure. While established architectures like FNO [1] and DeepONet [2] have demonstrated considerable success in learning PDE solution operators, their underlying assumptions Euclidean domain structure, isotropic regularity, and spectral uniformity—fundamentally limit their applicability to problems featuring intrinsic curvature, directional anisotropy, or modular symmetry.

The ONHSH framework systematically overcomes these limitations through three foundational innovations:

* **Geometric Adaptivity Beyond Euclidean Deformation:** Unlike approaches that deform Euclidean domains [4, 5], ONHSH employs curvature-driven kernels that intrinsically adapt to hyperbolic and anisotropic manifolds. This geometric awareness draws inspiration from function spaces on curved domains [24] and spectral geometry principles [25], enabling faithful representation of PDE dynamics on genuinely non-Euclidean spaces.

* **Spectral Modularity for Physical Consistency:** By embedding modular arithmetic directly into spectral filtering, ONHSH captures oscillatory patterns and aliasing phenomena that elude conventional FNO variants [13, 15]. This modular structure provides arithmetic-informed damping that naturally aligns with physical constraints and conservation laws, offering a principled approach to spectral regularization.

* **Function Space Theoretic Foundation:** Grounded in the approximation theory of anisotropic Besov and Triebel-Lizorkin spaces [16, 19], ONHSH provides rigorous convergence guarantees under directional smoothness assumptions. This theoretical grounding bridges neural architecture design with deep results in harmonic analysis [17, 22], ensuring mathematical consistency across scales and resolutions.

Methodologically, ONHSH represents a synthesis of neural operator design with advanced tools from approximation theory, spectral geometry, and noncommutative topology. The architecture incorporates spectral decompositions inspired by Chern characters from index theory [26], while leveraging statistical estimators from nonparametric analysis [27] to quantify approximation uncertainty. This integrated approach significantly extends the interpretability and applicability of operator learning to domains characterized by intrinsic curvature, modular arithmetic structure, and mixed anisotropic regularity.
## 1.2 Conceptual Diagram of the ONHSH Architecture

To clarify the interplay between geometric regularization, spectral symmetry, and functional approximation, we provide a schematic representation of the ONHSH operator pipeline. The architecture combines multiple stages including hyperbolic kernel convolution, symmetrized activation, modular spectral filtering, and global Fourier transforms into a cohesive operator learning flow.

Each module is designed to preserve or exploit a particular structural property: hyperbolic scaling governs spatial localization, modular filtering induces damping and asymmetry, while the spectral transform ensures global operator coherence. Together, these stages define an expressive operator capable of capturing directional smoothness, modular phenomena, and topologically curved input domains.

The diagram below illustrates the processing stages from functional input $f(x)$ to operator output $G(f)$, and highlights the key computational elements that compose the ONHSH framework.

<figure><img src="image_2.png" /><figcaption>Figure 1: Conceptual pipeline of the ONHSH operator. The framework applies a curved kernel, followed by symmetrized activation, modular spectral filtering, and final synthesis via spectral transform.</figcaption></figure>
# 2 Mathematical Background

This section rigorously establishes the mathematical foundations of Hypermodular Neural Operators with Hyperbolic Symmetry (ONHSH). We develop a comprehensive framework combining anisotropic function spaces, directional approximation theory, and spectral analysis with modular arithmetic structures. These tools provide the analytical bedrock for proving approximation, generalization, and invariance results throughout the paper.

## 2.1 Anisotropic Besov Spaces

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a measurable function. The anisotropic Besov space $B_{p,q}^s(\mathbb{R}^d)$, with directional smoothness vector $\mathbf{s} = (s_1, \dots, s_d) \in (0, \infty)^d$, integrability index $p \in [1, \infty]$, and summability index $q \in [1, \infty]$, is defined as the collection of functions $f \in L^p(\mathbb{R}^d)$ such that the anisotropic norm

$$ \|f\|_{B_{p,q}^s(\mathbb{R}^d)} := \|f\|_{L^p(\mathbb{R}^d)} + \left( \sum_{j=1}^{d} \int_{0}^{1} (t^{-s_j} \omega_{r,j}^p(f,t))^q \frac{dt}{t} \right)^{1/q} \quad (2.1) $$

is finite, with the usual modification when $q = \infty$. Here, the directional modulus of smoothness $\omega_{r,j}^p(f,t)$ of order $r \in \mathbb{N}$ in direction $x_j$ is defined via finite differences:

$$ \omega_{r,j}^p(f,t) := \sup_{|h| \le t} \|\Delta_h^{r,j} f\|_{L^p(\mathbb{R}^d)}, \quad \Delta_h^{r,j} f(x) := \sum_{k=0}^{r} (-1)^k \binom{r}{k} f(x + khe_j), \quad (2.2) $$

where $e_j \in \mathbb{R}^d$ is the canonical unit vector in the $j$-th coordinate direction.

**Geometric Interpretation and Physical Motivation.** The anisotropic Besov space $B_{p,q}^s(\mathbb{R}^d)$ characterizes mixed regularity profiles by permitting distinct smoothness orders $s_j$ along each coordinate axis. This directional sensitivity is crucial for modeling physical phenomena exhibiting intrinsic anisotropy—such as stratified turbulence in atmospheric flows, directional transport in porous media, and singularity propagation along characteristic curves in hyperbolic PDEs. The spaces naturally accommodate degenerate elliptic operators and multiscale dynamics where regularity varies substantially across different spatial directions.

**Functional Analytic Properties and Norm Structure.** The anisotropic Besov norm (2.1) provides a refined quantification of regularity by decomposing smoothness measurement along coordinate directions. Specifically:

- The term $t^{-s_j} \omega_{r,j}^p(f,t)$ acts as a directional Hölder quotient, penalizing lack of smoothness in the $x_j$-direction with weight proportional to $s_j$; - The integration $\int_0^1 (\cdot)^q \frac{dt}{t}$ captures the multiscale decay behavior of directional smoothness, with the $\frac{dt}{t}$ measure ensuring scale invariance; - The summation over $j = 1, \dots, d$ synthesizes directional regularities into a global mixed smoothness measure.
**Norm Equivalence via Peetre's K-Functional and Interpolation Theory.** For $0 < s_j < r$, $1 \le p, q \le \infty$, the norm (2.1) admits a powerful characterization through Peetre's K-functional. Define the directional K-functional:

$$ K_j(f, t^r)_{L^p, W_j^{r,p}} := \inf_{g \in W_j^{r,p}(\mathbb{R}^d)} \{\|f - g\|_{L^p} + t^r \|D_j^r g\|_{L^p}\}, \quad (2.3) $$

where $W_j^{r,p}(\mathbb{R}^d)$ denotes the Sobolev space with $r$-th order derivatives existing in $L^p$ along direction $x_j$. The fundamental equivalence:

$$ c_1 K_j(f, t^r)_{L^p, W_j^{r,p}} \le \omega_{r,j}^p(f, t) \le c_2 K_j(f, t^r)_{L^p, W_j^{r,p}}, \quad 0 < t < 1, \quad (2.4) $$

with constants $c_1, c_2 > 0$ depending only on $r$ and $d$, establishes that directional moduli of smoothness and K-functionals measure the same approximation-theoretic content. Integration over scales yields norm equivalence between (2.1) and the K-functional characterization.

**Characterization Theorem and Proof Details.** The following proposition provides the rigorous foundation for our anisotropic analysis:

**Proposition 2.1** (Moduli of Smoothness Characterize Anisotropic Besov Spaces). Let $r > \max_j s_j$ and $1 \le p, q \le \infty$. Then the following are equivalent:

$$ 1. f \in B_{p,q}^{s}(\mathbb{R}^d) $$

$$ 2. \|f\|_{L^p} + \left( \sum_{j=1}^{d} \int_{0}^{1} \left( t^{-s_j} \omega_{r,j}^{p}(f, t) \right)^q \frac{dt}{t} \right)^{1/q} < \infty $$

$$ 3. \text{ For all } j = 1, \dots, d, \omega_{r,j}^p(f, t) = \mathcal{O}(t^{s_j}) \text{ as } t \to 0^+ $$

Moreover, the quantities in (i) and (ii) define equivalent norms.

**Detailed Proof.** We provide a comprehensive proof establishing the norm equivalence.

(i) ⇒ (ii): Assume $f \in B_{p,q}^s(\mathbb{R}^d)$. Using the anisotropic Littlewood-Paley decomposition:

$$ f = \sum_{k \in \mathbb{N}_0^d} \Delta_k f, $$

where $\Delta_k f$ are directional dyadic blocks with Fourier support in $\prod_{j=1}^d \{\xi_j : |\xi_j| \asymp 2^{k_j}\}$. The anisotropic Besov norm satisfies:

$$ \|f\|_{B_{p,q}^s} \asymp \|f\|_{L^p} + \left( \sum_{k \in \mathbb{N}_0^d} \left( 2^{k \cdot s} \|\Delta_k f\|_{L^p} \right)^q \right)^{1/q}. $$

For the directional modulus, we employ the Jackson-type inequality:

$$ \omega_{r,j}^p(f, t) \le C \sum_{\ell=0}^{\infty} \min(1, (2^\ell t)^r) \|\Delta_\ell^{(j)} f\|_{L^p}, $$

where $\Delta_\ell^{(j)} f$ denotes the projection onto frequencies with $|\xi_\ell| \asymp 2^\ell$. Splitting the sum at $\ell_0$ with $2^{-\ell_0} \asymp t$ and using $\|\Delta_\ell^{(j)} f\|_{L^p} \lesssim 2^{-\ell s_j} \|f\|_{B_{p,q}^s}$ yields:

$$ \omega_{r,j}^p(f, t) \lesssim t^{s_j} \|f\|_{B_{p,q}^s}. $$
Substitution into (2.1) establishes the claim.

(ii) ⇒ (i): Assume the modulus condition holds. We construct approximants via convolution with anisotropic mollifiers $\varphi_t(x) = t^{-\sum_j 1/s_j} \varphi(D_s x/t)$, where $D_s = \text{diag}(t^{1/s_1}, \dots, t^{1/s_d})$. The Marchaud-type inequality:

$$ \|\Delta_k^{(j)} f\|_{L^p} \lesssim 2^{-ks_j} \left( \int_0^1 (u^{-s_j} \omega_{r,j}^p(f, u))^q \frac{du}{u} \right)^{1/q}, $$

combined with real interpolation between directional Sobolev spaces, yields the dyadic characterization of $B_{p,q}^s$.

**Equivalence of (i) and (iii):** The implication (i) ⇒ (iii) follows from the above Jackson inequality. For the converse, the assumption $\omega_{r,j}^p(f, t) \lesssim t^{s_j}$ permits the construction of a approximation family $\{f_\varepsilon\}$ with $\|f - f_\varepsilon\|_{L^p} \lesssim \varepsilon$ and $\|D_j^r f_\varepsilon\|_{L^p} \lesssim \varepsilon^{1-r/s_j}$, from which the Besov norm boundedness follows via the K-functional characterization and real interpolation. □

**Theorem 2.2 (Anisotropic Scaling and Invariance).** The anisotropic Besov space $B_{p,q}^s(\mathbb{R}^d)$ is invariant under the anisotropic scaling group:

$$ f(x) \mapsto f(D_{s,\lambda}x), \quad \text{where } D_{s,\lambda} := \text{diag}(\lambda^{1/s_1}, \dots, \lambda^{1/s_d}), \quad \lambda > 0. $$

Moreover, there exists $C > 0$ depending only on $s, p, q, d$ such that:

$$ \|f(D_{s,\lambda}\cdot)\|_{B_{p,q}^s} \le C\lambda^{\max_j(1/s_j)(1/p-1)}\|f\|_{B_{p,q}^s}. $$

*Proof.* The proof follows by direct computation of the norm under coordinate transformation. For the directional modulus:

$$ \omega_{r,j}^p(f(D_{s,\lambda}\cdot), t) = \lambda^{-1/s_j} \omega_{r,j}^p(f, \lambda^{1/s_j}t), $$

from which the scaling of the Besov norm follows by change of variables in (2.1). The exponent $\max_j(1/s_j)(1/p-1)$ arises from the Jacobian determinant of $D_{s,\lambda}$ and the homogeneity of the $L^p$-norm. □

**Remark 2.3 (Connection to Isotropic and Mixed-Norm Spaces).** When $s_1 = \dots = s_d = s$, we recover the classical isotropic Besov spaces $B_{p,q}^s(\mathbb{R}^d)$. The anisotropic framework interpolates between isotropic spaces and mixed-norm spaces $L_{x_1}^{p_1} \dots L_{x_d}^{p_d}$, providing a nuanced tool for directional regularity analysis. This versatility is particularly valuable in neural operator design, where different computational directions may exhibit distinct smoothness requirements.

**Remark 2.4 (Embedding Relations and Approximation Theory).** The anisotropic Besov spaces form a scale with continuous embeddings:

$$ B_{p,q}^{s+\varepsilon}(\mathbb{R}^d) \hookrightarrow B_{p,q}^s(\mathbb{R}^d) \hookrightarrow B_{p,r}^{s-\varepsilon}(\mathbb{R}^d), \quad \varepsilon > 0, $$

for $q \le r$. These embeddings, combined with the approximation properties characterized by the directional moduli of smoothness, provide the theoretical foundation for establishing convergence rates of neural operators in anisotropic function spaces.

**Theorem 2.5 (Anisotropic Embedding into Uniformly Continuous Functions).** Let $d \in \mathbb{N}$, $1 \le p < \infty$, $1 \le q \le \infty$, and let $\mathbf{s} = (s_1, \dots, s_d) \in (0, \infty)^d$. Suppose that

$$ s_j > \frac{1}{p}, \quad \text{for all } j = 1, \dots, d. \qquad (2.5) $$
Then the anisotropic Besov space $B_{p,q}^s(\mathbb{R}^d)$ is continuously embedded into the space of bounded, uniformly continuous functions:

$$B_{p,q}^s(\mathbb{R}^d) \hookrightarrow C_b^0(\mathbb{R}^d), \qquad (2.6)$$

i.e., there exists a constant $C > 0$, depending only on $d, p, q, \mathbf{s}$, such that

$$\|f\|_{L^\infty(\mathbb{R}^d)} \le C \|f\|_{B_{p,q}^s} \quad \text{for all } f \in B_{p,q}^s(\mathbb{R}^d). \qquad (2.7)$$

*Proof.* Let $f \in B_{p,q}^s(\mathbb{R}^d)$, and consider the anisotropic Littlewood-Paley characterization of the Besov norm

$$\|f\|_{B_{p,q}^s} \asymp \|f\|_{L^p} + \sum_{j=1}^{d} \left( \sum_{k=0}^{\infty} \left( 2^{ks_j} \|\psi_k^{(j)} * f\|_{L^p} \right)^q \right)^{1/q}, \qquad (2.8)$$

where $\{\psi_k^{(j)}\}_{k \ge 0} \subset S(\mathbb{R}^d)$ is a smooth dyadic partition of unity in the $j$-th direction, with

$$\supp \psi_k^{(j)} \subset \{\xi \in \mathbb{R}^d : 2^{k-1} \le |\xi_j| \le 2^{k+1}\}, \quad k \ge 1. \qquad (2.9)$$

By the anisotropic Bernstein inequality (derived from Nikol'skii-type estimates), for any $j = 1, \dots, d$ and $k \ge 0$, we have:

$$\|\psi_k^{(j)} * f\|_{L^\infty} \lesssim 2^{k/p} \|\psi_k^{(j)} * f\|_{L^p}, \qquad (2.10)$$

which holds because the Fourier transform of $\psi_k^{(j)} * f$ is supported in a frequency band of size $\sim 2^k$ in the $\xi_j$-direction, and $\psi_k^{(j)} \in S$ is band-limited.

Using (2.10) inside the anisotropic Besov expansion, we estimate:

$$\begin{aligned} \|f\|_{L^\infty} &\le \sum_{j=1}^{d} \sum_{k=0}^{\infty} \|\psi_k^{(j)} * f\|_{L^\infty} \\ &\lesssim \sum_{j=1}^{d} \sum_{k=0}^{\infty} 2^{k/p} \|\psi_k^{(j)} * f\|_{L^p} \\ &= \sum_{j=1}^{d} \sum_{k=0}^{\infty} \left( 2^{ks_j} \|\psi_k^{(j)} * f\|_{L^p} \right) \cdot 2^{-k(s_j-1/p)}. \end{aligned} \qquad (2.11)$$

Since $s_j > 1/p$, the exponential decay factor $2^{-k(s_j-1/p)}$ ensures that the above series converges. Applying Hölder's inequality (when $q < \infty$) or monotonicity (when $q = \infty$), and using (2.8), we obtain:

$$\|f\|_{L^\infty} \le C \|f\|_{B_{p,q}^s}, \qquad (2.12)$$

which completes the proof of the embedding into $L^\infty$.

To establish uniform continuity, we consider:

$$|f(x) - f(y)| \le \sum_{j=1}^{d} \sum_{k=0}^{\infty} \left| \psi_k^{(j)} * f(x) - \psi_k^{(j)} * f(y) \right|. \qquad (2.13)$$

Each summand $\psi_k^{(j)} * f$ is smooth and uniformly continuous due to the localization of $\psi_k^{(j)}$, with spatial diameter $\sim 2^{-k}$. Therefore, for any $\delta > 0$, there exists $\eta > 0$ such that

$$|x - y| \le \eta \Rightarrow \left| \psi_k^{(j)} * f(x) - \psi_k^{(j)} * f(y) \right| \le \delta \cdot 2^{-k}. \qquad (2.14)$$

Summing over $k$, the tail decays exponentially, and uniform continuity of $f$ follows. $\square$

<figure><img src="image_3.png" /></figure>
**Theorem 2.6** (Embedding Theorem on Bounded Lipschitz Domains). Let $\Omega \subset \mathbb{R}^d$ be a bounded Lipschitz domain. Suppose $1 \le p < \infty$, $1 \le q \le \infty$, and let the anisotropic smoothness vector $\mathbf{s} = (s_1, \dots, s_d) \in (0, \infty)^d$ satisfy

$$s_j > \frac{1}{p}, \quad j = 1, \dots, d. \qquad (2.15)$$

Then there exists a linear and bounded extension operator

$$E : B_{p,q}^{\mathbf{s}}(\Omega) \to B_{p,q}^{\mathbf{s}}(\mathbb{R}^d) \qquad (2.16)$$

such that $Ef|_{\Omega} = f$ a.e. and

$$\|Ef\|_{B_{p,q}^{\mathbf{s}}(\mathbb{R}^d)} \le C_{\Omega,s,p,q} \|f\|_{B_{p,q}^{\mathbf{s}}(\Omega)}. \qquad (2.17)$$

Consequently, $B_{p,q}^{\mathbf{s}}(\Omega)$ embeds continuously into $C^0(\bar{\Omega})$:

$$B_{p,q}^{\mathbf{s}}(\Omega) \hookrightarrow C^0(\bar{\Omega}), \quad \|f\|_{C^0(\bar{\Omega})} \le C \|f\|_{B_{p,q}^{\mathbf{s}}(\Omega)}. \qquad (2.18)$$

*Proof.* Since $\Omega$ is Lipschitz, there exists a finite covering of $\partial\Omega$ by coordinate patches $\{U_\alpha\}$ such that, in each patch, $\Omega$ is represented as the epigraph of a Lipschitz function $\varphi_\alpha : \mathbb{R}^{d-1} \to \mathbb{R}$. Let $\{\chi_\alpha\}$ be a smooth partition of unity subordinate to $\{U_\alpha\}$.

For a function $f \in B_{p,q}^{\mathbf{s}}(\Omega)$, define local extensions in each patch via reflection across the Lipschitz graph:

$$(E_{\alpha}f)(x', x_d) := \begin{cases} f(x', x_d), & x_d \ge \varphi_{\alpha}(x'), \\ f(x', 2\varphi_{\alpha}(x') - x_d), & x_d < \varphi_{\alpha}(x'). \end{cases}$$

Each $E_\alpha f$ preserves the anisotropic smoothness in the directions parallel to the boundary and in the normal direction, with norms controlled by the Lipschitz constant of $\varphi_\alpha$.

Combine the local extensions using the partition of unity:

$$Ef := \sum_{\alpha} \chi_{\alpha} E_{\alpha}(\chi_{\alpha}f) + \chi_{0}f,$$

where $\chi_0$ is supported in the interior away from the boundary. This defines a linear extension $E : B_{p,q}^{\mathbf{s}}(\Omega) \to B_{p,q}^{\mathbf{s}}(\mathbb{R}^d)$ with bounded norm, as shown in Triebel [16], Thm. 7.58.

Apply the global anisotropic Besov embedding theorem on $\mathbb{R}^d$:

$$B_{p,q}^{\mathbf{s}}(\mathbb{R}^d) \hookrightarrow C^0(\mathbb{R}^d), \quad s_j > \frac{1}{p}.$$

Then, for $f \in B_{p,q}^{\mathbf{s}}(\Omega)$:

$$\|f\|_{C^0(\bar{\Omega})} \le \|Ef\|_{C^0(\mathbb{R}^d)} \le C \|Ef\|_{B_{p,q}^{\mathbf{s}}(\mathbb{R}^d)} \le C' \|f\|_{B_{p,q}^{\mathbf{s}}(\Omega)}.$$

The construction preserves the anisotropic scaling: smoothness in each coordinate direction $s_j$ controls the modulus of continuity in that direction, and the reflection in Step 2 ensures that boundary regularity matches interior regularity.

This completes the proof of the embedding into $C^0(\bar{\Omega})$. $\square$
**Theorem 2.7** (Embedding Theorem on Compact Riemannian Manifolds). Let $(M, g)$ be a compact $d$-dimensional Riemannian manifold without boundary. Consider the anisotropic Besov space $B_{p,q}^{s}(M)$ defined via a finite atlas of coordinate charts and a partition of unity subordinate to this atlas. Suppose the smoothness parameters satisfy

$$s_j > \frac{1}{p}, \quad \text{for all } j = 1, \dots, d. \qquad (2.19)$$

Then there is a continuous embedding

$$B_{p,q}^{s}(M) \hookrightarrow C^{0}(M). \qquad (2.20)$$

In other words, every function in $B_{p,q}^{s}(M)$ admits a continuous representative on $M$.

*Proof.* Cover $M$ by a finite atlas of coordinate charts $\{(U_\alpha, \varphi_\alpha)\}_{\alpha \in A}$, where each $\varphi_\alpha : U_\alpha \to \mathbb{R}^d$ is a diffeomorphism. Using a smooth partition of unity $\{\rho_\alpha\}_{\alpha \in A}$ subordinate to this covering, define local Besov norms by pullback to $\mathbb{R}^d$:

$$\|f\|_{B_{p,q}^{s}(U_{\alpha})} := \|(f \circ \varphi_{\alpha}^{-1}) \cdot \rho_{\alpha}\|_{B_{p,q}^{s}(\mathbb{R}^{d})}. \qquad (2.21)$$

Then set the global norm on $M$ as

$$\|f\|_{B_{p,q}^{s}(M)} := \sum_{\alpha \in A} \|f\|_{B_{p,q}^{s}(U_{\alpha})}. \qquad (2.22)$$

By the Euclidean anisotropic Besov embedding theorem, the condition $s_j > \frac{1}{p}$ ensures that

$$B_{p,q}^{s}(\mathbb{R}^{d}) \hookrightarrow C^{0}(\mathbb{R}^{d}). \qquad (2.23)$$

Applying this embedding chart-wise, each $(f \circ \varphi_{\alpha}^{-1}) \cdot \rho_{\alpha}$ is continuous on $\varphi_{\alpha}(U_{\alpha})$, hence $f$ is continuous on each $U_{\alpha}$. Since $M$ is compact, these local continuities piece together to give continuity on all of $M$. $\square$

**Theorem 2.8** (Directional Approximation via Moduli of Smoothness). Let $f \in L^p(\mathbb{R}^d)$ with $1 \le p \le \infty$, $r \in \mathbb{N}$, and smoothness parameters $s = (s_1, \dots, s_d) \in (0, r)^d$. Define the directional modulus of smoothness of order $r$ in the coordinate direction $x_j$ by

$$\omega_{r,j}^{p}(f, t) := \sup_{|h| \le t} \left\| \Delta_{h}^{r,j} f \right\|_{L^{p}(\mathbb{R}^{d})}, \quad \text{where} \quad \Delta_{h}^{r,j} f(x) := \sum_{\ell=0}^{r} (-1)^{\ell} \binom{r}{\ell} f(x + \ell he_{j}). \qquad (2.24)$$

Then, for all $t > 0$, the modulus satisfies the following key properties:

1. **Seminorm property:** $\omega_{r,j}^{p}(f, t)$ is a seminorm in $f$.

2. **Equivalence to directional derivatives:** If $f$ is sufficiently smooth, say $f \in W^{r,p}(\mathbb{R}^d)$ with weak directional derivatives up to order $r$ in direction $x_j$, then

$$\omega_{r,j}^{p}(f, t) \le C_r t^r \left\| \frac{\partial^r f}{\partial x_j^r} \right\|_{L^p(\mathbb{R}^d)}, \qquad (2.25)$$

where $C_r > 0$ depends only on $r$.
3. Inverse inequality (Jackson-type estimate): There exists a constant $C > 0$ such that for any $f \in L^p(\mathbb{R}^d)$,

$$E_n^{(j)}(f)_p := \inf_{P \in \mathcal{P}_n^{(j)}} \|f - P\|_{L^p} \le C \omega_{r,j}^p(f, n^{-1}), \quad (2.26)$$

where $\mathcal{P}_n^{(j)}$ denotes the set of algebraic polynomials of degree at most $n$ in the variable $x_j$ only.

*Proof.* The operator $\Delta_h^{r,j}$ is linear in $f$ and satisfies

$$\Delta_h^{r,j}(f + g) = \Delta_h^{r,j} f + \Delta_h^{r,j} g, \quad (2.27)$$

and

$$\Delta_h^{r,j}(\alpha f) = \alpha \Delta_h^{r,j} f, \quad (2.28)$$

for any scalar $\alpha$. The supremum and $L^p$-norm preserve these linearities up to the triangle inequality. Moreover, $\omega_{r,j}^p(f, t) = 0$ if and only if $f$ is a polynomial of degree less than $r$ in the $j$-th variable, reflecting the fact that $\Delta_h^{r,j}$ annihilates such polynomials.

Assume $f \in W^{r,p}(\mathbb{R}^d)$ so that the $r$-th directional derivative $D_j^r f = \frac{\partial^r f}{\partial x_j^r}$ exists in $L^p$. By Taylor's theorem with integral remainder in direction $x_j$, for almost every $x \in \mathbb{R}^d$, we have

$$\Delta_h^{r,j} f(x) = h^r \int_0^1 \binom{r}{r} (1-t)^{r-1} D_j^r f(x + t \mathrm{tr} h e_j) dt. \quad (2.29)$$

Taking the $L^p$-norm and applying Minkowski's integral inequality yields

$$\|\Delta_h^{r,j} f\|_{L^p} \le |h|^r \int_0^1 (1-t)^{r-1} \|D_j^r f(\cdot + t \mathrm{tr} h e_j)\|_{L^p} dt = \frac{|h|^r}{r} \|D_j^r f\|_{L^p}, \quad (2.30)$$

since the $L^p$-norm is translation invariant. Thus,

$$\omega_{r,j}^p(f, t) = \sup_{|h| \le t} \|\Delta_h^{r,j} f\|_{L^p} \le C_r t^r \|D_j^r f\|_{L^p}. \quad (2.31)$$

For $f \in L^p(\mathbb{R}^d)$, define

$$E_n^{(j)}(f)_p := \inf_{P \in \mathcal{P}_n^{(j)}} \|f - P\|_{L^p}. \quad (2.32)$$

By classical approximation theory (see, e.g., DeVore and Lorentz (1993)), there exists a polynomial $P_n \in \mathcal{P}_n^{(j)}$ such that

$$\|f - P_n\|_{L^p} \le C \omega_{r,j}^p(f, n^{-1}). \quad (2.33)$$

This inequality shows that the modulus of smoothness controls the rate of approximation by polynomials in the $j$-th variable.

Collecting the above results establishes the role of the directional modulus of smoothness as a fundamental quantifier of directional regularity and approximation behavior. $\square$

**Theorem 2.9** (Directional Characterization of Anisotropic Besov Spaces). Let $0 < s_j < r$ for all $j \in \{1, \dots, d\}$, and let $1 \le p, q \le \infty$. Then the following norm equivalence holds:

$$\|f\|_{D_{p,q}^s(\mathbb{R}^d)} \asymp \|f\|_{L^p} + \sum_{j=1}^d \left( \int_0^1 \left( t^{-s_j} \omega_{r,j}^p(f,t) \right)^q \frac{dt}{t} \right)^{1/q}, \quad (2.34)$$
with the usual modification when $q = \infty$. In particular, for each direction $j$, if $f \in B_{p,q}^s$, then:

$$ \omega_{r,j}^p(f, t) = \mathcal{O}(t^{s_j}) \quad \text{as } t \to 0^+, \qquad (2.35) $$

and conversely, such estimates characterize the membership in $B_{p,q}^s$.

*Proof.* We sketch the main steps of the proof using classical approximation theory (cf. Triebel [16], Runst–Sickel [19]).

Suppose $f \in B_{p,q}^s$. By the Littlewood–Paley characterization, we have:

$$ \|\Delta_k^{(j)} f\|_{L^p} \lesssim 2^{-ks_j} \|f\|_{B_{p,q}^s}, \qquad (2.36) $$

where $\Delta_k^{(j)} f := \psi_k^{(j)} * f$ denotes the directional dyadic block. Using Jackson-type estimates in terms of moduli of smoothness, we obtain:

$$ \omega_{r,j}^p(f, t) \lesssim \sum_{k=0}^{\infty} \min\{1, (2^k t)^r\} \|\Delta_k^{(j)} f\|_{L^p}. \qquad (2.37) $$

Split the sum at $k_0$ such that $2^{-k_0} \le t < 2^{-k_0+1}$. Then for $k \ge k_0$, $(2^k t)^r \ge 1$, and for $k < k_0$, $(2^k t)^r \le 1$. Using (2.36), we conclude:

$$ \omega_{r,j}^p(f, t) \lesssim t^{s_j} \|f\|_{B_{p,q}^s}. $$

Conversely, assume that for some constant $C > 0$,

$$ \omega_{r,j}^p(f, t) \le C t^{s_j}, \quad \text{for all } 0 < t < 1. $$

Then one can construct smooth approximants $f_\delta$ via convolution with mollifiers of scale $\delta \sim t$, and show that directional differences decay like $\delta^{s_j}$, which implies:

$$ \|\Delta_k^{(j)} f\|_{L^p} \lesssim 2^{-ks_j}, \quad \text{uniformly in } k, \qquad (2.38) $$

and summing over $k$, we recover the anisotropic Besov norm. The proof is completed by establishing norm equivalence via interpolation and real-method arguments (cf. Peetre's K-functional and Marchaud inequalities). $\square$

*Remark 2.10.* The quantity $\omega_{r,j}^p(f, t)$ behaves analogously to a directional Hölder seminorm:

$$ \omega_{r,j}^p(f, t) \sim t^{s_j} \iff f \text{ is directionally } C^{s_j} \text{ in the } x_j \text{ variable.} $$

Thus, the anisotropic Besov norm naturally balances local $L^p$-integrability with directional Hölder-type behavior, allowing fine-grained control of regularity in multiscale operator learning frameworks.

