Effective Potential Classification Scheme

• Effective potential classification scheme is a methodological pattern that reduces complex system dynamics to a diagnostic potential for distinguishing different physical regimes.
• It is applied to delineate heavy-hole/light-hole behavior in semiconductors, classify photon-sphere topologies in gravitational physics, and drive binary decision rules in statistical learning.
• In quantum mechanics and quantum field theory, it organizes arbitrary potentials and loop contributions into tractable local segments, simplifying analysis and interpretation.

Searching arXiv for recent and topic-relevant papers on effective potential classification schemes. “Effective potential classification scheme” denotes, in the literature surveyed here, a family of procedures in which an effective potential, a potential-like score, or an effective band-offset profile is used not merely for dynamics or energetics but for regime discrimination. In layered III–V heterostructures, the scheme separates heavy-hole and light-hole behavior under increasing valence-band mixing (Flores-Godoy et al., 2013). In spherically symmetric spacetimes, the effective potential for null geodesics and the topology of photon spheres distinguish black-hole, naked-singularity, and forbidden parameter regions (Afshar et al., 2024). In statistical learning, potential functions support direct binary decision rules, potential-energy classifiers, and the potential-potential transform [(0812.3145); (Shi, 2012); (Pokotylo et al., 2016)]. In quantum field theory, effective potentials classify loop sectors, vacuum structure, and the consistency of renormalization prescriptions (Martin, 2017, Quinto et al., 2021, Funakubo et al., 2023, Anikin, 2023). This suggests that the expression is best understood as a methodological pattern rather than a single domain-specific formalism.

1. Valence-band classification in coupled-hole heterostructures

In the semiconductor setting, the scheme is formulated for coupled holes in layered III–V heterostructures of the form $A/L-B/M-A/R$, treated within the multiband effective-mass approximation using the two-band Kohn–Luttinger model. The control parameter is the in-plane wave vector

$\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$

which acts as the band-mixing parameter: as $\kappa_{\textsc t}$ increases, heavy-hole and light-hole states couple more strongly and the effective potential profile changes (Flores-Godoy et al., 2013).

The effective-potential operator for the hole subspace is written as

$$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$

with

$W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$

and the effective-hole envelope satisfies

$$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$

At zero mixing, $\kappa_{\textsc t}\approx 0$, the profile reduces to the standard rectangular step $V(z)=\Theta V_{\mathrm{eff}}$, namely the usual fixed-height barrier/well picture.

Assuming plane-wave dependence along the growth axis $z$, the coupled-mode system becomes a quadratic eigenvalue problem in $k_z$,

$\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$0

The paper introduces a root-locus-like procedure: $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$1 is varied, the eigenvalues $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$2 are solved from the quadratic eigenvalue problem, and their trajectories are plotted in the complex plane. Real $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$3 correspond to propagating or oscillatory modes, whereas pure imaginary or complex $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$4 correspond to evanescent modes. The resulting locus is used to diagnose both mode character and the effective-potential landscape.

The central classification result is the separation between heavy holes and light holes. For heavy holes, the standard rectangular potential is generally reliable even as mixing increases: the effective barrier or well height changes only modestly, the profile often shifts almost rigidly, and a fixed-height rectangular approximation remains a good reference over a broad range of $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$5. For light holes, the same approximation becomes unreliable once mixing grows: the effective band offset changes significantly with $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$6, the barrier height becomes mutable, and the scattering potential must be treated as dynamically evolving with the in-plane kinetic energy.

As $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$7 increases, valence-band mixing increases, the terms $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$8 become more important, the effective barrier height can be reduced, and in some cases the roles of wells and barriers can partially invert. The metamorphosis is stronger for light holes than for heavy holes. The paper also notes directional dependence in the in-plane Brillouin-zone orientation: behavior along $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$9 and $\kappa_{\textsc t}$0 is often nearly the same, whereas along $\kappa_{\textsc t}$1 more pronounced deviations may appear. In practical modeling, this yields a sharp rule: fixed rectangular barriers are usually sufficient for heavy-hole band-structure, tunneling, and bound-state estimates, but light-hole calculations require a mutable effective band offset that tracks $\kappa_{\textsc t}$2.

