A Taxonomy of Exchange Statistics from Configuration-Space Topology: Non-Abelian Flux Attachment and a Pre-Registered Thermal Hall Projection

#1. Introduction

The statistical behavior of identical particles is not an independent postulate of quantum mechanics but a consequence of the topology of the configuration space. When $N$ identical particles move on a manifold $M$, the physically distinguishable configurations form the orbit space

where $\Delta$ is the coincidence set, and quantization on this space requires a unitary representation of its fundamental group $\pi_1(C_N(M))$. In three or more dimensions $\pi_1(C_N(M)) \cong S_N$, whose abelianization is $\mathbb{Z}_2$, yielding exactly two exchange phases. In two dimensions $\pi_1(C_N(\mathbb{R}^2)) \cong B_N$, the braid group, whose abelianization is $\mathbb{Z}$, permitting any continuous exchange phase $e^{i\theta}$ and, through higher-dimensional representations, non-Abelian anyons whose braiding acts by matrices on a degenerate ground-state manifold.

This paper treats flux attachment as a homorphism from the braid group into a target group $G$ — possibly non-Abelian — and derives a taxonomy of the resulting exchange statistics from computable topological data. The program builds on the configuration-space-topology (CST) framework of Ref. [9], which proposes that exchange statistics be treated as the primary object derived from $\pi_1$, with the familiar $\pm 1$ boson/fermion dichotomy appearing as an abelianized "shadow" of the full representation data; Refs. [10] and [11] supply auxiliary machinery from the same program.

The practical motivation is predictive. Non-Abelian anyon phases support topologically protected degeneracies and, when the phase is chiral, a quantized thermal Hall conductance $\kappa_{xy}$ proportional to the chiral central charge $c_-$. Because $\kappa_{xy}$ is insensitive to disorder and couples only to the topological sector, it is the cleanest experimental handle on non-Abelian order. We derive explicit numerical targets for $\kappa_{xy}/T$ for candidate classes in the taxonomy and frame a falsifiable, pre-registered projection on an 18-month horizon.

#2. Background and Related Work

Configuration spaces of Artin and Coxeter groups [1]. This work reviews topological and algebraic aspects of Artin and Coxeter groups, computes the Schwartz genus of the covering associated to the orbit space for all affine Artin groups, and gives a partial computation of the cohomology of the braid group. The braid group $B_N$ is itself the Artin group of type $A_{N-1}$, so these cohomological computations constrain the possible characteristic classes of flat $G$-bundles over braid configuration spaces — precisely the objects that encode flux attachment. The Schwartz genus of the orbit covering bounds how many sheets of the unordered configuration space a flux-carrying excitation can traverse coherently.

Persistence modules and interleaving distances [2]. This work studies classes of persistence modules arising in topological data analysis and the interleaving distance between them. We invoke it methodologically: the taxonomy assigns to each flux-attachment class a persistence-like grading (the winding number), and the interleaving formalism provides a natural metric on the space of such gradings, stating when two attachment rules are topologically indistinguishable at finite braid length. It also supplies the language for a robustness criterion: the taxonomy axes are physical only insofar as $G_N$ is stable under sample perturbations.

Hawaiian groups and wild topology [3]. This work determines the structure of the $n$-dimensional Hawaiian group of the $m$-dimensional Hawaiian earring space for $1 \leq m \leq n$, and studies Hawaiian groups under products, cones, covering spaces, and locally trivial bundles. Although our physical configuration spaces are locally tame, the Hawaiian-group formalism is the correct diagnostic for a failure mode we must exclude: on configuration spaces with accumulating punctures (e.g., finite-density anyon gases in the thermodynamic limit), the fundamental group develops wild, non-discrete behavior, and flux attachment by ordinary homomorphisms breaks down. Section 6 treats this as a boundary of validity.

Classifying spaces for families of subgroups [4]. This work generalizes the Milnor, Gelfand–Fuks, and Segal existence theorems for classifying spaces of principal $G$-bundles to $G$-spaces with torsion, constructing universal spaces indexed by orbit types and cover cardinality. A flux-attachment rule is precisely a principal-$G$ bundle structure over the braid configuration space; the classifying-space machinery guarantees that, for any topological flux group $G$ and any admissible family of stabilizer subgroups, a universal flux-bundle space exists, so the taxonomy is not vacuous — every class it labels is realizable by some bundle.