## 2.2 Spectral Multipliers and Modular Damping

Let $f \in S(\mathbb{R}^d)$ and denote its Fourier transform by

$$ \hat{f}(\xi) := \mathcal{F}[f](\xi) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i x \cdot \xi} dx. \qquad (2.39) $$

We define a family of linear operators $T_n$ acting as smooth spectral projectors modulated by arithmetic damping:

$$ T_n(f)(x) := \mathcal{F}^{-1}[m_n(\xi) \cdot \hat{f}(\xi)](x), \qquad (2.40) $$
where $m_n(\xi)$ is a modular spectral multiplier of the form:

$$m_n(\xi) := \sum_{k \in \mathbb{Z}^d} q_n^{\|k\|^2} \chi_k(\xi), \quad \text{with} \quad q_n := e^{-\pi n^{-1/2}}. \qquad (2.41)$$

Here, the functions $\chi_k \in C_c^\infty(\mathbb{R}^d)$ form a smooth partition of unity over the frequency domain, each supported in a neighborhood of $k \in \mathbb{Z}^d$, typically satisfying:

$$\text{supp}(\chi_k) \subset B_\delta(k), \quad \sum_{k \in \mathbb{Z}^d} \chi_k(\xi) = 1 \quad \text{for all } \xi \in \mathbb{R}^d, \qquad (2.42)$$

where $B_\delta(k)$ denotes a ball of radius $\delta > 0$ centered at $k$.

**Modular damping.** The exponential weight

$$q_n^{\|k\|^2} = \exp(-\pi \|k\|^2 n^{-1/2}), \qquad (2.43)$$

ensures that components corresponding to large $\|k\|$ (i.e., high-frequency regions) are rapidly attenuated. This achieves spectral smoothing without sharp truncation, preserving analytical regularity.

**Functional-analytic properties.** The operator $T_n$ maps $L^p(\mathbb{R}^d) \to C^\infty(\mathbb{R}^d)$ compactly for any $1 \le p < \infty$. Moreover, the integral kernel of $T_n$ belongs to the Schwartz class $S(\mathbb{R}^{2d})$, as it is the inverse Fourier transform of a smooth compactly supported function:

$$K_n(x, y) := \mathcal{F}^{-1}[m_n](x - y) \in \mathcal{S}(\mathbb{R}^{2d}). \qquad (2.44)$$

**Spectral concentration and resolution.** As $n \to \infty$, we have:

$$q_n \to 1^{-}, \quad m_n(\xi) \to 1, \quad T_n(f) \to f \quad \text{in } L^p \text{ and pointwise almost everywhere.} \qquad (2.45)$$

Hence, $T_n$ acts as an approximate identity, with increasing spectral resolution. The rate of convergence can be quantified in terms of the decay of the Fourier tail of $f$, yielding:

$$\|T_n(f) - f\|_{L^p} \le C \cdot e^{-cn^{1/4}} \|f\|_{B_{p,q}^s}, \qquad (2.46)$$

for constants $C, c > 0$ depending on the directional smoothness $s$ and the bump system $\{\chi_k\}$.

**Comparison with classical multipliers.** Unlike sharp cutoff multipliers (e.g., $m(\xi) = 1_{|\xi| \le R}$), the modular multiplier $m_n$ is infinitely smooth and decays analytically, enabling better control over Gibbs-type artifacts and improved convergence in inverse problems or PDE solvers.

**Summary.** The modularly damped multiplier $m_n$ embeds both analytic decay and arithmetic structure into the spectral representation, making $T_n$ a powerful tool for compact approximation in anisotropic and non-uniformly regular settings. This mechanism will be refined to construct hyperbolically invariant neural operators.
## 2.3 Spectral Damping and Localization

The modular damping imposed by the coefficients $q_n^{\|k\|^2}$ ensures that the effective frequency content of $T_n(f)$ is concentrated near the origin in Fourier space. For each $n \in \mathbb{N}$, define the spectral cutoff region:

$$ \Omega_n := \{ \xi \in \mathbb{R}^d : |\xi - k| \le \delta \text{ for some } k \in \mathbb{Z}^d \text{ with } \|k\|^2 \le Cn^{1/2} \}, \quad (2.47) $$

where $\delta > 0$ depends on the support radius of $\chi_k$. Then $\text{supp}(\hat{T}_n f) \subset \Omega_n$, with exponential suppression outside this region.

To quantify the smoothing effect, we consider functions $f \in B_{p,q}^{s,\tau}(\mathbb{R}^d)$, an anisotropic Besov space with mixed smoothness $s \in (0, \infty)^d$, and establish the following decay estimate:

**Theorem 2.11 (Spectral Localization Estimate).** Let $f \in B_{p,q}^{s,\tau}(\mathbb{R}^d)$, with $s \in (0, \infty)^d$, $1 \le p < \infty$, and $1 \le q \le \infty$. Then, for all $n \in \mathbb{N}$, there exist constants $C, c > 0$, depending on $(p, q, s, d)$, such that

$$ \|T_n(f)\|_{L^p(\mathbb{R}^d)} \le C \cdot e^{-cn^{1/4}} \cdot \|f\|_{B_{p,q}^{s,\tau}(\mathbb{R}^d)}. \quad (2.48) $$

*Proof.* Let $\{\psi_k^{(j)}\}_{k \ge 0, 1 \le j \le d}$ be an anisotropic dyadic decomposition adapted to $s$. Using Littlewood-Paley theory and the partition of unity $\{\chi_k\}$, decompose:

$$ T_n(f) = \sum_{k \in \mathbb{Z}^d} q_n^{\|k\|^2} \cdot \mathcal{F}^{-1}[\chi_k \hat{f}] = \sum_{k \in \mathbb{Z}^d} q_n^{\|k\|^2} \cdot f_k, $$

where $f_k := \mathcal{F}^{-1}[\chi_k \hat{f}]$. Applying the triangle inequality and using the decay of $q_n^{\|k\|^2}$, we estimate:

$$ \|T_n(f)\|_{L^p} \le \sum_{k \in \mathbb{Z}^d} q_n^{\|k\|^2} \cdot \|f_k\|_{L^p} \quad (2.49) $$

$$ \le C \cdot \left( \sup_{\|k\| \le K(n)} \|f_k\|_{L^p} + \sum_{\|k\| > K(n)} e^{-\pi \|k\|^2 n^{-1/2}} \cdot \|f_k\|_{L^p} \right), $$

where $K(n) \sim n^{1/4}$. Since $\|f_k\|_{L^p} \lesssim \|f\|_{B_{p,q}^{s,\tau}} \cdot 2^{-k_j s_j}$ for appropriate dyadic levels $k$, and the tail decays super-exponentially, we obtain the bound (2.48). $\square$

This result shows that $T_n$ is a compact operator on $L^p$, with norm tending to zero exponentially as $n \to \infty$. This behavior is particularly beneficial in regularization and generalization analysis of learned operators, as it implies strong control over the bias-variance trade-off and robustness to high-frequency noise.

**Phase-Space Compactness.** The concentration of $T_n$ in both space and frequency yields compactness in phase space (i.e., joint localization in $x-\xi$ domains), allowing for stable sampling and reconstruction schemes, even in anisotropic settings. Such compactness properties play a central role in the convergence of operator networks based on hypermodular kernels, especially when applied to PDEs on non-Euclidean domains.
## 2.4 Symmetrized Hyperbolic Activation Kernels

A hallmark of the ONHSH framework is the incorporation of activation kernels that are simultaneously smooth, spectrally localized, and geometrically consistent with hyperbolic invariance. The base activation is defined as the symmetrized hyperbolic tangent function:

$$ \psi_{\lambda,q}(x) := \frac{1}{2} (\tanh(\lambda x) + \tanh(\lambda qx)), \quad \lambda > 0, 0 < q < 1, \qquad (2.50) $$

which satisfies the following key properties:

1. **Odd symmetry:** $\psi_{\lambda,q}(-x) = -\psi_{\lambda,q}(x)$,

2. **Lipschitz continuity:** $|\psi'_{\lambda,q}(x)| \le \lambda(1 + q)$,

3. **Hyperbolic symmetry:** as $q \to 0$, we recover a scaled tanh function: $\psi_{\lambda,q}(x) \to \tanh(\lambda x)/2$.

**Fourier decay.** The smoothness and rapid saturation of tanh ensure that the Fourier transform $\hat{\psi}_{\lambda,q} \in S(\mathbb{R})$ decays super-polynomially:

$$ |\hat{\psi}_{\lambda,q}(\xi)| \le \frac{C}{\lambda} (1 + |\xi|)^{-2}, \quad \forall \xi \in \mathbb{R}. \qquad (2.51) $$

This ensures that convolutional operators built from $\psi_{\lambda,q}$ act as smooth low-pass filters, controlling spectral leakage.

**Even moment structure.** Let

$$ \mu_{2m} := \int_{\mathbb{R}} x^{2m} \psi_{\lambda,q}(x) dx, \quad m \in \mathbb{N}_0, \qquad (2.52) $$

denote the even-order moments. Then:

$$ \mu_{2m} = \frac{\Gamma(2m + 1)}{\lambda^{2m}} \cdot \left( \frac{1 + q^{2m}}{2} \right) \cdot C_m, \qquad (2.53) $$

where $C_m$ are scaling constants depending on $m$. These moments play a central role in deriving Voronovskaya-type expansions (see Section 2.6).

**Kernel construction.** Define the hypermodular kernel induced by translations of $\psi_{\lambda,q}$:

$$ K_{\lambda,q,n}(x, y) := \sum_{k \in \mathbb{Z}^d} \alpha_k^{(n)} \psi_{\lambda,q}(x - y - k), \qquad (2.54) $$

where the coefficients satisfy modular decay of the form:

$$ |\alpha_k^{(n)}| \le C \cdot \exp \left( -\pi \frac{\|k\|^2}{n^{1/2}} \right), \qquad (2.55) $$

ensuring that the kernel is smooth and rapidly decaying in both physical and spectral domains.

**Geometric interpretation.** The invariance of $\psi_{\lambda,q}$ under hyperbolic reflections $x \mapsto -x$ reflects the embedding of hyperbolic symmetry at the level of operator kernels. Such symmetry is crucial for applications on curved or Lorentzian manifolds where standard Euclidean activations fail to respect geometric constraints.
## 2.5 Hyperbolic Symmetry Invariance

The ONHSH framework respects hyperbolic symmetry via compatibility with the Lorentz group $SO(1, d-1)$. This ensures that the resulting neural operators are invariant under transformations preserving Minkowski spacetime structure.

**Definition 2.12** (Lorentz Transformation). Let $\eta = \operatorname{diag}(-1, 1, \dots, 1) \in \mathbb{R}^{d \times d}$ be the Minkowski metric. Then:

$$ SO(1, d-1) := \{\Lambda \in GL(d, \mathbb{R}) \mid \Lambda^{\top}\eta\Lambda = \eta\} \qquad (2.56) $$

defines the connected component of the Lorentz group preserving orientation and time direction.

**Operator invariance.** Let $f \in L^2(\mathbb{R}^d)$, and consider a hypermodular kernel $K_{\lambda,q,n}(x, y)$ as in (2.54). Define the integral operator:

$$ T_n f(x) := \int_{\mathbb{R}^d} K_{\lambda,q,n}(x, y) f(y) dy. \qquad (2.57) $$

Then for any $\Lambda \in SO(1, d-1)$, we have:

$$ T_n(f \circ \Lambda^{-1})(x) = (T_n f)(\Lambda^{-1}x). \qquad (2.58) $$

This follows from the invariance $\psi_{\lambda,q}(\eta(x-y-k)) = \psi_{\lambda,q}(\eta(\Lambda(x-y-k)))$ and the modular symmetry of the coefficients.

**Representation-theoretic view.** The family $\{T_n\}$ forms a class of convolution-type integral operators whose kernels belong to induced unitary representations of $SO(1, d-1)$ on $L^2(\mathbb{R}^d)$, modulated by modular weights. These operators therefore live naturally within the framework of non-commutative harmonic analysis on symmetric spaces.

## 2.6 Spectral Variance and Voronovskaya-Type Expansions

To characterize the asymptotic behavior of the ONHSH operators, we derive a Voronovskaya-type expansion expressing the bias-variance decomposition induced by spectral smoothing.

**Theorem 2.13** (Voronovskaya Expansion for Modular Operators). Let $f \in B_{p,q}^{2s,\tau}(\mathbb{R}^d)$ for some smoothness vector $s \in (0, \infty)^d$, with $1 \le p, q, \tau \le \infty$. Let $T_n$ be the modular operator defined in (2.40), associated with a family of smoothing kernels $K_{\lambda,q,n}(x, y)$. Then, for every fixed $x \in \mathbb{R}^d$, the following pointwise expansion holds:

$$ T_n(f)(x) = f(x) + \frac{1}{2n} \sum_{j=1}^{d} \beta_j \frac{\partial^2 f}{\partial x_j^2}(x) + R_n(f)(x), \qquad (2.59) $$

where the spectral variance coefficients $\beta_j > 0$ are given by the second moments of the kernel in each coordinate direction,

$$ \beta_j = \int_{\mathbb{R}^d} (y_j - x_j)^2 K_{\lambda,q,n}(x, y) dy, \qquad (2.60) $$

and the remainder satisfies the norm bound

$$ \|R_n(f)\|_{L^p} \le Cn^{-\gamma}\|f\|_{B_{p,q}^{2s,\tau}}, \quad \text{for some } \gamma > 1, \qquad (2.61) $$

with $C > 0$ independent of $n$ and $f$.
Proof. The proof proceeds by a careful Taylor expansion of $f$ around the point $x$ and integration against the kernel $K_{\lambda,q,n}(x, y)$. Writing the Taylor expansion to second order yields:

$$f(y) = f(x) + \sum_{j=1}^{d} (y_j - x_j) \frac{\partial f}{\partial x_j}(x) + \frac{1}{2} \sum_{j,k=1}^{d} (y_j - x_j)(y_k - x_k) \frac{\partial^2 f}{\partial x_j \partial x_k}(x) + R_3(x, y), \quad (2.62)$$

where the remainder term $R_3(x, y)$ contains terms of order $\|y - x\|^3$.

By the symmetry and normalization properties of $K_{\lambda,q,n}(x, y)$, particularly its evenness in $y - x$, the first-order terms vanish upon integration:

$$\int_{\mathbb{R}^d} (y_j - x_j) K_{\lambda,q,n}(x, y) dy = 0, \quad \forall j = 1, \dots, d, \qquad (2.63)$$

(see, e.g., [20, 21]).

Furthermore, the kernel moments scale like $\beta_j/n$, reflecting the variance scaling in the smoothing operator:

$$\int_{\mathbb{R}^d} (y_j - x_j)(y_k - x_k) K_{\lambda,q,n}(x, y) dy = \frac{\beta_j}{n} \delta_{jk}, \qquad (2.64)$$

where $\delta_{jk}$ is the Kronecker delta [22].

Substituting into the integral operator expression for $T_n(f)(x)$, we obtain:

$$T_n(f)(x) = \int_{\mathbb{R}^d} f(y) K_{\lambda,q,n}(x, y) dy = f(x) + \frac{1}{2n} \sum_{j=1}^{d} \beta_j \frac{\partial^2 f}{\partial x_j^2}(x) + \int_{\mathbb{R}^d} R_3(x, y) K_{\lambda,q,n}(x, y) dy. \qquad (2.65)$$

The remainder integral is controlled by the smoothness of $f$. Using that $f \in B_{p,q}^{2s,\tau}$, combined with appropriate Sobolev embedding theorems and moment decay estimates of $K_{\lambda,q,n}$, one shows that

$$\left\| \int_{\mathbb{R}^d} R_3(\cdot, y) K_{\lambda,q,n}(\cdot, y) dy \right\|_{L^p} \le C n^{-\gamma} \|f\|_{B_{p,q}^{2s,\tau}}, \quad \text{for some } \gamma > 1, \qquad (2.66)$$

with $C > 0$ independent of $n$ and $f$ [23, 24].

The spectral variance coefficients $\beta_j$ are strictly positive due to the positive definiteness of the smoothing kernel and its construction via hyperbolic symmetry considerations [18].

This expansion rigorously quantifies the bias incurred by the smoothing $T_n$ as a second-order differential operator perturbation, with higher-order terms uniformly negligible in $L^p$ norm, completing the proof. $\square$

**Geometric Interpretation.** The spectral variance term $\sigma_{\text{spec}}^2(f)(x)$ arising in the Voronovskaya-type expansion (2.59) can be interpreted geometrically as a curvature-induced bias, analogous to the local contribution of a Laplace-type operator on a Riemannian manifold equipped with a connection. Specifically, let $(M, g)$ be a compact Riemannian manifold, and let $\nabla$ denote a compatible connection on a vector bundle $E \to M$. Then, for an elliptic pseudodifferential operator $D$ acting on sections of $E$, we have the local expression:

$$\sigma_{\text{spec}}^2(f)(x) \sim \text{Tr} (a_2(x) \cdot \nabla^2 f(x)), \qquad (2.67)$$

where $a_2(x)$ is the second term in the heat kernel expansion of $D$, often encoding curvature terms, and Tr is taken over the fiber of $E$ at point $x \in M$. This representation
establishes a bridge between functional approximation theory and the local index density in the Atiyah-Singer index theorem.

Furthermore, in the framework of noncommutative geometry, the operator $D$ can be replaced by a Dirac-type operator $\mathcal{D}$ affiliated with a spectral triple $(\mathcal{A}, \mathcal{H}, \mathcal{D})$, where $\mathcal{A}$ is a dense subalgebra of a von Neumann algebra $\mathcal{M} \subset \mathcal{B}(\mathcal{H})$. In this setting, the noncommutative Chern character $\mathrm{Ch}_n(\mathcal{D})$ appears via Connes-Moscovici cyclic cocycles [25, 26], and the spectral variance term becomes a local expression involving Dixmier traces:

$$ \sigma_{\mathrm{spec}}^2(f)(x) = \lim_{N \to \infty} \frac{1}{\log N} \sum_{\lambda_n \le N} \lambda_n^{-2} |\langle f, \psi_n \rangle|^2, \qquad (2.68) $$

where $\{\lambda_n, \psi_n\}$ are the eigenpairs of $\mathcal{D}$. This links the second-order approximation behavior of $f$ under neural-type kernel operators to traces on von Neumann algebras and noncommutative curvature.

Such an interpretation provides a geometric backbone to the asymptotic expansions derived in (2.59), highlighting that neural operators implicitly encode geometric information, such as scalar curvature or bundle torsion, when applied over manifolds or noncommutative spaces. This viewpoint enhances the spectral-theoretic and topological understanding of operator-based function approximation.

**Bias-Variance Trade-off.** The asymptotic expansion of nonlinear or kernel-based approximation operators $T_n$, such as neural or wavelet-type schemes, naturally separates the leading-order bias from the higher-order fluctuations. Explicitly, the expansion

$$ T_n f(x) = f(x) + \frac{1}{n} \mathcal{B}(f)(x) + R_n(f)(x), \qquad (2.69) $$

where $\mathcal{B}(f)$ denotes the principal bias operator and $R_n(f) = \mathcal{O}(n^{-\gamma})$, with $\gamma > 1$, offers direct insight into the interplay between approximation error and model complexity.

The bias term $\mathcal{B}(f)$ often admits a geometric interpretation, especially in manifold settings. If $f \in C^\infty(M)$, where $M$ is a compact Riemannian manifold endowed with Levi-Civita connection $\nabla$, then under mild assumptions on the operator family $T_n$, one may write:

$$ \mathcal{B}(f)(x) = \mathrm{Tr}_g(\nabla^2 f)(x) + \mathcal{K}(x)f(x), \qquad (2.70) $$

where $\mathrm{Tr}_g(\nabla^2 f)$ is the trace of the Hessian with respect to the Riemannian metric $g$, and $\mathcal{K}(x)$ is a curvature-related potential arising from the kernel moment asymmetry or commutator structure. This geometric term mirrors the behavior of Laplace-type pseudodifferential operators on smooth sections of bundles over $M$.

The variance is encapsulated in the decay of the remainder:

$$ \|R_n(f)\|_{L^p(M)} \le C n^{-\gamma} \|f\|_{W^{s,p}(M)}, \quad \text{for some } s > 0, \qquad (2.71) $$

highlighting the smoothing properties of $T_n$ and the regularity requirements on $f$.

Such bias-variance decomposition is central to the derivation of minimax optimal rates in statistical learning theory and approximation theory. In particular, given a target accuracy $\varepsilon > 0$, the optimal tuning $n^*(\varepsilon)$ satisfies:

$$ \frac{1}{n^*(\varepsilon)} \sim \left( \frac{1}{n^*(\varepsilon)} \right)^{\gamma} \Rightarrow n^*(\varepsilon) \sim \varepsilon^{-1/(\gamma-1)}, \qquad (2.72) $$

establishing the rate-optimal selection of model complexity to balance approximation fidelity and generalization.
Moreover, in noncommutative geometric frameworks, operators $T_n$ may be viewed as acting on modules over von Neumann algebras, and bias operators $\mathcal{B}(f)$ can be interpreted as traces of derivations or as the image under the Connes-Chern character map:

$$ \mathcal{B}(f) \simeq \tau([D, f]^2), \qquad (2.73) $$

where $D$ is a Dirac-type operator affiliated with a spectral triple $(\mathcal{A}, \mathcal{H}, D)$, and $\tau$ denotes the faithful normal trace on a von Neumann algebra $\mathcal{N}$ generated by $\mathcal{A}$. This perspective links the bias structure to index-theoretic invariants and spectral asymptotics.

## 2.7 Hyperbolic Symmetry Invariance

The study of invariance under non-compact symmetries is a cornerstone in both harmonic analysis and mathematical physics. In particular, the Lorentz group $SO(1, d-1)$, which encodes the isometries of Minkowski space, arises naturally in the analysis of hyperbolic partial differential equations, relativistic field theories, and automorphic structures on pseudo-Riemannian manifolds.

**Lorentz Group and Minkowski Geometry.** We consider the pseudo-inner product on $\mathbb{R}^d$ defined by

$$ \eta(x, y) := x^{\top} \eta y, \quad \text{with } \eta := \operatorname{diag}(-1, 1, \dots, 1), \qquad (2.74) $$

so that the associated pseudo-norm is given by $\eta(x) = \eta(x, x) = -x_0^2 + x_1^2 + \dots + x_{d-1}^2$. The Lorentz group is then defined as the set of linear isometries preserving this form:

$$ SO(1, d-1) := \{ \Lambda \in \operatorname{GL}(d, \mathbb{R}) \mid \Lambda^{\top} \eta \Lambda = \eta \}. \qquad (2.75) $$

This group acts naturally on functions $f: \mathbb{R}^d \to \mathbb{R}$ via pullbacks $f \mapsto f \circ \Lambda^{-1}$, inducing transformations in functional spaces compatible with the underlying spacetime geometry.

**Kernel-Level Invariance.** Let $K(x, y)$ be the integral kernel associated with the ONHSH operator, constructed via symmetrized hyperbolic activations. Suppose the kernel admits the structural form:

$$ K(x, y) := \psi_{\lambda,q}(\eta(x - y)), \qquad (2.76) $$

where $\psi_{\lambda,q}$ is an activation function with controlled decay and symmetry.

Then, for any Lorentz transformation $\Lambda \in SO(1, d-1)$, we have:

$$ K(\Lambda x, \Lambda y) = \psi_{\lambda,q}(\eta(\Lambda x - \Lambda y)) = \psi_{\lambda,q}(\eta(x - y)) = K(x, y), \qquad (2.77) $$

by invariance of the Minkowski bilinear form under Lorentz transformations. Thus, the ONHSH operator defined by

$$ (\mathcal{T}f)(x) := \int_{\mathbb{R}^d} K(x, y)f(y) dy \qquad (2.78) $$

commutes with the action of $SO(1, d-1)$, i.e., $\mathcal{T}(f \circ \Lambda^{-1}) = (\mathcal{T}f) \circ \Lambda^{-1}$. This places ONHSH within the class of equivariant integral operators on pseudo-Riemannian spaces.
**Modular-Hyperbolic Coupling.** The modular damping component interacts harmoniously with hyperbolic symmetry. Let $K_{\lambda,q,n}(x, y)$ be defined by:

$$K_{\lambda,q,n}(x, y) := \sum_{k \in \mathbb{Z}^d} \exp \left( -\pi \frac{\|k\|^2}{n^{1/2}} \right) \psi_{\lambda,q}(\eta(x - y - k)), \quad (2.79)$$

where the summation over $\mathbb{Z}^d$ introduces modular periodicity. The interplay between the Lorentz-invariant argument $\eta(x - y - k)$ and the modular filter ensures that the operator encodes both non-Euclidean geometric priors and arithmetic decay, providing a mechanism to regularize spectral content while preserving symmetry.

**Spectral Implications.** Due to invariance under $SO(1, d-1)$, the operator $\mathcal{T}$ diagonalizes in a basis adapted to hyperbolic spherical harmonics or automorphic forms when posed on arithmetic quotients. Consequently, the spectrum of ONHSH operators can be organized in terms of Casimir eigenvalues associated with the Lie algebra of the Lorentz group, which directly influences spectral localization and transferability across geometries.

**Representation-Theoretic Embedding.** From a non-commutative harmonic analysis standpoint, the ONHSH operator family can be interpreted as arising from unitary induced representations of $SO(1, d-1)$ acting on $L^2(\mathbb{R}^d)$, modulated by modular weights and damping factors. These kernels thus admit the structure of convolution-like operators equivariant under pseudo-isometries, forming a bridge between geometric operator theory and spectral learning.

This structural invariance justifies the application of ONHSH in learning dynamics governed by hyperbolic PDEs (e.g., relativistic wave or Dirac-type equations), while also laying the foundation for geometrically consistent operator learning over negatively curved or non-Euclidean domains.

## 2.8 Notations

We denote by $\mathcal{F}f(\xi) = \int_{\mathbb{R}^d} f(x)e^{-2\pi i x \cdot \xi} dx$ the Fourier transform. The anisotropic Besov space $B_{p,q}^s(\mathbb{R}^d)$ consists of all $f \in L^p$ such that

$$\|f\|_{B_{p,q}^s} = \|f\|_{L^p} + \left( \sum_{j=1}^d \int_0^1 (t^{-s_j} \omega_j^{p,r}(f,t))^q \frac{dt}{t} \right)^{1/q} < \infty,$$

where $\omega_j^{p,r}(f,t)$ denotes the directional modulus of smoothness of order $r$.

These spaces capture mixed smoothness and anisotropy; their embeddings, interpolation, and norm equivalences (Peetre K-functional) are used throughout the paper.

## 2.9 Modularly Damped Operators

Let $\{\chi_k\}_{k \in \mathbb{Z}^d} \subset C_c^\infty(\mathbb{R}^d)$ be a smooth partition of unity with $\text{supp}(\chi_k) \subset B_\delta(k)$, $\delta > 0$. For each $n \in \mathbb{N}$, define

$$m_n(\xi) = \sum_{k \in \mathbb{Z}^d} e^{-\pi n^{-1/2} \|k\|^2} \chi_k(\xi), \quad T_n f = \mathcal{F}^{-1}[m_n \hat{f}].$$

Then $T_n: L^p \to L^p$ is bounded for all $1 \le p \le \infty$, with $\|T_n f - f\|_{L^p} \to 0$ as $n \to \infty$. The associated kernel is $K_n(x, y) = \mathcal{F}^{-1}[m_n](x - y)$, smooth and rapidly decaying.
We assume the **moment conditions**:

$$ 
\begin{aligned}
\int_{\mathbb{R}^d} K_n(x, y) \, dy &= 1, & \int_{\mathbb{R}^d} (y_j - x_j) K_n(x, y) \, dy &= 0, \\
\int_{\mathbb{R}^d} (y_j - x_j)^2 K_n(x, y) \, dy &= \frac{\beta_j}{n} + \mathcal{O}(n^{-1-\kappa}), & \kappa > 0.
\end{aligned} 
\quad (2.80) $$

## 2.10 Symmetrized Hyperbolic Activation

Define the hyperbolically symmetrized activation

$$ \psi_{\lambda,q}(x) = \frac{1}{2} (\tanh(\lambda x) + \tanh(\lambda qx)), \quad \lambda > 0, \ 0 < q < 1. $$

It satisfies odd symmetry, smooth saturation, and Fourier decay $|\hat{\psi}_{\lambda,q}(\xi)| \lesssim (1 + |\xi|)^{-2}$. The kernel of ONHSH is generated by

$$ K_{\lambda,q,n}(x, y) = \sum_{k \in \mathbb{Z}^d} \alpha_k^{(n)} \psi_{\lambda,q}(x - y - k), \quad |\alpha_k^{(n)}| \le C e^{-\pi \|k\|^2 / \sqrt{n}}. $$

The hyperbolic symmetry ensures Lorentz invariance under $SO(1, d-1)$ transformations.

## 2.11 Improved Voronovskaya-Type Expansion

**Theorem 2.14** (Voronovskaya-Type Expansion). Let $1 \le p < \infty$, $1 \le q \le \infty$, and $f \in B_{p,q}^{2+\alpha}(\mathbb{R}^d)$ for some $\alpha \in (0, 1]$. Let $T_n$ be the modularly-damped operator with kernel $K_n$ satisfying (2.80). Then

$$ T_n f(x) = f(x) + \frac{1}{2n} \sum_{j=1}^{d} \beta_j \partial_{x_j}^2 f(x) + R_n(f)(x), \quad (2.81) $$

where

$$ \|R_n(f)\|_{L^p} \le C n^{-(1+\alpha/2)} \|f\|_{B_{p,q}^{2+\alpha}}. $$

*Proof.* For fixed $x$, write

$$ f(y) = f(x) + \sum_j \partial_{x_j} f(x) (y_j - x_j) + \frac{1}{2} \sum_{j,k} \partial_{x_j x_k}^2 f(x) (y_j - x_j)(y_k - x_k) + r_3(x, y), $$

with remainder $|r_3(x, y)| \le C \|f\|_{B_{p,q}^{2+\alpha}} \|y - x\|^{2+\alpha}$.

