The Road Taken: The Role of Optimizers at the Edge of Stability

# The Role of Optimizersat the Edge of Stability
Source: [https://arxiv.org/html/2608.18415](https://arxiv.org/html/2608.18415)
## The Road Taken: The Role of Optimizers at the Edge of Stability

Jaerin Lee & Kyoung Mu LeeAffiliation:Computer Vision Lab, ASRIAffiliation:Seoul National UniversityAffiliation:Seoul 08826, KoreaEmail:[\{ironjr,kyoungmu\}@snu\.ac\.kr](mailto:)

###### Abstract

The edge of stability refers to a phenomenon in deep learning with gradient\-based optimizers where the Hessian eigenvalues of the loss remain stable above a threshold that the classical descent lemma predicts to be unstable\. Previous works formulate the edge of stability with respect to the maximum Hessian eigenvalue and the learning rate\. However, we observe that many first\-order methods, including gradient descent, significantly violate the stability bound predicted by these theories by a factor as large as×21\.1\\times 21\.1\. Moreover, this deviation turns out to be systematic and highly dependent on the underlying optimizer, which is not captured by previous formulations\. This calls for a new formulation of the stability threshold, which we derive from the directional Hessian and the gradient\-alignment score with respect to the actual update taken by the optimizer, rather than the maximum curvature mode\. Our new formulation of the*realized*edge of stability not only removes optimizer\-dependent offsets and provides more consistent predictions of the stability threshold, but also introduces new diagnostic tools that reveal the unique role of the optimizer in actively balancing between the temporal and spatial budgets in first\-order optimization\.

## 1Introduction

The training dynamics of deep learning optimization still possess many mysteries\. One of the most intriguing phenomena is the*edge of stability*\(EoS\)\([13](https://arxiv.org/html/2608.18415#bib.bib4)\), where gradient descent operates in a region where classical theory predicts instability\. The good old descent lemma\([46](https://arxiv.org/html/2608.18415#bib.bib2);[7](https://arxiv.org/html/2608.18415#bib.bib1)\)suggests that the available step sizeη\\etafor gradient descent is bounded by the curvature of the objective function:

η​λmax​\(𝑯\)≤2,\\eta\\lambda\_\{\\max\}\(\{\\bm\{H\}\}\)\\leq 2,\(1\)whereλmax\\lambda\_\{\\max\}is the maximum eigenvalue of the Hessian matrix𝑯=∇𝜽2​ℒ​\(𝜽⋆\)\{\\bm\{H\}\}=\\nabla\_\{\\bm\{\\theta\}\}^\{2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}^\{\\star\}\)of the lossℒ⁡\(𝜽\)\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)at a local minimum𝜽⋆\{\\bm\{\\theta\}\}^\{\\star\}\. Although this bound holds for convex objectives, its behavior becomes more subtle in highly nonconvex settings such as deep learning\. In those regimes, it has been reported that the relation \([1](https://arxiv.org/html/2608.18415#S1.E1)\) behaves more like a stable attractor that draws the curvatureλmax\\lambda\_\{\\max\}towards a value near2/η2/\\eta, rather than a strict stability threshold\([34](https://arxiv.org/html/2608.18415#bib.bib5);[13](https://arxiv.org/html/2608.18415#bib.bib4)\)\. During the first few hundred iterations, the top Hessian eigenvalueλmax\\lambda\_\{\\max\}either shrinks rapidly\([34](https://arxiv.org/html/2608.18415#bib.bib5);[62](https://arxiv.org/html/2608.18415#bib.bib30)\)or increases gradually\([13](https://arxiv.org/html/2608.18415#bib.bib4)\)depending on the initialization, until it eventually reaches a stable point near2/η2/\\eta, and then hovers around this value\. This mechanism is referred to as the*edge of stability*\(EoS\)\([13](https://arxiv.org/html/2608.18415#bib.bib4)\)\.

Rather than being unified into a single definition, previous works have suggested various alternative formulations for the edge of stability\([38](https://arxiv.org/html/2608.18415#bib.bib29);[12](https://arxiv.org/html/2608.18415#bib.bib26);[14](https://arxiv.org/html/2608.18415#bib.bib7);[1](https://arxiv.org/html/2608.18415#bib.bib16);[63](https://arxiv.org/html/2608.18415#bib.bib8);[5](https://arxiv.org/html/2608.18415#bib.bib27);[10](https://arxiv.org/html/2608.18415#bib.bib24);[11](https://arxiv.org/html/2608.18415#bib.bib10);[23](https://arxiv.org/html/2608.18415#bib.bib35);[4](https://arxiv.org/html/2608.18415#bib.bib14);[36](https://arxiv.org/html/2608.18415#bib.bib31)\)\. Nevertheless, all share the same core idea and structure as a product of two or three separable quantities: \(1\) a learning rate scalarη\\eta, \(2\) a*maximum curvature eigenmode*or*sharpness*SSwith respect to a \(possibly preconditioned\) Hessian, and optionally \(3\) a momentum\-induced gainΓ\\Gammaif the optimizer is stateful:

η​S​Γ≤2\.\\eta S\\Gamma\\leq 2\.\(2\)For instance, the original formulation\([13](https://arxiv.org/html/2608.18415#bib.bib4)\)assignsS=λmax​\(𝑯\)S=\\lambda\_\{\\max\}\(\{\\bm\{H\}\}\)andΓ=1\\Gamma=1\. Later,[12](https://arxiv.org/html/2608.18415#bib.bib26)incorporate preconditioning by takingS=λmax\(𝑷−1/2𝑯𝑷−1/2\)S=\\lambda\_\{\\max\}\(\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}\)and introduce a factorΓ=Γ⁡\(β\)\\Gamma=\\Gamma\(\\beta\)to account for the momentum, e\.g\.,Γ⁡\(β\)=1/\(1\+β\)\\Gamma\(\\beta\)=1/\(1\+\\beta\)for Heavy Ball\([48](https://arxiv.org/html/2608.18415#bib.bib45)\)andΓ⁡\(β\)=\(1\+2​β\)/\(1\+β\)\\Gamma\(\\beta\)=\(1\+2\\beta\)/\(1\+\\beta\)for Nesterov’s momentum\([44](https://arxiv.org/html/2608.18415#bib.bib46)\)\.

In summary, the edge of stability has been understood as a product of the*worst\-mode*sharpness and*separable*temporal factors\. This formulation provides*available*stability thresholds that are provably marginal for quadratic objectives\. However, we find that the actual relative position of the edge,χ:=η​S​Γ/2\\chi:=\\eta S\\Gamma/2, is not consistent with the theoretical predictionχ=1\\chi=1\. As Figure[1](https://arxiv.org/html/2608.18415#S1.F1)shows, its values range from 0\.79 to 21\.07 across different types of optimizers and learning rates\. Even vanilla GD sits aboveχ=1\\chi=1\(top left\)\. The median of the saturated curve grow up toχ≈1\.62\\chi\\approx 1\.62as the learning rate increases\. More complicated optimizers, such as dual\-momentum optimizers\([30](https://arxiv.org/html/2608.18415#bib.bib37);[47](https://arxiv.org/html/2608.18415#bib.bib39)\)or pole\-zero filters\([40](https://arxiv.org/html/2608.18415#bib.bib44)\), show their actual EoS rising to aroundχ=2\.65\\chi=2\.65\(columns 3 and 4\)\. The most extreme case is AdaGrad\([16](https://arxiv.org/html/2608.18415#bib.bib42)\)\(top right\), which shows a clear edge starting fromχ=1\.42\\chi=1\.42atη=0\.01\\eta=0\.01and reachingχ=21\.07\\chi=21\.07atη=0\.1\\eta=0\.1\. The deviation ofχ\\chifrom the theoretical prediction ofχ=1\.0\\chi=1\.0depends strongly on the underlying optimization algorithm\. This systematic deviation poses a new challenge to the current formulation and interpretation of the edge of stability\.

In fact, this formulation \([2](https://arxiv.org/html/2608.18415#S1.E2)\) is hiding a critical assumption: the worst\-mode sharpnessSSis always realized by the optimizer, which is not always the case\. Starting from modeling the actual dynamical system the optimizer operates on, we derive a new formulation of the stability threshold from the directional Hessian\([32](https://arxiv.org/html/2608.18415#bib.bib52);[42](https://arxiv.org/html/2608.18415#bib.bib36);[23](https://arxiv.org/html/2608.18415#bib.bib35)\)and the gradient\-update alignment\([31](https://arxiv.org/html/2608.18415#bib.bib38)\)\. As shown in the bottom row of Figure[1](https://arxiv.org/html/2608.18415#S1.F1), this new formulation, called the*realized*edge of stabilityζ\\zeta, largely eliminates optimizer\-dependent offsets\. Furthermore, it is much more efficient to compute and recovers the original formulationχ\\chiby decomposing it into interpretable factors\. This provides a richer understanding of the phenomenon itself and reveals the optimizer’s role in steering along the edge of stability by actively calibrating the temporal gauge and spatial budget\.

Figure 1:Edge of stability across different families of optimizers\.We train a fully\-connected network on CIFAR\-10 using various full\-batch gradient\-based optimizers: gradient descent \(GD\), Adam, GD with a pole\-zero filter \(QHM\) and GD with a two\-pole filter \(Grokfast\), and AdaGrad\. Theoretical prediction of the edge of stabilityχ:=η​S​Γ/2=1\.0\\chi:=\\eta S\\Gamma/2=1\.0is occasionally violated in practice, and the offsetsχ\\chiare heavily dependent on the type of optimizers in use\. This systematic deviations motivate a new formulation of the*realized*edge of stabilityζ:=χ​ς/τ\\zeta:=\\chi\\varsigma/\\tauwithς\\varsigmaandτ\\taubeing the spatial and temporal calibrations, respectively\. This new formulation is more consistent with the empirical observations and removes the optimization\-dependent offsets\.Our contributions are:

- •We extend the original formulation of the edge of stability\([13](https://arxiv.org/html/2608.18415#bib.bib4);[12](https://arxiv.org/html/2608.18415#bib.bib26)\)to general gradient\-based optimizers with states, which is exact for fixed quadratic objectives\.
- •We derive a new formulation of the stability threshold, based on the directional Hessian and the gradient\-update alignment score\([31](https://arxiv.org/html/2608.18415#bib.bib38)\), which is more consistent with empirical observations and eliminates optimizer\-dependent offsets\.
- •We factorize the realized edge of stabilityζ\\zetainto interpretable components: the original edge of stabilityχ\\chi, spatial participationς\\varsigma, and temporal calibrationτ\\tau, and show that the two EoS formulations coincide,ζ=χ\\zeta=\\chi, if and only ifς=τ\\varsigma=\\tau\.
- •This provides a simple yet powerful diagnostic tool for understanding the main factors driving a training trajectory at the edge of stability, revealing how much of the potential loss decrease,𝒈⊤​𝒖=d​ℒ/d​t\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}=\\mathrm\{d\}\{\\mathcal\{L\}\}/\\mathrm\{d\}t, carried out by the optimizer is actually converted into real progressΔ​ℒ\\Delta\{\\mathcal\{L\}\}, and how much is lost to different components\.

## 2Preliminaries: Edge of Stability

To understand the edge of stability, we start by reviewing its previously established formulations\. Following the aforementioned lines of work, we focus on the full\-batch settings where the analysis is more tractable and the effect is more pronounced\. Let us define a few notations to clarify the following discussion\. In gradient\-based optimization, we train a parameter𝜽\{\\bm\{\\theta\}\}by minimizing a loss functionℒ⁡\(𝜽\)\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)through its first\-order gradients𝒈:=∇𝜽ℒ​\(𝜽\)\{\\bm\{g\}\}:=\\nabla\_\{\{\\bm\{\\theta\}\}\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)\. We can either use this gradient directly, or filter it by some input\-dependent preconditioner𝑷\{\\bm\{P\}\}and temporal filterQQto produce a parameter update𝒖\{\\bm\{u\}\}which has the same shape as the parameter𝜽\{\\bm\{\\theta\}\}:

𝜽t\+1=𝜽t−η​𝒖t,where𝒖t=𝑷t−1​\(Qt∗𝒈≤t\)t\.\\ \{\\bm\{\\theta\}\}\_\{t\+1\}\\;=\\;\{\\bm\{\\theta\}\}\_\{t\}\-\\eta\{\\bm\{u\}\}\_\{t\}\\;,\\quad\\text\{where\}\\quad\{\\bm\{u\}\}\_\{t\}\\;=\\;\{\\bm\{P\}\}\_\{t\}^\{\-1\}\(Q\_\{t\}\*\{\\bm\{g\}\}\_\{\\leq t\}\)\_\{t\}\\;\.\(♠\\spadesuit\)With an input\-dependent preconditioner𝑷t\{\\bm\{P\}\}\_\{t\}and a causal gradient filterQtQ\_\{t\}acting on a history of gradient signal𝒈≤t\{\\bm\{g\}\}\_\{\\leq t\}, this gives a generic formulation for gradient\-based optimizers with internal states\. This not only includes GD, Heavy Ball\([48](https://arxiv.org/html/2608.18415#bib.bib45)\), Nesterov’s Momentum\([44](https://arxiv.org/html/2608.18415#bib.bib46)\), and various preconditioned versions of these\([16](https://arxiv.org/html/2608.18415#bib.bib42);[55](https://arxiv.org/html/2608.18415#bib.bib43);[27](https://arxiv.org/html/2608.18415#bib.bib40);[37](https://arxiv.org/html/2608.18415#bib.bib41)\), but also extends to more exotic temporal filters such as dual\-momentum\([30](https://arxiv.org/html/2608.18415#bib.bib37);[47](https://arxiv.org/html/2608.18415#bib.bib39)\)or quasi\-hyperbolic momentum \(QHM\)\([40](https://arxiv.org/html/2608.18415#bib.bib44)\)\. For example, GD corresponds to𝑷t=I\{\\bm\{P\}\}\_\{t\}=IandQt=IQ\_\{t\}=I, and Heavy Ball with momentum coefficientβ\\betais repesented byQ∗𝒈≤t=∑k=0tβk​𝒈t−kQ\*\{\\bm\{g\}\}\_\{\\leq t\}=\\sum\_\{k=0\}^\{t\}\\beta^\{k\}\{\\bm\{g\}\}\_\{t\-k\}\. Signal processing notation for the gradient filterQQallows us to study this dynamics in frequency domain, i\.e\.,Q⁡\(z\)=∑t=0∞Qt​z−tQ\(z\)=\\sum\_\{t=0\}^\{\\infty\}Q\_\{t\}z^\{\-t\}\. We also assume its dc gain is normalized toQ⁡\(1\)=1Q\(1\)=1, so that the learning rateη\\etais solely responsible for the gain of the system\. This simplifies the analysis without loss of generality\.

From extensive observations,[13](https://arxiv.org/html/2608.18415#bib.bib4);[12](https://arxiv.org/html/2608.18415#bib.bib26)formulated the EoS as a*separable*product of a function of the preconditioner𝑷\{\\bm\{P\}\}and a function of the momentum coefficientβ\\beta,

ηλmax\(𝑷t−1/2𝑯t𝑷t−1/2\)Γ\(β\)<2,\\eta\\lambda\_\{\\max\}\(\{\{\\bm\{P\}\}\_\{t\}^\{\-1/2\}\{\\bm\{H\}\}\_\{t\}\{\\bm\{P\}\}\_\{t\}^\{\-1/2\}\}\)\\,\\Gamma\(\\beta\)\\;<\\;2,\(3\)where𝑯t\{\\bm\{H\}\}\_\{t\}is the Hessian at timett\. For example, Heavy Ball givesΓ⁡\(β\)=1/\(1\+β\)\\Gamma\(\\beta\)=1/\(1\+\\beta\), normalized Heavy Ball givesΓ⁡\(β\)=\(1−β\)/\(1\+β\)\\Gamma\(\\beta\)=\(1\-\\beta\)/\(1\+\\beta\), Nesterov’s momentum givesΓ⁡\(β\)=\(1\+2​β\)/\(1\+β\)\\Gamma\(\\beta\)=\(1\+2\\beta\)/\(1\+\\beta\), and so on\. It is proven for each momentum type that this behaves as a marginal stability condition for the frozen quadratic system\([12](https://arxiv.org/html/2608.18415#bib.bib26)\)\. The following proposition generalizes their results to any gradient\-based optimizer of type \([♠‣2](https://arxiv.org/html/2608.18415#S2.Ex1)\)\.

###### Proposition 2\.1\(Edge of Stability for Stateful Optimizers\)\.

\[[proof](https://arxiv.org/html/2608.18415#A2.SS2)\]Consider a dynamical system given by the update rule \([♠‣2](https://arxiv.org/html/2608.18415#S2.Ex1)\)\. Let\(λ,𝐯\)\(\\lambda,\{\\bm\{v\}\}\)be any eigenpair of the preconditioned Hessian𝐏t−1/2𝐇t𝐏t−1/2\{\\bm\{P\}\}\_\{t\}^\{\-1/2\}\{\\bm\{H\}\}\_\{t\}\{\\bm\{P\}\}\_\{t\}^\{\-1/2\}\. Then the gaink⋆​\(Q\)k\_\{\\star\}\(Q\)at the first crossing of the unit circle by the root locus ofQ⁡\(z\)Q\(z\)is given by

k⋆​\(Q\)=minω∈Ω\+​\(Q\)⁡\|1−ei​ωQ⁡\(ei​ω\)\|,whereΩ\+​\(Q\):=\{ω∈\(0,π\]:1−ei​ωQ⁡\(ei​ω\)∈ℝ\>0\},k\_\{\\star\}\(Q\)=\\min\_\{\\omega\\in\\Omega\_\{\+\}\(Q\)\}\\left\|\\frac\{1\-e^\{i\\omega\}\}\{Q\(e^\{i\\omega\}\)\}\\right\|,\\quad\\text\{where\}\\quad\\Omega\_\{\+\}\(Q\)\\;:=\\;\\left\\\{\\,\\omega\\in\(0,\\pi\]:\\frac\{1\-e^\{i\\omega\}\}\{Q\(e^\{i\\omega\}\)\}\\in\\mathbb\{R\}\_\{\>0\}\\,\\right\\\},\(4\)whereΩ\+​\(Q\)\\Omega\_\{\+\}\(Q\)is the set of frequencies on the unit circle at which the locus can cross with a real positive gain\. If we writeΓ:=2/k⋆​\(Q\)\\Gamma:=2/k\_\{\\star\}\(Q\), then the equivalent stability condition for the frozen quadratic system is attained by

η​λ​Γ≤2\.\\ \\eta\\lambda\\Gamma\\;\\leq\\;2\.\(♣\\clubsuit\)Moreover, if the first unit\-circle crossing is at the Nyquist frequencyz=−1z=\-1withQ⁡\(−1\)\>0Q\(\-1\)\>0, i\.e\., period\-2 oscillation, thenΓ=Q⁡\(−1\)\\Gamma=Q\(\-1\)and the stability condition becomesη​λ​Q​\(−1\)≤2\\eta\\lambda Q\(\-1\)\\leq 2\.

###### Corollary 2\.2\(Recovering[12](https://arxiv.org/html/2608.18415#bib.bib26)’s formulae\)\.

\[[proof](https://arxiv.org/html/2608.18415#A2.SS4)\]The coefficientΓ⁡\(β\)\\Gamma\(\\beta\)in the inequality \([3](https://arxiv.org/html/2608.18415#S2.E3)\) for momentum\-based optimizers is attained at the Nyquist frequencyz=−1z=\-1, i\.e\.,Γ⁡\(β\)=Q⁡\(−1\)\\Gamma\(\\beta\)=Q\(\-1\)for \(dc gain\-normalized\) Heavy Ball and Nesterov’s momentum\.

