Symmetry-Resolved Sensitivity Redistribution Under Tensor Lifting in Electromagnetic Sensing Architectures

Abstract

Symmetric electromagnetic sensing architectures induce representation-space decompositions that organize how measured fields respond to rotations, reflections, and programmable configurations. This paper develops a symmetry-resolved framework for analyzing how local parameter sensitivity is distributed across irreducible sectors and how this distribution changes under tensor lifting. Character-weighted Reynolds projectors decompose the derivatives of first-, second-, and fourth-order observables into orthogonal isotypic components, whose relative weights are quantified through normalized entropy, effective-sector occupancy, and dominant-sector concentration. The formulation distinguishes algebraic sector accessibility, determined by induced representations and tensor-product fusion, from the sensitivity profile realized by a specific physical observation model. The framework is validated using a narrowband far-field electromagnetic model of a two-ring $C_4$-symmetric receiving array and is further examined through matched cyclic and dihedral array ensembles. The results reveal a robust redistribution of sensitivity under tensor lifting in the tested cyclic architectures, while the dihedral configurations exhibit a different, order-dependent behavior associated with their richer representation structure. These findings do not imply a universal increase in information or estimation performance; rather, they show that tensorization reorganizes the symmetry channels through which local sensitivity is expressed. The proposed framework provides a diagnostic tool for comparing and designing symmetry-aware antenna arrays, metasurfaces, reconfigurable intelligent surfaces, and related programmable sensing architectures.

1. Introduction

Symmetry provides a compact description of how a physical architecture responds to transformations. In electromagnetic sensing, the relevant transformations may arise from rotations or reflections of an antenna layout, permutations of equivalent surface elements, polarization operations, or programmable configurations that preserve a subgroup of the underlying geometry. Once such an action is identified, the measured field belongs to a representation space that can be decomposed into irreducible symmetry sectors. This viewpoint has long supported modal classification and symmetry-based reduction in electromagnetics, and it has become increasingly relevant to arrays, metasurfaces, and reconfigurable intelligent surfaces (RIS), where geometry and controllable loading jointly determine the observable field structure [1,2,3].

Recent programmable and reconfigurable metasurface research has emphasized that electromagnetic control extends beyond independent phase adjustment. Contemporary architectures manipulate several field attributes, including amplitude, phase, polarization, frequency, and propagation direction, and are being developed for communication, sensing, imaging, and analog wave-domain processing [4,5,6,7]. These developments motivate analytical tools that retain the physical group structure of the architecture rather than treating the observation as an unstructured collection of complex samples. Symmetry breaking and symmetry selection are likewise established mechanisms for activating or suppressing electromagnetic responses in metasurfaces and photonic structures [8,9].

A separate line of research exploits higher-order observations. In array processing, such constructions are commonly motivated by non-Gaussian source separation, suppression of Gaussian contributions, virtual-aperture formation, or multilinear structure. Tensor lifting also appears more broadly in multilinear signal models, where repeated direct and conjugate factors carry induced group actions. However, increasing tensor order does not by itself specify how sensitivity to a physical parameter is distributed inside the resulting representation space.

Existing symmetry-aware signal-processing frameworks address related but distinct questions. Algebraic signal processing organizes signals, filters, and transforms through representations of algebraic structures [10,11], while equivariant methods construct operators that commute with prescribed group actions [12]. These approaches use symmetry to define transforms, processing rules, invariant features, or computational architectures. Conventional observability and identifiability analysis, by contrast, asks whether a state or parameter can be reconstructed from the available measurements, typically through rank, injectivity, conditioning, or information-based criteria. None of these perspectives directly quantifies how the local derivative of a parameterized electromagnetic observation is distributed among the irreducible sectors of the physical symmetry group, or how that distribution changes under tensor lifting.

This distinction is important because three levels of description need not coincide. The induced representation determines which irreducible labels and fusion pathways are algebraically available. The physical observation and its derivative determine which of those sectors are locally active and how much sensitivity they carry. Statistical estimation performance additionally depends on noise, nuisance parameters, sampling, and the chosen estimator. A broader occupation of symmetry sectors therefore cannot be identified automatically with larger Fisher information, a lower Cramér–Rao bound, or improved estimation accuracy.

The problem addressed in this paper is consequently narrower than generic higher-order inference and more structured than a global derivative norm. Given a symmetric electromagnetic architecture and a parameterized observable, we ask how the local parameter sensitivity is distributed across irreducible symmetry sectors and how that distribution changes under first-, second-, and fourth-order tensor lifts. We refer to this restricted, symmetry-resolved accessibility and weighting of local sensitivity as effective observability. The term does not replace classical observability; it identifies the symmetry channels through which an infinitesimal parameter perturbation is expressed.

The proposed framework combines character-weighted Reynolds projectors with normalized sector metrics. At observable order p, the local derivative is decomposed orthogonally into isotypic components, and the squared component norms are normalized to form sector-wise sensitivity fractions. Normalized entropy, effective-sector occupancy, and dominant-sector fraction then summarize the evenness and concentration of the resulting profile. Tensor-product fusion rules provide an algebraic upper bound on sector accessibility, while the actual sector weights remain dependent on the electromagnetic response and its derivative. This separation between fusion support and realized sensitivity redistribution is central to the analysis.

The use of cyclic and dihedral groups is motivated by both physical relevance and algebraic contrast. Cyclic groups describe architectures with discrete rotational symmetry but no common reflection axis, whereas dihedral groups add reflections to the same rotational structure. These families arise naturally in circular arrays, polygonal apertures, metasurface layouts, and ring-based programmable surfaces. They also provide a controlled comparison between purely one-dimensional cyclic irreducible representations and the mixed one- and two-dimensional representation structure of dihedral groups.

The main contributions are as follows:

• A symmetry-resolved formulation of local electromagnetic sensitivity is developed, with a precise distinction among represented sectors, locally active sectors, sector weights, total derivative magnitude, and statistical estimation performance.
• Character-weighted Reynolds projectors are used to obtain an orthogonal isotypic decomposition at each observable order. Sector-wise sensitivity fractions and three normalized profile metrics are defined from the corresponding Parseval identity.
• The effect of tensor lifting is characterized through induced representations and fusion support. For cyclic groups, the accessible sectors are expressed through modular sums and differences in the supports carried jointly by the observation and its derivative.
• A physics-based electromagnetic validation is performed using a two-ring $C_4$-symmetric receiving array under a narrowband far-field model. Across 200 physical realizations, the normalized sector profile becomes more even from $p=1$ to $p=2$ and $p=4$, while the four irreducible labels remain fixed.
• Matched $C_N$–$D_N$ ensembles are used to examine the dependence on group structure under controlled geometric conditions. The tested cyclic families exhibit monotonic entropy growth in all 1000 realizations, whereas the matched dihedral architectures display an order-dependent redistribution: their $1\to 2$ entropy increments are smaller and their $2\to 4$ increments are larger than those of the corresponding cyclic architectures across all tested symmetry orders.

These findings do not establish a universal monotonicity theorem. Rather, they show that tensor lifting induces a group-dependent reorganization of sector-wise sensitivity. The electromagnetic $C_4$ experiment demonstrates redistribution within a fixed irreducible-label set, while the dihedral results show that richer fusion structure does not guarantee monotonic diversification. The contribution is therefore a diagnostic representation-theoretic framework, not a new estimator or higher-order statistic.

A related research direction arises in mode-dependent observation models, including hidden-mode and fuzzy Markov-jump systems. These frameworks address state and mode estimation under uncertain or partially observed switching dynamics, which differs from the electromagnetic setting considered here. Nevertheless, symmetry-resolved sensitivity could be extended to mode-dependent observation maps or fuzzy local models by conditioning the sector profile on the active or estimated mode. This possibility is therefore regarded as a future extension rather than as part of the present formulation [13,14].

The remainder of the paper is organized as follows. Section 2 defines symmetric electromagnetic observation spaces, local parameter sensitivity, and effective observability in the sector-resolved sense. Section 3 develops the Reynolds decomposition and the sensitivity metrics. Section 4 analyzes induced tensor representations, fusion support, and the distinction between accessibility and redistribution. Section 5 presents the $C_4$ electromagnetic validation, and Section 6 compares matched cyclic and dihedral ensembles. Section 7 discusses the relation to higher-order and symmetry-aware processing, design implications, limitations, and research directions. Section 8 concludes the paper.

2. Symmetry-Resolved Sensitivity in Electromagnetic Sensing

The analysis begins by linking the physical symmetry of an electromagnetic architecture to the local variation in its measured response. The purpose of this section is deliberately foundational: it defines the observation model, the associated group action, and the restricted meaning of effective observability adopted in this work. The explicit irreducible-sector decomposition, its projectors, and the normalized sensitivity metrics are deferred to Section 3, where they can be introduced without presupposing undefined notation.

2.1. Symmetric Electromagnetic Observation Model

Symmetry enters electromagnetic sensing through the geometry, material configuration, boundary conditions, and sampling architecture of the physical system. When these ingredients are preserved by a finite set of transformations, the resulting group acts on both the physical parameter space and the measured field. Group-theoretic methods provide a systematic means of decomposing symmetric electromagnetic structures into modal subspaces and of constructing orthogonal excitations through projection operators [1]. In antenna arrays, metasurfaces, and reconfigurable intelligent surfaces, the same principle provides a natural description of how equivalent sensing or scattering elements transform under rotations and reflections [2,3,8].

Let $\mathsf{\Omega }$ denote the electromagnetic sensing architecture, including the element locations and any material, loading, or boundary parameters retained in the model. Its symmetry group is

$$G_{\mathsf{\Omega }}=\left\{g:g\mathsf{\Omega }=\mathsf{\Omega }\right\},$$

(1)

where equality means physical equivalence under the adopted electromagnetic description. The present paper considers finite groups and finite-dimensional observation spaces. This setting includes the cyclic and dihedral symmetries studied in Section 5 and Section 6 and applies directly to sampled fields, antenna-port responses, and finite arrays of controllable surface elements.

Let

$$y:\mathsf{\Phi }\to V$$

(2)

be the noiseless electromagnetic observation map, where $\mathsf{\Phi }$ is the parameter domain and

$$V\cong {\mathbb{C}}^M$$

(3)

is a finite-dimensional complex Hilbert space containing the M measured or modeled complex degrees of freedom. Depending on the application, $y\left(\varphi \right)$ may contain antenna-port voltages, sampled electric- or magnetic-field components, array steering responses, or coefficients obtained from a finite electromagnetic discretization.

Each transformation $g\in G_{\mathsf{\Omega }}$ acts on V through a representation

$$\rho \left(g\right):V\to V.$$

(4)

For identical uncoupled sensors, $\rho \left(g\right)$ is commonly a permutation matrix. identical uncoupled sensors, $\rho \left(g\right)$ is commonly a permutation matrix. More general models may also include phase, polarization, or modal transformations. Because $G_{\mathsf{\Omega }}$ is finite, an invariant inner product can be chosen so that the representation is unitary:

$$\rho {\left(g\right)}^H\rho \left(g\right)=I_V,g\in G_{\mathsf{\Omega }},$$

(5)

where $I_V$ is the identity operator on V. This unitarity ensures that symmetry transformations preserve the norm used later to resolve local sensitivity.

The physical parameter space may also carry a group action. We denote by

$$T_g:\mathsf{\Phi }\to \mathsf{\Phi }$$

(6)

the transformation induced by g on the parameter domain. For direction-of-arrival sensing, for example, $T_g\varphi$ is the incident angle obtained after applying the corresponding rotation or reflection. Compatibility between the physical symmetry and the observation model is expressed by the equivariance relation

$$y\left(T_g\varphi \right)=\rho \left(g\right)y\left(\varphi \right).$$

(7)

Thus, transforming the physical configuration and then evaluating the response is equivalent to transforming the observation coordinates.

Equation (7) should not be confused with invariance of an individual measurement. In general,

$$\rho \left(g\right)y\left(\varphi \right)\ne y\left(\varphi \right)$$

(8)

for a fixed $\varphi$ and a nontrivial g. A symmetric architecture therefore need not produce a response confined to the invariant subspace. The relevant property is that the family of responses is closed under the group action. This distinction permits a fixed observation and its derivative to contain nontrivial symmetry components even though the architecture itself is symmetric.

In programmable electromagnetic systems, the active control state may modify the effective symmetry group or the representation carried by the measured field. Electromagnetically consistent RIS models emphasize that the attainable response is determined by the coupled interaction among elements, loads, and propagation channels rather than by an unconstrained collection of independent phase shifts [2,3]. Likewise, deliberate symmetry breaking in metasurfaces can activate otherwise forbidden responses or alter polarization and resonance properties [8,9]. These examples motivate treating the group action as part of the physical observation model rather than as a purely formal post-processing device.