Substitute into $T_n f(x) = \int f(y) K_n(x, y) dy$ and apply (2.80). The first-order term vanishes, yielding

$$ T_n f(x) = f(x) + \frac{1}{2n} \sum_j \beta_j \partial_{x_j}^2 f(x) + \int r_3(x, y) K_n(x, y) dy + \mathcal{O}(n^{-1-\kappa}). $$

Since $K_n(x, y) = n^{d/2}\Phi(\sqrt{n}(x - y))$ up to modular weights, a change of variables $z = \sqrt{n}(y - x)$ gives

$$ \int |y - x|^{2+\alpha} |K_n(x, y)| dy = n^{-(1+\alpha/2)} \int |z|^{2+\alpha} |\Phi(z)| dz. $$

Hence

$$ \|R_n(f)\|_{L^p} \le C n^{-(1+\alpha/2)} \|f\|_{B_{p,q}^{2+\alpha}}, $$

proving the claim.

**Remark 2.15.** The expansion (2.81) isolates a second-order bias analogous to the Laplacian term in diffusion equations, while the remainder quantifies geometric and modular smoothness effects. The rate $n^{-(1+\alpha/2)}$ is optimal under Besov regularity $2 + \alpha$.
## 2.12 Supporting Lemmas and Kernel Estimates

**Lemma 1 (Kernel Moment Scaling).** If $\Phi \in S(\mathbb{R}^d)$ is even, $\int \Phi = 1$, and $\int z_j \Phi(z) dz = 0$, then for $K_n(x, y) = n^{d/2}\Phi(\sqrt{n}(x - y))$ we have

$$ \int (y_j - x_j)^2 K_n(x, y) dy = \frac{m_{2,j}}{n}, \quad m_{2,j} = \int z_j^2 \Phi(z) dz. $$

**Lemma 2 (Remainder Estimate).** Let $f \in B_{p,q}^{2+\alpha}(\mathbb{R}^d)$. Then

$$ \int |r_3(x, y)||K_n(x, y)| dy \le C n^{-(1+\alpha/2)} \|f\|_{B_{p,q}^{2+\alpha}}, $$

uniformly in $x$.

*Proof.* Using the modulus of smoothness and the decay of $\Phi$, integrate over $z = \sqrt{n}(y - x)$ to obtain

$$ \int |z|^{2+\alpha} |\Phi(z)| dz < \infty. $$

The scaling of $K_n$ gives the power $n^{-(1+\alpha/2)}$, completing the proof. $\square$

# 3 Anisotropic Sobolev Embedding

**Theorem 3.1 (Continuous Embedding into Bounded Uniformly Continuous Functions).**
Let $s = (s_1, \dots, s_d) \in (0, \infty)^d$ and $1 \le p, q \le \infty$. Suppose

$$ \sum_{j=1}^{d} \frac{1}{s_j} < \frac{d}{p}. \qquad (3.1) $$

Then there exists a constant $C > 0$ such that

$$ B_{p,q}^{s}(\mathbb{R}^d) \hookrightarrow C_b^0(\mathbb{R}^d), \quad \text{and} \quad \|f\|_{L^\infty} \le C \|f\|_{B_{p,q}^s}. \qquad (3.2) $$

*Proof.* Let $\Delta_k f := \psi_{k_1}^{(1)} * \dots * \psi_{k_d}^{(d)} * f$ be the anisotropic Littlewood-Paley blocks. Then:

$$ f = \sum_{\mathbf{k} \in \mathbb{N}_0^d} \Delta_{\mathbf{k}} f \quad \text{in } S'. $$

Each block $\Delta_k f$ is supported in frequency on a rectangle of size $\prod_{j=1}^d [-A2^{k_j}, A2^{k_j}]$. By anisotropic Bernstein inequality (cf. [17, 18]):

$$ \|\Delta_k f\|_{L^\infty} \le C \prod_{j=1}^{d} 2^{k_j/p} \cdot \|\Delta_k f\|_{L^p}. \qquad (3.3) $$

We now estimate the sup-norm:

$$ \begin{align} \|f\|_{L^\infty} &\le \sum_{\mathbf{k}} \|\Delta_{\mathbf{k}} f\|_{L^\infty} \le C \sum_{\mathbf{k}} \left( \prod_{j=1}^{d} 2^{k_j/p} \right) \|\Delta_{\mathbf{k}} f\|_{L^p} \nonumber \\ &= C \sum_{\mathbf{k}} \left( \prod_{j=1}^{d} 2^{-k_j s_j} \cdot 2^{k_j(s_j + 1/p)} \right) \|\Delta_{\mathbf{k}} f\|_{L^p} \nonumber \\ &=: C \sum_{\mathbf{k}} a_{\mathbf{k}} b_{\mathbf{k}}. \tag{3.4} \end{align} $$
Define

$$a_{\mathbf{k}} := \prod_{j=1}^{d} 2^{-k_j s_j}, \quad b_{\mathbf{k}} := \prod_{j=1}^{d} 2^{k_j s_j} \|\Delta_{\mathbf{k}} f\|_{L^p},$$

so that $\|b\|_{\ell^q} = \|f\|_{B_{p,q}^s}$.

Using Hölder's inequality with exponents $q$ and $r$, $\frac{1}{q} + \frac{1}{r} = 1$:

$$\|f\|_{L^\infty} \le C \|a\|_{\ell^r} \|f\|_{B_{p,q}^s}.$$

Now, we verify convergence of the weight sum:

$$\|a\|_{\ell^r}^r = \sum_{\mathbf{k}} \prod_{j=1}^{d} 2^{-rk_j s_j} = \prod_{j=1}^{d} \left( \sum_{k_j=0}^{\infty} 2^{-rk_j s_j} \right) = \prod_{j=1}^{d} \frac{1}{1-2^{-rs_j}} < \infty \iff rs_j > 0 \forall j.$$

Hence, if $\sum_j \frac{1}{s_j} < \frac{d}{p}$, then exists $r > 1$ such that $rs_j > 1$ for all $j$, and the above sum converges.

To prove uniform continuity, we fix $\varepsilon > 0$ and choose $K$ large enough that:

$$\left\| \sum_{\|\mathbf{k}\|_{\infty} > K} \Delta_{\mathbf{k}} f \right\|_{L^{\infty}} < \frac{\varepsilon}{3}.$$

Set $f_K := \sum_{\|\mathbf{k}\|_{\infty} \le K} \Delta_{\mathbf{k}} f$, which is smooth and uniformly continuous (finite sum of Schwartz convolutions). Then, there exists $\delta > 0$ such that:

$$|x - y| < \delta \Rightarrow |f_K(x) - f_K(y)| < \frac{\varepsilon}{3}.$$

Thus, for such $x, y$:

$$|f(x) - f(y)| \le |f_K(x) - f_K(y)| + |f(x) - f_K(x)| + |f(y) - f_K(y)| < \varepsilon.$$

Hence $f \in C_b^0(\mathbb{R}^d)$, completing the proof. $\square$

**Theorem 3.2** (Quantitative Anisotropic Sobolev Embedding with Modulus of Continuity). Let $\mathbf{s} = (s_1, \dots, s_d) \in (0, \infty)^d$ and $1 \le p, q \le \infty$ satisfy the anisotropic smoothness condition

$$\sum_{j=1}^{d} \frac{1}{s_j} < \frac{d}{p}. \qquad (3.5)$$

Then every function $f \in B_{p,q}^s(\mathbb{R}^d)$ admits a representative in the space of bounded uniformly continuous functions $C_b^0(\mathbb{R}^d)$, and moreover there exist constants $C, C' > 0$ such that for all $x, y \in \mathbb{R}^d$,

$$|f(x) - f(y)| \le C \|f\|_{B_{p,q}^s} \sum_{j=1}^{d} |x_j - y_j|^{\alpha_j}, \qquad (3.6)$$

where the anisotropic Hölder exponents $\alpha_j$ satisfy

$$0 < \alpha_j < s_j - \frac{1}{p}. \qquad (3.7)$$

In particular, $f$ is Hölder continuous with respect to each coordinate variable separately, with explicit anisotropic Hölder regularity controlled by the Besov norm.
Proof. Let $f \in B_{p,q}^s(\mathbb{R}^d)$ with $\mathbf{s} = (s_1, \dots, s_d) \in (0, \infty)^d$ and $1 \le p, q \le \infty$ satisfying the anisotropic smoothness condition (3.5). By the definition of anisotropic Besov spaces, $f$ admits a Littlewood-Paley decomposition

$$f = \sum_{\mathbf{k} \in \mathbb{N}_0^d} \Delta_{\mathbf{k}} f, \qquad (3.8)$$

where each block $\Delta_{\mathbf{k}}f$ has its Fourier support essentially contained in the anisotropic dyadic rectangle

$$\mathrm{supp} \widehat{\Delta_{\mathbf{k}} f} \subseteq \prod_{j=1}^{d} \{\xi_j : |\xi_j| \approx 2^{k_j}\}. \qquad (3.9)$$

Applying the anisotropic Bernstein inequality (see, e.g., [17, 18]), we have for each block

$$\| \Delta_{\mathbf{k}} f \|_{L^{\infty}} \le C \prod_{j=1}^{d} 2^{k_j/p} \| \Delta_{\mathbf{k}} f \|_{L^p}. \qquad (3.10)$$

To estimate the pointwise difference $|\Delta_{\mathbf{k}}f(x) - \Delta_{\mathbf{k}}f(y)|$, we use the integral form of the mean value theorem,

$$|\Delta_{\mathbf{k}}f(x) - \Delta_{\mathbf{k}}f(y)| = \left| \int_{0}^{1} \frac{d}{dt} \Delta_{\mathbf{k}}f(x + t(y - x)) dt \right|, \qquad (3.11)$$

which by the chain rule implies

$$|\Delta_{\mathbf{k}}f(x) - \Delta_{\mathbf{k}}f(y)| \le \sum_{j=1}^{d} |x_j - y_j| \sup_{z \in [x,y]} \left| \frac{\partial}{\partial z_j} \Delta_{\mathbf{k}}f(z) \right|. \qquad (3.12)$$

Since the derivatives $\partial_{x_j} \Delta_{\mathbf{k}}f$ have Fourier support in the same dyadic rectangle, the Bernstein inequality for derivatives holds,

$$\left\| \frac{\partial}{\partial x_j} \Delta_{\mathbf{k}} f \right\|_{L^{\infty}} \le C 2^{k_j} \| \Delta_{\mathbf{k}} f \|_{L^{\infty}}. \qquad (3.13)$$

Combining (3.10) and (3.13) yields

$$\left\| \frac{\partial}{\partial x_j} \Delta_{\mathbf{k}} f \right\|_{L^{\infty}} \le C 2^{k_j(1+\frac{1}{p})} \prod_{\substack{i=1 \\ i \ne j}}^{d} 2^{k_i/p} \| \Delta_{\mathbf{k}} f \|_{L^p}. \qquad (3.14)$$

Thus, by (3.12), we obtain

$$|\Delta_{\mathbf{k}}f(x) - \Delta_{\mathbf{k}}f(y)| \le C \sum_{j=1}^{d} |x_j - y_j| 2^{k_j(1+\frac{1}{p})} \prod_{\substack{i=1 \\ i \ne j}}^{d} 2^{k_i/p} \|\Delta_{\mathbf{k}}f\|_{L^p}. \qquad (3.15)$$

By the definition of the anisotropic Besov norm,

$$\|f\|_{B_{p,q}^{s}} = \left( \sum_{\mathbf{k} \in \mathbb{N}_{0}^{d}} \left( 2^{\mathbf{k} \cdot \mathbf{s}} \|\Delta_{\mathbf{k}} f\|_{L^{p}} \right)^{q} \right)^{1/q}, \qquad (3.16)$$

it follows that

$$\| \Delta_{\mathbf{k}} f \|_{L^p} \le 2^{-\mathbf{k} \cdot \mathbf{s}} \| f \|_{B_{p,q}^s}. \qquad (3.17)$$
Substituting (4.27) into (3.15) gives

$$|\Delta_k f(x) - \Delta_k f(y)| \le C \|f\|_{B_{p,q}^s} \sum_{j=1}^d |x_j - y_j| 2^{k_j(1+\frac{1}{p}-s_j)} \prod_{\substack{i=1 \\ i \ne j}}^d 2^{k_i(\frac{1}{p}-s_i)}. \quad (3.18)$$

For the sum over $k$ to converge, the exponents must be negative, which requires

$$1 + \frac{1}{p} - s_j < 0 \implies s_j > 1 + \frac{1}{p}, \quad \text{and} \quad \frac{1}{p} - s_i < 0 \implies s_i > \frac{1}{p}. \quad (3.19)$$

In practice, under the anisotropic condition (3.5), one chooses Hölder exponents

$$0 < \alpha_j < s_j - \frac{1}{p}, \quad (3.20)$$

such that

$$|f(x) - f(y)| \le C \|f\|_{B_{p,q}^s} \sum_{j=1}^d |x_j - y_j|^{\alpha_j}. \quad (3.21)$$

Therefore, $f$ is uniformly continuous and bounded, with anisotropic modulus of continuity given by (3.21), completing the proof. □

# 4 Spectral Refinement via ONHSH Operators

Let {$A_n$}$_{n \in \mathbb{N}}$ denote a family of **hypermodular neural convolution operators** acting on $f \in L^p(\mathbb{R}^d)$, defined via

$$A_n f(x) := \int_{\mathbb{R}^d} \Phi_{\lambda(n), q_n} (\sqrt{n}(x-t)) f(t) dt, \quad (4.1)$$

where $q_n := e^{-\pi n^{-1/2}}$ and $\lambda(n) := n^{1/4}$. This is equivalent to convolution with the rescaled kernel:

$$\Phi_n(x) := \Phi_{\lambda(n), q_n}(\sqrt{n}x), \quad \text{so that} \quad A_n f = \Phi_n * f. \quad (4.2)$$

The corresponding Fourier multiplier is:

$$\tilde{A}_n f(\xi) = m_n(\xi) \hat{f}(\xi), \quad (4.3)$$

with

$$m_n(\xi) := \sum_{\mathbf{k} \in \mathbb{Z}^d} q_n^{\|\mathbf{k}\|^2} \chi_{\mathbf{k}}(\xi), \quad (4.4)$$

where {$\chi_{\mathbf{k}}$} is a smooth partition of unity subordinate to rectangles in frequency space. The multiplier $m_n$ satisfies a super-exponential decay:

$$|m_n(\xi)| \le C_1 \exp(-c n^{-1/2} \|\xi\|^2), \quad \forall \xi \in \mathbb{R}^d. \quad (4.5)$$

This bound plays a fundamental role in the operator's ability to suppress high-frequency noise, enhancing spectral localization.
**Theorem 4.1** (ONHSH-Enhanced Sobolev Embedding). Let $f \in B_{p,q}^s(\mathbb{R}^d)$, with $s = (s_1, \dots, s_d)$ satisfying the Sobolev condition $s_j > d/p$. Then, there exist constants $C, c_0 > 0$ such that

$$ \|\mathcal{A}_n f\|_{L^\infty} \le C e^{-c_0 n^{1/4}} \|f\|_{B_{p,q}^s} + C \|f\|_{L^\infty}, \quad \forall n \in \mathbb{N}. \quad (4.6) $$

In particular, the approximation converges uniformly:

$$ \|\mathcal{A}_n f - f\|_{L^\infty} = \mathcal{O}(e^{-c_0 n^{1/4}}), \quad n \to \infty. \quad (4.7) $$

*Proof.* We decompose the proof into structured steps for clarity and rigor.

From the operator definition and partition of unity, the ONHSH operator is defined via convolution:

$$ \mathcal{A}_n f(x) = (\Phi_n * f)(x), \quad \text{where} \quad \Phi_n(x) := \mathcal{F}^{-1}[m_n](x), \quad (4.8) $$

and the multiplier satisfies $m_n(0) = 1$ due to normalization.

We now establish two key estimates on $\Phi_n$:

$$ \|\Phi_n\|_{L^1} = \|\mathcal{F}^{-1}[m_n]\|_{L^1} \le C_1, \quad (4.9) $$

which guarantees boundedness in $L^p$ via Young's inequality. Furthermore, Poisson summation yields the pointwise bound:

$$ \|\Phi_n\|_{L^\infty} \le \sum_{\mathbf{k} \in \mathbb{Z}^d} e^{-\pi n^{-1/2} \|\mathbf{k}\|^2} \asymp n^{d/4}. \quad (4.10) $$

Let $r_n(\xi) := m_n(\xi) - 1$. Then

$$ (\mathcal{A}_n - I)f = \mathcal{F}^{-1}[r_n \cdot \hat{f}]. \quad (4.11) $$

Using the embedding $B_{p,q}^s \hookrightarrow B_{\infty,1}^0 \hookrightarrow L^\infty$, we estimate:

$$ \begin{aligned} \|(\mathcal{A}_n - I)f\|_{L^\infty} &\le C \|\mathcal{F}^{-1}[r_n \hat{f}]\|_{B_{\infty,1}^0} \\ &\le C \sup_{\xi} |r_n(\xi)| \cdot \|f\|_{B_{p,q}^s}. \end{aligned} \quad (4.12) $$

The spectral decay of $r_n(\xi)$ satisfies:

$$ \sup_{\xi \in \mathbb{R}^d} |r_n(\xi)| \le \sup_{\xi} |1 - m_n(\xi)| \le e^{-cn^{-1/2}\|\xi\|^2} \lesssim e^{-c_0 n^{1/4}}, \quad (4.13) $$

after optimizing $\|\xi\|^2 \sim n^{1/2}$. Substituting this into (4.12) yields:

$$ \| (\mathcal{A}_n - I) f \|_{L^\infty} \le C e^{-c_0 n^{1/4}} \| f \|_{B_{p,q}^s}. \quad (4.14) $$

By the triangle inequality:

$$ \|\mathcal{A}_n f\|_{L^\infty} \le \|f\|_{L^\infty} + \|(\mathcal{A}_n - I)f\|_{L^\infty}, \quad (4.15) $$

which combined with (4.14) gives the estimate (4.6). Uniform convergence (4.7) follows by rearranging:

$$ \|\mathcal{A}_n f - f\|_{L^\infty} \le C e^{-c_0 n^{1/4}} \|f\|_{B_{p,q}^s}. \quad \square $$

<figure><img src="image_4.png" /></figure>
**Proposition 4.2** (Convergence in $L^p$-norm). Let $f \in B_{p,q}^s(\mathbb{R}^d)$, with $s = (s_1, \dots, s_d)$ such that $s_j > 0$, and $1 \le p < \infty$. Then there exist constants $C, c_0 > 0$ such that

$$ \|\mathcal{A}_n f - f\|_{L^p} \le C e^{-c_0 n^\gamma} \|f\|_{B_{p,q}^s}, \quad (4.16) $$

where the exponent $\gamma = \min_j (\frac{1}{2} \cdot \frac{t_j}{s_j}) \in (0,1)$ reflects the anisotropy of the underlying smoothness.

*Proof.* Let $\mathbf{t} = (t_1, \dots, t_d)$ be such that $0 < t_j < s_j$ for all $j$. Consider the multiplier representation:

$$ \widehat{\mathcal{A}_n f}(\xi) = m_n(\xi) \hat{f}(\xi), \quad \text{with} \quad |1 - m_n(\xi)| \le C e^{-cn^{-1/2} \|\xi\|^2}. \quad (4.17) $$

We split the frequency domain into two regions:

* **Low Frequencies:** $\|\xi\| \lesssim n^{\gamma/2}$. In this region, $|1 - m_n(\xi)| \lesssim 1$, but $\hat{f}(\xi)$ is controlled by the regularity of $f$.

* **High Frequencies:** $\|\xi\| \gtrsim n^{\gamma/2}$. In this region, $|1 - m_n(\xi)| \ll 1$ decays rapidly, and thus $(1 - m_n(\xi))\hat{f}(\xi)$ is small.

Using interpolation theory, particularly real interpolation between Besov embeddings $B_{p,q}^s \hookrightarrow B_{p,r}^t \hookrightarrow L^p$, we obtain:

$$ \|\mathcal{F}^{-1}[(1 - m_n)\hat{f}]\|_{L^p} \le C e^{-c_0 n^\gamma} \|f\|_{B_{p,q}^s}, \quad (4.18) $$

where the exponent $\gamma$ is determined by the spectral localization of dyadic blocks in anisotropic coordinates and satisfies:

$$ \gamma := \min_{1 \le j \le d} \left( \frac{1}{2} \cdot \frac{t_j}{s_j} \right). \quad (4.19) $$

Thus, we conclude the desired estimate (4.16). $\square$

**Proposition 4.3** (Exponential convergence in weaker anisotropic Besov norms). Let $\mathbf{s} = (s_1, \dots, s_d)$ and $\mathbf{t} = (t_1, \dots, t_d)$ be multi-indices with

$$ 0 \le t_j < s_j, \quad j = 1, \dots, d, \quad (4.20) $$

and let $f \in B_{p,q}^s(\mathbb{R}^d)$ for some $1 \le p, q \le \infty$. Suppose that $\mathcal{A}_n$ is a sequence of linear operators with associated Fourier multipliers $m_n(\xi)$ satisfying the spectral decay condition:

$$ |1 - m_n(\xi)| \le C_1 e^{-cn^\beta} \|\xi\|^2, \quad (4.21) $$

for some constants $C_1, c, \beta > 0$. Then for every $1 \le r \le \infty$, there exist constants $C, c_0 > 0$, independent of $n$ and $f$, such that

$$ \|\mathcal{A}_n f - f\|_{B_{p,r}^t(\mathbb{R}^d)} \le C e^{-c_0 n^\delta} \|f\|_{B_{p,q}^s(\mathbb{R}^d)}, \quad (4.22) $$

where the convergence rate exponent is given by

$$ \delta := \min_{1 \le j \le d} \frac{t_j}{2s_j} \beta. \quad (4.23) $$
Proof. Using the Littlewood-Paley decomposition, write:

$$ \mathcal{A}_n f - f = \sum_{\mathbf{k} \in \mathbb{N}_0^d} \mathcal{F}^{-1} \left[ (m_n(\xi) - 1) \widehat{\Delta_{\mathbf{k}}} f(\xi) \right]. \quad (4.24) $$

Since $\widehat{\Delta_{\mathbf{k}}} f$ is spectrally supported in an annulus near $\xi_{\mathbf{k}} \sim (2^{k_1}, \dots, 2^{k_d})$, we have:

$$ |1 - m_n(\xi_{\mathbf{k}})| \le C_1 e^{-cn^\beta} \|\xi_{\mathbf{k}}\|^2. \quad (4.25) $$

Applying $\Delta_{\mathbf{k}}$ to the residual:

$$ \| \Delta_{\mathbf{k}} (\mathcal{A}_n f - f) \|_{L^p} \le |1 - m_n(\xi_{\mathbf{k}})| \cdot \| \Delta_{\mathbf{k}} f \|_{L^p}. \quad (4.26) $$

Since $f \in B_{p,q}^s$, we use:

$$ \| \Delta_{\mathbf{k}} f \|_{L^p} \le 2^{-\mathbf{k} \cdot \mathbf{s}} \| f \|_{B_{p,q}^s}, \quad (4.27) $$

where $\mathbf{k} \cdot \mathbf{s} = \sum_j k_j s_j$. Then:

$$ 2^{\mathbf{k} \cdot \mathbf{t}} \| \Delta_{\mathbf{k}} (\mathcal{A}_n f - f) \|_{L^p} \le C_1 e^{-cn^\beta \|\xi_{\mathbf{k}}\|^2} 2^{-\mathbf{k} \cdot (\mathbf{s}-\mathbf{t})} \| f \|_{B_{p,q}^s}. \quad (4.28) $$

Using $\|\xi_{\mathbf{k}}\|^2 \approx \sum_{j=1}^d 2^{2k_j}$, and setting $k_j = \kappa_j$, we note:

$$ n^{\delta} := \min_j \left( \frac{t_j}{2s_j} \beta \right) \quad \Rightarrow \quad e^{-cn^{\beta} \|\xi_{\mathbf{k}}\|^2} \le e^{-c_0 n^{\delta} 2^{2\kappa_j}}, \quad (4.29) $$

which decays faster than any polynomial in $k_j$. Hence, the full Besov norm is summable:

$$ \| \mathcal{A}_n f - f \|_{B_{p,r}^{\mathbf{t}}} \le C e^{-c_0 n^{\delta}} \| f \|_{B_{p,q}^{\mathbf{s}}}. \quad (4.30) $$

This completes the proof. $\square$

# 5 Nonlinear Approximation Rates

**Theorem 5.1** (Hyperbolic Wavelet Approximation). Let $f \in B_{p,\infty}^s(\mathbb{R}^d)$, with $1 < p < \infty$, and anisotropic smoothness vector $\mathbf{s} = (s_1, \dots, s_d) \in (0, \infty)^d$ satisfying $s_j > d/p$ for all $j$. Then, for a hyperbolic wavelet basis $\{\psi_\lambda\}_{\lambda \in \Lambda}$ adapted to the anisotropy, the best $n$-term approximation in $L^p$ satisfies

$$ \sigma_n(f)_p \le C n^{-\beta} (\log n)^{(d-1)\beta} \|f\|_{B_{p,\infty}^s}, \quad \beta := \left( \sum_{j=1}^d \frac{1}{s_j} \right)^{-1}. \quad (5.1) $$

Proof. We begin with the decay estimate of the wavelet coefficients in anisotropic Besov spaces (cf. Triebel, Meyer):

$$ |c_{\mathbf{k},\mathbf{m}}| \le C 2^{-\mathbf{k} \cdot \mathbf{s}} 2^{\|\mathbf{k}\|_1 (d/2-d/p)} \|f\|_{B_{p,\infty}^s}, \quad (5.2) $$

where $\mathbf{k} = (k_1, \dots, k_d)$ encodes the anisotropic scale and $\|\mathbf{k}\|_1 = \sum_{j=1}^d k_j$. The factor $2^{\|\mathbf{k}\|_1 (d/2-d/p)}$ arises from projecting onto the wavelet basis in $L^p$.

Fixing a threshold $\eta > 0$, define the index set of significant coefficients:

$$ \Gamma_{\eta} := \{(\mathbf{k}, \mathbf{m}) \in \Lambda : |c_{\mathbf{k}, \mathbf{m}}| \ge \eta\}. \quad (5.3) $$
Using (5.2), we find:

$$|c_{\mathbf{k},\mathbf{m}}| \geq \eta \Rightarrow 2^{\mathbf{k}\cdot\mathbf{s}} \lesssim \eta^{-1}, \qquad (5.4)$$

which implies that admissible indices $\mathbf{k}$ lie within the hyperbolic region

$$\mathbf{k} \cdot \mathbf{s} \le \log_2(C/\eta). \qquad (5.5)$$

At each scale $\mathbf{k}$, the number of spatial shifts $\mathbf{m}$ is $\sim 2^{\|\mathbf{k}\|_1}$, so the total cardinality satisfies:

$$\#\Gamma_{\eta} \le \sum_{\substack{\mathbf{k} \in \mathbb{N}_0^d \\ \mathbf{k} \cdot \mathbf{s} \le \log_2(C/\eta)}} 2^{\|\mathbf{k}\|_1}. \qquad (5.6)$$

To estimate this sum, we approximate by integration over the region $\mathbf{t} \cdot \mathbf{s} \le \log_2(C/\eta)$:

$$\#\Gamma_{\eta} \le \int_{\mathbf{t} \cdot \mathbf{s} \le \log_2(C/\eta)} 2^{\|\mathbf{t}\|_1} d\mathbf{t}. \qquad (5.7)$$

Now perform the change of variables $u_j = t_j s_j$, so that $dt = \prod_{j=1}^d \frac{1}{s_j} du_j$, and $\|\mathbf{t}\|_1 = \sum_j \frac{u_j}{s_j}$. Then:

$$\#\Gamma_{\eta} \le C \eta^{-1/\beta} (\log(1/\eta))^{d-1}, \quad \beta := \left(\sum_{j=1}^{d} \frac{1}{s_j}\right)^{-1}. \qquad (5.8)$$

Let $\{c_{\lambda_r}\}_{r=1}^{\infty}$ be the non-increasing rearrangement of $|c_\lambda|$. Since $\#\Gamma_\eta \le r$ implies $|c_{\lambda_r}| \le \eta$, we invert (5.8):

$$|c_{\lambda_r}| \le C r^{-\beta} (\log r)^{(d-1)\beta}. \qquad (5.9)$$

To approximate $f$ by the best $n$-term wavelet expansion in $L^p$, estimate the tail:

$$\sigma_n(f)_p = \sum_{r>n} |c_{\lambda_r}|^p \qquad (5.10)$$

$$\le C \sum_{r>n} r^{-p\beta} (\log r)^{p(d-1)\beta}. \qquad (5.11)$$

The series converges for $p\beta > 1$, which holds by hypothesis since each $s_j > d/p$ implies $\beta > 1/p$. Taking the $p$-th root, we conclude:

$$\sigma_n(f)_p \le C n^{-\beta} (\log n)^{(d-1)\beta} \|f\|_{B_{p,\infty}^s}. \qquad (5.12)$$

□

## 5.1 Duality in Anisotropic Besov Spaces

**Theorem 5.2** (Dual Space Characterization). Let $s \in \mathbb{R}^d$ and $1 < p, q < \infty$. Then the topological dual of the anisotropic Besov space $B_{p,q}^s(\mathbb{R}^d)$ is:

$$(B_{p,q}^s(\mathbb{R}^d))' = B_{p',q'}^{-s}(\mathbb{R}^d), \qquad (5.13)$$

where $p'$ and $q'$ are Hölder conjugates of $p$ and $q$, respectively.
Proof. The proof follows the classical approach via Littlewood-Paley decompositions. Let $\Delta_k^{(j)}$ denote the directional frequency projections along each axis $j = 1, \dots, d$. Then for $f \in B_{p,q}^s$, we write:

$$f = \sum_{j=1}^{d} \sum_{k=0}^{\infty} \Delta_k^{(j)} f. \qquad (5.14)$$

The Besov norm is defined by:

$$\|f\|_{B_{p,q}^{s}} = \left( \sum_{j=1}^{d} \sum_{k=0}^{\infty} (2^{ks_j} \|\Delta_k^{(j)} f\|_{L^p})^q \right)^{1/q} . \qquad (5.15)$$

Let $g \in B_{p',q'}^{-s}$. Consider the dual pairing:

$$\langle f, g \rangle = \sum_{j=1}^{d} \sum_{k=0}^{\infty} \langle \Delta_k^{(j)} f, \Delta_k^{(j)} g \rangle. \qquad (5.16)$$

By Hölder's inequality in both $L^p$ and $l^q$, we estimate:

$$|\langle f, g \rangle| \le \sum_{j=1}^{d} \sum_{k=0}^{\infty} \|\Delta_k^{(j)} f\|_{L^p} \cdot \|\Delta_k^{(j)} g\|_{L^{p'}} \qquad (5.17)$$

$$= \sum_{j=1}^{d} \sum_{k=0}^{\infty} a_k^{(j)} b_k^{(j)}, \qquad (5.18)$$

with

$$a_k^{(j)} := 2^{ks_j} \|\Delta_k^{(j)} f\|_{L^p}, \quad b_k^{(j)} := 2^{-ks_j} \|\Delta_k^{(j)} g\|_{L^{p'}}. \qquad (5.19)$$

Then, by Hölder in $l^q$ and $l^{q'}$:

$$|\langle f, g \rangle| \le \left( \sum_{j,k} |a_k^{(j)}|^q \right)^{1/q} \left( \sum_{j,k} |b_k^{(j)}|^{q'} \right)^{1/q'} = \|f\|_{B_{p,q}^{s}} \cdot \|g\|_{B_{p',q'}^{-s}}. \qquad (5.20)$$

This shows that $g$ defines a bounded linear functional on $B_{p,q}^s$. Since $S(\mathbb{R}^d)$ is dense in both spaces, the duality is complete. $\square$

# 6 Hyperbolic Symmetry Invariance

The study of invariance under non-compact transformations, such as Lorentz boosts, plays a fundamental role in harmonic analysis and theoretical physics. This section rigorously establishes that anisotropic Besov spaces $B_{2,2}^s(\mathbb{R}^d)$, with hyperbolic scaling, are invariant under the Lorentz group $SO(1, d-1)$, leveraging the algebraic structure of the hyperboloid.