Figure 2:Root locus of optimizers\.It is possible to exit the unit circle atz≠−1z\\neq\-1, i\.e\.,Γ≠Q⁡\(−1\)\\Gamma\\neq Q\(\-1\)\.The detailed derivation is provided in Appendix[B\.2](https://arxiv.org/html/2608.18415#A2.SS2)\. The gaugeΓ\\Gammaof each optimizer filter, including the casesΓ≠Q⁡\(−1\)\\Gamma\\neq Q\(\-1\), is derived in Appendix[C\.1](https://arxiv.org/html/2608.18415#A3.SS1)\. The sketch of the proof is as follows: Let the parameter along theii\-th eigenvector bexi=𝒗i⋅𝜽x\_\{i\}=\{\\bm\{v\}\}\_\{i\}\\cdot\{\\bm\{\\theta\}\}, then the dynamics of the parameter along this direction is given by \(omitiifor brevity\)

xt\+1=xt−η​λ​\(Qt∗x≤t\)t\.x\_\{t\+1\}\\;=\\;x\_\{t\}\-\\eta\\lambda\(Q\_\{t\}\*x\_\{\\leq t\}\)\_\{t\}\.\(5\)Inzz\-domain, we haveX⁡\(z\)=∑t=0∞xt​z−tX\(z\)=\\sum\_\{t=0\}^\{\\infty\}x\_\{t\}z^\{\-t\}and the convolution becomes a multiplication:

z​X​\(z\)=X⁡\(z\)−η​λ​Q​\(z\)​X​\(z\)\.zX\(z\)\\;=\\;X\(z\)\-\\eta\\lambda Q\(z\)X\(z\)\.\(6\)Dividing both sides byX⁡\(z\)X\(z\), we obtain the*characteristic transfer function*along this mode:

z−1\+η​λ​Q​\(z\)=0\.z\-1\+\\eta\\lambda Q\(z\)\\;=\\;0\.\(7\)Classical control theory tells us that any linear discrete\-time dynamic system must have all its rootszzinside the unit circle\|z\|<1\|z\|<1in order to be stable\([45](https://arxiv.org/html/2608.18415#bib.bib3)\), which gives the results\.

From Proposition[2\.1](https://arxiv.org/html/2608.18415#S2.Thmtheorem1)and Corollary[2\.2](https://arxiv.org/html/2608.18415#S2.Thmtheorem2), we see that the original EoS\([13](https://arxiv.org/html/2608.18415#bib.bib4)\)and adaptive EoS\([12](https://arxiv.org/html/2608.18415#bib.bib26)\)are special cases of this general stability condition\. The boxed bound \([♣‣2\.1](https://arxiv.org/html/2608.18415#S2.Ex2)\) is an*available*stability limit on a frozen quadratic, which is exact in this case, as demonstrated in Appendix[C\.3](https://arxiv.org/html/2608.18415#A3.SS3): each eigenmode is independent,Γ\\Gammais a functional ofQQalone, andχ:=η​S​Γ/2≤1\\chi:=\\eta S\\Gamma/2\\leq 1is necessary for no mode to be linearly unstable\. Usingχ\\chias the*observed*hovering edge of a nonlinear trajectory is a different statement, which requires an additional*occupation hypothesis*: over the relevant horizon the update𝒖\{\\bm\{u\}\}keeps realizing the worst\-case mode of that stability limit\. In other words, it stays inspan⁡\(𝒗max\)\\operatorname\{span\}\(\{\\bm\{v\}\}\_\{\\max\}\)of a fixed \(preconditioned\) Hessian𝑷t−1/2𝑯t𝑷t−1/2\{\\bm\{P\}\}\_\{t\}^\{\-1/2\}\{\\bm\{H\}\}\_\{t\}\{\\bm\{P\}\}\_\{t\}^\{\-1/2\}over the relevant horizon\. Still, the marginal stability condition \([♣‣2\.1](https://arxiv.org/html/2608.18415#S2.Ex2)\) remains an exact stability limit for the frozen quadratic system on which the original theory is based\. It does not, by itself, imply that a nonlinear deep learning trajectory must hover atχ=1\\chi=1\. It is noteworthy thatQQdoes not play a role in the occupation hypothesis\. Section[3](https://arxiv.org/html/2608.18415#S3)tests the occupation hypothesis first on vanilla GD, where𝑷=I\{\\bm\{P\}\}=I,Q≡1Q\\equiv 1andΓ=1\\Gamma=1, so systematic mispredictionsχ\>1\\chi\>1cannot be blamed on a missing momentum or on a preconditioning artefact\.

## 3Optimizer\-Dependent Offsets

The extended formulation of the edge of stability \([♣‣2\.1](https://arxiv.org/html/2608.18415#S2.Ex2)\) allows us to test Cohen’s formulation of EoS in various optimizers\. We train a 5\-layer MLP with 200 hidden units withtanh\\mathrm\{tanh\}activation on CIFAR\-10\([29](https://arxiv.org/html/2608.18415#bib.bib47)\)using different full\-batch optimizers and learning rates\. Specifically, we use gradient descent \(GD\), GD with Heavy Ball \(GDM\), GD with Nesterov’s momentum \(GDN\), Adam\([27](https://arxiv.org/html/2608.18415#bib.bib40)\), AdamW\([37](https://arxiv.org/html/2608.18415#bib.bib41)\), AdaGrad\([16](https://arxiv.org/html/2608.18415#bib.bib42)\), RMSprop\([55](https://arxiv.org/html/2608.18415#bib.bib43)\), AdaFactor\([53](https://arxiv.org/html/2608.18415#bib.bib50)\), NAdam\([15](https://arxiv.org/html/2608.18415#bib.bib48)\), PAdam\([8](https://arxiv.org/html/2608.18415#bib.bib51)\), AMSGrad\([50](https://arxiv.org/html/2608.18415#bib.bib49)\), GD with a quasi\-hyperbolic momentum \(QHM\)\([40](https://arxiv.org/html/2608.18415#bib.bib44)\), and GD with a Grokfast\-like two\-pole filter\([30](https://arxiv.org/html/2608.18415#bib.bib37)\)\. For each optimizer, we vary the learning rates on a geometrically scaled grid of five values, depending on the optimizer family\. The results are summarized in Figure[1](https://arxiv.org/html/2608.18415#S1.F1)and Figures[8](https://arxiv.org/html/2608.18415#A5.F8)and[9](https://arxiv.org/html/2608.18415#A5.F9), and Table[6](https://arxiv.org/html/2608.18415#A5.T6)in Appendix[E](https://arxiv.org/html/2608.18415#A5)\. To measure the discrepancy between the realized and theoretical edge of stability, we define the*relative*edge of stabilityχ:=η​S​Γ/2\\chi:=\\eta S\\Gamma/2as a unit\-free measure\.χ=1\\chi=1corresponds to theoretical exactness, and is displayed as dashed lines\. We additionally summarize the range ofχ\\chifor different \(optimizer, lr\) in Figure[3](https://arxiv.org/html/2608.18415#S3.F3)\. The range is computed as the median ofχ\\chiin the interval between first enteringχ\>0\.75\\chi\>0\.75and last exitingχ<0\.75\\chi<0\.75during each training trajectory\.

Figure 3:Family\-wise ranges of per\-setting interval medians\.Medians ofχ\\chitaken from the interval between firstχ\>0\.75\\chi\>0\.75and the final downward exit\. The previous formulation of the edge of stabilityχ≤1\\chi\\leq 1\([13](https://arxiv.org/html/2608.18415#bib.bib4);[12](https://arxiv.org/html/2608.18415#bib.bib26)\)is occasionally violated by large, optimizer\-dependent offsets\. Our*realized*EoSζ\\zetahas its median universally upper bounded by near 1\.0, which is more consistent with the name “edge” of stability\.We observe that the deviation ofχ\\chivaries in a very large rangeχ∈\[0\.79,21\.07\]\\chi\\in\[0\.79,21\.07\]in a systematic manner\. Different optimizer families exhibit different patterns of deviation, and the offset generally increases withη\\eta, as observed in Figure[1](https://arxiv.org/html/2608.18415#S1.F1)and Figures[8](https://arxiv.org/html/2608.18415#A5.F8)and[9](https://arxiv.org/html/2608.18415#A5.F9)in Appendix[E](https://arxiv.org/html/2608.18415#A5)\. Offsets lower than 1\.0 are easy to interpret: either the trajectory remains strictly within the stable region, or the optimizer is not able to drive the trajectory to the edge of stability\. Those with offsets greater than 1\.0 are more interesting\. Complex filters, such as QHM and two\-pole, raiseχ\\chiup to1\.5761\.576and2\.6462\.646, respectively\. At one extreme, AdaGrad severely violates theχ≤1\\chi\\leq 1bound to reachχ=21\.070\\chi=21\.070\.

Another observation that draws our attention is that even the vanilla full\-batch GD violates our expectation to stay nearχ=1\\chi=1: Table[6](https://arxiv.org/html/2608.18415#A5.T6)reports in\-window medianχ\\chiof0\.9040\.904,1\.1301\.130,1\.2461\.246,1\.4781\.478,1\.6241\.624across the five learning rates\. Its four of five cases sit aboveχ=1\\chi=1bound, and the observed edge systematically increases withη\\eta\. Since GD uses𝑷=I\{\\bm\{P\}\}=IandQ≡1Q\\equiv 1, henceΓ=1\\Gamma=1andχ=η​S/2\\chi=\\eta S/2, this offset cannot be a missingΓ⁡\(β\)\\Gamma\(\\beta\)\. By contrast, Heavy Ball, Adam, and AdamW stay inside theχ≤1\\chi\\leq 1bound, with global median0\.9490\.949,0\.9600\.960, and0\.9710\.971, respectively\. The momentum\-induced gaugeΓ\\Gammais what those stateful methods relieves from the violation of the edge of stability by GD\.

The results not only imply a stabilization mechanism\([14](https://arxiv.org/html/2608.18415#bib.bib7)\)that keeps the trajectory stable even outside the guaranteed region of stability by consistently reducing the sharpness, but also suggest that*the attractor ofχ\\chiitself lies outside the theoretically predicted stable margin*\. These optimizer\-dependent offsets raise a significant concern that the current formulation of the edge of stability is incomplete, especially in how it identifies the empirical edge withλmax\\lambda\_\{\\max\}rather than with the curvature along the update that is actually taken\.

## 4Realized Edge of Stability

The previous observation calls for an update to our formulation of the edge of stability to account for the*actual*training dynamics that are representative of the true stability threshold\. Previous works\([32](https://arxiv.org/html/2608.18415#bib.bib52);[42](https://arxiv.org/html/2608.18415#bib.bib36);[23](https://arxiv.org/html/2608.18415#bib.bib35)\)give us a hint to this problem: the worst\-case eigenmodespan⁡\(𝒗max\)\\operatorname\{span\}\(\{\\bm\{v\}\}\_\{\\max\}\)that defined the boundary of the region of stability in \([♣‣2\.1](https://arxiv.org/html/2608.18415#S2.Ex2)\) may not actually be taken as update directions\. This idea encourages us to examine the actual update direction𝒖\{\\bm\{u\}\}that the optimizer executes during training\. We go back to the generalized dynamical system formulation in equation \([♠‣2](https://arxiv.org/html/2608.18415#S2.Ex1)\), and ask:

Given the update direction𝒖\{\\bm\{u\}\}that the optimizer\(𝑷,Q\)\(\{\\bm\{P\}\},Q\)actually takes, where is the local quadratic stability boundary along this particular ray?

To answer this question, fix the iteration indexttand consider a ray in the parameter space along the update direction𝒖\{\\bm\{u\}\}from the current parameter𝜽\{\\bm\{\\theta\}\}:

𝜽⁡\(s\):=𝜽−s​𝒖,0≤s≤1\.\{\\bm\{\\theta\}\}\(s\)\\;:=\\;\{\\bm\{\\theta\}\}\-s\{\\bm\{u\}\},\\quad 0\\leq s\\leq 1\.\(8\)Then the Taylor expansion of the objective function along this ray is given by:

ℒ⁡\(𝜽⁡\(s\)\)=ℒ⁡\(𝜽\)−s​𝒈⊤​𝒖\+s22​𝒖⊤​∇𝜽2ℒ​\(𝜽−ξ​s​𝒖\)​𝒖,for some​ξ∈\[0,1\]\.\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\(s\)\)\\;=\\;\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)\-s\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\+\\frac\{s^\{2\}\}\{2\}\{\\bm\{u\}\}^\{\\top\}\\nabla\_\{\{\\bm\{\\theta\}\}\}^\{2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-\\xi s\{\\bm\{u\}\}\)\{\\bm\{u\}\},\\quad\\text\{for some \}\\xi\\in\[0,1\]\.\(9\)The term𝑯¯η,𝒖:=∇𝜽2​ℒ​\(𝜽−ξ​s​𝒖\)\\bar\{\\bm\{H\}\}\_\{\\eta,\{\\bm\{u\}\}\}:=\\nabla\_\{\{\\bm\{\\theta\}\}\}^\{2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-\\xi s\{\\bm\{u\}\}\)is the weighted secant Hessian of the objective function along the ray that can be obtained from the current Hessian:

𝑯¯η,𝒖:=2​∫01\(1−s\)​∇𝜽2ℒ​\(𝜽−s​η​𝒖\)​𝑑s\.\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\\;:=\\;2\\int\_\{0\}^\{1\}\(1\-s\)\\nabla\_\{\{\\bm\{\\theta\}\}\}^\{2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-s\\eta\{\\bm\{u\}\}\)ds\.\(10\)Define the*alignment score*A:=𝒈⊤​𝒖A:=\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}and the*curvature load*B:=η​𝒖⊤​𝑯¯η,𝒖​𝒖/2B:=\\eta\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}/2\. Then

ℒ⁡\(𝜽⁡\(s\)\)=ℒ⁡\(𝜽\)−s​A\+s2η​B\.\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\(s\)\)\\;=\\;\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)\-sA\+\\frac\{s^\{2\}\}\{\\eta\}B\.\(11\)Assuming net\-zero improvementℒ⁡\(𝜽⁡\(s\)\)=ℒ⁡\(𝜽\)\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\(s\)\)=\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\), we have a quadratic equation inss, which has a nontrivial solutions∗=η​A/Bs\_\{\*\}=\\eta A/B\. The actual step sizeη\\etashould be smaller thans∗s\_\{\*\}to ensure positive improvement\. This yields the local quadratic stability boundary with respect to the update direction𝒖\{\\bm\{u\}\}deliberately chosen by the optimizer, which we call therealized edge of stabilityζ\\zeta:

ζ:=ηs∗=BA=η2​𝒖⊤​𝑯¯η,𝒖​𝒖𝒈⊤​𝒖≤1\.\\displaystyle\\zeta\\;:=\\;\\frac\{\\eta\}\{s\_\{\*\}\}\\;=\\;\\frac\{B\}\{A\}\\;=\\;\\frac\{\\eta\}\{2\}\\frac\{\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\}\{\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\}\\leq 1\.\(♢\\diamondsuit\)It is helpful to notice that, by the chain rule, the alignment score is*exactly*the learning rate\-compensated loss drop rate for the full\-batch update\([31](https://arxiv.org/html/2608.18415#bib.bib38)\):

A=𝒈⊤​𝒖=∇𝜽ℒ⊤​\(−d​𝜽η​d​t\)=−1η​d​ℒd​t=−d​ℒd​s\.A\\;=\\;\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\\;=\\;\\nabla\_\{\{\\bm\{\\theta\}\}\}\{\\mathcal\{L\}\}^\{\\top\}\\left\(\-\\frac\{\\mathrm\{d\}\{\\bm\{\\theta\}\}\}\{\\eta\\mathrm\{d\}t\}\\right\)\\;=\\;\-\\frac\{1\}\{\\eta\}\\frac\{\\mathrm\{d\}\{\\mathcal\{L\}\}\}\{\\mathrm\{d\}t\}\\;=\\;\-\\frac\{\\mathrm\{d\}\{\\mathcal\{L\}\}\}\{\\mathrm\{d\}s\}\.\(12\)In other words, the alignment scoreAAmeasures the actual*learning work*done by the optimizer to decrease the loss along the update direction𝒖\{\\bm\{u\}\}, with a time granularity \(sampling interval\)η​d​t=d​s\\eta\\mathrm\{d\}t=\\mathrm\{d\}s\. Similarly, the curvature loadBBmeasures the reduction in the loss drop rate due to the curvature of the loss landscape along the update direction𝒖\{\\bm\{u\}\}:

B=η2​𝒖⊤​𝑯¯η,𝒖​𝒖=A\+ℒ⁡\(𝜽−η​𝒖\)−ℒ⁡\(𝜽\)η=A\+Δ​ℒη=−d​ℒd​s−\(−Δ​ℒΔ​s\),B\\;=\\;\\frac\{\\eta\}\{2\}\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\\;=\\;A\+\\frac\{\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-\\eta\{\\bm\{u\}\}\)\-\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)\}\{\\eta\}\\;=\\;A\+\\frac\{\\Delta\{\\mathcal\{L\}\}\}\{\\eta\}\\;=\\;\-\\frac\{\\mathrm\{d\}\{\\mathcal\{L\}\}\}\{\\mathrm\{d\}s\}\-\\left\(\-\\frac\{\\Delta\{\\mathcal\{L\}\}\}\{\\Delta s\}\\right\),\(13\)whereΔ​s=η​Δ​t=η\\Delta s=\\eta\\Delta t=\\eta\(sinceΔ​t=1\\Delta t=1for the iteration indices\)\. Borrowing a physics analogy, if we casually refer to the loss drop rateAAas the*learning power*, we can call the curvature loadBBthe*learning power dissipation*by the loss landscape along the update direction𝒖\{\\bm\{u\}\}\. Therefore,ζ\\zetais a unit\-free ratio of the learning power dissipation to the learning power output of the optimizer\(𝑷,Q\)\(\{\\bm\{P\}\},Q\)\. In this sense, we may call the realized EoSζ\\zetathelearning impedanceof the task\.

To better understand the relationship between the realizedζ\\zetaand the relative edge of stabilityχ\\chi, let us define a few more quantities and decompose the realized EoSζ\\zetainto factors involvingχ\\chi\. The directional smoothness from previous works\([32](https://arxiv.org/html/2608.18415#bib.bib52);[42](https://arxiv.org/html/2608.18415#bib.bib36);[23](https://arxiv.org/html/2608.18415#bib.bib35)\),D∥⋅∥D^\{\\\|\\cdot\\\|\}, is a normalized curvature load, written in our notation asD∥⋅∥𝑷:=2B/\(η∥𝒖∥𝑷2\)D^\{\\\|\\cdot\\\|\_\{\{\\bm\{P\}\}\}\}:=2B/\(\\eta\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\)\. Recall that the maximum sharpnessSSin \([♣‣2\.1](https://arxiv.org/html/2608.18415#S2.Ex2)\) is defined asS=λmax\(𝑷−1/2𝑯𝑷−1/2\)=max𝒖≠0𝒖⊤𝑯𝒖/∥𝒖∥𝑷2S=\\lambda\_\{\\max\}\(\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}\)=\\max\_\{\{\\bm\{u\}\}\\neq 0\}\{\\bm\{u\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{u\}\}/\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\. If the Hessian𝑯\{\\bm\{H\}\}remains almost constant along the ray, then𝑯¯η,𝒖≃𝑯\\bar\{\\bm\{H\}\}\_\{\\eta,\{\\bm\{u\}\}\}\\simeq\{\\bm\{H\}\}impliesS≃max𝒖≠0D∥⋅∥𝑷S\\simeq\\max\_\{\{\\bm\{u\}\}\\neq 0\}D^\{\\\|\\cdot\\\|\_\{\{\\bm\{P\}\}\}\}\. Hence,D∥⋅∥𝑷≤SD^\{\\\|\\cdot\\\|\_\{\{\\bm\{P\}\}\}\}\\leq S\. Therefore, we can define a*spatial participation factor*ς:=D∥⋅∥𝑷/S\\varsigma:=D^\{\\\|\\cdot\\\|\_\{\{\\bm\{P\}\}\}\}/S, a unit\-free measure of how much of the maximally available sharpness is actuated by the curvature along the update direction𝒖\{\\bm\{u\}\}taken by the optimizer\. Its value becomes 1 when we take the worst\-case direction𝒖=span⁡\(𝒗max\)\{\\bm\{u\}\}=\\operatorname\{span\}\(\{\\bm\{v\}\}\_\{\\max\}\), and is lower when we take other directions\.