2.2. Local Sensitivity and Sector-Resolved Effective Observability

Let $\varphi \in \mathsf{\Phi }$ be a scalar physical parameter. Its first-order local sensitivity is the derivative

$$J_{\varphi }^{\left(1\right)}={\displaystyle \frac{\partial y\left(\varphi \right)}{\partial \varphi }}\in V.$$

(9)

For an infinitesimal perturbation $\Delta \varphi$,

$$y(\varphi +\Delta \varphi )=y\left(\varphi \right)+J_{\varphi }^{\left(1\right)}\Delta \varphi +o\left(|\Delta \varphi |\right),\Delta \varphi \to 0.$$

(10)

Hence, $J_{\varphi }^{\left(1\right)}$ describes the local change in the noiseless electromagnetic response produced by the parameter variation, and

$$∥J_{\varphi }^{\left(1\right)}∥$$

(11)

measures the magnitude of that variation in the chosen Hilbert-space geometry.

The symmetry of the observation model also constrains the sensitivity. Differentiating Equation (7) with respect to $\varphi$ gives

$$J_{T_g\varphi }^{\left(1\right)}{\displaystyle \frac{\partial T_g\varphi }{\partial \varphi }}=\rho \left(g\right)J_{\varphi }^{\left(1\right)}.$$

(12)

For a parameter shift $T_g\varphi =\varphi +{\alpha }_g$, such as a planar rotation acting on an angular coordinate,

$${\displaystyle \frac{\partial T_g\varphi }{\partial \varphi }}=1,$$

(13)

and therefore

$$J_{T_g\varphi }^{\left(1\right)}=\rho \left(g\right)J_{\varphi }^{\left(1\right)}.$$

(14)

For reflections or nonlinear reparameterizations, the Jacobian factor in Equation (12) must be retained.

A vanishing derivative,

$$J_{\varphi }^{\left(1\right)}=0,$$

(15)

means that the selected observation is locally insensitive to $\varphi$ at the operating point. Conversely,

$$J_{\varphi }^{\left(1\right)}\ne 0$$

(16)

establishes first-order variation in the noiseless response. Neither condition alone establishes global identifiability or statistical estimation performance. Distinct parameter values may still generate the same observation, and any inferential conclusion additionally depends on noise, nuisance parameters, sampling, and the estimator.

The present work refines the deterministic condition in Equation (16) by asking how the local variation is organized relative to the irreducible structure of $G_{\mathsf{\Omega }}$. Section 3 constructs the orthogonal isotypic decomposition of the relevant observation space and projects $J_{\varphi }^{\left(p\right)}$ onto the sectors represented at observable order p. In this restricted setting, a sector is locally accessible when the corresponding projected sensitivity is nonzero, and the relative squared norms of the projected components define the sector-wise sensitivity profile.

We use the term effective observabilityfor this symmetry-resolved accessibility and weighting of local sensitivity. The adjective effective emphasizes that the analysis is conditioned on the selected architecture, observable, parameter, operating point, and symmetry action. It does not refer to all electromagnetic degrees of freedom of the underlying system, nor does it replace classical observability in dynamical systems.

The distinction among three levels is fundamental:

$$\begin{array}{cc}\hfill \mathrm{local}\mathrm{sensitivity}& ⟶\mathrm{first}-\mathrm{order}\mathrm{variation}\mathrm{in}\mathrm{the}\mathrm{noiseless}\mathrm{observation},\hfill \\ \hfill \mathrm{sector}-\mathrm{resolved}\mathrm{sensitivity}& ⟶\mathrm{allocation}\mathrm{of}\mathrm{that}\mathrm{variation}\mathrm{among}\mathrm{symmetry}\mathrm{channels},\hfill \\ \hfill \mathrm{statistical}\mathrm{information}& ⟶\mathrm{noise}-\mathrm{and}\mathrm{model}-\mathrm{dependent}\mathrm{estimation}\mathrm{content}.\hfill \end{array}$$

(17)

The first two levels are developed in this paper. The third requires a probabilistic observation model and is intentionally kept separate.

For example, under a complex Gaussian model with covariance matrix $\Sigma$, a Fisher-information contribution may involve

$${\left(J_{\varphi }^{\left(1\right)}\right)}^H{\Sigma }^{-1}{J}_{\varphi }^{\left(1\right)}.$$

(18)

This quantity differs from both the unweighted norm in Equation (11) and the normalized sector profile introduced in Section 3. A more even sector distribution therefore does not imply a larger derivative norm, greater Fisher information, a lower Cramér–Rao bound, or improved estimation accuracy.

The same notation is later applied to tensor observables. At order p, the observable is denoted by $C^{\left(p\right)}\left(\varphi \right)$, its space by ${\mathcal{V}}^{\left(p\right)}$, and its local sensitivity by

$$J_{\varphi }^{\left(p\right)}={\displaystyle \frac{\partial C^{\left(p\right)}\left(\varphi \right)}{\partial \varphi }}\in {\mathcal{V}}^{\left(p\right)}.$$

(19)

The explicit first-, second-, and fourth-order constructions are introduced in Section 4. At this stage, Equation (19) serves only to establish consistent notation for the sector decomposition developed next.

For a vector parameter

$$\mathsf{\theta }={\left({\theta }_1,\dots ,{\theta }_d\right)}^T,$$

(20)

the corresponding local object would be the Jacobian

$${\mathcal{J}}_{\mathsf{\theta }}^{\left(p\right)}={\displaystyle \frac{\partial C^{\left(p\right)}}{\partial {\mathsf{\theta }}^T}}.$$

(21)

The present paper restricts attention to a scalar parameter so that redistribution can be analyzed for a single sensitivity vector without introducing coordinate-dependent choices within a multidimensional parameter subspace.

The electromagnetic examples in Section 5 and Section 6 use direction of arrival as the scalar parameter. There, $J_{\varphi }^{\left(1\right)}$ is the derivative of the array steering response with respect to the incident angle. Section 3 now introduces the Reynolds projectors, the represented-sector set, and the normalized quantities required to determine how this angular sensitivity is distributed across irreducible symmetry channels.

3. Irreducible-Sector Decomposition and Sensitivity Metrics

Once the electromagnetic observation and its local derivative have been defined, the next step is to resolve that sensitivity with respect to the irreducible structure of the symmetry group. This section constructs the orthogonal isotypic decomposition using character-weighted Reynolds projectors and derives normalized sector quantities from the associated Parseval identity. The resulting metrics provide a common language for comparing sensitivity allocation across observable orders, physical realizations, and symmetry groups without conflating redistribution with total sensitivity magnitude.

3.1. Reynolds Projectors and Orthogonal Isotypic Components

Let G be a finite group and let

$${\rho }^{\left(p\right)}:G\to U\left({\mathcal{V}}^{\left(p\right)}\right)$$

(22)

be the unitary representation carried by the observable space ${\mathcal{V}}^{\left(p\right)}$ at order p. The first-order case corresponds to the representation on V, whereas the higher-order actions are the induced tensor representations introduced in Section 4.1.

The space ${\mathcal{V}}^{\left(p\right)}$ is equipped with the natural Hermitian inner product. For vector-valued observables this is the standard inner product,

$$\langle x,z\rangle =x^{H}z,$$

(23)

whereas matrix- and tensor-valued observables are identified with their vectorized forms and endowed with the corresponding Hilbert–Schmidt inner product,

$${\langle X,Z\rangle }_{\mathrm{HS}}=tr\left(X^{H}Z\right).$$

(24)

The associated norm is denoted by

$${\parallel X\parallel }^2=\langle X,X\rangle .$$

(25)

For notational simplicity, the same brackets and norm symbols are used below for all observable orders.

Let $\lambda$ index an irreducible representation of G, with dimension $d_{\lambda }$ and character ${\chi }_{\lambda }$. The orthogonal projector onto the $\lambda$-isotypic component of ${\mathcal{V}}^{\left(p\right)}$ is

$${\mathsf{\Pi }}_{\lambda }^{\left(p\right)}={\displaystyle \frac{d_{\lambda }}{\left|G\right|}}\sum _{g\in G}{\chi }_{\lambda }{\left(g\right)}^{\ast }{\rho }^{\left(p\right)}\left(g\right).$$

(26)

Equation (26) is the character-weighted Reynolds projector associated with the irreducible label $\lambda$.

The term isotypic component is used deliberately. If an irreducible representation $V_{\lambda }$ occurs with multiplicity $m_{\lambda }^{\left(p\right)}$, then

$${\mathcal{V}}_{\lambda }^{\left(p\right)}\cong m_{\lambda }^{\left(p\right)}{V}_{\lambda }$$

(27)

denotes the direct sum of all copies of $V_{\lambda }$ contained in ${\mathcal{V}}^{\left(p\right)}$. The projector ${\mathsf{\Pi }}_{\lambda }^{\left(p\right)}$ isolates this entire isotypic component; it does not select an individual copy when $m_{\lambda }^{\left(p\right)}>1$. The present framework resolves sensitivity by irreducible label and therefore does not require a basis-dependent decomposition inside the multiplicity space.

Because the representation is unitary, the projectors are Hermitian,

$${\left({\mathsf{\Pi }}_{\lambda }^{\left(p\right)}\right)}^H={\mathsf{\Pi }}_{\lambda }^{\left(p\right)},$$

(28)

idempotent,

$${\left({\mathsf{\Pi }}_{\lambda }^{\left(p\right)}\right)}^2={\mathsf{\Pi }}_{\lambda }^{\left(p\right)},$$

(29)

and mutually orthogonal:

$${\mathsf{\Pi }}_{\lambda }^{\left(p\right)}{\mathsf{\Pi }}_{\mu }^{\left(p\right)}={\delta }_{\lambda \mu }{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}.$$

(30)

Here, ${\delta }_{\lambda \mu }$ denotes the Kronecker delta. Summing over all irreducible labels represented in ${\mathcal{V}}^{\left(p\right)}$ gives

$$\sum _{\lambda \in {\mathcal{A}}^{\left(p\right)}}{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}=I_{{\mathcal{V}}^{\left(p\right)}},$$

(31)

where

$${\mathcal{A}}^{\left(p\right)}=\left\{\lambda :{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}\ne 0\right\}$$

(32)

is the set of irreducible labels represented at observable order p.

Equations (30) and (31) yield the orthogonal isotypic decomposition

$${\mathcal{V}}^{\left(p\right)}=\underset{\lambda \in {\mathcal{A}}^{\left(p\right)}}{⨁}{\mathcal{V}}_{\lambda }^{\left(p\right)}.$$

(33)

Accordingly, any local sensitivity

$$J_{\varphi }^{\left(p\right)}\in {\mathcal{V}}^{\left(p\right)}$$

(34)

admits the decomposition

$$J_{\varphi }^{\left(p\right)}=\sum _{\lambda \in {\mathcal{A}}^{\left(p\right)}}{J}_{\varphi ,\lambda }^{\left(p\right)},J_{\varphi ,\lambda }^{\left(p\right)}={\mathsf{\Pi }}_{\lambda }^{\left(p\right)}{J}_{\varphi }^{\left(p\right)}.$$

(35)

For $\lambda \ne \mu$, the projected components satisfy

$$〈J_{\varphi ,\lambda }^{\left(p\right)},J_{\varphi ,\mu }^{\left(p\right)}〉=0.$$

(36)

The corresponding Parseval identity is therefore

$${∥J_{\varphi }^{\left(p\right)}∥}^2=\sum _{\lambda \in {\mathcal{A}}^{\left(p\right)}}{∥{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}{J}_{\varphi }^{\left(p\right)}∥}^2.$$

(37)

Equation (37) is the basis of the normalized sensitivity measures introduced in the next subsection. It permits the total local sensitivity energy to be partitioned exactly among mutually orthogonal symmetry sectors. No statistical interpretation is required at this stage: the decomposition is determined by the unitary group action and the Hilbert-space geometry of the observable.

The represented label set ${\mathcal{A}}^{\left(p\right)}$ may change with observable order because tensor lifting changes the induced representation and its multiplicities. The projectors in Equation (26) must therefore be constructed from ${\rho }^{\left(p\right)}$ at the corresponding order. This order dependence is essential in Section 4.1 and Section 4.2, where tensorization modifies the available representation structure. By contrast, once ${\rho }^{\left(p\right)}$ is fixed, the projectors provide a canonical, orthogonal resolution of $J_{\varphi }^{\left(p\right)}$ by irreducible label.

The next subsection uses the sector energies in Equation (37) to define normalized sensitivity fractions. These fractions quantify how the local sensitivity is distributed across the represented isotypic components without conflating sector occupancy with total sensitivity magnitude.

3.2. Sector-Wise Sensitivity Fractions

The orthogonal decomposition in Equation (35) permits the local sensitivity to be resolved into contributions associated with the represented irreducible labels. For each observable order p and sector $\lambda \in {\mathcal{A}}^{\left(p\right)}$, define

$$J_{\varphi ,\lambda }^{\left(p\right)}={\mathsf{\Pi }}_{\lambda }^{\left(p\right)}{J}_{\varphi }^{\left(p\right)}.$$

(38)

The squared norm

$$E_{\lambda }^{\left(p\right)}={∥J_{\varphi ,\lambda }^{\left(p\right)}∥}^2$$

(39)

measures the local sensitivity energy carried by the $\lambda$-isotypic component.