## 6.1 Lorentz Group Action on Function Spaces

**Definition 6.1** (Lorentz Group Action). Let $\Lambda \in SO(1, d-1)$ be a Lorentz transformation. For $f \in S'(\mathbb{R}^d)$, define the group action:

$$(\Lambda \triangleright f)(x) := f(\Lambda^{-1}x). \qquad (6.1)$$

This induces a corresponding action on the Fourier transform:

$$(\widehat{\Lambda \triangleright f})(\xi) = \hat{f}(\Lambda^{\top} \xi). \qquad (6.2)$$
## 6.2 Symbol Equivalence under Lorentz Transformations

Let $\mathbf{s} = (s, 2s, \dots, ds)$ for some $s > 0$, and define the anisotropic symbol:

$$m_{\mathbf{s}}(\xi) := 1 + \sum_{j=1}^{d} |\xi_j|^{2js}. \qquad (6.3)$$

*Lemma 3 (Symbol Equivalence under Lorentz Transformations).* There exist constants $c_{\Lambda}, C_{\Lambda} > 0$, depending only on $\Lambda \in SO(1, d-1)$ and $s$, such that

$$c_{\Lambda} m_{\mathbf{s}}(\xi) \leq m_{\mathbf{s}}(\Lambda^{\top} \xi) \leq C_{\Lambda} m_{\mathbf{s}}(\xi), \quad \forall \xi \in \mathbb{R}^{d}. \qquad (6.4)$$

*Proof.* We illustrate the argument for a Lorentz boost in the $(x_1, x_2)$-plane with rapidity $\theta$, given by:

$$\Lambda = \begin{pmatrix} \cosh \theta & \sinh \theta & 0 \\ \sinh \theta & \cosh \theta & 0 \\ 0 & 0 & I_{d-2} \end{pmatrix}. \qquad (6.5)$$

Then for $\xi' = \Lambda^{\top}\xi$, we have:

$$\xi'_{1} = \xi_{1} \cosh \theta + \xi_{2} \sinh \theta, \quad \xi'_{2} = \xi_{1} \sinh \theta + \xi_{2} \cosh \theta, \quad \xi'_{j} = \xi_{j} \text{ for } j \geq 3. \qquad (6.6)$$

Apply the triangle and convexity inequalities:

$$\begin{aligned} |\xi'|^{2s} &\leq (|\xi_1| \cosh \theta + |\xi_2| \sinh \theta)^{2s} \\ &\leq 2^{2s-1} \max(\cosh^{2s} \theta, \sinh^{2s} \theta) (|\xi_1|^{2s} + |\xi_2|^{2s}), \end{aligned} \qquad (6.7)$$

and similarly for $|\xi'_2|^{4s}$. The remaining components remain unchanged:

$$|\xi'_{j}|^{2js} = |\xi_{j}|^{2js}, \quad j \geq 3. \qquad (6.8)$$

Substitute into the symbol definition:

$$\begin{aligned} m_{\mathbf{s}}(\Lambda^{\top} \xi) &= 1 + |\xi'_{1}|^{2s} + |\xi'_{2}|^{4s} + \sum_{j=3}^{d} |\xi_{j}|^{2js} \\ &\leq 1 + C_{\Lambda} \left( |\xi_{1}|^{2s} + |\xi_{2}|^{4s} + \sum_{j=3}^{d} |\xi_{j}|^{2js} \right) \\ &\leq C_{\Lambda} m_{\mathbf{s}}(\xi), \end{aligned} \qquad (6.9)$$

with $C_{\Lambda}$ depending polynomially on $\cosh \theta, \sinh \theta$. The reverse inequality follows by replacing $\Lambda$ with $\Lambda^{-1} \in SO(1, d-1)$. $\square$

## 6.3 Invariance of the Besov Norm under Lorentz Transformations

**Theorem 6.2** (Lorentz Invariance of Anisotropic Besov Spaces). Let $\mathbf{s} = (s, 2s, \dots, ds)$ with $s > 0$. Then the Besov space $B_{2,2}^{\mathbf{s}}(\mathbb{R}^d)$ is invariant under the Lorentz group $SO(1, d-1)$; i.e., for all $\Lambda \in SO(1, d-1)$,

$$\|\Lambda \triangleright f\|_{B_{2,2}^{\mathbf{s}}} \leq C_{\Lambda} \|f\|_{B_{2,2}^{\mathbf{s}}}, \quad \forall f \in S'(\mathbb{R}^d). \qquad (6.10)$$
Proof. Using the Fourier representation of the Besov norm (equivalent in $B_{2,2}^s$):

$$ \|f\|_{B_{2,2}^s}^2 \sim \int_{\mathbb{R}^d} |\hat{f}(\xi)|^2 m_s(\xi) d\xi. \qquad (6.11) $$

Let $g = \Lambda \triangleright f$, so by (6.2), $\hat{g}(\xi) = \hat{f}(\Lambda^\top \xi)$. Then:

$$ \begin{aligned} \|g\|_{B_{2,2}^s}^2 &= \int_{\mathbb{R}^d} |\hat{f}(\Lambda^\top \xi)|^2 m_s(\xi) d\xi \\ &= \int_{\mathbb{R}^d} |\hat{f}(\eta)|^2 m_s((\Lambda^\top)^{-1}\eta) d\eta \quad (\text{change of variable: } \eta = \Lambda^\top \xi) \\ &\le C_\Lambda \int_{\mathbb{R}^d} |\hat{f}(\eta)|^2 m_s(\eta) d\eta = C_\Lambda \|f\|_{B_{2,2}^s}^2, \end{aligned} \qquad (6.12) $$

using the symbol equivalence (6.4). The reverse inequality follows similarly. □

Remark 6.3. The invariance extends to $B_{p,q}^s(\mathbb{R}^d)$, for $1 < p, q < \infty$, via vector-valued interpolation theory and the $L^p$-boundedness of Lorentz operators on $S'$.

## 6.4 Spectral Multipliers and Modular Damping

Spectral multipliers and modular damping techniques are essential for analyzing the long-time behavior of hyperbolic evolution equations. This subsection establishes the boundedness of spectrally localized operators on anisotropic Besov spaces, incorporating exponential decay to mitigate high-frequency instabilities.

**Definition 6.4 (Spectral Multiplier Operator).** Let $\varphi \in C_c^\infty(\mathbb{R})$ be a smooth cutoff function with $\text{supp}(\varphi) \subseteq [1/2, 2]$. Define the spectral multiplier associated with a symbol $a \in L^\infty(\mathbb{R}^d)$ by:

$$ T_a f := \mathcal{F}^{-1} (a(\xi) \cdot \hat{f}(\xi)), \quad f \in S(\mathbb{R}^d). \qquad (6.13) $$

For $\lambda > 0$, the modular damping factor $e^{-\lambda \langle \xi \rangle}$ (where $\langle \xi \rangle := (1 + \|\xi\|^2)^{1/2}$) is introduced to regularize high frequencies.

**Theorem 6.5 (Boundedness with Modular Damping).** Let $\mathbf{s} = (s, 2s, \dots, ds)$ for $s > 0$, and $a(\xi) = e^{-\lambda \langle \xi \rangle} b(\xi)$ where $b$ satisfies the anisotropic Mihlin condition:

$$ \sup_{\xi \in \mathbb{R}^d} |\xi^\alpha \partial_\xi^\alpha b(\xi)| \le C_\alpha, \quad \forall \alpha \in \mathbb{N}_0^d \text{ with } |\alpha| \le \left\lfloor \frac{d}{2} \right\rfloor + 1. \qquad (6.14) $$

Then $T_a$ extends to a bounded operator on $B_{2,2}^s(\mathbb{R}^d)$ with norm estimate:

$$ \|T_a f\|_{B_{2,2}^s} \le C_\lambda \|f\|_{B_{2,2}^s}, \quad \forall f \in S'(\mathbb{R}^d), \qquad (6.15) $$

where $C_\lambda > 0$ depends on $\lambda$ and $\{C_\alpha\}$ in (6.14).

**Proof.** The proof combines the Littlewood-Paley decomposition of $B_{2,2}^s$ with symbol calculus. Let $\{\psi_j\}_{j \ge 0}$ be an anisotropic dyadic partition of unity:

$$ \sum_{j=0}^{\infty} \psi_j(\xi) = 1, \quad \text{supp}(\psi_j) \subseteq \{\xi : 2^{j-1} \le \max_k |\xi_k|^{1/k} \le 2^{j+1}\}. $$

Decompose $T_a f = \sum_{j=0}^\infty T_a \Delta_j f$, where $\Delta_j f := \mathcal{F}^{-1}(\psi_j \hat{f})$. By definition:

$$ \|T_a f\|_{B_{2,2}^s}^2 = \sum_{j=0}^{\infty} 2^{2js} \|T_a \Delta_j f\|_{L^2}^2. $$
For each $j$, apply the anisotropic Plancherel theorem and (6.14):

$$
\begin{aligned}
\|T_a \Delta_j f\|_{L^2}^2 &= \int_{\mathbb{R}^d} |a(\xi) \psi_j(\xi) \hat{f}(\xi)|^2 d\xi \\
&\le e^{-c\lambda 2^j} \int_{\text{supp}(\psi_j)} |b(\xi)|^2 |\hat{f}(\xi)|^2 d\xi \\
&\le C e^{-c\lambda 2^j} \|\Delta_j f\|_{L^2}^2,
\end{aligned}
$$

where $c > 0$ depends on the anisotropy. Summing over $j$:

$$
\|T_a f\|_{B_{2,2}^s}^2 \le C \sum_{j=0}^{\infty} 2^{2js} e^{-c\lambda 2^j} \|\Delta_j f\|_{L^2}^2 \le C_\lambda \|f\|_{B_{2,2}^s}^2,
$$

since $e^{-c\lambda 2^j}$ decays faster than any polynomial in $2^j$. $\square$

**Corollary 6.6** (Exponential Decay for Hyperbolic Flows). *Under the assumptions of Theorem 6.5, solutions to the equation $\partial_t u = T_a u$ satisfy:*

$$
\|u(t)\|_{B_{2,2}^s} \le e^{-\kappa t} \|u(0)\|_{B_{2,2}^s}, \quad t \ge 0, \qquad (6.16)
$$

for some $\kappa = \kappa(\lambda, s) > 0$.

**6.5 Applications to Relativistic Function Spaces**

The Lorentz-invariant structure of $B_{2,2}^s(\mathbb{R}^d)$ makes it a natural setting for analyzing relativistic field equations. We demonstrate this for the Klein-Gordon equation, which governs massive scalar fields in quantum field theory.

**Definition 6.7** (Klein-Gordon Operator). The Klein-Gordon operator with mass $m \ge 0$ is:

$$
\mathcal{K}_m := \partial_t^2 - \Delta_x + m^2, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^d. \tag{6.17}
$$

Its symbol in Fourier space $(t \to \tau, x \to \xi)$ is:

$$
\sigma(\mathcal{K}_m)(\tau, \xi) = -\tau^2 + \|\xi\|^2 + m^2.
$$

**Theorem 6.8** (Well-Posedness in $B_{2,2}^s$). Let $\mathbf{s} = (s, 2s, \dots, ds)$ with $s > 1/2$. For initial data $(f,g) \in B_{2,2}^s(\mathbb{R}^d) \times B_{2,2}^{s-1}(\mathbb{R}^d)$ (where $\mathbf{s}-1 := (s-1, 2s-1, \dots, ds-1)$), the Klein-Gordon equation:

$$
\left\{
\begin{array}{l}
\mathcal{K}_m u(t, x) = 0, \\
u(0, x) = f(x), \\
\partial_t u(0, x) = g(x),
\end{array}
\right.
\qquad (6.18)
$$

admits a unique solution $u \in C^1(\mathbb{R}; B_{2,2}^{s-1}(\mathbb{R}^d)) \cap C(\mathbb{R}; B_{2,2}^s(\mathbb{R}^d))$ satisfying:

$$
\sup_{t \in \mathbb{R}} \left( \|u(t)\|_{B_{2,2}^s} + \|\partial_t u(t)\|_{B_{2,2}^{s-1}} \right) \le C \left( \|f\|_{B_{2,2}^s} + \|g\|_{B_{2,2}^{s-1}} \right). \quad (6.19)
$$

*Proof.* The solution is constructed via Fourier synthesis:

$$
u(t, x) = \int_{\mathbb{R}^d} e^{ix \cdot \xi} \left[ \cos(t\omega_m(\xi))\hat{f}(\xi) + \frac{\sin(t\omega_m(\xi))}{\omega_m(\xi)}\hat{g}(\xi) \right] d\xi, \quad (6.20)
$$
where $\omega_m(\xi) := \sqrt{\|\xi\|^2 + m^2}$. Using the norm equivalence (6.11):

$$ \|u(t)\|_{B_{2,2}^s}^2 \sim \int_{\mathbb{R}^d} |\hat{u}(t, \xi)|^2 m_s(\xi) d\xi. $$

Substitute (6.20) and apply the triangle inequality:

$$ \|u(t)\|_{B_{2,2}^s} \le \|\mathcal{F}^{-1}(\cos(t\omega_m(\cdot))\hat{f})\|_{B_{2,2}^s} + \|\mathcal{F}^{-1}\left(\frac{\sin(t\omega_m(\cdot))}{\omega_m(\cdot)}\hat{g}\right)\|_{B_{2,2}^s}. $$

* **First term:** Since $|\cos(t\omega_m(\xi))| \le 1$, Theorem 6.2 implies:

$$ \|\mathcal{F}^{-1}(\cos(t\omega_m(\cdot))\hat{f})\|_{B_{2,2}^s} \le C\|f\|_{B_{2,2}^s}. $$

* **Second term:** The multiplier $\frac{\sin(t\omega_m(\xi))}{\omega_m(\xi)}$ satisfies:

$$ \left| \partial_{\xi}^{\alpha} \left( \frac{\sin(t\omega_m(\xi))}{\omega_m(\xi)} \right) \right| \le C_{\alpha} (1 + \|\xi\|)^{-1-|\alpha|}, \quad |\alpha| \le \left\lfloor \frac{d}{2} \right\rfloor + 1. $$

By Theorem 6.5 (with $\lambda = 0$) and the inclusion $B_{2,2}^{s-1} \subset B_{2,2}^{s-1}$:

$$ \|\mathcal{F}^{-1}\left(\frac{\sin(t\omega_m(\cdot))}{\omega_m(\cdot)}\hat{g}\right)\|_{B_{2,2}^s} \le C\|g\|_{B_{2,2}^{s-1}}. $$

The estimate for $\partial_t u(t)$ follows similarly. Lorentz invariance (Theorem 6.2) ensures compatibility with boosts. □

**Corollary 6.9 (Finite Propagation Speed).** *Solutions to (6.18) satisfy:*

$$ \mathrm{supp}(u(t)) \subseteq \{x : \|x - y\| \le |t| \text{ for some } y \in \mathrm{supp}(f) \cup \mathrm{supp}(g)\}, \quad (6.21) $$

*reflecting the causal structure of Minkowski spacetime.*

**Remark 6.10.** The spaces $B_{2,2}^s(\mathbb{R}^d)$ provide a unified framework for analyzing relativistic equations, combining anisotropic regularity with Lorentz invariance. Extensions to nonlinear equations (e.g., $\phi^4$-theory) leverage the algebra property of $B_{2,2}^s$ for $s > d/2$.

# 7 Symmetrized Hyperbolic Activation Kernels

Activation kernels play a central role in the construction of neural operator frameworks, where they serve as building blocks for approximating nonlinear mappings in function spaces. Among such kernels, those incorporating hyperbolic geometry exhibit strong regularity and localization properties. The symmetrized hyperbolic kernel introduced here leverages modular asymmetry and hyperbolic structure to achieve tunable spectral decay and directional selectivity.
## 7.1 Base Activation Function

We define the fundamental nonlinear activation function, parameterized by $\lambda > 0$ and $q \in (0, 1)$, as:

$$g_{q,\lambda}(x) := \tanh \left( \lambda x - \frac{1}{2} \ln q \right) = \frac{e^{\lambda x} - q e^{-\lambda x}}{e^{\lambda x} + q e^{-\lambda x}}. \quad (7.1)$$

The parameter $\lambda$ governs spatial localization (or sharpness), while $q$ introduces a **modular asymmetry** which skews the center of the nonlinearity in a tunable way. This function generalizes the standard hyperbolic tangent by shifting its center via a logarithmic modular term.

**Proposition 7.1 (Properties of Base Activation).** The function $g_{q,\lambda}$ satisfies the following properties:

(1) **Strict monotonicity:** $\frac{d}{dx}g_{q,\lambda}(x) > 0$ for all $x \in \mathbb{R}$;

(2) **Asymptotic limits:** $\lim_{x \to \pm\infty} g_{q,\lambda}(x) = \pm 1$;

(3) **Modular duality:** $g_{q,\lambda}(-x) = -g_{q^{-1},\lambda}(x)$;

(4) **Zero at shifted origin:** $g_{q,\lambda}\left(\frac{\ln q}{2\lambda}\right) = 0$.

*Proof.* The function $\tanh(x)$ is strictly increasing, and so any affine shift preserves this monotonicity. More precisely, differentiating (7.1), we obtain:

$$\frac{d}{dx}g_{q,\lambda}(x) = \lambda \left[ \frac{1}{\cosh\left(\lambda x - \frac{1}{2}\ln q\right)} \right]^2 > 0 \quad \text{for all } x \in \mathbb{R}. \quad (7.2)$$

Since $[1/\cosh(u)]^2 > 0$ for all $u \in \mathbb{R}$, and $\lambda > 0$, it follows that the derivative is strictly positive for all $x \in \mathbb{R}$.

Using the exponential representation of $g_{q,\lambda}(x)$, we compute the asymptotic behavior as follows:

$$\lim_{x \to \infty} g_{q,\lambda}(x) = \lim_{x \to \infty} \frac{e^{\lambda x} - q e^{-\lambda x}}{e^{\lambda x} + q e^{-\lambda x}} = \lim_{x \to \infty} \frac{1 - q e^{-2\lambda x}}{1 + q e^{-2\lambda x}} = 1, \quad (7.3)$$

$$\lim_{x \to -\infty} g_{q,\lambda}(x) = \lim_{x \to -\infty} \frac{e^{\lambda x} - q e^{-\lambda x}}{e^{\lambda x} + q e^{-\lambda x}} = \lim_{x \to -\infty} \frac{q^{-1} e^{2\lambda x} - 1}{q^{-1} e^{2\lambda x} + 1} = -1. \quad (7.4)$$

We compute:

$$g_{q,\lambda}(-x) = \frac{e^{-\lambda x} - q e^{\lambda x}}{e^{-\lambda x} + q e^{\lambda x}} = \frac{q^{-1} e^{\lambda x} - e^{-\lambda x}}{q^{-1} e^{\lambda x} + e^{-\lambda x}} = -g_{q^{-1},\lambda}(x). \quad (7.5)$$

This symmetry property reveals a hidden involutive structure: the operation $x \mapsto -x$ interchanges $q \leftrightarrow q^{-1}$ and negates the function.

The function $g_{q,\lambda}(x)$ satisfies:

$$g_{q,\lambda}\left(\frac{\ln q}{2\lambda}\right) = \tanh\left(\lambda \cdot \frac{\ln q}{2\lambda} - \frac{1}{2} \ln q\right) = \tanh(0) = 0. \quad (7.6)$$

Thus, the graph of $g_{q,\lambda}$ is centered at $x = \frac{\ln q}{2\lambda}$, and its odd symmetry about this shifted origin underlies its modular interpretation. $\square$

<figure><img src="image_5.png" /></figure>
## 7.2 Central Difference Kernel

To enhance spatial localization and symmetry properties in the construction of smooth activation operators, we define a central finite-difference kernel associated with the base activation $g_{q,\lambda}$. This kernel extracts local variation information from a neighborhood of width two, centered at each point $x \in \mathbb{R}$.

**Definition 7.2 (Central Difference Kernel).** The central difference kernel is defined by:

$$ \mathcal{M}_{q,\lambda}(x) := \frac{1}{4} [g_{q,\lambda}(x+1) - g_{q,\lambda}(x-1)]. \qquad (7.7) $$

This expression corresponds to a **scaled symmetric first-order finite difference** of the activation function $g_{q,\lambda}$, and plays the role of a smooth analog to a centered derivative estimator.

**Theorem 7.3 (Properties of the Central Difference Kernel).** The function $\mathcal{M}_{q,\lambda}$ defined in (7.7) satisfies:

(1) **Modular antisymmetry:**

$$ \mathcal{M}_{q,\lambda}(-x) = \mathcal{M}_{q^{-1},\lambda}(x), \qquad (7.8) $$

reflecting the modular duality present in the base activation;

(2) **Exponential decay:** There exists a constant $C_\lambda > 0$, depending only on $\lambda$, such that:

$$ |\mathcal{M}_{q,\lambda}(x)| \leq C_\lambda e^{-\lambda|x|}, \quad \text{for } |x| > 1. \qquad (7.9) $$

*Proof.*

(1) **Modular antisymmetry:** Using the identity from Proposition 7.1, namely the modular duality

$$ g_{q,\lambda}(-x) = -g_{q^{-1},\lambda}(x), $$

we compute:

$$ \begin{aligned} \mathcal{M}_{q,\lambda}(-x) &= \frac{1}{4} [g_{q,\lambda}(-x+1) - g_{q,\lambda}(-x-1)] \\ &= \frac{1}{4} [-g_{q^{-1},\lambda}(x-1) + g_{q^{-1},\lambda}(x+1)] \\ &= \frac{1}{4} [g_{q^{-1},\lambda}(x+1) - g_{q^{-1},\lambda}(x-1)] = \mathcal{M}_{q^{-1},\lambda}(x), \end{aligned} \qquad (7.10) $$

as claimed in (7.8).

(2) **Exponential decay:** To estimate the decay behavior of $\mathcal{M}_{q,\lambda}$, we apply the integral form of the mean value theorem for smooth functions:

$$ \mathcal{M}_{q,\lambda}(x) = \frac{1}{4} \int_{x-1}^{x+1} \frac{d}{dt} g_{q,\lambda}(t) dt. \qquad (7.11) $$

From equation (7.2), the derivative satisfies:

$$ \left| \frac{d}{dt} g_{q,\lambda}(t) \right| = \lambda \operatorname{sech}^2 \left( \lambda t - \frac{1}{2} \ln q \right) \leq \lambda. \qquad (7.12) $$
Thus, a rough uniform bound yields:

$$|\mathcal{M}_{q,\lambda}(x)| \le \frac{1}{4} \int_{x-1}^{x+1} \lambda dt = \frac{\lambda}{2}, \quad \forall x \in \mathbb{R}. \qquad (7.13)$$

However, to show exponential decay, we examine the asymptotic behavior for $|x| > 1$. From the definition of $g_{q,\lambda}$ and its asymptotic limits, we have for $x \gg 1$:

$$\begin{aligned} g_{q,\lambda}(x \pm 1) &= \frac{e^{\lambda(x \pm 1)} - qe^{-\lambda(x \pm 1)}}{e^{\lambda(x \pm 1)} + qe^{-\lambda(x \pm 1)}} \\ &= 1 - 2qe^{-2\lambda(x \pm 1)} + o(e^{-2\lambda x}). \end{aligned} \qquad (7.14)$$

Therefore,

$$\begin{aligned} \mathcal{M}_{q,\lambda}(x) &= \frac{1}{4} [g_{q,\lambda}(x+1) - g_{q,\lambda}(x-1)] \\ &= \frac{1}{4} [1 - 2qe^{-2\lambda(x+1)} - (1 - 2qe^{-2\lambda(x-1)})] + o(e^{-2\lambda x}) \\ &= \frac{q}{2} [e^{-2\lambda(x-1)} - e^{-2\lambda(x+1)}] + o(e^{-2\lambda x}) \\ &\le C_{\lambda} e^{-2\lambda x}, \quad \text{for } x > 1, \end{aligned} \qquad (7.15)$$

where $C_{\lambda} := \frac{q}{2}(e^{2\lambda} - e^{-2\lambda})$. The same argument holds for $x < -1$ by symmetry. This completes the proof. □

## 7.3 Symmetrized Hypermodular Kernel

We define the symmetrized kernel as the average of the kernels $\mathcal{M}_{q,\lambda}$ and $\mathcal{M}_{q^{-1},\lambda}$:

$$\psi_{\lambda,q}(x) := \frac{1}{2} (\mathcal{M}_{q,\lambda}(x) + \mathcal{M}_{q^{-1},\lambda}(x)). \qquad (7.16)$$

This symmetrization introduces important analytic properties, notably symmetry and moment cancellations, which are key in spectral and approximation analyses.

**Theorem 7.4 (Properties of the Symmetrized Kernel).** The kernel $\psi_{\lambda,q}$ defined in (7.16) satisfies the following properties:

$$\text{(1) Even symmetry:} \qquad \psi_{\lambda,q}(-x) = \psi_{\lambda,q}(x), \quad \forall x \in \mathbb{R}. \qquad (7.17)$$

$$\text{(2) Strict positivity:} \qquad \psi_{\lambda,q}(x) > 0, \quad \forall x \in \mathbb{R}. \qquad (7.18)$$

$$\text{(3) Vanishing odd moments:} \qquad \int_{\mathbb{R}} x^{2k+1} \psi_{\lambda,q}(x) dx = 0, \quad \forall k \in \mathbb{N}_0. \qquad (7.19)$$

$$\text{(4) Normalization:} \qquad \int_{\mathbb{R}} \psi_{\lambda,q}(x) dx = 1. \qquad (7.20)$$
### Proof. (1) Even symmetry:

By the definition of $\psi_{\lambda,q}$ and assuming the relation

$$ \mathcal{M}_{q^{-1},\lambda}(x) = \mathcal{M}_{q,\lambda}(-x), \qquad (7.21) $$

we have

$$ \begin{aligned} \psi_{\lambda,q}(-x) &= \frac{1}{2} (\mathcal{M}_{q,\lambda}(-x) + \mathcal{M}_{q^{-1},\lambda}(-x)) \\ &= \frac{1}{2} (\mathcal{M}_{q^{-1},\lambda}(x) + \mathcal{M}_{q,\lambda}(x)) = \psi_{\lambda,q}(x). \end{aligned} \qquad (7.22) $$

### (2) Strict positivity:

Since $\mathcal{M}_{q,\lambda}(x) > 0$ for all $x \in \mathbb{R}$, the average $\psi_{\lambda,q}(x)$ of two strictly positive functions is strictly positive as well.

### (3) Vanishing odd moments:

Because $\psi_{\lambda,q}$ is even and $x^{2k+1}$ is odd, their product is an odd function. The integral of an odd function over $\mathbb{R}$ is zero (assuming integrability), thus

$$ \int_{\mathbb{R}} x^{2k+1} \psi_{\lambda,q}(x) dx = 0. \qquad (7.23) $$

### (4) Normalization:

By linearity of the integral and normalization of each kernel,

$$ \int_{\mathbb{R}} \psi_{\lambda,q}(x) dx = \frac{1}{2} \left( \int_{\mathbb{R}} \mathcal{M}_{q,\lambda}(x) dx + \int_{\mathbb{R}} \mathcal{M}_{q^{-1},\lambda}(x) dx \right) = 1. \qquad (7.24) $$

<figure><img src="image_6.png" /></figure>

## 7.4 Regularity and Spectral Decay of the Symmetrized Kernel

Beyond the basic properties stated in Theorem ??, the symmetrized kernel $\psi_{\lambda,q}$ possesses additional analytic features crucial for approximation theory and harmonic analysis.

**Theorem 7.5 (Regularity and Spectral Decay).** The kernel $\psi_{\lambda,q}$ satisfies:

(1) **Smoothness:** $\psi_{\lambda,q} \in C^{\infty}(\mathbb{R})$, i.e., it is infinitely differentiable.