Permutation statistics for block designs [5]. This work introduces a nonparametric permutation test statistic for complete block designs, characterizes its region of existence, and derives saddlepoint approximations for tail probabilities. We cite it for its method of boundary analysis — our taxonomy has an analogous boundary, the Abelian/non-Abelian transition at representation dimension $r = 2$ — and, following one draft's usage, for the inferential framework of the pre-registered projection: a permutation-based test statistic with an explicitly stated null region, used in Section 5 to define confirmation versus refutation of the measured $\kappa_{xy}/T$.

Tight contact structures on $\Sigma_2 \times I$ [6]. This work classifies tight contact structures on the closed genus-2 surface times an interval with a boundary condition forcing the contact Euler class to vanish on each boundary component, proving uniqueness of the non-chiral product structure. Genus-2 surfaces are the natural stage for doubled non-Abelian anyon theories, and the cut-and-paste uniqueness result suggests that the topological sector relevant to our $c_- = 3/2$ candidate class is rigid: there is essentially one way to glue the flux sectors, strengthening the robustness of the thermal Hall prediction. The contact-topological handedness classification also serves as the low-dimensional analogue of chirality: orientation reversal of $M$ flips the sign of the thermal Hall response.

Minimal topological groups and the Roelcke uniformity [7]. This work studies minimal topological groups — those admitting no strictly coarser Hausdorff group topology — and Roelcke-precompact non-Abelian groups. Compact flux groups such as $SU(2)$ and its finite subgroups are Roelcke-precompact, and minimality ensures that the flux-attachment homomorphism $\varphi: B_N \to G$ cannot be degraded by coarsening the topology of $G$: the discrete quantum-statistics content of the attachment is topologically locked. This underwrites the claim that the taxonomy is stable under physical regularization, and informs the discussion of the $N \to \infty$ limit, where the discrete taxonomy must be replaced by a coarse structure.

String topology and homotopy invariance [8]. This work proves that the Chas–Sullivan string topology loop product and string bracket on the homology of the free loop space $LM$ are homotopy invariants. Worldline braiding of anyons is naturally a loop-space phenomenon: the loop product is the homological shadow of concatenating excitation worldlines, and its homotopy invariance guarantees that the statistics classification — which depends only on the homotopy class of braids — is invariant under continuous deformations of the anyon trajectories, as physics requires. The loop space $LC_N(M)$ also carries the transgressed form of the exchange data: holonomies around loops in $C_N(M)$ become integrated observables on the loop space, the mechanism by which the group-theoretic axes couple to the field-theoretic axis (D).

The CST derivation program [9]–[11]. Ref. [9] formulates exchange statistics as an "exchange scalar" derived from configuration-space topology, with the $\pm 1$ boson/fermion shadow in $d \geq 3$ as a special case, and pre-registers a derivation program for extending the calculus to non-Abelian settings. Ref. [10] supplies a derivation checklist (each axis derived from a stated presentation, each response coefficient from a stated field theory), and Ref. [11] contributes a self-referential consistency requirement: the taxonomy must classify its own derivation procedure. This paper executes one branch of that program.

#3. Methods

#3.1 Configuration spaces and their fundamental groups

Let $M$ be a connected surface and define the ordered and unordered configuration spaces

The covering $F_N(M) \to C_N(M)$ has deck group $S_N$. For $M = \mathbb{R}^d$:

where $B_N$ is generated by elementary braids $\sigma_1, \dots, \sigma_{N-1}$ subject to

For $M$ a general surface, $G_N$ is a surface braid group, modified by $\pi_1(M)$ and by boundaries.

#3.2 Flux attachment as a homomorphism

A flux-attachment rule is a homomorphism

where $G$ is the flux group (e.g., $U(1)$ for Abelian anyons, $SU(2)$ or a finite quotient for non-Abelian anyons). Because $B_N$ is generated by the $\sigma_i$, $\varphi$ is determined by the tuple $(g_1, \dots, g_{N-1})$, $g_i = \varphi(\sigma_i) \in G$, subject to the braid relations. The exchange statistics of two particles in flux sector $a$ is the conjugacy class $[a]$ of $a = \varphi(\sigma_i)$ in $G$; the statistics is Abelian iff $[a]$ is central, i.e., $\varphi(\sigma_i) \in Z(G)$.

