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and the theory of pseudo-Hermitian operators. It can be seen as an algebraic dual to the traditional set-theoretic framework of the dynamical systems theory, with its added algebraic structure and an inherent [[Supersymmetry#Supersymmetry in dynamical systems|topological supersymmetry]] (TS) enabling the generalization of certain concepts from [[Deterministic system|deterministic]] to [[Stochastic process|stochastic]] models.
Using tools of [[Topological quantum field theories|topological field theory]] originally developed in [[Particle physics|high-energy physics]], STS seeks to give a rigorous mathematical derivation to several [[Universality class|universal]] phenomena of [[Stochastic process|stochastic dynamical systems]]. Particularly, the theory identifies dynamical chaos as a [[Spontaneous symmetry breaking|spontaneous order]] originating from the [[supersymmetry|TS]] hidden in all stochastic models. STS also provides the lowest level classification of stochastic chaos which has a potential to explain [[
== Overview ==
The traditional approach to stochastic dynamics focuses on the [[
From an [[algebraic topology]] perspective, the wavefunctions are [[differential forms]]<ref name=":6"/> and [[dynamical systems theory]] defines their dynamics by the generalized transfer operator (GTO)<ref name=":0"/><ref name=":19"/>
The presence of TS arises from the fact that continuous-time dynamics preserves the [[Topological space|topology]] of the [[Phase space|phase]]/[[State-space representation|state]] space: trajectories originating from close initial conditions remain close over time for any noise configuration. If TS is [[Spontaneous symmetry breaking|spontaneously broken]], this property no longer holds on average in the limit of infinitely long evolution, meaning the system is chaotic because it exhibits a stochastic variant of the butterfly effect. In modern theoretical nomenclature, chaos, along with other realizations of [[spontaneous symmetry breaking]], is an [[spontaneous symmetry breaking#Generalisation and technical usage|ordered phase]]
| last = Uthamacumaran
| first = Abicumaran
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| volume = 2
| issue = 4
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| year = 2021
| doi = 10.1016/j.patter.2021.100226
| pmc = 8085613
}}</ref>▼
:''... chaos is counter-intuitively the "ordered" phase of dynamical systems. Moreover, a pioneer of complexity, [[Ilya Prigogine|Prigogine]], would define chaos as a spatiotemporally complex form of order...'' ▼
The [[Goldstone theorem]] necessitates the long-range response, which may account for [[pink noise|1/f noise]]. The [[Edge of Chaos]] is interpreted as noise-induced
▲}}</ref>
▲:''... chaos is counter-intuitively the "ordered" phase of dynamical systems. Moreover, a pioneer of complexity, Prigogine, would define chaos as a spatiotemporally complex form of order...''
▲The [[Goldstone theorem]] necessitates the long-range response, which may account for [[pink noise|1/f noise]]. The [[Edge of Chaos]] is interpreted as noise-induced chaos -- a distinct phase where TS is broken in a specific manner and dynamics is dominated by noise-induced instantons. In the deterministic limit, this phase collapses onto the critical boundary of conventional chaos.