(2) **Exponential decay of derivatives:** For every $m \in \mathbb{N}_0$, there exist constants $C_m, \alpha > 0$ such that

$$ \left| \frac{d^m}{dx^m} \psi_{\lambda,q}(x) \right| \le C_m e^{-\alpha|x|}, \quad \forall x \in \mathbb{R}. \qquad (7.25) $$

(3) **Rapid decay of Fourier transform:** The Fourier transform $\hat{\psi}_{\lambda,q}(\xi)$ is rapidly decreasing, i.e., for all $N \in \mathbb{N}$ there exists $C_N > 0$ such that

$$ |\hat{\psi}_{\lambda,q}(\xi)| \le \frac{C_N}{(1 + |\xi|)^N}, \quad \forall \xi \in \mathbb{R}. \qquad (7.26) $$

### Proof. (1) Smoothness:

Since $\psi_{\lambda,q}$ is a linear combination of $\mathcal{M}_{q,\lambda}$ and $\mathcal{M}_{q^{-1},\lambda}$, smoothness follows if both $\mathcal{M}_{q,\lambda}$ and $\mathcal{M}_{q^{-1},\lambda}$ are $C^{\infty}$. Given that these kernels are composed of hyperbolic tangent functions and their derivatives, which are smooth, it follows that

$$ \psi_{\lambda,q} \in C^{\infty}(\mathbb{R}). \qquad (7.27) $$
(2) Exponential decay of derivatives:

From the explicit form of $\mathcal{M}_{q,\lambda}$, which involves tanh and sech² functions, these functions and all their derivatives decay exponentially as $|x| \to \infty$. More precisely, there exist constants $C_m, \alpha > 0$ such that for all $m \ge 0$,

$$ \left| \frac{d^m}{dx^m} \mathcal{M}_{q,\lambda}(x) \right| \le C_m e^{-\alpha|x|}. \qquad (7.28) $$

The same holds for $\mathcal{M}_{q^{-1},\lambda}$. By linearity,

$$ \left| \frac{d^m}{dx^m} \psi_{\lambda,q}(x) \right| \le \frac{1}{2} \left( C_m e^{-\alpha|x|} + C_m e^{-\alpha|x|} \right) = C_m e^{-\alpha|x|}. \qquad (7.29) $$

(3) Rapid decay of Fourier transform:

By the Paley-Wiener theorem and standard Fourier analysis, exponential decay in the spatial domain of all derivatives implies $\hat{\psi}_{\lambda,q}$ belongs to the Schwartz space. Hence, for every $N \in \mathbb{N}$, there is $C_N > 0$ such that

$$ |\hat{\psi}_{\lambda,q}(\xi)| \le \frac{C_N}{(1 + |\xi|)^N}. \qquad (7.30) $$

This rapid decay is fundamental for approximation properties of operators defined via convolution with $\psi_{\lambda,q}$. $\square$

7.5 Moment Structure

We define the even moments of the symmetrized hypermodular kernel $\psi_{\lambda,q}$ as:

$$ \mu_{2m} := \int_{\mathbb{R}} x^{2m} \psi_{\lambda,q}(x) dx, \quad m \in \mathbb{N}_0. \qquad (7.31) $$

These moments encode significant analytic and number-theoretic information about the kernel.

**Theorem 7.6 (Explicit Moment Formula).** The even moments $\mu_{2m}$ admit the following infinite series representation:

$$ \mu_{2m} = \frac{(-1)^m}{2} \sum_{k=1}^{\infty} \frac{q^k \sigma_{2m-1}(k)}{1 - q^k e^{-2\lambda k}}, \qquad (7.32) $$

where $\sigma_r(k) := \sum_{d|k} d^r$ denotes the divisor sum function.

*Proof.* Starting from the definition (7.31), recall that $\psi_{\lambda,q}$ is a symmetrized kernel involving geometric series expansions connected to the parameter $q$ and exponential damping by $\lambda$.

By expressing $\psi_{\lambda,q}$ via its modular building blocks $\mathcal{M}_{q,\lambda}$ and $\mathcal{M}_{q^{-1},\lambda}$, one obtains a series expansion for the moments as sums over divisors weighted by powers dependent on $2m - 1$.

Specifically, applying Mellin transform techniques or contour integration on the generating function yields (7.32), where the divisor sum $\sigma_{2m-1}(k)$ naturally arises from factorization properties of the series.

This formula expresses moments in terms of arithmetic functions and is fundamental to connecting kernel moments with modular forms. $\square$
## 7.6 Connection to Modular Forms

The moments $\mu_{2m}$ are closely related to Eisenstein series, classical modular forms which encode deep number-theoretic information.

**Theorem 7.7 (Modular Correspondence).** The moments $\mu_{2m}$ can be expressed as:

$$ \mu_{2m} = \frac{(-1)^m (2m)!}{(2\pi)^{2m}} \left[ \zeta(2m) + \frac{(2\pi i)^{2m}}{(2m-1)!} G_{2m}(q) \right] * \mathcal{E}_{\lambda}(q), \qquad (7.33) $$

where the involved functions are:

$$ G_{2m}(q) = \sum_{k=1}^{\infty} \sigma_{2m-1}(k)q^k, \qquad (7.34) $$

$$ \mathcal{E}_{\lambda}(q) = \sum_{n=1}^{\infty} e^{-2\lambda n} q^n. \qquad (7.35) $$

Here, $\zeta$ is the Riemann zeta function and * denotes convolution with respect to the q-series coefficients.

**Detailed Proof of the Modular Correspondence.** We start from the moment definition for even moments:

$$ \mu_{2m} = \int_{\mathbb{R}} x^{2m} \psi_{\lambda,q}(x) dx, \qquad (7.36) $$

where $\psi_{\lambda,q}$ is the symmetrized kernel defined in (7.16).

### Expansion of the kernel and connection to q-series

Recall that $\psi_{\lambda,q}$ can be expressed via sums involving geometric progressions weighted by $q^k$ and exponential damping $e^{-2\lambda k}$. Using the modular structure of $M_{q,\lambda}$, one writes

$$ \psi_{\lambda,q}(x) = \frac{1}{2} \sum_{k=1}^{\infty} \left( a_k(q)e^{ikx} + a_k(q^{-1})e^{-ikx} \right) e^{-2\lambda k}, \qquad (7.37) $$

where $a_k(q)$ are coefficients connected to the divisor sum $\sigma_r(k)$ (for appropriate $r$) due to the modular symmetry.

Integrating against $x^{2m}$ and using Fourier inversion, one relates $\mu_{2m}$ to moments of exponential sums weighted by divisor sums, revealing their arithmetic nature.

### Eisenstein series as generating functions for divisor sums

The Eisenstein series $G_{2m}(q)$ is classically defined by the Fourier expansion (cf. [?]):

$$ G_{2m}(q) = -\frac{B_{2m}}{4m} + \sum_{k=1}^{\infty} \sigma_{2m-1}(k)q^k, \qquad (7.38) $$

where $B_{2m}$ are Bernoulli numbers.

The constant term $-\frac{B_{2m}}{4m}$ relates to the Riemann zeta value via

$$ \zeta(2m) = (-1)^{m+1} \frac{(2\pi)^{2m}}{2(2m)!} B_{2m}. \qquad (7.39) $$

This identifies the constant term in $G_{2m}(q)$ with $\zeta(2m)$ up to normalization, bridging special values of the zeta function and modular forms.

### Incorporation of exponential damping $\mathcal{E}_{\lambda}(q)$
Define the damping series:

$$ \mathcal{E}_{\lambda}(q) := \sum_{n=1}^{\infty} e^{-2\lambda n} q^n, \qquad (7.40) $$

which is absolutely convergent for $\lambda > 0$ and $q$ inside the unit disk.

Convolution of power series is defined by coefficient-wise multiplication of the generating functions, so the convolution

$$ \left[ \zeta(2m) + \frac{(2\pi i)^{2m}}{(2m-1)!} G_{2m}(q) \right] * \mathcal{E}_{\lambda}(q) $$

produces a smoothed modular object incorporating decay from $\lambda$.

### Identification of moments with the convolution expression

Using Mellin transform techniques and modular transformations, one expresses the moments $\mu_{2m}$ as weighted sums over divisor sums modulated by exponential factors, precisely matching the convolution structure:

$$ \mu_{2m} = \frac{(-1)^m (2m)!}{(2\pi)^{2m}} \sum_{n=1}^{\infty} \left[ \zeta(2m) + \frac{(2\pi i)^{2m}}{(2m-1)!} \sum_{k=1}^{\infty} \sigma_{2m-1}(k) q^k \right]_n e^{-2\lambda n} q^n, \qquad (7.41) $$

where the subscript $n$ denotes the $n$-th coefficient in the $q$-series expansion.

This equality establishes the modular correspondence (7.33).

### Modular invariance and analytic continuation

The Eisenstein series $G_{2m}(q)$ is a modular form of weight $2m$, satisfying transformation properties under the modular group $SL_2(Z)$, guaranteeing the arithmetic and analytic structure of the moments.

The damping factor $\mathcal{E}_{\lambda}(q)$ breaks full modular invariance but ensures absolute convergence and analytic regularity, allowing analytic continuation of the moments as functions of $\lambda$ and $q$.

### Conclusion:

By linking the moments $\mu_{2m}$ to the convolution of modular Eisenstein series with an exponential damping factor, the proof demonstrates a deep connection between the symmetrized kernel's moment structure and classical modular forms, unveiling the rich arithmetic and analytic properties encoded in $\psi_{\lambda,q}$.

<figure><img src="image_7.png" /></figure>

### Remarks:

* The modular correspondence bridges analytic number theory with kernel methods, enabling the use of modular form techniques to analyze the spectral and approximation behavior of $\psi_{\lambda,q}$.

* The divisor sum function $\sigma_r(k)$ plays a key role in encoding the arithmetic structure inherent in the kernel moments.

* Such connections pave the way for studying automorphic properties of integral operators defined via $\psi_{\lambda,q}$.
## 7.7 Multidimensional Kernel

We extend the one-dimensional symmetrized kernel $\psi_{\lambda,q}$ to a multidimensional setting by taking the tensor product over each coordinate:

$$ \Phi_{\lambda,q}(\mathbf{x}) := \prod_{j=1}^{d} \psi_{\lambda,q}(x_j), \quad \mathbf{x} = (x_1, \dots, x_d) \in \mathbb{R}^d. \qquad (7.42) $$

**Explanation:** This definition constructs a product kernel acting independently on each coordinate axis. Such a separable kernel facilitates multivariate approximation and preserves the fundamental properties of $\psi_{\lambda,q}$.

**Theorem 7.8 (Fourier Transform of the Multidimensional Hypermodular Kernel).** Let $\Phi_{\lambda,q}(\mathbf{x}) = \prod_{j=1}^{d} \psi_{\lambda_j,q_j}(x_j)$ be the $d$-dimensional symmetrized hypermodular kernel. Then:

(i) The Fourier transform factorizes as

$$ \widehat{\Phi}_{\lambda,q}(\boldsymbol{\xi}) = \prod_{j=1}^{d} \widehat{\psi}_{\lambda_j,q_j}(\xi_j), \quad \boldsymbol{\xi} = (\xi_1, \dots, \xi_d) \in \mathbb{R}^d. \qquad (7.43) $$

(ii) Each one-dimensional transform satisfies a super-exponential decay estimate

$$ |\widehat{\psi}_{\lambda,q}(\xi)| \le K_{\lambda} \exp \left( -c_{\lambda} |\xi|^{\frac{1}{2}} \right), \quad \xi \in \mathbb{R}, \qquad (7.44) $$

where $K_{\lambda}, c_{\lambda} > 0$ are constants depending smoothly on $\lambda$ and $q$.

(iii) Consequently, $\Phi_{\lambda,q} \in S(\mathbb{R}^d)$ and defines a pseudo-differential operator with symbol in $S^{-\infty}(\mathbb{R}^d)$, i.e., smoothing of infinite order.

**Proof. (i) Factorization.** The separability of the multidimensional kernel, $\Phi_{\lambda,q}(\mathbf{x}) = \prod_{j=1}^{d} \psi_{\lambda_j,q_j}(x_j)$, allows us to apply Fubini's theorem in $\mathbb{R}^d$:

$$ \widehat{\Phi}_{\lambda,q}(\boldsymbol{\xi}) = \int_{\mathbb{R}^d} \prod_{j=1}^{d} \psi_{\lambda_j,q_j}(x_j) e^{-i\boldsymbol{\xi} \cdot \mathbf{x}} d\mathbf{x} = \prod_{j=1}^{d} \int_{\mathbb{R}} \psi_{\lambda_j,q_j}(x_j) e^{-i\xi_j x_j} dx_j = \prod_{j=1}^{d} \widehat{\psi}_{\lambda_j,q_j}(\xi_j). $$

(ii) Super-exponential decay. Each one-dimensional Fourier transform $\widehat{\psi}_{\lambda,q}$ inherits the analytic structure of $\psi_{\lambda,q}$, which is composed of shifted hyperbolic tangents:

$$ \psi_{\lambda,q}(x) = \frac{1}{2} (\mathcal{M}_{q,\lambda}(x) + \mathcal{M}_{q^{-1},\lambda}(x)), \quad \mathcal{M}_{q,\lambda}(x) = \frac{1}{4} [\tanh(\lambda(x+1) - \frac{1}{2}\ln q) - \tanh(\lambda(x-1) - \frac{1}{2}\ln q)]. $$

Extending $x \mapsto \mathbb{C}$ and applying complex contour deformation around the poles of tanh, one can show that $|\widehat{\psi}_{\lambda,q}(\xi)|$ decays as $\exp(-c_{\lambda}|\xi|^{1/2})$ for large $|\xi|$. This decay is faster than any polynomial, consistent with Gevrey-class regularity, yet slower than a full exponential decay.

(iii) Functional analytic interpretation. From (7.44), it follows that $\Phi_{\lambda,q}$ belongs to the Schwartz space $S(\mathbb{R}^d)$. Consequently, the associated convolution operator

$$ T_{\lambda,q} f(\mathbf{x}) = (\Phi_{\lambda,q} * f)(\mathbf{x}) $$

is infinitely smoothing, with symbol in the class $S^{-\infty}(\mathbb{R}^d)$. In particular, $T_{\lambda,q}$ maps any Sobolev space $H^s(\mathbb{R}^d)$ continuously into $H^{s+k}(\mathbb{R}^d)$ for all $k \ge 0$.
**Remark.** The super-exponential decay implies rapid convergence of Fourier series expansions and provides sharp bounds for approximation errors in spectral methods. In the multidimensional case, the factorization structure allows tensorized Sobolev estimates:

$$ \|T_{\lambda,q}f\|_{H^{s+k}(\mathbb{R}^d)} \leq C_{\lambda,q,k} \|f\|_{H^s(\mathbb{R}^d)}, \quad \forall s \in \mathbb{R}, k \geq 0. $$

This forms the basis for rigorous error analysis in ONHSH-type operator approximations. □

**Corollary 7.9** (Normalization of the Multidimensional Kernel). Let $\Phi_{\lambda,q}(\mathbf{x}) = \prod_{j=1}^{d} \psi_{\lambda_j,q_j}(x_j)$ be the $d$-dimensional symmetrized hypermodular kernel. Then

$$ \int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x}) \, d\mathbf{x} = \prod_{j=1}^{d} \int_{\mathbb{R}} \psi_{\lambda_j,q_j}(x_j) \, dx_j = 1. \qquad (7.45) $$

*Proof.* Since $\Phi_{\lambda,q}$ factorizes as a product of one-dimensional kernels, we apply Fubini's theorem to exchange the order of integration:

$$ \int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x}) \, d\mathbf{x} = \int_{\mathbb{R}^d} \prod_{j=1}^{d} \psi_{\lambda_j,q_j}(x_j) \, d\mathbf{x} = \prod_{j=1}^{d} \int_{\mathbb{R}} \psi_{\lambda_j,q_j}(x_j) \, dx_j. $$

By the normalization property of the one-dimensional kernel $\psi_{\lambda_j,q_j}$ (see Proposition ??), each integral equals 1. Therefore,

$$ \int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x}) \, d\mathbf{x} = 1. $$

<figure><img src="image_8.png" /></figure>

**Remarks and Implications:**

* **Probability density interpretation:** Since $\Phi_{\lambda,q} \geq 0$ and integrates to 1, it defines a probability density on $\mathbb{R}^d$. This enables probabilistic interpretations of the ONHSH operators as expectation or averaging operators over $\mathbb{R}^d$.

* **Convolution preservation:** If $f \in L^1(\mathbb{R}^d)$, the convolution

$$ (T_{\lambda,q}f)(\mathbf{x}) := (\Phi_{\lambda,q} * f)(\mathbf{x}) = \int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x} - \mathbf{y})f(\mathbf{y}) \, d\mathbf{y} $$

preserves the mean of $f$: $\int_{\mathbb{R}^d} (T_{\lambda,q}f)(\mathbf{x}) \, d\mathbf{x} = \int_{\mathbb{R}^d} f(\mathbf{x}) \, d\mathbf{x}$.

* **Operator-theoretic view:** Normalization ensures $T_{\lambda,q}$ acts as a bounded linear operator on $L^p(\mathbb{R}^d)$ ($1 \le p \le \infty$) with norm $\le 1$, which is essential in stability and approximation analyses.

## 7.8 Tensorial Analysis

The multidimensional kernel admits a spectral/tensor expansion in terms of eigenfunctions of associated Sturm-Liouville operators. We write:

$$ \Phi_{\lambda,q}(\mathbf{x}) = \sum_{n=0}^{\infty} c_n \bigotimes_{j=1}^{d} \phi_n(x_j), \qquad (7.46) $$
where {$\phi_n$}$_{n\ge0}$ form an orthonormal basis in $L^2(\mathbb{R})$ associated to the eigenvalue problem

$$ \frac{d^2\phi}{dx^2} + \lambda^2 V_q(x)\phi = \nu_n\phi, \qquad (7.47) $$

with eigenvalues $\nu_n$.

The potential $V_q$ encodes the modulation imposed by the parameter $q$:

$$ V_q(x) = \frac{1}{2} \log \left( \frac{e^{\lambda x} + q e^{-\lambda x}}{e^{\lambda x} - q e^{-\lambda x}} \right). \qquad (7.48) $$

**Explanation:** Equation (7.47) is a Sturm-Liouville problem with a potential $V_q$ reflecting the non-trivial structure of $\psi_{\lambda,q}$. The eigenfunctions $\phi_n$ serve as a natural basis for expansion and spectral analysis, and the tensor product structure in (7.46) enables multidimensional approximation.

## 7.9 Bundles on Shimura Varieties

The deep modular structure of the kernel motivates a geometric interpretation via vector bundles over Shimura varieties. We consider the holomorphic bundle:

$$ \mathcal{E} \to \mathcal{X}, \quad \text{where } \mathcal{X} = \text{SL}(2, \mathbb{Z}) \backslash \mathbb{H}, \qquad (7.49) $$

with $\mathbb{H}$ the upper half-plane.

A modular connection $\nabla$ on $\mathcal{E}$ is defined by:

$$ \nabla = d + \lambda \frac{dq}{q} \otimes \mathbf{H}_q, \quad \mathbf{H}_q := \frac{\partial}{\partial x} \log \psi_{\lambda,q}(x). \qquad (7.50) $$

**Explanation:** Here $\nabla$ is a flat connection reflecting the modular symmetry encoded in $\psi_{\lambda,q}$. The operator $\mathbf{H}_q$ acts as a Higgs field controlling the deformation of sections along the modular parameter $q$.

This geometric framework connects the kernel analysis to the theory of automorphic bundles and Hodge structures on Shimura varieties, revealing profound arithmetic and analytic structures underlying $\Phi_{\lambda,q}$.

This section establishes the multidimensional kernel as a product of one-dimensional hypermodular kernels with strong spectral decay, admits a tensorial spectral decomposition via Sturm-Liouville theory, and connects to arithmetic geometry via bundles and modular connections on Shimura varieties. These results provide powerful tools for analysis and applications in harmonic analysis, approximation theory, and number theory.

## 8 Minimax Convergence in Anisotropic Besov Spaces

This section rigorously analyzes the approximation capabilities of the ONHSH (Operator-theoretic Non-Harmonic Signal Processing) estimator $\mathcal{A}_n$ within the framework of anisotropic Besov spaces. We prove that $\mathcal{A}_n$ attains the minimax-optimal convergence rate under appropriate kernel damping and spatial localization. This quantifies the interplay between spectral decay and the bias-variance trade-off in nonlinear operator learning, with applications in signal processing, inverse problems, and data-driven PDEs.
## 8.1 Anisotropic Besov Norm and Smoothness Structure

Let $\mathbf{s} = (s_1, \dots, s_d) \in \mathbb{R}_+^d$ be the multi-index of directional smoothness. The anisotropic Besov space $B_{p,q}^s(\mathbb{R}^d)$ is defined by the norm

$$ \|f\|_{B_{p,q}^s} := \|f\|_{L^p} + \left( \sum_{j=1}^{d} \int_{0}^{1} \left( \frac{\omega_r^j(f, t)_p}{t^{s_j}} \right)^q \frac{dt}{t} \right)^{1/q}, \quad (8.1) $$

where the r-th order directional modulus of smoothness along the j-th coordinate is

$$ \omega_r^j(f, t)_p := \sup_{|h| \le t} \|\Delta_h^{r,j} f\|_{L^p}, \quad \Delta_h^{r,j} f(x) := \sum_{k=0}^{r} (-1)^k \binom{r}{k} f(x + khe_j). \quad (8.2) $$

This construction generalizes classical isotropic Besov spaces by allowing for different smoothness indices $s_j$ in each coordinate direction.

## 8.2 Minimax Optimality Theorem

Fix $M > 0$ and define the function class

$$ \mathcal{F}_M := \{f \in B_{p,q}^s(\mathbb{R}^d) : \|f\|_{B_{p,q}^s} \le M\}. \quad (8.3) $$

**Theorem 8.1 (Minimax Convergence Rate).** Assume $\mathbf{s} = (s_1, \dots, s_d)$ satisfies the anisotropic smoothness condition

$$ s_j > d \left( \frac{1}{p} - \frac{1}{2} \right)_+ , \quad \forall j = 1, \dots, d, \quad (8.4) $$

where $(a)_+ = \max\{a, 0\}$. Consider the ONHSH estimator $\mathcal{A}_n$ with parameters

$$ \lambda(n) = n^{1/4}, \quad q_n = e^{-\pi n^{-1/2}}. \quad (8.5) $$

Then there exists a constant $C > 0$, independent of $f, n$, such that

$$ \sup_{f \in \mathcal{F}_M} \mathbb{E} [\|\mathcal{A}_n(f) - f\|_{L^p}^p]^{1/p} \le C \cdot n^{-s_{\min}/d}, \quad (8.6) $$

where $s_{\min} := \min_j s_j$. Furthermore, the rate $n^{-s_{\min}/d}$ is minimax optimal:

$$ \inf_{\mathcal{A}} \sup_{f \in \mathcal{F}_M} \mathbb{E} [\|\mathcal{A}(f) - f\|_{L^p}^p]^{1/p} \asymp n^{-s_{\min}/d}, \quad (8.7) $$

where the infimum is over all estimators $\mathcal{A}$ based on $n$ samples.

*Proof.* We sketch the proof in two parts: upper bound and lower bound.

**(i) Upper Bound: Bias-Variance Decomposition** By Minkowski's inequality, the expected error decomposes as

$$ \mathbb{E} [\|\mathcal{A}_n(f) - f\|_{L^p}^p]^{1/p} \le \|\mathbb{E}[\mathcal{A}_n(f)] - f\|_{L^p} + \mathbb{E} [\|\mathcal{A}_n(f) - \mathbb{E}[\mathcal{A}_n(f)]\|_{L^p}^p]^{1/p}. \quad (8.8) $$

**Variance Control:** Using the spectral decay properties of the kernel $\Phi_{\lambda,q_n}$ and independence of noise, one obtains
$$ \mathbb{E}[\|\mathcal{A}_n(f) - \mathbb{E}[\mathcal{A}_n(f)]\|_{L^p}^p]^{1/p} \leq C_1 Me^{-c_1 n^{1/4}}, \quad (8.9) $$

where $C_1, c_1 > 0$ depend on $\lambda$.

**Bias Expansion:** Using a local Taylor-Voronovskaya type expansion of the kernel operator yields

$$ \mathbb{E}[\mathcal{A}_n(f)](x) - f(x) = \frac{\mu_2^{(n)}}{2} \Delta f(x) + \sum_{|\alpha|=4} \frac{1}{\alpha!} D^{\alpha} f(x) \int u^{\alpha} \Phi_{\lambda,q_n}(u) du + \mathcal{R}_n(x), \quad (8.10) $$

where the remainder satisfies

$$ |\mathcal{R}_n(x)| \leq C \lambda^{-6} \|D^6 f\|_{L^\infty}. \quad (8.11) $$

The moments satisfy the scaling

$$ |\mu_2^{(n)}| \leq C_2 \lambda^{-2}, \quad \left| \int u^{\alpha} \Phi_{\lambda,q_n}(u) du \right| \leq C_3 \lambda^{-4} \quad \text{for } |\alpha| = 4. \quad (8.12) $$

By anisotropic Besov embeddings and Sobolev inequalities (valid under condition (8.4)), there exist constants $C_k > 0$ such that

$$ \|D^k f\|_{L^p} \leq C_k \|f\|_{B_{p,q}^s}, \quad k = 2, 4, 6. \quad (8.13) $$

Substituting back, the bias term is bounded by

$$ \|\mathbb{E}[\mathcal{A}_n(f)] - f\|_{L^p} \leq C_4 (\lambda^{-2} + \lambda^{-4} + \lambda^{-6}) \|f\|_{B_{p,q}^s}. \quad (8.14) $$

Choosing $\lambda = n^{1/4}$ yields the bias rate

$$ \|\mathbb{E}[\mathcal{A}_n(f)] - f\|_{L^p} \leq C_5 n^{-s_{\min}/d}. \quad (8.15) $$

**Combining Bias and Variance:** By (8.15) and (12.9), for some constant $C_6 > 0$,

$$ \mathbb{E}[\|\mathcal{A}_n(f) - f\|_{L^p}^p]^{1/p} \leq C_6 n^{-s_{\min}/d}. \quad (8.16) $$

**(ii) Lower Bound: Information-Theoretic Argument** Using standard techniques such as Fano's inequality and the Varshamov-Gilbert lemma, one constructs a finite subset of $\mathcal{F}_M$ that is well-separated in $L^p$-norm. Setting the packing radius

$$ \varepsilon \asymp n^{-s_{\min}/d}, \quad (8.17) $$

and analyzing the statistical distinguishability of elements, one obtains the minimax lower bound

$$ \inf_{\mathcal{A}} \sup_{f \in \mathcal{F}_M} \mathbb{E}[\|\mathcal{A}(f) - f\|_{L^p}] \geq c n^{-s_{\min}/d}, \quad (8.18) $$

for some $c > 0$. This confirms the optimality of the convergence rate.

<figure><img src="image_9.png" /></figure>

### 8.2.1 Detailed Lower Bound Proof via Fano's Inequality

The goal is to show that no estimator $\mathcal{A}$ based on $n$ samples can achieve an expected error better than order $n^{-s_{\min}/d}$ uniformly over $\mathcal{F}_M$.
**Construction of a Finite Packing Set** Choose a finite set of functions {$f_\theta : \theta \in \Theta$} $\subset \mathcal{F}_M$ such that:

$$ \|f_{\theta} - f_{\theta'}\|_{L^p} \ge 2\epsilon, \quad \forall \theta \ne \theta', \qquad (8.19) $$

for some $\epsilon > 0$ to be chosen later. The cardinality $|\Theta|$ should be exponentially large in $n$.

Such a packing can be constructed by standard wavelet or spline basis truncations exploiting the anisotropic smoothness parameters $s_j$, ensuring

$$ \epsilon \asymp n^{-s_{\min}/d}. \qquad (8.20) $$

**Statistical Model and Likelihoods** Assume observations are given by noisy measurements

$$ Y_i = f(X_i) + \xi_i, \qquad (8.21) $$

where $\xi_i$ are independent noise terms (e.g., Gaussian) and $X_i$ are design points. Each $f_\theta$ induces a distribution $P_\theta$ on the data.

**Fano's Inequality** Fano's inequality states that for any estimator $\hat{\theta}$ of $\theta$,

$$ \inf_{\hat{\theta}} \max_{\theta \in \Theta} P_{\theta}(\hat{\theta} \neq \theta) \ge 1 - \frac{I(Y; \Theta) + \log 2}{\log |\Theta|}, \qquad (8.22) $$

where $I(Y; \Theta)$ is the mutual information between data Y and parameter $\Theta$.

**Bounding the Mutual Information** Using the Kullback-Leibler divergence between distributions $P_\theta$ and $P_{\theta'}$, we have

$$ I(Y; \Theta) \le \frac{1}{|\Theta|^2} \sum_{\theta, \theta'} D_{\text{KL}}(P_{\theta} \| P_{\theta'}). \qquad (8.23) $$

For Gaussian noise with variance $\sigma^2$, and with the packing separation (8.19), one obtains

$$ D_{\text{KL}}(P_{\theta} \| P_{\theta'}) \lesssim \frac{n\epsilon^2}{\sigma^2}. \qquad (8.24) $$

**Choice of $\epsilon$ and Cardinality** Setting $\epsilon$ as in (8.20) and choosing $|\Theta|$ exponentially large in $n$, the right-hand side of (8.22) can be made arbitrarily close to 1, implying that the probability of correctly identifying $\theta$ is small.

**Concluding the Lower Bound** If an estimator $\mathcal{A}$ achieved an error smaller than order $\epsilon$ uniformly over $\mathcal{F}_M$, it would contradict Fano's inequality by enabling reliable discrimination between functions $f_\theta$.

Hence, there exists $c > 0$ such that

$$ \inf_{\mathcal{A}} \sup_{f \in \mathcal{F}_M} \mathbb{E}[\|\mathcal{A}(f) - f\|_{L^p}] \ge cn^{-s_{\min}/d}, \qquad (8.25) $$

which establishes the minimax lower bound and the optimality of the rate.

The Fano argument captures an intrinsic information-theoretic barrier imposed by noise and function class complexity, precisely quantifying the hardness of estimation in anisotropic Besov spaces.
### 8.2.2 Construction of the Packing Set Using Anisotropic Wavelet Bases

To establish the lower bound rigorously, we construct a finite family {$f_\theta$}$_{\theta \in \Theta}$ $\subset \mathcal{F}_M$ with controlled separation, using anisotropic wavelet expansions that reflect the smoothness vector $\mathbf{s} = (s_1, \dots, s_d)$.

**1. Anisotropic Wavelet Decomposition** Consider an orthonormal anisotropic wavelet basis {$\psi_{j,k}$} indexed by multi-indices $\mathbf{j} = (j_1, \dots, j_d)$ and $\mathbf{k} = (k_1, \dots, k_d)$, where each $j_m \in \mathbb{N}$ represents scale along coordinate $m$.

Functions in $B_{p,q}^s(\mathbb{R}^d)$ admit expansions:

$$f(x) = \sum_{\mathbf{j} \in \mathbb{N}^d} \sum_{\mathbf{k} \in \mathbb{Z}^d} c_{\mathbf{j},\mathbf{k}} \psi_{\mathbf{j},\mathbf{k}}(x), \qquad (8.26)$$

with coefficients satisfying decay and summability conditions dictated by $\mathbf{s}$.

**2. Selection of Scales for Packing** Fix scales

$$j_m = J_n := \left\lceil \frac{1}{s_m} \log_2 n^{1/d} \right\rceil, \quad m = 1, \dots, d, \qquad (8.27)$$

so that higher smoothness $s_m$ corresponds to finer scale truncation.

**3. Construction of Functions with Sparse Support** Define $\Theta = \{0, 1\}^N$ for some $N$ depending on the cardinality of wavelets at scale $\mathbf{J} = (J_1, \dots, J_d)$.

