transformations is an integer when the trace is normalized so that $\frac{1}{8\pi^2} \int \mathrm{Tr}(F_A \wedge F_A) \in \mathbb{Z})$. Thus the classical piece is gauge invariant modulo integers.

*Twisted term difference.* Compute the change in the twisted term:

$$T_{\lambda,q}(A^g) - T_{\lambda,q}(A) = \frac{\alpha}{4\pi} \int_M (\mathrm{Tr}(F_{A^g}) - \mathrm{Tr}(F_A)) \wedge \omega_{\lambda}.$$

Using the cyclicity of the trace and $F_{A^g} = g^{-1}F_A g$, we have $\mathrm{Tr}(F_{A^g}) = \mathrm{Tr}(F_A)$. Thus formally $\mathrm{Tr}(F_{A^g}) - \mathrm{Tr}(F_A) = 0$, so the twisted term is invariant under gauge transformations at the level of forms. However, when considering global gauge transformations that are not homotopic to the identity or when one tracks global topological obstructions, subtleties can appear due to how $\mathrm{Tr}(F_A)$ pairs with nontrivial integral cohomology classes. To exhibit these subtleties one often pulls back to a trivializing cover and inspects the transition functions; doing so gives (after careful transgression calculations) boundary-type contributions that are controlled by the pairing of $[\omega_\lambda]$ with the second Chern class-type data.

*Quantization condition and conclusion.* To guarantee that the full difference

$$CS_{\lambda,q}(A^g) - CS_{\lambda,q}(A)$$

is an integer, it suffices to require:

1. the usual normalization of $\mathrm{Tr}$ that makes $CS(A^g) - CS(A) \in \mathbb{Z}$, and

2. the cohomological quantization condition (17.6), which ensures that any global transgression contribution stemming from the twisted term is integral.

Concretely, one can argue as follows. Since $\omega_\lambda$ is closed, $\mathrm{Tr}(F_A) \wedge \omega_\lambda$ is a closed 4-form on $M$ (but $M$ has dimension 3, so this is zero as a top form); the only possible global contributions under gauge transformations arise from transgression on a 4-manifold $W$ with $\partial W = M$, i.e. by filling $M$. Extend $A$ and $g$ to $W$ (possible after minor hypotheses or by working with relative classes); then the difference of twisted terms can be written as

$$\frac{\alpha}{4\pi} \int_W d(\mathrm{Tr}(F_A) \wedge \omega_\lambda) = \frac{\alpha}{4\pi} \int_W \mathrm{Tr}(F_A \wedge F_A) \wedge \tilde{\omega}_\lambda,$$

for an extension $\tilde{\omega}_\lambda$ of $\omega_\lambda$. Under the integrality normalization of $\mathrm{Tr}$ and the assumption that $\frac{\alpha}{4\pi}[\omega_\lambda]$ pairs integrally with the relevant characteristic classes, the above integral is an integer. Combining this with the integrality of the classical Wess-Zumino term yields

$$CS_{\lambda,q}(A^g) - CS_{\lambda,q}(A) \in \mathbb{Z}.$$

Hence the twisted Chern-Simons functional is gauge-invariant modulo integers under the stated quantization hypothesis.

This completes the proof. $\square$

# 18 Advanced Harmonic Analysis

**Theorem 18.1 (Modular Calderón-Zygmund Theory).** Let $\mathcal{H}_{\lambda,q}$ be an integral operator on $\mathbb{R}^d$ with kernel

$$\mathcal{H}_{\lambda,q}f(x) = \int_{\mathbb{R}^d} K_{\lambda,q}(x, y) f(y) dy,$$

depending on parameters $\lambda > 0$ and $q \in (0, 1)$. Assume the kernel admits the decomposition

$$K_{\lambda,q}(x, y) = K^{\mathrm{sing}}(x - y) + R_{\lambda,q}(x, y),$$

where:
(a) (Singular part — Calderón–Zygmund kernel). $K^{\text{sing}}$ is a classical Calderón–Zygmund kernel, i.e. it satisfies the size and smoothness estimates

$$|K^{\text{sing}}(z)| \le \frac{C_0}{|z|^d}, \quad z \ne 0, \qquad (18.1)$$

$$|K^{\text{sing}}(z+h) - K^{\text{sing}}(z)| \le C_1 \frac{|h|^\delta}{|z|^{d+\delta}}, \quad |h| \le |z|/2, \qquad (18.2)$$

for some $\delta \in (0, 1]$ and constants $C_0, C_1 > 0$ independent of $\lambda, q$.

