Gauging and Disentangling Procedure

• Gauging and disentangling procedure is a method that uses auxiliary variables and local constraints to separate mixed components or transition from global to local symmetry.
• It is applied across domains such as representation learning, spectroscopy, quantum field theory, and quantum codes to reframe complex systems.
• Practical implementations leverage metrics like the β-VAE score and residual analysis to quantitatively assess and optimize disentanglement.

The expression gauging and disentangling procedure is used in several technical literatures to denote a class of constructions that either separate mixed components of a signal or representation, or promote a global symmetry to a local one and then reorganize the degrees of freedom by local constraints. In representation learning, it refers to quantifying and enforcing factor separation in latent codes; in spectroscopy, it refers to recovering component spectra, velocities, and flux ratios from composite observations; in quantum field theory, gravity, and quantum information, it refers to introducing gauge variables, imposing Gauss-law constraints, and then moving physical information from matter variables into gauge or dual variables by local unitaries, projections, or field integration. This suggests a common structural motif—auxiliary variables, local constraints, and a re-expression of the same system in a more factorized or dual form—even though the mathematical objects differ substantially across domains (Peychev et al., 2017, Roose et al., 29 Sep 2025, Seeburger et al., 2024).

1. General pattern and domain-specific meanings

In continuum field theory, gauging means promoting a global symmetry to a local symmetry by introducing gauge fields and minimally coupling charged operators, while disentangling means using Gauss’s law to move charge from matter into gauge degrees of freedom by a local unitary in the Hamiltonian framework or by integrating out matter or gauge fields in the path integral. After disentangling, the matter is fixed to its neutral sector, and the dynamics is captured entirely by gauge or dual variables; the outcome is an equivalent theory with the same spectra, correlators, and operator content, up to degeneracies on trivial action sectors (Roose et al., 29 Sep 2025).

In fault-tolerant quantum computation, the same phrase is used operationally. A logical operator representative $L$ is treated as a global $\mathbb{Z}_2$ symmetry of the code Hamiltonian, auxiliary gauge qubits are added on the edges of a graph whose vertices are the qubits in the support of $L$, and local “Gauss’s law” checks

$$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$

are introduced. Measuring all $A_v$ projects into a simultaneous eigenspace of these local symmetries, and the product of the local outcomes equals the logical measurement outcome; a subsequent ungauging step disentangles the edge qubits and restores the original code space (Williamson et al., 2024).

In spectroscopic work, the same phrase has a more data-analytic meaning. The procedure begins by preparing spectra on a logarithmic wavelength grid so that Doppler shifts are additive, fitting a composite-spectrum forward model with component templates, velocities, and flux ratios, and then gauging the solution by residual statistics, reconstruction error, cross-correlation sharpness, and bootstrap uncertainty estimates (Seeburger et al., 2024). In this usage, “gauging” is not gauge theory; it denotes quantitative assessment of the disentangled solution.

2. Variational autoencoders, $\beta$-regularization, and reliability

For variational autoencoders, the baseline objective is the evidence lower bound

$$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$

and the $\beta$-VAE objective replaces the KL term by a $\beta$-weighted penalty,

$$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$

with $\mathbb{Z}_2$0. Increasing $\mathbb{Z}_2$1 enforces the posterior to be close to an isotropic normal with uncorrelated components, thereby encouraging statistically independent latent dimensions, but also shifts learning pressure away from reconstruction. In the reported experiments, $\mathbb{Z}_2$2 ranged from $\mathbb{Z}_2$3 to $\mathbb{Z}_2$4 in steps of $\mathbb{Z}_2$5 on a synthetic shapes dataset with generative factors shape, position $\mathbb{Z}_2$6, position $\mathbb{Z}_2$7, scale, and rotation, and from $\mathbb{Z}_2$8 to $\mathbb{Z}_2$9 in steps of $L$0 for MNIST classification via an SVM on latent codes. Disentanglement generally improved as $L$1 increased, especially for convolutional denoising autoencoders, but the gain plateaued and larger $L$2 reduced reconstruction fidelity and downstream discriminative performance; on MNIST there was a small regularization gain from $L$3 to $L$4, after which accuracy generally deteriorated (Peychev et al., 2017).

The main evaluation protocol in that study is the $L$5-VAE disentanglement metric. One samples two images differing in exactly one ground-truth factor, encodes their mean codes $L$6 and $L$7, forms the normalized latent-difference feature

$L$8

and trains a low-capacity linear classifier to predict which factor changed. The reported disentanglement score is the classifier accuracy. Five models were trained per $L$9 with different random initializations, and a central empirical result was consistently high variance of this metric across runs with identical $$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$0. The paper attributes that variance to imperfect alignment between the metric and “factor independence,” to rotated or curved but still orthogonal positional latent coordinate systems, and to different local minima induced by stochastic optimization and random initialization (Peychev et al., 2017).

