Relative Stability Threshold

• Relative stability threshold is a concept defining critical boundary values where a system’s behavior shifts, determined by comparing perturbation scales or control parameters against stability limits.
• It is applied across disciplines—from hydrodynamic flows with anisotropic norms to optimization algorithms and spectral separation in neural operators—establishing exact criteria for system transitions.
• The framework underpins practical methods in analyzing stability, leveraging techniques like scaling laws, slope inequalities, and geometric decompositions to predict qualitative changes across varied systems.

“Relative stability threshold” denotes a family of threshold constructions that compare a perturbation scale, control parameter, or degenerating family against a stability boundary. In the cited literature, the expression does not refer to a single universal invariant. In hydrodynamic stability it is the critical perturbation size as a function of viscosity; in stochastic optimization and softmax feedback it is the exact parameter value below which linear or global fixed-point stability persists; in mechanical and stochastic dynamical systems it is the boundary where linear stability, basin escape, or inferential resolvability changes qualitatively; and in algebraic geometry and categorical stability it is tied to slope inequalities, valuation-theoretic ratios, or relative extensions across a fibration or semiorthogonal decomposition (Liang et al., 31 Aug 2025, Mulayoff et al., 2023, Asselle et al., 9 Apr 2026, Wang et al., 2021, Blum et al., 7 Oct 2025).

1. Fluid-mechanical thresholds and transition criteria

For two-dimensional Navier–Stokes Couette flow in the infinite channel $\mathbb{R}\times[-1,1]$ with Navier slip boundary conditions, the relative stability threshold is the perturbation size

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$

measured in an anisotropic Sobolev norm $X$ built from $x$-regularity and a scaled $y$-derivative. This improves the earlier Arbon–Bedrossian threshold

$\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$

by removing the logarithmic loss. The anisotropy is encoded by $\langle \partial_x\rangle^m$, $(\nu^{1/3}\partial_y)^j$, and the low-frequency correction $$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$, while the decay rate

$$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$

captures both enhanced dissipation and inviscid damping. The paper identifies $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$0 as the natural hydrodynamic scaling and argues that the previous logarithmic correction was an artifact of low-frequency analysis (Liang et al., 31 Aug 2025).

For three-dimensional compressible Couette flow in the isentropic compressible Navier–Stokes equations, the corresponding nonlinear threshold is

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$1

The proof separates diffusion waves, acoustic waves, and the lift-up mechanism, and combines zero-mode/nonzero-mode decompositions with multiplier estimates. In that setting the exponent $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$2 is tied to the strongest interaction among non-normal transient growth, the 3D lift-up effect, compressible acoustic and diffusive waves, and density-driven couplings (Li et al., 8 May 2026).

A different fluid-dynamical use of threshold language appears in disturbance-based transition analysis. There the stability threshold is the largest velocity perturbation magnitude compatible with a small-gain bound. In the unstructured case the criterion takes the form

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$3

while structured variants replace the $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$4 gain by a structured singular-value quantity. The paper emphasizes a hierarchy in which the unstructured approach gives the strictest bound, and it uses this framework to explain finite-amplitude transition in Couette, plane Poiseuille, and Blasius flows (Frank-Shapir et al., 7 Jul 2025).

For transitional plane Couette flow, the global stability threshold is instead a Reynolds number $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$5: for $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$6 turbulence is sustained, whereas for $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$7 it is transient and eventually decays. The operational criterion is extreme-value-theoretic: $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$8 is identified by the Reynolds number at which the GEV shape parameter for perturbation-energy minima changes sign from negative to positive. This use of “threshold” is global and statistical rather than perturbative (Faranda et al., 2012).

2. Exact thresholds in optimization, neural operators, and softmax feedback

Near a twice-differentiable minimum $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$9, the exact mean-square linear stability threshold for SGD is a closed-form step-size bound. In the interpolating case, the paper proves

$X$0

with

$X$1

where

$X$2

The threshold is monotonically non-decreasing in $X$3, equals the GD threshold at full batch, and is exactly the threshold of a mixture process that takes a full-batch step with probability $X$4 and a single-sample step with probability $X$5 when $X$6 (Mulayoff et al., 2023).

For manifold neural networks, the relative stability threshold is the spectral separation parameter $X$7, together with the perturbation condition

$X$8

for a relative perturbation $X$9 of the Laplace–Beltrami operator. The spectrum is partitioned into $x$0-separated groups, and $x$1-FRT filters are approximately constant on each group. Under $x$2-integral Lipschitz filters and normalized Lipschitz activations, the paper proves

$x$3

Larger $x$4 yields coarser grouping and greater stability, but less discriminability; smaller $x$5 yields finer grouping and potentially less stability (Wang et al., 2021).

For affine logit feedback systems on the simplex,

$x$6

the sharp Euclidean threshold is

$x$7

Below this threshold, $x$8 is a contraction, has a unique fixed point, and both Picard iteration and continuous-time logit adjustment converge globally. The factor $x$9 comes from the sharp covariance bound

$y$0

In the two-action example $y$1, the true bifurcation occurs at $y$2, showing that the older condition $y$3 misses the full pre-bifurcation stable regime $y$4 (Wang, 15 May 2026).