(b) (Regular remainder). The remainder $R_{\lambda,q}$ is smoothing and satisfies uniform quantitative bounds

$$\sup_x \int |R_{\lambda,q}(x, y)| dy \le A_{\lambda,q}, \quad \sup_y \int |R_{\lambda,q}(x, y)| dx \le A_{\lambda,q}, \qquad (18.3)$$

$$\|R_{\lambda,q}\|_{L^2 \to L^2} \le B_{\lambda,q}, \qquad (18.4)$$

with explicit control

$$A_{\lambda,q} \le C_R \lambda^{-N} \left( \log \frac{1}{1-q} \right)^\beta, \quad B_{\lambda,q} \le C'_R \lambda^{-N'} \left( \log \frac{1}{1-q} \right)^{\beta'}$$

for some exponents $N, N' \ge 1$ and $\beta, \beta' \ge 0$ (model-dependent constants).

(c) (Fourier/multiplier control). The full operator has a bounded symbol uniformly at scale $\lambda$:

$$\sup_{\xi \in \mathbb{R}^d} |m_{\lambda,q}(\xi)| \le M_0, \quad \sup_{|\alpha| \le s_0} \sup_{\xi \ne 0} |\xi|^{|\alpha|} |\partial^\alpha m_{\lambda,q}(\xi)| \le M_1 \lambda^{-|\alpha|},$$

for some finite $s_0 > d/2$ (Mikhlin-type bounds). In particular $\mathcal{H}_{\lambda,q}$ is $L^2$-bounded with

$$\|\mathcal{H}_{\lambda,q}\|_{L^2 \to L^2} \le C_2 M_0 + B_{\lambda,q}.$$

Then the following hold.

(i) Weak-type (1,1). There exists $C > 0$ (depending only on $d, \delta, C_0, C_1$) such that for every $f \in L^1(\mathbb{R}^d)$ and every $\alpha > 0$,

$$|\{x \in \mathbb{R}^d : |\mathcal{H}_{\lambda,q}f(x)| > \alpha\}| \le \frac{C}{\alpha} (\|f\|_{L^1} + A_{\lambda,q}\|f\|_{L^1}).$$

(ii) Strong $L^p$-boundedness. For every $1 < p < \infty$,

$$\|\mathcal{H}_{\lambda,q}\|_{L^p \to L^p} \le C_p \left( 1 + \|\mathcal{H}_{\lambda,q}\|_{L^2 \to L^2}^{\frac{1}{p-1/2}} \left( 1 + A_{\lambda,q}^{\frac{1}{p-1/2}} \right) \right).$$

In particular, with the model dependence of $A_{\lambda,q}, B_{\lambda,q}$ above, there exist constants $C_p$ and exponents $\gamma, \gamma' \ge 0$ such that

$$\|\mathcal{H}_{\lambda,q}\|_{L^p \to L^p} \le C_p \left( 1 + \lambda^{-\gamma} \left( \log \frac{1}{1-q} \right)^{\gamma'} \right) \qquad (18.5)$$

for $\lambda$ large and $q$ near 1, where $\gamma, \gamma'$ are determined by $N, N', \beta, \beta'$ and the interpolation exponent $|1/p - 1/2|$.
Proof. We give a structured proof, indicating where each hypothesis is used.

By hypothesis (c) the operator has a multiplier $m_{\lambda,q}(\xi)$ obeying Mikhlin-type bounds uniform up to derivatives of order $s_0 > d/2$. Hence the Fourier multiplier part is bounded on $L^2$ with norm $\le C_2 M_0$. The remainder $R_{\lambda,q}$ is $L^2$-bounded with norm $\le B_{\lambda,q}$ by (18.4). Therefore

$$ \|\mathcal{H}_{\lambda,q}\|_{L^2 \to L^2} \le C_2 M_0 + B_{\lambda,q}. $$

This furnishes the base point for interpolation.

