Tipping points—abrupt, potentially irreversible reorganizations of a system's statistical state—are commonly anticipated through critical slowing down: recovery slows, autocorrelation and variance rise, and spectra redden. This paradigm is powerful near simple equilibrium bifurcations but is not a general theory for stochastic, multistable, or metastable systems. We develop such a theory from the Ruelle–Pollicott (RP) spectrum of Kolmogorov generators of hypoelliptic Itô diffusions. Resolving the sensitivity of invariant statistics on RP spectral blocks shows that each contribution factorizes into a spectral denominator and a residue coupling the block to both the observable and perturbation direction. Small denominators permit large responses; residues produce them. Thus a closing RP gap is not sufficient for an early warning, while growing residues can generate the classical signature with no gap closure. An early-warning signal is therefore a property of a triple: RP block, observable, and perturbation direction. We demonstrate this on a stochastic non-normal system whose RP spectrum is exactly frozen, yet classical indicators become more alarming than during genuine gap closure. The RP decomposition attributes this to residue growth and yields an index N(f) satisfying N(f)≤1 for reversible dynamics; hence N(f)>1 certifies residue-driven amplification. For metastable systems, one killed problem yields two complementary spectral objects: the Doob Q-process isolates in-well recovery, while escape clocks and committor-weighted destination probabilities govern interwell transitions. In a one-dimensional fold, these scale as (εc−ε)1/2 and (εc−ε)3/2, separating bifurcation-induced from noise-induced tipping. Doob drift also connects to optimal Girsanov sampling.
Significance Statement:This work advances tipping-point theory by replacing the usual gap-centered view of early-warning signals with a fully Ruelle–Pollicott spectral framework. Its central originality is to show that an early warning is not determined by a closing spectral gap alone: it depends jointly on the critical spectral block, the measured observable, and the perturbation direction through spectral residues. We demonstrate that classical warning indicators can become strongly amplified even when the entire RP spectrum is frozen, and introduce a dimensionless diagnostic that certifies such residue-driven amplification. For metastable systems, we further show that loss of local recovery and increasing transition risk are governed by distinct spectral objects derived from the same killed process, thereby placing bifurcation-induced and noise-induced tipping on a common mathematical footing. By connecting these constructions to Doob transforms, committors, and optimal Girsanov sampling, the framework also provides a practical route from operator theory to data analysis and rare-event computation.
Introduction
Tipping points and the critical-slowing-down paradigm
Many natural and engineered systems can undergo abrupt, and often irreversible, reorganizations of their statistical state under a (possibly) slow change of external conditions. In the climate sciences such tipping points have been identified in the Atlantic meridional overturning circulation, in polar ice sheets, in monsoon systems and in vegetation cover [1, 2], and analogous transitions occur in ecology, in neuroscience and in finance [3, 4]. The literature on the subject is extensive, on the theoretical as well as on the empirical side [5, 6, 7, 8, 9, 10, 11, 12].
Abrupt transitions are commonly sorted by what causes them [13]. In bifurcation-induced (B-)tipping the state the system occupies ceases to exist, or ceases to be stable, because a control parameter has been pushed past a bifurcation; the transition would occur without noise, which only advances it. In noise-induced (N-)tipping the state remains stable, but a sufficiently large random fluctuation carries the system out of its basin anyway; the transition is then a rare event whose likelihood depends on how stable the state is, not on whether it is about to disappear. Further classes are tied to nonautonomous forcing [14]: in rate-induced (R-)tipping the transition is caused by how fast the forcing changes rather than by where it ends up [15]; in phase-induced (P-)tipping it occurs preferentially in a particular phase of a periodically modulated system [16]; in shock-induced (S-)tipping a single finite perturbation sends the system from one basin of attraction to a competing one [17]. This paper is concerned with B- and N-tipping, and with a common spectral language in which the two can be described and told apart.
The prospect of anticipating tipping events from observational or model time series has motivated an extensive literature on early-warning signals (EWS). The dominant conceptual framework for EWS is critical slowing down (CSD). If the observed state fluctuates around a stable equilibrium whose linearization has a leading eigenvalue approaching zero, then recovery from perturbations becomes slower, lag-one autocorrelation increases, variance may grow, and spectral power shifts toward low frequencies [3, 4, 18]. In the climate context this idea underlies degenerate fingerprinting [19] and its many descendants, and it has been used to argue that the Greenland ice sheet [20] and the AMOC [21, 22, 23] are approaching thresholds. Its mathematical backbone is the normal-form theory of local bifurcations perturbed by small noise [5, 24].