== History and relation to other theories ==
The first relation between supersymmetry and stochastic dynamics was established in two papers in 1979 and 1982 by [[Giorgio Parisi]] and Nicolas Sourlas,<ref name=":9">{{Cite journal|last1=Parisi|first1=G.|last2=Sourlas|first2=N.|date=1979|title=Random Magnetic Fields, Supersymmetry, and Negative Dimensions|journal=Physical Review Letters|volume=43|issue=11|pages=744–745|doi=10.1103/PhysRevLett.43.744|bibcode=1979PhRvL..43..744P}}</ref><ref name=":15">{{Cite journal|last=Parisi|first=G.|title=Supersymmetric field theories and stochastic differential equations|journal=Nuclear Physics B|language=en|volume=206|issue=2|pages=321–332|doi=10.1016/0550-3213(82)90538-7|year=1982|bibcode=1982NuPhB.206..321P}}</ref> where [[Langevin equation|Langevin SDEs]]
The Parisi-Sourlas method has been extended to several other classes of dynamical systems, including [[classical mechanics]],<ref name=":1">{{Cite journal|last1=Gozzi|first1=E.|last2=Reuter|first2=M.|title=Classical mechanics as a topological field theory|journal=Physics Letters B|language=en|volume=240|issue=1–2|pages=137–144|doi=10.1016/0370-2693(90)90422-3|year=1990|bibcode=1990PhLB..240..137G|url=https://cds.cern.ch/record/204132|url-access=subscription}}</ref><ref name=":16">{{Cite journal|last=Niemi|first=A. J.|title=A lower bound for the number of periodic classical trajectories|journal=Physics Letters B|language=en|volume=355|issue=3–4|pages=501–506|doi=10.1016/0370-2693(95)00780-o|year=1995|bibcode=1995PhLB..355..501N}}</ref> its stochastic generalization,<ref name=":13">{{Cite journal|last1=Tailleur|first1=J.|last2=Tănase-Nicola|first2=S.|last3=Kurchan|first3=J.|date=2006-02-01|title=Kramers Equation and Supersymmetry|journal=Journal of Statistical Physics|language=en|volume=122|issue=4|pages=557–595|doi=10.1007/s10955-005-8059-x|issn=0022-4715|bibcode=2006JSP...122..557T|arxiv=cond-mat/0503545|s2cid=119716999}}</ref> and higher-order Langevin SDEs.<ref name=":14">{{Cite journal|last1=Kleinert|first1=H.|last2=Shabanov|first2=S. V.|date=1997-10-27|title=Supersymmetry in stochastic processes with higher-order time derivatives|journal=Physics Letters A|volume=235|issue=2|pages=105–112|doi=10.1016/s0375-9601(97)00660-9|bibcode=1997PhLA..235..105K|arxiv=quant-ph/9705042|s2cid=119459346}}</ref>
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further enabled the extension of the theory to SDEs of arbitrary form and the identification of the spontaneous BRST supersymmetry breaking as a stochastic generalization of chaos.<ref name=":10">{{Cite journal|last=Ovchinnikov|first=I. V.|date=2016-03-28|title=Introduction to Supersymmetric Theory of Stochastics|journal=Entropy|language=en|volume=18|issue=4|pages=108|doi=10.3390/e18040108|bibcode=2016Entrp..18..108O|arxiv=1511.03393|s2cid=2388285|doi-access=free}}</ref>
In parallel, the concept of the generalized [[transfer operator]] have been introduced in the [[dynamical systems theory]].<ref name=":0">{{cite journal|date=2002|title=Dynamical Zeta Functions and Transfer Operators|url=http://www.ams.org/notices/200208/fea-ruelle.pdf|journal=Notices of the AMS|volume=49|issue=8|pages=887|author=Reulle, D.}}</ref><ref name=":19">{{Cite journal|last=Ruelle|first=D.|date=1990-12-01|title=An extension of the theory of Fredholm determinants|journal=Publications Mathématiques de l'Institut des Hautes Études Scientifiques|language=en|volume=72|issue=1|pages=175–193|doi=10.1007/bf02699133|s2cid=121869096|issn=0073-8301|url=http://www.numdam.org/item/PMIHES_1990__72__175_0/}}</ref> This concept underlies the stochastic evolution operator of STS and provides it with a solid and natural mathematical meaning. Similar constructions were studied in the theory of SDEs.<ref>{{Cite book|title=Stochastic differential geometry at Saint-Flour|last1=Ancona|first1=A.|last2=Elworthy|first2=K. D.|last3=Emery|first3=M.|last4=Kunita|first4=H.|date=2013|publisher=Springer|isbn=9783642341700|oclc=811000422}}</ref><ref>{{Cite book|title=Stochastic flows and stochastic differential equations|last=Kunita|first=H.|date=1997|publisher=Cambridge University Press|isbn=978-0521599252|oclc=36864963}}</ref>