To compare sector occupancy independently of the total sensitivity magnitude, we normalize these energies and define the sector-wise sensitivity fraction

$$R_{\lambda }^{\left(p\right)}={\displaystyle \frac{{∥{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}{J}_{\varphi }^{\left(p\right)}∥}^2}{{∥J_{\varphi }^{\left(p\right)}∥}^2}},\lambda \in {\mathcal{A}}^{\left(p\right)}.$$

(40)

The definition is meaningful whenever

$$∥J_{\varphi }^{\left(p\right)}∥>0.$$

(41)

Parameter values at which $J_{\varphi }^{\left(p\right)}=0$ correspond to a locally insensitive observable and are excluded from the normalized sector analysis.

By construction,

$$R_{\lambda }^{\left(p\right)}\ge 0$$

(42)

for every represented sector. Moreover, the Parseval identity in Equation (37) gives

$$\sum _{\lambda \in {\mathcal{A}}^{\left(p\right)}}{R}_{\lambda }^{\left(p\right)}=1.$$

(43)

Thus,

$${\mathbf{R}}^{\left(p\right)}={\left(R_{\lambda }^{\left(p\right)}\right)}_{\lambda \in {\mathcal{A}}^{\left(p\right)}}$$

(44)

is a normalized distribution over the irreducible labels represented at order p.

The word distribution is used here in a normalized geometric sense. The quantities $R_{\lambda }^{\left(p\right)}$ are not probabilities of physical events, posterior probabilities, or source-occurrence probabilities. They are fractions of the squared local sensitivity norm assigned to mutually orthogonal isotypic components. Their normalization follows from the Hilbert-space decomposition rather than from a stochastic model.

This interpretation is particularly useful because the fractions separate the organization of sensitivity from its overall scale. If

$$J_{\varphi }^{\left(p\right)}\mapsto \alpha J_{\varphi }^{\left(p\right)},\alpha \in \mathbb{C}\setminus \left\{0\right\},$$

(45)

then

$$R_{\lambda }^{\left(p\right)}\mapsto R_{\lambda }^{\left(p\right)}.$$

(46)

Hence, the sector fractions are invariant under a global rescaling of the local sensitivity. This property permits observable orders with different total sensitivity norms to be compared at the level of their symmetry-resolved organization.

The normalization also defines the scope of the metric. A large value of $R_{\lambda }^{\left(p\right)}$ indicates that a large fraction of the local sensitivity energy lies in the $\lambda$-isotypic component. It does not imply that the corresponding sector provides large Fisher information or superior estimation performance. Those quantities depend on the statistical observation model, the noise structure, and the parameterization of the estimator. The present measure addresses only the symmetry-resolved geometry of the derivative $J_{\varphi }^{\left(p\right)}$.

Similarly, a sector can be represented in ${\mathcal{V}}^{\left(p\right)}$ while carrying zero local sensitivity at a particular parameter value. In the notation of Section 4.3,

$$\lambda \in {\mathcal{A}}^{\left(p\right)}\nRightarrow R_{\lambda }^{\left(p\right)}>0.$$

(47)

The represented label set is determined by the induced representation, whereas the nonzero fractions are determined by the physical sensitivity tensor. This distinction is essential when interpreting tensorization: an accessible sector need not be populated, and a change in the sector fractions need not require an expansion of the represented label set.

The sector fractions can also be averaged over a parameter domain or an ensemble of physical realizations. If ${\varphi }_1,\dots ,{\varphi }_Q$ denote sampled parameter values, a realization-level angular average may be formed as

$${\overline{R}}_{\lambda }^{\left(p\right)}={\displaystyle \frac{1}{Q}}{\displaystyle \sum _{q=1}^Q}{R}_{\lambda }^{\left(p\right)}\left({\varphi }_q\right).$$

(48)

This operation should not be confused with projecting an averaged sensitivity tensor. In general,

$${\displaystyle \frac{1}{Q}}{\displaystyle \sum _{q=1}^Q}{R}_{\lambda }^{\left(p\right)}\left({\varphi }_q\right)\ne {\displaystyle \frac{{∥{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}\left(Q^{-1}{\sum }_{q=1}^Q{J}_{{\varphi }_q}^{\left(p\right)}\right)∥}^2}{{∥Q^{-1}{\sum }_{q=1}^Q{J}_{{\varphi }_q}^{\left(p\right)}∥}^2}}.$$

(49)

The numerical experiments in Section 5 and Section 6 use the former procedure: sector metrics are evaluated at each sampled direction and subsequently averaged at the realization level. This preserves the local interpretation of $J_{\varphi }^{\left(p\right)}$ before statistical aggregation.

The normalized vector ${\mathbf{R}}^{\left(p\right)}$ is the basic object used to quantify sensitivity redistribution. The next subsection introduces three scalar summaries of this vector: normalized sector entropy, normalized effective-sector occupancy, and dominant-sector fraction. Together, these metrics describe the evenness and concentration of the symmetry-resolved sensitivity profile without reintroducing the total sensitivity norm.

3.3. Entropy, Effective-Sector Occupancy, and Dominant-Sector Fraction

The normalized sector-fraction vector

$${\mathbf{R}}^{\left(p\right)}={\left(R_{\lambda }^{\left(p\right)}\right)}_{\lambda \in {\mathcal{A}}^{\left(p\right)}}$$

(50)

contains the complete symmetry-resolved distribution of local sensitivity at observable order p. To compare redistribution patterns across parameter values, physical realizations, and group structures, we use three scalar summaries of this vector: sector entropy, effective-sector occupancy, and dominant-sector fraction.

Let

$$K^{\left(p\right)}=\left|{\mathcal{A}}^{\left(p\right)}\right|$$

(51)

denote the number of irreducible labels represented in ${\mathcal{V}}^{\left(p\right)}$. Importantly, $K^{\left(p\right)}$ is determined by the induced representation and is not defined as the number of sectors with numerically nonzero sensitivity in a particular realization. A represented sector remains part of ${\mathcal{A}}^{\left(p\right)}$ even when its local sensitivity fraction vanishes.

The sector entropy is defined as

$$H^{\left(p\right)}=-\sum _{\lambda \in {\mathcal{A}}^{\left(p\right)}}{R}_{\lambda }^{\left(p\right)}log{R}_{\lambda }^{\left(p\right)},$$

(52)

with the standard convention

$$0log0=0.$$

(53)

The entropy satisfies

$$0\le H^{\left(p\right)}\le log{K}^{\left(p\right)}.$$

(54)

The lower bound is attained when all local sensitivity is concentrated in one represented sector, whereas the upper bound is attained for the uniform distribution

$$R_{\lambda }^{\left(p\right)}={\displaystyle \frac{1}{K^{\left(p\right)}}}\mathrm{for}\mathrm{all}\lambda \in {\mathcal{A}}^{\left(p\right)}.$$

(55)

Because the number of represented irreducible labels may depend on observable order or group structure, direct comparison of $H^{\left(p\right)}$ can be misleading. We therefore use the normalized entropy

$$H_{\mathrm{norm}}^{\left(p\right)}={\displaystyle \frac{H^{\left(p\right)}}{log{K}^{\left(p\right)}}},K^{\left(p\right)}>1.$$

(56)

For the degenerate case $K^{\left(p\right)}=1$, we define

$$H_{\mathrm{norm}}^{\left(p\right)}=0.$$

(57)

Hence,

$$0\le H_{\mathrm{norm}}^{\left(p\right)}\le 1.$$

(58)

A larger value indicates a more even distribution of the normalized sensitivity fractions across the represented irreducible labels. It does not indicate a larger total sensitivity norm.

A complementary quantity is the effective-sector occupancy,

$$N_{\mathrm{eff}}^{\left(p\right)}=exp\left(H^{\left(p\right)}\right).$$

(59)

This is the exponential Shannon effective number associated with the sector-fraction vector. It satisfies

$$1\le N_{\mathrm{eff}}^{\left(p\right)}\le K^{\left(p\right)}.$$

(60)

For example, if the sensitivity is uniformly distributed over exactly q sectors and zero in the remaining represented sectors, then

$$N_{\mathrm{eff}}^{\left(p\right)}=q.$$

(61)

Thus, $N_{\mathrm{eff}}^{\left(p\right)}$ expresses the entropy of the sector profile on an effective-count scale.

To compare architectures with different values of $K^{\left(p\right)}$, we also define

$$N_{\mathrm{eff},\mathrm{norm}}^{\left(p\right)}={\displaystyle \frac{N_{\mathrm{eff}}^{\left(p\right)}}{K^{\left(p\right)}}}.$$

(62)

Its range is

$${\displaystyle \frac{1}{K^{\left(p\right)}}}\le N_{\mathrm{eff},\mathrm{norm}}^{\left(p\right)}\le 1.$$

(63)

The upper bound corresponds to equal sensitivity fractions over all represented sectors.

The entropy and effective-sector occupancy are related deterministically:

$$N_{\mathrm{eff}}^{\left(p\right)}=exp\left(H^{\left(p\right)}\right).$$

(64)

They should therefore be interpreted as complementary parameterizations of the same sector-evenness information rather than as statistically independent indicators. We report both because normalized entropy provides a bounded diversity scale, whereas effective-sector occupancy gives an intuitive equivalent number of populated sectors.

To quantify concentration directly, we define the dominant-sector fraction as

$$R_{max}^{\left(p\right)}=\underset{\lambda \in {\mathcal{A}}^{\left(p\right)}}{max}{R}_{\lambda }^{\left(p\right)}.$$

(65)

Its bounds are

$${\displaystyle \frac{1}{K^{\left(p\right)}}}\le R_{max}^{\left(p\right)}\le 1.$$

(66)

The upper bound corresponds to complete concentration in one sector, while the lower bound is achieved by the uniform sector distribution. In the present work, $R_{max}^{\left(p\right)}$ is used as a complementary concentration descriptor. A decrease in $R_{max}^{\left(p\right)}$ that accompanies an increase in $H_{\mathrm{norm}}^{\left(p\right)}$ is consistent with reduced dominance by a single symmetry sector, but it is not treated as independent evidence of a separate phenomenon.

The three metrics therefore characterize the shape of the normalized sector profile:

$$H_{\mathrm{norm}}^{\left(p\right)}\mathrm{and}{N}_{\mathrm{eff},\mathrm{norm}}^{\left(p\right)}⟶\mathrm{sector}\mathrm{evenness},$$

(67)

whereas

$$R_{max}^{\left(p\right)}⟶\mathrm{dominant}-\mathrm{sector}\mathrm{concentration}.$$

(68)

They do not retain the total sensitivity magnitude

$${∥J_{\varphi }^{\left(p\right)}∥}^2.$$

(69)

Consequently, these quantities should not be interpreted as Fisher information, signal-to-noise ratio, estimation accuracy, or a Cramér–Rao bound.

The choice of $K^{\left(p\right)}$ in Equations (56) and (62) is particularly relevant to the experiments in Section 5 and Section 6. In the $C_4$ electromagnetic experiment, the same four irreducible labels are represented at every observable order. Changes in the normalized metrics therefore describe redistribution within a fixed sector-label set. In the broader $C_N$–$D_N$ comparison, $K^{\left(p\right)}$ is determined separately from the corresponding induced representation, allowing the normalized quantities to compare sector evenness without defining the normalization from realization-dependent numerical support.

For the numerical experiments, the metrics in Equations (56), (62), and (65) are evaluated locally at each sampled parameter value and then averaged over the parameter grid for each physical realization. Ensemble means and confidence intervals are subsequently computed from these realization-level averages. This ordering preserves the local sensitivity interpretation established in Section 3.2.

These metrics provide the quantitative language used in Section 5 and Section 6 to describe sensitivity redistribution. The following section examines how tensor lifting changes the representation carried by the observable and distinguishes algebraic sector accessibility from the realized sector-weight profile.

4. Tensor Lifting and Symmetry-Sector Redistribution

Section 3 established how a local sensitivity vector is decomposed into orthogonal irreducible sectors at a fixed observable order. The present section explains what changes when the same electromagnetic response is represented through tensor observables. The central point is that tensor lifting modifies the representation carried by the observable and therefore changes the sector couplings permitted by the group. This algebraic change constrains, but does not determine, the sensitivity fractions introduced in Section 3.

