Iterative Perturbation Theory (IPT) Overview

• Iterative Perturbation Theory (IPT) is an iterative method that approximates many-body quantum problems by interpolating between weak- and strong-coupling limits.
• In DMFT, IPT constructs the impurity self-energy using second-order perturbation theory, ensuring alignment with atomic, weak-coupling, and high-frequency regimes.
• Variations like MO-IPT and IPT+parquet balance low computational cost with improved accuracy, though they require benchmarking against controlled methods.

Searching arXiv for recent and foundational papers on Iterative Perturbation Theory across its main usages. Iterative Perturbation Theory (IPT) denotes a family of perturbative schemes in which an approximate solution is obtained through an iterative or interpolative construction rather than through a bare low-order expansion alone. In contemporary many-body physics, the term most commonly refers to impurity solvers used within dynamical mean-field theory (DMFT), where the self-energy is built from second-order perturbation theory and constrained to reproduce exact limits such as the weak-coupling, atomic, and high-frequency regimes (Mizuno et al., 2021). In other parts of quantum mechanics and numerical analysis, the same label has been used for fixed-point or non-power-series perturbative iterations for eigenvalue problems (Smerlak, 2021, Kenmoe et al., 2020, 2020.14702). Because these usages are not identical, precise context is essential.

1. Terminological scope and historical uses

Within DMFT, IPT is a low-cost approximate impurity solver that constructs the self-energy by interpolating between the weak-coupling (perturbative) and strong-coupling (atomic) limits (Yamada et al., 4 Jun 2026). This condensed-matter usage includes the single-orbital formulation, the Kajueter-Kotliar extension away from half filling, multi-orbital generalizations, and parquet-enhanced variants (Mazzocchi et al., 17 Apr 2026, Dasari et al., 2015, Mizuno et al., 2021).

Outside DMFT, the term has also been used for iterative solution methods in time-independent quantum-mechanical perturbation theory. In one line of work, the Schrödinger equation is rewritten in a matrix form that leads to a non-series iterative update for expansion coefficients, with explicit treatment of two-fold degeneracies and optional use of a synthetic Hamiltonian (Kerley, 2013). In another, a relaxed fixed-point iterative scheme is introduced to obtain convergent expressions for challenging ground-state wavefunctions at arbitrarily strong coupling, explicitly contrasting such constructions with Rayleigh-Schrödinger power series (Smerlak, 2021). A related numerical linear-algebra formulation, also termed IPT, interprets near-diagonal eigenvalue problems through a perturbatively inspired fixed-point map acting on eigenvectors or the full eigenvector matrix (Kenmoe et al., 2020).

A separate cosmological framework, integrated Perturbation Theory, is abbreviated as iPT rather than IPT and concerns biased tracers and large-scale structure rather than impurity models or quantum eigenproblems (Yokoyama et al., 2013). A common misconception is therefore to treat all appearances of “IPT” as instances of a single formalism; the literature instead contains several method families linked by perturbative iteration but differing in object, domain, and convergence rationale.

2. IPT as a DMFT impurity solver

In DMFT, the lattice problem is mapped onto an effective impurity model embedded in a self-consistent bath. IPT enters at the impurity-solver stage by approximating the impurity self-energy with a rational expression built from the second-order self-energy computed from a non-interacting Green’s function $G_0$. In the conventional form summarized for multiband systems, the correlation part of the self-energy, after subtracting the static Hartree-Fock contribution, is written as

$$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$

with $A$ and $B$ fixed to satisfy exact limits and with a pseudo chemical potential $\mu_0$ chosen to enforce consistency with either total or orbital-resolved filling (Yamada et al., 4 Jun 2026).

The Kajueter-Kotliar formulation extends IPT away from half filling. In the nonequilibrium extension, the retarded self-energy Ansatz is

$$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$

with

$$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$

for $\mathrm{X}=\{\mathrm{R},\mathrm{K}\}$, and

$$A = \frac{n(1-n)}{n_0(1-n_0)}, \qquad B = \frac{(1-n)U+\epsilon_f+\mu_0}{n_0(1-n_0)U^2}.$$

Here $n$ is the interacting occupation, $$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$0 is the Weiss-field occupation, and $$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$1 is an auxiliary chemical potential adjusted to match impurity and bath occupations (Mazzocchi et al., 17 Apr 2026).

The single-orbital half-filled IPT self-energy also appears in particularly compact form in studies of two-particle correlations: $$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$2 which at particle-hole symmetry becomes

$$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$3

This expression is simple enough to permit analytical treatment of the associated two-particle vertex and the DMFT Jacobian (Loon, 2021).