The Parisi-Sourlas method has been recognized <ref name=":Baulieu_Grossman"/><ref name=":1"/> as a member of Witten-type or cohomological [[topological quantum field theory|topological field theory]],<ref name=":3">{{Cite journal|last1=Birmingham|first1=D|last2=Blau|first2=M.|last3=Rakowski|first3=M.|last4=Thompson|first4=G.|title=Topological field theory|journal=Physics Reports|language=en|volume=209|issue=4–5|pages=129–340|doi=10.1016/0370-1573(91)90117-5|year=1991|bibcode=1991PhR...209..129B|url=https://cds.cern.ch/record/218572}}
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=== Generalized transfer operator ===
The [[
<math display="block"> \dot x(t) = F(x(t))+(2\Theta)^{1/2}G_a(x(t))\xi^a(t)\equiv{\mathcal F}(\xi(t)),</math>
where <math display="inline">x\in X </math> is a point in a [[Closed manifold|closed]] [[smooth manifold]], <math display="inline">X</math>, called in dynamical systems theory a [[State-space representation|state space]] while in physics, where <math>X</math> is often a [[symplectic manifold]] with half of variables having the meaning of momenta, it is called the [[phase space]]. Further, <math> F\in TX </math> is a sufficiently smooth flow [[vector field]] from the [[tangent space]] of <math> X</math> having the meaning of deterministic law of evolution, and <math> G_a \in TX, a=1, \ldots, D_\xi </math> is a set of sufficiently smooth vector fields that specify how the system is coupled to the time-dependent noise, <math>\xi(t)\in\mathbb{R}^{D_\xi}</math>, which is called [[Additive noise|additive]]/[[Multiplicative noise|multiplicative]] depending on whether <math> G_a </math>'s are independent/dependent on the position on <math>X</math>.
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According to the example above, the temporal evolution of k-forms is given by,
<math display="block">|\psi(t)\rangle = \hat M(\xi)_{t't}^*|\psi(t')\rangle,</math>
where <math>|\psi\rangle\in\Omega(X)=\bigoplus\nolimits_{k=0}^D\Omega^{(k)}(X)</math> is a time-dependent "wavefunction", adopting the terminology of quantum theory.
Unlike, say, trajectories or positions in <math>X</math>, pullbacks are linear objects even for nonlinear <math>X</math>. As a linear object, the pullback can be averaged over the noise configurations leading to the generalized transfer operator (GTO)
<ref name=":0">{{cite journal|date=2002|title=Dynamical Zeta Functions and Transfer Operators|url=http://www.ams.org/notices/200208/fea-ruelle.pdf|journal=Notices of the AMS|volume=49|issue=8|pages=887|author=Reulle, D.}}</ref>
<ref name=":19">{{Cite journal|last=Ruelle|first=D.|date=1990-12-01|title=An extension of the theory of Fredholm determinants|journal=Publications Mathématiques de l'Institut des Hautes Études Scientifiques|language=en|volume=72|issue=1|pages=175–193|doi=10.1007/bf02699133|s2cid=121869096|issn=0073-8301|url=http://www.numdam.org/item/PMIHES_1990__72__175_0/}}</ref>—the [[dynamical systems theory]] counterpart of the stochastic evolution operator of the theory of SDEs and/or the Parisi-Sourlas approach. For [[Gaussian noise|Gaussian]] [[white noise]], <math> \langle \xi^a(t) \rangle_{\text{noise}} =0, \langle\xi^a(t)\xi^b(t')\rangle_{\text{noise}} = \delta^{ab}\delta(t-t')</math>..., the GTO is
<math display="block" > \hat{\mathcal M }_{tt'} = \langle \hat M(\xi)_{t't}^*\rangle_{\text{noise}} = e^{-(t-t')\hat H}
Here, the ''infinitesimal'' GTO is the stochastic evolution operator in the [[Stratonovich integral|Stratonovich interpretation]] in the traditional approach to SDEs,<ref>
▲<math display="block" > \hat{\mathcal M }_{tt'} = \langle \hat M(\xi)_{t't}^*\rangle_{\text{noise}} = e^{-(t-t')\hat H}, </math>
<ref>{{Cite journal▼
| last = Ruelle▼
| first = David▼
| title = Ergodic theory of chaos and strange attractors▼
| journal = Reviews of Modern Physics▼
| volume = 57▼
| issue = 3▼
| pages = 617–656▼
| date = July 1985▼
| doi = 10.1103/RevModPhys.57.617▼
| bibcode = 1985RvMP...57..617E▼
}}</ref> <ref>▼
{{cite book
| last = Kunita
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}}
</ref>
▲<ref>{{Cite journal
▲ | last = Ruelle
▲ | first = David
▲ | title = Ergodic theory of chaos and strange attractors
▲ | journal = Reviews of Modern Physics
▲ | volume = 57
▲ | issue = 3
▲ | pages = 617–656
▲ | date = July 1985
▲ | doi = 10.1103/RevModPhys.57.617
▲| bibcode = 1985RvMP...57..617E
<math display="block" > \hat H = \hat L_F - \Theta \hat L_{G_a}\hat L_{G_a},</math>
where <math> \hat L_F</math> is the [[Lie derivative]] along the vector field specified in the subscript.