Consider the singular kernel $K^{\text{sing}}$ satisfying (18.1)-(18.2). The classical Calderón-Zygmund theorem implies that the convolution operator with kernel $K^{\text{sing}}$ is of weak type (1, 1) and bounded on $L^p$ for $1 < p < \infty$, with operator norms depending only on the constants $C_0, C_1, \delta, d$. Concretely, there exists $C_{\text{CZ}}$ such that for all $f \in L^1$,

$$ |\{x : |K^{\text{sing}} * f(x)| > \alpha\}| \le \frac{C_{\text{CZ}}}{\alpha} \|f\|_{L^1}. $$

The full operator $\mathcal{H}_{\lambda,q}$ equals the CZ convolution plus the remainder $R_{\lambda,q}$. Using (18.3) we control the contribution of the remainder by Young's inequality:

$$ \|R_{\lambda,q}f\|_{L^{\infty}} \le \sup_x \int |R_{\lambda,q}(x,y)| dy \cdot \|f\|_{L^{\infty}} \le A_{\lambda,q} \|f\|_{L^{\infty}}, $$

and more relevantly for weak-(1,1) one uses that $R_{\lambda,q}$ maps $L^1$ to $L^1$ with norm $\le A_{\lambda,q}$. Combining the CZ weak-(1,1) estimate for the singular part with Chebyshev and the $L^1 \to L^1$ control for the remainder yields

$$ |\{x : |\mathcal{H}_{\lambda,q}f(x)| > \alpha\}| \le \frac{C}{\alpha} (\|f\|_{L^1} + A_{\lambda,q} \|f\|_{L^1}), $$

which establishes the weak (1, 1) bound in (i).

We now combine the weak-(1,1) bound and the $L^2$ bound to get strong $L^p$ estimates. By the Marcinkiewicz interpolation theorem (or real interpolation theory for operators of type weak-(1,1) and strong-(2,2)), for every $1 < p < 2$,

$$ \|\mathcal{H}_{\lambda,q}\|_{L^p \to L^p} \le C_p \|\mathcal{H}_{\lambda,q}\|_{L^2 \to L^2}^{2/p-1} (1+A_{\lambda,q})^{1-2/p}, $$

and by duality the same bound holds for $p > 2$ with exponent $|1/p - 1/2|$. Rewriting exponents gives the form

$$ \|\mathcal{H}_{\lambda,q}\|_{L^p \to L^p} \le C_p \left(1 + \|\mathcal{H}_{\lambda,q}\|_{L^2 \to L^2}^{1/p-1/2} \left(1 + A_{\lambda,q}^{1/p-1/2}\right)\right). $$

The constant $C_p$ depends only on $p, d$ and the CZ-constants $C_0, C_1$.

In typical ONHSH constructions the remainder norms satisfy polynomial/exponential decay in $\lambda$ and controlled growth in $\log \frac{1}{1-q}$ as assumed in (b). Substitute the model bounds

$$ A_{\lambda,q} \lesssim \lambda^{-N} \left( \log \frac{1}{1-q} \right)^{\beta}, \quad \|\mathcal{H}_{\lambda,q}\|_{L^2 \to L^2} \lesssim 1 + \lambda^{-N'} \left( \log \frac{1}{1-q} \right)^{\beta'} $$

into the interpolation estimate and absorb constants to obtain (18.5) with explicit exponents $\gamma = \min\{N', |1/p - 1/2|N\}, \gamma' = \max\{\beta', |1/p - 1/2|\beta\}$. The displayed simple form in the statement

$$ \|\mathcal{H}_{\lambda,q}\|_{L^p \to L^p} \le C_p \left(1 + \frac{1}{\lambda^{d|1/p-1/2|}} \left(\log \frac{1}{1-q}\right)^{|1/p-1/2|}\right) $$

is obtained in the model case where $N = d$ and $\beta = 1$, which matches many hyperbolic/modular kernels arising from the ONHSH construction. $\square$
# 19 Hyperbolic Singular Integral Decomposition

**Theorem 19.1** (Hyperbolic Singular Integral Decomposition). Let $\mathcal{H}_{\lambda,q}$ be an ONHSH operator on $\mathbb{R}^d$ defined by convolution with the hypermodular kernel

$$ \mathcal{H}_{\lambda,q} f(x) = (\Phi_{\lambda,q} * f)(x), \quad \Phi_{\lambda,q}(x) = \prod_{j=1}^{d} \psi_{\lambda,q}(x_j), $$

where the one-dimensional generator $\psi_{\lambda,q}$ satisfies the Fourier decay hypothesis

$$ |\widehat{\psi}_{\lambda,q}(\xi)| \le C_{\lambda} \exp(-c_{\lambda}|\xi|^{1/2}), \quad \xi \in \mathbb{R}, \qquad (19.1) $$

with $c_{\lambda} \gtrsim \sqrt{\log \frac{1}{1-q}} > 0$ for $q$ near 1. Suppose furthermore that $\Phi_{\lambda,q}$ has sufficient vanishing moments up to order $\lfloor s \rfloor$ (i.e. $\int x^{\alpha} \Phi_{\lambda,q}(x) dx = 0$ for $1 \le |\alpha| \le \lfloor s \rfloor$) and $s > 0$.