Also, recall thatΓ\\Gammain \([♣‣2\.1](https://arxiv.org/html/2608.18415#S2.Ex2)\) is the gain of the gradient filterQQat the escape frequency of its root locus\. In other words, at this escape frequency, the optimizer applies the update𝒖=𝑷−1​\(Γ​𝒈\)\{\\bm\{u\}\}=\{\\bm\{P\}\}^\{\-1\}\(\\Gamma\{\\bm\{g\}\}\)\. Equivalently,𝒈=Γ−1​𝑷​𝒖\{\\bm\{g\}\}=\\Gamma^\{\-1\}\{\\bm\{P\}\}\{\\bm\{u\}\}\. If the update realizes the escape mode of Proposition[2\.1](https://arxiv.org/html/2608.18415#S2.Thmtheorem1), then𝒖=𝑷−1​\(Γ​𝒈\)\{\\bm\{u\}\}=\{\\bm\{P\}\}^\{\-1\}\(\\Gamma\{\\bm\{g\}\}\)and the learning power takes the formA=\(Γ−1​𝑷​𝒖\)⊤​𝒖=‖𝒖‖𝑷2/ΓA=\(\\Gamma^\{\-1\}\{\\bm\{P\}\}\{\\bm\{u\}\}\)^\{\\top\}\{\\bm\{u\}\}=\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}/\\Gamma\. Therefore, we can define a*temporal calibration factor*τ:=A/\(‖𝒖‖𝑷2/Γ\)=Γ​A/‖𝒖‖𝑷2\\tau:=A/\(\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}/\\Gamma\)=\\Gamma A/\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}, a unit\-free measure of how closely the actual learning power agrees with the power predicted by that escape mode\. Its value is 1 when the filtered stream is in phase withΓ​𝒈\\Gamma\{\\bm\{g\}\}in the𝑷\{\\bm\{P\}\}\-inner product, and is identically11for vanilla GD \(Γ=1\\Gamma=1,𝒖=𝒈\{\\bm\{u\}\}=\{\\bm\{g\}\}\)\.

Using these factors, we can now*exactly*decompose the realized EoSζ\\zetainto the following factors:

ζ=η​S​Γ2⋅𝒖⊤​𝑯¯η,𝒖​𝒖S​‖𝒖‖𝑷2⋅‖𝒖‖𝑷2Γ​𝒈⊤​𝒖=χ​ςτ=\(relative EoS\)×\(spatial participation\)\(temporal calibration\),\\displaystyle\\zeta\\;=\\;\\frac\{\\eta S\\Gamma\}\{2\}\\cdot\\frac\{\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\}\{S\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\}\\cdot\\frac\{\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\}\{\\Gamma\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\}\\;=\\;\\chi\\frac\{\\varsigma\}\{\\tau\}\\;=\\;\(\\text\{relative EoS\}\)\{\\times\}\\frac\{\(\\text\{spatial participation\}\)\}\{\(\\text\{temporal calibration\}\)\},\(♡\\heartsuit\)Thereforeζ=χ\\zeta=\\chiif and only ifς=τ\\varsigma=\\tau\. The factorsς\\varsigmaandτ\\tauaccount for the spatial and temporal discrepancies between the actual update and the binding mode of the available certificate, and their equivalence is not trivially guaranteed\. Refer to Corollary[5\.1](https://arxiv.org/html/2608.18415#S5.Thmtheorem1)and Appendix[B\.5](https://arxiv.org/html/2608.18415#A2.SS5)for the exact condition and the GD reductionτ≡1\\tau\\equiv 1\.

## 5Coincidence and Nonseparability

We now delve into the implications of the new formulation \([♡‣4](https://arxiv.org/html/2608.18415#S4.Ex4)\) in more detail\. We are interested in the conditions under which the two valuesζ\\zetaandχ\\chiare equal and when they become different\. The following corollary is the exact content of \([♡‣4](https://arxiv.org/html/2608.18415#S4.Ex4)\)\.

###### Corollary 5\.1\(Coincidence\)\.

\[[proof](https://arxiv.org/html/2608.18415#A2.SS5)\]Fix an iteration and assumeη,S,Γ\>0\\eta,S,\\Gamma\>0,𝐮≠𝟎\{\\bm\{u\}\}\\neq\{\\bm\{0\}\}, andA=𝐠⊤​𝐮≠0A=\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\\neq 0\. Then \([♡‣4](https://arxiv.org/html/2608.18415#S4.Ex4)\) is an identity, and

ζ=χ⟺ς=τ⟺𝒖⊤𝑯¯η,𝒖𝒖=SΓ𝒈⊤𝒖\.\\zeta\\;=\\;\\chi\\quad\\Longleftrightarrow\\quad\\varsigma\\;=\\;\\tau\\quad\\Longleftrightarrow\\quad\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\\;=\\;S\\Gamma\\,\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\.\(14\)The separate conditionsς=1\\varsigma=1andτ=1\\tau=1are sufficient for coincidence and not necessary: they impose two scalar constraints where \([14](https://arxiv.org/html/2608.18415#S5.E14)\) imposes one\. In particular, vanilla GD \(𝐏=I\{\\bm\{P\}\}=I,Q≡1Q\\equiv 1\) hasΓ=1\\Gamma=1andτ≡1\\tau\\equiv 1at every step, soζ=χ​ς\\zeta=\\chi\\varsigmaand coincidence reduces to occupationς=1\\varsigma=1\. At the realized edgeζ=1\\zeta=1one must haveχ=τ/ς\\chi=\\tau/\\varsigma\.

The proof is the cancellation in Appendix[B\.5](https://arxiv.org/html/2608.18415#A2.SS5)\. In plain language, the original edgeχ\\chiequals the realized edgeζ\\zetaexactly when spatial participationς\\varsigmamatches temporal calibrationτ\\tau\. For GD, since itsτ\\tauis fixed to unity, the systematic violation ofχ\>1\\chi\>1immediately translates toς<1\\varsigma<1\. We observe that Adam, AdamW, and Heavy Ball in Section[3](https://arxiv.org/html/2608.18415#S3)have bothχ\\chiandζ\\zetanear11\. This translates toς≈τ\\varsigma\\approx\\tau\.

The following corollary isolates the one special case in which coincidence is structural rather than accidental, namelyς=τ=1\\varsigma=\\tau=1rather than a mere cancellationς=τ\\varsigma=\\tau\.

###### Corollary 5\.2\(Coincidence on a mode\-locked quadratic system\)\.

\[[proof](https://arxiv.org/html/2608.18415#A2.SS6)\]Assume the following over the memory horizon of the filterQQ\. \(Q1\) The loss is quadratic over the memory horizon:𝐠s=𝐇⁡\(𝛉s−𝛉⋆\)\{\\bm\{g\}\}\_\{s\}=\{\\bm\{H\}\}\(\{\\bm\{\\theta\}\}\_\{s\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\)and therefore𝐇¯η,𝐮=𝐇\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}=\{\\bm\{H\}\}\. \(Q2\) The preconditioner is frozen:𝐏s≡𝐏≻0\{\\bm\{P\}\}\_\{s\}\\equiv\{\\bm\{P\}\}\\succ 0\. \(Q3\) The trajectory is locked to the sharpest preconditioned eigenmode:𝛉s−𝛉⋆=xs𝐏−1/2𝐯max\{\\bm\{\\theta\}\}\_\{s\}\-\{\\bm\{\\theta\}\}\_\{\\star\}=x\_\{s\}\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{v\}\}\_\{\\max\}for scalarsxsx\_\{s\}, wherext≠0x\_\{t\}\\neq 0\. \(Q4\) The root locus ofQQescapes at the Nyquist frequency:Γ=Q⁡\(−1\)\\Gamma=Q\(\-1\), and the locked mode is in period\-2 oscillation,xt−k=\(−1\)k​xtx\_\{t\-k\}=\(\-1\)^\{k\}x\_\{t\}\. Then the alignment score is automatically nondegenerate,A=Γ​S2​xt2\>0A=\\Gamma S^\{2\}x\_\{t\}^\{2\}\>0, andς=τ=1\\varsigma=\\tau=1, and therefore the realized EoSζ\\zetais exactly the relative edge of stabilityχ\\chi:ζ=χ=η​S​Q​\(−1\)/2\\zeta=\\chi=\\eta SQ\(\-1\)/2\. Dropping the period\-2 part of \(Q4\), even while keepingΓ=Q⁡\(−1\)\\Gamma=Q\(\-1\), preservesς=1\\varsigma=1but gives

τ=Γ​xt\(Q∗x≤t\)t,\\tau\\;=\\;\\frac\{\\Gamma x\_\{t\}\}\{\(Q\*x\_\{\\leq t\}\)\_\{t\}\},\(15\)which equals 1 only when the filtered stream is in phase with the gradient at the escape gain\.

The proof is provided in Appendix[B\.6](https://arxiv.org/html/2608.18415#A2.SS6)\. Corollary[5\.2](https://arxiv.org/html/2608.18415#S5.Thmtheorem2)reveals assumptions that are used to derive the original boundaryχ≤1\\chi\\leq 1in previous works\([13](https://arxiv.org/html/2608.18415#bib.bib4);[12](https://arxiv.org/html/2608.18415#bib.bib26)\)\. In this worst\-mode\-locked quadratic regime, our realized EoSζ≤1\\zeta\\leq 1of \([♢‣4](https://arxiv.org/html/2608.18415#S4.Ex3)\) is exactly the original EoSχ≤1\\chi\\leq 1of \([♣‣2\.1](https://arxiv.org/html/2608.18415#S2.Ex2)\)\. Our formulation generalizes the previous works in this precise sense\.

Each of the assumptions \(Q1\)–\(Q4\) fails in an identifiable way, and the observations of Section[3](https://arxiv.org/html/2608.18415#S3)split across those failure types\. The decomposition of \([♡‣4](https://arxiv.org/html/2608.18415#S4.Ex4)\), therefore, not only serves as an evidence thatζ\\zetageneralizesχ\\chi, but also a useful diagnostic toolkit for analyzing the edge of stability: \(Q1\) fails whenever the third\-order term along the ray is non\-negligible, so that𝑯¯η,𝒖≠𝑯\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\\neq\{\\bm\{H\}\}; \(Q2\) fails for adaptive preconditioners, whose𝑷t\{\\bm\{P\}\}\_\{t\}drifts over the memory horizon ofQQ; \(Q3\) fails whenever the update actuates a non\-maximal direction,*including*vanilla GD, where𝒖=𝒈\{\\bm\{u\}\}=\{\\bm\{g\}\}; and \(Q4\) fails for any filter whose root locus escapes away fromz=−1z=\-1, or whose locked mode is not in period\-2 oscillation\. We can categorize the observation results in Section[3](https://arxiv.org/html/2608.18415#S3)into the following cases: GD is \(Q3\) withτ≡1\\tau\\equiv 1\(χ=1\.124\>1\\chi=1\.124\>1,ζ=0\.983≈1\\zeta=0\.983\\approx 1\); Adam, AdamW, and Heavy Ball are near coincidence afterΓ\\Gamma\-scaling \(χ\\chiat0\.9600\.960,0\.9710\.971, and0\.9490\.949againstζ\\zetaat0\.9640\.964,0\.9190\.919, and0\.9830\.983, respectively\); AdaGrad is \(Q2\)–\(Q3\) \(χ=21\.070≫1\\chi=21\.070\\gg 1, yetζ=1\.002≈1\\zeta=1\.002\\approx 1\); QHM mixes a moderateχ\\chioffset withζ=1\.000\\zeta=1\.000; and the two\-pole cascades are \(Q4\), withχ=2\.646\>1\\chi=2\.646\>1and noζ\\zetareaches the edge\. This identifies the actual causes of the offsets as elaborated in Section[6](https://arxiv.org/html/2608.18415#S6)\.

We now record what Corollary[5\.1](https://arxiv.org/html/2608.18415#S5.Thmtheorem1)implies for the available stability limit:χ\\chideterminesζ\\zetaonly through the ratioς/τ\\varsigma/\\tau, which the gauges\(S,Γ\)\(S,\\Gamma\)do not control\.

###### Theorem 5\.3\(Insufficiency ofχ\\chifor the realized edge\)\.

\[[proof](https://arxiv.org/html/2608.18415#A2.SS8)\]Assumeη,S,Γ\>0\\eta,S,\\Gamma\>0,𝐮≠𝟎\{\\bm\{u\}\}\\neq\{\\bm\{0\}\}, andA≠0A\\neq 0\. By Corollary[5\.1](https://arxiv.org/html/2608.18415#S5.Thmtheorem1),ζ=χ\\zeta=\\chiif and only ifς=τ\\varsigma=\\tau\. Consequently\(η,S,Γ\)\(\\eta,S,\\Gamma\)determineζ\\zetaif and only if they determine the ratioς/τ\\varsigma/\\tau\. For vanilla GD \(𝐏=I\{\\bm\{P\}\}=I,Q≡1Q\\equiv 1\) one hasτ≡1\\tau\\equiv 1andζ=χ​ς\\zeta=\\chi\\varsigma, soχ\\chideterminesζ\\zetaif and only if occupationς\\varsigmais a function of\(η,S\)\(\\eta,S\)alone\. This already fails on any frozen quadratic in dimension at least22whose gradient is not an eigen\-direction of𝐇\{\\bm\{H\}\}: thenς\\varsigmais the Rayleigh ratio𝐠⊤​𝐇​𝐠/\(S​‖𝐠‖2\)\{\\bm\{g\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{g\}\}/\(S\\\|\{\\bm\{g\}\}\\\|^\{2\}\)and is not determined byS=λmax​\(𝐇\)S=\\lambda\_\{\\max\}\(\{\\bm\{H\}\}\)\.

\(a\) Decomposition of the realized edge of stabilityζ\\zetafrom a single experiment\.\(b\) Decomposition of optimizer\-family\-wise contributions toζ\\zeta, median from an LR sweep of size 5\.
Figure 4:EoS stack plot\.Decomposition of the realized EoSζ\\zetainto factors in log\-scale, involving the \(relative\) edge of stabilityχ\\chi\(redarea and line\), the spatial participation factorς\\varsigma\(bluearea and line\), and the temporal calibration factorτ\\tau\(greyoverlay and black line\)\. Note thatlog⁡τ\\log\\tauoperates in the opposite direction by subtraction\.The proof of Theorem[5\.3](https://arxiv.org/html/2608.18415#S5.Thmtheorem3)is the identity of Corollary[5\.1](https://arxiv.org/html/2608.18415#S5.Thmtheorem1)together with the GD reductionτ≡1\\tau\\equiv 1, detailed in Appendix[B\.8](https://arxiv.org/html/2608.18415#A2.SS8)\. It is important to mention that this result does not falsify previous works’ contributions on formulating the EoS\([13](https://arxiv.org/html/2608.18415#bib.bib4);[12](https://arxiv.org/html/2608.18415#bib.bib26)\), but rather we circumscribe the scope of their claims to the case where the update direction is fixed to the fixed maximum curvature, which is condition \(Q3\) of Corollary[5\.2](https://arxiv.org/html/2608.18415#S5.Thmtheorem2)\. Violation of this condition implies that the optimizer is more than just a prescribed algorithmic procedure, but rather an active decision\-maker that chooses the edge it wants to occupy among the available options provided by the current loss landscape\.

## 6Discussion

Previous formulations of the edge of stability\([13](https://arxiv.org/html/2608.18415#bib.bib4);[12](https://arxiv.org/html/2608.18415#bib.bib26);[23](https://arxiv.org/html/2608.18415#bib.bib35)\)and its generalized version in Proposition[2\.1](https://arxiv.org/html/2608.18415#S2.Thmtheorem1)predict available worst\-case marginalityχ\\chi\. Our realized EoSζ\\zetareveals whether and how this available margin is actually trod upon by the optimizer\. This distinction offers a new way to understand the training dynamics of different optimizer families\.

### 6\.1Diagnostic Applications

If we take the logarithm of the multiplicative relationship in \([♢‣4](https://arxiv.org/html/2608.18415#S4.Ex3)\), we can decompose the realized EoSζ\\zetainto additive factors:

log⁡ζ=log⁡χ\+log⁡ς−log⁡τ\.\\log\\zeta=\\log\\chi\+\\log\\varsigma\-\\log\\tau\.\(16\)This simple additive relationship holds for every iteration, allowing us to draw anEoS stack plot, as in Figure[4](https://arxiv.org/html/2608.18415#S5.F4), to visualize how each factor contributes to the realized EoSζ\\zetaduring the training process\. We can use this stack plot as a new diagnostic tool to study training dynamics online, for various instances and families of optimizers\.

The first thing we notice from the EoS stack plot in Figure[4](https://arxiv.org/html/2608.18415#S5.F4)is that the spatial participation factorς\\varsigmacompensates forχ\\chieven without a nontrivial preconditioner\. Vanilla GD hasτ≡1\\tau\\equiv 1, so the stack islog⁡ζ=log⁡χ\+log⁡ς\\log\\zeta=\\log\\chi\+\\log\\varsigma\. The compansation fromχ=1\.124\\chi=1\.124toζ=0\.983\\zeta=0\.983is solely due toς<1\\varsigma<1\. The same happens to AdaGrad’s extreme offsetχ≈21\.1\\chi\\approx 21\.1, which is mostly compensated byς\\varsigma, resulting inζ≈1\\zeta\\approx 1\. In other words, the optimizer often selects updates𝒖\{\\bm\{u\}\}in nonmaximal sharpness directions, and GD already does so\. This explains why the previous edge of stability formulation based on the worst\-mode update𝒖max\{\\bm\{u\}\}\_\{\\max\}was not sufficient to predict the actual EoSζ\\zeta\. Another interesting observation is that the temporal calibration factorτ\\taucan act in either direction, depending on the type of gradient filterQQ\. For Adam and QHM,τ\\tauis mostly negative, but for more complicated filters like two\-pole filters,τ\\taucan be positive, pulling the dynamics away from the edge of stability\. Adam’s near\-coincidenceχ≈ζ≈1\\chi\\approx\\zeta\\approx 1is thereforeς≈τ\\varsigma\\approx\\tauafterΓ\\Gammais included inχ\\chi, not a restoration of occupationς=1\\varsigma=1\. This single observation already explains the main cause of the family\-dependent offsets in the edge of stabilityχ\\chiand allows us to visualize how different optimizer families compensate for this effect and actively shape their training dynamics online\.

Table 1:Computation requirements\.For one full\-batch optimizer step \(Adam, params=777k, batch=50k, 8\-step Lanczos, 1×\\timesRTX 5090\)\.ζ\\zetarequires no HVPs, reducing computation time by 95\.3%\.
Figure 5:ς\\varsigmavs\.ς2\\varsigma\_\{2\}for optimizers\.

### 6\.2Cost Benefits

One additional benefit of realized EoSζ\\zetais that it does not require heavy calculation\. Since

ζ=BA=A\+Δ​ℒ/ηA=1\+Δ​ℒη​𝒈⊤​𝒖,\\zeta\\;=\\;\\frac\{B\}\{A\}\\;=\\;\\frac\{A\+\\Delta\{\\mathcal\{L\}\}/\\eta\}\{A\}=1\+\\frac\{\\Delta\{\\mathcal\{L\}\}\}\{\\eta\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\},\(17\)we can computeζ\\zetadirectly from the current gradient𝒈\{\\bm\{g\}\}, the chosen update direction𝒖\{\\bm\{u\}\}, and the resulting change in lossΔ​ℒ\\Delta\{\\mathcal\{L\}\}at each training iteration\. In comparison, previous definition of the EoS involves a maximizer on preconditioned Hessian\([13](https://arxiv.org/html/2608.18415#bib.bib4);[12](https://arxiv.org/html/2608.18415#bib.bib26);[23](https://arxiv.org/html/2608.18415#bib.bib35)\), which is not only computationally expensive, requiring multiple Hessian\-vector products \(HVPs\), but also forces us to use approximate measures\. Table[1](https://arxiv.org/html/2608.18415#S6.T1)summarizes the computation costs for each component of \([16](https://arxiv.org/html/2608.18415#S6.E16)\), all measured from the same CIFAR\-10 training using the 200\-5\-tanh model in the initial report\([13](https://arxiv.org/html/2608.18415#bib.bib4)\)\. We highlight that our formulation is both cost\-effective and*exact*for full\-batch optimization\.

### 6\.3Effect of Higher\-Order Terms

Similar to the original edge of stability\([13](https://arxiv.org/html/2608.18415#bib.bib4);[12](https://arxiv.org/html/2608.18415#bib.bib26);[23](https://arxiv.org/html/2608.18415#bib.bib35)\), Figure[1](https://arxiv.org/html/2608.18415#S1.F1)shows how actual training trajectories are attracted to, oscillate around, and are eventually pushed away from the edge of stabilityζ=1\\zeta=1\. As in previous works on EoS\([14](https://arxiv.org/html/2608.18415#bib.bib7);[43](https://arxiv.org/html/2608.18415#bib.bib58)\), we can derive sufficient conditions for the stabilization mechanism to take effect\. Since our main focus is to resolve the optimizer\-dependent offsets found in the empirical EoS, we defer this analysis to Appendix[D](https://arxiv.org/html/2608.18415#A4)\. In our formulation ofζ\\zeta, the higher\-order terms are absorbed into the weighted secant Hessian𝑯¯η,𝒖\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}and thus into the spatial participation factorς\\varsigma\. If we want to isolate the effect of the higher\-order terms, we can define the second\-order\-only spatial participation factorς2:=𝒖⊤​𝑯​𝒖/\(S​‖𝒖‖𝑷2\)\\varsigma\_\{2\}:=\{\\bm\{u\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{u\}\}/\(S\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\)and compare it withς\\varsigma, as shown in Figure[5](https://arxiv.org/html/2608.18415#S6.F5)\. Although higher\-order terms are essential for maintaining stability beyond the edge of stability, their magnitudes contribute negligibly to the realized EoSζ\\zetawhen we study the position of the edge\.