This picture is powerful precisely because it is simple. It is also, for the same reason, fragile. It presupposes (i) that the reference state is a fixed point, so that a single linear rate exists; (ii) that whatever slows down is visible in the particular observable that happens to be measured; and (iii) that the loss of stability, rather than a noise-induced escape over a still-finite barrier, is what actually terminates the metastable state. Each of these assumptions can fail, and each failure has been documented. Regime shifts can occur with no detectable warning [25]; non-catastrophic transitions can produce warnings that are not followed by tipping [26]; and nonstationary correlated noise generates spurious CSD [27]. When tipping is associated with high-dimensional boundary crises, the possibility of long chaotic transients of arbitrary duration can make it extremely hard to define any useful EWS [28]. Most strikingly, stochastic non-normal systems generate the entire canonical EWS battery — rising variance, rising autocorrelation, dimensional collapse — through transient amplification alone, while remaining uniformly far from any bifurcation [29, 30, 31]. We take this failure as the sharpest available test of any proposed early-warning theory, and return to it in Sec. {#sec:IV}. Conversely, boundary crises of chaotic attractors can be approached without the decay of correlations flagging anything at all [32]. In high-dimensional systems the question of which observable carries the signal becomes acute, and recent work shows that observables aligned with the relevant edge state are dramatically more informative than generic ones [33].
Three questions therefore remain open, and they are the questions this paper is written to answer. What exactly is the spectral object that slows down, when the reference state is not a fixed point? Through which observables is that object visible, and through which is it invisible? And how is the loss of local recovery to be separated, operationally, from the growth of escape risk? In other words: is there a single spectral language in which B- and N-tipping can both be described, and told apart?
Ruelle–Pollicott resonances as the right spectral object
Ruelle-Pollicott (RP) resonances, which were initially introduced for deterministic chaotic systems [34, 35, 36, 37, 38, 39], have since been shown to extend seamlessly to stochastic systems, where they provide a powerful tool for analyzing temporal correlations, spectral densities, and bifurcations of complex systems subject to inherent stochasticity [40, 41, 42, 43], such as those encountered in climate modeling [6, 42, 44], turbulent flows [45], and biological systems [9]. These resonances reveal the fundamental time scales and oscillatory modes of the system, acting as characteristic frequencies and decay rates that govern the relaxation of any correlations, even in the presence of significant noise. Theoretically compelling, RP resonances bridge the gap between observable behavior and the underlying governing equations: they can be extracted from time series data through methods like Markov chain modeling [46, 47] and are mathematically defined as the eigenvalues of the generator of the stochastic process, the Kolmogorov operator [6, 40, 48].
Crucially, RP resonances characterize the system’s relaxation towards its statistical equilibrium and their corresponding eigenfunctions, often referred to as Kolmogorov modes, provide a natural basis for understanding and quantifying the response of complex systems to perturbations [6, 9, 43, 44, 47, 49, 50, 51].
Two features of the RP spectrum make it the natural carrier of an early-warning theory. First, it is defined for the invariant dynamics rather than for a distinguished trajectory: the reference state may be a noisy equilibrium, a limit cycle, a chaotic invariant set, or a metastable region, and in every case the resonances are the eigenvalues of one and the same object, the Kolmogorov generator [40, 43]. The classical CSD eigenvalue is recovered as the special case of a noisy fixed point. Second, the RP spectrum comes equipped with spectral projectors and, when resonances collide, with nilpotent Jordan parts [52]. These are exactly the objects needed to say through which observable a slow mode is visible, and they are systematically discarded when one summarizes the spectrum by its gap alone. Transfer-operator and Markov-state estimation of RP spectra from data is by now standard [40, 53, 54, 55, 56], and RP-based early-warning indicators have already been explored empirically [32, 57, 58].