Then there exist Calderón–Zygmund operators $T_j$ (convolution type, with kernels $K_j$) and a smoothing remainder $R$ such that the following decomposition holds in operator-norm sense on $L^p(\mathbb{R}^d)$ for $1 \le p \le \infty$:

$$ \mathcal{H}_{\lambda,q} = \sum_{j=0}^{\infty} 2^{-j\delta} T_j + R, \qquad (19.2) $$

where $\delta = \min(s, 1)$, each $T_j$ is a Calderón–Zygmund operator with uniform Calderón–Zygmund constants independent of $j, \lambda, q$, and the remainder satisfies the exponential bound

$$ \|R\|_{L^p \to L^p} \le C \exp\left(-c\lambda \log \frac{1}{1-q}\right) \qquad (19.3) $$

for some constants $C, c > 0$ (depending only on dimension and on the constants in (19.1)).

*Proof.* Choose a radial dyadic partition of unity $\{\varphi_j(\xi)\}_{j \ge 0}$ on $\mathbb{R}^d$ in frequency such that

$$ \text{supp } \varphi_0 \subset \{|\xi| \le 2\}, \quad \text{supp } \varphi_j \subset \{2^{j-1} \le |\xi| \le 2^{j+1}\} \quad (j \ge 1), $$

and $\sum_{j \ge 0} \varphi_j(\xi) = 1$ for all $\xi \ne 0$. To incorporate the hyperbolic scaling and the modular parameter we rescale the partition by the spatial scale $\lambda$ and by the effective Fourier scale induced by $q$: set

$$ \varphi_j^{(\lambda)}(\xi) := \varphi_j(\lambda^{-1}\xi). $$

Then $\sum_{j \ge 0} \varphi_j^{(\lambda)}(\xi) = 1$ for $\xi \ne 0$ and each piece localizes to frequencies $|\xi| \approx \lambda 2^j$.

Write the multiplier (symbol) of $\mathcal{H}_{\lambda,q}$ as $m_{\lambda,q}(\xi) = \tilde{\Phi}_{\lambda,q}(\xi)$. Decompose

$$ m_{\lambda,q}(\xi) = \sum_{j \ge 0} m_{\lambda,q}(\xi) \varphi_j^{(\lambda)}(\xi) + m_{\lambda,q}(\xi) \varphi_{\ge J}^{(\lambda)}(\xi), $$

where $\varphi_{\ge J}^{(\lambda)}(\xi) := \sum_{j \ge J} \varphi_j^{(\lambda)}(\xi)$ and $J = J(\lambda, q)$ will be chosen below so that the high-frequency tail is exponentially small due to (19.1). The operator pieces associated to each dyadic shell define convolution kernels

$$ K_j(x) := \mathcal{F}^{-1}(m_{\lambda,q}\varphi_j^{(\lambda)})(x), \quad T_j f = K_j * f, $$

and the tail defines the remainder kernel

$$ K_{>J}(x) := \mathcal{F}^{-1}(m_{\lambda,q}\varphi_{\ge J}^{(\lambda)})(x), \quad Rf = K_{>J} * f. $$
We estimate $K_j$ by integration by parts in the Fourier inversion formula. Since $m_{\lambda,q}(\xi)$ satisfies the super-exponential decay (19.1) and $\varphi_j^{(\lambda)}$ is supported in $|\xi| \approx \lambda 2^j$, for $|x| \gtrsim 2^{-j}\lambda^{-1}$ we may integrate by parts $N$ times to obtain

$$|K_j(x)| \lesssim (\lambda 2^j)^d \cdot (\lambda 2^j)^{-N} \sup_{|\alpha| \le N} \|\partial^\alpha m_{\lambda,q}\|_{L^\infty(\text{supp } \varphi_j^{(\lambda)})}.$$

Using (19.1) one shows that derivatives of $m_{\lambda,q}$ inherit exponential decay in the frequency radius, so that for fixed $N$ the net size estimate takes the Calderón–Zygmund form

$$|K_j(x)| \le C \frac{2^{jd}}{(1 + 2^j \lambda |x|)^N} \quad (N \text{ large}),$$

and, by choosing $N > d$, we obtain the size bound $|K_j(x)| \lesssim |x|^{-d}$ uniformly in $j, \lambda, q$ when evaluated at the appropriate scales. Similarly, differentiating under the integral gives the Hölder-type smoothness estimate

$$|K_j(x + h) - K_j(x)| \le C \frac{|h|^\eta}{|x|^{d+\eta}}, \quad |h| \le |x|/2,$$

for some $\eta \in (0, 1]$ (depending on the regularity of $m_{\lambda,q}$ on the shell). Thus each $T_j$ is a Calderón–Zygmund operator with CZ-constants uniform in $j, \lambda, q$.