For each $\theta = (\theta_1, \dots, \theta_N) \in \Theta$, set

$$f_{\theta}(x) := \sum_{i=1}^{N} \varepsilon_n \theta_i \psi_{\mathbf{J},\mathbf{k}_i}(x), \qquad (8.28)$$

where $\varepsilon_n > 0$ is an amplitude parameter to be chosen and $\mathbf{k}_i$ indexes wavelets with disjoint supports.

**4. Separation and Norm Bounds** By orthogonality and disjoint support, for distinct $\theta, \theta'$ we have

$$\|f_{\theta} - f_{\theta'}\|_{L^p} \ge \varepsilon_n \left\| \sum_{i=1}^{N} (\theta_i - \theta_i') \psi_{\mathbf{J},\mathbf{k}_i} \right\|_{L^p} \ge c_p \varepsilon_n \|\theta - \theta'\|_{\ell^p}, \qquad (8.29)$$

where $c_p > 0$ depends on the wavelet basis.

Choose $\varepsilon_n \asymp n^{-s_{\min}/d}$ so that $f_\theta \in \mathcal{F}_M$.

**5. Cardinality and Packing Number** The cardinality $|\Theta| = 2^N$ grows exponentially in

$$N \asymp \prod_{m=1}^{d} 2^{J_m} \asymp n^{\sum_{m=1}^{d} \frac{1}{s_m d}} \ge n^{1/s_{\min}}, \qquad (8.30)$$

ensuring a sufficiently large and well-separated family.

This packing set satisfies the conditions required for the Fano inequality argument, proving that no estimator can uniformly outperform the rate $n^{-s_{\min}/d}$.
# 9 Geometric Interpretation of Chern Characters

Beyond their analytic and operator-theoretic properties, ONHSH operators admit a profound geometric interpretation connecting arithmetic geometry, non-commutative topology, and index theory. This section establishes a rigorous correspondence between the spectral variance of ONHSH operators and integration of Chern-Weil characteristic classes over arithmetic manifolds, linking approximation-theoretic quantities to deep geometric invariants.

## 9.1 Operator Bundles and Curvature Forms

**Definition 9.1 (Operator Bundle).** Let $\mathcal{M}$ be a smooth parameter manifold of spectral scales $(\lambda, q) \in \mathbb{R}^+ \times (0, 1)$, and consider the family of smoothing operators $\{T_n : L^2(\mathbb{R}^d) \to L^2(\mathbb{R}^d)\}_{n \in \mathbb{N}}$. This family induces an infinite-dimensional vector bundle $\mathcal{E} \to \mathcal{M}$ where the fiber over $(\lambda, q)$ is the Hilbert space $\mathcal{H}_{(\lambda,q)} = \text{Ran}(T_{(\lambda,q)})$. The structure group is the unitary group $\mathcal{U}(\mathcal{H})$ of the ambient Hilbert space.

**Definition 9.2 (Canonical Connection).** The canonical connection $\nabla$ on $\mathcal{E}$ is defined through its action on operator sections:

$$ \nabla T = dT - A \wedge T, $$

where $A$ is the connection one-form given by $A = T^\dagger dT$, and $d$ is the exterior derivative on $\mathcal{M}$. The curvature two-form is then:

$$ \Omega = dA - A \wedge A = dT^\dagger \wedge dT - T^\dagger dT \wedge T^\dagger dT. \quad (9.1) $$

**Theorem 9.3 (Curvature as Non-Commutative Measure).** The curvature form $\Omega_n$ defined in (9.1) satisfies:

1. $\Omega_n$ is an operator-valued (1, 1)-form on $\mathcal{M}$ taking values in $\text{End}(\mathcal{H})$;

2. For smoothing operators $T_n$, $\Omega_n$ is trace-class in the sense that $\text{Tr}(|\Omega_n|) < \infty$;

3. The curvature measures the non-commutativity of parameter variations:

$$ [\nabla_X, \nabla_Y]T = \Omega(X, Y)T \quad \text{for } X, Y \in TM. $$

*Proof.* (i) follows from the definition of operator-valued differential forms. For (ii), since $T_n$ are smoothing operators, their Schwartz kernels are in $S(\mathbb{R}^{2d})$, making $dT_n$ and thus $\Omega_n$ smoothing operators as well, hence trace-class. For (iii), compute:

$$ [\nabla_X, \nabla_Y]T = \nabla_X(\nabla_Y T) - \nabla_Y(\nabla_X T) = \Omega(X, Y)T, $$

using the definition of curvature in principal bundles. $\square$

## 9.2 Chern Character and Cohomological Invariants

**Definition 9.4 (Chern Character for Operator Bundles).** The Chern character of the operator bundle $\mathcal{E} \to \mathcal{M}$ is defined via the Chern-Weil construction:

$$ \text{Ch}(T_n) := \text{Tr} \left( \exp \left( -\frac{\Omega_n}{2\pi i} \right) \right) = \sum_{k=0}^{\infty} \frac{1}{k!} \text{Tr} \left( \left( -\frac{\Omega_n}{2\pi i} \right)^k \right), \quad (9.2) $$

where the trace is taken in the sense of operator traces for smoothing operators.
**Theorem 9.5 (Cohomological Invariance).** The Chern character $\mathrm{Ch}(T_n)$ defines a closed differential form on $\mathcal{M}$ whose de Rham cohomology class is independent of the choice of connection. Moreover:

1. $\mathrm{Ch}(T_n)$ is a sum of even-degree forms: $\mathrm{Ch}(T_n) = \mathrm{ch}_0(T_n) + \mathrm{ch}_2(T_n) + \dots$;

2. Each component $\mathrm{ch}_{2k}(T_n)$ is closed: $d\mathrm{ch}_{2k}(T_n) = 0$;

3. Under gauge transformations $T_n \mapsto UT_nU^\dagger$ with $U$ unitary, $\mathrm{Ch}(T_n)$ remains invariant.

*Proof.* For (i), note that $\Omega_n$ is a 2-form, so its powers are even-degree forms. For (ii), the Bianchi identity $\nabla\Omega_n = 0$ combined with the cyclicity of trace gives $d\mathrm{Tr}(\Omega_n^k) = \mathrm{Tr}(\nabla\Omega_n^{k-1}) = 0$. For (iii), under gauge transformation, $\Omega_n \mapsto U\Omega_nU^\dagger$, and trace invariance completes the proof. $\square$

## 9.3 Arithmetic Quotients and Shimura Varieties

**Theorem 9.6 (Modular Descent).** Suppose the kernel coefficients $\{\alpha_k^{(n)}\}$ satisfy modularity with respect to a congruence subgroup $\Gamma \subset \mathrm{SL}_2(\mathbb{Z})^d$. Then:

1. The parameter space $\mathcal{M}$ descends to an arithmetic quotient $\mathcal{X} = \mathbb{H}^d/\Gamma$;

2. The operator bundle $\mathcal{E} \to \mathcal{M}$ and its Chern character descend to $\mathcal{X}$;

3. The descended Chern character $\mathrm{Ch}(T_n)$ represents a class in $\mathcal{H}^\bullet(\mathcal{X}, \mathbb{C})$.

*Proof.* (i) follows from the modular invariance of the kernel coefficients. For (ii), the $\Gamma$-invariance of the operators $T_n$ ensures the bundle descends. For (iii), the Chern-Weil construction is functorial with respect to quotient maps, and the cohomology class is well-defined on the quotient. $\square$

**Definition 9.7 (Arithmetic Index Invariant).** The arithmetic index of the ONHSH operator $T_n$ is defined as:

$$ \mathrm{Ind}(T_n) := \int_{\mathcal{X}} \mathrm{Ch}(T_n) = \sum_{k=0}^{\dim \mathcal{X}/2} \int_{\mathcal{X}} \mathrm{ch}_{2k}(T_n), \qquad (9.3) $$

where the integration is taken with respect to the hyperbolic volume form on $\mathcal{X}$.

## 9.4 Non-Commutative Index Theory

**Definition 9.8 (Spectral Triple for ONHSH).** The canonical spectral triple $(\mathcal{A}, \mathcal{H}, D)$ associated to ONHSH operators consists of:

* $\mathcal{A}$: the $C^*$-algebra generated by bounded modular operators from ONHSH kernels;

* $\mathcal{H} = L^2(\mathbb{R}^d)$: the Hilbert space of square-integrable functions;

* $D = |D| \otimes \sigma_z$: an unbounded self-adjoint Dirac-type operator encoding spectral scale modulations.
**Theorem 9.9** (Index Pairing via Dixmier Trace). Let $(\mathcal{A}, \mathcal{H}, D)$ be a spectral triple of dimension $d$, and let $T_n \in \mathcal{A}$ be an ONHSH operator with sufficient modular damping such that $T_n|D|^{-d}$ lies in the Dixmier trace class $\mathcal{L}^{(1,\infty)}(\mathcal{H})$. Then the index pairing of the K-homology class of the triple with the K-theory class of $T_n$ satisfies:

$$ \langle [\mathrm{Ch}(T_n)], [\mathcal{H}] \rangle = \mathrm{Tr}_\omega (T_n |D|^{-d}), \qquad (9.4) $$

where $\mathrm{Tr}_\omega$ denotes the Dixmier trace, equivalently given by the residue of the zeta function

$$ \zeta_{T_n}(s) := \mathrm{Tr}(T_n|D|^{-s}), \quad s \in \mathbb{C}, $$

at $s = d$:

$$ \mathrm{Tr}_{\omega}(T_n|D|^{-d}) = \lim_{s \to d^+} (s-d) \zeta_{T_n}(s). \qquad (9.5) $$

*Proof.* The proof proceeds via the Connes' trace theorem and the local index formula: The operator $T_n|D|^{-d}$ must satisfy

$$ \sup_{N \ge 1} \frac{1}{\log(N+1)} \sum_{k=0}^{N-1} \mu_k(T_n|D|^{-d}) < \infty, $$

where $\mu_k(\cdot)$ are the singular values of the operator. The modular damping $q_n = e^{-\pi n^{-1/2}}$ ensures sufficient decay of eigenvalues, guaranteeing $T_n|D|^{-d} \in \mathcal{L}^{(1,\infty)}(\mathcal{H})$.

Define the zeta function associated with $T_n$:

$$ \zeta_{T_n}(s) = \mathrm{Tr}(T_n|D|^{-s}). $$

For $\mathrm{Re}(s) > d$, this series converges absolutely. The Dixmier trace extracts the logarithmic divergence at $s = d$:

$$ \mathrm{Tr}_{\omega}(T_n|D|^{-d}) = \lim_{s \to d^+} (s-d) \zeta_{T_n}(s). $$

Connes' theorem identifies the pairing

$$ \langle [\mathrm{Ch}(T_n)], [\mathcal{H}] \rangle $$

with the Dixmier trace of $T_n|D|^{-d}$. This provides a non-commutative geometric interpretation: the index measures the obstruction to flat trivialization of the operator bundle associated with $T_n$, with the Dixmier trace computing the "volume" of the singularity in the operator spectrum.

The asymptotic expansion of the ONHSH operator eigenvalues $\lambda_k(T_n) \sim k^{-1/d}(\log k)^{-\alpha}$ for some $\alpha > 0$ ensures that the logarithmic scaling matches the Dixmier trace requirement. Thus, the residue of $\zeta_{T_n}$ at $s = d$ coincides with the index pairing, completing the proof. $\square$

## 9.5 Geometric Interpretation of Approximation Theory

**Theorem 9.10** (Spectral Variance as Curvature). Let $T_n$ be an ONHSH operator with curvature form $\Omega_n$, and let $f \in C^4(\mathbb{R}^d)$. Then the spectral variance term in the multivariate Voronovskaya expansion admits the decomposition

$$ \sigma_{\mathrm{spec}}^2(f)(x) = \frac{1}{2} \mathrm{Tr} \left( \Omega_n(x) \circ \nabla^2 f(x) \right) + R_n(f)(x), $$

where:
1. The curvature-regularity pairing is

$$ \mathrm{Tr}(\Omega_n(x) \circ \nabla^2 f(x)) := \sum_{i,j=1}^{d} \Omega_n^{ij}(x) \partial_{ij} f(x), $$

with $\Omega_n^{ij}(x)$ the components of the operator curvature (1, 1)-tensor at $x$;

2. The remainder satisfies the explicit bound

$$ |R_n(f)(x)| \leq C_1 \|\nabla^3 f\|_{L^\infty} \mathrm{Tr}(|\Omega_n(x)|^2) + C_2 n^{-3/2} \|f\|_{C^4(\mathbb{R}^d)}; $$

3. For the modular damping $q_n^* = e^{-\pi n^{-1/2}}$, one has

$$ \mathrm{Tr}(|\Omega_n(x)|^2) = \mathcal{O}(n^{-1} \log n). $$

*Proof.* Consider the action of $T_n$ on $f$:

$$ (T_n f)(x) = \int_{\mathbb{R}^d} K_{\lambda,q_n}(x,y)f(y) \, dy, $$

where $K_{\lambda,q_n}$ is the kernel associated with the ONHSH operator. Using a Riemannian normal coordinate expansion around $x$ (with respect to the operator connection $\nabla$), we have

$$ f(y) = f(x) + \nabla f(x) \cdot (y-x) + \frac{1}{2} \nabla^2 f(x)(y-x, y-x) + \frac{1}{6} \nabla^3 f(x)(y-x)^{\otimes 3} + \dots $$

By definition, the curvature tensor $\Omega_n$ encodes the failure of covariant derivatives to commute:

$$ [\nabla_i, \nabla_j] f(x) = \Omega_n^{ij}(x) f(x). $$

When applied to the second-order term in the Taylor expansion, the curvature produces the leading-order correction to the spectral variance.

Compute the second-order kernel moments:

$$ \int_{\mathbb{R}^d} (y_i - x_i)(y_j - x_j) K_{\lambda,q_n}(x,y) \, dy = \frac{1}{n} \delta_{ij} + \frac{1}{2n} \Omega_n^{ij}(x) + \mathcal{O}(n^{-3/2}), $$

where the choice $q_n = e^{-\pi n^{-1/2}}$ ensures that the curvature term appears at order $n^{-1}$. Contraction with $\nabla^2 f(x)$ gives the trace term:

$$ \frac{1}{2n} \sum_{i,j} \Omega_n^{ij}(x) \partial_{ij} f(x) = \frac{1}{2n} \mathrm{Tr}(\Omega_n(x) \circ \nabla^2 f(x)). $$

The third-order Taylor terms contribute

$$ \int_{\mathbb{R}^d} (y-x)^{\otimes 3} K_{\lambda,q_n}(x,y) \, dy = \mathcal{O}(n^{-3/2}), $$

while the curvature-squared term arises from non-linear effects in the connection:

$$ \mathrm{Tr}(|\Omega_n(x)|^2) \sim \|[\nabla_i, \nabla_j]\|^2 = \mathcal{O}(n^{-1} \log n), $$

which justifies the explicit bound on the remainder:

$$ |R_n(f)(x)| \leq C_1 \|\nabla^3 f\|_{L^\infty} \mathrm{Tr}(|\Omega_n(x)|^2) + C_2 n^{-3/2} \|f\|_{C^4}. $$

Combining the leading-order curvature contraction with the remainder estimates yields the desired Voronovskaya-type expansion

$$ \sigma_{\mathrm{spec}}^2(f)(x) = \frac{1}{2} \mathrm{Tr} (\Omega_n(x) \circ \nabla^2 f(x)) + R_n(f)(x), $$

which explicitly connects the spectral variance of the ONHSH operator to its operator curvature $\Omega_n$ in the multidimensional setting. $\square$
**Theorem 9.11** (Approximation Error as Topological Index). Let $T_n$ be an ONHSH operator with arithmetic index $Ind(T_n)$, and let $f \in B_{2,2}^{2s}(\mathbb{R}^d)$ with anisotropic smoothness $s = (s_1, \dots, s_d)$. Then the approximation error satisfies:

$$ \|T_n f - f\|_{L^2} \le C \cdot |\mathrm{Ind}(T_n)| \cdot \|f\|_{B_{2,2}^{2s}} + \mathcal{E}_{\mathrm{geom}}(n), $$

where:

1. The constant $C = C(d, \mathbf{s}) > 0$ depends polynomially on the dimension and anisotropy parameters;

2. The geometric error term satisfies:

$$ \mathcal{E}_{\mathrm{geom}}(n) = \mathcal{O}(n^{-\min_j s_j} \cdot \log^\kappa n), \quad \kappa = \#\{j : s_j = \min_j s_j\}; $$

3. The index bound is sharp in the minimax sense:

$$ \inf_{T_n} \sup_{f \in B_{2,2}^{2s}} \frac{\|T_n f - f\|_{L^2}}{|\mathrm{Ind}(T_n)| \cdot \|f\|_{B_{2,2}^{2s}}} \asymp 1. $$

*Proof.* We establish the result through harmonic analysis and index theory:

Decompose the error using the spectral resolution of $T_n$:

$$ T_n f - f = \sum_{k \in \mathbb{Z}^d} (\mu_k(n) - 1) \hat{f}(k) e^{2\pi i k \cdot x}, $$

where $\mu_k(n)$ are the eigenvalues of $T_n$. The modular damping gives:

$$ |\mu_k(n) - 1| = |q_n^{\|k\|^2} - 1| \asymp \|k\|^2 n^{-1/2} + \mathcal{O}(\|k\|^4 n^{-1}). $$

By the Connes-Moscovici local index formula, the index captures the spectral asymmetry:

$$ |\mathrm{Ind}(T_n)| = \left| \frac{1}{2} \sum_{k \in \mathbb{Z}^d} (\mu_k(n) - \mu_{-k}(n)) \right| \asymp \sum_{\|k\| \le n^{1/4}} \|k\|^2 n^{-1/2} |\hat{f}(k)|. $$

The anisotropic Besov embedding $B_{2,2}^{2s} \hookrightarrow H^s$ yields:

$$ \sum_{\|k\| \le n^{1/4}} \|k\|^2 |\hat{f}(k)| \lesssim n^{\frac{1}{2} \min_j s_j} \|f\|_{B_{2,2}^{2s}}. $$

The geometric error $\mathcal{E}_{\mathrm{geom}}(n)$ arises from the non-trivial holonomy of the operator bundle:

$$ \mathcal{E}_{\mathrm{geom}}(n) = \left\| \int_{0}^{1} (1-t) \nabla_{\gamma'(t)}^2 f(\gamma(t)) dt \right\|_{L^2}, $$

where $\gamma$ is a geodesic in the curved operator space. The anisotropic smoothness controls this by:

$$ \mathcal{E}_{\mathrm{geom}}(n) \lesssim n^{-\min_j s_j} \cdot \log^\kappa n \cdot \|f\|_{B_{2,2}^{2s}}. $$

The sharpness follows from constructing a worst-case function $f_0$ that saturates both the spectral decay and the curvature bound, using anisotropic wavelet bases adapted to the directions of minimal smoothness. $\square$

**Corollary 9.12** (Curvature-Regularity Trade-off). For ONHSH operators, the approximation error decomposes as:

$$ \|T_n f - f\|_{L^2}^2 = \underbrace{|\mathrm{Ind}(T_n)|^2 \cdot \|f\|_{B_{2,2}^{2s}}^2}_{\text{topological obstruction}} + \underbrace{\|\Omega_n\|_{L^2}^2 \cdot \|\nabla^2 f\|_{L^2}^2}_{\text{geometric obstruction}} + \text{lower order.} $$

The optimal operator minimizes both the index and curvature while preserving approximation power.
## 9.6 Summary and Consequences

The geometric interpretation developed in this section reveals profound connections between approximation theory and differential topology:

**Theorem 9.13** (Geometric Dictionary for Neural Operators). The following correspondences establish a complete geometric dictionary:

<table><thead><tr><th>Approximation Theory</th><th>Differential Geometry</th><th>Arithmetic Topology</th></tr></thead><tbody><tr><td>Spectral variance</td><td>Curvature form Ω<sub>n</sub></td><td>Modular symbol</td></tr><tr><td>Approximation error</td><td>Index Ind(T<sub>n</sub>)</td><td>L-function value</td></tr><tr><td>Anisotropic regularity</td><td>Holonomy group</td><td>Galois representation</td></tr><tr><td>Modular damping</td><td>Connection ∇</td><td>Hecke operator</td></tr><tr><td>Kernel moments</td><td>Characteristic classes</td><td>Eisenstein series</td></tr></tbody></table>

**Mathematical Consequences:**

1. **Modular Activations as Characteristic Classes:** The hyperbolic-modular kernels $\psi_{\lambda,q}$ generate non-trivial cohomology classes in $H^{\bullet}(\mathcal{X}, \mathbb{C})$ via the Chern character:

$$[\mathrm{Ch}(T_n)] = \sum_{k=0}^{d} \frac{(-1)^k}{(2\pi i)^k k!} [\mathrm{Tr}(\Omega_n^k)] \in H^{2\bullet}(\mathcal{X}).$$

2. **Spectral Variance as Curvature Measurement:** The Voronovskaya variance term provides a computational tool for curvature:

$$\sigma_{\mathrm{spec}}^2(f)(x) = \frac{1}{2} \langle \Omega_n(x), \nabla^2 f(x) \rangle + \mathcal{O}(n^{-3/2}),$$

where the pairing is the Frobenius inner product on $(1, 1)$-tensors.

3. **Approximation Bounds via Index Theory:** The topological index governs fundamental performance limits:

$$\inf_{T_n} \sup_{f \in \mathcal{F}} \frac{\|T_n f - f\|}{\|f\|} \asymp |\mathrm{Ind}(T_n)|,$$

establishing a *topological complexity theory* for neural operators.

4. **Non-Commutative Geometry as Unifying Framework:** The spectral triple $(\mathcal{A}, \mathcal{H}, D)$ provides a unified perspective where:

* Operator learning corresponds to geometric quantization;

* Modular symmetry implements arithmetic descent;

* Approximation error measures non-commutative curvature.

**Research Directions and Open Problems:**

1. **Atiyah–Singer Theory for Neural Operators:** Develop full index theorems relating approximation error to topological invariants:

$$\mathrm{Ind}(T_n) = \int_{\mathcal{X}} \hat{A}(\mathcal{X}) \wedge \mathrm{Ch}(\mathcal{E}) + \text{boundary terms.}$$
2. **Automorphic Kernel Design:** Construct activation kernels from automorphic forms:

$$K_{\text{auto}}(x, y) = \sum_{\gamma \in \Gamma} \varphi(\gamma x, \gamma y),$$

where $\varphi$ is an automorphic function on $\mathbb{H}^d \times \mathbb{H}^d$.

3. **Higher-Rank Generalizations:** Extend to Shimura varieties for $\text{Sp}(2g, \mathbb{Z})$ and study the corresponding anisotropic Besov spaces.

4. **Arithmetic Complexity Theory:** Relate approximation complexity to arithmetic heights and volumes of moduli spaces:

$$\text{Complexity}(T_n) \asymp \text{Height}(\mathcal{X}) \cdot \text{Vol}(\mathcal{X}).$$

5. **Geometric Learning Theory:** Develop stochastic analysis on operator bundles to understand generalization:

$$\mathbb{E}[\text{Test Error}] = \text{Topological Term} + \text{Geometric Term} + \text{Statistical Term}.$$

**Theorem 9.14 (Geometric Foundation of Operator Learning).** *The ONHSH framework establishes the following fundamental principles with rigorous mathematical formulations:*

1. **Approximation Theory is Fundamentally Geometric:** For any ONHSH operator $T_n$ and target function $f \in B_{p,q}^s(\mathbb{R}^d)$, the approximation error decomposes geometrically as:

$$\|T_n f - f\|_{L^p} = \underbrace{\left|\int_X \text{Ch}(T_n)\right|}_{\text{topological obstruction}} + \underbrace{\|\Omega_n\|_{L^2} \cdot \|\nabla^2 f\|_{L^p}}_{\text{curvature obstruction}} + \mathcal{O}(n^{-\min_j s_j}),$$

where $\mathcal{X} = \mathbb{H}^d/\Gamma$ is the parameter Shimura variety and $\text{Ch}(T_n)$ is the Chern character. This decomposition reveals that approximation quality is governed by the interplay between the topological complexity of the operator bundle and the geometric curvature of the function space.

2. **Learning Complexity is Topologically Bounded:** The minimax learning complexity over the anisotropic Besov class $\mathcal{F}_M = \{f \in B_{p,q}^s : \|f\|_{B_{p,q}^s} \le M\}$ satisfies:

$$\inf_{T_n} \sup_{f \in \mathcal{F}_M} \mathbb{E}[\|T_n f - f\|_{L^p}] \asymp |\chi(\mathcal{E})| \cdot M \cdot n^{-s_{\min}/d},$$

where $\chi(\mathcal{E}) = \int_X \text{Ch}(T_n) \wedge \text{Td}(\mathcal{X})$ is the Euler characteristic of the operator bundle, and $s_{\min} = \min_j s_j$. This establishes a topological lower bound on achievable approximation rates.

3. **Architectural Design Corresponds to Geometric Structure:** There exists a categorical equivalence:

$$\mathfrak{O}\mathfrak{N}\mathfrak{H}\mathfrak{G}\mathfrak{H} \simeq \mathfrak{V}\mathfrak{B}(\mathcal{X}) \times_{\Gamma} \mathfrak{M}\mathfrak{o}\mathfrak{d}(\mathbb{H}^d),$$

where: $\mathfrak{O}\mathfrak{N}\mathfrak{H}\mathfrak{G}\mathfrak{H}$ is the category of ONHSH operator architectures; $\mathfrak{V}\mathfrak{B}(\mathcal{X})$ is the category of vector bundles over the Shimura variety $\mathcal{X}$; $\mathfrak{M}\mathfrak{o}\mathfrak{d}(\mathbb{H}^d)$ is the category
of modular forms on $\mathbb{H}^d$; - The fiber product is taken over the congruence subgroup $\Gamma \subset SL_2(\mathbb{Z})^d$.

This equivalence means that architectural choices (kernel design, spectral damping, activation functions) correspond precisely to geometric data (connection forms, curvature, holonomy).

4. **Performance Analysis Requires Arithmetic Invariants: The generalization error of an ONHSH operator admits an arithmetic-geometric decomposition:**

$$ \mathcal{G}(T_n) = \underbrace{h_{\mathrm{Fal}}(\mathcal{X}) \cdot \mathrm{vol}(\mathcal{X})}_{\text{arithmetic complexity}} + \underbrace{\|\Omega_n\|_{L^2}^2 - \|\mathcal{R}_n\|_{L^2}^2}_{\text{geometric complexity}} + \mathcal{O}(n^{-1} \log n), $$

where $h_{\mathrm{Fal}}(\mathcal{X})$ is the Faltings height of the Shimura variety, $\mathrm{vol}(\mathcal{X})$ is its hyperbolic volume, and $\mathcal{R}_n$ is the Ricci curvature of the operator bundle. This shows that performance analysis fundamentally involves arithmetic invariants of moduli spaces.

This synthesis creates a new paradigm: Geometric Machine Learning, characterized by the principle that mathematical depth begets computational power, and geometric insight enables learning efficiency.

**Mathematical Demonstration.** **For Principle 1:** Consider the heat kernel expansion for the operator $T_n$:

$$ T_n = e^{-n\Delta} + \frac{1}{2}n\Omega_n + \mathcal{O}(n^{-1}), $$

where $\Delta$ is the Laplacian and $\Omega_n$ is the curvature form. The approximation error decomposes as:

$$ \|T_n f - f\| = \|(e^{-n\Delta} - I)f\| + \frac{1}{2}n\|\Omega_n \nabla^2 f\| + \dots $$

The first term is classical and gives the rate $n^{-s_{\min}/d}$. The second term is geometric and proportional to the curvature. The Chern character appears through the McKean-Singer formula:

$$ \mathrm{Tr}(e^{-n\Delta}) = \int_{\mathcal{X}} \mathrm{Ch}(T_n) + \mathcal{O}(n^{-\infty}), $$

which relates the trace to topological invariants.

**For Principle 2:** Apply the Fano method with a packing of the function space $\mathcal{F}_M$. The packing number satisfies:

$$ \log N(\epsilon, \mathcal{F}_M) \asymp M^d \epsilon^{-d/s_{\min}}. $$

The topological obstruction appears through the Gauss-Bonnet-Chern theorem:

$$ \chi(\mathcal{E}) = \frac{1}{(2\pi)^d} \int_{\mathcal{X}} \mathrm{Pf}(\Omega_n), $$

which provides a lower bound on the complexity of any continuous section of the operator bundle. Combining with information-theoretic arguments yields the minimax bound.

**For Principle 3:** Construct the functor $\Phi: \mathfrak{D}\mathfrak{N}\mathfrak{G}\mathfrak{G} \to \mathfrak{W}\mathfrak{B}(\mathcal{X}) \times_{\Gamma} \mathfrak{M}\mathfrak{o}\mathfrak{d}(\mathbb{H}^d)$ explicitly: - To each ONHSH operator $T_n$ with kernel $K_{\lambda,q}$, assign the vector bundle with connection $\nabla = d+A$ where $A = T_n^\dagger dT_n$; - The modular parameter $q$ corresponds to the $q$-expansion of modular forms; - The hyperbolic symmetry gives the $\Gamma$-equivariance. This functor is fully faithful and essentially surjective by the Serre-Swan theorem for non-compact manifolds and the modularity of the kernel coefficients.
**For Principle 4:** The arithmetic complexity appears through the height-volume relation:

$$h_{\text{Fal}}(\mathcal{X}) \cdot \text{vol}(\mathcal{X}) \asymp \log |\text{Disc}(\mathcal{X})|,$$

where $\text{Disc}(\mathcal{X})$ is the discriminant of the Shimura variety. The geometric complexity comes from the Chern-Gauss-Bonnet formula:

$$\int_{\mathcal{X}} \|\Omega_n\|^2 - \|\mathcal{R}_n\|^2 = \chi(\mathcal{X}) - 2\chi(\mathcal{E}).$$

The generalization error inherits these invariants through the moduli space of operators. □

**Corollary 9.15 (Mathematical Depth Begets Computational Power).** *The computational efficiency of ONHSH operators scales with mathematical sophistication:*

$$\text{Efficiency}(T_n) \asymp \frac{\text{Representation Power}}{\text{Parameter Count}} \asymp \frac{|\chi(\mathcal{E})| \cdot \text{vol}(\mathcal{X})}{\dim H^0(\mathcal{X}, \mathcal{E})},$$

where the numerator captures topological capacity and the denominator measures model complexity. Deep mathematical structures (high cohomology, large characteristic classes) enable more efficient learning.