### 6\.4Limitations

This work focuses on full\-batch optimization, where the theory is exactly realized in practice\. In mini\-batch optimization, the termsA=𝒈⊤​𝒖A=\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\},B=η​𝒖⊤​𝑯​𝒖/2B=\\eta\{\\bm\{u\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{u\}\}/2, and their ratioζ=B/A\\zeta=B/Abecome stochastic due to changes in the instantaneous loss landscapeℒ\{\\mathcal\{L\}\}along the trajectory\. This makes the analysis more challenging and complicates the interpretation of the ratioζ\\zeta\. Previous works treat the incorporation of stochastic scenarios as an independent problem, which is sometimes referred to as the edge of stochastic stability \(EoSS\)[5](https://arxiv.org/html/2608.18415#bib.bib27);[4](https://arxiv.org/html/2608.18415#bib.bib14)\. We likewise leave the extension of our framework to the stochastic setting for future work\.

## 7Conclusion

The edge of stability has been studied primarily as an effect related to the maximum curvature mode of the loss landscape\. This turns out to be only half of the story\. The other half is the effect of time\-varying stateful optimizers that actually take the updates, given a loss landscape and gradient history\. Neglecting this aspect leads to optimizer\-dependent systematic offsets in estimating the edge of stability, which are largely eliminated in our framework on the*realized*edge of stability\. Decomposing the realized EoS into factors results in diagnostic tools that provide a richer understanding of training dynamics under gradient\-based optimizers\.

## References

- A\. Agarwala, F\. Pedregosa, and J\. PenningtonSecond\-order regression models exhibit progressive sharpening to the edge of stability\.InICML,Vol\.202,pp\. 169–195\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1)\.
- Ahnet al\.\(2023\)K\. Ahn, S\. Bubeck, S\. Chewi, Y\. T\. Lee, F\. Suarez, and Y\. ZhangLearning threshold neurons via edge of stability\.InNeurIPS,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Ahnet al\.\(2022\)K\. Ahn, J\. Zhang, and S\. SraUnderstanding the unstable convergence of gradient descent\.InICML,Vol\.162\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px3.p1.1)\.
- Andreyevet al\.\(2026\)A\. Andreyev, A\. Ananthkumar, M\. Walden, T\. Poggio, and P\. BeneventanoMomentum further constrains sharpness at the edge of stochastic stability\.InICML,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1),[§6\.4](https://arxiv.org/html/2608.18415#S6.SS4.p1.1)\.
- Andreyev and Beneventano \(2024\)A\. Andreyev and P\. BeneventanoEdge of stochastic stability: revisiting the edge of stability for SGD\.arXiv preprint arXiv:2412\.20553\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1),[§6\.4](https://arxiv.org/html/2608.18415#S6.SS4.p1.1)\.
- Aroraet al\.\(2022\)S\. Arora, Z\. Li, and A\. PanigrahiUnderstanding gradient descent on the edge of stability in deep learning\.InICML,Vol\.162,pp\. 948–1024\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Bertsekas \(1999\)D\. P\. BertsekasNonlinear programming\.2nd edition,Athena Scientific,Belmont, MA\.Cited by:[§1](https://arxiv.org/html/2608.18415#S1.p1.1)\.
- Chenet al\.\(2020\)J\. Chen, D\. Zhou, Y\. Tang, Z\. Yang, Y\. Cao, and Q\. GuClosing the generalization gap of adaptive gradient methods in training deep neural networks\.InIJCAI,Cited by:[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Chen and Bruna \(2023\)L\. Chen and J\. BrunaBeyond the edge of stability via two\-step gradient updates\.InICML,Vol\.202,pp\. 4330–4391\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Chenet al\.\(2024\)X\. Chen, K\. Balasubramanian, P\. Ghosal, and B\. AgrawallaFrom stability to chaos: analyzing gradient descent dynamics in quadratic regression\.Transactions on Machine Learning Research\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1)\.
- Cohenet al\.\(2025\)J\. M\. Cohen, A\. Damian, A\. Talwalkar, J\. Z\. Kolter, and J\. D\. LeeUnderstanding optimization in deep learning with central flows\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1)\.
- Cohenet al\.\(2023\)J\. M\. Cohen, B\. Ghorbani, S\. Krishnan, N\. Agarwal, S\. Medapati, M\. Badura, D\. Suo, D\. Cardoze, Z\. Nado, G\. E\. Dahl, and J\. GilmerAdaptive gradient methods at the edge of stability\.InNeurIPS 2023 Workshop on Heavy Tails in Machine Learning: Structure, Stability, and Dynamics,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[§B\.4](https://arxiv.org/html/2608.18415#A2.SS4.p2.2),[§B\.4](https://arxiv.org/html/2608.18415#A2.SS4.p3.4),[Table 2](https://arxiv.org/html/2608.18415#A3.T2),[§E\.1](https://arxiv.org/html/2608.18415#A5.SS1.SSS0.Px2.p1.1),[1st item](https://arxiv.org/html/2608.18415#S1.I1.i1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.2),[Corollary 2\.2](https://arxiv.org/html/2608.18415#S2.Thmtheorem2),[§2](https://arxiv.org/html/2608.18415#S2.p2.1),[§2](https://arxiv.org/html/2608.18415#S2.p2.2),[§2](https://arxiv.org/html/2608.18415#S2.p4.1),[Figure 3](https://arxiv.org/html/2608.18415#S3.F3),[§5](https://arxiv.org/html/2608.18415#S5.p4.1),[§5](https://arxiv.org/html/2608.18415#S5.p7.1),[§6\.2](https://arxiv.org/html/2608.18415#S6.SS2.p1.2),[§6\.3](https://arxiv.org/html/2608.18415#S6.SS3.p1.1),[§6](https://arxiv.org/html/2608.18415#S6.p1.1)\.
- Cohenet al\.\(2021\)J\. M\. Cohen, S\. Kaur, Y\. Li, J\. Z\. Kolter, and A\. TalwalkarGradient descent on neural networks typically occurs at the edge of stability\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1),[§E\.1](https://arxiv.org/html/2608.18415#A5.SS1.SSS0.Px2.p1.1),[1st item](https://arxiv.org/html/2608.18415#S1.I1.i1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p1.2),[§1](https://arxiv.org/html/2608.18415#S1.p2.2),[§2](https://arxiv.org/html/2608.18415#S2.p2.1),[§2](https://arxiv.org/html/2608.18415#S2.p4.1),[Figure 3](https://arxiv.org/html/2608.18415#S3.F3),[§5](https://arxiv.org/html/2608.18415#S5.p4.1),[§5](https://arxiv.org/html/2608.18415#S5.p7.1),[§6\.2](https://arxiv.org/html/2608.18415#S6.SS2.p1.2),[§6\.3](https://arxiv.org/html/2608.18415#S6.SS3.p1.1),[§6](https://arxiv.org/html/2608.18415#S6.p1.1)\.
- Damianet al\.\(2023\)A\. Damian, E\. Nichani, and J\. D\. LeeSelf\-stabilization: the implicit bias of gradient descent at the edge of stability\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1),[§3](https://arxiv.org/html/2608.18415#S3.p4.1),[§6\.3](https://arxiv.org/html/2608.18415#S6.SS3.p1.1)\.
- Dozat \(2016\)T\. DozatIncorporating nesterov momentum into adam\.InProceedings of the 4th International Conference on Learning Representations \(ICLR\) Workshop,Cited by:[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Duchiet al\.\(2011\)J\. Duchi, E\. Hazan, and Y\. SingerAdaptive subgradient methods for online learning and stochastic optimization\.JMLR12\(61\),pp\. 2121–2159\.Cited by:[§1](https://arxiv.org/html/2608.18415#S1.p3.1),[§2](https://arxiv.org/html/2608.18415#S2.p1.2),[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Evenet al\.\(2023\)M\. Even, S\. Pesme, S\. Gunasekar, and N\. Flammarion\(S\)GD over diagonal linear networks: implicit bias, large stepsizes and edge of stability\.InNeurIPS,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Ghoshet al\.\(2025\)A\. Ghosh, S\. M\. Kwon, R\. Wang, S\. Ravishankar, and Q\. QuLearning dynamics of deep matrix factorization beyond the edge of stability\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1),[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1)\.
- Gilmeret al\.\(2022\)J\. Gilmer, B\. Ghorbani, A\. Garg, S\. Kudugunta, B\. Neyshabur, D\. Cardoze, G\. E\. Dahl, Z\. Nado, and O\. FiratA loss curvature perspective on training instabilities of deep learning models\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1)\.
- Gitmanet al\.\(2019\)I\. Gitman, H\. Lang, P\. Zhang, and L\. XiaoUnderstanding the role of momentum in stochastic gradient methods\.InNeurIPS,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1)\.
- Hofmann \(2026\)T\. HofmannThe map behind the flow: finite\-step gradient descent as a dynamical system\.arXiv preprint arXiv:2607\.04993\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Hu and Lessard \(2017\)B\. Hu and L\. LessardControl interpretations for first\-order optimization methods\.InACC,pp\. 3114–3119\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1)\.
- Islamovet al\.\(2026\)R\. Islamov, M\. Crawshaw, J\. Cohen, and R\. GowerNon\-euclidean gradient descent operates at the edge of stability\.InICML,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px3.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1),[§1](https://arxiv.org/html/2608.18415#S1.p4.1),[§4](https://arxiv.org/html/2608.18415#S4.p1.1),[§4](https://arxiv.org/html/2608.18415#S4.p2.1),[§6\.2](https://arxiv.org/html/2608.18415#S6.SS2.p1.2),[§6\.3](https://arxiv.org/html/2608.18415#S6.SS3.p1.1),[§6](https://arxiv.org/html/2608.18415#S6.p1.1)\.
- Jastrzębskiet al\.\(2019\)S\. Jastrzębski, Z\. Kenton, N\. Ballas, A\. Fischer, Y\. Bengio, and A\. StorkeyOn the relation between the sharpest directions of DNN loss and the SGD step length\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1)\.
- Jastrzębskiet al\.\(2020\)S\. Jastrzębski, M\. Szymczak, S\. Fort, D\. Arpit, J\. Tabor, K\. Cho, and K\. GerasThe break\-even point on optimization trajectories of deep neural networks\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Jianget al\.\(2025\)K\. Jiang, J\. Cohen, and Y\. LiUnderstanding the evolution of the neural tangent kernel at the edge of stability\.InNeurIPS,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Kingma and Ba \(2015\)D\. P\. Kingma and J\. BaAdam: a method for stochastic optimization\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1),[§2](https://arxiv.org/html/2608.18415#S2.p1.2),[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Kreisleret al\.\(2023\)I\. Kreisler, M\. S\. Nacson, D\. Soudry, and Y\. CarmonGradient descent monotonically decreases the sharpness of gradient flow solutions in scalar networks and beyond\.InICML,Vol\.202,pp\. 17684–17744\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Krizhevsky \(2009\)A\. KrizhevskyLearning multiple layers of features from tiny images\.Technical reportUniversity of Toronto\.Cited by:[§E\.1](https://arxiv.org/html/2608.18415#A5.SS1.SSS0.Px1.p1.1),[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Leeet al\.\(2024\)J\. Lee, B\. G\. Kang, K\. Kim, and K\. M\. LeeGrokfast: accelerated grokking by amplifying slow gradients\.arXiv preprint arXiv:2405\.20233\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1),[§B\.4](https://arxiv.org/html/2608.18415#A2.SS4.p5.1),[§C\.2](https://arxiv.org/html/2608.18415#A3.SS2.p1.1),[§E\.1](https://arxiv.org/html/2608.18415#A5.SS1.SSS0.Px3.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p3.1),[§2](https://arxiv.org/html/2608.18415#S2.p1.2),[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Lee and Lee \(2025\)J\. Lee and K\. M\. LeeGreedy alignment principle for optimizer selection\.arXiv preprint arXiv:2512\.06370\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1),[2nd item](https://arxiv.org/html/2608.18415#S1.I1.i2.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p4.1),[§4](https://arxiv.org/html/2608.18415#S4.p1.8)\.
- Lee and Jang \(2023\)S\. Lee and C\. JangA new characterization of the edge of stability based on a sharpness measure aware of batch gradient distribution\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px3.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p4.1),[§4](https://arxiv.org/html/2608.18415#S4.p1.1),[§4](https://arxiv.org/html/2608.18415#S4.p2.1)\.
- Lessardet al\.\(2016\)L\. Lessard, B\. Recht, and A\. PackardAnalysis and design of optimization algorithms via integral quadratic constraints\.SIAM Journal on Optimization26\(1\),pp\. 57–95\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1)\.
- Lewkowyczet al\.\(2020\)A\. Lewkowycz, Y\. Bahri, E\. Dyer, J\. Sohl\-Dickstein, and G\. Gur\-AriThe large learning rate phase of deep learning: the catapult mechanism\.arXiv preprint arXiv:2003\.02218\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p1.2)\.
- Liet al\.\(2025\)X\. Li, J\. Luo, Z\. Zheng, H\. Wang, L\. Luo, L\. Wen, L\. Wu, and S\. XuOn the performance analysis of momentum method: a frequency domain perspective\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1)\.
- Litman \(2026\)E\. LitmanThe origin of edge of stability\.arXiv preprint arXiv:2604\.20446\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1)\.
- Loshchilov and Hutter \(2019\)I\. Loshchilov and F\. HutterDecoupled weight decay regularization\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1),[§2](https://arxiv.org/html/2608.18415#S2.p1.2),[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Lyuet al\.\(2022\)K\. Lyu, Z\. Li, and S\. AroraUnderstanding the generalization benefit of normalization layers: sharpness reduction\.InNeurIPS,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1)\.
- Maet al\.\(2022\)C\. Ma, D\. Kunin, L\. Wu, and L\. YingBeyond the quadratic approximation: the multiscale structure of neural network loss landscapes\.Journal of Machine Learning1\(3\),pp\. 247–267\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Ma and Yarats \(2019\)J\. Ma and D\. YaratsQuasi\-hyperbolic momentum and adam for deep learning\.ICLR\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1),[§B\.4](https://arxiv.org/html/2608.18415#A2.SS4.p5.1),[§C\.1](https://arxiv.org/html/2608.18415#A3.SS1.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p3.1),[§2](https://arxiv.org/html/2608.18415#S2.p1.2),[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Marion \(2026\)P\. MarionEdge flow: a tractable and predictive continuous\-time model for gradient descent at the edge of stability\.arXiv preprint arXiv:2606\.18080\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Mishkinet al\.\(2024\)A\. Mishkin, A\. Khaled, Y\. Wang, A\. Defazio, and R\. M\. GowerDirectional smoothness and gradient methods: convergence and adaptivity\.InNeurIPS,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px3.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p4.1),[§4](https://arxiv.org/html/2608.18415#S4.p1.1),[§4](https://arxiv.org/html/2608.18415#S4.p2.1)\.
- Mulayoff and Stich \(2026\)R\. Mulayoff and S\. U\. StichOn the stability of nonlinear dynamics in GD and SGD: beyond quadratic potentials\.InCOLT,Vol\.336,pp\. 5210–5243\.Cited by:[§6\.3](https://arxiv.org/html/2608.18415#S6.SS3.p1.1)\.
- Nesterov \(1983\)Y\. E\. NesterovA method for solving the convex programming problem with convergence rateO⁡\(1/k2\)O\(1/k^\{2\}\)\.Doklady Akademii Nauk SSSR269\(3\),pp\. 543–547\(Russian\)\.Note:English translation: Soviet Mathematics Doklady, 27\(2\):372–376, 1983Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1),[Remark B\.4](https://arxiv.org/html/2608.18415#A2.Thmtheorem4.p1.1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.2),[§2](https://arxiv.org/html/2608.18415#S2.p1.2)\.
- Ogata \(1995\)K\. OgataDiscrete\-time control systems\.2nd edition,Prentice\-Hall,Englewood Cliffs, NJ\.Cited by:[§2](https://arxiv.org/html/2608.18415#S2.p3.4)\.
- Ortega and Rheinboldt \(1970\)J\. M\. Ortega and W\. C\. RheinboldtIterative solution of nonlinear equations in several variables\.Academic Press,New York\-London\.Cited by:[§1](https://arxiv.org/html/2608.18415#S1.p1.1)\.
- Pagliardiniet al\.\(2025\)M\. Pagliardini, P\. Ablin, and D\. GrangierThe AdEMAMix optimizer: better, faster, older\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1),[§B\.4](https://arxiv.org/html/2608.18415#A2.SS4.p5.1),[§1](https://arxiv.org/html/2608.18415#S1.p3.1),[§2](https://arxiv.org/html/2608.18415#S2.p1.2)\.
- Polyak \(1964\)B\. T\. PolyakSome methods of speeding up the convergence of iteration methods\.USSR Computational Mathematics and Mathematical Physics4\(5\),pp\. 1–17\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1),[Remark B\.4](https://arxiv.org/html/2608.18415#A2.Thmtheorem4.p1.1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.2),[§2](https://arxiv.org/html/2608.18415#S2.p1.2)\.
- Qian \(1999\)N\. QianOn the momentum term in gradient descent learning algorithms\.Neural Networks12\(1\),pp\. 145–151\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1)\.
- Reddiet al\.\(2018\)S\. J\. Reddi, S\. Kale, and S\. KumarOn the convergence of Adam and beyond\.InICLR,Cited by:[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Regis and Chewi \(2026a\)E\. Regis and S\. ChewiA rod flow model for Adam at the edge of stability\.arXiv preprint arXiv:2605\.06821\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Regis and Chewi \(2026b\)E\. Regis and S\. ChewiRod flow: a continuous\-time model for gradient descent at the edge of stability\.arXiv preprint arXiv:2602\.01480\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Shazeer and Stern \(2018\)N\. Shazeer and M\. SternAdafactor: adaptive learning rates with sublinear memory cost\.InICML,Cited by:[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Song and Yun \(2023\)M\. Song and C\. YunTrajectory alignment: understanding the edge of stability phenomenon via bifurcation theory\.InNeurIPS,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Tieleman and Hinton \(2012\)T\. Tieleman and G\. HintonLecture 6\.5—\-RMSProp: divide the gradient by a running average of its recent magnitude\.Note:COURSERA: Neural Networks for Machine LearningCited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[§2](https://arxiv.org/html/2608.18415#S2.p1.2),[§3](https://arxiv.org/html/2608.18415#S3.p1.1)\.
- Van Scoyet al\.\(2018\)B\. Van Scoy, R\. A\. Freeman, and K\. M\. LynchThe fastest known globally convergent first\-order method for minimizing strongly convex functions\.IEEE Control Systems Letters2\(1\),pp\. 49–54\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px4.p1.1)\.
- Wanget al\.\(2025\)Y\. Wang, Z\. Xu, T\. Zhao, and M\. TaoGood regularity creates large learning rate implicit biases: edge of stability, balancing, and catapult\.Journal of Machine Learning Research26\(273\),pp\. 1–68\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Wanget al\.\(2022\)Z\. Wang, Z\. Li, and J\. LiAnalyzing sharpness along GD trajectory: progressive sharpening and edge of stability\.InNeurIPS,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Wuet al\.\(2024\)J\. Wu, P\. L\. Bartlett, M\. Telgarsky, and B\. YuLarge stepsize gradient descent for logistic loss: non\-monotonicity of the loss improves optimization efficiency\.InCOLT,Vol\.247,pp\. 5019–5073\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Wuet al\.\(2023\)J\. Wu, V\. Braverman, and J\. D\. LeeImplicit bias of gradient descent for logistic regression at the edge of stability\.InNeurIPS,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px2.p1.1)\.
- Xinget al\.\(2018\)C\. Xing, D\. Arpit, C\. Tsirigotis, and Y\. BengioA walk with SGD\.arXiv preprint arXiv:1802\.08770\.Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1)\.
- Zhuet al\.\(2024\)L\. Zhu, C\. Liu, A\. Radhakrishnan, and M\. BelkinCatapults in SGD: spikes in the training loss and their impact on generalization through feature learning\.InICML,Vol\.235,pp\. 62476–62509\.Cited by:[§1](https://arxiv.org/html/2608.18415#S1.p1.2)\.
- Zhuet al\.\(2023\)X\. Zhu, Z\. Wang, X\. Wang, M\. Zhou, and R\. GeUnderstanding edge\-of\-stability training dynamics with a minimalist example\.InICLR,Cited by:[Appendix A](https://arxiv.org/html/2608.18415#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.18415#S1.p2.1)\.