=== Topological supersymmetry ===
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=== Eigensystem of GTO ===
GTO is a pseudo-Hermitian operator.<ref name=":2">{{Cite journal|last=Mostafazadeh|first=A.|date=2002-07-19|title=Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries|journal=Journal of Mathematical Physics|volume=43|issue=8|pages=3944–3951|doi=10.1063/1.1489072|issn=0022-2488|bibcode=2002JMP....43.3944M|arxiv=math-ph/0203005|s2cid=7096321}}</ref> It has a complete bi-orthogonal eigensystem with the left and right eigenvectors, or the bras and the kets, related nontrivially. The eigensystems of GTO have a certain set of universal properties that limit the possible spectra of the physically meaningful
* The eigenvalues are either real or come in complex conjugate pairs called in dynamical systems theory Reulle-Pollicott resonances. This form of spectrum implies the presence of pseudo-time-reversal symmetry.
* Each eigenstate has a well-defined degree.
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In dynamical systems theory, a system can be characterized as chaotic if the spectral radius of the finite-time GTO is larger than unity. Under this condition, the partition function,
<math display="block" > Z_{tt'} = Tr \hat{\mathcal M }_{tt'} = \sum\nolimits_{\alpha}e^{-(t-t')H_\alpha}, </math>
grows exponentially in the limit of infinitely long evolution signaling the exponential growth of the number of closed
<math display="block" > \Delta = - \min_\alpha \text{Re }H_\alpha > 0, </math>
where <math > \Delta </math> is the rate of the exponential growth which is known as "pressure", a member of the family of dynamical entropies such as [[topological entropy]]. Spectra b and c in the figure satisfy this condition.
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Another object of interest is the ''sharp'' trace of the GTO,
<math display="block" > W = Tr (-1)^{\hat k} \hat{\mathcal M }_{tt'} = \sum\nolimits_\alpha (-1)^{k_\alpha}e^{-(t-t')H_\alpha}, </math>
where <math> \hat k |\psi_\alpha\rangle = k_\alpha |\psi_\alpha\rangle</math> with <math>\hat k</math> being the operator of the degree of the differential form. This is a fundamental object of topological nature known in physics as the [[Witten index]]. From the properties of the eigensystem of GTO, only supersymmetric singlets contribute to the Witten index, <math>W=\sum\nolimits_{k=0}^D (-1)^k B_k=Eu.Ch(X)</math>, where <math>Eu.Ch.</math> is the [[Euler characteristic]] and ''B'' 's
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The idea of the Parisi–Sourlas method is to rewrite the partition function of the noise in terms of the dynamical variables of the model using [[BRST quantization|BRST]] gauge-fixing procedure.<ref name=":Baulieu_Grossman"/><ref name=":3"/> The resulting expression is the Witten index, whose physical meaning is (up to a topological factor) the partition function of the noise.
