Symmetry-Loss in ML and Physics

• Symmetry-Loss is a paradigm that treats symmetry as a structural constraint, using explicit regularizers or inherent loss landscape invariances.
• It underpins methods in machine learning that bolster invariance and equivariance, yielding enhanced robustness and connected optimization landscapes with modest overhead.
• In physics, symmetry-loss formulations analyze gain–loss compensation and non-Hermitian effects, guiding steady-state behavior and symmetry-protected transport.

Searching arXiv for recent and foundational papers on symmetry-related loss formulations and symmetry-loss phenomena across machine learning and physics. “Symmetry-Loss” denotes a family of research programs in which symmetry is treated as a structurally informative constraint on an objective, a dynamical invariant of a loss landscape, or a balance law between physical gain and dissipation. Across the works considered here, the phrase covers at least three distinct but related usages: auxiliary penalties that encourage invariance or equivariance in learned representations; analyses of how symmetries of a loss function generate degeneracy, critical manifolds, collapse, or optimization constraints; and non-Hermitian or open-system settings in which loss and gain determine whether a symmetry is broken, restored, or compensated in steady states, spectra, or transport channels (Hebbar et al., 3 Nov 2025, Zhao et al., 16 Jun 2025, Kepesidis et al., 2015, Klimov et al., 2017).

1. Scope and conceptual meanings

In machine learning, the most direct meaning of symmetry-loss is a regularizer added to a training objective so that the learned map respects a target transformation law. A representative abstract form is

$$\min_{\theta} \mathbb{E}[\mathcal{L}(f_{\theta}(x), y) + \lambda\Gamma(f_{\theta}(x), y)],$$

where $\Gamma$ penalizes symmetry violation and $\lambda$ controls its strength (Hebbar et al., 3 Nov 2025). Closely related formulations replace explicit transformed-input comparisons by infinitesimal generator constraints or by invariant-coordinate matching (Akhound-Sadegh et al., 2023, Dönmez, 4 Dec 2025).

A second usage concerns symmetries of the loss function itself. In this setting, parameter transformations preserve the objective,

$L(w)=L(T(w)),$

or, for grouped parameters in neural networks,

$$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$

This viewpoint treats symmetry as a generator of orbits of equivalent parameters, flat directions, replicated minima, and critical manifolds rather than as an external regularizer (Zhao et al., 16 Jun 2025, Şimşek et al., 2021). A more dynamical version sharpens this by asserting that mirror symmetries induce linear stationary constraints such as

$O^T\theta=0,$

with weight decay or gradient noise favoring these constrained solutions (Ziyin, 2023).

In non-Hermitian and open-system physics, “loss” refers to dissipation, while symmetry-loss refers to the breaking or compensation of parity-time, loss-compensation, or related symmetries in the presence of balanced gain and loss. Here the central object is not a training objective but a spectral, steady-state, or transport problem: real eigenvalues versus complex-conjugate pairs, symmetry-preserving versus symmetry-broken stationary distributions, or protected versus attenuated transport channels (Kepesidis et al., 2015, Klimov et al., 2017, Hooper et al., 2024).

This suggests a common abstract pattern: symmetry-loss formulations compare an actual state—parameter, representation, field, or stationary ensemble—to an orbit or invariant structure prescribed by a group action. What differs across domains is whether the penalty is imposed explicitly, emerges from the geometry of the objective, or is read off from dissipative dynamics.