4.1. Observable Orders and Induced Tensor Actions

Let $y\left(\varphi \right)\in V$ be the first-order observation defined in Section 2, and let $\rho :G\to U\left(V\right)$ be the unitary representation induced by the physical symmetry group G. We consider the observable orders

$$p\in \{1,2,4\},$$

(70)

with associated spaces

$${\mathcal{V}}^{\left(1\right)}=V,{\mathcal{V}}^{\left(2\right)}=V\otimes V^{\ast },{\mathcal{V}}^{\left(4\right)}=V^{\otimes 2}\otimes {\left(V^{\ast }\right)}^{\otimes 2}.$$

(71)

Here, $V^{\ast }$ denotes the conjugate dual of V. The order index p refers to the number of direct and conjugate observation factors in the corresponding tensor construction; it is used consistently throughout the paper and should not be confused with the order of a statistical moment or cumulant.

The three observables are

$$C^{\left(1\right)}\left(\varphi \right)=y\left(\varphi \right),$$

(72)

$$C^{\left(2\right)}\left(\varphi \right)=y\left(\varphi \right)\otimes y{\left(\varphi \right)}^{\ast },$$

(73)

and

$$C^{\left(4\right)}\left(\varphi \right)=y\left(\varphi \right)\otimes y\left(\varphi \right)\otimes y{\left(\varphi \right)}^{\ast }\otimes y{\left(\varphi \right)}^{\ast }.$$

(74)

The second- and fourth-order quantities are deterministic tensor lifts of the same physical response. No stochastic averaging is implied, and $C^{\left(4\right)}$ is not assumed to be a fourth-order cumulant.

The group action on V induces actions on the lifted spaces:

$${\rho }^{\left(1\right)}\left(g\right)=\rho \left(g\right),$$

(75)

$${\rho }^{\left(2\right)}\left(g\right)=\rho \left(g\right)\otimes \rho {\left(g\right)}^{\ast },$$

(76)

and

$${\rho }^{\left(4\right)}\left(g\right)=\rho {\left(g\right)}^{\otimes 2}\otimes {\left(\rho {\left(g\right)}^{\ast }\right)}^{\otimes 2},$$

(77)

for every $g\in G$. Since $\rho \left(g\right)$ is unitary, each ${\rho }^{\left(p\right)}\left(g\right)$ is unitary with respect to the natural Hilbert or Hilbert–Schmidt inner product on ${\mathcal{V}}^{\left(p\right)}$. Therefore, the Reynolds projectors introduced in Section 3 apply directly at every observable order.

The corresponding local sensitivities are

$$J_{\varphi }^{\left(p\right)}={\displaystyle \frac{\partial C^{\left(p\right)}\left(\varphi \right)}{\partial \varphi }}\in {\mathcal{V}}^{\left(p\right)}.$$

(78)

For the quadratic lift, differentiation gives

$$J_{\varphi }^{\left(2\right)}=J_{\varphi }^{\left(1\right)}\otimes y{\left(\varphi \right)}^{\ast }+y\left(\varphi \right)\otimes J_{\varphi }^{\left(1\right)\ast }.$$

(79)

For the quartic lift,

$$\begin{array}{cc}\hfill J_{\varphi }^{\left(4\right)}=& J_{\varphi }^{\left(1\right)}\otimes y\otimes y^{\ast }\otimes y^{\ast }+y\otimes J_{\varphi }^{\left(1\right)}\otimes y^{\ast }\otimes y^{\ast }\hfill \\ \hfill & +y\otimes y\otimes J_{\varphi }^{\left(1\right)\ast }\otimes y^{\ast }+y\otimes y\otimes y^{\ast }\otimes J_{\varphi }^{\left(1\right)\ast },\hfill \end{array}$$

(80)

where all factors on the right-hand side are evaluated at $\varphi$. Equations (79) and (80) show that the lifted sensitivity is assembled jointly from the sector content of the observation and that of its first-order derivative. Tensor order alone is therefore insufficient to predict the resulting sector weights.

4.2. Fusion Support and Sector Accessibility

To describe which irreducible labels may occur after tensor lifting, let

$$V\cong \underset{\alpha \in \widehat{G}}{⨁}{m}_{\alpha }{W}_{\alpha }$$

(81)

be the irreducible decomposition of the first-order observation space. Here, $\widehat{G}$ is the set of equivalence classes of irreducible representations of G, $W_{\alpha }$ is a representative irrep with label $\alpha$, and $m_{\alpha }$ is its multiplicity in V.

For two irreducible representations $W_{\alpha }$ and $W_{\beta }$, their tensor product decomposes as

$$W_{\alpha }\otimes W_{\beta }\cong \underset{\gamma \in \widehat{G}}{⨁}{N}_{\alpha \beta }^{\gamma }{W}_{\gamma },$$

(82)

where the nonnegative integer $N_{\alpha \beta }^{\gamma }$ is the fusion multiplicity of $W_{\gamma }$ in $W_{\alpha }\otimes W_{\beta }$. These coefficients determine the irreducible labels permitted by the induced tensor representation.

The representation support of a vector x is the set

$$\mathcal{S}\left(x\right)=\left\{\alpha \in \widehat{G}:{\mathsf{\Pi }}_{\alpha }x\ne 0\right\},$$

(83)

where ${\mathsf{\Pi }}_{\alpha }$ denotes the isotypic projector associated with $W_{\alpha }$. We use

$${\mathcal{S}}_y=\mathcal{S}\left(y\left(\varphi \right)\right),{\mathcal{S}}_J=\mathcal{S}\left(J_{\varphi }^{\left(1\right)}\right)$$

(84)

for the supports of the first-order observation and first-order sensitivity, respectively.

A label $\gamma \in \widehat{G}$ is said to be accessible at order p if it can occur in at least one tensor-product term contributing to $J_{\varphi }^{\left(p\right)}$. Accessibility is therefore a support-level statement. It indicates that symmetry does not forbid a sector, but it does not imply that the corresponding projected derivative has nonzero numerical weight for a specific physical realization.

From Equation (79), the second-order accessible support satisfies

$${\mathcal{S}}^{\left(2\right)}\subseteq \left({\mathcal{S}}_J\otimes {\mathcal{S}}_y^{\ast }\right)\cup \left({\mathcal{S}}_y\otimes {\mathcal{S}}_J^{\ast }\right),$$

(85)

where $\mathcal{A}\otimes \mathcal{B}$ denotes the set of irreducible labels $\gamma$ for which $N_{\alpha \beta }^{\gamma }>0$ for at least one $\alpha \in \mathcal{A}$ and $\beta \in \mathcal{B}$. The superscript * denotes the support of the conjugate representation.

Likewise, Equation (80) gives

$$\begin{array}{cc}\hfill {\mathcal{S}}^{\left(4\right)}\subseteq & {\mathcal{S}}_J\otimes {\mathcal{S}}_y\otimes {\mathcal{S}}_y^{\ast }\otimes {\mathcal{S}}_y^{\ast }\hfill \\ \hfill & \cup {\mathcal{S}}_y\otimes {\mathcal{S}}_J\otimes {\mathcal{S}}_y^{\ast }\otimes {\mathcal{S}}_y^{\ast }\hfill \\ \hfill & \cup {\mathcal{S}}_y\otimes {\mathcal{S}}_y\otimes {\mathcal{S}}_J^{\ast }\otimes {\mathcal{S}}_y^{\ast }\hfill \\ \hfill & \cup {\mathcal{S}}_y\otimes {\mathcal{S}}_y\otimes {\mathcal{S}}_y^{\ast }\otimes {\mathcal{S}}_J^{\ast }.\hfill \end{array}$$

(86)

This expression makes explicit that quartic accessibility is generated by one derivative factor and three observation factors in each term.

For the cyclic group $C_N$, every irreducible representation is one-dimensional and can be indexed by an element $k\in {\mathbb{Z}}_N$. Its character is

$${\chi }_k\left(r^m\right)=exp\left({\displaystyle \frac{2\pi \mathrm{i}km}{N}}\right),k,m\in {\mathbb{Z}}_N,$$

(87)

where r is the generator of $C_N$. The fusion rule is additive:

$${\chi }_k\otimes {\chi }_{\ell }={\chi }_{k+\ell },k,\ell \in {\mathbb{Z}}_N,$$

(88)

with all indices understood modulo N. Conjugation maps k to $-k$.

For subsets $\mathcal{A},\mathcal{B}\subseteq {\mathbb{Z}}_N$, define their modular difference set by

$$\mathcal{A}-\mathcal{B}=\left\{a-b(modN):a\in \mathcal{A},b\in \mathcal{B}\right\}.$$

(89)

Equation (85) then reduces to

$${\mathcal{S}}^{\left(2\right)}\subseteq \left({\mathcal{S}}_J-{\mathcal{S}}_y\right)\cup \left({\mathcal{S}}_y-{\mathcal{S}}_J\right).$$

(90)

As a concrete support calculation, suppose

$${\mathcal{S}}_J=\{1,3\},{\mathcal{S}}_y=\{0,1\}$$

(91)

in $C_8$. Then

$${\mathcal{S}}_J-{\mathcal{S}}_y=\{0,1,2,3\},{\mathcal{S}}_y-{\mathcal{S}}_J=\{0,5,6,7\},$$

(92)

so that sector 4 is inaccessible at second order. The remaining labels are only algebraically admissible: their actual sensitivity fractions may still vanish or be strongly unequal.

For non-Abelian groups, including the dihedral groups used in Section 6, accessibility is governed by the full fusion coefficients in Equation (82). In particular, higher-dimensional irreducible representations may appear, and reflection symmetry changes the tensor-product couplings relative to the additive cyclic rule. This provides the algebraic basis for expecting different redistribution trajectories in cyclic and dihedral architectures, without predetermining their direction or magnitude.

4.3. From Accessibility to Sensitivity Redistribution

At order p, let ${\mathcal{A}}^{\left(p\right)}\subseteq \widehat{G}$ denote the set of irreducible labels represented in ${\mathcal{V}}^{\left(p\right)}$, as defined in Section 3. The Reynolds projectors ${\mathsf{\Pi }}_{\lambda }^{\left(p\right)}$ decompose the lifted sensitivity as

$$J_{\varphi }^{\left(p\right)}=\sum _{\lambda \in {\mathcal{A}}^{\left(p\right)}}{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}{J}_{\varphi }^{\left(p\right)}.$$

(93)

The corresponding normalized sector fractions $R_{\lambda }^{\left(p\right)}$ are those defined in Section 3.

The fusion analysis and the sector fractions describe two distinct levels:

$$\begin{array}{cc}\hfill \mathrm{induced}\mathrm{representation}\mathrm{and}\mathrm{factor}\mathrm{supports}& ⟶\mathrm{sector}\mathrm{accessibility},\hfill \\ \hfill J_{\varphi }^{\left(p\right)}& ⟶\mathrm{sector}\mathrm{weights}.\hfill \end{array}$$

(94)

The first line is algebraic. It determines which irreducible labels are permitted by the tensor-product structure. The second is physical and parameter dependent. It determines how much of the local sensitivity lies in each permitted sector.

This distinction has several consequences. First, a sector may be represented in ${\mathcal{V}}^{\left(p\right)}$ but absent from $J_{\varphi }^{\left(p\right)}$. Second, a tensor lift may activate additional fusion pathways without producing a more uniform sensitivity profile. Third, the sector fractions may be redistributed among an unchanged set of irreducible labels. The $C_4$ experiment in Section 5 realizes precisely this third situation: the same four labels are present at all observable orders, while their normalized weights change substantially.

Multiplicity and sensitivity weight must also be distinguished. If an irrep $W_{\lambda }$ occurs several times in ${\mathcal{V}}^{\left(p\right)}$, the isotypic projector ${\mathsf{\Pi }}_{\lambda }^{\left(p\right)}$ collects all equivalent copies. A larger multiplicity increases the dimension available to that irrep but does not force a larger value of $R_{\lambda }^{\left(p\right)}$. The projected weight depends on the orientation of $J_{\varphi }^{\left(p\right)}$ within the full tensor space.

The entropy and concentration metrics of Section 3 therefore quantify realized redistribution, not the size of the tensor space or the richness of its fusion algebra. In particular, no general ordering of $H_{\mathrm{norm}}^{\left(1\right)}$, $H_{\mathrm{norm}}^{\left(2\right)}$, and $H_{\mathrm{norm}}^{\left(4\right)}$ follows from representation theory alone. Any monotonic pattern must be established analytically under additional assumptions or demonstrated empirically for a specified physical ensemble.

Finally, the sector profile is not a direct estimator-performance measure. Tensor lifting may change the distribution of local sensitivity without increasing its total norm, Fisher information, or statistical identifiability. The quantities $R_{\lambda }^{\left(p\right)}$, $H_{\mathrm{norm}}^{\left(p\right)}$, $N_{\mathrm{eff},\mathrm{norm}}^{\left(p\right)}$, and $R_{max}^{\left(p\right)}$ describe how sensitivity is organized relative to symmetry. They complement, rather than replace, noise-aware observability and estimation criteria.

Section 5 and Section 6 use this two-level interpretation to separate the algebraic mechanism of tensor lifting from the redistribution actually produced by electromagnetic array responses. The first experiment holds the group and irreducible-label set fixed, whereas the second compares cyclic and dihedral architectures to determine how the underlying group structure modifies the observed sector-weight trajectories.