What this paper does
We develop, in an Lμ2 spectral setting for hypoelliptic Itô diffusions, a systematic theory of the sensitivity of invariant statistics in terms of RP blocks, and we use it to reformulate the early-warning problem; none of the results depends on the reference state being a fixed point. Four results structure the paper. Figure 1 collects them, together with their physical
Figure 1. Map of the paper for the physics-oriented reader. (a) Anatomy of an early warning (Secs. II–III). The sensitivity of any invariant statistic μϵ(f) is a sum over the Ruelle–Pollicott (RP) resonances λj of the generator, each entering through a small denominator, the distance of λj to the origin, times a residue Rjf(B) that pairs the block with the measured observable f and with the direction B in which the control parameter perturbs the dynamics. A closing gap permits a large response; only a nonzero residue produces it. An early-warning signal is therefore a property of the triple (block, observable, perturbation), not of the dynamics alone. (b) Same alarm, two mechanisms (Sec. IV). Rising variance, rising autocorrelation and a reddening spectrum are produced identically by a pole moving toward the imaginary axis (genuine critical slowing down) and by non-normal growth of the residues at frozen poles, when the RP eigenmodes become nearly parallel and the spectral projectors ill-conditioned. Two RP-based tools separate the mechanisms: the index N(f), which compares the observed relaxation rate with the one implied by the variance budget and cannot exceed one for reversible dynamics, and residue selection, which measures the Kolmogorov mode of the critical block and is blind to the spurious alarm. (c) Metastable tipping (Secs. V–VII). Near a metastable state A0, let q0(x) be the committor, the probability that a trajectory started at x returns to a small core around A0 before reaching a competing state; the metastable neighborhood W0={q0>1−η}, with 0<η≪1, is the region of high commitment to A0, and its boundary ∂W0 is a smooth, noise-adapted interface that lies uphill of A0 but below the barrier top. One kills the process on ∂W0. Passing to the Doob Q-process removes escape and isolates the local critical block, whose gap Δloc measures in-well recovery: its closure is B-tipping. Retaining the escape clock and weighting it by the committor yields instead the reduced generator Qϵ on metastable labels, whose transition rates are controlled by the quasipotential barrier ΔV: its decrease raises the N-tipping risk. In a one-dimensional fold the two carry different exponents, (ϵc−ϵ)1/2 and (ϵc−ϵ)3/2. The same Girsanov/Doob control serves both purposes: with its weights it estimates rates and committors, without them it simulates the conditioned in-well dynamics.
reading, in one page; the reader more interested in the consequences for tipping than in the spectral machinery may use it as a map of what follows.
(i) A residue-resolved sensitivity formula. In Sec. II we express the Poisson corrector uf of an observable f through the RP blocks of the generator, including spectral projectors and nilpotent components, and obtain the first-order sensitivity of μϵ(f) under a perturbation B of the generator as a sum of block contributions weighted by (−λj)−(k+1) and by RP response residues Rj,kf(B)=μ(BNjkf). Small denominators permit large sensitivity; nonzero residues produce it. Because nonsemisimple blocks are retained, the attainable singular order is ∣λj∣−m rather than ∣λj∣−1, and it is reached only if the top nilpotent residue survives—the stochastic analogue of the amplification associated with exceptional points of non-Hermitian operators [31, 52, 59]. This is also the spectral mechanism through which the non-normal “pseudo-bifurcations” of [29] become legible: transient amplification without gap closure is amplification carried by projectors and nilpotents, not by denominators.
(ii) A triple criterion for early warnings. In Sec. III we argue that an EWS is not a property of the dynamics alone, but of a triple: a critical RP block, a measured observable, and a perturbation direction. Variance, autocorrelation time, integrated autocorrelation, parameter susceptibility, and spectral power are shown to be different contractions of the same RP expansion, which may select different blocks and weight them by different residues. In particular, spectral reddening is not the general frequency-domain mechanism of warning; it is the zero-frequency, real-resonance, residue-visible corner of a broader RP-resolved theory, the general case involving pairs of blocks through two resolvents.