2. Auxiliary symmetry penalties in representation learning

A major contemporary usage of symmetry-loss is as an auxiliary training term for invariance or equivariance. In “SEAL - A Symmetry EncourAging Loss for High Energy Physics” (Hebbar et al., 3 Nov 2025), the generic penalty is

$$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$

with two practical approximations. The first, GSEAL, uses sampled finite transformations,

$$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$

while the second, $\delta$SEAL, imposes local generator consistency,

$$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$

For invariant scalar outputs, $\Gamma$0SEAL reduces to suppressing directional derivatives along symmetry orbits. In top-tagging experiments, the reported gain is primarily robustness rather than ordinary in-distribution accuracy: balanced accuracy remains $\Gamma$1-$\Gamma$2, AUC remains $\Gamma$3, while background rejection at fixed signal efficiency improves by a factor of roughly $\Gamma$4–$\Gamma$5 over baseline in unseen high-$\Gamma$6 regions; training-time overhead rises from $\Gamma$7 s per batch for the baseline to $\Gamma$8 s for both GSEAL and $\Gamma$9SEAL, with unchanged evaluation cost of $\lambda$0 s (Hebbar et al., 3 Nov 2025).

“SymFace: Additional Facial Symmetry Loss for Deep Face Recognition” (Prakash et al., 2024) instantiates the same general idea using approximate bilateral facial symmetry. A frontalness coefficient,

$\lambda$1

selects images for which left and right hemi-faces are compared in embedding space. The total loss is

$\lambda$2

The paper sets $\lambda$3 and $\lambda$4, and interprets the penalty as reducing nuisance asymmetries due to expression and illumination. The reported improvements are frequent but not universal: SymFace outperforms the baseline around $\lambda$5 of the time for lightweight networks and $\lambda$6 of the time for ResNet settings (Prakash et al., 2024).

For PDE solvers, “Lie Point Symmetry and Physics Informed Networks” (Akhound-Sadegh et al., 2023) derives a symmetry-loss from the infinitesimal criterion

$\lambda$7

and uses

$\lambda$8

inside the total objective

$\lambda$9

The important technical qualification is that many exact PDE symmetries are useless as training signals because $L(w)=L(T(w)),$0 may vanish identically or reduce to a constant multiple of the PDE residual. Even so, the low-collocation gains are substantial: for the heat equation at $L(w)=L(T(w)),$1, average test MSE improves from $L(w)=L(T(w)),$2 to $L(w)=L(T(w)),$3 (Akhound-Sadegh et al., 2023).

A more abstract formulation appears in “Developmental Symmetry-Loss: A Free-Energy Perspective on Brain-Inspired Invariance Learning” (Dönmez, 4 Dec 2025). Here the quotient map

$L(w)=L(T(w)),$4

is built from invariant generators $L(w)=L(T(w)),$5, and the proposed loss matches invariant signatures relative to an anchor $L(w)=L(T(w)),$6: $L(w)=L(T(w)),$7 with $L(w)=L(T(w)),$8. The paper interprets this as minimizing “structural surprise” and organizes learning as iterative refinement of an effective symmetry group

$L(w)=L(T(w)),$9

The Free-Energy connection is explicitly interpretive rather than a full variational derivation (Dönmez, 4 Dec 2025).

Taken together, these formulations show three technical routes to symmetry-aware objectives: direct transformed-input consistency, infinitesimal generator constraints, and invariant-coordinate matching. This suggests that “symmetry-loss” is best understood as a family of orbit-consistency penalties rather than a single canonical formula.

3. Symmetry-structured loss landscapes in neural networks and optimization

A second major literature studies not penalties added to a loss, but symmetries already present in the loss landscape. In “Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances” (Şimşek et al., 2021), permutation symmetry of hidden units is formalized by

$$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$0

and shown to generate invariant subspaces, connected equal-loss manifolds, and symmetry-induced critical affine subspaces. A central theorem states that adding one extra neuron in each hidden layer connects previously discrete permutation-related minima into a connected manifold of global minima; in the two-layer case, the manifold is a connected union of $$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$1 affine subspaces, while critical manifolds inherited from narrower subnetworks occur in $$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$2 affine subspaces (Şimşek et al., 2021).

“Symmetry in Neural Network Parameter Spaces” (Zhao et al., 16 Jun 2025) generalizes this viewpoint into a survey language of group actions, orbits,

$$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$3

and fibers of the realization map $$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$4. The main implication is that parameter-space minima frequently overcount function-space solutions: orbits create flat directions, high-dimensional minima, mode connectivity, and non-identifiability. The survey also emphasizes that same-loss orbit points can have different gradients, curvature, sharpness, and subsequent optimization behavior under ordinary Euclidean GD or SGD (Zhao et al., 16 Jun 2025).