5. Electromagnetic Validation in a ${\mathit{C}}_{\mathbf{4}}$-Symmetric Array

The preceding sections define the symmetry-resolved sensitivity framework and identify the algebraic mechanisms through which tensorization can modify sector accessibility. We now test whether these mechanisms produce a measurable redistribution in a concrete electromagnetic setting. The validation uses a physically interpretable $C_4$-symmetric receiving array and a narrowband far-field model, with the geometry, parameter, and symmetry group held fixed while only the observable order is changed. This design isolates tensorization-induced redistribution from changes in the underlying architecture.

5.1. Physical Array Geometry and Propagation Model

We consider the two-ring planar receiving array shown in Figure 1. The architecture is designed to possess exact fourfold rotational symmetry while avoiding any reflection axis common to both rings. Its purpose is to provide a physically interpretable electromagnetic realization whose nominal symmetry group is $C_4$, rather than the dihedral group $D_4$ associated with a conventional square, a single four-element ring, or two aligned four-element rings. The displayed geometry is one realization drawn directly from the Monte Carlo ensemble used in the numerical study.

The array consists of two concentric rings, each containing four identical sensing elements. The inner-ring positions are

$${\mathbf{r}}_{1,m}=r_1\left[\begin{array}{c}cos\left(2\pi m/4\right)\\ sin\left(2\pi m/4\right)\end{array}\right],m=0,\dots ,3,$$

(95)

whereas the outer-ring positions are

$${\mathbf{r}}_{2,m}=r_2\left[\begin{array}{c}cos\left(\delta +2\pi m/4\right)\\ sin\left(\delta +2\pi m/4\right)\end{array}\right],m=0,\dots ,3.$$

(96)

Here, $r_1$ and $r_2$ are the inner and outer radii, respectively, and $\delta$ is the relative angular offset indicated in Figure 1. The reference elements $s_{1,0}$ and $s_{2,0}$ identify the two radii whose angular separation is $\delta$.

A rotation by $\pi /2$ maps each ring onto itself and cyclically permutes its four elements. For a generic nonaligned value of $\delta$, however, a reflection that preserves one ring does not preserve the other. The complete geometry is therefore invariant under the four rotations generated by r, but not under a common reflection. Its symmetry group is $G=C_4=\{e,r,r^2,r^3\}$, where e is the identity and r denotes rotation by $\pi /2$.

The action of G on the eight sensors induces a unitary permutation representation $\rho :G\to U\left({\mathbb{C}}^8\right)$. The same physical group is used at every observable order. Consequently, the sensitivity is always resolved with respect to the four irreducible labels of $C_4$. Tensor lifting changes the induced representation and the multiplicities carried by the observable space, but it does not change the underlying group or its irreducible-label set.

We adopt a narrowband far-field model for a monochromatic plane wave incident in the array plane. The sensing elements are assumed to be identical and isotropic, the propagation medium is homogeneous, and mutual coupling is neglected. These assumptions preserve the equivalence among elements related by the $C_4$ action and isolate the interaction between physical symmetry and observable order.

Let $\varphi$ denote the direction of arrival and let $\mathbf{u}\left(\varphi \right)={[cos\varphi ,sin\varphi ]}^T$ be the corresponding unit propagation vector. For a sensor located at ${\mathbf{r}}_n$, the noiseless complex response is

$$a_n\left(\varphi \right)=exp\left(\mathrm{j}{k}_0{\mathbf{r}}_n^T\mathbf{u}\left(\varphi \right)\right),$$

(97)

where $k_0=2\pi /\lambda$ is the wavenumber and $\lambda$ is the wavelength. Collecting the eight responses gives the steering vector

$$\mathbf{a}\left(\varphi \right)={\left[\begin{array}{ccc}{a}_1\left(\varphi \right)& \cdots & a_8\left(\varphi \right)\end{array}\right]}^T\in {\mathbb{C}}^8.$$

(98)

The first-order angular sensitivity is obtained analytically as

$${\displaystyle \frac{\partial \mathbf{a}\left(\varphi \right)}{\partial \varphi }}=\mathrm{j}{k}_{0}diag\left(\mathbf{R}{\mathbf{u}}^{\prime }\left(\varphi \right)\right)\mathbf{a}\left(\varphi \right),$$

(99)

where ${\mathbf{u}}^{\prime }\left(\varphi \right)={[-sin\varphi ,cos\varphi ]}^T$, and $\mathbf{R}\in {\mathbb{R}}^{8\times 2}$ is the matrix whose n-th row is ${\mathbf{r}}_n^T$. Equation (99) makes the physical origin of the analyzed sensitivity explicit: an infinitesimal variation in $\varphi$ changes the propagation phase accumulated at each sensor according to its position in the array.

For this electromagnetic model, the first-order observation $y\left(\varphi \right)$ introduced in Section 2 is identified with $\mathbf{a}\left(\varphi \right)$. The first-, second-, and fourth-order tensor observables and their derivatives are then obtained directly from the general definitions in Section 4. They are deterministic lifts of the same steering response; in particular, the fourth-order observable is not interpreted as a fourth-order cumulant and introduces no additional source statistics.

This construction isolates the phenomenon studied in the paper. For each physical realization, the geometry, direction parameter, and group $C_4$ are held fixed, while only the observable order is varied. The subsequent sector-resolved comparison therefore tests whether tensor lifting reorganizes local angular sensitivity among the same four symmetry labels without conflating that effect with changes in aperture, parameter space, or architectural symmetry. The Monte Carlo generation of the radii and angular offsets, together with the numerical validation checks, is described in Section 5.2.

5.2. Numerical Protocol and Validation Checks

The numerical study was designed to test whether the redistribution of angular sensitivity persists across a family of physically distinct arrays sharing the same nominal $C_4$ symmetry. To this end, we generated 200 independent realizations of the two-ring architecture defined in Section 5.1. For each realization, the inner and outer radii were drawn independently from

$${\displaystyle \frac{r_1}{\lambda }}\sim \mathcal{U}(0.30,0.48)$$

(100)

and

$${\displaystyle \frac{r_2}{\lambda }}\sim \mathcal{U}(0.62,0.95),$$

(101)

respectively. The relative angular offset was generated as

$$\delta =\eta {\displaystyle \frac{2\pi }{4}},\eta \sim \mathcal{U}(0.14,0.36).$$

(102)

The interval in Equation (102) was selected to avoid the special alignments that restore a common reflection symmetry while retaining the fourfold rotational action. A fixed pseudorandom seed of 20260715 was used throughout the experiment.

For each physical realization, the angular response was evaluated at 48 uniformly spaced directions over a complete $2\pi$ interval. A realization-specific random angular offset was added to the sampling grid to prevent systematic alignment between the tested directions and the array axes. If ${\varphi }_0$ denotes this offset, the sampled directions were

$${\varphi }_q={\varphi }_0-\pi +{\displaystyle \frac{2\pi q}{48}},q=0,\dots ,47.$$

(103)

The same geometry and the same angular grid were used for observable orders $p=1$, $p=2$, and $p=4$. The resulting comparison is therefore paired at the realization level: changes in the sector metrics are produced by the observable lift rather than by differences in array geometry or angular sampling.

All sensitivities $J_{\varphi }^{\left(p\right)}$ were evaluated from the analytic derivatives of the corresponding observables. As a numerical verification, one randomly selected direction in each realization was also evaluated by a centered finite-difference approximation,

$${\widehat{J}}_{\varphi }^{\left(p\right)}={\displaystyle \frac{C^{\left(p\right)}(\varphi +h)-C^{\left(p\right)}(\varphi -h)}{2h}},h={10}^{-7}.$$

(104)

The relative derivative error was measured as

$${\epsilon }_{\mathrm{FD}}^{\left(p\right)}={\displaystyle \frac{∥J_{\varphi }^{\left(p\right)}-{\widehat{J}}_{\varphi }^{\left(p\right)}∥}{∥J_{\varphi }^{\left(p\right)}∥}}.$$

(105)

Across all realizations and observable orders, the maximum value of Equation (105) was $2.37\times {10}^{-9}$, confirming the consistency of the analytic sensitivity expressions with the numerical differentiation check.

The $C_4$ sector components were computed using the character projectors defined in Section 3. For each sensitivity tensor, two additional numerical checks were performed. First, projector completeness was quantified by

$${\epsilon }_{\mathrm{comp}}={\displaystyle \frac{∥J_{\varphi }^{\left(p\right)}-{\sum }_{\lambda }{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}{J}_{\varphi }^{\left(p\right)}∥}{∥J_{\varphi }^{\left(p\right)}∥}}.$$

(106)

Second, orthogonal sector-energy conservation was assessed through

$${\epsilon }_{\mathrm{en}}={\displaystyle \frac{\left|{∥J_{\varphi }^{\left(p\right)}∥}^2-{\sum }_{\lambda }{∥{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}{J}_{\varphi }^{\left(p\right)}∥}^2\right|}{{∥J_{\varphi }^{\left(p\right)}∥}^2}}.$$

(107)

The largest residuals observed over the complete experiment were

$$max{\epsilon }_{\mathrm{comp}}=3.11\times {10}^{-16}$$

(108)

and

$$max{\epsilon }_{\mathrm{en}}=4.83\times {10}^{-15}.$$

(109)

These values are at the level of floating-point roundoff and verify, numerically, both the completeness of the sector decomposition and the energy identity underlying the sensitivity fractions.

Because the intended architecture is cyclic rather than dihedral, each generated array was also checked for accidental reflection symmetry. For every realization, the sensor set was transformed by each of the four reflection operations associated with $D_4$. The reflection residual was defined as the smallest, over those transformations, of the maximum nearest-neighbor mismatch between the transformed and original sensor sets. The minimum residual observed over the 200 realizations was $0.206\lambda$. Hence, none of the tested geometries was numerically compatible with an additional reflection symmetry.

The sector fractions $R_{\lambda }^{\left(p\right)}$, normalized entropy $H_{\mathrm{norm}}^{\left(p\right)}$, normalized effective-sector occupancy $N_{\mathrm{eff},\mathrm{norm}}^{\left(p\right)}$, and dominant-sector fraction $R_{max}^{\left(p\right)}$ were computed at each sampled direction and subsequently averaged over the 48 directions of each realization. Statistical comparisons between observable orders were then performed on the resulting realization-level averages. In particular, entropy increments were evaluated as paired quantities,

$$\Delta H_{1\to 2}=H_{\mathrm{norm}}^{\left(2\right)}-H_{\mathrm{norm}}^{\left(1\right)}$$

(110)

and

$$\Delta H_{2\to 4}=H_{\mathrm{norm}}^{\left(4\right)}-H_{\mathrm{norm}}^{\left(2\right)}.$$

(111)

Sample means and two-sided $95\%$ confidence intervals were computed from these paired realization-level increments. The corresponding redistribution results are presented in Section 5.3 and Section 5.4.

5.3. Sector Redistribution Under Tensor Lifting

The first question is whether the tensor lifts defined in Section 5.1 alter the distribution of angular sensitivity among the four irreducible sectors of $C_4$. Figure 2 shows the sector fractions for a representative realization. At first order, the sensitivity is concentrated mainly in the $k=1$ and $k=3$ sectors, while the $k=0$ component carries only a small fraction of the total sensitivity. The second- and fourth-order lifts produce substantially flatter profiles: the initially weak sectors gain relative weight and the dominant first-order sectors become less prominent.

The ensemble statistics confirm that the representative behavior in Figure 2 is not an isolated case.

Table 1. Sector-resolved sensitivity metrics for the $C_4$-symmetric electromagnetic array. Values are reported as ensemble mean $[95\%\mathrm{CI}]$ over 200 physical realizations.

Observable Order p	${\mathit{H}}_{\mathbf{norm}}$	${\mathit{N}}_{\mathbf{eff},\mathbf{norm}}$	${\mathit{R}}_{max}$
1	$0.8593[0.8554,0.8632]$	$0.8311[0.8269,0.8353]$	$0.4267[0.4231,0.4302]$
2	$0.9597[0.9579,0.9615]$	$0.9481[0.9460,0.9503]$	$0.3354[0.3334,0.3374]$
4	$0.9914[0.9903,0.9925]$	$0.9888[0.9875,0.9901]$	$0.2747[0.2738,0.2757]$

reports the realization-level means and two-sided $95\%$ confidence intervals of the normalized sector entropy, normalized effective-sector occupancy, and dominant-sector fraction. Each realization-level metric was first averaged over the 48 sampled directions, as described in Section 5.2; the values in Table 1 summarize the resulting ensemble of 200 physical arrays.

The normalized sector entropy increases from $0.8593$ at $p=1$ to $0.9597$ at $p=2$ and $0.9914$ at $p=4$. The normalized effective-sector occupancy follows the same progression, increasing from $0.8311$ to $0.9481$ and then to $0.9888$. These two metrics quantify complementary aspects of the same redistribution: the sector weights become more even, and the sensitivity distribution approaches the occupancy that would be obtained from four equally populated sectors.