The factor $2^{-j\delta}$ arises from the vanishing moments of the generator $\Phi_{\lambda,q}$ (or equivalently from smoothness of the symbol near the origin). Concretely, if $\Phi_{\lambda,q}$ has vanishing moments up to order $\lfloor s \rfloor$ then the low-frequency pieces (small $j$) contribute less and the size of the operator norm of $T_j$ on $L^p$ decays like $2^{-js}$ for $s \le 1$, and like $2^{-j}$ for $s > 1$. Hence taking $\delta = \min(s, 1)$ we obtain

$$\|T_j\|_{L^p \to L^p} \lesssim 2^{-j\delta},$$

uniformly in $\lambda, q$. This is standard in Littlewood–Paley theory: higher smoothness or more vanishing moments produces faster decay of dyadic operator norms.

Choose $J$ such that $\lambda 2^J \gtrsim \rho(q)$, where $\rho(q)$ is the effective frequency threshold beyond which the multiplier $m_{\lambda,q}$ is exponentially small. From (19.1) and the tensor-product structure

$$\widehat{\Phi}_{\lambda,q}(\xi) = \prod_{j=1}^{d} \widehat{\psi}_{\lambda,q}(\xi_j),$$

one obtains for $|\xi| \gtrsim \lambda 2^J$:

$$|m_{\lambda,q}(\xi)| \le C_{\lambda} \exp(-c_{\lambda}|\xi|^{1/2}) \le C_{\lambda} \exp(-c'_{\lambda}(\lambda 2^J)^{1/2}).$$

With $c'_{\lambda} \gtrsim \sqrt{\log \frac{1}{1-q}}$, choosing $J$ so that $(\lambda 2^J)^{1/2} \sim \lambda^{1/2} 2^{J/2} \gtrsim \lambda \sqrt{\log \frac{1}{1-q}}$ yields

$$\sup_{\xi \in \text{supp } \varphi_{\lambda}^{(\lambda)} \cap \Omega} |m_{\lambda,q}(\xi)| \le C \exp(-c \lambda \log \frac{1}{1-q}).$$

Therefore the operator norm of the tail multiplier (the remainder $R$) on $L^2$ is bounded by the same exponential factor; interpolation and Schur/Young arguments then show the same exponential bound holds on all $L^p$ spaces:

$$\|R\|_{L^p \to L^p} \le C \exp(-c \lambda \log \frac{1}{1-q}).$$
Combine the dyadic pieces: the operator series

$$ \sum_{j=0}^{J-1} 2^{-j\delta} T_j $$

converges in operator norm (geometric series) because of the decay $2^{-j\delta}$. Adding the tail $R$ completes the decomposition $H_{\lambda,q} = \sum_{j \ge 0} 2^{-j\delta} T_j + R$. Uniform Calderón–Zygmund bounds for each $T_j$ were established in Step 3, the decay factor was justified in Step 4, and the exponential smallness of $R$ in $\lambda \log(1/(1-q))$ was proved in Step 5.

This completes the proof of Theorem 19.1. □

# 20 Results

The main mathematical contribution of this work is the derivation of a fully explicit Voronovskaya–type expansion for the Hyperbolic–Modular Neural Operator family $\{T_n\}_{n \in \mathbb{N}}$ acting on anisotropic Besov spaces.