A later development replaces direct total-correlation minimization by Dual Total Correlation and estimates one-dimensional conditional densities instead of the full latent joint distribution. In the Gaussian Channel Autoencoder, the latent variable is

$$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$1

and the training objective is

$$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$2

with

$$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$3

The method trains $$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$4 lightweight discriminators to estimate $$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$5 via a density-ratio trick with a uniform reference on $$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$6, thereby avoiding full high-dimensional joint estimation. Reported results show that $$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$7 and $$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$8 yield stable, high scores on Beamsynthesis and dSprites, that performance remains stable when the latent dimension increases from $$A_v = X_v \prod_{e\ni v} X_e,\qquad B_p = \prod_{e\in p} Z_e$$9 to $A_v$0, and that the correlation between $A_v$1 and MIG is $A_v$2, so the regularizer itself functions as an unsupervised gauge of disentanglement (Yeats et al., 2023).

3. Quantitative evaluation: unsupervised, structured, and logic-derived metrics

One line of work addresses model selection without access to ground-truth factors by ranking generative models through training dynamics. For each realization of a model specification, decoder activations are logged across epochs on a fixed trace set, normalized per epoch, assembled into a multislice kernel over epoch–sample pairs, converted into a Markov diffusion operator, and then embedded jointly across random seeds by singular value decomposition. Similarity between realizations is measured by Maximum Mean Discrepancy in the joint space, and the specification score is the mean pairwise MMD,

$A_v$3

Lower values indicate more stable, aligned dynamics across seeds. On dSprites, the reported Spearman rank correlations between this unsupervised ranking and supervised metrics increased with the number of seeds: for MIG they were $A_v$4, $A_v$5, and $A_v$6 at $A_v$7, and the method was unaffected by label noise because it did not use labels at all (Holtz et al., 2022).

A second line of work considers structured latent representations, especially object-centric models with permutation-invariant slots. The central object is an affinity matrix $A_v$8 relating latent coordinates $A_v$9 and factors $\beta$0, normalized so that $\beta$1. Hierarchy-sensitive projections $\beta$2 group latent indices and factor indices, and completeness and disentanglement are defined by normalized conditional entropies

$\beta$3

A key technical component is an EM-like permutation-invariant probing algorithm that alternates fitting a predictor $\beta$4 and permuting slots per sample so as to minimize

$\beta$5

This yields object-level, property-level, and intrinsic/extrinsic disentanglement scores that are invariant to slot permutations. In the reported experiments on Multi-dSprites, object-level disentanglement/completeness reached $\beta$6 for GENESIS, while ablations without disentanglement regularization reduced the structured metrics even when ARI could increase, showing that sharp masks and disentangled structured representations are not equivalent (Dang-Nhu, 2021).

A third line of work derives metrics directly from logical definitions. The construction replaces equality by a strict premetric, replaces the Heyting algebra of binary truth values by a Lawvere quantale on $\beta$7, and replaces quantifiers by aggregators. For a product-function notion of disentanglement, the resulting metric can be written as a distance to the best product approximation,

$\beta$8

or, equivalently, as a constancy-of-curry score

$\beta$9

For informativeness, an implication-based injectivity surrogate is

$$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$0

The formal guarantee is that if the converted quantity is zero, then the original predicate holds, and if the predicate contains no implication, then the converse also holds. This gives a direct logical interpretation to modularity and informativeness metrics that are often only heuristically motivated (Zhang et al., 2023).

4. Spectral disentangling in astronomy

For multi-epoch stellar spectroscopy, the basic forward model writes the observed spectrum at epoch $$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$1 as a sum of Doppler-shifted component spectra convolved with the instrumental line-spread function,

$$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$2

On a logarithmic wavelength grid $$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$3, non-relativistic Doppler shifts become approximately additive, with $$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$4. In survey-grade applications, spectra are resampled to a common log-wavelength grid, bad pixels and tellurics are masked, continua are corrected by a broad running median and shifted to zero mean, and all epochs are homogenized to a common LSF. The paper focusing on low-resolution surveys reports $$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$5, so $$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$6, with synthetic tests at $$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$7 epochs and $$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$8; recovery was robust when velocity amplitudes were at least comparable to the resolution element and when the flux ratio satisfied roughly $$\mathcal{L}(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$9 (Seeburger et al., 2024).