The dominant-sector fraction evolves in the opposite direction. Its ensemble mean decreases from $0.4267$ at first order to $0.3354$ at second order and $0.2747$ at fourth order, yielding

$$R_{max}^{\left(1\right)}>R_{max}^{\left(2\right)}>R_{max}^{\left(4\right)}.$$

(112)

The reduction in $R_{max}$ is consistent with the entropy-based measures: the most sensitive sector carries a progressively smaller share of the total local sensitivity as the observable order increases. We use this quantity as a complementary concentration measure rather than as independent evidence of diversification.

The interpretation of Table 1 is particularly direct because the sensitivity is resolved with respect to the same four irreducible $C_4$ labels at every observable order. The sector-label set is

$${\widehat{C}}_4=\{0,1,2,3\},$$

(113)

for $p=1$, $p=2$, and $p=4$. Tensor lifting changes the induced representation and the multiplicities with which irreducible components occur in the tensor space, but it does not create additional irreducible labels for $C_4$. The increase in $H_{\mathrm{norm}}$ and $N_{\mathrm{eff},\mathrm{norm}}$ therefore cannot be explained by a larger number of sector labels at higher observable order.

Instead, the measured effect is a redistribution of the normalized sensitivity fractions

$${\left\{R_k^{\left(p\right)}\right\}}_{k=0}^3$$

(114)

within a fixed symmetry-sector set. The simultaneous increase in sector entropy and effective-sector occupancy, together with the reduction in dominant-sector concentration, shows that the angular sensitivity becomes progressively less localized in a small subset of symmetry channels. In the present electromagnetic model, tensor lifting therefore reorganizes the sector-wise geometry of local direction-of-arrival sensitivity.

This conclusion is intentionally narrower than a claim of improved estimation performance. The total sensitivity magnitude is not used to rank observable orders, and the experiment introduces neither an estimator nor a performance criterion such as estimation error or a Cramér–Rao bound. The result concerns the distribution of local sensitivity across irreducible symmetry sectors. Section 5.4 next tests whether the observed ordering persists for individual physical realizations rather than only at the level of ensemble means.

5.4. Statistical Robustness over Physical Array Realizations

The ensemble averages reported in Section 5.3 indicate a clear increase in sector diversity with observable order. We next examine whether this trend is driven by a subset of favorable geometries or whether it persists at the level of individual physical realizations.

Because each array realization is evaluated at all three observable orders, the relevant comparison is paired. For realization i, define

$$\Delta H_{1\to 2}^{\left(i\right)}=H_{\mathrm{norm}}^{(2,i)}-H_{\mathrm{norm}}^{(1,i)}$$

(115)

and

$$\Delta H_{2\to 4}^{\left(i\right)}=H_{\mathrm{norm}}^{(4,i)}-H_{\mathrm{norm}}^{(2,i)}.$$

(116)

Positive values of Equations (115) and (116) indicate that the normalized sector entropy increases under the corresponding tensor lift for the same physical architecture.

The paired entropy increments are summarized in Figure 3. Both mean increments are strictly positive:

$${\overline{\Delta H}}_{1\to 2}=0.1004,$$

(117)

and

$${\overline{\Delta H}}_{2\to 4}=0.03173.$$

(118)

The associated two-sided $95\%$ confidence intervals are

$${\mathrm{CI}}_{95\%}\left(\Delta H_{1\to 2}\right)=[0.0978,0.1031]$$

(119)

and

$${\mathrm{CI}}_{95\%}\left(\Delta H_{2\to 4}\right)=[0.03055,0.03290].$$

(120)

Neither interval approaches zero, showing that the increase in normalized sector entropy is not a consequence of uncertainty in the ensemble mean.

More strongly, the ordering

$$H_{\mathrm{norm}}^{(1,i)}<H_{\mathrm{norm}}^{(2,i)}<H_{\mathrm{norm}}^{(4,i)}$$

(121)

was observed for every realization $i=1,\dots ,200$. The empirical monotonicity probability is therefore

$$\widehat{P}\left[H_{\mathrm{norm}}^{\left(1\right)}<H_{\mathrm{norm}}^{\left(2\right)}<H_{\mathrm{norm}}^{\left(4\right)}\right]=1.$$

(122)

Within the tested ensemble, no physical geometry produced an entropy reversal between consecutive observable orders.

The smallest realization-level increments were

$$\underset{i}{min}\Delta H_{1\to 2}^{\left(i\right)}=0.0513$$

(123)

and

$$\underset{i}{min}\Delta H_{2\to 4}^{\left(i\right)}=0.0135.$$

(124)

Thus, the monotonic ordering in Equation (121) is not produced by increments that fluctuate around numerical zero. Even the least pronounced redistribution observed in the ensemble remains clearly positive.

These results substantially strengthen the electromagnetic validation, but their scope should be stated precisely. Equation (122) is an empirical result for the family of two-ring $C_4$-symmetric arrays and parameter ranges specified in Section 5.2. It does not establish a universal theorem for all cyclic representations, all tensor observables, or all electromagnetic architectures. In particular, the representation-theoretic support relations discussed in Section 4 identify sector-coupling pathways permitted by tensor products, but they do not by themselves imply monotonic growth of sector entropy.

The present experiment instead establishes a robust regularity within a concrete physics-based sensing model. Across 200 independently generated arrays, tensor lifting from $p=1$ to $p=2$ and from $p=2$ to $p=4$ consistently reduced sector concentration and increased the diversity of the normalized sensitivity distribution. Combined with the fixed $C_4$ sector-label set discussed in Section 5.3, this result provides direct evidence that higher-order tensor observables can reorganize local electromagnetic sensitivity among existing symmetry channels rather than merely enlarge the ambient observation space.

The next section uses controlled cyclic and dihedral ensembles to determine whether the regularity observed in this electromagnetic $C_4$ family persists when the underlying group structure is varied.

6. Dependence on Group Structure

The $C_4$ experiment in Section 5 isolates the effect of observable order while keeping the physical symmetry group fixed. The next question is whether the resulting redistribution pattern persists when the group structure itself changes. To address this issue, the present section compares matched cyclic and dihedral array families over several symmetry orders. The analysis preserves the same electromagnetic model, tensor observables, and sector metrics, while introducing reflections in a controlled manner. This comparison separates the robust behavior observed for cyclic symmetry from the order-dependent effects produced by the richer representation structure of dihedral groups.

6.1. Controlled Cyclic and Dihedral Ensembles

The electromagnetic experiment in Section 5 establishes a robust monotonic redistribution pattern for a concrete family of $C_4$-symmetric arrays. To determine whether that behavior is specific to the fourfold architecture or reflects a broader dependence on group structure, we next compare matched cyclic and dihedral ensembles. The objective is not to rank the arrays by estimation performance, but to examine how the addition of reflection symmetry modifies the redistribution of local angular sensitivity under the same tensor lifts.

The tested symmetry orders are $N\in \{3,4,5,6,8\}$. For each N, 200 matched pairs of two-ring planar arrays were generated. The two members of each pair share the same inner radius $r_1$, outer radius $r_2$, wavelength normalization, direction-of-arrival grid, and Monte Carlo index. Their only structural difference is the relative angular alignment of the rings.

For the cyclic member, the sensor positions are

$$\begin{array}{cc}\hfill {\mathbf{r}}_{1,m}^C& =r_1\left[\begin{array}{c}cos\left(2\pi m/N\right)\\ sin\left(2\pi m/N\right)\end{array}\right],\hfill \\ \hfill {\mathbf{r}}_{2,m}^C& =r_2\left[\begin{array}{c}cos\left(\delta +2\pi m/N\right)\\ sin\left(\delta +2\pi m/N\right)\end{array}\right],m=0,\dots ,N-1.\hfill \end{array}$$

(125)

The relative offset is $\delta =\eta 2\pi /N$, where $\eta \sim \mathcal{U}(0.14,0.36)$. For generic values in this interval, the geometry is invariant under the rotational action of $C_N$ but has no reflection axis common to both rings.

The dihedral member uses the same radii but aligns the two rings:

$$\begin{array}{cc}\hfill {\mathbf{r}}_{1,m}^D& =r_1\left[\begin{array}{c}cos\left(2\pi m/N\right)\\ sin\left(2\pi m/N\right)\end{array}\right],\hfill \\ \hfill {\mathbf{r}}_{2,m}^D& =r_2\left[\begin{array}{c}cos\left(2\pi m/N\right)\\ sin\left(2\pi m/N\right)\end{array}\right],m=0,\dots ,N-1.\hfill \end{array}$$

(126)

This alignment preserves both the N-fold rotations and the associated reflections, so the complete geometry carries the action of $D_N$.

The matched construction reduces geometric variability in the comparison. Each $C_N$–$D_N$ pair contains the same number of sensors, has the same radial scale, and is evaluated on the same angular grid. Nevertheless, restoring reflection symmetry requires a geometric realignment. The experiment should therefore be interpreted as a controlled comparison between matched symmetry classes, not as a formal causal isolation of reflection symmetry alone.

The radii are generated from the same distributions used in Section 5, namely $r_1/\lambda \sim \mathcal{U}(0.30,0.48)$ and $r_2/\lambda \sim \mathcal{U}(0.62,0.95)$. For both members of each pair, the narrowband far-field steering model and the first-, second-, and fourth-order tensor observables are those defined in Section 4 and Section 5.1. The same 48-direction grid, including a realization-specific random angular offset, is used for both symmetry classes and for all observable orders. A fixed pseudorandom seed of 20260716 was used for the complete experiment.

The symmetry sectors are evaluated by character projection. For an irreducible representation $\lambda$ of a finite group G, with dimension $d_{\lambda }$, character ${\chi }_{\lambda }$, and induced representation ${\rho }^{\left(p\right)}$, the projected sector energy can be evaluated without explicitly assembling the projector:

$${∥{\mathsf{\Pi }}_{\lambda }^{\left(p\right)}{J}_{\varphi }^{\left(p\right)}∥}^2={\displaystyle \frac{d_{\lambda }}{\left|G\right|}}\sum _{g\in G}{\chi }_{\lambda }{\left(g\right)}^{\ast }〈J_{\varphi }^{\left(p\right)},{\rho }^{\left(p\right)}\left(g\right)J_{\varphi }^{\left(p\right)}〉.$$

(127)

This expression is equivalent to applying the Reynolds projector defined in Section 3. It is computationally preferable at fourth order because it avoids constructing dense projector matrices in the full tensor space.

Inner products between pure tensor factors are evaluated through

$$〈{\displaystyle \underset{j=1}{\overset{q}{⨂}}}{x}_j,{\displaystyle \underset{j=1}{\overset{q}{⨂}}}{y}_j〉={\displaystyle \prod _{j=1}^q}〈x_j,y_j〉,$$

(128)

where q is the number of tensor factors. By linearity, this identity also applies to the sums of tensor-product terms appearing in the lifted derivatives. It preserves the exact character-projection formula while avoiding unnecessary high-dimensional matrix assembly.

The numerical implementation was checked independently. Analytic derivatives were compared with centered finite differences at randomly selected angular samples, giving a maximum relative error of $2.50\times {10}^{-9}$ over all groups, realizations, and observable orders. The largest relative residual in the sector-energy decomposition was $6.20\times {10}^{-16}$, which is consistent with floating-point roundoff.

The intended symmetry class of each geometry was also verified directly. For cyclic realizations, the minimum mismatch obtained after testing all candidate reflections was $0.1431\lambda$, excluding accidental dihedral symmetry. For dihedral realizations, the maximum reflection residual was $3.14\times {10}^{-16}$, confirming numerical invariance under the prescribed reflections.

For every realization and symmetry class, the normalized entropy was first averaged over the 48 sampled directions. The paired increments between consecutive observable orders are

$$\Delta H_{1\to 2}^G=H_{\mathrm{norm}}^{(2,G)}-H_{\mathrm{norm}}^{(1,G)},\Delta H_{2\to 4}^G=H_{\mathrm{norm}}^{(4,G)}-H_{\mathrm{norm}}^{(2,G)},$$

(129)

where G denotes either $C_N$ or $D_N$.

Because the ensembles are matched, the change associated with replacing the cyclic geometry by its dihedral counterpart is summarized by

$$E_{1\to 2}^{\left(\mathrm{ref}\right)}=\Delta H_{1\to 2}^{D_N}-\Delta H_{1\to 2}^{C_N},E_{2\to 4}^{\left(\mathrm{ref}\right)}=\Delta H_{2\to 4}^{D_N}-\Delta H_{2\to 4}^{C_N}.$$

(130)

These quantities are descriptive paired differences. Negative values indicate that the corresponding entropy increment is smaller for the dihedral member, whereas positive values indicate that it is larger. They do not by themselves establish a universal causal effect of reflection symmetry.