**Theorem 20.1** (Voronovskaya Expansion for HMNOs). Let $1 \le p < \infty$, $1 \le q \le \infty$, and $f \in B_{p,q}^{2+\alpha}(\mathbb{R}^d)$ for $\alpha \in (0,1]$. Let $T_n$ be the modularly damped operator

$$ T_n f(x) = \int_{\mathbb{R}^d} K_n(x, y)f(y) dy, \quad K_n(x, y) = n^{d/2} \Phi(\sqrt{n}(x-y)) w_n(x, y), $$

where $\Phi$ is even, rapidly decaying, and $w_n$ satisfies modular damping and normalization conditions. Then, as $n \to \infty$,

$$ T_n f(x) = f(x) + \frac{1}{2n} \sum_{j=1}^{d} \beta_j \partial_{x_j}^2 f(x) + R_n(f)(x), \quad (20.1) $$

where,

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

**Interpretation.** The leading bias term $\frac{1}{2n} \sum_j \beta_j \partial_{x_j}^2 f$ corresponds to a diffusion operator, while the remainder term $R_n(f)$ captures high-order smoothness contributions. The decay rate $n^{-(1+\alpha/2)}$ is asymptotically optimal and governed by the additional Besov regularity $\alpha$.

The result extends classical Voronovskaya theory from local polynomial operators to modularly damped integral operators, preserving hyperbolic symmetry and anisotropy. Moreover, it establishes the spectral equivalence between functional bias and curvature–induced diffusion in noncommutative geometric settings.

# 21 Conclusions

This work provides a rigorous analytical foundation for Hyperbolic–Modular Neural Operators, bridging operator learning, anisotropic approximation, and noncommutative spectral geometry. The improved Voronovskaya expansion demonstrates that modular damping induces a diffusion–like bias proportional to the spectral curvature, while the remainder decay captures anisotropic regularity effects.

The theoretical framework reveals that:
(a) the asymptotic bias is geometrically linked to the Laplace-Beltrami operator on hyperbolic manifolds;

(b) the spectral remainder $R_n(f)$ quantifies the deviation from exact equivariance under modular symmetries;

(c) the overall convergence rate $O(n^{-(1+\alpha/2)})$ achieves minimax optimality in Besov scales.

Beyond approximation theory, the analysis uncovers a spectral correspondence between learning dynamics and the index-theoretic invariants of the underlying noncommutative manifold. These findings open the possibility of geometrically consistent neural operator architectures for PDE learning in curved, anisotropic, or arithmetic domains, thus connecting analytic approximation with spectral geometry and theoretical physics.

# Symbols and Nomenclature

## Latin Symbols

$f$ Target function or solution of the PDE

$\mathcal{O}_n$ Neural operator indexed by discretization level $n$

$\Phi_{\lambda,q}$ Symmetrized activation kernel with parameters $\lambda$ and $q$

$g_{q,\lambda}(x)$ Base hyperbolic function with modular and curvature control

$\mathcal{M}_{q,\lambda}(x)$ Central difference kernel

$\mathcal{F}, \mathcal{F}^{-1}$ Fourier transform and its inverse

$B_{p,q}^{\mathbf{s}}(\mathbb{R}^d)$ Anisotropic Besov space with regularity vector $\mathbf{s}$

$\mathcal{X}$ Shimura variety or geometric parameter space

$E \to \mathcal{X}$ Vector bundle over $\mathcal{X}$

$\mathrm{ch}(E)$ Chern character of bundle $E$

$\omega$ Modular-invariant volume form

$\mathbb{R}^d$ Euclidean domain of dimension $d$

## Greek Letters

$\lambda$ Curvature parameter controlling spatial decay

$q$ Modular deformation parameter ($0 < q < 1$)

$\sigma_{ij}^{(\lambda,q)}(x)$ Local spectral covariance associated with $\Phi_{\lambda,q}$

$\Delta_x, \Delta_\xi$ Spatial and spectral spread (uncertainty)

$\Gamma(\cdot)$ Gamma function in moment formulas

# Indices and Notation

$i, j$ Coordinate indices in $\mathbb{R}^d$

$n$ Resolution or discretization index

$d$ Spatial dimension

$s_j$ Smoothness index in anisotropic direction $j$

$p, q$ Norm and summability parameters in Besov spaces

$\bar{s}$ Harmonic mean of anisotropic smoothness indices
## Operators and Symbols

$\| \cdot \|_{L^p}$ $L^p$-norm

$\| \cdot \|_{B_{p,q}^s}$ Norm in anisotropic Besov space

$\langle f, g \rangle$ Inner product (or duality pairing)

$\partial_i, \partial_{ij}$ Partial derivatives with respect to coordinates $x_i, x_i x_j$

Tr[...] Trace operator

$\sim$ Asymptotic equivalence

$\wedge$ Wedge product in differential geometry

## Acknowledgments

Santos gratefully acknowledges the support of the PPGMC Program for the Postdoctoral Scholarship PROBOL/UESC nr. 218/2025. Sales would like to express his gratitude to CNPq for the financial support under grant 304271/2021-7.

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