The
{{Equation box 1
|indent=:
|title='''supersymmetric
|equation=<math>
W = \langle \iint_{p.b.c} J(\xi) \left(\prod\nolimits_\tau \delta ^D (\dot x(\tau) - {\mathcal F}(x(\tau),\xi(\tau)))\right) {\mathcal D}x\rangle_{\text{noise}} = \iint_{p.b.c.} e^{(Q,\Psi(\Phi))}{\mathcal D}\Phi.
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|background colour = #DCDCDC
}}
Here, the noise is assumed Gaussian white, p.b.c. signifies periodic boundary conditions, <math> \textstyle J(\xi) </math> is the Jacobian compensating (up to a sign) the Jacobian from the <math>\delta</math>-functional, <math> \Phi</math> is the collection of fields that includes, besides the original field <math> x</math>, the [[
<math display="block"> Q = \textstyle \int d\tau(\chi^i(\tau)\delta/\delta x^i(\tau) + B_i(\tau)\delta/\delta \bar\chi_i(\tau)), </math>
that can be looked upon as a
<math display="inline"> \Psi = \int d\tau (\imath_{\dot x} - \bar d )</math> with
<math display="inline"> \textstyle \bar d = \textstyle \imath_F - \Theta \imath_{G_a} L_{G_a}, \text{ and } L_{G_a}=(Q,\imath_{G_a})</math> and <math display="inline"> \imath_A = i\bar\chi A</math> being the
=== STS as a topological field theory ===
The Parisi-Sourlas method is peculiar in that sense that it looks like gauge fixing of an empty
and as a TFT it has got objects that are topological invariants.
The Parisi-Sourlas functional is one of them. It is essentially a
<math display="block">W =
\langle \iint_{p.b.c} J(\xi) \left(\prod\nolimits_\tau \delta ^D (\dot x(\tau) - {\mathcal F}(x(\tau),\xi(\tau)))\right) {\mathcal D}x \rangle_\text{noise} = \textstyle \left \langle I_N(\xi)\right \rangle_\text{noise},</math>
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| access-date = 2025-06-05
}}</ref>
which is the algebraic representation of the [[Morse–Smale system|Morse-Smale complex]]. In fact, cohomological TFTs are often called intersection theory on instantons. From the STS viewpoint, instantons refers to quanta of transient dynamics, such as neuronal avalanches or solar flares, and complex or composite instantons represent nonlinear dynamical processes that occur in response to
=== Operator representation ===
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=== Effective field theory ===
The fermions of STS represent the differentials of the wavefunctions understood as differential forms.<ref name=":6"/> These differentials and/or fermions are intrinsically linked to stochastic Lyapunov exponents<ref name=":Graham"/> that define the [[butterfly effect]].
{{Equation box 1
|indent=:
|equation=<math>
G(\eta) = - \lim_{T\to\infty} log \langle g | \hat M_{T/2, -T/2}(\eta)| g \rangle / \langle g | \hat M_{T/2, -T/2}(0)| g \rangle,
</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #DCDCDC
}}
where <math> \eta </math> are external probing fields coupled to the system and <math>g</math> is the [[ground state]], i.e., an eigenstate of the GTO picked up from the set of eigenstates with the lowest real part of their eigenvalues—a requirement needed to ensure stability of the response. The ground state represents the system which has been allowed to evolve for a long time without perturbations. The generating functional describes how the ground state responses to external perturbations.