For each N and symmetry class, ensemble means and two-sided $95\%$ confidence intervals were computed from the 200 realization-level values. The resulting cyclic regularity and the order-selective behavior of the matched dihedral ensembles are examined in the following subsections.

6.2. Robust Cyclic Diversification

The cyclic ensembles exhibit a consistent redistribution pattern across all tested symmetry orders. For every tested $C_N$ family, the mean entropy increments associated with both tensor lifts are positive. Figure 4 compares the cyclic and dihedral results; the cyclic curves are considered first.

For the first tensor lift, the mean cyclic increment satisfies

$${\overline{\Delta H}}_{1\to 2}^{C_N}>0\mathrm{for}\mathrm{all}N\in \{3,4,5,6,8\}.$$

(131)

The corresponding values and confidence intervals are reported in

Table 2. Paired normalized-entropy increments for the cyclic ensembles. Values are reported as ensemble mean $[95\%\mathrm{CI}]$ over 200 realizations for each symmetry order.

Group	$\mathsf{\Delta }{\mathit{H}}_{1\to 2}$	$\mathsf{\Delta }{\mathit{H}}_{2\to 4}$
$C_3$	$0.1169[0.1125,0.1213]$	$0.02265[0.02136,0.02395]$
$C_4$	$0.1034[0.1007,0.1061]$	$0.03019[0.02918,0.03120]$
$C_5$	$0.1177[0.1127,0.1227]$	$0.01237[0.01118,0.01357]$
$C_6$	$0.1215[0.1139,0.1291]$	$0.00943[0.00844,0.01041]$
$C_8$	$0.2167[0.2103,0.2231]$	$0.02270[0.02123,0.02418]$

. The smallest mean increment occurs for $C_4$, with

$${\overline{\Delta H}}_{1\to 2}^{C_4}=0.1034,{\mathrm{CI}}_{95\%}=[0.1007,0.1061],$$

(132)

whereas the largest occurs for $C_8$,

$${\overline{\Delta H}}_{1\to 2}^{C_8}=0.2167,{\mathrm{CI}}_{95\%}=[0.2103,0.2231].$$

(133)

Thus, the first tensor lift produces a positive diversification increment throughout the cyclic family, although its magnitude depends on N.

The second tensor lift exhibits the same sign regularity:

$${\overline{\Delta H}}_{2\to 4}^{C_N}>0\mathrm{for}\mathrm{all}\mathrm{tested}N.$$

(134)

Its mean increments are smaller than those observed for the $1\to 2$ lift, ranging from $0.00943$ for $C_6$ to $0.03019$ for $C_4$. Importantly, every associated $95\%$ confidence interval remains strictly above zero.

The ensemble means alone do not establish whether the ordering persists for individual architectures. We therefore evaluated the realization-level condition

$$H_{\mathrm{norm}}^{(1,i)}<H_{\mathrm{norm}}^{(2,i)}<H_{\mathrm{norm}}^{(4,i)}$$

(135)

for each realization i. Figure 5 reports the corresponding empirical probabilities. For every cyclic group in the experiment,

$${\widehat{P}}_{C_N}\left[H_{\mathrm{norm}}^{\left(1\right)}<H_{\mathrm{norm}}^{\left(2\right)}<H_{\mathrm{norm}}^{\left(4\right)}\right]=1.$$

(136)

Hence, the ordering in Equation (135) is observed in all 1000 cyclic realizations comprising the five $C_N$ ensembles.

The cyclic result extends the electromagnetic $C_4$ observation of Section 5 in two respects. First, monotonic diversification is recovered for cyclic groups with different symmetry orders and different numbers of sensing elements. Second, the regularity persists under independently generated physical radii and angular offsets throughout the controlled ensemble. The effect is therefore not restricted to the particular $C_4$ family used for the primary electromagnetic validation.

The data nevertheless do not support a universal theorem stating that tensor lifting must increase sector entropy for every cyclic representation or every cyclic observable. The present experiment considers a specific class of two-ring far-field sensing models and the observable lifts defined in Section 5. Within that class, however, the empirical regularity is exceptionally stable: both entropy increments are positive at the ensemble level, their confidence intervals exclude zero for every tested N, and the realization-level monotonic ordering holds without exception.

The contrast with the dihedral members of the matched ensembles is immediate in Figure 4 and Figure 5. Unlike the cyclic families, the dihedral architectures do not exhibit a uniform monotonic regime across symmetry orders. The next subsection analyzes this difference through the paired reflection effects defined in Equation (130).

6.3. Order-Selective Effect of Reflection Symmetry

The cyclic ensembles analyzed in Section 6.2 exhibit monotonic entropy growth at every tested symmetry order. The matched dihedral ensembles do not follow the same uniform pattern. The relevant difference is not a global suppression or enhancement of diversification. Instead, the matched dihedral architectures exhibit different entropy-increment changes at the two tensor-lifting stages.

This effect is quantified by the paired differences introduced in Equation (130). Figure 6 reports these matched reflection effects as functions of the symmetry order N.

For the first tensor lift, the matched reflection effect is negative at every tested symmetry order:

$$E_{1\to 2}^{\left(\mathrm{ref}\right)}<0\mathrm{for}\mathrm{all}N\in \{3,4,5,6,8\}.$$

(137)

The corresponding ensemble means range from $-0.1445$ for $N=3$ to $-0.0598$ for $N=6$. As reported in

Table 3. Matched reflection effects on the normalized-entropy increments. Each entry is the ensemble mean $[95\%\mathrm{CI}]$ of the dihedral-minus-cyclic paired difference over 200 matched realizations.

N	${\mathit{E}}_{1\to 2}^{\left(\mathbf{ref}\right)}$	${\mathit{E}}_{2\to 4}^{\left(\mathbf{ref}\right)}$
3	$-0.1445[-0.1510,-0.1380]$	$0.03381[0.02990,0.03771]$
4	$-0.09678[-0.09972,-0.09383]$	$0.03892[0.03617,0.04167]$
5	$-0.1042[-0.1058,-0.1025]$	$0.01549[0.01210,0.01887]$
6	$-0.05979[-0.06166,-0.05792]$	$0.02021[0.01809,0.02232]$
8	$-0.07997[-0.08218,-0.07776]$	$0.03573[0.03457,0.03688]$

, the upper endpoint of every $95\%$ confidence interval remains below zero. Hence, within the matched ensemble, the $1\to 2$ entropy increment is systematically smaller for the dihedral architecture than for the cyclic architecture sharing the same radii and angular sampling grid.

The second tensor lift exhibits the opposite sign:

$$E_{2\to 4}^{\left(\mathrm{ref}\right)}>0\mathrm{for}\mathrm{all}\mathrm{tested}N.$$

(138)

The mean paired effects range from $0.01549$ for $N=5$ to $0.03892$ for $N=4$, and every associated $95\%$ confidence interval lies strictly above zero. Thus, the same matched dihedral architectures that exhibit a reduced entropy increment from $p=1$ to $p=2$ exhibit a larger increment from $p=2$ to $p=4$.

The sign reversal between Equations (137) and (138) is the central group-structure result of the controlled experiment. Reflection symmetry does not produce a uniform decrease in sector diversification. Rather, its effect is selective with respect to observable order: relative to the matched cyclic architectures, the dihedral cases exhibit less entropy growth in the first lift and more entropy growth in the second.

This order selectivity also explains the non-uniform monotonicity probabilities shown in Figure 5. For $D_3$, the mean first-lift entropy increment is negative, and only $3\%$ of the realizations satisfy the complete ordering in Equation (135). The corresponding probabilities increase to $0.57$ for $D_4$ and $0.58$ for $D_5$, before reaching 1 for $D_6$ and $D_8$. These results indicate that the dihedral families do not belong to a single qualitative redistribution regime over the tested values of N.

A particularly clear example is $D_3$. Its ensemble-mean normalized entropy decreases from first to second order,

$${\overline{\Delta H}}_{1\to 2}^{D_3}=-0.0276,$$

(139)

and subsequently increases from second to fourth order,

$${\overline{\Delta H}}_{2\to 4}^{D_3}=0.0565.$$

(140)

The corresponding mean entropy sequence is

$$0.7339\to 0.7063\to 0.7628.$$

(141)

This example demonstrates directly that tensor lifting does not, by itself, imply monotonic entropy growth. The representation-theoretic availability of additional fusion pathways and the redistribution of the associated sensitivity weights must be distinguished.

The observed sign pattern is consistent across all tested values of N, but its interpretation should remain limited to the matched two-ring ensembles studied here. In particular, we do not claim that reflections universally delay, suppress, or enhance diversification. Such statements would require analytical conditions linking the fusion algebra, representation multiplicities, and the parameter-dependent sector weights. The present result is empirical and more specific: within the controlled ensemble, changing from $C_N$ to the matched $D_N$ architecture reduces the $1\to 2$ entropy increment and increases the $2\to 4$ increment for every tested symmetry order.

Accordingly, group structure enters the redistribution process in an order-dependent manner. The cyclic and dihedral architectures are subjected to the same observable lifts, yet their sector-weight trajectories differ. The final subsection examines the implications of this observation for the interpretation of tensorization-induced redistribution.

6.4. Consequences for Tensorization-Induced Redistribution

The controlled comparison between cyclic and dihedral architectures leads to a more precise interpretation of tensorization-induced redistribution. Tensor lifting does not impose a universal direction of change on sector entropy. Instead, it modifies the representation carried by the observable and thereby changes the symmetry couplings available to the lifted sensitivity. The sector weights that are actually realized remain determined by the electromagnetic response, the parameter value, and the derivative structure of the tensor observable.

The cyclic ensembles exhibit an exceptionally regular empirical pattern. For every tested value of N, both the $1\to 2$ and $2\to 4$ entropy increments are positive, and the complete monotonic ordering is observed in all cyclic realizations. The matched dihedral ensembles show that this regularity cannot be attributed to tensor order alone. Their first lifting stage exhibits a smaller entropy increment than the corresponding cyclic case, whereas their second lifting stage exhibits a larger one. The opposite signs of the paired effects reported in Equations (137) and (138) therefore show that the group dependence cannot be summarized by a single concentration or suppression factor.

This behavior is consistent with the representation-theoretic mechanism developed in Section 4. In the cyclic case, the one-dimensional irreducible representations obey the additive fusion rule in Equation (88), so the lifted derivative is supported through modular sums and differences in the labels carried jointly by the observation and its first-order sensitivity. In the dihedral case, reflections alter the fusion structure and introduce higher-dimensional irreducible components for the values of N considered here. The same tensor lift may therefore redistribute sensitivity differently at successive observable orders.

The results support a two-level interpretation. At the algebraic level, tensorization modifies sector accessibility through the fusion structure of the induced representation. At the physical level, the electromagnetic parameterization determines how strongly the accessible sectors are populated. The first level constrains the admissible redistribution pathways; the second determines the realized sector profile measured by $R_{\lambda }^{\left(p\right)}$, $H_{\mathrm{norm}}^{\left(p\right)}$, and $R_{max}^{\left(p\right)}$.

This distinction also explains why support expansion and entropy growth are not equivalent. A tensor product may make additional couplings available without producing a more even sensitivity distribution. Conversely, entropy may increase through redistribution among an unchanged set of irreducible labels, as demonstrated by the $C_4$ electromagnetic experiment in Section 5.

The broad conclusion is therefore not that tensorization intrinsically promotes diversification, but that it induces a group-dependent reorganization of symmetry-resolved sensitivity. Monotonic diversification is a robust empirical regularity for the tested cyclic ensembles, whereas the matched dihedral results show that richer representation structure can produce order-selective trajectories. The electromagnetic validation in Section 5 and the controlled group comparison in the present section provide complementary evidence for this interpretation.

7. Discussion

The results support a two-level interpretation of tensorization-induced redistribution. At the algebraic level, tensor lifting changes the representation carried by the observable and therefore modifies the sector couplings permitted by the group. At the physical level, the electromagnetic response and its local derivative determine how strongly those admissible sectors are populated. The experiments in Section 5 and Section 6 show that these levels interact, but neither determines the other in isolation.

The observed trends should therefore be interpreted as symmetry-resolved reorganization rather than as a generic benefit of increasing observable order. The tested cyclic ensembles exhibit a highly regular progression toward less concentrated sector profiles, whereas the matched dihedral ensembles show that reflection symmetry modifies the two lifting stages differently. This section places those findings in relation to established signal-processing approaches, examines their implications for electromagnetic design, and states the limits of the conclusions.

7.1. Relation to Higher-Order, Symmetry-Aware, and Observability Methods

Higher-order signal processing commonly uses moments, cumulants, polyspectra, and multilinear constructions to reveal statistical structure that is absent from lower-order descriptions. Such methods are especially useful for non-Gaussian source models, blind identification, phase coupling, and array-processing problems in which higher-order statistics suppress Gaussian contributions or generate virtual apertures [15]. The tensor observables studied here share a multilinear structure with those methods, but their role is different. They are deterministic lifts of the same parameterized electromagnetic response, not empirical moments or cumulants estimated from repeated data.