#3.3 The four-axis taxonomy

We assign to any physical exchange behavior a quadruple $(G_N, r, \mathrm{factor}, c_-)$:

• Axis (A): Exchange group. $G_N = \pi_1(C_N(M))$: $S_N$ for $\dim M \geq 3$, $B_N$ for $M = \mathbb{R}^2$.
• Axis (B): Representation dimension. A statistics assignment is $\rho: G_N \to U(r)$. For $r = 1$, $\rho$ factors through the abelianization $G_N^{\mathrm{ab}}$; the particle is an Abelian anyon with exchange phase $\theta = \arg \rho(\sigma_i) \in \mathbb{R}/2\pi\mathbb{Z}$. For $r \gt 1$ with non-commuting image, exchanges act by matrices on a degenerate internal space: a non-Abelian anyon.
• Axis (C): Finite factoring. If $\rho$ factors through a finite quotient $q: G_N \to F$ (e.g., $B_N \to S_N$ via $\sigma_i \mapsto (i\ i{+}1)$), the statistics is parastatistics-like: exchanges permute finite internal types with no braiding phase. If $\rho$ does not factor finitely and $r \gt 1$, the statistics is genuinely braided non-Abelian. Axis (C) exists precisely to separate these two $r \gt 1$ cases.
• Axis (D): Chiral central charge. If $(G_N, \rho)$ is realized by a gapped topological phase with topological order $\mathcal{C}$, the chiral central charge $c_-$ of $\mathcal{C}$ fixes the thermal Hall response

Axis (D) is not determined by (A)–(C) alone: distinct modular tensor categories can share the same braid group representation but differ in $c_-$. The taxonomy is therefore a surjection, not a bijection, from phases to quadruples; it classifies exchange behavior, not topological order.

#3.4 Thermal Hall conductance as the observable

For a chiral topological phase with chiral central charge $c_-$,

with $k_B = 1.380649 \times 10^{-23}\ \mathrm{J\,K^{-1}}$ and $h = 6.62607015 \times 10^{-34}\ \mathrm{J\,s}$, both exact 2019-SI defined values. Each family in the taxonomy predicts a value of $c_-$ via its flux-fusion structure; we compute these in Section 4.

#4. Analysis

#4.1 Axis (A): explicit computations

$N = 2$, $M = \mathbb{R}^2$. $C_2(\mathbb{R}^2) \simeq \mathbb{R}^2 \times (\mathbb{R}^2 \setminus \{0\})$, so

generated by the single braid $\sigma_1$ with no relations. The square $\sigma_1^2$ is the double braid (monodromy of one particle around the other). This is the entire group-theoretic content of anyonic statistics in two dimensions.

$N = 2$, $\dim M \geq 3$. $\pi_1(\mathbb{R}^d \setminus \{0\}) \cong \pi_1(S^{d-1})$, trivial for $d \geq 3$; the exchange group is $S_2 \cong \mathbb{Z}/2\mathbb{Z}$. The relation $\sigma_1^2 = 1$ collapses the anyonic continuum $\theta \in [0, 2\pi)$ to the two points $\theta = 0, \pi$ — the topological origin of the boson/fermion dichotomy in three and higher dimensions.

$N = 3$, $M = \mathbb{R}^2$. $G_3 = B_3 = \langle \sigma_1, \sigma_2 \mid \sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2 \rangle$. The abelianization is $B_3^{\mathrm{ab}} \cong \mathbb{Z}$ (both generators map to $1$; the braid relation is homogeneous and vanishes in the abelianization). The center is $Z(B_3) = \langle (\sigma_1 \sigma_2)^3 \rangle \cong \mathbb{Z}$. The finite quotient $B_3 \to S_3$ exists because transpositions satisfy the braid relation: $(12)(23)(12) = (13) = (23)(12)(23)$.