(iii) A counterexample that decides the matter. Section IV exhibits a two-dimensional linear diffusion, taken from [29], whose RP resonances are exactly independent of a non-normality parameter κ while its spectral projectors become ill-conditioned like κ. Every classical early-warning indicator then grows like κ2 with the spectrum frozen to machine precision, and along a slow ramp the variance rises by three orders of magnitude and the lag-one autocorrelation reaches 0.9997—a more alarming record than a genuine approach to a fold produces. The RP decomposition attributes all of it to the numerators, and supplies two repairs: a residue-selected observable, the Kolmogorov mode itself, whose normalized autocorrelation is exactly κ-independent; and a dimensionless index
N(f)=E(f,f)ΔRPVarμ(f).(1)
Here ΔRP is the leading RP gap, i.e. the rate at which the slowest mode actually decays; Varμ(f) is the stationary variance of the measured observable, i.e. how much fluctuation it carries; and E(f,f) is the Dirichlet form [60], i.e. the rate at which the noise supplies that variance, equal to minus the initial slope of the autocovariance of f. The ratio Varμ(f)/E(f,f) is thus the relaxation time one would infer from a variational gap bound alone—how long the observable would need to dissipate what the noise puts in—and multiplying by ΔRP compares it with the
@page 3/37 starts_mid=1 ends_mid=1 loss of local recovery to be separated, operationally, from the growth of escape risk? In other words: is there a single spectral language in which B- and N-tipping can both be described, and told apart?
Ruelle–Pollicott resonances as the right spectral object
Ruelle-Pollicott (RP) resonances, which were initially introduced for deterministic chaotic systems [34, 35, 36, 37, 38, 39], have since been shown to extend seamlessly to stochastic systems, where they provide a powerful tool for analyzing temporal correlations, spectral densities, and bifurcations of complex systems subject to inherent stochasticity [40, 41, 42, 43], such as those encountered in climate modeling [6, 42, 44], turbulent flows [45], and biological systems [9]. These resonances reveal the fundamental time scales and oscillatory modes of the system, acting as characteristic frequencies and decay rates that govern the relaxation of any correlations, even in the presence of significant noise. Theoretically compelling, RP resonances bridge the gap between observable behavior and the underlying governing equations: they can be extracted from time series data through methods like Markov chain modeling [46, 47] and are mathematically defined as the eigenvalues of the generator of the stochastic process, the Kolmogorov operator [6, 40, 48].
Crucially, RP resonances characterize the system’s relaxation towards its statistical equilibrium and their corresponding eigenfunctions, often referred to as Kolmogorov modes, provide a natural basis for understanding and quantifying the response of complex systems to perturbations [6, 9, 43, 44, 47, 49, 50, 51].
Two features of the RP spectrum make it the natural carrier of an early-warning theory. First, it is defined for the invariant dynamics rather than for a distinguished trajectory: the reference state may be a noisy equilibrium, a limit cycle, a chaotic invariant set, or a metastable region, and in every case the resonances are the eigenvalues of one and the same object, the Kolmogorov generator [40, 43]. The classical CSD eigenvalue is recovered as the special case of a noisy fixed point. Second, the RP spectrum comes equipped with spectral projectors and, when resonances collide, with nilpotent Jordan parts [52]. These are exactly the objects needed to say through which observable a slow mode is visible, and they are systematically discarded when one summarizes the spectrum by its gap alone. Transfer-operator and Markov-state estimation of RP spectra from data is by now standard [40, 53, 54, 55, 56], and RP-based early-warning indicators have already been explored empirically [32, 57, 58].
What this paper does
We develop, in an Lμ2 spectral setting for hypoelliptic Itô diffusions, a systematic theory of the sensitivity of invariant statistics in terms of RP blocks, and we use it to reformulate the early-warning problem; none of the results depends on the reference state being a fixed point. Four results structure the paper. Figure 1 collects them, together with their physical
reading, in one page; the reader more interested in the consequences for tipping than in the spectral machinery may use it as a map of what follows.
(i) A residue-resolved sensitivity formula. In Sec. II we express the Poisson corrector uf of an observable f through the RP blocks of the generator, including spectral projectors and nilpotent components, and obtain the first-order sensitivity of μϵ(f) under a perturbation B of the generator as a sum of block contributions weighted by (−λj)−(k+1) and by RP response residues Rj,kf(B)=μ(BNjkf). Small denominators permit large sensitivity; nonzero residues produce it. Because nonsemisimple blocks are retained, the attainable singular order is ∣λj∣−m rather than ∣λj∣−1, and it is reached only if the top nilpotent residue survives—the stochastic analogue of the amplification associated with exceptional points of non-Hermitian operators [31, 52, 59]. This is also the spectral mechanism through which the non-normal “pseudo-bifurcations” of [29] become legible: transient amplification without gap closure is amplification carried by projectors and nilpotents, not by denominators.