The principal technical appeal of IPT in DMFT is unchanged across these variants: low numerical cost, direct access to real-frequency quantities in some implementations, and an interpolative structure intended to retain physically important limits (Dasari et al., 2015).

3. Multi-orbital, parquet, and nonequilibrium extensions

The need for extensions arises because standard IPT is often insufficient in multiband settings or away from equilibrium. A multi-orbital iterative perturbation theory (MO-IPT) was developed for $$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$4-fold degenerate and non-degenerate Anderson impurity models and then combined with DMFT for lattice models and DFT+DMFT calculations. Its self-energy Ansatz for orbital $$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$5 is

$$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$6

with $$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$7 and $$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$8 fixed from high-frequency and atomic constraints, and with $$\Sigma^{\mathrm{cr}}(\omega_n) = \left[I - B \Sigma^{\mathrm{2nd}}(\omega_n)\right]^{-1} A \Sigma^{\mathrm{2nd}}(\omega_n),$$9 fixed by Luttinger’s theorem at $A$0 (Dasari et al., 2015).

A conceptually distinct extension is IPT+parquet, introduced by combining IPT with a simplified parquet treatment of two-particle scattering channels. In this construction, the correlation self-energy is upgraded from a bare-$A$1 second-order diagram to one built from a parquet-dressed local two-particle vertex: $A$2 with

$A$3

and

$A$4

The simplified parquet approach is used to keep computations tractable, and the pseudo chemical potential is made orbital-dependent through the constraint $A$5 (Yamada et al., 4 Jun 2026).

For nonequilibrium steady states, the Kajueter-Kotliar construction has been reformulated in Keldysh space. The key additional object is the Keldysh component of the self-energy,

$A$6

together with a self-consistency loop enforcing the “IPT-$A$7” approximation $A$8 (Mazzocchi et al., 17 Apr 2026).

A steady-state nonequilibrium IPT has also been used for Mott insulators in static electric fields with optical phonons and electronic baths. There the local Hubbard self-energy is constructed directly from Keldysh lesser and greater Weiss functions,

$A$9

with retarded and Keldysh components obtained in the standard way (Mazzocchi et al., 2023).

4. Accuracy, benchmarks, and known failure modes

The empirical status of IPT is strongly regime dependent. In multiband DMFT, conventional IPT cannot correctly describe the competition between intra- and inter-orbital interactions $B$0 and $B$1, because the $B$2- and $B$3-dependent terms in the self-energy differ only by constant factors; it therefore cannot capture orbital fluctuations or the atomic-configuration physics associated with $B$4 or $B$5 (Yamada et al., 4 Jun 2026). This limitation motivated IPT+parquet, which was shown to capture competition between orbital fluctuation channels that conventional IPT cannot capture (Yamada et al., 4 Jun 2026).

Benchmarking against numerically exact or controlled impurity solvers is correspondingly central. The 2026 validation study compares IPT+parquet, two IPT variants, CT-QMC, and exact diagonalization in two-orbital problems. For a two-orbital square lattice, IPT+parquet significantly outperforms conventional IPT, especially for strong $B$6 and near half-filling, whereas IPT with orbital-resolved $B$7 already improves substantially and can be close to numerically exact results in weak coupling (Yamada et al., 4 Jun 2026). For a two-orbital Bethe lattice, exact diagonalization shows a dome-like stabilization of the metallic state near $B$8; conventional IPT completely fails to reproduce this stabilization, whereas IPT+parquet captures the qualitative and partly quantitative effect, including cases with $B$9 apart from some convergence issues for $\mu_0$0 (Yamada et al., 4 Jun 2026).

MO-IPT has been benchmarked extensively against hybridization-expansion CTQMC. The reported pattern is that agreement becomes better as one moves away from particle-hole symmetry; in degenerate multi-orbital models, performance is worse near half-filling and/or particle-hole symmetry, while away from these conditions the agreement is described as excellent (Dasari et al., 2015). In DFT+DMFT calculations for $\mu_0$1, the calculated spectral function shows the expected three-peak structure and the photoemission spectra agree closely near the Fermi level and lower Hubbard band, with discrepancies mostly in the upper Hubbard band (Dasari et al., 2015).

Away from half-filling at strong coupling, the standard IPT-$\mu_0$2 closure can fail. On the 3d FCC lattice, it was shown that the standard implementation fails when the interaction strength is much larger than the bandwidth, motivating IPT-$\mu_0$3, which replaces the closure condition $\mu_0$4 by the requirement that double occupancy be correct (Arsenault et al., 2012). The modified scheme recovers the Fermi liquid ground state away from half-filling and yields benchmarked Fermi liquid parameters, density of states, chemical potential, energy, specific heat, resistivity, and optical conductivity in good agreement with CTQMC on the FCC lattice (Arsenault et al., 2012).