#4.2 Axis (B): abelianization and the exchange phase

Step 1. Abelianization imposes commutativity: the braid relation becomes $\sigma_i^2 \sigma_{i+1} = \sigma_{i+1} \sigma_i^2$, which is trivial; far-commutativity is already trivial. Hence no relation survives, and

Step 2. A one-dimensional unitary representation is $\rho: B_N \to U(1)$ with $\rho(\sigma_i) = e^{i\theta}$ for all $i$ (the braid relations force equal phases). The exchange of two identical anyons multiplies the wavefunction by $e^{i\theta}$; the monodromy $\sigma_1^2$ gives $e^{2i\theta}$. For $\theta = \pi \chi_1$ with $\chi_1 \in \mathbb{Z}$ we recover bosons ($\chi_1$ even) and fermions ($\chi_1$ odd); for general real $\theta$ we obtain Abelian anyons. In $d \geq 3$, $S_N^{\mathrm{ab}} \cong \mathbb{Z}_2$ forces $\chi_1 \in \{0, 1\}$ — the CST exchange-scalar dichotomy of [9], i.e., the $\pm 1$ shadow.

Non-Abelian anyons. For $r \gt 1$, the image $\rho(B_N) \subset U(r)$ need not commute. The minimal illustrative case is a representation of $B_3$ in which $\sigma_1, \sigma_2$ act on a two-dimensional fusion space by matrices whose products depend on order: $\rho(\sigma_1)\rho(\sigma_2) \neq \rho(\sigma_2)\rho(\sigma_1)$. The physical consequence is order-dependent fusion: bringing a third particle to a pair can yield different outcome channels depending on the braid history — the operational definition of non-Abelian statistics.

Axis (C) in action. The representation $\rho: B_3 \to S_3 \to U(3)$ (permutation matrices) has $r = 3 \gt 1$ but factors finitely; exchanges permute internal types without phase. This is parastatistics, topologically distinct from genuine non-Abelian braiding even though both have $r \gt 1$.

#4.3 Axis (D): the thermal Hall conductance quantum

Input numbers and sources. $k_B = 1.380649 \times 10^{-23}\ \mathrm{J\,K^{-1}}$ and $h = 6.62607015 \times 10^{-34}\ \mathrm{J\,s}$, exact 2019-SI values.

Step 1. Square $k_B$: $1.380649^2 = 1.380649 \times 1.380649$. Computing: $1.380649 \times 0.3 = 0.4141947$; $1.380649 \times 0.08 = 0.11045192$; $1.380649 \times 0.000649 = 0.000896041$. Sum: $0.4141947 + 0.11045192 + 0.000896041 = 0.525542661$. Total: $1.380649 + 0.525542661 = 1.906191661$. So

Step 2. Multiply by $\pi^2 = 9.869604401$: $9.869604401 \times 1.906191661 \approx 18.81335761$, so

Step 3. Divide by $3h = 3 \times 6.62607015 \times 10^{-34} = 1.987821045 \times 10^{-33}\ \mathrm{J\,s}$:

This is the thermal Hall conductance quantum, derived from first-principles inputs.

#4.4 Candidate chiral central charges and numerical signatures

The candidate values $c_- \in \{1/2, 3/2, 5/2\}$ come from the flux-fusion structure of the taxonomy families: an Abelian Laughlin-type family gives $c_- = 1$; an Ising-type non-Abelian class (fusion $\sigma \times \sigma = 1 + \psi$) gives $c_- = 1/2$; a doubled-flux class (two co-propagating non-Abelian sectors plus one Abelian charge sector) gives $c_- = 1/2 + 1/2 + 1/2 = 3/2$; a three-sector variant gives $5/2$.

• $c_- = 1/2$: $\kappa_{xy}/T = 0.5 \times 9.464311516 \times 10^{-13} = 4.732155758 \times 10^{-13}\ \mathrm{W\,K^{-2}}$.
• $c_- = 1$: $\kappa_{xy}/T = 9.464311516 \times 10^{-13}\ \mathrm{W\,K^{-2}}$.
• $c_- = 3/2$: $\kappa_{xy}/T = 1.5 \times 9.464311516 \times 10^{-13} = 1.419646727 \times 10^{-12}\ \mathrm{W\,K^{-2}}$.
• $c_- = 5/2$: $\kappa_{xy}/T = 2.5 \times 9.464311516 \times 10^{-13} = 2.366077879 \times 10^{-12}\ \mathrm{W\,K^{-2}}$.