2. Photon-sphere topology and black-hole parameter classification

A second use of effective-potential classification appears in gravitational physics, where the effective potential for null geodesics is used to categorize spacetime parameters. For a static, spherically symmetric metric, photon motion is described by

$\kappa_{\textsc t}$3

with

$\kappa_{\textsc t}$4

A circular null geodesic occurs at an extremum of the effective potential,

$\kappa_{\textsc t}$5

equivalently

$\kappa_{\textsc t}$6

The paper emphasizes that this effective potential depends only on spacetime geometry, not on the detailed properties of the incoming particle, and therefore serves as a classifier of the background itself (Afshar et al., 2024).

The physical interpretation is standard but operationally central. An unstable photon sphere corresponds to a maximum of $\kappa_{\textsc t}$7, whereas a stable photon sphere corresponds to a minimum. The proposed classification scheme combines horizon data, photon-sphere extrema, and topological charge. If the spacetime has an event horizon and only an unstable photon sphere outside it, the solution is classified as a black hole, typically with total topological charge $\kappa_{\textsc t}$8. If a stable photon sphere appears, especially as a minimum in the effective potential, the spacetime is classified as horizonless or as a naked singularity, with $\kappa_{\textsc t}$9 or sometimes $$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$0. If neither the metric function nor the effective potential yields a meaningful root or critical structure in the studied parameter range, the region is treated as forbidden.

The topological part of the construction follows Wei’s topological-photon-sphere method. A potential function

$$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$1

is used to define a two-component vector field $$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$2, a normalized field $$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$3, an antisymmetric superpotential $$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$4, and the topological current

$$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$5

With Duan’s $$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$6-mapping identity, the current is nonzero only at the zeros of $$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$7, namely at photon-sphere locations. The total topological charge is

$$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$8

with $$\widehat{W}_{\mathrm{eff}}= \begin{bmatrix} W_{11} & W_{12} & 0 & 0 \ W_{12}^{*} & W_{22} & 0 & 0 \ 0 & 0 & W_{22} & W_{12} \ 0 & 0 & W_{12}^{*} & W_{11} \end{bmatrix},$$9 the winding number of the $W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$0-th zero point. The classification is therefore simultaneously dynamical, via $W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$1, and topological, via the sum of winding numbers.

The paper applies this logic to Perfect Fluid Dark Matter models, charged AdS black holes with PFDM, Euler–Heisenberg black holes with and without PFDM, and nonlinear electrodynamics models with multiple horizons. PFDM can move a system from black-hole behavior to naked-singularity behavior in the simpler PFDM metric, whereas charge and AdS curvature can stabilize black-hole behavior over the studied range. The method also remains applicable in three-horizon and four-horizon cases, where photon-sphere topology and effective-potential shape continue to identify black-hole, naked-singularity, or forbidden regimes.

3. Potential-based statistical classification

In supervised learning, potential-based classification uses potential functions as decision scores rather than as physical energies. A simple binary classifier is defined by

$W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$2

where $W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$3 are positive training points, $W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$4 negative training points, $W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$5 positive weights, and $W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$6 distance functions. The rule is immediate: $W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$7 assigns $W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$8 to the positive class, $W_{11}=A_1 \kappa_{\textsc t}^{2}+V(z), \qquad W_{22}=A_2 \kappa_{\textsc t}^{2}+V(z),$9 to the negative class, and the decision surface is $$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$0. The method is described as computationally trivial because test-time evaluation requires only distance computations, inverse powers, and summation over training samples; the terms are mutually independent and therefore highly parallelizable (0812.3145).

The distance can be the $$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$1-norm,

$$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$2

or the weighted distance

$$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$3

with feature weights chosen from supervised relevance measures such as absolute univariate correlations, $$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$4 univariate $$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$5-values, or more sophisticated multivariate criteria. The paper reports that $$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$6-values works quite well. Training-point weights can be chosen as distances to the nearest opposite-class point, which smooths the decision surface by reducing the influence of borderline examples. In the limit $$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$7 and $$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$8, the procedure approaches nearest-neighbor classification.

The reported experiments make the scheme concrete. On the synthetic 4-by-4 checkerboard problem, a version without boundary weights gave 95% correct, while the weighted variant

$$\left[\widehat W_{\mathrm{eff}}-V_{\mathrm{eff}} I_4\right]\Psi(z)=O_4.$$9

produced 96.2% accuracy, compared with 97% correct after 100,000 iterations for the cited SVM benchmark. On DLBCL, the method reached 98.7% under LOOCV; on Prostate_Tumor, it reached 89.2%, improving to 96.1% with 20 selected features (0812.3145).