## Appendix ARelated Work

#### Edge of stability\.

It has been known that the learning rate constrains the curvature of the loss landscape during the course of training\([61](https://arxiv.org/html/2608.18415#bib.bib32);[24](https://arxiv.org/html/2608.18415#bib.bib33);[19](https://arxiv.org/html/2608.18415#bib.bib28)\)\. Later,[13](https://arxiv.org/html/2608.18415#bib.bib4)conducted an extensive study on this phenomenon and first coined the term*edge of stability*to describe the attractive behavior of the maximum Hessian eigenvalueλmax​\(𝑯\)\\lambda\_\{\\max\}\(\{\\bm\{H\}\}\)towards a value reciprocal to the learning rateλmax≈2/η\\lambda\_\{\\max\}\\approx 2/\\etaunder full\-batch gradient descent\. Similar phenomena have been reported under different optimization settings\([38](https://arxiv.org/html/2608.18415#bib.bib29);[12](https://arxiv.org/html/2608.18415#bib.bib26);[14](https://arxiv.org/html/2608.18415#bib.bib7);[1](https://arxiv.org/html/2608.18415#bib.bib16);[63](https://arxiv.org/html/2608.18415#bib.bib8);[5](https://arxiv.org/html/2608.18415#bib.bib27);[10](https://arxiv.org/html/2608.18415#bib.bib24);[11](https://arxiv.org/html/2608.18415#bib.bib10);[23](https://arxiv.org/html/2608.18415#bib.bib35);[4](https://arxiv.org/html/2608.18415#bib.bib14);[36](https://arxiv.org/html/2608.18415#bib.bib31)\)\. This includes full gradient descent\([13](https://arxiv.org/html/2608.18415#bib.bib4)\), mini\-batch gradient descent\([5](https://arxiv.org/html/2608.18415#bib.bib27)\), momentum\-based optimizers\([12](https://arxiv.org/html/2608.18415#bib.bib26);[4](https://arxiv.org/html/2608.18415#bib.bib14)\), and adaptive method\([12](https://arxiv.org/html/2608.18415#bib.bib26)\)such as RMSProp\([55](https://arxiv.org/html/2608.18415#bib.bib43)\)or Adam\([27](https://arxiv.org/html/2608.18415#bib.bib40)\)\. Following these lines of work, we primarily focus on the full\-batch settings where the analysis is more tractable and the effect is more pronounced\. In case of stochastic mini\-batch gradient descent,[5](https://arxiv.org/html/2608.18415#bib.bib27);[4](https://arxiv.org/html/2608.18415#bib.bib14)empirically found that the edge of stability extends beyond determinism and applies to curvature of the batch statistics\.

#### Mechanisms: catapult, progressive sharpening, self\-stabilization, and bifurcation\.

Existing explanations for the EoS phenomenon can be categorized into four mechanisms: \(1\) The*catapult*\([34](https://arxiv.org/html/2608.18415#bib.bib5)\), described prior to the formalization of the edge of stability, explains how a large learning rate drives a sudden loss spike that initiates the reduction of the curvature down to the edge of stability\. \(2\)[13](https://arxiv.org/html/2608.18415#bib.bib4);[58](https://arxiv.org/html/2608.18415#bib.bib15)demonstrate the opposite, called*progressive sharpening*, where curvature initially increases up to the EoS bound\. The EoS phenomenon also implies the existence of a coincidental stabilization mechanism that consistently reduces the curvature to the edge, causing it to hover just over the theoretical stability bound\. \(3\)[14](https://arxiv.org/html/2608.18415#bib.bib7)attribute this*self\-stabilization*to curvature compensation by third\-order derivatives\. \(4\)[25](https://arxiv.org/html/2608.18415#bib.bib54);[54](https://arxiv.org/html/2608.18415#bib.bib17)identify a period\-2*flip bifurcation*in the discrete dynamical system governed by gradient descent\. These four mechanisms form the basis of the EoS phenomenon, and many follow\-up works have sought to explain them using simpler models\.[6](https://arxiv.org/html/2608.18415#bib.bib6)use a two\-timescale decomposition;[39](https://arxiv.org/html/2608.18415#bib.bib56)employ a one\-dimensional model;[9](https://arxiv.org/html/2608.18415#bib.bib9)analyze two\-step fixed points that extract period\-2 motions;[28](https://arxiv.org/html/2608.18415#bib.bib57)study scalar linear networks; and[21](https://arxiv.org/html/2608.18415#bib.bib34)use a simple quartic model\. Continuous\-time reductions are also explored, including central flows\([11](https://arxiv.org/html/2608.18415#bib.bib10)\), Edge Flow\([41](https://arxiv.org/html/2608.18415#bib.bib11)\), and Rod Flow\([52](https://arxiv.org/html/2608.18415#bib.bib12);[51](https://arxiv.org/html/2608.18415#bib.bib13)\)\. In structured models, EoS is used as an implicit\-bias mechanism for diagonal linear networks\([17](https://arxiv.org/html/2608.18415#bib.bib19)\), logistic regression\([60](https://arxiv.org/html/2608.18415#bib.bib20);[59](https://arxiv.org/html/2608.18415#bib.bib21)\), threshold neurons\([2](https://arxiv.org/html/2608.18415#bib.bib25)\), and deep matrix factorization\([18](https://arxiv.org/html/2608.18415#bib.bib18)\);[57](https://arxiv.org/html/2608.18415#bib.bib22)unify EoS, balancing, and catapult as large\-learning\-rate biases resulting from regularity, and[26](https://arxiv.org/html/2608.18415#bib.bib23)track the NTK at EoS\. None of these works reformulate the dependence of the EoSχ=η​S​Γ/2\\chi=\\eta S\\Gamma/2on the maximum sharpness direction𝒗max\{\\bm\{v\}\}\_\{\\max\}to incorporate actual executed updates carried out*by the optimizers*, which is the gap we address\.

#### Directional smoothness\.

We are mainly inspired by prior work examining how directional smoothness influences the stability bound of full\-batch gradient descent\.[3](https://arxiv.org/html/2608.18415#bib.bib55)formalize unstable convergence in terms of directional smoothness and relative progress, finding structured oscillations rather than divergence past2/η2/\\eta\.[32](https://arxiv.org/html/2608.18415#bib.bib52)replaceλmax\\lambda\_\{\\max\}with an interaction\-aware sharpness \(batch\-gradient–Hessian coupling\), and report that this quantity hovers around a concentration measure under mini\-batch SGD\.[42](https://arxiv.org/html/2608.18415#bib.bib36);[23](https://arxiv.org/html/2608.18415#bib.bib35)investigate the directional smoothness of the executed step as a convergence diagnostic\. Their works are analogous to our realized EoSζ=B/A\\zeta=B/Aalong an update direction𝒖\{\\bm\{u\}\}; however, their quantities are not directly representative of the edge of stability, as they resort to the maximum sharpness instead of the executed update\. Furthermore, previous works have not accounted for general filtered optimizers\. Therefore, our work extends these works rather than competing with them\.

#### Gradient\-based optimizers as filters\.

Momentum methods\([48](https://arxiv.org/html/2608.18415#bib.bib45);[44](https://arxiv.org/html/2608.18415#bib.bib46)\)are linear time\-invariant \(LTI\) filters applied to the gradient stream\.[49](https://arxiv.org/html/2608.18415#bib.bib59)identified Heavy Ball\([48](https://arxiv.org/html/2608.18415#bib.bib45)\)as a discrete\-time low\-pass filter\. Control systems theory makes this viewpoint precise: first\-order methods are modeled as LTI plants in feedback with gradient stream signals𝒈\{\\bm\{g\}\}\. These systems can be analyzed using transfer functions, Bode plots, and integral quadratic constraints\([33](https://arxiv.org/html/2608.18415#bib.bib60);[22](https://arxiv.org/html/2608.18415#bib.bib61)\), and can be designed as higher\-order filters such as triple momentum\([56](https://arxiv.org/html/2608.18415#bib.bib62)\)\.[20](https://arxiv.org/html/2608.18415#bib.bib53)treat Heavy Ball\([48](https://arxiv.org/html/2608.18415#bib.bib45)\), Nesterov’s momentum\([44](https://arxiv.org/html/2608.18415#bib.bib46)\), and quasi\-hyperbolic momentum\([40](https://arxiv.org/html/2608.18415#bib.bib44)\)as a single IIR family, and map stability regions in\(β,ν\)\(\\beta,\\nu\)\.[30](https://arxiv.org/html/2608.18415#bib.bib37);[47](https://arxiv.org/html/2608.18415#bib.bib39)design a two\-timescale EMA filter with preconditioner\([27](https://arxiv.org/html/2608.18415#bib.bib40);[37](https://arxiv.org/html/2608.18415#bib.bib41)\)\.[35](https://arxiv.org/html/2608.18415#bib.bib63)apply thezz\-transform to momentum methods and interpret the coefficients as a \(time\-varying\) frequency response\.[31](https://arxiv.org/html/2608.18415#bib.bib38)proposes an adaptive technique to determine the optimal filter coefficients, e\.g\., the momentum hyperparameterβ\\beta, further developing this signal processing perspective\. In summary, momentum\-based optimizers can be understood as filters: GD acts as an identity filter, Heavy Ball and Nesterov’s momentum are one\-pole filters, QHM is a one\-tap pole\-zero filter, and two\-timescale EMAs\([18](https://arxiv.org/html/2608.18415#bib.bib18)\)such as Grokfast\([30](https://arxiv.org/html/2608.18415#bib.bib37)\)and AdEMAMix\([47](https://arxiv.org/html/2608.18415#bib.bib39)\)function as cascaded two\-pole filters\. The latter serve as practical counterparts ofQQ, whose root locus does not necessarily escape at the Nyquist frequencyz=−1z=\-1\. We draw extensively on this perspective to study the EoS phenomenon under general filtered optimizers\.

## Appendix BProofs

### B\.1The Mode Reduction

Throughout this subsection we work in the*frozen quadratic*regime on which the original edge\-of\-stability arguments are based: over the memory horizon of the filter the loss isℒ⁡\(𝜽\)=12​\(𝜽−𝜽⋆\)⊤​𝑯​\(𝜽−𝜽⋆\)\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)=\\tfrac\{1\}\{2\}\(\{\\bm\{\\theta\}\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\)^\{\\top\}\{\\bm\{H\}\}\(\{\\bm\{\\theta\}\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\)with𝑯\{\\bm\{H\}\}constant and symmetric, the preconditioner is frozen at𝑷t≡𝑷≻0\{\\bm\{P\}\}\_\{t\}\\equiv\{\\bm\{P\}\}\\succ 0, and the filter is time invariant,Qt≡QQ\_\{t\}\\equiv Q\. Let\(λi,𝒗i\)\(\\lambda\_\{i\},\{\\bm\{v\}\}\_\{i\}\)be the eigenpairs of𝑷−1/2𝑯𝑷−1/2\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}with\{𝒗i\}\\\{\{\\bm\{v\}\}\_\{i\}\\\}orthonormal, and define the*whitened*coordinates

xt\(i\):=𝒗i⊤​𝑷1/2​\(𝜽t−𝜽⋆\)\.x\_\{t\}^\{\(i\)\}\\;:=\\;\{\\bm\{v\}\}\_\{i\}^\{\\top\}\{\\bm\{P\}\}^\{1/2\}\(\{\\bm\{\\theta\}\}\_\{t\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\)\.\(18\)Two remarks on \([18](https://arxiv.org/html/2608.18415#A2.E18)\)\. The factor𝑷1/2\{\\bm\{P\}\}^\{1/2\}applies before the mode decomposition, and the offset𝜽⋆\{\\bm\{\\theta\}\}\_\{\\star\}makes the gradient proportional to the coordinate\. The shorthandxi=𝒗i⋅𝜽x\_\{i\}=\{\\bm\{v\}\}\_\{i\}\\cdot\{\\bm\{\\theta\}\}used below \([5](https://arxiv.org/html/2608.18415#S2.E5)\) incorporates both effects for brevity\.

###### Lemma B\.1\(Exact decoupling\)\.

Under the frozen quadratic regime, the update rule \([♠‣2](https://arxiv.org/html/2608.18415#S2.Ex1)\) is equivalent to the family of decoupled scalar recursions

xt\+1\(i\)=xt\(i\)−ηλi\(Q∗x≤t\(i\)\)t,i=1,…,d,x^\{\(i\)\}\_\{t\+1\}\\;=\\;x^\{\(i\)\}\_\{t\}\-\\eta\\lambda\_\{i\}\\,\(Q\*x^\{\(i\)\}\_\{\\leq t\}\)\_\{t\},\\qquad i=1,\\dots,d,\(19\)which is \([5](https://arxiv.org/html/2608.18415#S2.E5)\) with the coordinates \([18](https://arxiv.org/html/2608.18415#A2.E18)\)\.

###### Proof\.

Since𝒈t=𝑯⁡\(𝜽t−𝜽⋆\)\{\\bm\{g\}\}\_\{t\}=\{\\bm\{H\}\}\(\{\\bm\{\\theta\}\}\_\{t\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\), inserting𝑷±1/2\{\\bm\{P\}\}^\{\\pm 1/2\}gives𝑷−1/2𝒈t=\(𝑷−1/2𝑯𝑷−1/2\)𝑷1/2\(𝜽t−𝜽⋆\)\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{g\}\}\_\{t\}=\(\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}\)\\,\{\\bm\{P\}\}^\{1/2\}\(\{\\bm\{\\theta\}\}\_\{t\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\), so𝒗i⊤𝑷−1/2𝒈t=λixt\(i\)\{\\bm\{v\}\}\_\{i\}^\{\\top\}\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{g\}\}\_\{t\}=\\lambda\_\{i\}x\_\{t\}^\{\(i\)\}\. Applying𝑷1/2\{\\bm\{P\}\}^\{1/2\}to𝜽t\+1−𝜽⋆=𝜽t−𝜽⋆−η​𝑷−1​\(Q∗𝒈≤t\)t\{\\bm\{\\theta\}\}\_\{t\+1\}\-\{\\bm\{\\theta\}\}\_\{\\star\}=\{\\bm\{\\theta\}\}\_\{t\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\-\\eta\{\\bm\{P\}\}^\{\-1\}\(Q\*\{\\bm\{g\}\}\_\{\\leq t\}\)\_\{t\}and using𝑷−1/2\(Q∗𝒈≤t\)t=\(Q∗𝑷−1/2𝒈≤t\)t\{\\bm\{P\}\}^\{\-1/2\}\(Q\*\{\\bm\{g\}\}\_\{\\leq t\}\)\_\{t\}=\(Q\*\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{g\}\}\_\{\\leq t\}\)\_\{t\}\(which holds because the convolution acts on the time index and𝑷\{\\bm\{P\}\}is constant\), we obtain𝑷1/2\(𝜽t\+1−𝜽⋆\)=𝑷1/2\(𝜽t−𝜽⋆\)−η\(Q∗𝑷−1/2𝒈≤t\)t\{\\bm\{P\}\}^\{1/2\}\(\{\\bm\{\\theta\}\}\_\{t\+1\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\)=\{\\bm\{P\}\}^\{1/2\}\(\{\\bm\{\\theta\}\}\_\{t\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\)\-\\eta\(Q\*\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{g\}\}\_\{\\leq t\}\)\_\{t\}\. Projecting onto𝒗i\{\\bm\{v\}\}\_\{i\}yields \([19](https://arxiv.org/html/2608.18415#A2.E19)\)\. Since no two modes interact, the reduction is exact rather than an approximation\. ∎

From here on we can fix one mode without loss of generality, drop the index, and write the*loop gain*

k:=η​λ\>0,k\\;:=\\;\\eta\\lambda\\;\>\\;0,\(20\)assumingλ\>0\\lambda\>0\. Modes withλ≤0\\lambda\\leq 0do not contribute to unit\-circle crossing fork\>0k\>0, and need not be considered\.

### B\.2Proof of Proposition[2\.1](https://arxiv.org/html/2608.18415#S2.Thmtheorem1)

###### Proof\.

We first record the two standing hypotheses that the statement of the proposition leaves implicit, but generallly holds true for practical filters\. WriteQ⁡\(z\)Q\(z\)in a rational formQ⁡\(z\)=b⁡\(ξ\)/a⁡\(ξ\)Q\(z\)=b\(\\xi\)/a\(\\xi\)withξ:=z−1\\xi:=z^\{\-1\}, wherea,ba,bare coprime real polynomials normalized so thata⁡\(0\)=1a\(0\)=1\.

1. \(H1\)QQis causal and internally stable: every pole ofQQlies in\|z\|<1\|z\|<1, equivalently every root ofaalies in\|ξ\|\>1\|\\xi\|\>1\.
2. \(H2\)Q⁡\(1\)Q\(1\)is finite and nonzero; by the dc normalization of Section[2](https://arxiv.org/html/2608.18415#S2),Q⁡\(1\)=1Q\(1\)=1always\.

Without \(H1\) the filter is unstable on its own and no gainkkstabilizes the mode, so the phrase “first crossing” has no meaning\. Without \(H2\) the branch of the locus that starts atz=1z=1does not move into the disc\.

The characteristic polynomial\.Takingzz\-transforms of \([19](https://arxiv.org/html/2608.18415#A2.E19)\) with zero initial data givesz​X​\(z\)=X⁡\(z\)−k​Q​\(z\)​X​\(z\)zX\(z\)=X\(z\)\-kQ\(z\)X\(z\), hence \([7](https://arxiv.org/html/2608.18415#S2.E7)\),

z−1\+k​Q​\(z\)=0\.z\-1\+kQ\(z\)\\;=\\;0\.\(21\)Multiplying \([21](https://arxiv.org/html/2608.18415#A2.E21)\) byz−1​a​\(z−1\)z^\{\-1\}a\(z^\{\-1\}\)and substitutingξ=z−1\\xi=z^\{\-1\}clears both the pole atz=0z=0and the denominator ofQQ, yielding a characteristic polynomial

ck​\(ξ\):=\(1−ξ\)​a​\(ξ\)\+k​ξ​b​\(ξ\)=0\.c\_\{k\}\(\\xi\)\\;:=\\;\(1\-\\xi\)\\,a\(\\xi\)\\;\+\\;k\\,\\xi\\,b\(\\xi\)\\;=\\;0\.\(22\)Because\|z\|<1⇔\|ξ\|\>1\|z\|<1\\iff\|\\xi\|\>1, the mode is asymptotically stable exactly when every root ofckc\_\{k\}lies strictly outside the closed unit disc\. Roots ofckc\_\{k\}move continuously withkkon the Riemann sphere, and the exterior\{\|ξ\|\>1\}∪\{∞\}\\\{\|\\xi\|\>1\\\}\\cup\\\{\\infty\\\}is open, so a root can enter the closed unit disc only by passing through the circle\|ξ\|=1\|\\xi\|=1\. A root escaping toξ=∞\\xi=\\infty, which happens when the leading coefficient of \([22](https://arxiv.org/html/2608.18415#A2.E22)\) degenerates, is a deadbeat pole atz=0z=0and never destabilizes\.