Template recovery is formulated as a regularized least-squares problem. For two components, the component spectra are stacked into a vector $\beta$0, the shifted observation operator is assembled into a sparse block matrix $\beta$1, and the solution is obtained from an augmented system

$\beta$2

where $\beta$3 encodes second-derivative curvature regularization. Velocities are initialized by cross-correlation or TIRAVEL, spectra are updated by sparse iterative solvers such as LSMR, and velocities and flux ratios are then refined in an alternating scheme. Quality is gauged by weighted residuals, residual structure versus wavelength, cross-correlation peak sharpness, and bootstrap or jackknife uncertainty estimates (Seeburger et al., 2024).

A related wavelength-domain implementation, Spectangular, uses singular value decomposition coupled to a downhill simplex optimizer and supports variable line profiles, telluric contamination, and per-spectrum flux-ratio optimization. On a logarithmic grid, the forward model becomes

$\beta$4

with $\beta$5 and $\beta$6. Residuals are a central diagnostic: tellurics can be isolated by disentangling once, spline-fitting residual telluric features, correcting the original spectra, and re-running the disentangling. For eclipsing binaries, phase-dependent flux ratios can be optimized directly from spectroscopy, and in the artificial example with $\beta$7 spectra at $\beta$8 the recovered flux fractions matched the model values within RMS $\beta$9 (Sablowski et al., 2019).

5. Field theory, sigma models, and gravity

In continuum duality constructions, the compact-boson example makes the basic mechanism explicit. Starting from

$\beta$0

with $\beta$1, one gauges the shift symmetry by introducing a worldsheet gauge field $\beta$2 and a Lagrange multiplier $\beta$3,

$\beta$4

Gauge-fixing $\beta$5 and integrating out $\beta$6 yields the dual scalar action and the T-duality map $\beta$7. In the Hamiltonian formulation, the corresponding disentangler is the local Gaussian unitary

$\beta$8

which implements Gauss’s law at the operator level and identifies the dual bosonic variables with the gauge electric field and potential (Roose et al., 29 Sep 2025).

A different generalization appears in Dirac sigma models. Starting from a sigma model with target $\beta$9 and Wess–Zumino form $$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$0, the gauging is adapted from Lie algebras to Lie algebroids $$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$1. One introduces an algebroid-valued one-form $$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$2 and minimally couples

$$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$3

The construction is encoded by an equivariantly closed extension $$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$4 on the graded manifold $$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$5 satisfying $$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$6 and $$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$7 for the lifted symmetry algebra. Dirac sigma models then arise from pulling back $$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$8 by a Q-bundle map and include both twisted Poisson sigma models and the $$\mathcal{L}_\beta(\theta,\phi; x) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - \beta\,\mathrm{KL}\big(q_\phi(\mathbf{z}|\mathbf{x})\,\|\, p(\mathbf{z})\big),$$9 WZW model as special cases (Salnikov et al., 2013).

In Cartan-geometric gravity, the gauging procedure starts from a Klein pair $\mathbb{Z}_2$00 associated with a homogeneous space $\mathbb{Z}_2$01 and produces a Cartan connection

$\mathbb{Z}_2$02

on a principal $\mathbb{Z}_2$03-bundle. In reductive cases, $\mathbb{Z}_2$04 splits into an $\mathbb{Z}_2$05-valued connection and an $\mathbb{Z}_2$06-valued soldering form. Gauging Minkowski spacetime reproduces the four-dimensional first-order Palatini–Holst–$\mathbb{Z}_2$07 action, while gauging Carroll, de Sitter–Carroll, anti-de Sitter–Carroll, and lightcone geometries yields unified first-order Carrollian gravity Lagrangians. A central point is that torsion-like constraints arise as equations of motion rather than being imposed by hand; for the reductive Carrollian models, $\mathbb{Z}_2$08 and $\mathbb{Z}_2$09 follow from the field equations when $\mathbb{Z}_2$10 (Figueroa-O'Farrill et al., 2022).

Related gauge constructions extend to topological phases and anomalous symmetries. In $\mathbb{Z}_2$11 dimensions, gauging a global $\mathbb{Z}_2$12 symmetry of a modular tensor category is gapped iff the Hall conductance $\mathbb{Z}_2$13; in that case the gauged theory is an MTC with anyon data modified by flux attachment, while for $\mathbb{Z}_2$14 gauging is dynamically equivalent to condensing the bosonic vison created by inserting $\mathbb{Z}_2$15 $\mathbb{Z}_2$16 flux (Cheng et al., 2022). In four-dimensional QED, the anomalous axial $\mathbb{Z}_2$17 can be promoted to an exact non-invertible $\mathbb{Z}_2$18 by defining a conserved but non-gauge-invariant current, summing over gauge transformations on the defect, and coupling the resulting non-invertible current to a gauge field; the gauging then imposes the local constraint

$\mathbb{Z}_2$19

which removes anomalous configurations and defines a “non-invertible gauge theory” (Karasik, 2022).