A related literature compares K-nearest neighbors with the Potential Energy method, where a class score is

$\kappa_{\textsc t}\approx 0$0

The paper studies Yukawa and Gaussian potentials,

$\kappa_{\textsc t}\approx 0$1

and concludes that KNN and PE methods have similar performance in general, while PE with Yukawa potential has worse performance than KNN when the density of the data is higher in the distribution of the database; PE with Gaussian potential shows similar behavior to KNN. The indicators used are correlation coefficients and information gain, with McNemar’s test as a paired-error comparison (Shi, 2012).

A more formal development is the pot-pot plot. For class $\kappa_{\textsc t}\approx 0$2, the potential is defined as the kernel density estimate multiplied by the class prior,

$\kappa_{\textsc t}\approx 0$3

and each data point is mapped to the vector

$\kappa_{\textsc t}\approx 0$4

For $\kappa_{\textsc t}\approx 0$5, the ordinary KDE classifier corresponds to the diagonal separator $\kappa_{\textsc t}\approx 0$6, but the paper instead applies either $\kappa_{\textsc t}\approx 0$7-nearest neighbors or the $\kappa_{\textsc t}\approx 0$8-procedure on the pot-pot plot. Under standard bandwidth conditions, $\kappa_{\textsc t}\approx 0$9-NN on the pot-pot plot is strongly Bayes-consistent; under the additional assumption $V(z)=\Theta V_{\mathrm{eff}}$0, the $V(z)=\Theta V_{\mathrm{eff}}$1-procedure is also strongly Bayes-consistent. Separate scaling generally improves performance over joint scaling, and regression-based separate scaling often retains most of the benefit of full separate search at much lower cost (Pokotylo et al., 2016).

4. Representation of arbitrary one-dimensional potentials

In one-dimensional quantum mechanics, the classification function of an effective potential is not primarily to separate external classes, but to encode an arbitrary $V(z)=\Theta V_{\mathrm{eff}}$2 as a structured sequence of solvable local elements. The proposed scheme replaces an arbitrary potential profile by a collection of ultra-short, piecewise-localized potentials. In each short segment, the exact profile is replaced by its zeroth-order value, the wavefunction is taken as nearly constant inside the segment, and the full problem is reconstructed through a transfer-matrix product (Rajendran et al., 2018).

For the ultra-short region, the simplifying assumption is

$V(z)=\Theta V_{\mathrm{eff}}$3

For bound states, integrating the Schrödinger equation across the narrow region yields the derivative-jump condition

$V(z)=\Theta V_{\mathrm{eff}}$4

which leads to

$V(z)=\Theta V_{\mathrm{eff}}$5

For scattering, the transmission and reflection probabilities are given explicitly; for the barrier case,

$V(z)=\Theta V_{\mathrm{eff}}$6

The general arbitrary-potential problem is then represented as a chain of such ultra-short segments with local parameters $V(z)=\Theta V_{\mathrm{eff}}$7, $V(z)=\Theta V_{\mathrm{eff}}$8, and breakpoints $V(z)=\Theta V_{\mathrm{eff}}$9. The total transfer matrix is written schematically as

$z$0

from which the transmission coefficient is

$z$1

while bound-state energies are obtained from the zeros of $z$2. For arbitrary $z$3 and position-dependent effective mass $z$4, the midpoint assignments

$z$5

supply the local data.

A key validation is that the rectangular barrier is reproduced in the short-width limit. The standard rectangular transmission formula reduces, for small $z$6, to

$z$7

which the paper identifies as exactly the ultra-short result. The computational claim is that for $z$8 regions the present method needs about $z$9 matrix multiplications, whereas piecewise-constant methods can require up to $k_z$0 and piecewise-linear methods $k_z$1. The scheme is therefore a classification or representation of potentials by a sequence of local effective elements with analytic matching rules.

5. Effective-potential organization in quantum field theory

In renormalizable quantum field theory, effective-potential classification becomes an organizing principle for loop structure, anomalous-dimension evolution, vacuum multiplicity, and renormalization consistency. At three loops, the effective potential of a general renormalizable theory in the $k_z$2 scheme and Landau gauge is expanded as

$k_z$3

The three-loop contribution is organized into the sectors

$k_z$4

and the 1PI vacuum diagrams are classified into topological families labeled $k_z$5. The paper identifies a finite set of 89 distinct loop integral functions spanning all three-loop contributions in a general renormalizable theory, thereby providing a universal dictionary from field content and couplings to the three-loop effective potential (Martin, 2017).