(ii) A triple criterion for early warnings. In Sec. III we argue that an EWS is not a property of the dynamics alone, but of a triple: a critical RP block, a measured observable, and a perturbation direction. Variance, autocorrelation time, integrated autocorrelation, parameter susceptibility, and spectral power are shown to be different contractions of the same RP expansion, which may select different blocks and weight them by different residues. In particular, spectral reddening is not the general frequency-domain mechanism of warning; it is the zero-frequency, real-resonance, residue-visible corner of a broader RP-resolved theory, the general case involving pairs of blocks through two resolvents.
(iii) A counterexample that decides the matter. Section IV exhibits a two-dimensional linear diffusion, taken from [29], whose RP resonances are exactly independent of a non-normality parameter κ while its spectral projectors become ill-conditioned like κ. Every classical early-warning indicator then grows like κ2 with the spectrum frozen to machine precision, and along a slow ramp the variance rises by three orders of magnitude and the lag-one autocorrelation reaches 0.9997—a more alarming record than a genuine approach to a fold produces. The RP decomposition attributes all of it to the numerators, and supplies two repairs: a residue-selected observable, the Kolmogorov mode itself, whose normalized autocorrelation is exactly κ-independent; and a dimensionless index
N(f)=E(f,f)ΔRPVarμ(f).(2)
Here ΔRP is the leading RP gap, i.e. the rate at which the slowest mode actually decays; Varμ(f) is the stationary variance of the measured observable, i.e. how much fluctuation it carries; and E(f,f) is the Dirichlet form [60], i.e. the rate at which the noise supplies that variance, equal to minus the initial slope of the autocovariance of f. The ratio Varμ(f)/E(f,f) is thus the relaxation time one would infer from a variational gap bound alone—how long the observable would need to dissipate what the noise puts in—and multiplying by ΔRP compares it with the relaxation time the spectrum actually exhibits. If a single mode carries the observable the two agree and N=1. We prove that N(f)≤1 for every observable whenever the generator is self-adjoint in Lμ2, so that a measured value above one means the system relaxes more slowly than its own variance budget permits, which can only happen through cancellation between signed residues. The index is therefore a certificate that amplification is residue-driven rather than gap-driven. This is, to our knowledge, the first quantitative discriminator between critical slowing down and non-normal transient amplification that is formulated at the level of the generator rather than of a fitted indicator.
(iv) Local recovery versus escape risk in metastable systems. Section V contains the main application: a unified spectral treatment of B- and N-tipping. In a metastable regime, the dominant nonzero RP block of the full generator is typically an interwell transition block, which measures residence times, not recovery inside a well. To isolate local recovery we kill the process at a committor-defined, noise-regularized boundary and pass to the associated Doob Q-process [61, 62, 63, 64]. This removes the survival decay carried by the principal Dirichlet eigenvalue and yields a conservative in-well dynamics whose leading nonzero RP block we call the local critical block. The complementary reduction keeps the killed escape clocks and weights them by committor-based destination probabilities, producing a continuous-time Markov generator Qϵ on metastable labels whose entries carry the Freidlin–Wentzell/Eyring–Kramers asymptotics [65, 66, 67, 68, 69, 70, 71, 72]. The two reductions use the same killed problem in complementary ways, and they answer different questions.
(v) B-tipping and N-tipping as two spectral objects. The distinction just described maps precisely onto the tipping taxonomy recalled in Sec. I A [13, 15, 16]. The two mechanisms are usually described in dynamical terms; the point of what follows is that each has a clean spectral counterpart. Bifurcation-induced (B-)tipping is the closure of the conditioned local RP gap; noise-induced (N-)tipping is a property of the transition blocks, i.e. of escape clocks, committors and quasipotential barriers. Both mechanisms produce long memory and enhanced low-frequency power, and a global lag-one autocorrelation estimate mixes them. The RP framework makes the separation operational, and Sec. VII makes it quantitative on a scalar metastable fold, where the local conditioned gap closes like (ϵc−ϵ)1/2 while the quasipotential barrier collapses like (ϵc−ϵ)3/2. The same example shows that variance, correlation time and zero-frequency power carry three different exponents of the same closing block, and that the committor weighting is not decorative: it is exactly the classical factor two between the mean first-passage time to the barrier top and the Kramers transition rate [66, 73].