6. Quantum codes, topological order, and iterated gauging

Gauging procedures play a central role in lattice quantum phases. For subsystem symmetries on cubic lattices, one first identifies minimal symmetric coupling terms, places gauge degrees of freedom on the corresponding cells, defines Gauss-law operators, minimally couples matter terms, and then adds magnetic flux operators. For planar $\mathbb{Z}_2$20 subsystem symmetries, gauging a cubic paramagnet yields an X-cube–type theory. In the face-centered convention, the Gauss law is

$\mathbb{Z}_2$21

and the flux terms are four-face products such as

$\mathbb{Z}_2$22

The paper emphasizes that the species of symmetry charges in the ungauged model determine the resulting foliated fracton order: charges acted on by one family of planes become planons, by two families become lineons, and by three become fractons (Shirley et al., 2018).

A systematic higher-dimensional version starts from a $\mathbb{Z}_2$23-dimensional abelian symmetry on a lattice, introduces gauge qudits for a family of commuting checks, and alternates a gauging map $\mathbb{Z}_2$24 with a dual gauging map $\mathbb{Z}_2$25. For a global $\mathbb{Z}_2$26-form symmetry, the elementary projector is

$\mathbb{Z}_2$27

and the gauging map is

$\mathbb{Z}_2$28

Iterating this construction across layers produces $\mathbb{Z}_2$29-dimensional commuting-projector Hamiltonians with vertex-like and plaquette- or cube-like stabilizers and boundary conditions classified by subgroups $\mathbb{Z}_2$30. The paper gives explicit realizations for Clifford-deformed surface codes, foliated type-I fracton orders from linear subsystem symmetries, and type-I fracton order from Sierpinski fractal symmetry (Cuiper et al., 2024).

Regional gauging and disentangling also provide explicit ribbon operators for anyons and defects. In the $\mathbb{Z}_2$31 toric code, the charge-conjugation defect line is obtained by ungauging along a one-dimensional strip, applying a boundary circuit of controlled-$\mathbb{Z}_2$32 gates to perpendicular edges, and then regauging:

$\mathbb{Z}_2$33

The same regional procedure produces non-Abelian ribbon operators for $\mathbb{Z}_2$34 by decorating the $\mathbb{Z}_2$35 ribbons with a $\mathbb{Z}_2$36 gauging circuit (Lyons et al., 2024).

For non-invertible symmetries on spin chains, the lattice gauging prescription uses a Frobenius algebra internal to the symmetry category. In the Rep$\mathbb{Z}_2$37 example generated by the Kennedy–Tasaki transformation, two qubits are introduced around each link, the physical subspace is selected by commuting projectors

$\mathbb{Z}_2$38

with

$\mathbb{Z}_2$39

and the gauging map satisfies $\mathbb{Z}_2$40. The same construction is related to a continuous non-invertible cosine symmetry, with $\mathbb{Z}_2$41 realizing the gauging map (Seifnashri et al., 4 Mar 2025).

Finally, for arbitrary finite groups, repeated gauging of boundary Rep$\mathbb{Z}_2$42 symmetries reconstructs the full non-Abelian quantum double $\mathbb{Z}_2$43. The key ingredient is the commutative Frobenius algebra of functions on $\mathbb{Z}_2$44, internal to Rep$\mathbb{Z}_2$45, with multiplication and comultiplication

$\mathbb{Z}_2$46

Iterated gauging then yields a PEPS stabilized by the standard quantum-double projectors

$\mathbb{Z}_2$47

with anyons labeled by pairs $\mathbb{Z}_2$48, where $\mathbb{Z}_2$49 is a conjugacy class of $\mathbb{Z}_2$50 and $\mathbb{Z}_2$51 is an irrep of the corresponding centralizer (Blanik et al., 9 Dec 2025).

Across these literatures, gauging and disentangling procedures serve different immediate purposes—latent-factor separation, spectral recovery, duality derivation, logical measurement, or topological-state construction—but they repeatedly rely on the same technical pattern: encode symmetry or mixture structure into auxiliary variables, impose local consistency constraints, and then rewrite the problem in variables for which the relevant factors, sectors, or excitations are explicit.