This distinction determines the scope of the contribution. Conventional higher-order processing typically asks whether a higher-order statistic reveals additional source information, increases an effective aperture, or improves identifiability under a stochastic model. The present framework instead asks how the local sensitivity already carried by the observation is reorganized across irreducible symmetry sectors when the observable is lifted. The comparison is therefore between symmetry-resolved sensitivity profiles, not between competing estimators.

The work is also related to algebraic and equivariant signal processing. Algebraic signal-processing theory represents signals, filters, and transforms through algebraic structures [10,11], while equivariant methods construct maps that commute with prescribed group actions [12]. These approaches exploit symmetry to define transforms, processing operators, invariant features, or computational architectures. By contrast, the present method assumes that the group action arises from the physical sensing architecture and uses the resulting irreducible decomposition as a diagnostic coordinate system for local sensitivity.

The framework is therefore complementary to, rather than a replacement for, existing symmetry-aware methods. It does not introduce a new group transform, equivariant estimator, invariant descriptor, or signal-processing architecture. Its specific contribution is to compare how the derivative of a parameterized observation is distributed across the isotypic sectors induced at different observable orders.

The same distinction applies to classical observability and identifiability analysis. Traditional observability asks whether an internal state or parameter can be recovered from available measurements, usually through rank, injectivity, conditioning, or information criteria. The sector-resolved perspective developed here does not establish global recoverability. It refines the local differential response by identifying the symmetry channels through which an infinitesimal parameter perturbation becomes visible.

This refinement provides information that is absent from both the total derivative norm and the ambient tensor-space dimension. Two observables may have comparable total local sensitivity but allocate it differently across sectors. Conversely, a larger tensor space may contain more irrep copies or additional fusion pathways without producing a broader realized sensitivity profile. The $C_4$ experiment demonstrates redistribution within an unchanged set of four irreducible labels, whereas the $D_3$ result shows that tensor lifting need not increase normalized sector entropy monotonically.

Accordingly, broader sector occupancy should not be interpreted as larger Fisher information, lower estimation error, or a smaller Cramér–Rao bound. Those quantities depend on noise statistics, nuisance parameters, sampling, and the estimator. The present metrics describe the organization of local sensitivity relative to symmetry. They complement conventional observability and statistical estimation analysis by retaining the sector allocation that scalar norms or global rank conditions suppress.

7.2. Implications for Electromagnetic Sensing and Programmable Architectures

The results suggest that symmetry can be treated as a design variable in electromagnetic sensing rather than solely as a descriptive property of the hardware. Array geometry, element arrangement, material configuration, and programmable loading determine the group action carried by the measured field. Observable order then determines the induced tensor representation through which local sensitivity is expressed. The relevant design problem is therefore joint: architecture, symmetry class, and observable order should be evaluated together.

This perspective extends conventional design criteria based on aperture, beamwidth, sidelobe level, conditioning, or total Fisher information. Those criteria remain essential, but they do not describe how a parameter perturbation is distributed among irreducible symmetry channels. Two architectures with similar total sensitivity may have markedly different sector profiles, and a tensor lift may change that profile without altering the physical aperture.

The electromagnetic $C_4$ experiment illustrates this point directly. The geometry, physical parameter, and four irreducible labels remain fixed while the observable order changes. The resulting diversification is therefore not caused by additional sensors or by the appearance of new labels. It arises from the tensorial coupling of the observation and derivative factors within the induced representation.

The cyclic–dihedral comparison adds a second design implication. Architectures with the same element count and comparable radial extent can follow different redistribution trajectories when their group structures differ. In the controlled ensembles, the matched dihedral architectures exhibit a smaller entropy increment from first to second order and a larger increment from second to fourth order than their cyclic counterparts. This order-dependent contrast shows that dihedral symmetry cannot be assigned a universal role such as promoting or suppressing sector diversification; the observed behavior depends jointly on the group structure, the observable order, and the physical parameterization.

For programmable electromagnetic systems, this order dependence is particularly relevant. A reconfigurable intelligent surface or tunable metasurface may preserve a nominal group, reduce it to a subgroup, or deliberately break selected rotations or reflections. When the programmed state remains compatible with a well-defined group action, the sector-resolved framework can compare how candidate configurations allocate local sensitivity [2,3].

A practical symmetry-aware design objective could therefore target a prescribed sector profile rather than only a total field metric. Depending on the application, a configuration might be selected to emphasize a chosen irrep, reduce dominant-sector concentration, or maintain sensitivity across several symmetry channels. Such an objective would have to be combined with electromagnetic constraints, including passivity, mutual coupling, quantized loads, losses, bandwidth, and calibration.

The framework also provides a language for evaluating deliberate symmetry breaking. Removing a reflection or perturbing a rotationally symmetric layout may activate previously inaccessible channels, but support expansion alone does not guarantee a useful redistribution. The physically realized sector weights must still be computed. In practical terms, symmetry breaking should be assessed by the resulting sensitivity profile, not inferred solely from the enlarged algebraic support.

The present analysis does not establish that sector diversification improves receiver performance or reduces computational cost. Those consequences depend on the downstream estimator, the noise covariance, and the implementation architecture. Its practical role is diagnostic: it reveals how the same physical parameter is expressed across symmetry channels and thereby supports the joint evaluation of sensing geometry, programmable state, and observable order.

7.3. Scope, Limitations, and Research Directions

The framework is intentionally restricted to local, deterministic sensitivity under exact finite symmetry. The electromagnetic experiments use narrowband far-field propagation, identical isotropic elements, a homogeneous medium, and no mutual coupling. These assumptions isolate the interaction between tensor lifting and symmetry-sector redistribution, but they limit direct quantitative transfer to measured arrays, metasurfaces, or RIS implementations.

Real hardware generally exhibits only approximate symmetry because of manufacturing tolerances, element-pattern differences, coupling asymmetries, loading errors, environmental scattering, and calibration uncertainty. Under such perturbations, the nominal isotypic sectors are no longer perfectly decoupled and sensitivity can leak between them. A natural extension is to develop perturbation bounds that relate physical asymmetry to sector leakage and to determine when the nominal group decomposition remains a useful approximation.

The analysis is also local. A nonzero derivative indicates first-order variation in the observation at a specified operating point, but it does not exclude global ambiguities or guarantee stable inversion over an extended parameter domain. Global identifiability, ambiguity resolution, and conditioning must therefore be studied separately.

Only a scalar parameter is considered. For a vector parameter, the local object is a Jacobian whose range is a multidimensional sensitivity subspace. A sector-resolved generalization should characterize the projected Jacobian subspaces or their Gram operators in a basis-independent manner. This extension would make it possible to compare how different parameter directions share or compete for the same symmetry channels.

The normalized sector fractions intentionally remove the total derivative magnitude. This normalization isolates redistribution, but it prevents the proposed metrics from ranking observable orders by absolute information content. A statistical extension should introduce noise-weighted sector quantities and determine when the noise covariance is compatible with the group action. If the covariance does not commute with the representation, nominally distinct sectors may couple in the noise-weighted geometry, and a simple additive decomposition of Fisher information may no longer hold.

The quadratic and quartic observables are deterministic tensor lifts rather than empirical higher-order statistics. In an experimental implementation, such quantities would be estimated from noisy finite data and would exhibit order-dependent variance, bias, and computational cost. The robustness of the sector metrics to finite-sample estimation, regularization, outliers, and model mismatch remains to be established.

The numerical conclusions are empirical and ensemble-specific. Monotonic diversification is observed in every tested cyclic realization, but the dihedral results show that no universal entropy ordering follows from tensorization alone. Deriving sufficient conditions for monotonic redistribution would require explicit assumptions on fusion coefficients, multiplicities, factor supports, and parameter-dependent tensor amplitudes.

The paired $C_N$–$D_N$ construction controls radii, element count, and angular sampling, but restoring reflection symmetry also changes the ring alignment. The comparison therefore demonstrates an order-selective difference between matched architecture families rather than a universal causal law for reflections. Alternative geometries, continuously tunable symmetry breaking, and fixed electromagnetic transfer models would help separate geometric and group-theoretic effects more sharply.

Programmable surfaces provide a natural experimental direction. A measured RIS or metasurface could be configured to preserve, reduce, or break a prescribed symmetry, allowing the predicted sector redistribution to be tested under coupling, losses, quantized control, and calibration constraints. The observable order could then be selected jointly with the programmable state through a cost-aware criterion that balances sector allocation, sample complexity, and estimator performance.

The framework may also be extended to mode-dependent or nonlinear observation models. For example, fuzzy affine or Markov-jump descriptions associate different local observation maps with different operating modes. A symmetry-resolved analysis could be applied conditionally to each mode and then combined with hidden-mode probabilities or asynchronous mode estimates [13,14]. This connection is conceptually plausible but remains outside the electromagnetic model and validation considered here.

Finally, symmetry sectors may provide structured inputs, regularizers, or modular channels for downstream learning and inference. Such combinations should preserve the distinction established throughout the paper: sector occupancy is a structural descriptor of local sensitivity, not a guarantee of improved inference.

The present contribution should therefore be viewed as a diagnostic foundation. It establishes a controlled connection among electromagnetic symmetry, induced tensor representations, and local sensitivity redistribution. Extending that connection to approximate symmetry, vector parameters, noise-weighted information, measured hardware, and performance-oriented design defines the next stage of the research program.

8. Conclusions

This paper has developed a representation-theoretic framework for analyzing how local parameter sensitivity is distributed across irreducible symmetry sectors and how that distribution changes under tensor lifting. The central object is not the ambient dimension of the lifted observation space, but the symmetry-resolved profile of the local derivative. Character-weighted Reynolds projectors provide an orthogonal decomposition of the sensitivity into isotypic components, while normalized sector fractions, entropy, effective-sector occupancy, and dominant-sector fraction quantify the resulting allocation.

A principal conceptual result is the separation between algebraic accessibility and realized sensitivity redistribution. The induced tensor representation and the sector supports of the observation and its derivative determine which symmetry channels are admissible under fusion. They do not determine the numerical weight carried by those channels. The actual profile depends on the electromagnetic response, the parameter value, and the derivative structure of the lifted observable. Consequently, support expansion, multiplicity growth, and sector diversification are related but non-equivalent effects.

The electromagnetic validation demonstrates this distinction in a concrete $C_4$-symmetric receiving array. Across 200 independently generated two-ring geometries, the normalized sector entropy and effective-sector occupancy increase from first- to second- and fourth-order observables, while the dominant-sector fraction decreases. The same four irreducible labels are represented at every order. The observed effect is therefore a redistribution of local angular sensitivity within a fixed sector-label set rather than the appearance of additional symmetry labels.

The broader cyclic–dihedral comparison shows that this behavior depends on group structure. For the tested cyclic ensembles, the ordering

$$H_{\mathrm{norm}}^{\left(1\right)}<H_{\mathrm{norm}}^{\left(2\right)}<H_{\mathrm{norm}}^{\left(4\right)}$$

(142)

holds in all 1000 realizations. The matched dihedral ensembles do not exhibit a uniform monotonic regime. Instead, the addition of reflection symmetry has an order-selective effect: relative to the cyclic counterparts, it reduces the entropy increment from $p=1$ to $p=2$ and increases the increment from $p=2$ to $p=4$. The $D_3$ case provides a direct counterexample to any universal claim that tensor lifting must increase sector entropy.

These findings support a restricted but robust conclusion. Tensorization is not intrinsically a mechanism for increasing information, estimation accuracy, or observable degrees of freedom. Its general role in the present framework is to reorganize the symmetry-resolved pathways through which local parameter sensitivity is expressed. Whether that reorganization produces diversification, concentration, or an order-selective trajectory depends jointly on the group, the observable order, and the physical sensing model.

For electromagnetic sensing and programmable architectures, this suggests that geometry, symmetry group, and tensor observable should be considered jointly. Arrays, metasurfaces, and RIS configurations may be designed not only to control total field response, but also to shape the allocation of sensitivity across symmetry channels. The framework provides a diagnostic basis for such co-design, although a performance-optimal configuration requires additional noise, estimator, and hardware models.

The present results are intentionally limited to exact finite symmetries, deterministic local sensitivities, scalar parameters, and idealized narrowband far-field models. Extensions to approximate symmetry, sector leakage, vector-parameter Jacobians, noise-weighted sector information, finite-sample higher-order observables, and experimentally measured programmable surfaces remain open. These directions would connect the symmetry-resolved sensitivity framework developed here with statistical estimation and practical electromagnetic design.

Overall, the work establishes a mathematically controlled link among electromagnetic symmetry, tensor representations, and effective observability. Its main contribution is a framework for distinguishing which sectors are algebraically available from how physical sensitivity is actually distributed among them. That distinction provides a basis for analyzing and eventually designing higher-order sensing architectures in which symmetry is treated as an active component of observability rather than as a purely geometric attribute.