A final, more practical thread runs through Sec. VI. The Doob drift of the Q-process is identified with the long-horizon limit of the logarithmic-gradient control that solves the optimal Girsanov importance-sampling problem for Koopman expectations [74, 75, 76]. This yields a dual computational prescription: keep the Girsanov weights and one estimates escape probabilities, exit distributions and transition rates; discard them and use the exact Doob drift and one samples the conditioned in-well dynamics, hence the local critical block. Confusing these two uses is, we argue, one concrete route by which transition risk gets misreported as local critical slowing down. Rare-event samplers such as adaptive multilevel splitting [77, 78, 79] occupy the first branch; Fleming–Viot and Doob simulations occupy the second. Both of the latter are ways of simulating a process conditioned not to have escaped, which is what the local theory needs. A Fleming–Viot scheme does this with an ensemble and no extra modelling: one runs many copies of the original dynamics and, whenever a copy leaves the region, immediately restarts it at the current position of another randomly chosen surviving copy, so that the population size is held fixed and its empirical distribution converges to the quasi-stationary distribution [80, 81, 82]. A Doob simulation does the same job with a single trajectory, by adding to the drift the extra term that repels the process from the boundary; it requires knowing the principal Dirichlet eigenfunction, but returns unweighted samples of the conditioned dynamics.
Relation to the companion study and outline
This paper is the theoretical member of a pair. Its companion [83] analyzes the approach to a warm-to-cold transition in a stochastic Ghil–Sellers energy balance model through reduced RP resonances and Green’s functions, extreme-value statistics, and full-field data-adaptive harmonic modes. That study is deliberately empirical, and it raises — without resolving — exactly the questions addressed here: near tipping, several decay rates compress toward the origin rather than one, response amplification is conditional on nonvanishing residues, extremes probing the escape direction towards the competing state organize differently from bulk relaxation, and full-field phase coherence degrades while reduced modes harmonize. The present framework supplies the spectral vocabulary in which those observations are statements about distinct RP blocks, distinct contractions, and distinct residues, rather than about a single scalar indicator.
The paper is organized as follows. Section II sets up the Lμ2 RP framework, derives the residue-resolved sensitivity formula and its frequency-resolved two-resolvent counterpart. Section III formulates the triple criterion and contrasts it with classical CSD. Section IV tests the whole construction on a non-normal counterexample with a frozen spectrum, and introduces the non-normality index. Section V develops the metastable theory: committor-defined neighborhoods, killed dynamics and the Q-process, local critical blocks, transition blocks and the reduced generator Qϵ, and the metastable fold scenario. Section VI develops the dual Girsanov–Doob view and the unified account of B- and N-tipping. Section VII works out a scalar metastable fold in closed form. Section VIII discusses scope, limitations and open problems.
Ruelle-Pollicott Sensitivity Formulas
Unperturbed Stochastic Dynamics in the L2-Spectral Setting
We consider an unperturbed Itô diffusion of the form
dXt=F(Xt)dt+σD(Xt)dWt,Xt∈Rd,(3)
where Wt is an Rq-valued Brownian motion whose components are mutually independent and D is a matrix valued function from Rd to MatR(d×q). The drift part is provided by a vector field F on Rd. We denote by a(x)=D(x)D(x)T the unperturbed diffusion tensor. Here σ>0 is a scalar noise amplitude that has been factored out of D, so that D carries only the (normalized) noise geometry and the full diffusion tensor of (3) is σ2a. This convention is kept throughout: σ is the small parameter of the metastable/Freidlin–Wentzell asymptotics of Secs. V–VII, whereas ϵ always denotes the control parameter driving the system toward tipping. The Markov semigroup acting on observables is Ptφ(x)=E[φ(Xt)], where Xtx denotes the stochastic process solving (3). The Kolmogorov generator is
Kφ=F⋅∇φ+2σ2a:∇2φ,(4)
where
a:∇2φ=i,j=1∑daij∂ij2φ.(5)
We assume that the unperturbed diffusion is non-explosive and satisfies a Lyapunov dissipativity condition ensuring the existence of an invariant probability measure [e.g. [40, 84, 85]]. We assume furthermore that the diffusion is hypoelliptic, in the sense of Hörmander [86, 87] and that the Markov process is irreducible [e.g. [40, 88, 89]].
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