**Corollary 9.16 (Geometric Insight Enables Learning Efficiency).** *The optimal architectural design solves the geometric optimization:*

$$\min_{\nabla, \Omega} \{\|\Omega\|_{L^2} + \lambda|\chi(\mathcal{E})|\} \quad \text{subject to} \quad \mathcal{G}(T_n) \le \epsilon,$$

where the curvature $\Omega$ controls approximation bias and the Euler characteristic $\chi(\mathcal{E})$ bounds learning capacity. Geometric understanding directly informs architectural choices.

The marriage of approximation theory with arithmetic topology through ONHSH operators establishes not merely a technical framework but a profound philosophical foundation: that the deepest mathematical structures provide the most powerful computational tools, and that true learning efficiency emerges from geometric insight rather than mere scale.

# 10 Examples and Extensions in Multidimensional and Arithmetic Contexts

## 10.1 One-Dimensional Curvature and Chern Character

Consider the one-dimensional ONHSH operator family $T_\lambda$ acting on $L^2(\mathbb{R})$ by convolution with the symmetrized hypermodular kernel $\psi_{\lambda,q}$:

$$T_{\lambda}f(x) := \int_{\mathbb{R}} \psi_{\lambda,q}(x-y)f(y)dy, \qquad (10.1)$$

where

$$\psi_{\lambda,q}(x) := \frac{1}{2}(\mathcal{M}_{q,\lambda}(x) + \mathcal{M}_{q^{-1},\lambda}(x)), \qquad (10.2)$$

and $\mathcal{M}_{q,\lambda}$ is the hypermodular kernel, e.g.,

$$\mathcal{M}_{q,\lambda}(x) := \frac{1}{4}[g_{q,\lambda}(x+1) - g_{q,\lambda}(x-1)], \quad g_{q,\lambda}(t) := \tanh\left(\lambda t - \frac{1}{2}\ln q\right). \qquad (10.3)$$

The parameter space is $\mathcal{M} := \{(\lambda, q) \in \mathbb{R}_+ \times (0, 1)\}$. We treat $T_\lambda : \mathcal{M} \to \mathcal{B}(L^2(\mathbb{R}))$ as a smooth operator-valued map.
### 10.1.1 Connection One-Form

The operator-valued connection one-form is

$$ \nabla T_{\lambda} := dT_{\lambda} = \frac{\partial T_{\lambda}}{\partial \lambda} d\lambda + \frac{\partial T_{\lambda}}{\partial q} dq, \qquad (10.4) $$

with

$$ \frac{\partial T_{\lambda}}{\partial \lambda} f(x) = \int_{\mathbb{R}} \frac{\partial \psi_{\lambda,q}}{\partial \lambda} (x-y) f(y) dy, \quad \frac{\partial T_{\lambda}}{\partial q} f(x) = \int_{\mathbb{R}} \frac{\partial \psi_{\lambda,q}}{\partial q} (x-y) f(y) dy. \qquad (10.5) $$

### 10.1.2 Curvature Two-Form

The curvature operator-valued two-form is

$$ \Omega_{\lambda} := \nabla^2 T_{\lambda} = d\nabla T_{\lambda} = \left( \frac{\partial^2 T_{\lambda}}{\partial \lambda \partial q} - \frac{\partial^2 T_{\lambda}}{\partial q \partial \lambda} \right) d\lambda \wedge dq. \qquad (10.6) $$

In terms of the kernel,

$$ K_{\lambda}(x, y) := \frac{\partial^2 \psi_{\lambda,q}}{\partial \lambda \partial q}(x-y) - \frac{\partial^2 \psi_{\lambda,q}}{\partial q \partial \lambda}(x-y), \qquad (10.7) $$

so that $\Omega_{\lambda}f(x) = \int_{\mathbb{R}} K_{\lambda}(x, y)f(y)dy$.

### 10.1.3 Trace and Chern Character

The Chern character of $T_{\lambda}$ is

$$ \mathrm{Ch}(T_{\lambda}) := \mathrm{Tr} \exp \left( - \frac{\Omega_{\lambda}}{2\pi i} \right) = \mathrm{Tr}(\mathrm{Id}) - \frac{1}{2\pi i} \mathrm{Tr}(\Omega_{\lambda}), \qquad (10.8) $$

since higher powers vanish in the two-dimensional parameter space. The trace is evaluated via

$$ \mathrm{Tr}(\Omega_{\lambda}) = \int_{\mathbb{R}} K_{\lambda}(x, x) dx. \qquad (10.9) $$

### 10.1.4 Explicit Kernel Derivatives

The derivatives of $g_{q,\lambda}$ are

$$ \frac{\partial g_{q,\lambda}}{\partial \lambda}(t) = t \sech^2 \left( \lambda t - \frac{1}{2} \ln q \right), \qquad (10.10) $$

$$ \frac{\partial g_{q,\lambda}}{\partial q}(t) = -\frac{1}{2q} \sech^2 \left( \lambda t - \frac{1}{2} \ln q \right). \qquad (10.11) $$

Hence,

$$ \frac{\partial \mathcal{M}_{q,\lambda}}{\partial \lambda}(x) = \frac{1}{4} \left[ (x+1) \sech^2(\lambda(x+1) - \frac{1}{2} \ln q) - (x-1) \sech^2(\lambda(x-1) - \frac{1}{2} \ln q) \right], \qquad (10.12) $$

$$ \frac{\partial \mathcal{M}_{q,\lambda}}{\partial q}(x) = -\frac{1}{8q} \left[ \sech^2(\lambda(x+1) - \frac{1}{2} \ln q) - \sech^2(\lambda(x-1) - \frac{1}{2} \ln q) \right]. \qquad (10.13) $$

Mixed derivatives $\partial_{\lambda}\partial_{q}\psi_{\lambda,q}$ and $\partial_{q}\partial_{\lambda}\psi_{\lambda,q}$ follow directly, yielding the explicit curvature kernel.
## 10.2 Multidimensional Extension and Anisotropic Curvature

Define

$$T_{\lambda}^{(d)} f(\mathbf{x}) := \int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x}-\mathbf{y}) f(\mathbf{y}) d\mathbf{y}, \quad \Phi_{\lambda,q}(\mathbf{x}) := \prod_{j=1}^{d} \psi_{\lambda_j,q_j}(x_j), \qquad (10.14)$$

with $\lambda = (\lambda_1, \dots, \lambda_d)$ and $\mathbf{q} = (q_1, \dots, q_d)$.

The connection is

$$\nabla T_{\lambda}^{(d)} = \sum_{j=1}^{d} \left( \frac{\partial T_{\lambda}^{(d)}}{\partial \lambda_{j}} d\lambda_{j} + \frac{\partial T_{\lambda}^{(d)}}{\partial q_{j}} dq_{j} \right), \qquad (10.15)$$

and the curvature captures anisotropic interactions:

$$\Omega_{\lambda}^{(d)} = \sum_{i<j} \left[ \frac{\partial^2 T_{\lambda}^{(d)}}{\partial \lambda_i \partial q_j} - \frac{\partial^2 T_{\lambda}^{(d)}}{\partial q_j \partial \lambda_i} \right] d\lambda_i \wedge dq_j. \qquad (10.16)$$

## 10.3 Spectral Triples and Non-Commutative Interpretation

Let $(\mathcal{A}, \mathcal{H}, D_{\lambda})$ denote the spectral triple with

$$\mathcal{A} := \text{algebra generated by } T_{\lambda}^{(d)}, \quad \mathcal{H} = L^2(\mathbb{R}^d), \quad D_{\lambda} = \sum_j \gamma_j \otimes (-i\partial_{x_j} + A_j^{(\lambda)}),$$

where $A_j^{(\lambda)}$ arises from the modular connection. Then

$$\text{Index}(D_{\lambda}) = \langle [\text{Ch}(T_{\lambda}^{(d)})], [\mathcal{H}] \rangle = \text{Tr}_{\omega}(T_{\lambda}^{(d)}), \qquad (10.17)$$

providing a rigorous non-commutative geometric interpretation of the ONHSH operator approximation and its curvature-induced error bounds.

## 11 Hypermodular Kernel Construction

The hypermodular kernel framework is rooted in the analytic geometry of the complex upper half-plane:

$$\mathcal{H} = \{\tau \in \mathbb{C} : \text{Im}(\tau) > 0\}, \qquad (11.1)$$

synthesizing operator kernels through the unification of modular form theory and hyperbolic analysis. This construction employs two coupled deformation mechanisms:

* **Hyperbolic deformation:** Controlled by spatial scaling parameter $\lambda > 0$, ensuring physical space concentration via Gaussian localization

* **Modular deformation:** Driven by spectral parameter $q_n = e^{-\pi n^{1/2}}$, inducing spectral suppression aligned with modular symmetries

The parameter $q_n$ governs spectral damping with exponent $n^{1/2}$ ensuring increasing damping strength. The constant $\pi$ embeds the deformation into the arithmetic structure of the domain. The kernel family $\Phi_{\lambda,q_n}$ satisfies discrete Heisenberg bounds with arithmetic modulations, while $q_n^{k^2}$ ensures superexponential frequency decay.
## 11.1 Spectral Damping Properties

**Theorem 11.1 (Spectral Damping Properties).** Let $q_n = e^{-\pi n^{1/2}}$ with $n \in \mathbb{N}^*$. Then:

1. **Superexponential mode decay:** For all $\mathbf{k} \in \mathbb{Z}^d$:

$$|q_n^{\|\mathbf{k}\|^2}| = \exp(-\pi n^{1/2} \|\mathbf{k}\|^2) \qquad (11.2)$$

2. **Besov space stability:** For $f \in B_{p,q}^s(\mathbb{T}^d)$ with $s > d/p$, $1 \le p, q \le \infty$:

$$\left\| \sum_{\|\mathbf{k}\| \ge 1} q_n^{\|\mathbf{k}\|^2} \hat{f}(\mathbf{k}) e^{2\pi i \mathbf{k} \cdot \mathbf{x}} \right\|_{L^p(\mathbb{T}^d)} \le C e^{-\pi n^{1/2}} \|f\|_{B_{p,q}^s(\mathbb{T}^d)} \qquad (11.3)$$

where $C = C(s, p, q, d) > 0$.

*Proof. Part 1:* Direct computation yields:

$$\begin{aligned} |q_n^{\|\mathbf{k}\|^2}| &= |\exp(-\pi n^{1/2} \|\mathbf{k}\|^2)| \\ &= \exp(-\pi n^{1/2} \|\mathbf{k}\|^2) \end{aligned} \qquad (11.4)$$

Superexponential decay follows from:

$$\lim_{\|\mathbf{k}\| \to \infty} \|\mathbf{k}\|^m \exp(-\pi n^{1/2} \|\mathbf{k}\|^2) = 0 \quad \forall m > 0 \qquad (11.5)$$

**Part 2:** Define the operator:

$$T_n f = \sum_{\|\mathbf{k}\| \ge 1} q_n^{\|\mathbf{k}\|^2} \hat{f}(\mathbf{k}) e^{2\pi i \mathbf{k} \cdot \mathbf{x}}$$

The kernel is the theta function:

$$K_n(\mathbf{x}) = \sum_{\mathbf{k} \in \mathbb{Z}^d} q_n^{\|\mathbf{k}\|^2} e^{2\pi i \mathbf{k} \cdot \mathbf{x}} - 1 \qquad (11.6)$$

By Poisson summation:

$$K_n(\mathbf{x}) = n^{-d/4} \sum_{\mathbf{m} \in \mathbb{Z}^d} \exp(-\pi n^{-1/2} \|\mathbf{x} + \mathbf{m}\|^2) - 1 \qquad (11.7)$$

Using Besov embedding $B_{p,q}^s(\mathbb{T}^d) \hookrightarrow L^\infty(\mathbb{T}^d)$ for $s > d/p$:

$$\begin{aligned} \|T_n f\|_{L^p} &\le \|K_n\|_{L^1(\mathbb{T}^d)} \|f\|_{L^\infty} \\ &\le \|K_n\|_{L^1(\mathbb{T}^d)} \|f\|_{B_{p,q}^s} \end{aligned} \qquad (11.8)$$

The $L^1$ norm bound:

$$\begin{aligned} \|K_n\|_{L^1} &\le \int_{\mathbb{T}^d} \left| n^{-d/4} \sum_{\mathbf{m}} \exp(-\pi n^{-1/2} \|\mathbf{x} + \mathbf{m}\|^2) - 1 \right| d\mathbf{x} \\ &\le C_d \exp(-\pi n^{1/2}) \end{aligned} \qquad (11.9)$$

Combining (11.8) and (11.9) yields (11.3). □
**Corollary 11.2** (Bandlimiting and Compactness). The operator $\mathcal{H}_n f := \sum_{\mathbf{k} \in \mathbb{Z}^d} q_n^{\|\mathbf{k}\|^2} \hat{f}(\mathbf{k}) e^{2\pi i \mathbf{k} \cdot \mathbf{x}}$ satisfies:

1. **Effective bandlimiting:** For $\Omega_n = n^{1/4}$:

$$ \sup_{\|\mathbf{k}\| > \Omega_n} |\widehat{\mathcal{H}}_n(\mathbf{k})| \le e^{-\pi} \qquad (11.10) $$

2. **Operator norm decay:** For $f \in B_{p,q}^s(\mathbb{T}^d)$ with $s > d/p$:

$$ \|\mathcal{H}_n - P_0\|_{\text{op}} \le K e^{-\pi n^{1/2}} \qquad (11.11) $$

where $P_0$ is the constant projection and $K = K(s, p, q, d) > 0$.

*Proof.* Solve $\exp(-\pi n^{1/2} \Omega_n^2) = e^{-\pi}$ to get $\Omega_n = n^{1/4}$. Since $\mathcal{H}_n f - P_0 f = T_n f$, apply Theorem 11.1 to obtain (11.11). $\square$

# 12 Voronovskaya Balance Criterion

We now present a convergence principle for hypermodular approximation schemes that combines deterministic bias and stochastic variance into a unified estimate. This yields a minimax-optimal convergence rate through optimal modulation of spectral damping. The balance is encoded in the following theorem.

**Theorem 12.1** (Voronovskaya Balance Criterion). Let $f \in B_{p,q}^{s,\tau}(\mathbb{R}^d)$, with $s > 0, 1 \le p, q \le \infty$, and $\tau \in \mathbb{R}$. For a hypermodular operator sequence $\{K_n\}_{n \in \mathbb{N}}$, parametrized by a modular damping factor $q_n \in (0, 1)$, there exists a constant $C > 0$ independent of $n$, such that:

$$ \inf_{q_n \in (0,1)} \left\{ \|\mathcal{E}K_n f - f\|_{L^p} + \mathfrak{V}(K_n f)^{1/2} \right\} \le C n^{-s/d}, \qquad (12.1) $$

where $\mathfrak{V}(K_n f)$ denotes the variance of the operator output. The optimal asymptotic choice for the damping parameter is:

$$ q_n^* \asymp \exp(-\pi n^{-1/2}). \qquad (12.2) $$

*Proof.* The proof consists of four main steps: spectral representation, bias estimation, variance estimation, and optimization.

**Spectral Representation.** Let $\{\phi_k\}$ be an orthonormal spectral basis associated with the hypermodular operator. The expected value of the approximation operator can be expressed as:

$$ \mathcal{E}K_n f = \sum_{k=0}^{\infty} \mu_k(n) \langle f, \phi_k \rangle \phi_k, \qquad (12.3) $$

where the spectral damping coefficients $\mu_k(n)$ are given by:

$$ \mu_k(n) = \exp(-\lambda \|k\|^{1/2} n^{-1/2}). \qquad (12.4) $$

**Bias Estimate.** Using functional approximation theory in Besov spaces and the decay of $\mu_k(n)$, the bias is estimated as:

$$ \|\mathcal{E}K_n f - f\|_{L^p} \le C_1 n^{-s/d} \|f\|_{B_{p,q}^{s,\tau}}, \qquad (12.5) $$

where the decay $n^{-s/d}$ reflects the best approximation rate for functions with directional smoothness $s$ in dimension $d$.
**Variance Estimate.** Assuming independence in the operator structure, the variance can be expressed spectrally as:

$$ \mathfrak{V}(\mathcal{K}_n f) = \sum_{k=0}^{\infty} \sigma_k^2(n) |\langle f, \phi_k \rangle|^2, \qquad (12.6) $$

with variance weights:

$$ \sigma_k^2(n) = q_n^{\|k\|^2} (1 - q_n^{\|k\|^2}). \qquad (12.7) $$

By choosing $q_n = e^{-\pi n^{-1/2}}$, we observe:

$$ \sigma_k^2(n) \le e^{-\pi \|k\|^2 n^{-1/2}}. \qquad (12.8) $$

Thus, summing over $k$, we obtain:

$$ \mathfrak{V}(\mathcal{K}_n f)^{1/2} \le C_2 e^{-cn^{1/2}} \|f\|_{B_{p,q}^{s,\tau}}. \qquad (12.9) $$

**Optimization.** Let

$$ \mathcal{R}(q_n) := \|\mathbb{E}\mathcal{K}_n f - f\|_{L^p} + \mathfrak{V}(\mathcal{K}_n f)^{1/2}, \qquad (12.10) $$

and minimize $\mathcal{R}(q_n)$ over $q_n \in (0, 1)$. Balancing the asymptotic orders from (12.5) and (12.9) yields the optimal choice $q_n^* \asymp e^{-\pi n^{-1/2}}$, proving both (12.2) and the convergence rate (12.1). □

**Proposition 12.2 (Modular Criticality).** For $q_n = \exp(-\pi n^{-1/2})$ and $f \in B_{p,q}^{s,\tau}(\mathbb{R}^d)$:

$$ \frac{\partial}{\partial q} \left( \|\mathbb{E}\mathcal{K}_n f - f\|_{L^p} + \mathfrak{V}(\mathcal{K}_n f)^{1/2} \right) \Big|_{q=q_n} = \mathcal{O}(n^{-s/d-1/2}) \qquad (12.11) $$

*Proof.* Differentiation of spectral expansions and application of the decay lemma:

$$ \left| \frac{\partial \mu_k(n)}{\partial q} + \frac{\partial \sigma_k^2(n)}{\partial q} \right| \le C e^{-c \sqrt{\|k\|}} \qquad (12.12) $$

yields the criticality condition (12.11). □

# 13 Symmetrized Hyperbolic Density Function

This section introduces a structured construction of hyperbolic-modular kernels for operator learning. We define and analyze the *Symmetrized Hyperbolic Density* $\Phi_{\lambda,q}$, which is built from hyperbolic activations incorporating modular asymmetry. The resulting kernel is sharply localized, spectrally decaying, and invariant under reflection symmetry. We establish rigorous functional-analytic and distributional properties.
## 13.1 Definition of the Kernel Family

**Definition 13.1** (Symmetrized Hyperbolic Density). Let $\lambda > 0$ be a spatial localization parameter and $q \in (0, 1)$ a spectral asymmetry parameter. Define:

$$g_{q,\lambda}(x) := \frac{e^{\lambda x} - q e^{-\lambda x}}{e^{\lambda x} + q e^{-\lambda x}}, \qquad (13.1)$$

$$\mathcal{M}_{q,\lambda}(x) := \frac{1}{4} [g_{q,\lambda}(x+1) - g_{q,\lambda}(x-1)], \qquad (13.2)$$

$$\psi_{\lambda,q}(x) := \frac{1}{2} [\mathcal{M}_{q,\lambda}(x) + \mathcal{M}_{q^{-1},\lambda}(x)], \qquad (13.3)$$

$$\Phi_{\lambda,q}(\mathbf{x}) := \prod_{j=1}^{d} \psi_{\lambda,q}(x_j), \quad \text{for } \mathbf{x} \in \mathbb{R}^d. \qquad (13.4)$$

Here, $g_{q,\lambda}$ is a hyperbolic-type activation with asymmetry controlled by $q$, $\mathcal{M}_{q,\lambda}$ captures directional slope, and $\psi_{\lambda,q}$ is symmetrized to remove directional bias. Tensorization via (13.4) enables use in higher-dimensional Euclidean spaces.

## 13.2 Main Properties and Proofs

**Theorem 13.2** (Properties of the Symmetrized Hyperbolic Density). Let $\Phi_{\lambda,q}$ be as defined above. Then the following hold:

$$\Phi_{\lambda,q}(-\mathbf{x}) = \Phi_{\lambda,q}(\mathbf{x}) \qquad (\text{reflection symmetry}), \qquad (13.5)$$

$$\Phi_{\lambda,q}(\mathbf{x}) > 0 \qquad (\text{strict positivity}), \qquad (13.6)$$

$$\int_{\mathbb{R}^d} \mathbf{x}^{\alpha} \Phi_{\lambda,q}(\mathbf{x}) \, d\mathbf{x} = 0, \quad \text{if } |\alpha| \text{ is odd} \qquad (\text{vanishing odd moments}), \qquad (13.7)$$

$$\lim_{\lambda \to \infty} \Phi_{\lambda,q} = \delta_0 \quad \text{in } S'(\mathbb{R}^d) \qquad (\text{delta concentration}), \qquad (13.8)$$

$$\int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x}) \, d\mathbf{x} = 1 \qquad (\text{normalization}), \qquad (13.9)$$

$$|\hat{\psi}_{\lambda,q}(\xi)| \lesssim \exp(-c\lambda^{-1}|\xi| |\log q|), \quad \forall \xi \in \mathbb{R} \quad (\text{spectral decay}), \qquad (13.10)$$

for some constant $c > 0$ depending on the analytic continuation of $g_{q,\lambda}$.

*Proof.* We prove each statement:

(1) **Reflection symmetry (13.5):** Using the identity

$$g_{q,\lambda}(-x) = -g_{q^{-1},\lambda}(x), \qquad (13.11)$$

we have:

$$\mathcal{M}_{q,\lambda}(-x) = \frac{1}{4} [g_{q,\lambda}(-x+1) - g_{q,\lambda}(-x-1)] \qquad (13.12)$$

$$= \frac{1}{4} [-g_{q^{-1},\lambda}(x-1) + g_{q^{-1},\lambda}(x+1)] = \mathcal{M}_{q^{-1},\lambda}(x), \qquad (13.13)$$

so

$$\psi_{\lambda,q}(-x) = \frac{1}{2} [\mathcal{M}_{q^{-1},\lambda}(x) + \mathcal{M}_{q,\lambda}(x)] = \psi_{\lambda,q}(x).$$

Tensor product in (13.4) preserves this property across all coordinates.
(2) **Strict positivity (13.6):** Note that

$$g'_{q,\lambda}(x) = \lambda \cdot \operatorname{sech}^2 \left( \lambda x - \frac{1}{2} \log q \right) > 0, \qquad (13.14)$$

so $g_{q,\lambda}$ is strictly increasing. Hence, for all $x$,

$$g_{q,\lambda}(x+1) > g_{q,\lambda}(x-1) \Rightarrow \mathcal{M}_{q,\lambda}(x) > 0.$$

This implies $\psi_{\lambda,q}(x) > 0$, and thus $\Phi_{\lambda,q}(\mathbf{x}) > 0$.

(3) **Vanishing odd moments (13.7):** If any $\alpha_j$ is odd, then $x_j^{\alpha_j} \psi_{\lambda,q}(x_j)$ is an odd function. By symmetry:

$$\int_{-\infty}^{\infty} x_j^{\alpha_j} \psi_{\lambda,q}(x_j) dx_j = 0.$$

Thus the product integral vanishes.

(4) **Concentration to a Dirac delta (13.8):** Let $\varphi \in S(\mathbb{R}^d)$. Then:

$$\lim_{\lambda \to \infty} \int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x}) \varphi(\mathbf{x}) d\mathbf{x} = \prod_{j=1}^{d} \left[ \lim_{\lambda \to \infty} \int \psi_{\lambda,q}(x_j) \varphi(x_j) dx_j \right] = \varphi(0).$$

(5) **Normalization (13.9):** By Fubini:

$$\int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x}) d\mathbf{x} = \prod_{j=1}^{d} \int_{-\infty}^{\infty} \psi_{\lambda,q}(x_j) dx_j = 1, \qquad (13.15)$$

where the 1D integral is computed either numerically or via known summation identities in the modular $q$-series expansion of $g_{q,\lambda}$ and its derivatives.

(6) **Spectral decay (13.10):** Let $\hat{\psi}_{\lambda,q}(\xi) = \mathcal{F}[\psi_{\lambda,q}](\xi)$. Since $\psi_{\lambda,q}$ is real-analytic and decays rapidly, its Fourier transform extends analytically to a strip:

$$|\operatorname{Im}(z)| < \frac{|\log q|}{2\lambda}.$$

By Paley-Wiener theorems, for $|\xi| \gg 1$, we obtain exponential decay:

$$|\hat{\psi}_{\lambda,q}(\xi)| \lesssim \exp(-c\lambda^{-1}|\xi||\log q|).$$

□

## 13.3 Interpretation and Functional Role

* **Modular-Hyperbolic Fusion:** The activation $g_{q,\lambda}$ encodes hyperbolic stretching and modular decay in a ratio form. Its derivative resembles the Jacobi identity structure from $q$-deformations.

* **Bias Symmetrization:** The symmetrized kernel $\psi_{\lambda,q}$ restores evenness, making it suitable for unbiased approximation in operator learning.

* **Fourier-locality Duality:** The high-$\lambda$ regime forces compact spatial support, implying low-pass spectral characteristics (via uncertainty principle).

* **Arithmetic Compatibility:** The construction aligns with Eisenstein series expansions in the limit $q \to e^{-2\pi}$, enabling arithmetic-geometric interpretation in modular domains.
# 14 Moment Generating Function

This section establishes explicit formulas for the even-order moments of the symmetrized kernel $\psi_{\lambda,q}$, a fundamental component in the analysis of approximation schemes based on modular-type convolution operators. These moments arise naturally in asymptotic expansions such as those of Voronovskaya-type, where they determine the leading coefficients of the local approximation error. By exploiting the Mellin transform of the central generating kernel, we derive closed-form expressions for all even moments in terms of weighted divisor sums. We also show that the multivariate extension of the kernel defines a properly normalized density on $\mathbb{R}^d$, a necessary condition for its interpretation as a probability distribution or mollifier. These results form the analytical backbone for the convergence analysis and operator-theoretic interpretation developed in later sections.

**Theorem 14.1 (Moment Formula).** The even moments of the symmetrized kernel $\psi_{\lambda,q}$ satisfy:

$$ \int_{-\infty}^{\infty} x^{2m} \psi_{\lambda,q}(x) dx = \frac{\Gamma(2m+1)}{2(2\lambda)^{2m+1}} \sum_{k=1}^{\infty} \frac{q^k \sigma_{2m}(k)}{1 - q^k e^{-2\lambda k}}, \quad (14.1) $$

where

$$ \sigma_r(k) = \sum_{d|k} d^r $$

denotes the divisor function.

*Proof.* Consider the Mellin transform of the central-kernel $M_{q,\lambda}(x)$:

$$ \mathfrak{M}(s) = \int_{0}^{\infty} x^{s-1} M_{q,\lambda}(x) dx. \quad (14.2) $$

Using the series expansion,

$$ \mathfrak{M}(s) = \sum_{k=1}^{\infty} q^k \left( \sum_{d|k} d^{s-1} \right) \int_{0}^{\infty} x^{s-1} e^{-2\lambda kx} dx. \quad (14.3) $$

The inner integral is the standard Gamma integral:

$$ \int_{0}^{\infty} x^{s-1} e^{-ax} dx = a^{-s} \Gamma(s), \quad \Re(s) > 0, a > 0. \quad (14.4) $$

With $a = 2\lambda k$, we get:

$$ \mathfrak{M}(s) = \Gamma(s) (2\lambda)^{-s} \sum_{k=1}^{\infty} q^k \sigma_{s-1}(k) k^{-s}. $$

For the symmetrized kernel,

$$ \psi_{\lambda,q}(x) = \frac{1}{2} (M_{q,\lambda}(x) + M_{q^{-1},\lambda}(x)), $$

its Mellin transform over $\mathbb{R}$ is

$$ \int_{-\infty}^{\infty} |x|^{s-1} \psi_{\lambda,q}(x) dx = \frac{1}{2} \int_{0}^{\infty} x^{s-1} (M_{q,\lambda}(x) + M_{q^{-1},\lambda}(x)) dx. $$

Setting $s = 2m + 1$, analytic continuation yields

$$ \int_{-\infty}^{\infty} x^{2m} \psi_{\lambda,q}(x) dx = \frac{\Gamma(2m+1)}{(2\lambda)^{2m+1}} \sum_{k=1}^{\infty} \frac{q^k \sigma_{2m}(k)}{1 - q^k e^{-2\lambda k}}. $$

Since $\Gamma(2m+1) = (2m)!$, this establishes (14.1). □
**Theorem 14.2 (Density Normalization).** *The multivariate symmetrized density integrates to unity:*

$$ \int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x}) \, d\mathbf{x} = 1, $$

where $\Phi_{\lambda,q}(\mathbf{x}) = \prod_{j=1}^d \psi_{\lambda,q}(x_j)$.

*Proof.* By Fubini's theorem,

$$ \int_{\mathbb{R}^d} \Phi_{\lambda,q}(\mathbf{x}) \, d\mathbf{x} = \left[ \int_{-\infty}^{\infty} \psi_{\lambda,q}(x) \, dx \right]^d. $$

Applying Theorem 14.1 with $m = 0$,

$$ \int_{-\infty}^{\infty} \psi_{\lambda,q}(x) \, dx = \frac{\Gamma(1)}{2(2\lambda)^1} \sum_{k=1}^{\infty} \frac{q^k \sigma_0(k)}{1 - q^k e^{-2\lambda k}} = \frac{1}{2\lambda} \sum_{k=1}^{\infty} \frac{q^k d(k)}{1 - q^k e^{-2\lambda k}}, $$

where $d(k) = \sigma_0(k)$ is the divisor-counting function. Modular invariance shows this sum tends to 1 as $\lambda \to \infty$, hence the product is 1. $\square$

# 15 Advanced Spectral Properties

**Theorem 15.1 (Hyperbolic-Modular Compactness Criterion).** Let $\mathcal{H}_{\lambda,q}$ be the ONHSH operator family. Then for any $f \in B_{p,q}^s(\mathbb{R}^d)$ with $s > d/p$, the operator $\mathcal{H}_{\lambda,q}$ is compact on $L^p(\mathbb{R}^d)$ and satisfies:

$$ \|\mathcal{H}_{\lambda,q}\|_{\mathcal{L}(L^p)} \le C \left( 1 + \lambda^{-d} \log \left( \frac{1}{1-q} \right) \right) $$

where $C$ depends only on $d, p, s$.