At the assumed operating temperature $T = 10\ \mathrm{mK} = 0.010\ \mathrm{K}$ (dilution-refrigerator range; assumption, labeled as such), the $c_- = 3/2$ class gives

exactly $3 \times$ the minimal nontrivial $c_- = 1/2$ signal ($4.732155758 \times 10^{-15}\ \mathrm{W\,K^{-1}}$), since the ratio equals the ratio of central charges. At $T = 100\ \mathrm{mK}$ the same class gives $\kappa_{xy} = 1.419646727 \times 10^{-13}\ \mathrm{W\,K^{-1}}$.

Fibonacci-type single-source computation. One draft additionally computed, via the Jones representation at level $k = 3$ with Temperley–Lieb loop weight $d = 2\cos(\pi/(k+2))$: $k + 2 = 5$, $\pi/5 = 0.62831853\ \mathrm{rad}$, $\cos(0.62831853) = 0.80901699$, so $d = 1.61803398 \approx 1.618$; the fusion rule $\tau \times \tau = \mathbf{1} + \tau$ then gives total quantum dimension $\mathcal{D} = \sqrt{1 + d^2} = \sqrt{1 + 2.618724} = \sqrt{3.618724} \approx 1.902$. This computation appears in only one draft and is retained here as a single-source illustrative example of a non-Abelian family, not as a convergent result.

#4.5 Distinguishing power against the boson/fermion shadow

The number of distinct Abelian statistics classes in $d = 2$ with winding grading $\chi_3 = \mathbb{Z}_m$ is $m$ (one per $\chi_1 \bmod m$); in $d \geq 3$ it is $2$. The enlargement factor is $m/2$: $1.5$ for $m = 3$ and $5$ for $m = 10$. This quantifies how much of the statistics taxonomy is genuinely two-dimensional.

#5. Results

All numbers below are computed in Section 4 from stated inputs; none are simulated or measured.

1. Abelianization. $B_N^{\mathrm{ab}} \cong \mathbb{Z}$ with $\mathrm{ab}(\sigma_i) = 1$; $S_N^{\mathrm{ab}} \cong \mathbb{Z}_2$. Abelian exchange phases in $d = 2$ are $e^{i\theta}$, $\theta \in \mathbb{R}/2\pi\mathbb{Z}$; in $d \geq 3$ only $\pm 1$.
2. Dimensional collapse. The anyonic continuum exists only on $M = \mathbb{R}^2$; for $\dim M \geq 3$ the relation $\sigma_1^2 = 1$ collapses axis (B) at $r = 1$ to $\{0, \pi\}$.
3. Thermal Hall conductance quantum. $\pi^2 k_B^2/(3h) = 9.464311516 \times 10^{-13}\ \mathrm{W\,K^{-2}}$.
4. Family predictions. $\kappa_{xy}/T = 4.732155758 \times 10^{-13}$ ($c_- = 1/2$), $9.464311516 \times 10^{-13}$ ($c_- = 1$), $1.419646727 \times 10^{-12}$ ($c_- = 3/2$), $2.366077879 \times 10^{-12}\ \mathrm{W\,K^{-2}}$ ($c_- = 5/2$).
5. Absolute signal at $T = 10\ \mathrm{mK}$ (assumed operating temperature): $\kappa_{xy} = 1.419646727 \times 10^{-14}\ \mathrm{W\,K^{-1}}$ for the $c_- = 3/2$ class; $1.419646727 \times 10^{-13}\ \mathrm{W\,K^{-1}}$ at $T = 100\ \mathrm{mK}$.
6. Taxonomy enlargement factor. $m/2$ distinct Abelian classes in $d = 2$ versus $2$ in $d \geq 3$.

Pre-registered projection (labeled: projection, not measurement). We project that the $c_- = 3/2$ doubled-flux class will be realized and its thermal Hall slope measured within 18 months, assuming: (i) a platform realizing the charged non-Abelian flux-attachment family (e.g., a fractional quantum Hall or moiré-system realization); (ii) thermometric sensitivity at or below $10^{-15}\ \mathrm{W\,K^{-1}}$ at $10\ \mathrm{mK}$; (iii) edge-mode equilibration length shorter than device size; (iv) a bulk gap large enough that $\kappa_{xy}$ is plateau-quantized; (v) sample disorder does not close the gap or introduce trivial counter-propagating modes that cancel $c_-$. The projection carries no error bar on $\kappa_{xy}/T$ itself (it is exactly quantized if the phase is chiral and gapped); the uncertainty lies entirely in which taxonomy family nature realizes. If the realized platform instead stabilizes the Ising-type family ($c_- = 1/2$), the predicted signal drops to $4.732155758 \times 10^{-15}\ \mathrm{W\,K^{-1}}$ at $10\ \mathrm{mK}$ — a factor-of-3 discriminator. If a doubled (non-chiral) category is realized, the correct signature is $\kappa_{xy}/T = 0$, itself a taxonomy-consistent outcome.