In nonequilibrium impurity transport, nonequilibrium KK-IPT was benchmarked against the auxiliary master equation approach (AMEA). In equilibrium it reproduces AMEA results for different fillings with high accuracy at the level of both spectral properties and electron densities; out of equilibrium it shows very good agreement for moderate temperatures and biases, while remaining numerically stable in low-temperature, low-bias regions where AMEA becomes less reliable (Mazzocchi et al., 17 Apr 2026). For Mott insulators in a static electric field with optical phonons, IPT yields results qualitatively in good agreement with AMEA but fails to reproduce some correlation effects, including subtle resonances and an accurate Mott gap (Mazzocchi et al., 2023).

The following summary, restricted to formulations explicitly reported in the source material, captures the comparative status of several DMFT impurity-solver variants.

Method	Captures orbital fluctuations?	Computational cost
CT-QMC, ED	Yes	High
IPT (global $\mu_0$5)	No	Low
IPT ($\mu_0$6)	Only weak orbital fluctuations	Very low
IPT+parquet	Yes	Moderate

This suggests that the phrase “IPT” in the DMFT literature often denotes not a single approximation but a hierarchy of closely related closures whose reliability depends on the fluctuation channels one expects to dominate.

5. Mathematical structure of IPT-DMFT

Recent work has subjected IPT-DMFT to explicit mathematical analysis. For the finite Hubbard model, the DMFT equations can be formulated using one-body time-ordered Green’s functions, self-energies, and Anderson impurity models, with the DMFT approximation taking the full self-energy to be block diagonal across clusters (Cancès et al., 2024). In the Matsubara formalism, the impurity IPT map is written as

$\mu_0$7

where

$\mu_0$8

in the half-filled setting summarized in that analysis (Cancès et al., 2024).

A central structural result is that the relevant Green’s functions, self-energies, and hybridization functions are Pick functions, and that the single-site paramagnetic translation-invariant DMFT loop can be reformulated as an operator on probability measures on $\mu_0$9 (Cancès et al., 2024). Within that measure-theoretic setting, the IPT-DMFT map is continuous, compact, and maps the admissible set into itself, allowing a fixed-point argument to prove existence of a solution for any set of physical parameters under the stated assumptions (Cancès et al., 2024). The same work also proves that, except in trivial limits, the single-site translation-invariant IPT-DMFT equations do not admit solutions for finite-rank hybridization functions; solutions must therefore be sought among genuine infinite-dimensional Pick functions (Cancès et al., 2024).

A further step concerns discretization on a finite Matsubara grid $$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$0, $$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$1. The discretized equations read

$$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$2

with $$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$3 a rational approximation of the continuous IPT functional (Cancès et al., 27 May 2025). Existence of solutions to the discretized system is proved only in a parameter range depending on $$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$4, and uniqueness is proved in a smaller range using a contraction argument (Cancès et al., 27 May 2025). For bipartite systems with particle-hole symmetry, the discretized equations have purely imaginary solutions and reduce to a real algebraic system of $$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$5 equations in $$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$6 variables; the $$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$7 and $$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$8 cases can be analyzed explicitly for the Hubbard dimer (Cancès et al., 27 May 2025).

These results clarify a practical issue in numerical DMFT. The continuous formulation admits general existence results in the Pick-function setting, whereas the discretized formulation can violate the causal structure outside its admissible parameter range or for too small $$\Sigma^{\mathrm{R}}(\omega) = nU + (1-\Sigma^{\mathrm{R}}_{\mathrm{B}}(\omega))^{-1} \Sigma^{\mathrm{R}}_{\mathrm{A}}(\omega),$$9 (Cancès et al., 27 May 2025). A plausible implication is that some numerical pathologies attributed informally to “IPT failure” may arise from discretization and analyticity issues rather than from the perturbative Ansatz alone.

6. Two-particle structure, reinterpretations, and other perturbative iterations

IPT has also been revisited at the two-particle level. Because the half-filled IPT self-energy is an explicit cubic functional of $$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$0, one can compute the functional derivative $$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$1 analytically. At particle-hole symmetry,

$$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$2

and the DMFT Jacobian can be written explicitly in Matsubara space (Loon, 2021). This enables analytical studies of the metal-insulator transition in terms of the eigenstructure of the susceptibility and of the forward-iteration stability. At the same time, the approximate nature of IPT precludes an interpretation of the transition in terms of a Landau free energy functional, because the IPT vertex in Matsubara frequency space is not symmetric under exchange of frequencies, unlike the exact impurity vertex (Loon, 2021).