The dynamical role of symmetry is sharpened in “Symmetry Induces Structure and Constraint of Learning” (Ziyin, 2023). For an $$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$5-mirror symmetry with reflection operator $$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$6, the paper proves that every such symmetry induces a linear stationary condition

$$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$7

It further shows that for sufficiently large weight decay, all minima satisfy this constraint, and that large SGD noise or large learning rate can stabilize these symmetry-constrained subspaces through the criterion

$$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$8

The paper’s concrete corollaries are that rescaling symmetry leads to sparsity, rotation symmetry leads to low rankness, and permutation symmetry leads to homogeneous ensembling (Ziyin, 2023).

“Remove Symmetries to Control Model Expressivity and Improve Optimization” (Ziyin et al., 2024) turns this into an intervention. It argues that symmetry creates low-capacity invariant subspaces and proves two mechanisms: feature masking in the kernel regime and exact dimension reduction of training dynamics when initialized in a symmetric subspace. The proposed method, syre, replaces the original objective by

$$L^m(\theta)=L^m(P_\pi\theta)\qquad \text{for all }\pi\in S_m.$$9

with a fixed random bias $O^T\theta=0,$0, and in the more general form

$O^T\theta=0,$1

Under mild assumptions, reflection symmetries are removed almost surely, and at formerly symmetric points one obtains a nonzero symmetry-breaking gradient $O^T\theta=0,$2. Empirically, the paper reports improvements in dead-neuron settings, low-rank supervised representations, posterior collapse in $O^T\theta=0,$3-VAEs, low-rank SSL heads, continual learning, and reinforcement learning (Ziyin et al., 2024).

A broader empirical program appears in “Ubiquitous Symmetry at Critical Points Across Diverse Optimization Landscapes” (Schneider, 4 May 2025), which studies invariant losses on neural-network, projective, graph, matching, and particle-interaction spaces. The direct symmetry measure is the stabilizer $O^T\theta=0,$4, while a stronger interaction-based measure is

$O^T\theta=0,$5

The paper’s claim is empirical rather than classificatory: across all examined examples, all observed critical points had nontrivial symmetry under $O^T\theta=0,$6 or, where $O^T\theta=0,$7 became trivial, under $O^T\theta=0,$8 (Schneider, 4 May 2025).

This body of work makes one misconception untenable: symmetry in a loss landscape is not merely a bookkeeping redundancy. It can connect minima, create critical manifolds, reduce effective model dimension, bias implicit regularization, and generate collapse-inducing invariant subspaces.

4. Symmetry constraints can also eliminate solutions

A less common but technically important usage arises when symmetry is enforced in a self-consistent approximation and the result becomes unsolvable. “Loss of solution in the symmetry improved Phi-derivable expansion scheme” (Markó et al., 2016) studies the two-loop $O^T\theta=0,$9PI/$$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$0-derivable approximation for the $$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$1-symmetric scalar model in the broken phase, with the Goldstone constraint

$$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$2

This is intended to enforce Goldstone’s theorem by replacing the field equation with the symmetry-imposed condition.

The paper’s main conclusion is negative. For smooth ultraviolet regulators, solutions of the regularized equations disappear below a nonzero infrared scale $$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$3. The mechanism is structural: once $$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$4 is imposed, the transverse propagator becomes massless, the longitudinal equation contains the bubble $$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$5, and infrared regularity would require anomalous transverse scaling. But the symmetry-improved transverse equation produces only

$$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$6

not the anomalous behavior required to tame the infrared singularity (Markó et al., 2016).