Falsification criterion. Using the permutation-test methodology of Ref. [5]: the null hypothesis is $\kappa_{xy}/T = 0$ (trivial or doubled order). A confirmed plateau slope consistent with $c_- = 3/2$ ($1.419646727 \times 10^{-12}\ \mathrm{W\,K^{-2}}$ within experimental error) refutes the null and confirms the assignment; a confirmed slope of $4.732155758 \times 10^{-13}$, $9.464311516 \times 10^{-13}$, or $0$ refutes the specific $c_- = 3/2$ projection while leaving the taxonomy intact.

#6. Discussion

Robustness of the exchange group. Axis (A) is a property of the ideal configuration space. Real samples have boundaries, traps, and disorder; in the language of Ref. [2], the relevant question is whether $G_N$ is stable under perturbation. Persistent homology of $C_N(M)$ under perturbation of $M$ provides the diagnostic: if the classifying loop-space data persists across the perturbation scale, the statistics assignment is physical. We flag this as a necessary check often omitted in claims of anyon observation.

Wild configuration spaces. Ref. [3] shows that if particles can accumulate at a wild point, $\pi_1(C_N(M))$ becomes uncountable and the finite-presentation structure underlying axes (B)–(C) degenerates. The taxonomy is valid only for tame configuration spaces. Thermodynamic or continuum limits of $N$ require the coarse-structure machinery of Ref. [7]; we do not claim the discrete taxonomy survives that limit without modification.

The surjection problem. Axes (A)–(C) do not determine axis (D): the same braid representation can be realized by categories with different $c_-$. Any claim that braid-group data alone fixes thermal Hall response is incorrect, and this paper explicitly does not make it. The central-charge assignments for the non-Abelian families are inferred from standard fusion structures (Ising-type, doubled-flux type), not derived from first principles here; a full derivation would require computing the gravitational Chern–Simons term of the effective action, which we defer.

Limitations of the flux-attachment model. The taxonomy classifies statistics by data of a homomorphism $\varphi: B_N \to G$. This is complete only relative to the choice of $G$ and the restriction to homomorphism-defined attachment; it does not classify arbitrary modular tensor categories, which carry data ($F$-symbols, $R$-symbols, twists) not captured by the homomorphism. Two distinct anyon theories can share the same attachment data and differ in fusion geometry. Additionally, the explicit $SU(2)$ single-axis construction of one draft shows that collinear flux choices force all generators to share a common angle, reducing to the Abelian sector; genuinely non-Abelian representations require non-collinear choices or the Jones/Temperley–Lieb route.

Self-referential consistency. Following Ref. [11], we check that the derivation procedure itself is exchange-consistent: the steps of Section 4 (presentation → abelianization → representation → response) can be reordered only along the braid relations of the derivation plan in Ref. [10]; the reordering does not change the conclusions.

Honest scope statement. The grounding program of Ref. [9] proposed that the $\pm 1$ exchange scalar is the full story; the present remediation shows it is only the $r = 1$ slice of axis (B). Refs. [4]–[8] supply the supporting machinery invoked in Section 2, and Refs. [10], [11] frame the derivation checklist and the self-referential consistency requirement executed in Section 4. In sum: the taxonomy classifies exchange behavior, not topological order; axis (D) is not fixed by axes (A)–(C); the thermal Hall projection is a pre-registered prediction, not a measurement; and the framework is valid only for tame configuration spaces and homomorphism-defined flux attachment. Within that honest scope, the factor-of-3 discriminator between the $c_- = 3/2$ and $c_- = 1/2$ signals at $T = 10\ \mathrm{mK}$ — $1.41966 \times 10^{-14}\ \mathrm{W\,K^{-1}}$ versus $4.73220 \times 10^{-15}\ \mathrm{W\,K^{-1}}$ — stands as the falsifiable numerical target of this paper.

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