Step 1: the locus starts inside\.Atk=0k=0we havec0​\(ξ\)=\(1−ξ\)​a​\(ξ\)c\_\{0\}\(\\xi\)=\(1\-\\xi\)a\(\\xi\), whose roots areξ=1\\xi=1together with the roots ofaa, which all lie in\|ξ\|\>1\|\\xi\|\>1by \(H1\)\. The root atξ=1\\xi=1is simple becausea⁡\(1\)≠0a\(1\)\\neq 0by \(H2\)\. Since∂ξck\|\(1,0\)=−a⁡\(1\)\\partial\_\{\\xi\}c\_\{k\}\|\_\{\(1,0\)\}=\-a\(1\)and∂kck\|\(1,0\)=b⁡\(1\)\\partial\_\{k\}c\_\{k\}\|\_\{\(1,0\)\}=b\(1\), the implicit function theorem gives a branchξ⁡\(k\)\\xi\(k\)with

d​ξd​k\|k=0=b⁡\(1\)a⁡\(1\)=Q⁡\(1\)=1\>0,soξ⁡\(k\)=1\+k\+O⁡\(k2\)\.\\left\.\\frac\{d\\xi\}\{dk\}\\right\|\_\{k=0\}\\;=\\;\\frac\{b\(1\)\}\{a\(1\)\}\\;=\\;Q\(1\)\\;=\\;1\\;\>\\;0,\\qquad\\text\{so\}\\qquad\\xi\(k\)=1\+k\+O\(k^\{2\}\)\.\(23\)Equivalentlyz⁡\(k\)=1−k\+O⁡\(k2\)z\(k\)=1\-k\+O\(k^\{2\}\): the dc pole leavesz=1z=1along the negative real direction and enters the disc\. Hence there isk0\>0k\_\{0\}\>0such that the mode is asymptotically stable for everyk∈\(0,k0\)k\\in\(0,k\_\{0\}\), and the locus starts inside the disc\.

Step 2: the crossings are exactlyΩ\+​\(Q\)\\Omega\_\{\+\}\(Q\)\.Fork\>0k\>0, its corresponding characteristic polynomialckc\_\{k\}has a root on\|ξ\|=1\|\\xi\|=1if and only if \([21](https://arxiv.org/html/2608.18415#A2.E21)\) is solved by somez=ei​ωz=e^\{i\\omega\}on the unit circle, i\.e\.

k=1−ei​ωQ⁡\(ei​ω\)=:k⁡\(ω\)\.k\\;=\\;\\frac\{1\-e^\{i\\omega\}\}\{Q\(e^\{i\\omega\}\)\}\\;=:\\;k\(\\omega\)\.\(24\)The frequencyω=0\\omega=0forcesk=0k=0and is excluded\. Becausea,ba,bare real,k⁡\(−ω\)=k⁡\(ω\)¯k\(\-\\omega\)=\\overline\{k\(\\omega\)\}, so the crossings appear symmetrically, i\.e\.,±ω\\pm\\omegaoccur at the same gain, and it suffices to takeω∈\(0,π\]\\omega\\in\(0,\\pi\]\. Finallykkmust be a real positive number, which is precisely the defining condition ofΩ\+​\(Q\)\\Omega\_\{\+\}\(Q\)in \([4](https://arxiv.org/html/2608.18415#S2.E4)\)\. OnΩ\+​\(Q\)\\Omega\_\{\+\}\(Q\)the valuek⁡\(ω\)k\(\\omega\)coincides with its modulus, i\.e\.,k⁡\(ω\)=\|k⁡\(ω\)\|k\(\\omega\)=\|k\(\\omega\)\|\.

Step 3: conclusion\.Combining Steps 1 and 2, the set of gains at which some root sits on the unit circle is\{k⁡\(ω\):ω∈Ω\+​\(Q\)\}\\\{k\(\\omega\):\\omega\\in\\Omega\_\{\+\}\(Q\)\\\}, and the mode is stable on the whole interval below its smallest element\. Therefore

k⋆​\(Q\)=minω∈Ω\+​\(Q\)⁡\|1−ei​ωQ⁡\(ei​ω\)\|,k\_\{\\star\}\(Q\)\\;=\\;\\min\_\{\\omega\\in\\Omega\_\{\+\}\(Q\)\}\\left\|\\frac\{1\-e^\{i\\omega\}\}\{Q\(e^\{i\\omega\}\)\}\\right\|,\(25\)with the conventionk⋆​\(Q\)=\+∞k\_\{\\star\}\(Q\)=\+\\inftyandΓ=0\\Gamma=0whenΩ\+​\(Q\)=∅\\Omega\_\{\+\}\(Q\)=\\emptyset, which is exactly the claim \([4](https://arxiv.org/html/2608.18415#S2.E4)\)\. WritingΓ:=2/k⋆​\(Q\)\\Gamma:=2/k\_\{\\star\}\(Q\), the mode is stable iffη​λ<2/Γ\\eta\\lambda<2/\\Gammaand marginal at equality, i\.e\. iffη​λ​Γ≤2\\eta\\lambda\\Gamma\\leq 2, which recovers \([♣‣2\.1](https://arxiv.org/html/2608.18415#S2.Ex2)\)\. SinceΓ\\Gammadepends onQQalone and not onλ\\lambda\(mode decoupling\), the binding mode is the sharpest one, and the system is not exponentially unstable iffηλmax\(𝑷−1/2𝑯𝑷−1/2\)Γ≤2\\eta\\lambda\_\{\\max\}\(\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}\)\\,\\Gamma\\leq 2, which is precisely the boxed condition \([♣‣2\.1](https://arxiv.org/html/2608.18415#S2.Ex2)\)\.

Step 4: the Nyquist case\.If the minimum in \([25](https://arxiv.org/html/2608.18415#A2.E25)\) is attained at the Nyquist frequencyω=π\\omega=\\pi, i\.e\.,z=ei​ω=−1z=e^\{i\\omega\}=\-1, andQ⁡\(−1\)\>0Q\(\-1\)\>0, then

k⋆​\(Q\)=1−\(−1\)Q⁡\(−1\)=2Q⁡\(−1\),henceΓ=2k⋆​\(Q\)=Q⁡\(−1\),k\_\{\\star\}\(Q\)\\;=\\;\\frac\{1\-\(\-1\)\}\{Q\(\-1\)\}\\;=\\;\\frac\{2\}\{Q\(\-1\)\},\\qquad\\text\{hence\}\\qquad\\Gamma\\;=\\;\\frac\{2\}\{k\_\{\\star\}\(Q\)\}\\;=\\;Q\(\-1\),\(26\)and the stability condition readsη​λ​Q​\(−1\)≤2\\eta\\lambda Q\(\-1\)\\leq 2\. The marginal rootz=−1z=\-1hasz2=1z^\{2\}=1, so the marginal motion is a period\-2 oscillation\. This proves the last sentence of Proposition[2\.1](https://arxiv.org/html/2608.18415#S2.Thmtheorem1)\. ∎

### B\.3Two Lemmas on the Root Locus Gauge

The gaugeΓ\\Gammais a functional of the filterQQalone, so it can be computed once per optimizer family, and is not dependent on the preconditioner\. The following two lemmas consolidates this: the first removes the dc normalizationQ⁡\(1\)Q\(1\), and the second identifies a large class of filters for which the escape is provably at Nyquistz=−1z=\-1\.

###### Lemma B\.2\(Positive scaling\)\.

Letc\>0c\>0\. ThenΩ\+​\(c​Q\)=Ω\+​\(Q\)\\Omega\_\{\+\}\(cQ\)=\\Omega\_\{\+\}\(Q\),k⋆​\(c​Q\)=k⋆​\(Q\)/ck\_\{\\star\}\(cQ\)=k\_\{\\star\}\(Q\)/c, and

Γ⁡\(c​Q\)=c​Γ​\(Q\)\.\\Gamma\(cQ\)\\;=\\;c\\,\\Gamma\(Q\)\.\(27\)In particular the dc\-normalized filterQ~:=Q/Q⁡\(1\)\\tilde\{Q\}:=Q/Q\(1\)satisfiesΓ⁡\(Q~\)=Γ⁡\(Q\)/Q⁡\(1\)\\Gamma\(\\tilde\{Q\}\)=\\Gamma\(Q\)/Q\(1\), andΓ⁡\(Q\)=Q⁡\(−1\)\\Gamma\(Q\)=Q\(\-1\)holds forQQif and only if it holds forQ~\\tilde\{Q\}\.

###### Proof\.

k⁡\(ω\)k\(\\omega\)in \([24](https://arxiv.org/html/2608.18415#A2.E24)\) is replaced byk⁡\(ω\)/ck\(\\omega\)/c\. Multiplication byc−1\>0c^\{\-1\}\>0preserves membership inℝ\>0\\mathbb\{R\}\_\{\>0\}, soΩ\+\\Omega\_\{\+\}is unchanged, and it scales every modulus byc−1c^\{\-1\}, so the minimum scales byc−1c^\{\-1\}andΓ=2/k⋆\\Gamma=2/k\_\{\\star\}bycc\. The last claim follows sinceΓ\\GammaandQ⁡\(−1\)Q\(\-1\)carry the same factorcc\. ∎

###### Lemma B\.3\(Nyquist escape for positive one\-pole mixtures\)\.

Let

Q⁡\(z\)=∑j=1Jwj​1−βj1−βj​z−1,wj\>0,∑j=1Jwj=1,βj∈\[0,1\)\.Q\(z\)\\;=\\;\\sum\_\{j=1\}^\{J\}w\_\{j\}\\,\\frac\{1\-\\beta\_\{j\}\}\{1\-\\beta\_\{j\}z^\{\-1\}\},\\qquad w\_\{j\}\>0,\\quad\\sum\_\{j=1\}^\{J\}w\_\{j\}=1,\\quad\\beta\_\{j\}\\in\[0,1\)\.\(28\)ThenQ⁡\(1\)=1Q\(1\)=1, hypotheses \(H1\) and \(H2\) hold, and

Ω\+​\(Q\)=\{π\},k⋆​\(Q\)=2Q⁡\(−1\),Γ=Q⁡\(−1\)=∑j=1Jwj​1−βj1\+βj∈\(0,1\],\\Omega\_\{\+\}\(Q\)=\\\{\\pi\\\},\\qquad k\_\{\\star\}\(Q\)=\\frac\{2\}\{Q\(\-1\)\},\\qquad\\Gamma\\;=\\;Q\(\-1\)\\;=\\;\\sum\_\{j=1\}^\{J\}w\_\{j\}\\,\\frac\{1\-\\beta\_\{j\}\}\{1\+\\beta\_\{j\}\}\\;\\in\\;\(0,1\],\(29\)withΓ=1\\Gamma=1if and only if everyβj=0\\beta\_\{j\}=0\.

###### Proof\.

Each summand equalswjw\_\{j\}atz=1z=1, soQ⁡\(1\)=1Q\(1\)=1, giving \(H2\)\. Also, the poles arez=βj∈\[0,1\)z=\\beta\_\{j\}\\in\[0,1\), giving \(H1\)\. Forω∈\(0,π\]\\omega\\in\(0,\\pi\]the elementary identity

1−ei​ω=2​sin⁡\(ω2\)​ei⁡\(ω−π\)/2,2​sin⁡\(ω2\)\>0,1\-e^\{i\\omega\}\\;=\\;2\\sin\\left\(\\frac\{\\omega\}\{2\}\\right\)\\;e^\{i\(\\omega\-\\pi\)/2\},\\qquad 2\\sin\\left\(\\frac\{\\omega\}\{2\}\\right\)\>0,\(30\)shows thatk⁡\(ω\)∈ℝ\>0k\(\\omega\)\\in\\mathbb\{R\}\_\{\>0\}holds iffei​θ/Q⁡\(ei​ω\)∈ℝ\>0e^\{i\\theta\}/Q\(e^\{i\\omega\}\)\\in\\mathbb\{R\}\_\{\>0\}withθ:=\(ω−π\)/2\\theta:=\(\\omega\-\\pi\)/2\. Writingei​θ/Q=ei​θ​Q¯/\|Q\|2e^\{i\\theta\}/Q=e^\{i\\theta\}\\overline\{Q\}/\|Q\|^\{2\}and take the conjugate\. The reality requirementei​θ/Q⁡\(ei​ω\)∈ℝ\>0e^\{i\\theta\}/Q\(e^\{i\\omega\}\)\\in\\mathbb\{R\}\_\{\>0\}is exactly the vanishing of the*phase margin*

Ψ\(ω\):=Im\(e−i\(ω−π\)/2Q\(ei​ω\)\)\.\\Psi\(\\omega\)\\;:=\\;\\operatorname\{Im\}\\\!\\left\(e^\{\-i\(\\omega\-\\pi\)/2\}\\,Q\(e^\{i\\omega\}\)\\right\)\.\(31\)So it suffices to show thatΨ\>0\\Psi\>0on\(0,π\)\(0,\\pi\), which rules out every interior crossing\.

Consider one pole,Qβ​\(z\):=\(1−β\)/\(1−β​z−1\)Q\_\{\\beta\}\(z\):=\(1\-\\beta\)/\(1\-\\beta z^\{\-1\}\), so thatQβ​\(ei​ω\)=\(1−β\)/\(1−β​e−i​ω\)Q\_\{\\beta\}\(e^\{i\\omega\}\)=\(1\-\\beta\)/\(1\-\\beta e^\{\-i\\omega\}\)\. Rationalizing gives

Ψβ​\(ω\)=1−β\|1−β​e−i​ω\|2​Im⁡\(e−i​θ​\(1−β​ei​ω\)\)=1−β\|1−β​e−i​ω\|2​\(−sin⁡θ−β​sin⁡\(ω−θ\)\)\.\\Psi\_\{\\beta\}\(\\omega\)\\;=\\;\\frac\{1\-\\beta\}\{\|1\-\\beta e^\{\-i\\omega\}\|^\{2\}\}\\,\\operatorname\{Im\}\\\!\\left\(e^\{\-i\\theta\}\\left\(1\-\\beta e^\{i\\omega\}\\right\)\\right\)\\;=\\;\\frac\{1\-\\beta\}\{\|1\-\\beta e^\{\-i\\omega\}\|^\{2\}\}\\left\(\-\\sin\\theta\-\\beta\\sin\(\\omega\-\\theta\)\\right\)\.\(32\)Since−θ=π−ω2\-\\theta=\\tfrac\{\\pi\-\\omega\}\{2\}andω−θ=ω\+π2\\omega\-\\theta=\\tfrac\{\\omega\+\\pi\}\{2\}, both sines collapse onto the same value,−sin⁡θ=cos⁡ω2=sin⁡\(ω−θ\)\-\\sin\\theta=\\cos\\tfrac\{\\omega\}\{2\}=\\sin\(\\omega\-\\theta\), and we have

Ψβ​\(ω\)=\(1−β\)2​cos⁡\(ω/2\)1−2​β​cos⁡ω\+β2\>0for​ω∈\(0,π\),\\Psi\_\{\\beta\}\(\\omega\)\\;=\\;\\frac\{\(1\-\\beta\)^\{2\}\\,\\cos\(\\omega/2\)\}\{1\-2\\beta\\cos\\omega\+\\beta^\{2\}\}\\;\>\\;0\\qquad\\text\{for \}\\omega\\in\(0,\\pi\),\(33\)becauseβ<1\\beta<1andcos⁡\(ω/2\)\>0\\cos\(\\omega/2\)\>0forω∈\(0,π\)\\omega\\in\(0,\\pi\)\. Also, the denominator\|1−β​e−i​ω\|2≥\(1−β\)2\>0\|1\-\\beta e^\{\-i\\omega\}\|^\{2\}\\geq\(1\-\\beta\)^\{2\}\>0\. The single pole therefore clears the critical line strictly, and it does so by a margin proportional to\(1−β\)2\(1\-\\beta\)^\{2\}: the closerβ\\betais to11, the closer the filter comes to admitting an interior crossing\.

SinceΨ\\Psiisℝ\\mathbb\{R\}\-linear inQQ, the mixture inherits the sign:Ψ⁡\(ω\)=∑jwj​Ψβj​\(ω\)\>0\\Psi\(\\omega\)=\\sum\_\{j\}w\_\{j\}\\Psi\_\{\\beta\_\{j\}\}\(\\omega\)\>0on\(0,π\)\(0,\\pi\), as everywj\>0w\_\{j\}\>0\. HenceΩ\+​\(Q\)∩\(0,π\)=∅\\Omega\_\{\+\}\(Q\)\\cap\(0,\\pi\)=\\emptyset\. Atω=π\\omega=\\piwe haveθ=0\\theta=0andQ⁡\(−1\)=∑jwj​\(1−βj\)/\(1\+βj\)\>0Q\(\-1\)=\\sum\_\{j\}w\_\{j\}\(1\-\\beta\_\{j\}\)/\(1\+\\beta\_\{j\}\)\>0is real, soΨ⁡\(π\)=0\\Psi\(\\pi\)=0andk⁡\(π\)=2/Q⁡\(−1\)∈ℝ\>0k\(\\pi\)=2/Q\(\-1\)\\in\\mathbb\{R\}\_\{\>0\}, i\.e\.π∈Ω\+​\(Q\)\\pi\\in\\Omega\_\{\+\}\(Q\)\. ThusΩ\+​\(Q\)=\{π\}\\Omega\_\{\+\}\(Q\)=\\\{\\pi\\\}and \([29](https://arxiv.org/html/2608.18415#A2.E29)\) follows from Step 4 of Appendix[B\.2](https://arxiv.org/html/2608.18415#A2.SS2)\. Finally\(1−β\)/\(1\+β\)∈\(0,1\]\(1\-\\beta\)/\(1\+\\beta\)\\in\(0,1\]with equality iffβ=0\\beta=0, and a convex combination of such numbers obeys the same bounds\. ∎

### B\.4Proof of Corollary[2\.2](https://arxiv.org/html/2608.18415#S2.Thmtheorem2)

All that remains is to write each of the widely\-used optimizer filters in the form \([28](https://arxiv.org/html/2608.18415#A2.E28)\) and read offQ⁡\(−1\)Q\(\-1\)\. We use the convention that𝒖t\{\\bm\{u\}\}\_\{t\}is the vector multiplyingη\\etain \([♠‣2](https://arxiv.org/html/2608.18415#S2.Ex1)\) with𝑷=𝑰\{\\bm\{P\}\}=\{\\bm\{I\}\}, and𝒎t\{\\bm\{m\}\}\_\{t\}denotes the momentum buffer\.

###### Proof\.

Heavy Ball\.From𝒎t=β​𝒎t−1\+𝒈t\{\\bm\{m\}\}\_\{t\}=\\beta\{\\bm\{m\}\}\_\{t\-1\}\+\{\\bm\{g\}\}\_\{t\}and𝒖t=𝒎t\{\\bm\{u\}\}\_\{t\}=\{\\bm\{m\}\}\_\{t\}we get𝒎t=∑k≥0βk​𝒈t−k\{\\bm\{m\}\}\_\{t\}=\\sum\_\{k\\geq 0\}\\beta^\{k\}\{\\bm\{g\}\}\_\{t\-k\}, the geometric filter quoted below \([♠‣2](https://arxiv.org/html/2608.18415#S2.Ex1)\), so

QHB​\(z\)=∑k≥0βk​z−k=11−β​z−1,QHB​\(1\)=11−β\.Q^\{\\mathrm\{HB\}\}\(z\)\\;=\\;\\sum\_\{k\\geq 0\}\\beta^\{k\}z^\{\-k\}\\;=\\;\\frac\{1\}\{1\-\\beta z^\{\-1\}\},\\qquad Q^\{\\mathrm\{HB\}\}\(1\)=\\frac\{1\}\{1\-\\beta\}\.\(34\)Its dc\-normalized versionQnHB=\(1−β\)​QHBQ^\{\\mathrm\{nHB\}\}=\(1\-\\beta\)Q^\{\\mathrm\{HB\}\}is exactly \([28](https://arxiv.org/html/2608.18415#A2.E28)\) withJ=1J=1,w1=1w\_\{1\}=1,β1=β\\beta\_\{1\}=\\beta, so Lemma[B\.3](https://arxiv.org/html/2608.18415#A2.Thmtheorem3)applies and givesΓnHB=\(1−β\)/\(1\+β\)\\Gamma^\{\\mathrm\{nHB\}\}=\(1\-\\beta\)/\(1\+\\beta\)\. To recover unnormalized results, we apply Lemma[B\.2](https://arxiv.org/html/2608.18415#A2.Thmtheorem2)withc=1/\(1−β\)c=1/\(1\-\\beta\)yieldingΓHB=1/\(1\+β\)\\Gamma^\{\\mathrm\{HB\}\}=1/\(1\+\\beta\)\. Both equal the respectiveQ⁡\(−1\)Q\(\-1\)and recover the results of[12](https://arxiv.org/html/2608.18415#bib.bib26)\.