The paper distinguishes smooth regulators from non-analytic ones such as a sharp cutoff. A sharp cutoff can induce a spurious linear small-$$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$7 term in the mixed bubble and thereby fake the needed infrared behavior, apparently rescuing a solution at fixed finite cutoff. The authors argue that such solutions are regulator artifacts that should disappear in a renormalized treatment (Markó et al., 2016).

This case is a useful counterpoint to regularization-based symmetry losses in machine learning. It shows that externally imposing a symmetry condition can overconstrain an approximation if the approximation lacks the dynamical infrared structure needed to support that symmetry. A plausible implication is that “symmetry-loss” methods are safest when the model class can realize the relevant symmetry sector without singular compensation mechanisms.

5. Gain-loss symmetry breaking and compensation in non-Hermitian systems

In non-Hermitian photonics and open-system mechanics, symmetry-loss is governed by the interplay of gain, dissipation, nonlinear saturation, and noise. “PT-symmetry breaking in the steady state of microscopic gain-loss systems” (Kepesidis et al., 2015) reframes $$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$8-symmetry breaking as a steady-state phenomenon rather than merely a spectral exceptional-point transition. For coupled gain and loss modes with amplitudes $$\Gamma =\sum_{x\in X}\int_{g\in G}dg\left|f(g\odot x)-{g\odot f}(x)\right|^2,$$9,

$$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$0

with

$$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$1

The linearized Hamiltonian has eigenvalues

$$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$2

so the usual exceptional point occurs at $$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$3. But the deterministic steady-state analysis reveals a richer sequence: a zero-amplitude phase $$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$4, a nonzero $$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$5-symmetric finite-amplitude phase $$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$6, a weakly broken limit-cycle phase $$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$7 for $$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$8, and a fully symmetry-broken phase $$\Gamma_{G} = \frac{1}{N}\sum_{i=1}^N|f(g_i \odot x_i) - {g_i\odot f}(x_i)|^2,$$9 for $\delta$0. With strong thermal noise, phases $\delta$1 and $\delta$2 are washed out and replaced by a high-noise thermal phase $\delta$3, followed by an abrupt transition to a low-fluctuation symmetry-broken phase. Symmetry of the stationary ensemble is quantified by

$\delta$4

with $\delta$5 for a thermal distribution and $\delta$6 in the fully symmetry-broken state (Kepesidis et al., 2015).

Two companion photonics works broaden the notion of symmetry beyond standard $\delta$7 balance. “Loss compensation symmetry in dimers made of gain and lossy nanoparticles” (Klimov et al., 2017) studies a quasistatic two-cylinder dimer with permittivities

$\delta$8

The exact dispersion relation

$\delta$9

supports both the ordinary $$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$0-symmetric branch and a broader Loss Compensation Symmetry (LCS) branch. LCS requires

$$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$1

and is encoded in a field relation

$$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$2

The point is that exact compensation can occur even when gain and loss magnitudes are unequal, because the mode localizes asymmetrically and satisfies the global balance

$$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$3

“Multimode parity-time and loss-compensation symmetries in coupled waveguides with loss and gain” (Hlushchenko et al., 2021) extends this from identical single-mode waveguides to asymmetric cylindrical waveguides of different radii and to dissimilar TM, TE, HE, and EH mode pairings. In these systems, exact loss compensation occurs at LC thresholds where $$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$4, but, unlike standard identical-waveguide $$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$5 settings, compensation in the asymmetric case occurs at a single point rather than across an extended exact phase (Hlushchenko et al., 2021).

Nonlinearity can also restore bounded symmetry-compatible dynamics after linear symmetry breaking. “Dimer with gain and loss: Integrability and $$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$6-symmetry restoration” (Barashenkov et al., 2015) studies cubic gain-loss dimers whose linearized form,

$$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$7

breaks $$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$8 symmetry at $$\Gamma_{\delta} =\frac{1}{Nn}\sum^N_{i=1}\sum^n_{a=1} |\nabla f\cdot L_x^a\cdot x_i - L_f^a\cdot f(x_i)|^2.$$9, with growth rate

$\Gamma$00

The paper constructs two four-parameter families of cubic $\Gamma$01-symmetric dimers and proves that they are completely integrable Hamiltonian systems. For broad parameter regimes, nonlinear coupling diverts energy from the gaining to the losing site, trapping all trajectories in a finite phase-space region regardless of $\Gamma$02. This is the paper’s notion of spontaneous $\Gamma$03-symmetry restoration (Barashenkov et al., 2015).