3. Dynamical systems, mechanical stability, and critical inference

In the electromagnetic reformulation of reduced $y$5-body dynamics, the relative stability threshold is most explicit in the two-dimensional model with quadratic potential

$y$6

For the equilibrium $y$7, the paper proves the equivalence

$y$8

The threshold is therefore

$y$9

At that boundary the topology of the zero set of the electromagnetic curvature changes: below threshold the boundary of the positive-curvature region is a hyperbola, above threshold it is an ellipse, and at threshold it degenerates into a pair of parallel lines. In the planar three-body Lagrange case this criterion becomes exactly Routh’s classical condition

$\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$0

The same blockwise curvature test yields a sufficient instability criterion when the reduced linearized dynamics splits into invariant symplectic planes (Asselle et al., 9 Apr 2026).

A more geometric dynamical-systems threshold is the stability threshold

$\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$1

defined as the minimal distance from an attractor $\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$2 to the boundary $\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$3 of its basin of attraction. The minimizing vector $\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$4 is the most dangerous perturbation direction. The computational scheme proceeds by locating basin-boundary points and converging to local threshold points (LOCT points), after which the global threshold is the minimum of the local values. The paper does not define a separate formal quantity called “relative stability threshold,” but it explicitly treats comparison of $\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$5 across parameters or systems as a relative robustness metric (Klinshov et al., 2015).

In high-dimensional stochastic dynamics, the threshold itself may be well defined while its estimation becomes statistically singular. For the multivariate Ornstein–Uhlenbeck process

$\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$6

the distance to instability is

$\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$7

with criticality at $\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$8. The paper shows that the relative uncertainty of an estimate of $\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$9 diverges as $\langle \partial_x\rangle^m$0 vanishes, that temporal correlations reduce the effective number of independent samples, and that inference breaks down when

$\langle \partial_x\rangle^m$1

It also derives an optimal sampling interval

$\langle \partial_x\rangle^m$2

which diverges near criticality. This gives a threshold theory for inference itself rather than for the underlying dynamics alone (Costa et al., 22 Jun 2026).

The compartmental voter-flow model introduces a local stability threshold

$\langle \partial_x\rangle^m$3

under $\langle \partial_x\rangle^m$4. With

$\langle \partial_x\rangle^m$5

the regime $\langle \partial_x\rangle^m$6 implies $\langle \partial_x\rangle^m$7, so the mobilised component contracts locally, whereas $\langle \partial_x\rangle^m$8 allows transient amplification. The paper couples this threshold to an impulse-controlled leaky reservoir, proves that the scalar reservoir is a conservative envelope of the full nonlinear dynamics, and derives explicit safe-capacity frontiers such as

$\langle \partial_x\rangle^m$9

for a fixed horizon $(\nu^{1/3}\partial_y)^j$0 (Omelchenko, 22 May 2026).

4. Algebraic and birational geometry

For relative hypersurfaces

$(\nu^{1/3}\partial_y)^j$1

inside a projective bundle over a curve, the threshold parameter is the ratio $(\nu^{1/3}\partial_y)^j$2, where $(\nu^{1/3}\partial_y)^j$3. The comparison quantity is the slope

$(\nu^{1/3}\partial_y)^j$4

The paper proves a dichotomy: if

$(\nu^{1/3}\partial_y)^j$5

then the relevant $(\nu^{1/3}\partial_y)^j$6-positivity and slope inequalities hold, whereas if

$(\nu^{1/3}\partial_y)^j$7

every fibre is Chow unstable. In that unstable regime one obtains the singularity bound

$(\nu^{1/3}\partial_y)^j$8

and, for the total space, $(\nu^{1/3}\partial_y)^j$9; in particular, if $$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$0, the pair is not log canonical. Nakayama’s Zariski decomposition explains this threshold through fixed components determined by the Harder–Narasimhan filtration (Barja et al., 2014).

Relative K-stability for Kähler manifolds is formulated via test configurations together with projection away from a torus $$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$1. If $$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$2 is an orthogonal basis of the Lie algebra of $$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$3, the relative Donaldson–Futaki invariant is

$$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$4

A Kähler manifold is relatively K-stable if $$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$5 for every test configuration of positive norm. The paper proves that existence of an extremal Kähler metric implies relative K-stability, and states that in the projective case this notion is stronger than the usual definition due to Székelyhidi (Dervan, 2016).

For relative flag varieties $$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$6, the destabilizing data arise from subbundles $$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$7. Over a curve, the Donaldson–Futaki invariant of the natural degeneration satisfies

$$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$8

with $$\left\langle \frac{1}{\partial_x}\right\rangle^\varepsilon$$9. Thus slope instability of $$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$0 implies K-unstability of the flag bundle, and proper semistability of $$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$1 implies proper K-semistability. Over higher-dimensional bases with adiabatic polarization $$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$2, the leading asymptotic term is

$$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$3

again with $$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$4 (Isopoussu, 2013).