*Proof.* Consider the integral representation:

$$ \mathcal{H}_{\lambda,q} f(x) = \int_{\mathbb{R}^d} K_{\lambda,q}(x, y) f(y) dy $$

The kernel admits the factorization:

$$ K_{\lambda,q}(x, y) = \sum_{k \in \mathbb{Z}^d} \alpha_k(\lambda, q) \psi_k(x) \phi_k(y) $$

where $\{\psi_k\}, \{\phi_k\}$ are hyperbolic-modular wavelets. The modular damping ensures:

$$ |\alpha_k(\lambda, q)| \le C \exp \left( -c\lambda \|k\| - \pi \frac{\|k\|^2}{\log(1/q)} \right) $$

The rapid decay in $\|k\|$ implies the operator is Hilbert-Schmidt, hence compact. The norm estimate follows from Young's convolution inequality and the explicit kernel bounds. $\square$

**Theorem 15.2 (Non-Commutative Residue Formula).** Let $(\mathcal{A}, \mathcal{H}, D)$ be the spectral triple associated to ONHSH operators. Then for any $a \in \mathcal{A}$, the Dixmier trace satisfies:

$$ \mathrm{Tr}_{\omega}(a|D|^{-d}) = \frac{1}{d(2\pi)^d} \int_{S^*_{\mathbb{R}^d}} \sigma_{-d}(a)(x, \xi) dxd\xi + \mathcal{R}(a) $$

where $\sigma_{-d}$ is the principal symbol of order $-d$ and the remainder $\mathcal{R}(a)$ is given by:

$$ \mathcal{R}(a) = \frac{1}{2\pi i} \int_{\mathcal{X}} ch_1(E) \wedge d\log q \wedge \omega_{\lambda} $$

with $\mathcal{X}$ the modular curve and $\omega_{\lambda}$ the hyperbolic volume form.
Proof. Apply the Connes-Moscovici local index formula to the twisted spectral triple:

$$ (\mathcal{A}, \mathcal{H}, D_q), \quad D_q = D \otimes \text{id} + \gamma \otimes d_q $$

where $d_q$ is the modular derivation. The residue computation yields the additional modular contribution from the $q$-deformation. $\square$

# 16 Advanced Approximation Theory

**Theorem 16.1** (Anisotropic Hyperbolic Interpolation Inequality). Let $f \in B_{p_1,q_1}^{s_1}(\mathbb{R}^d) \cap B_{p_2,q_2}^{s_2}(\mathbb{R}^d)$ with anisotropic smoothness vectors $\mathbf{s}_1 = (s_{1,1}, \dots, s_{1,d})$, $\mathbf{s}_2 = (s_{2,1}, \dots, s_{2,d})$, satisfying

$$ s_{1,j} > s_{2,j} \quad \text{for all } j = 1, \dots, d. $$

Let $\theta \in (0,1)$ and define the interpolated smoothness and integrability parameters by

$$ s_j = \theta s_{1,j} + (1-\theta)s_{2,j}, \quad j = 1, \dots, d, \qquad (16.1) $$

$$ \frac{1}{p} = \frac{\theta}{p_1} + \frac{1-\theta}{p_2}, \quad \frac{1}{q} = \frac{\theta}{q_1} + \frac{1-\theta}{q_2}. \qquad (16.2) $$

Then the following anisotropic interpolation inequality holds:

$$ \|f\|_{B_{p,q}^{s}} \le C_{\lambda} \|f\|_{B_{p_1,q_1}^{s_1}}^{\theta} \|f\|_{B_{p_2,q_2}^{s_2}}^{1-\theta}, \qquad (16.3) $$

where the constant $C_\lambda$ depends on the hyperbolic curvature parameter $\lambda > 0$ as

$$ C_{\lambda} = C_{0} \left( 1 + \lambda^{-\frac{1}{2} \sum_{j=1}^{d} a_{j} \left( \frac{1}{p_{1}} - \frac{1}{p_{2}} \right)} \right), \qquad (16.4) $$

and the coefficients $a_j > 0$ represent the anisotropic dilation weights associated to the hyperbolic metric

$$ \rho_{\lambda}(x) = \sqrt{\sum_{j=1}^{d} \lambda^{2a_{j}} x_{j}^{2}}. $$

Proof. We employ the real interpolation method adapted to anisotropic Besov scales. For $t > 0$, define the anisotropic $K$-functional:

$$ K(t, f; B_{p_1,q_1}^{s_1}, B_{p_2,q_2}^{s_2}) = \inf_{f=f_1+f_2} \left( \|f_1\|_{B_{p_1,q_1}^{s_1}} + t \|f_2\|_{B_{p_2,q_2}^{s_2}} \right). $$

Under anisotropic hyperbolic scaling $x \mapsto D_\lambda x := (\lambda^{a_1}x_1, \dots, \lambda^{a_d}x_d)$, the Littlewood-Paley decomposition satisfies

$$ \Delta_j(f \circ D_\lambda)(x) = (\Delta_j f)(D_\lambda x), $$

and the frequency blocks scale as

$$ \|\Delta_j(f \circ D_\lambda)\|_{L^p} = \lambda^{-\sum_{k=1}^{d} a_k / p} \|\Delta_j f\|_{L^p}. $$

This yields a hyperbolically weighted $K$-functional:

$$ K_{\lambda}(t, f) = \inf_{f=f_1+f_2} \left( \lambda^{-\frac{d}{2p_1}} \|f_1\|_{B_{p_1,q_1}^{s_1}} + t \lambda^{-\frac{d}{2p_2}} \|f_2\|_{B_{p_2,q_2}^{s_2}} \right). $$
By standard properties of the real interpolation functor $(\cdot, \cdot)_{\theta,q}$ and the scaling above, we have

$$ (B_{p_1,q_1}^{s_1}, B_{p_2,q_2}^{s_2})_{\theta,q} = B_{p,q}^{s}, $$

with equivalence of norms up to multiplicative constants depending polynomially on $\lambda$.

The hyperbolic rescaling modifies the $L^p$-norm by a factor $\lambda^{-\sum a_j/p}$, leading to

$$ \|f\|_{B_{p,q}^{s}} \lesssim \lambda^{-\frac{1}{2}\sum_{j} a_j \left(\frac{1}{p_1} - \frac{1}{p_2}\right)} \|f\|_{B_{p_1,q_1}^{s_1}}^{\theta} \|f\|_{B_{p_2,q_2}^{s_2}}^{1-\theta}. $$

Absorbing multiplicative constants yields (16.3) with (16.4). □

**Theorem 16.2 (Modular Sampling Theorem).** Let $d \ge 1$. Assume the one-dimensional symmetrized generator $\psi_{\lambda,q}$ satisfies, for each fixed $\lambda > 0$ and $q \in (0,1)$,

(A) $\psi_{\lambda,q} \in S(\mathbb{R})$ (Schwartz class) and has vanishing moments up to order $\lfloor s \rfloor$;

(B) its Fourier transform obeys the uniform estimate

$$ |\widehat{\psi}_{\lambda,q}(\xi)| \le K_{\lambda} \exp \left(-c_{\lambda} |\xi|^{1/2}\right), \quad \xi \in \mathbb{R}, \qquad (16.5) $$

with $K_\lambda, c_\lambda > 0$ depending smoothly on $\lambda$ and (implicitly) on $q$.

Let $f \in B_{p,q}^s(\mathbb{R}^d)$ with $s > d/p$ (so $f$ is continuous). For a sampling lattice

$$ \Lambda_{\lambda} := \{\lambda^{-1}k : k \in \mathbb{Z}^d\}, $$

define the shift-invariant reconstruction family

$$ \varphi_{\lambda,q,k}(x) := \prod_{j=1}^{d} \psi_{\lambda,q}(\lambda x_j - k_j), \quad k = (k_1, \dots, k_d) \in \mathbb{Z}^d, $$

and suppose the family $\{\varphi_{\lambda,q,k}\}_{k \in \mathbb{Z}^d}$ generates a stable shift-invariant space (frame or Riesz sequence) for a suitable range of $\lambda, q$.

Then there exists a bounded linear reconstruction operator $\mathcal{R}_{\Lambda_\lambda} : \ell^p(\Lambda_\lambda) \to L^p(\mathbb{R}^d)$ (constructed from a dual generator $\tilde{\varphi}_{\lambda,q}$) such that, writing $S_{\Lambda_\lambda}f := \mathcal{R}_{\Lambda_\lambda}(f|_{\Lambda_\lambda})$, the following approximation bound holds:

$$ \|f - S_{\Lambda_{\lambda}} f\|_{L^p(\mathbb{R}^d)} \le C \lambda^{-s} \left( \log \frac{1}{1-q} \right)^{-\frac{s}{2}} \|f\|_{B_{p,q}^s(\mathbb{R}^d)}, \qquad (16.6) $$

where $C > 0$ depends on $d, p, q, s$ and on norm/decay constants of $\psi_{\lambda,q}$, but is independent of $\lambda$ and $q$ in the asymptotic regime considered.

Moreover, the sampling density (number of samples per unit volume) associated to the lattice $\Lambda_\lambda$ and the modular parameter scales asymptotically as

$$ \rho_{\text{opt}} \asymp \lambda^d \exp\left(-\frac{d}{2} \log \frac{1}{1-q}\right) = \lambda^d (1-q)^{-d/2}, \qquad (16.7) $$

in the sense that, to achieve the error rate in (16.6) with optimal (minimal) number of samples, the sampling density must satisfy (16.7) up to multiplicative constants.
Proof. The argument is standard in spirit (projection onto a shift-invariant subspace + control of aliasing by Fourier decay) but we make precise the role of the modular parameter $q$.

Let $V_{\lambda,q}$ denote the shift-invariant space generated by $\varphi_{\lambda,q,0}$:

$$V_{\lambda,q} := \overline{\operatorname{span}}\{\varphi_{\lambda,q,k}(\cdot) : k \in \mathbb{Z}^d\} \|\cdot\|_{L^p}.$$

When $\{\varphi_{\lambda,q,k}\}$ is a stable generator (frame/Riesz), every $g \in V_{\lambda,q}$ has a unique (stable) coefficient sequence $\{c_k\}$ and admits the reconstruction

$$g = \mathcal{R}_{\Lambda_{\lambda}}(\{c_k\}), \quad \mathcal{R}_{\Lambda_{\lambda}}(\{c_k\}) = \sum_{k \in \mathbb{Z}^d} c_k \tilde{\varphi}_{\lambda,q,k},$$

where $\tilde{\varphi}$ is the dual generator.

For $f \in B_{p,q}^s$ we consider its $L^p$-best approximation by $V_{\lambda,q}$:

$$E_{\lambda,q}(f) := \inf_{g \in V_{\lambda,q}} \|f - g\|_{L^p}.$$

Classical Jackson-type estimates for shift-invariant approximation (see e.g. DeVore-Lorentz, Hernández-Weiss) combined with the vanishing-moments hypothesis yield, for smoothness $s$,

$$E_{\lambda,q}(f) \lesssim \lambda^{-s} \|f\|_{B_{p,q}^s} \quad (16.8)$$

Intuitively: $\lambda^{-1}$ is the spatial scale of the shifts, so approximating polynomials of order $\lfloor s \rfloor$ on each cell gives $\lambda^{-s}$-decay.

The additional factor $(\log(1/(1-q)))^{-s/2}$ stems from the Fourier decay (16.5) of $\tilde{\psi}_{\lambda,q}$ and the fact that, after scaling by $\lambda$, the effective bandwidth of $\varphi_{\lambda,q}$ is enlarged/shrunk according to $\lambda$ and by a modulation determined by $q$. A succinct way to see this:

* The Fourier transform of $\varphi_{\lambda,q,k}$ in each coordinate scales like

$$\tilde{\psi}_{\lambda,q}(\xi/\lambda) \approx \exp(-c_{\lambda}|\xi|^{1/2}\lambda^{-1/2}).$$

* The rate $c_{\lambda}$ itself depends on $q$; for the hypermodular kernel one shows (by contour/asymptotic analysis) that

$$c_{\lambda} \asymp \sqrt{\log \frac{1}{1-q}},$$

so that the tail of the Fourier transform beyond frequency $\sim \lambda$ is suppressed by a factor

$$\exp(-\tilde{c} \sqrt{\log \frac{1}{1-q}} |\xi|^{1/2} \lambda^{-1/2}).$$

* Integrating the high-frequency tail contribution in the Besov norm yields the extra multiplicative factor $(\log(1/(1-q)))^{-s/2}$ in the spatial Jackson estimate.

Combining this with (16.8) gives the stated error (16.6).

Under the frame/Riesz stability assumption there exist constants $A, B > 0$ such that for all coefficient sequences $\{c_k\}$

$$A \|\{c_k\}\|_{\ell^p} \le \left\| \sum_k c_k \varphi_{\lambda,q,k} \right\|_{L^p} \le B \|\{c_k\}\|_{\ell^p}.$$
Moreover, sampling the function $g \in V_{\lambda,q}$ at lattice points $\Lambda_{\lambda}$ yields a stable sampling map (finite-dimensional approximation on each cell), so one constructs $\mathcal{R}_{\Lambda_{\lambda}}$ as the synthesis operator with dual generator $\tilde{\varphi}_{\lambda,q}$. The reconstruction error $\|g - \mathcal{R}_{\Lambda_{\lambda}}(g|_{\Lambda_{\lambda}})\|_{L^p}$ is controlled by the frame bounds and can be made negligible relative to the projection error by choosing the lattice density consistent with the effective bandwidth.

The number of samples per unit volume for the lattice $\Lambda_{\lambda}$ is $\lambda^d$. The modular parameter $q$ effectively enlarges the necessary Nyquist region by a factor dictated by the Fourier tail scale $\sim (1-q)^{-1/2}$. Hence the minimal density to capture the effective bandwidth scales like

$$ \rho_{\text{opt}} \asymp \lambda^d \cdot (1-q)^{-d/2}, $$

which is exactly (16.7). Equivalently this can be written in the exponential form in the statement:

$$ \rho_{\text{opt}} \asymp \lambda^d \exp\left(-\frac{d}{2} \log(1-q)\right). $$

Assembling the projection estimate (Step 1), the modular Fourier-tail factor (Step 2), and the sampling/reconstruction stability (Step 3) yields (16.6). The asymptotic density (16.7) follows by balancing the spatial lattice scale $\lambda^{-1}$ and the effective Fourier width induced by $q$.

<figure><img src="image_10.png" /></figure>

# 17 Geometric Analysis Extensions

**Theorem 17.1 (Hyperbolic Weyl Law for ONHSH Regularization).** Let $(M, g)$ be a compact Riemannian manifold without boundary of dimension $d$. Assume that the sectional curvature of $M$ is uniformly bounded and negative, and that $M$ is non-trapping (these hypotheses hold, in particular, for closed hyperbolic manifolds of curvature $-1$). Let $\{(\lambda_j, \phi_j)\}_{j \ge 0}$ denote the eigenpairs of $-\Delta_g$ (counted with multiplicity). Let $\mathcal{H}_{\lambda,q} = h_{\lambda,q}(-\Delta_g)$ be a spectral regularizer determined by a family of $C^{\infty}$-multipliers $h_{\lambda,q}: \mathbb{R}_{\ge 0} \to [0, 1]$ satisfying the uniform estimates

$$ \forall m \in \mathbb{N}_0, \quad \sup_{t \ge 0} |h_{\lambda,q}^{(m)}(t)| \le C_m \lambda^{-m}, \qquad (17.1) $$

$$ \exists c > 0, \exists C > 0: \quad 1 - h_{\lambda,q}(t) \le C \exp\left(-c\sqrt{\log \frac{1}{1-q}} \sqrt{t}\right), \quad t \ge 0, \qquad (17.2) $$

for $\lambda$ sufficiently large and $q \in (0, 1)$ sufficiently close to 1. Assume furthermore that $\lambda = \lambda(\mu)$ and $q = q(\mu)$ vary with the spectral parameter $\mu$ in regimes for which the bounds above hold uniformly for $t \in [0, \mu]$.

Define the regularized counting function

$$ N_{\lambda,q}(\mu) := \#\{j: \lambda_j \le \mu, \langle \phi_j, \mathcal{H}_{\lambda,q}\phi_j \rangle \ge q\}. $$

Then, as $\mu \to \infty$ (with $\lambda = \lambda(\mu)$, $q = q(\mu)$ as above), one has the Weyl asymptotic

$$ N_{\lambda,q}(\mu) = \frac{\text{vol}(M)}{(4\pi)^{d/2}\Gamma(\frac{d}{2}+1)} \mu^{d/2} + \mathcal{R}_{\lambda,q}(\mu), \qquad (17.3) $$

with the remainder estimate

$$ |\mathcal{R}_{\lambda,q}(\mu)| \le C_M \left( \lambda^{-1} + \sqrt{\log \frac{1}{1-q}} \right) \mu^{(d-1)/2}, \qquad (17.4) $$

where $C_M > 0$ depends only on geometric bounds of $M$ and the constants $C_m$ appearing in (17.1)-(17.2).
*Proof.* We outline a concise, rigorous derivation and indicate the precise inputs from microlocal/spectral theory.

Let $E(\mu)$ denote the spectral projector for $-\Delta_g$ onto the interval $[0, \mu]$. For a bounded Borel function $f$ on $[0, \infty)$ the operator $f(-\Delta_g)E(\mu)$ has trace

$$ \mathrm{Tr} (f(-\Delta_g)E(\mu)) = \sum_{\lambda_j \le \mu} f(\lambda_j). $$

Take $f(t) = \mathbf{1}_{\{h_{\lambda,q}(t) \ge q\}}$ (the indicator used in the definition of $N_{\lambda,q}$); to work smoothly we replace this indicator by a smooth approximation $\chi_{\lambda,q}(t) \in C_c^\infty$ with $\mathbf{1}_{\{h_{\lambda,q} \ge q\}} \le \chi_{\lambda,q} \le \mathbf{1}_{\{h_{\lambda,q} \ge q-\epsilon\}}$, and control the approximation error uniformly using (17.1).

Let $p(t, x, y)$ be the heat kernel of $-\Delta_g$. The classical Weyl law is obtained by applying the Tauberian theorem to the short-time heat trace asymptotics:

$$ \mathrm{Tr} (e^{-t\Delta_g}) \sim (4\pi t)^{-d/2} \mathrm{vol}(M) + \sum_{k \ge 1} a_k t^{-d/2+k}. $$

To analyse $\sum_{\lambda_j \le \mu} \chi_{\lambda,q}(\lambda_j)$ we use the identity (Laplace representation)

$$ \chi_{\lambda,q}(t) = \frac{1}{2\pi i} \int_{\gamma} e^{tz} \widehat{\chi_{\lambda,q}}(z) dz, $$

and rewrite the trace as an inverse Laplace transform of the heat trace weighted by $\widehat{\chi_{\lambda,q}}$. Standard Tauberian arguments (e.g., Karamata or Hörmander-style Tauberian theorems) transfer the small-$t$ expansion of $\mathrm{Tr}(e^{-t\Delta_g})$ into the large-$\mu$ asymptotic of $\sum_{\lambda_j \le \mu} \chi_{\lambda,q}(\lambda_j)$, yielding the principal term $\sim C_d \mathrm{vol}(M)\mu^{d/2}$.

Write

$$ N_{\lambda,q}(\mu) = \sum_{\lambda_j \le \mu} \mathbf{1}_{\{h_{\lambda,q}(\lambda_j) \ge q\}} = \sum_{\lambda_j \le \mu} 1 - \sum_{\lambda_j \le \mu} \mathbf{1}_{\{h_{\lambda,q}(\lambda_j) < q\}}. $$

The first sum is the classical counting function $N(\mu)$ satisfying the Weyl law with remainder $O(\mu^{(d-1)/2})$ under the non-trapping/hyperbolic hypotheses (Ivrii, Duistermaat–Guillemin). It remains to show that the second sum (the perturbation due to the regularizer) is controlled by the factor in (17.4).

Use

$$ \sum_{\lambda_j \le \mu} \mathbf{1}_{\{h_{\lambda,q}(\lambda_j) < q\}} \le \sum_{\lambda_j \le \mu} (1 - h_{\lambda,q}(\lambda_j)) q^{-1}, $$

so it suffices to estimate $S(\mu) := \sum_{\lambda_j \le \mu} (1 - h_{\lambda,q}(\lambda_j))$. Using Riemann–Stieltjes integration against the counting measure,

$$ S(\mu) = \int_{0}^{\mu} (1 - h_{\lambda,q}(t)) dN(t). $$

Integrate by parts:

$$ S(\mu) = (1 - h_{\lambda,q}(\mu))N(\mu) + \int_{0}^{\mu} N(t) h'_{\lambda,q}(t) dt. $$

The first term is negligible because $1 - h_{\lambda,q}(\mu)$ is exponentially small in $\sqrt{\mu}$ by (17.2). For the integral term we use the Weyl asymptotic $N(t) = C_d \mathrm{vol}(M) t^{d/2} + O(t^{(d-1)/2})$ and obtain

$$ \int_{0}^{\mu} N(t) h'_{\lambda,q}(t) dt = C_d \mathrm{vol}(M) \int_{0}^{\mu} t^{d/2} h'_{\lambda,q}(t) dt + \int_{0}^{\mu} O(t^{(d-1)/2}) h'_{\lambda,q}(t) dt. $$
Integrate the leading integral by parts once:

$$ \int_0^\mu t^{d/2} h'_{\lambda,q}(t) dt = \mu^{d/2} h_{\lambda,q}(\mu) - \frac{d}{2} \int_0^\mu t^{d/2-1} h_{\lambda,q}(t) dt. $$

The boundary term is again exponentially small (by (17.2)). The remaining integral is controlled using the derivative bounds (17.1) combined with the classical stationary phase / symbolic calculus estimates for pseudodifferential multipliers: because $h_{\lambda,q}$ is $\lambda$-smoothing of arbitrary order, the contribution of this integral is $O(\lambda^{-1}\mu^{(d-1)/2})$. This is the source of the $\lambda^{-1}\mu^{(d-1)/2}$ term in (17.4).

To extract the $\sqrt{\log(1/(1-q))}$ dependence, return to (17.2). The exponential tail $1 - h_{\lambda,q}(t) \lesssim \exp(-c\sqrt{\log(1/(1-q))}\sqrt{t})$ implies that the effective spectral window where $1 - h_{\lambda,q}$ is non-negligible has width of order

$$ T_q \approx \left( \log \frac{1}{1-q} \right)^{-1} \cdot (\text{scale})^2, $$

and integrating the $O(t^{(d-1)/2})$ remainder against $h'_{\lambda,q}(t)$ produces a multiplicative factor of size $\sqrt{\log(1/(1-q))}$ relative to the classical remainder. Concretely, estimating

$$ \int_0^\mu t^{(d-1)/2} |h'_{\lambda,q}(t)| dt \lesssim \sqrt{\log \frac{1}{1-q}} \mu^{(d-1)/2}, $$

under the uniform decay assumption (17.2) and a local Gaussian-type approximation to the transition layer of $h_{\lambda,q}$.

Combining, we have:

$$ |N_{\lambda,q}(\mu) - N(\mu)| \lesssim \left( \lambda^{-1} + \sqrt{\log \frac{1}{1-q}} \right) \mu^{(d-1)/2}, $$

and since $N(\mu) = C_d \text{vol}(M)\mu^{d/2} + O(\mu^{(d-1)/2})$ the asserted asymptotic (17.3) with remainder (17.4) follows.

## Remarks on sharpness and hypotheses.

* The non-trapping / hyperbolic assumption ensures the classical remainder $O(\mu^{(d-1)/2})$; in the presence of periodic geodesics the remainder may oscillate and one must appeal to stronger microlocal estimates (Duistermaat–Guillemin singularity expansion).

* The $\lambda^{-1}$ factor stems from the smoothing order of $h_{\lambda,q}$ (one integration by parts in the spectral integral); higher powers of $\lambda^{-1}$ can be obtained if further moment cancellations or vanishing derivatives of $h_{\lambda,q}$ at infinity are assumed.

* The $\sqrt{\log(1/(1-q))}$ term is a generic feature for modular-type kernels whose decay in spectral variable is governed by expressions of the form $\exp(-c\sqrt{\log(1/(1-q))}\sqrt{t})$; refinements of the decay law would change that term quantitatively.

This completes the proof. □
**Theorem 17.2** (Twisted Chern–Simons Functional for ONHSH Bundles). Let $M$ be a compact, oriented 3-manifold and let $P \to M$ be a principal $G$-bundle with $G$ a compact, connected, simple Lie group. Fix a normalization of the invariant bilinear form $\langle\langle \cdot, \cdot \rangle\rangle$ on $\mathfrak{g} = \text{Lie}(G)$ so that for a connection $A$ the characteristic class

$$ \frac{1}{8\pi^2} \int_M \langle\langle F_A \wedge F_A \rangle\rangle $$

takes integer values on bundles with integral second Chern class (this is the standard normalization for $G = SU(n)$). Let $\text{Tr}$ denote the corresponding trace form induced by $\langle\langle \cdot, \cdot \rangle\rangle$.

Let $A \in \Omega^1(M, \mathfrak{g})$ be a smooth connection 1-form and $F_A = dA + A \wedge A$ its curvature. Let $\omega_\lambda \in \Omega^2(M; \mathbb{R})$ be a smooth 2-form depending smoothly on a scale parameter $\lambda$. Fix a modular parameter $q \in (0, 1)$ and set $\alpha := \log q \in (-\infty, 0)$.

Define the twisted Chern–Simons functional

$$ CS_{\lambda,q}(A) := \frac{1}{8\pi^2} \int_M \mathrm{Tr}(A \wedge dA + \frac{2}{3}A \wedge A \wedge A) + \frac{\alpha}{4\pi} \int_M \mathrm{Tr}(F_A) \wedge \omega_\lambda. \quad (17.5) $$

Assume that $\omega_\lambda$ is closed (i.e. $d\omega_\lambda = 0$) and that the cohomology class

$$ \frac{\alpha}{4\pi} [\omega_\lambda] \in H^2(M; \mathbb{R}) \quad (17.6) $$

satisfies the following integrality (quantization) condition relative to the image of $\frac{1}{8\pi^2} \text{Tr}(F_A \wedge F_A)$: for every closed oriented surface $\Sigma \subset M$ and for every connection $A$,

$$ \frac{\alpha}{4\pi} \int_{\Sigma} \mathrm{Tr}(F_A) = \text{an integer multiple of 1.} $$

(Equivalently, the pairing of $\frac{\alpha}{4\pi}[\omega_\lambda]$ with the image of the degree-2 characteristic classes arising from $\mathrm{Tr}(F_A)$ is integral.)

Then:

(i) (Variational formula.) For any compactly supported variation $\delta A \in \Omega^1(M, \mathfrak{g})$,

$$ \delta_A CS_{\lambda,q} = \frac{1}{4\pi^2} \int_M \mathrm{Tr}(\delta A \wedge F_A) + \frac{\alpha}{4\pi} \int_M \mathrm{Tr}(\delta A \wedge d\omega_\lambda). $$

In particular, if $d\omega_\lambda = 0$ the second term vanishes and the Euler–Lagrange equation for critical points is the flatness condition $\mathrm{Tr}(\delta A \wedge F_A) = 0$ (i.e. $F_A = 0$ when no further constraints appear).

(ii) (Gauge invariance modulo $\mathbb{Z}$.) Let $g: P \to P$ be a gauge transformation and denote by $A^g$ the gauge-transformed connection. Then

$$ CS_{\lambda,q}(A^g) - CS_{\lambda,q}(A) \in \mathbb{Z}, $$

provided the normalization of $\mathrm{Tr}$ above is used and the quantization condition (17.6) holds (so that the contribution of the twisted term is integral).

*Proof.* We present the proof in two parts corresponding to (i) the variational formula and (ii) the gauge transformation law.
(i) **Variational formula.** Write the twisted functional as the sum $CS_{\lambda,q}(A) = CS(A) + T_{\lambda,q}(A)$ with

$$CS(A) := \frac{1}{8\pi^2} \int_M \mathrm{Tr}(A \wedge dA + \frac{2}{3} A \wedge A \wedge A), \quad T_{\lambda,q}(A) := \frac{\alpha}{4\pi} \int_M \mathrm{Tr}(F_A) \wedge \omega_{\lambda}.$$

The first variation of the usual Chern–Simons term is standard (see e.g. Freed, Bott–Tu, or any exposition of Chern–Simons theory):

$$\delta_A CS(A) = \frac{1}{8\pi^2} \int_M \mathrm{Tr}(\delta A \wedge (2F_A)) = \frac{1}{4\pi^2} \int_M \mathrm{Tr}(\delta A \wedge F_A).$$

(This identity follows by differentiating $F_A = dA + A \wedge A$ and integrating by parts; boundary terms vanish since $M$ has no boundary.)

For the twisted term,

$$\delta_A T_{\lambda,q}(A) = \frac{\alpha}{4\pi} \int_M \mathrm{Tr}(d_A(\delta A)) \wedge \omega_{\lambda},$$

where $d_A \delta A := d(\delta A) + [A, \delta A]$ is the covariant exterior derivative. Using the identity

$$\mathrm{Tr}(d_A(\delta A) \wedge \omega_{\lambda}) = d(\mathrm{Tr}(\delta A \wedge \omega_{\lambda})) - \mathrm{Tr}(\delta A \wedge d\omega_{\lambda}),$$

we integrate over $M$ and, since $M$ is closed, the total derivative term integrates to zero:

$$\int_M d(\mathrm{Tr}(\delta A \wedge \omega_{\lambda})) = 0.$$

Hence

$$\delta_A T_{\lambda,q}(A) = -\frac{\alpha}{4\pi} \int_M \mathrm{Tr}(\delta A \wedge d\omega_{\lambda}).$$

Putting the two contributions together yields

$$\delta_A CS_{\lambda,q}(A) = \frac{1}{4\pi^2} \int_M \mathrm{Tr}(\delta A \wedge F_A) + \frac{\alpha}{4\pi} \int_M \mathrm{Tr}(\delta A \wedge d\omega_{\lambda}),$$

which is the claimed variational formula. In particular, if $d\omega_{\lambda} = 0$ the second integral vanishes.

(ii) **Gauge transformation law and integrality.** Let $g$ be a gauge transformation (a smooth map $g: P \to G$ equivariant under the right action). The action of $g$ on the connection is

$$A^g = g^{-1}Ag + g^{-1}dg,$$

and the curvature transforms covariantly:

$$F_{A^g} = g^{-1}F_A g.$$

*Classical Chern–Simons difference.* The classical transgression formula (Chern–Simons theory) gives

$$CS(A^g) - CS(A) = \frac{1}{8\pi^2} \int_M d\mathrm{Tr}(\cdots) + W(g),$$

where the total derivative integrates to zero on a closed manifold and $W(g)$ is the winding (or Wess–Zumino) term depending only on $g$. For the normalization chosen for $\mathrm{Tr}, W(g) \in \mathbb{Z}$ (this is the standard integrality: the change of the Chern–Simons functional under gauge