For periodic waveguide arrays with richer unit cells, “$\Gamma$04 symmetry breaking in waveguide arrays with competing loss/gain pairs” (Kalozoumis et al., 2016) shows that a quadrimer cell with

$\Gamma$05

supports several distinct broken phases rather than a single transition. The paper derives a symmetry-adapted nonlocal current,

$\Gamma$06

satisfying

$\Gamma$07

and argues that its site average acts as an order parameter: it vanishes in the unbroken phase and is nonzero in the broken phase (Kalozoumis et al., 2016).

A recurrent misconception in this domain is that the exceptional point alone determines long-time behavior. These works show instead that steady-state energy distributions, nonlinear saturation, noise-activated escape, modal overlap, and field localization can all separate symmetry loss from the simplest spectral threshold narrative.

6. Symmetry-protected transport, lossless modes, and general lessons

A complementary strand studies how symmetry can protect transport or generate lossless propagation in explicitly lossy media. “Symmetry-protected transport through a lattice with a local particle loss” (Visuri et al., 2022) considers a finite fermionic chain connected to reservoirs and subjected to local loss at the center site. For the coherent chain

$\Gamma$08

the crucial fact is that antisymmetric eigenstates of the isolated reflection-symmetric chain satisfy

$\Gamma$09

at the symmetry center. Such modes therefore decouple from the strictly local loss. In transport, this produces conductance resonances that remain almost unaffected, while symmetric resonances are strongly suppressed. At $\Gamma$10, the conductance is

$\Gamma$11

The paper emphasizes that the protection is exact only for the isolated symmetric lattice; finite coupling to reservoirs weakly breaks the symmetry and turns exact protection into approximate protection (Visuri et al., 2022).

“Symmetry-Protected Lossless Modes in Dispersive Time-Varying Media” (Hooper et al., 2024) identifies a different mechanism. For a real physical field, the Fourier spectrum obeys $\Gamma$12, leading to a combined frequency-reflection and complex-conjugation symmetry, $\Gamma$13, for the propagation operator: $\Gamma$14 As a result, allowed propagation constants are either real or appear in complex-conjugate pairs. In the truncated $\Gamma$15 ladder, the eigenvalues are

$\Gamma$16

so lossless eigenpulses exist whenever

$\Gamma$17

These $\Gamma$18-unbroken modes are dissipation-free even in a lossy Drude medium, but they are adirectional: they do not possess a well-defined propagation direction. In a finite slab this can produce divergent transmission coefficients at specific lengths (Hooper et al., 2024).

Across the transport and time-modulation settings, the shared lesson is that loss is filtered by symmetry-selective channel structure. A local absorber can become nearly irrelevant if symmetry forces a node at the dissipative site; a temporally modulated lossy medium can support dissipation-free modes if the reality symmetry of the field remains unbroken at the mode level. This suggests that symmetry-loss in physics is often less about globally minimizing dissipation than about redistributing fields, frequencies, or currents so that dissipation has vanishing matrix element on the active mode.

What unifies the machine-learning and physical literatures is therefore not a single formula but a structural principle: symmetry-loss methods identify a preferred quotient, invariant, or channel decomposition and then penalize, exploit, or diagnose deviations from it. In learning, this yields auxiliary regularizers, critical manifolds, or collapse-inducing constraints; in non-Hermitian and open systems, it yields steady-state order parameters, loss-compensation branches, and symmetry-protected transport. The main open distinction is methodological. Some works assume the symmetry and enforce it softly or hard; others remove harmful symmetries; still others treat symmetry as emergent and observable only through spectral or steady-state signatures.