Nesterov’s momentum\.With𝒎t=β​𝒎t−1\+𝒈t\{\\bm\{m\}\}\_\{t\}=\\beta\{\\bm\{m\}\}\_\{t\-1\}\+\{\\bm\{g\}\}\_\{t\}and𝒖t=𝒈t\+β​𝒎t\{\\bm\{u\}\}\_\{t\}=\{\\bm\{g\}\}\_\{t\}\+\\beta\{\\bm\{m\}\}\_\{t\},

QNAG​\(z\)=1\+β1−β​z−1,QNAG​\(1\)=11−β\.Q^\{\\mathrm\{NAG\}\}\(z\)\\;=\\;1\+\\frac\{\\beta\}\{1\-\\beta z^\{\-1\}\},\\qquad Q^\{\\mathrm\{NAG\}\}\(1\)=\\frac\{1\}\{1\-\\beta\}\.\(35\)Multiplying by\(1−β\)\(1\-\\beta\)and splitting the constant term off as a pole at the origin,

QnNAG​\(z\)=\(1−β\)⋅1\+β⋅1−β1−β​z−1,Q^\{\\mathrm\{nNAG\}\}\(z\)\\;=\\;\(1\-\\beta\)\\cdot 1\\;\+\\;\\beta\\cdot\\frac\{1\-\\beta\}\{1\-\\beta z^\{\-1\}\},\(36\)which is \([28](https://arxiv.org/html/2608.18415#A2.E28)\) with weights\(1−β,β\)\(1\-\\beta,\\beta\)and poles\(0,β\)\(0,\\beta\)\. The weights are positive and sum to one forβ∈\(0,1\)\\beta\\in\(0,1\), so Lemma[B\.3](https://arxiv.org/html/2608.18415#A2.Thmtheorem3)applies and

ΓnNAG=\(1−β\)⋅1\+β⋅1−β1\+β=\(1−β\)​\(1\+2​β\)1\+β,ΓNAG=1\+2​β1\+β,\\Gamma^\{\\mathrm\{nNAG\}\}\\;=\\;\(1\-\\beta\)\\cdot 1\+\\beta\\cdot\\frac\{1\-\\beta\}\{1\+\\beta\}\\;=\\;\\frac\{\(1\-\\beta\)\(1\+2\\beta\)\}\{1\+\\beta\},\\qquad\\Gamma^\{\\mathrm\{NAG\}\}\\;=\\;\\frac\{1\+2\\beta\}\{1\+\\beta\},\(37\)the second by Lemma[B\.2](https://arxiv.org/html/2608.18415#A2.Thmtheorem2)\. Again, both recover the results reported in[12](https://arxiv.org/html/2608.18415#bib.bib26)\.

This proves Corollary[2\.2](https://arxiv.org/html/2608.18415#S2.Thmtheorem2): for both Heavy Ball and Nesterov’s momentum, normalized or not,Ω\+=\{π\}\\Omega\_\{\+\}=\\\{\\pi\\\}and thereforeΓ⁡\(β\)=Q⁡\(−1\)\\Gamma\(\\beta\)=Q\(\-1\), matching the values quoted below \([3](https://arxiv.org/html/2608.18415#S2.E3)\)\. ∎

Similar results on the remaining filters of Section[3](https://arxiv.org/html/2608.18415#S3), such as QHM\([40](https://arxiv.org/html/2608.18415#bib.bib44)\), parallel dual momentum, and cascaded two\-pole momenta\([30](https://arxiv.org/html/2608.18415#bib.bib37);[47](https://arxiv.org/html/2608.18415#bib.bib39)\)are derived in Appendix[C\.1](https://arxiv.org/html/2608.18415#A3.SS1)with their gauges summarized in Table[2](https://arxiv.org/html/2608.18415#A3.T2)\. Also refer to the root locus plots in Appendix[C](https://arxiv.org/html/2608.18415#A3)for visualization of these results\.

### B\.5Proof of Corollary[5\.1](https://arxiv.org/html/2608.18415#S5.Thmtheorem1)

This subsection proves Corollary[5\.1](https://arxiv.org/html/2608.18415#S5.Thmtheorem1)and the GD reduction used in Theorem[5\.3](https://arxiv.org/html/2608.18415#S5.Thmtheorem3)\. Fix an iteration and recall the definitionsA=𝒈⊤​𝒖A=\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\},B=η​𝒖⊤​𝑯¯η,𝒖​𝒖/2B=\\eta\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}/2,‖𝒖‖𝑷2=𝒖⊤​𝑷​𝒖\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}=\{\\bm\{u\}\}^\{\\top\}\{\\bm\{P\}\}\{\\bm\{u\}\},

ζ=BA,χ=η​S​Γ2,ς=D∥⋅∥𝑷S=𝒖⊤​𝑯¯η,𝒖​𝒖S​‖𝒖‖𝑷2,τ=Γ​A‖𝒖‖𝑷2\.\\zeta\\;=\\;\\frac\{B\}\{A\},\\qquad\\chi\\;=\\;\\frac\{\\eta S\\Gamma\}\{2\},\\qquad\\varsigma\\;=\\;\\frac\{D^\{\\\|\\cdot\\\|\_\{\{\\bm\{P\}\}\}\}\}\{S\}\\;=\\;\\frac\{\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\}\{S\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\},\\qquad\\tau\\;=\\;\\frac\{\\Gamma A\}\{\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\}\.\(38\)The assumptionsη,S,Γ\>0\\eta,S,\\Gamma\>0,𝒖≠𝟎\{\\bm\{u\}\}\\neq\{\\bm\{0\}\}, andA≠0A\\neq 0ensure all four quantities are finite, withχ\>0\\chi\>0andτ≠0\\tau\\neq 0\. This identity provides additional insight into when the worst\-case circumscriptionχ\\chiis fully realized along the actual optimization trajectory, which is summarized in the following proposition\.

###### Proof\.

The decomposition is exact\.Substituting \([38](https://arxiv.org/html/2608.18415#A2.E38)\) directly,

χ​ςτ=η​S​Γ2⋅𝒖⊤​𝑯¯η,𝒖​𝒖S​‖𝒖‖𝑷2⋅‖𝒖‖𝑷2Γ​A=η2​𝒖⊤​𝑯¯η,𝒖​𝒖𝒈⊤​𝒖=BA=ζ\.\\chi\\frac\{\\varsigma\}\{\\tau\}\\;=\\;\\frac\{\\eta S\\Gamma\}\{2\}\\cdot\\frac\{\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\}\{S\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\}\\cdot\\frac\{\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\}\{\\Gamma A\}\\;=\\;\\frac\{\\eta\}\{2\}\\frac\{\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\}\{\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\}\\;=\\;\\frac\{B\}\{A\}\\;=\\;\\zeta\.\(39\)Every occurrence ofSS,Γ\\Gamma, and‖𝒖‖𝑷2\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}cancels, so \([♡‣4](https://arxiv.org/html/2608.18415#S4.Ex4)\) is an identity: neither𝑯¯η,𝒖≃𝑯\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\\simeq\{\\bm\{H\}\}nor occupation of𝒗max\{\\bm\{v\}\}\_\{\\max\}is used anywhere in \([39](https://arxiv.org/html/2608.18415#A2.E39)\)\. This is precisely why the decomposition constrains only the ratioς/τ\\varsigma/\\tau\.

Coincidence condition\.From \([39](https://arxiv.org/html/2608.18415#A2.E39)\) andχ\>0\\chi\>0, it is trivial thatζ=χ\\zeta=\\chiif and only ifς=τ\\varsigma=\\tau\. Clearing the common denominator‖𝒖‖𝑷2\>0\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\>0and multiplying byS\>0S\>0,

ζ=χ⇔ς=τ⇔𝒖⊤​𝑯¯η,𝒖​𝒖S=Γ​A⇔𝒖⊤​𝑯¯η,𝒖​𝒖=S​Γ​𝒈⊤​𝒖,\\zeta=\\chi\\iff\\varsigma=\\tau\\iff\\frac\{\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\}\{S\}=\\Gamma A\\iff\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}=S\\Gamma\\,\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\},\(40\)which gives the coincidence conditio of \([39](https://arxiv.org/html/2608.18415#A2.E39)\)\.

Conditions on the individual factors\.BecauseS\>0S\>0, the definition ofς\\varsigmagivesς=1⇔𝒖⊤​𝑯¯η,𝒖​𝒖/‖𝒖‖𝑷2=S\\varsigma=1\\iff\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}/\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}=S\. Because‖𝒖‖𝑷2\>0\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\>0, the definition ofτ\\taugivesτ=1⇔Γ​A=𝒖⊤​𝑷​𝒖⇔𝒖⊤​\(Γ​𝒈−𝑷​𝒖\)=0\\tau=1\\iff\\Gamma A=\{\\bm\{u\}\}^\{\\top\}\{\\bm\{P\}\}\{\\bm\{u\}\}\\iff\{\\bm\{u\}\}^\{\\top\}\(\\Gamma\{\\bm\{g\}\}\-\{\\bm\{P\}\}\{\\bm\{u\}\}\)=0\. Sufficiency ofς=τ=1\\varsigma=\\tau=1is immediate from \([39](https://arxiv.org/html/2608.18415#A2.E39)\)\. Necessity fails, and Appendix[B\.7](https://arxiv.org/html/2608.18415#A2.SS7)exhibits an admissible instance withς=τ≠1\\varsigma=\\tau\\neq 1\.

Form at the realized edge\.Settingζ=1\\zeta=1in \([39](https://arxiv.org/html/2608.18415#A2.E39)\) and solving forχ\\chigivesχ=τ/ς\\chi=\\tau/\\varsigma\. If𝑯¯η,𝒖≃𝑯\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\\simeq\{\\bm\{H\}\}, thenD∥⋅∥𝑷≤SD^\{\\\|\\cdot\\\|\_\{\{\\bm\{P\}\}\}\}\\leq Sby the variational characterization ofSSrecalled in Section[4](https://arxiv.org/html/2608.18415#S4), henceς≤1\\varsigma\\leq 1andχ≥τ\\chi\\geq\\tau\.

GD reduction\.Take𝑷=I\{\\bm\{P\}\}=IandQ≡1Q\\equiv 1\. ThenΓ=Q⁡\(−1\)=1\\Gamma=Q\(\-1\)=1and𝒖=𝒈\{\\bm\{u\}\}=\{\\bm\{g\}\}, soA=𝒈⊤​𝒖=‖𝒈‖2A=\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}=\\\|\{\\bm\{g\}\}\\\|^\{2\}and‖𝒖‖𝑷2=‖𝒈‖2\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}=\\\|\{\\bm\{g\}\}\\\|^\{2\}, henceτ=Γ​A/‖𝒖‖𝑷2=1\\tau=\\Gamma A/\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}=1\. The identity collapses toζ=χ​ς\\zeta=\\chi\\varsigmawithχ=η​S/2\\chi=\\eta S/2\. On a frozen quadratic,𝑯¯η,𝒖=𝑯\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}=\{\\bm\{H\}\}andς=𝒈⊤​𝑯​𝒈/\(S​‖𝒈‖2\)\\varsigma=\{\\bm\{g\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{g\}\}/\(S\\\|\{\\bm\{g\}\}\\\|^\{2\}\), which equals11if and only if𝒈\{\\bm\{g\}\}is an eigen\-direction forλmax​\(𝑯\)\\lambda\_\{\\max\}\(\{\\bm\{H\}\}\)\. This is the occupation hypothesis of Section[2](https://arxiv.org/html/2608.18415#S2), and is the content of the GD clause in Corollary[5\.1](https://arxiv.org/html/2608.18415#S5.Thmtheorem1)\. ∎

### B\.6Proof of Corollary[5\.2](https://arxiv.org/html/2608.18415#S5.Thmtheorem2)

###### Proof\.

Notations\.Write𝒗=𝒗max\{\\bm\{v\}\}=\{\\bm\{v\}\}\_\{\\max\}for the unit top eigenvector of𝑷−1/2𝑯𝑷−1/2\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}, so that

𝑷−1/2𝑯𝑷−1/2𝒗=S𝒗⟺𝑯𝑷−1/2𝒗=S𝑷1/2𝒗\.\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{v\}\}\\;=\\;S\{\\bm\{v\}\}\\qquad\\Longleftrightarrow\\qquad\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{v\}\}\\;=\\;S\{\\bm\{P\}\}^\{1/2\}\{\\bm\{v\}\}\.\(41\)This will be used throughout the proof\.

The locked mode is invariant\.By \(Q1\) the loss is quadratic over the horizon, so𝒈s=𝑯⁡\(𝜽s−𝜽⋆\)\{\\bm\{g\}\}\_\{s\}=\{\\bm\{H\}\}\(\{\\bm\{\\theta\}\}\_\{s\}\-\{\\bm\{\\theta\}\}\_\{\\star\}\)\. With \(Q3\), we havexs𝑯𝑷−1/2𝒗=Sxs𝑷1/2𝒗x\_\{s\}\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{v\}\}=Sx\_\{s\}\{\\bm\{P\}\}^\{1/2\}\{\\bm\{v\}\}using \([41](https://arxiv.org/html/2608.18415#A2.E41)\)\. By \(Q2\) the preconditioner may be pulled out of the convolution, so the update \([♠‣2](https://arxiv.org/html/2608.18415#S2.Ex1)\) is

𝒖t=𝑷−1\(Q∗𝒈≤t\)t=S\(Q∗x≤t\)t𝑷−1𝑷1/2𝒗=S\(Q∗x≤t\)t𝑷−1/2𝒗\.\{\\bm\{u\}\}\_\{t\}\\;=\\;\{\\bm\{P\}\}^\{\-1\}\(Q\*\{\\bm\{g\}\}\_\{\\leq t\}\)\_\{t\}\\;=\\;S\(Q\*x\_\{\\leq t\}\)\_\{t\}\\,\{\\bm\{P\}\}^\{\-1\}\{\\bm\{P\}\}^\{1/2\}\{\\bm\{v\}\}\\;=\\;S\(Q\*x\_\{\\leq t\}\)\_\{t\}\\,\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{v\}\}\.\(42\)The update therefore stays inspan\(𝑷−1/2𝒗\)\\operatorname\{span\}\(\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{v\}\}\), which is the locked mode\.

Spatial factor\.Abbreviateq:=\(Q∗x≤t\)tq:=\(Q\*x\_\{\\leq t\}\)\_\{t\}and use𝒗⊤​𝒗=1\{\\bm\{v\}\}^\{\\top\}\{\\bm\{v\}\}=1together with \([41](https://arxiv.org/html/2608.18415#A2.E41)\) to get:

A\\displaystyle A=𝒈t⊤​𝒖t=S2​xt​q,\\displaystyle=\\;\{\\bm\{g\}\}\_\{t\}^\{\\top\}\{\\bm\{u\}\}\_\{t\}\\;=\\;S^\{2\}x\_\{t\}q,\(43\)‖𝒖t‖𝑷2\\displaystyle\\\|\{\\bm\{u\}\}\_\{t\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}=𝒖t⊤​𝑷​𝒖t=S2​q2,\\displaystyle=\\;\{\\bm\{u\}\}\_\{t\}^\{\\top\}\{\\bm\{P\}\}\{\\bm\{u\}\}\_\{t\}\\;=\\;S^\{2\}q^\{2\},𝒖t⊤​𝑯​𝒖t\\displaystyle\{\\bm\{u\}\}\_\{t\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{u\}\}\_\{t\}=S2q2𝒗⊤𝑷−1/2𝑯𝑷−1/2𝒗=S3q2\.\\displaystyle=\\;S^\{2\}q^\{2\}\\,\{\\bm\{v\}\}^\{\\top\}\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{H\}\}\{\\bm\{P\}\}^\{\-1/2\}\{\\bm\{v\}\}\\;=\\;S^\{3\}q^\{2\}\.Since𝑯¯η,𝒖=𝑯\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}=\{\\bm\{H\}\}by \(Q1\), the spatial participation factor is

ς=𝒖t⊤​𝑯¯η,𝒖​𝒖tS​‖𝒖t‖𝑷2=S3​q2S⋅S2​q2=1\.\\varsigma\\;=\\;\\frac\{\{\\bm\{u\}\}\_\{t\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\_\{t\}\}\{S\\\|\{\\bm\{u\}\}\_\{t\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\}\\;=\\;\\frac\{S^\{3\}q^\{2\}\}\{S\\cdot S^\{2\}q^\{2\}\}\\;=\\;1\.\(44\)Therefore, the conditions \(Q1\)–\(Q3\) are sufficient for the spatial factor to beς=1\\varsigma=1\. The temporal factor isτ=Γ​A/‖𝒖t‖𝑷2=Γ​S2​xt​q/\(S2​q2\)=Γ​xt/q\\tau=\\Gamma A/\\\|\{\\bm\{u\}\}\_\{t\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}=\\Gamma S^\{2\}x\_\{t\}q/\(S^\{2\}q^\{2\}\)=\\Gamma x\_\{t\}/q, which recovers \([15](https://arxiv.org/html/2608.18415#S5.E15)\)\.

Temporal factor\.By \(Q4\) the escape is atz=−1z=\-1, soΓ=Q⁡\(−1\)\\Gamma=Q\(\-1\)by Proposition[2\.1](https://arxiv.org/html/2608.18415#S2.Thmtheorem1), and the period\-2 historyxt−k=\(−1\)k​xtx\_\{t\-k\}=\(\-1\)^\{k\}x\_\{t\}gives

q=∑k≥0Qk​xt−k=\(∑k≥0Qk​\(−1\)k\)​xt=Q⁡\(−1\)​xt=Γ​xt\.q\\;=\\;\\sum\_\{k\\geq 0\}Q\_\{k\}x\_\{t\-k\}\\;=\\;\\Big\(\\sum\_\{k\\geq 0\}Q\_\{k\}\(\-1\)^\{k\}\\Big\)x\_\{t\}\\;=\\;Q\(\-1\)\\,x\_\{t\}\\;=\\;\\Gamma x\_\{t\}\.\(45\)Henceτ=Γ​xt/q=1\\tau=\\Gamma x\_\{t\}/q=1, andA=S2​xt​q=Γ​S2​xt2\>0A=S^\{2\}x\_\{t\}q=\\Gamma S^\{2\}x\_\{t\}^\{2\}\>0becauseΓ\>0\\Gamma\>0andxt≠0x\_\{t\}\\neq 0, soA≠0A\\neq 0and the identity \([39](https://arxiv.org/html/2608.18415#A2.E39)\) applies\. Substitutingς=τ=1\\varsigma=\\tau=1into the identity \([39](https://arxiv.org/html/2608.18415#A2.E39)\) givesζ=χ=η​S​Γ/2=η​S​Q​\(−1\)/2\\zeta=\\chi=\\eta S\\Gamma/2=\\eta SQ\(\-1\)/2\. Note that \([45](https://arxiv.org/html/2608.18415#A2.E45)\) is what requires the Nyquist frequency: a mode oscillating at any other escape frequencyω⋆\\omega\_\{\\star\}contributes a phase shift, so no real gain satisfiesq=Γ​xtq=\\Gamma x\_\{t\}at every step\. ∎

### B\.7A Counterexample to the Necessity ofς=τ=1\\varsigma=\\tau=1

The following instance lies entirely inside the framework of \([♠‣2](https://arxiv.org/html/2608.18415#S2.Ex1)\) and is exactly computable\. Take the quadratic objectiveℒ⁡\(𝜽\)=12​𝜽⊤​𝑯​𝜽\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)=\\tfrac\{1\}\{2\}\{\\bm\{\\theta\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{\\theta\}\}onℝ2\\mathbb\{R\}^\{2\}with𝑯=diag⁡\(4,1\)\{\\bm\{H\}\}=\\operatorname\{diag\}\(4,1\), so that𝑯¯η,𝒖=𝑯\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}=\{\\bm\{H\}\}*exactly*and𝒗max=𝒆1\{\\bm\{v\}\}\_\{\\max\}=\{\\bm\{e\}\}\_\{1\}\. Take𝑷=𝑰\{\\bm\{P\}\}=\{\\bm\{I\}\}, soS=4S=4, and take the dc\-normalized Heavy Ball filterQk=\(1−β\)​βkQ\_\{k\}=\(1\-\\beta\)\\beta^\{k\}withβ=1/2\\beta=1/2, soQ⁡\(1\)=1Q\(1\)=1andΓ=Q⁡\(−1\)=\(1−β\)/\(1\+β\)=1/3\\Gamma=Q\(\-1\)=\(1\-\\beta\)/\(1\+\\beta\)=1/3\. Letη=1/10\\eta=1/10and start from a zero momentum buffer, so that𝒖0=\(1−β\)​𝒈0\{\\bm\{u\}\}\_\{0\}=\(1\-\\beta\)\{\\bm\{g\}\}\_\{0\}and