The 2021 IPT+parquet development proposed a reinterpretation of conventional IPT itself: rather than viewing IPT only as an interpolation between weak- and strong-coupling limits, it can be regarded as mimicking particular frequency structures of the exact full vertex, notably the “cross” and “central” structures important in the strong-correlation regime (Mizuno et al., 2021). On that reading, conventional IPT misses the “diagonal” structures associated with two-particle fluctuations, which explains why parquet-derived corrections improve multiband performance (Mizuno et al., 2021).

Beyond many-body impurity theory, non-DMFT perturbative iterations are relevant because they illuminate what “iterative perturbation theory” can mean more generally. Kerley’s reformulation of time-independent perturbation theory leads to an iterative update for coefficients

$$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$3

with a corresponding iterative energy update, and can also be combined with a synthetic Hamiltonian $$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$4 to improve convergence (Kerley, 2013). Relaxed fixed-point iterations of the form

$$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$5

have been used to obtain convergent expressions for wavefunctions and energies in problems whose Rayleigh-Schrödinger series diverge, including quartic, sextic and octic anharmonic oscillators and the Herbst-Simon Hamiltonian (Smerlak, 2021). For near-diagonal matrices, a perturbatively inspired fixed-point map

$$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$6

defines an eigenvalue algorithm termed IPT, for which a sufficient condition for linear convergence is

$$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$7

(Kenmoe et al., 2020).

These quantum-mechanical and numerical usages are not interchangeable with DMFT-IPT. They do, however, share a family resemblance: perturbative information is reorganized into an iterative map whose fixed point is intended to capture behavior beyond a naive truncated series.

7. Applications and common points of debate

The main application domain of IPT in current condensed-matter research is DMFT for correlated electrons. It has been used for single-band and multi-band Hubbard models, Bethe and square lattices, bilayer systems, frustrated lattices such as the FCC lattice, and real-material calculations through DFT+DMFT, including $$\Sigma^{\mathrm{X}}_{\mathrm{A}}(\omega)\equiv A \tilde{\Sigma}^{\mathrm{X}}(\omega),\qquad \Sigma^{\mathrm{X}}_{\mathrm{B}}(\omega)\equiv B \tilde{\Sigma}^{\mathrm{X}}(\omega),$$8 [(Dasari et al., 2015); (Mizuno et al., 2021); (Arsenault et al., 2012)]. More recently, IPT has been adapted to topological heavy-fermion models of twisted bilayer graphene and twisted symmetric trilayer graphene, where it enables momentum- and energy-resolved spectral calculations over broad temperature and filling ranges and captures Mott-Hubbard bands, Kondo-like resonance, and finite lifetime broadening more faithfully than the Hubbard-I approximation (Călugăru et al., 22 Sep 2025).

Three recurring debates organize the literature.

First, there is the question of validity away from half-filling or in multiorbital settings. The record is mixed rather than uniformly favorable: standard implementations can fail badly near half-filling at strong coupling or under strong orbital competition, while orbital-resolved chemical-potential constraints, double-occupancy closures, or parquet-dressed vertices materially improve performance [(Arsenault et al., 2012); (Yamada et al., 4 Jun 2026)].

Second, there is the issue of computational efficiency versus control. IPT remains attractive because it is computationally inexpensive and often produces real-frequency information directly, whereas CT-QMC, ED, and AMEA are numerically exact or controlled but substantially more costly (Dasari et al., 2015, Mazzocchi et al., 17 Apr 2026). The literature consistently presents IPT not as a controlled expansion in the rigorous sense, but as an interpolation scheme whose success must be benchmarked in the regime of interest (Yamada et al., 4 Jun 2026).

Third, there is the relation between one-particle accuracy and two-particle consistency. IPT can reproduce many one-particle observables well in favorable regimes, yet its approximate vertex may violate symmetries that matter for response functions, susceptibilities, and free-energy interpretations (Loon, 2021). This suggests that “good spectra” do not automatically imply a faithful two-particle theory.

Taken together, the modern literature presents IPT as a technically diverse class of perturbative iterations. In DMFT, it remains a central approximate impurity solver because of its favorable cost profile and extensibility, but its reliability is sharply problem dependent and increasingly tied to how vertex feedback, filling constraints, and symmetry structure are incorporated (Mizuno et al., 2021, Mazzocchi et al., 17 Apr 2026, Yamada et al., 4 Jun 2026).