A valuation-theoretic relative stability threshold appears for a family of polarized pairs over a DVR: $$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$5 Under the hypothesis

$$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$6

the infimum is computed by a divisorial valuation. The paper then replaces the special fibre by the divisor computing instability, obtaining a new family whose special fibre has strictly larger threshold, and iterates this birational improvement to prove properness of K-moduli (Blum et al., 7 Oct 2025).

5. Relative stability conditions in Fukaya and triangulated categories

For the partially wrapped Fukaya category of a marked surface, a relative stability condition is defined with respect to a boundary arc $$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$7 by attaching a disk $$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$8 and passing to the extended surface

$$\lambda_k=\begin{cases} \nu^{1/3}|k|^{2/3}, & |k|\ge \nu,\ \nu, & |k|\le \nu, \end{cases}$$9

An ordinary stability condition on $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$00 is then regarded as a stability condition on $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$01 relative to $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$02. The key point is functoriality under cutting and gluing: if

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$03

then a stability condition on $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$04 can be cut into compatible relative stability conditions on $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$05 and $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$06, and conversely glued back. The gluing construction is governed by unobstructed lozenges of stable intervals. This relative framework reduces the classification of stability conditions on fully stopped surfaces to the disk, annulus, and punctured torus, and yields that the HKK map from flat surfaces to $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$07 is a global homeomorphism in the fully stopped case (Takeda, 2018).

For triangulated categories with a left admissible subcategory $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$08, a relative stability condition is a pair $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$09 consisting of a relative central charge and a slicing on $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$10, subject to extendability to a Bridgeland stability condition on $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$11. The gluing condition is expressed by a phase window

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$12

ensuring that suitable hearts on the semiorthogonal components glue to a bounded heart on $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$13. The paper proves that $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$14 is a complex manifold and that the forgetful map to central charges is a local isomorphism. Here the threshold is not a single scalar invariant but an extendability regime controlled by phase inequalities and deformation bounds (Liu et al., 2024).

6. Discrete, probabilistic, and statistical thresholds

In random graph weak saturation, the stability property

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$15

admits a threshold probability $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$16. The paper proves existence of such a threshold and bounds it between a lower scale

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$17

and an upper scale around

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$18

Below the lower scale, uncovered edges force failure of $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$19; above the upper scale, the extension property EXT and Hamilton-power property HAM allow an explicit weakly saturated construction of complete-graph size (Bidgoli et al., 2020).

In two-dimensional threshold cellular automata on a torus, the threshold parameter is the local update rule itself. Threshold-1 and Threshold-5 have trivial stable configurations, whereas Threshold-2, Threshold-3, and Threshold-4 admit nontrivial geometric structure. The paper characterizes stable configurations for Threshold-2, Threshold-4, and Threshold-3, and gives a testing algorithm that distinguishes Threshold-2 stability from being $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$20-far from any stable configuration with query complexity independent of configuration size and depending quadratically on $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$21. In this setting, “relative stability threshold” describes the sharp change in both geometry and testability as the update threshold varies (Nakar et al., 19 Jul 2025).

In stability selection, the threshold parameter is the selection-probability cutoff $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$22. The paper argues that fixed choices of $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$23 are not universally appropriate, and replaces them with data-adaptive procedures. ATS orders the empirical maximum selection probabilities

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$24

and chooses an elbow $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$25 by maximizing a profile log-likelihood under a two-segment normal model, yielding $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$26. EATS first estimates a null exclusion threshold

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$27

from a shuffled dataset, restricts to $\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$28, and then applies ATS. The resulting threshold remains compatible with the classical false-selection bound

$\|\omega_{\mathrm{in}}\|_{X}\lesssim \nu^{1/2},$29

(Huang et al., 28 May 2025).

7. Comparative interpretation

Across these works, a relative stability threshold is consistently a boundary value defined only relative to a chosen geometry, norm, decomposition, or observable. In hydrodynamic PDE it is a perturbation amplitude measured in an anisotropic Sobolev norm or via an input-output gain; in optimization it is an exact step-size or inverse-temperature cutoff tied to second-moment or contraction geometry; in mechanical dynamics it is a spectral, curvature, or basin-boundary boundary; in algebraic geometry it is a slope or valuation ratio controlling stability, singularities, or degeneration; and in categorical settings it is an extendability or gluing regime rather than a single number (Liang et al., 31 Aug 2025, Mulayoff et al., 2023, Asselle et al., 9 Apr 2026, Barja et al., 2014, Takeda, 2018).

This suggests a common structural pattern. A threshold becomes “relative” when stability is not tested absolutely on the ambient system, but against auxiliary data: viscosity scaling, batch size, tangent-space projection, electromagnetic reduction, a base morphism, a torus of automorphisms, a boundary arc, a left admissible subcategory, or an empirically estimated null baseline. The mathematical content then lies in identifying the sharp boundary, proving whether it is necessary, sufficient, or exact, and describing what changes qualitatively when that boundary is crossed (Wang, 15 May 2026, Blum et al., 7 Oct 2025, Wang et al., 2021, Huang et al., 28 May 2025).