When TS is spontaneously broken, the ground state is degenerate and the system can be effortlessly excited. In higher-dimensional theories, this degeneracy evolves into a gapless branch of excitations above the ground state called [[goldstino]]s. Due to gaplessness of goldstinos, the resulting effective field theory must be scale-invariant, or, a [[conformal field theory]]
<ref>{{cite book
| last = Brauner
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| access-date = 2025-06-05
}}</ref>
| last1 = Di Francesco
| first1 = Philippe
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| publisher = Springer
| year = 1997
| doi = 10.1007/978-1-4612-2256-9
| isbn = 978-0-387-94785-3
| url = https://link.springer.com/book/10.1007/978-1-4612-2256-9
| access-date = 2025-06-03
}}</ref>
This qualitatively explains the widespread occurrence of long-range behavior in chaotic dynamics known as [[pink noise|1/f noise]].<ref name =":10"
== Applications ==
=== Self-organized criticality and instantonic chaos ===
[[File:STS Phase Diagram.png|thumb|STS provides basic classification of stochastic dynamics based on whether the topological supersymmetry (TS) present in all stochastic models is spontaneously broken or unbroken (
Since the late 80's,<ref name=EOC-T-30>{{cite book|last=A. Bass|first=Thomas|title = The Predictors : How a Band of Maverick Physicists Used Chaos Theory to Trade Their Way to a Fortune on Wall Street|url =https://books.google.com/books?id=MQ-xGC7BdS0C&pg=PA138|publisher = Henry Holt and Company |year =1999|isbn =9780805057560 |page =[https://books.google.com/books?id=MQ-xGC7BdS0C&pg=PA138 138] |access-date=12 November 2020}}</ref><ref name=EOC-T-20>{{cite web|last=H. Packard|first=Norman|title = Adaptation Toward the Edge of Chaos|url =https://books.google.com/books?id=8prgtgAACAAJ|publisher = University of Illinois at Urbana-Champaign, Center for Complex Systems Research |year =1988|access-date=12 November 2020}}</ref>
the concept of the [[Edge of chaos]] has
<ref>{{cite book|first=Markus|last=Aschwanden|title=Self-Organized Criticality in Astrophysics|year=2011|publisher =Springer|bibcode=2011soca.book.....A }}</ref>
This phase has also been recognized as potentially significant for information processing.<ref name=EOC-T-17>{{cite journal|last1=Langton|first1=Christopher.|title=Studying artificial life with cellular automata|journal=Physica D|date=1986|volume=22|issue=1–3|pages=120–149|doi=10.1016/0167-2789(86)90237-X|bibcode=1986PhyD...22..120L |hdl=2027.42/26022|hdl-access=free}}</ref><ref name=EOC-T-28>{{cite web|last2=Young|first2=Karl|last1=P. Crutchfleld|first1=James|title=Computation at the Onset of Chaos|url=http://csc.ucdavis.edu/~cmg/papers/CompOnset.pdf|year=1990|access-date=11 November 2020}}</ref>
Its phenomenological understanding is largely based on the concepts of [[Adaptive system|self-adaptation]] and [[self-organized criticality|self-organization]].<ref>{{Cite journal|last1=Watkins|first1=N. W.|last2=Pruessner|first2=G.|last3=Chapman|first3=S. C.|last4=Crosby|first4=N. B.|last5=Jensen|first5=H. J.|date=2016-01-01|title=25 Years of Self-organized Criticality: Concepts and Controversies|journal=Space Science Reviews|language=en|volume=198|issue=1–4|pages=3–44|doi=10.1007/s11214-015-0155-x|issn=0038-6308|bibcode=2016SSRv..198....3W|arxiv=1504.04991|s2cid=34782655}}</ref><ref>{{Cite journal|last1=Bak|first1=P.|last2=Tang|first2=C.|last3=Wiesenfeld|first3=K.|date=1987|title=Self-organized criticality: An explanation of the 1/f noise|journal=Physical Review Letters|volume=59|issue=4|pages=381–384|doi=10.1103/PhysRevLett.59.381|pmid=10035754|bibcode=1987PhRvL..59..381B|s2cid=7674321 }}</ref>
STS offers the following explanation for the [[Edge of chaos]] (see figure on the right).,<ref name=":10"/>
<ref>
Under this condition, the dynamics must be dominated by instantons with power-law distributions, as dictated by the Goldstone theorem. In the deterministic limit, the noise-induced instantons vanish, causing the phase hosting this type of noise-induced dynamics to collapse onto the boundary of the deterministic chaos (see figure on
== See also ==
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