A different classification appears in massive $k_z$6-theory, where the mass term is treated as an interaction in an asymptotic expansion so that propagators remain massless and conformal methods remain applicable. The effective vertices are

$k_z$7

and connected vacuum diagrams are grouped into four classes: standard diagrams containing only $k_z$8-vertices, Type-I non-standard diagrams containing only $k_z$9, Type-II containing only $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$00, and Type-III mixed diagrams of the form $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$01. The vacuum $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$02-operation maps non-local operator Green functions to the corresponding vacuum integrations that generate the effective potential, providing an algebraic route from conformal-recursion data to effective-potential anomalous dimensions (Anikin, 2023).

Vacuum-structure classification is explicit in the massless Abelian Higgs model with an $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$03-component complex scalar field. Using four-loop RG functions and leading-log recursion, the paper constructs a five-loop approximation to the effective potential and maps the parameter space $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$04 into three qualitative regions: a yellow region with three real positive solutions for $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$05 and therefore three possible non-symmetric vacua, a brown region with one solution and one broken-symmetry vacuum, and a blue region with no acceptable solution so that dynamical symmetry breaking does not occur. For

$\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$06

the reported positive solutions are

$\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$07

The effective potential is thus used as a classifier of vacuum multiplicity in a classically scale-invariant gauge theory (Quinto et al., 2021).

At finite temperature, effective-potential classification becomes a classification of renormalization prescriptions. The cited work distinguishes fixed-order, unresummed potentials; thermally resummed potentials with ordinary $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$08 RG; thermally resummed RG-improved potentials with temperature-dependent beta functions; and further RG-improved thermal potentials with optimized $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$09. The key point is that thermal resummation changes the divergence structure, so ordinary $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$10 running is not fully consistent order by order. By defining counterterms and beta functions in the resummed perturbation theory itself, the authors restore RG invariance order by order and then use the running potential with optimized

$\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$11

to incorporate important higher-order thermal effects (Funakubo et al., 2023).

6. Recurrent structure, interpretive cautions, and comparative summary

Across these disparate settings, the same structural pattern recurs: a complicated object is projected onto a reduced potential-like quantity, and classification is then performed in that reduced space. In semiconductor heterostructures, the reduced object is the effective band offset seen by coupled holes. In the gravitational case, it is the null-geodesic effective potential together with the winding structure of photon-sphere zeros. In statistical learning, it is a potential score, class potential, or potential-potential coordinate. In quantum mechanics, it is a chain of local effective potentials. In quantum field theory, it is the loop-organized or RG-improved effective potential that classifies vacuum sectors or renormalization regimes. This suggests a common logic: the effective potential is not merely an output of the theory but a compressed diagnostic of admissible behavior.

Domain	Potential quantity	Resulting classes or regimes
Coupled holes	$\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$12, mutable band offset	heavy-hole rectangular regime; light-hole mutable-offset regime
Photon spheres	$\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$13 for null geodesics, TTC	black hole; naked singularity; forbidden region
Statistical learning	$\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$14, $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$15, $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$16	class labels or pot-pot separators
Abelian Higgs / thermal QFT	RG-improved $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$17	no DSB; one broken vacuum; multiple broken vacua; consistent or inconsistent RG classes

Several misconceptions are explicitly contradicted by the literature. A fixed rectangular barrier is not universally valid in valence-band transport: it remains reliable for heavy holes over a broad range of $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$18, but it is no longer valid for light holes once mixing grows. In the photon-sphere classification, a minimum of $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$19 is not a marker of an ordinary black-hole exterior; it signals a stable photon sphere and is associated with horizonless or naked-singularity behavior. In potential-based statistical classification, summing contributions from all training samples does not eliminate dependence on metric choice, bandwidth, interaction radius, or feature weighting; the cited studies emphasize parameter tuning, density sensitivity for Yukawa PE, and the importance of weighted distances or separate scaling. In thermally resummed quantum field theory, ordinary $\vec{\kappa}_{\textsc t} = k_x \hat e_x + k_y \hat e_y,$20 running is not automatically compatible with the resummed divergence structure.

The comparative significance of these schemes lies in how they transform classification into a problem of geometry in a reduced space. In some domains, such as the coupled-hole problem and black-hole parameter classification, the reduced geometry is directly physical: the shape of the effective potential and the location of extrema determine the operative regime. In statistical learning, the reduced geometry becomes an algorithmic decision surface. In quantum field theory, it becomes a classification of vacuum branches, loop sectors, and renormalization consistency. The term “effective potential classification scheme” therefore names a recurring research strategy: classify the system by the structure of the effective potential rather than by the full original dynamics.