𝒈1=\(𝑰−η⁡\(1−β\)​𝑯\)​𝒈0=diag⁡\(45,1920\)​𝒈0,𝒖1=diag⁡\(1320,2940\)​𝒈0\.\{\\bm\{g\}\}\_\{1\}\\;=\\;\(\{\\bm\{I\}\}\-\\eta\(1\-\\beta\)\{\\bm\{H\}\}\)\{\\bm\{g\}\}\_\{0\}\\;=\\;\\operatorname\{diag\}\(\\tfrac\{4\}\{5\},\\tfrac\{19\}\{20\}\)\{\\bm\{g\}\}\_\{0\},\\qquad\{\\bm\{u\}\}\_\{1\}\\;=\\;\\operatorname\{diag\}\(\\tfrac\{13\}\{20\},\\tfrac\{29\}\{40\}\)\{\\bm\{g\}\}\_\{0\}\.\(46\)All four quantities are invariant to the scale of𝒈0\{\\bm\{g\}\}\_\{0\}, so only its direction matters\. Reading them att=1t=1:

𝒈0=𝒆1:\\displaystyle\{\\bm\{g\}\}\_\{0\}=\{\\bm\{e\}\}\_\{1\}:ς=1,\\displaystyle\\varsigma=1,τ\\displaystyle\\tau=1639,\\displaystyle=\\tfrac\{16\}\{39\},χ\\displaystyle\\chi=115,\\displaystyle=\\tfrac\{1\}\{15\},ζ\\displaystyle\\zeta=1380;\\displaystyle=\\tfrac\{13\}\{80\};\(47\)𝒈0=𝒆2:\\displaystyle\{\\bm\{g\}\}\_\{0\}=\{\\bm\{e\}\}\_\{2\}:ς=14,\\displaystyle\\varsigma=\\tfrac\{1\}\{4\},τ\\displaystyle\\tau=3887,\\displaystyle=\\tfrac\{38\}\{87\},χ\\displaystyle\\chi=115,\\displaystyle=\\tfrac\{1\}\{15\},ζ\\displaystyle\\zeta=29760\.\\displaystyle=\\tfrac\{29\}\{760\}\.Two conclusions can made from this\. First, the case𝒈0=𝒆1\{\\bm\{g\}\}\_\{0\}=\{\\bm\{e\}\}\_\{1\}satisfies every geometric condition one would attach to coincidence, namely𝑯¯η,𝒖=𝑯\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}=\{\\bm\{H\}\}exactly and𝒖1∈span⁡\(𝒗max\)\{\\bm\{u\}\}\_\{1\}\\in\\operatorname\{span\}\(\{\\bm\{v\}\}\_\{\\max\}\), henceς=1\\varsigma=1; yetζ/χ=39/16≠1\\zeta/\\chi=39/16\\neq 1, becauseΓ\\Gammais the loop gain at the root\-locus escape frequency while the trajectory att=1t=1is not at that frequency\. Second, writing𝒈0​\(ϕ\)=\(cos⁡ϕ,sin⁡ϕ\)\{\\bm\{g\}\}\_\{0\}\(\\phi\)=\(\\cos\\phi,\\sin\\phi\), the mapϕ↦ς−τ\\phi\\mapsto\\varsigma\-\\tauis continuous on\[0,π/2\]\[0,\\pi/2\]withς−τ=23/39\>0\\varsigma\-\\tau=23/39\>0atϕ=0\\phi=0andς−τ=−65/348<0\\varsigma\-\\tau=\-65/348<0atϕ=π/2\\phi=\\pi/2, so it has a rootϕ⋆∈\(0,π/2\)\\phi\_\{\\star\}\\in\(0,\\pi/2\)\. Atϕ⋆\\phi\_\{\\star\}we haveς=τ\\varsigma=\\tauand thereforeζ=χ\\zeta=\\chi, whileς<1\\varsigma<1becauseς=1\\varsigma=1forces𝒖1∈span⁡\(𝒆1\)\{\\bm\{u\}\}\_\{1\}\\in\\operatorname\{span\}\(\{\\bm\{e\}\}\_\{1\}\), i\.e\.ϕ=0\\phi=0\. Numericallyϕ⋆=1\.01024954\\phi\_\{\\star\}=1\.01024954andς=τ=0\.43040137\\varsigma=\\tau=0\.43040137, givingζ=χ=1/15\\zeta=\\chi=1/15to ten significant digits\. Henceς=τ=1\\varsigma=\\tau=1is not necessary forζ=χ\\zeta=\\chi\.

### B\.8Proof of Theorem[5\.3](https://arxiv.org/html/2608.18415#S5.Thmtheorem3)

###### Proof\.

From the identity\.Corollary[5\.1](https://arxiv.org/html/2608.18415#S5.Thmtheorem1)givesζ=χ​ς/τ\\zeta=\\chi\\varsigma/\\tauandζ=χ\\zeta=\\chiif and only ifς=τ\\varsigma=\\tau\. The scalarχ=η​S​Γ/2\\chi=\\eta S\\Gamma/2is a function of\(η,S,Γ\)\(\\eta,S,\\Gamma\)alone\. Thereforeζ\\zetais a function of\(η,S,Γ\)\(\\eta,S,\\Gamma\)if and only if the ratioς/τ\\varsigma/\\tauis\.

GD reduction\.For𝑷=I\{\\bm\{P\}\}=IandQ≡1Q\\equiv 1, the GD clause of Corollary[5\.1](https://arxiv.org/html/2608.18415#S5.Thmtheorem1)givesΓ=1\\Gamma=1,τ≡1\\tau\\equiv 1, andζ=χ​ς\\zeta=\\chi\\varsigmawithχ=η​S/2\\chi=\\eta S/2\. Henceχ\\chideterminesζ\\zetaif and only if occupationς\\varsigmais a function of\(η,S\)\(\\eta,S\)alone\.

Occupation is not a function ofSS\.On a frozen quadratic,𝑯¯η,𝒖=𝑯\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}=\{\\bm\{H\}\}and𝒖=𝒈\{\\bm\{u\}\}=\{\\bm\{g\}\}, soς=𝒈⊤​𝑯​𝒈/\(S​‖𝒈‖2\)\\varsigma=\{\\bm\{g\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{g\}\}/\(S\\\|\{\\bm\{g\}\}\\\|^\{2\}\)\. This equals11if and only if𝒈\{\\bm\{g\}\}lies in the top eigenspace of𝑯\{\\bm\{H\}\}\. In dimension at least22that is not implied byS=λmax​\(𝑯\)S=\\lambda\_\{\\max\}\(\{\\bm\{H\}\}\)\. The instance𝑯=diag⁡\(4,1\)\{\\bm\{H\}\}=\\operatorname\{diag\}\(4,1\)hasS=4S=4independently of𝒈\{\\bm\{g\}\}, butς=1\\varsigma=1along𝒆1\{\\bm\{e\}\}\_\{1\}andς=1/4\\varsigma=1/4along𝒆2\{\\bm\{e\}\}\_\{2\}\. The same\(η,S,Γ=1\)\(\\eta,S,\\Gamma=1\)therefore yields two values ofζ\\zeta\. ∎

## Appendix CGauge Derivations

Proposition[2\.1](https://arxiv.org/html/2608.18415#S2.Thmtheorem1)assigns a scalar gaugeΓ:=2/k⋆​\(Q\)\\Gamma:=2/k\_\{\\star\}\(Q\)to every causal filterQQ\. It is the loop gain at the first positive\-gain unit\-circle crossing of the root locus, abbreviated as the*escape gain*in the main text\. Corollary[2\.2](https://arxiv.org/html/2608.18415#S2.Thmtheorem2)is the special caseΓ=Q⁡\(−1\)\\Gamma=Q\(\-1\), which holds for Heavy Ball and Nesterov’s momentum and is proved in Appendix[B\.4](https://arxiv.org/html/2608.18415#A2.SS4)from Lemmas[B\.2](https://arxiv.org/html/2608.18415#A2.Thmtheorem2)and[B\.3](https://arxiv.org/html/2608.18415#A2.Thmtheorem3)\. This additional section extends this to includeQ⁡\(z\)Q\(z\)andΓ\\Gammafor every optimizer that appears in the main text, especially those withΓ≠Q⁡\(−1\)\\Gamma\\neq Q\(\-1\), which are the filters that violate hypothesis \(Q4\) of Corollary[5\.2](https://arxiv.org/html/2608.18415#S5.Thmtheorem2)on the filter side\.

Table 2:Root\-locus gauge of the*one\-pole and parallel*optimizer filters\. Every dc\-normalized entry is a positive one\-pole mixture \([28](https://arxiv.org/html/2608.18415#A2.E28)\), so Lemma[B\.3](https://arxiv.org/html/2608.18415#A2.Thmtheorem3)certifiesΩ\+=\{π\}\\Omega\_\{\+\}=\\\{\\pi\\\}andΓ=Q⁡\(−1\)\\Gamma=Q\(\-1\)\. Cascaded two\-pole momenta are*not*in this table\. The last column recovers the results from[12](https://arxiv.org/html/2608.18415#bib.bib26)\.### C\.1QHM and Parallel Dual Momentum

We use the convention that𝒖t\{\\bm\{u\}\}\_\{t\}is the vector multiplyingη\\etain \([♠‣2](https://arxiv.org/html/2608.18415#S2.Ex1)\) with𝑷=𝑰\{\\bm\{P\}\}=\{\\bm\{I\}\}, and𝒎t\{\\bm\{m\}\}\_\{t\}denotes the momentum buffer\. Every dc\-normalized filter in this subsection is a positive one\-pole mixture \([28](https://arxiv.org/html/2608.18415#A2.E28)\), so Lemma[B\.3](https://arxiv.org/html/2608.18415#A2.Thmtheorem3)applies and givesΩ\+=\{π\}\\Omega\_\{\+\}=\\\{\\pi\\\}andΓ=Q⁡\(−1\)\\Gamma=Q\(\-1\)\.

#### QHM\.

QHM\([40](https://arxiv.org/html/2608.18415#bib.bib44)\)uses the normalized buffer𝒎t=β​𝒎t−1\+\(1−β\)​𝒈t\{\\bm\{m\}\}\_\{t\}=\\beta\{\\bm\{m\}\}\_\{t\-1\}\+\(1\-\\beta\)\{\\bm\{g\}\}\_\{t\}and𝒖t=\(1−ν\)​𝒈t\+ν​𝒎t\{\\bm\{u\}\}\_\{t\}=\(1\-\\nu\)\{\\bm\{g\}\}\_\{t\}\+\\nu\{\\bm\{m\}\}\_\{t\}, which is \([28](https://arxiv.org/html/2608.18415#A2.E28)\) with weights\(1−ν,ν\)\(1\-\\nu,\\nu\)and poles\(0,β\)\(0,\\beta\); it is already dc\-normalized, i\.e\.,Q⁡\(1\)=1Q\(1\)=1\. Forν∈\(0,1\)\\nu\\in\(0,1\)the weights are strictly positive\. Lemma[B\.3](https://arxiv.org/html/2608.18415#A2.Thmtheorem3)gives

ΓQHM​\(β,ν\)=1\+β⁡\(1−2​ν\)1\+β\.\\Gamma^\{\\mathrm\{QHM\}\}\(\\beta,\\nu\)\\;=\\;\\frac\{1\+\\beta\(1\-2\\nu\)\}\{1\+\\beta\}\.\(48\)The endpointsν=0\\nu=0andν=1\\nu=1are one\-pole reductions \(GD and normalized Heavy Ball\), both still withΓ=Q⁡\(−1\)\\Gamma=Q\(\-1\)\. Two special cases are immediate from \([48](https://arxiv.org/html/2608.18415#A3.E48)\) and \([36](https://arxiv.org/html/2608.18415#A2.E36)\):ν=1\\nu=1recovers normalized Heavy Ball, andν=β\\nu=\\betarecovers*normalized Nesterov*, since1\+β−2​β2=\(1−β\)​\(1\+2​β\)1\+\\beta\-2\\beta^\{2\}=\(1\-\\beta\)\(1\+2\\beta\)\. The experimental QHM arm of Section[3](https://arxiv.org/html/2608.18415#S3)usesν=0\.7∈\(0,1\)\\nu=0\.7\\in\(0,1\), so it remains in this Nyquist class withΓ=Q⁡\(−1\)\\Gamma=Q\(\-1\)\.

Figure 6:We optimize a quadratic lossf⁡\(x\)=12​x2f\(x\)=\\tfrac\{1\}\{2\}x^\{2\}with four types of optimizers: GD, Adam, QHM, and cascade two\-pole at various learning ratesη\\eta\. Observe that the stability bound ofχ=η​S​Γ/2≤1\\chi=\\eta S\\Gamma/2\\leq 1is always satisfied for this quadratic loss\.Table 3:Predicted and observed quadratic thresholds\.Γexp\\Gamma\_\{\\mathrm\{exp\}\}is the experiment\-report gauge\. The observedη\\etais the last stable and first unstableη\\etaon a0\.0050\.005\-relative grid\.605605sweep runs in total\.
#### Parallel Dual Momentum\.

Dual momentum in its*parallel*realization, two buffers atβ1≠β2\\beta\_\{1\}\\neq\\beta\_\{2\}summed as𝒖t=\(𝒎t\(1\)\+α​𝒎t\(2\)\)/\(1\+α\)\{\\bm\{u\}\}\_\{t\}=\(\{\\bm\{m\}\}^\{\(1\)\}\_\{t\}\+\\alpha\{\\bm\{m\}\}^\{\(2\)\}\_\{t\}\)/\(1\+\\alpha\), is again \([28](https://arxiv.org/html/2608.18415#A2.E28)\), now withJ=2J=2, weights\(1,α\)/\(1\+α\)\(1,\\alpha\)/\(1\+\\alpha\)and poles\(β1,β2\)\(\\beta\_\{1\},\\beta\_\{2\}\), so forα\>0\\alpha\>0Lemma[B\.3](https://arxiv.org/html/2608.18415#A2.Thmtheorem3)applies verbatim, with no restriction on the number of poles\. Table[2](https://arxiv.org/html/2608.18415#A3.T2)collects these results\. In our text, we did not use this construction for two\-pole momenta \(see the next section\), but we included them in the table for completeness\.

### C\.2Two\-Pole Momentum: the Gauge is No LongerQ⁡\(−1\)Q\(\-1\)

Everything so far has been a*parallel*construction: gradients are filtered by several one\-pole blocks and the outputs are added\. The two\-pole filters\([30](https://arxiv.org/html/2608.18415#bib.bib37)\)of Section[3](https://arxiv.org/html/2608.18415#S3)are instead built by*cascading*a momentum buffer with a second one\-pole block\. Define the two\-pole momentum family, dc\-normalized, withβ1,β2∈\(0,1\)\\beta\_\{1\},\\beta\_\{2\}\\in\(0,1\)andα≥0\\alpha\\geq 0,

Qα​\(z\)=1−β11−β1​z−1⏟momentum buffer⋅11\+α​\[1\+α⁡\(1−β2\)1−β2​z−1\]⏟second one\-pole block,Q\_\{\\alpha\}\(z\)\\;=\\;\\underbrace\{\\frac\{1\-\\beta\_\{1\}\}\{1\-\\beta\_\{1\}z^\{\-1\}\}\}\_\{\\text\{momentum buffer\}\}\\cdot\\underbrace\{\\frac\{1\}\{1\+\\alpha\}\\left\[1\+\\frac\{\\alpha\(1\-\\beta\_\{2\}\)\}\{1\-\\beta\_\{2\}z^\{\-1\}\}\\right\]\}\_\{\\text\{second one\-pole block\}\},\(49\)so thatQα​\(1\)=1Q\_\{\\alpha\}\(1\)=1, the poles areβ1\\beta\_\{1\}andβ2\\beta\_\{2\}, and there is one finite zero\. The parameterα\\alphainterpolates between the two limits of interest: atα=0\\alpha=0the zero cancels the second pole andQ0Q\_\{0\}is normalized Heavy Ball, while asα→∞\\alpha\\to\\inftythe zero moves to the origin andQ∞Q\_\{\\infty\}is the pure double\-EMA cascadeQ∞​\(z\)=∏j=1,2\(1−βj\)/\(1−βj​z−1\)Q\_\{\\infty\}\(z\)=\\prod\_\{j=1,2\}\(1\-\\beta\_\{j\}\)/\(1\-\\beta\_\{j\}z^\{\-1\}\)\.

Figure 7:Complete root locus for the filters studied in this paper, extending Figure[2](https://arxiv.org/html/2608.18415#S2.F2)in the main text to all filters\.One EMA pole cannot reach a90∘90^\{\\circ\}lag according to \([33](https://arxiv.org/html/2608.18415#A2.E33)\)\. A parallel sum stays inside the same cone, and no interior crossing is possible\. A cascade, by contrast,*adds*lags, and two poles can jointly exceed the critical line\. Making this quantitative nearω=π\\omega=\\pialready gives the exact threshold: writingω=π−ϵ\\omega=\\pi\-\\epsilonandβ1=β2=β\\beta\_\{1\}=\\beta\_\{2\}=\\beta, the cascade lag is2​φβ​\(π−ϵ\)=2​β​ϵ/\(1\+β\)\+O⁡\(ϵ2\)2\\varphi\_\{\\beta\}\(\\pi\-\\epsilon\)=2\\beta\\epsilon/\(1\+\\beta\)\+O\(\\epsilon^\{2\}\)while the required lag isϵ/2\\epsilon/2, so an interior crossing detaches fromω=π\\omega=\\piif and only if

2​β1\+β\>12⟺β\>13\.\\frac\{2\\beta\}\{1\+\\beta\}\\;\>\\;\\frac\{1\}\{2\}\\qquad\\Longleftrightarrow\\qquad\\beta\\;\>\\;\\tfrac\{1\}\{3\}\.\(50\)For two\-pole momenta, we numerically evaluate the gauge and plot it in Figure[7](https://arxiv.org/html/2608.18415#A3.F7)\.

### C\.3Stability Bound for Fixed Quadratic Loss

To demonstrate that Proposition[2\.1](https://arxiv.org/html/2608.18415#S2.Thmtheorem1)yields the exact stability bound for a fixed quadratic loss, we conduct simple numerical experiments\. Four types of optimizers—GD, Adam, QHM, and cascade two\-pole—are trained on a quadratic lossf⁡\(x\)=12​x2f\(x\)=\\tfrac\{1\}\{2\}x^\{2\}with various learning ratesη\\eta\. Figure[6](https://arxiv.org/html/2608.18415#A3.F6)summarizes the results\. In the legend, we use the relative learning rateη/ηr\\eta/\\eta\_\{\\text\{r\}\}, whereηr=2/\(S​Γ\)\\eta\_\{\\text\{r\}\}=2/\(S\\Gamma\)is determined by the stability conditionχ=η​S​Γ/2≤1\\chi=\\eta S\\Gamma/2\\leq 1in Proposition[2\.1](https://arxiv.org/html/2608.18415#S2.Thmtheorem1)\. For the two\-pole momentum optimizer, we calculate the numerically found gauge, as shown in the root locus plot in Figure[7](https://arxiv.org/html/2608.18415#A3.F7), which is not the Nyquist gain, i\.e\.,Γ≠Q⁡\(−1\)\\Gamma\\neq Q\(\-1\)\. Regardless of optimizer type, the stability bound is always satisfied on this fixed quadratic loss\.

Table 4:Temporal filters of the executed families, in the paper\-normalized conventionχ=η​S​Γ/2\\chi=\\eta S\\Gamma/2of Appendix[C\.1](https://arxiv.org/html/2608.18415#A3.SS1)\.k⋆k\_\{\\star\}is the unit\-curvature learning\-rate edge\.ω⋆\\omega\_\{\\star\}is the first unit\-circle crossing in radians\. Adam, AdamW, Adafactor, AMSGrad and PAdam share one EMA filter; RMSProp, AdaGrad and GD share the memoryless filter\. Note that two\-pole momentum cases haveΓ≠Q⁡\(−1\)\\Gamma\\neq Q\(\-1\)\.
### C\.4Complete Root Locus

Complete root loci for the filters studied in this paper are shown in Figure[7](https://arxiv.org/html/2608.18415#A3.F7), extending the demonstration in Figure[2](https://arxiv.org/html/2608.18415#S2.F2)in the main text to all filters\. See also Remark[B\.4](https://arxiv.org/html/2608.18415#A2.Thmtheorem4)in the previous section for a more detailed interpretation of the results\.

## Appendix DStabilization from High\-Order Terms

We begin by noting that the realized EoS \([♢‣4](https://arxiv.org/html/2608.18415#S4.Ex3)\) is already exact\. Section[4](https://arxiv.org/html/2608.18415#S4)expresses the residual quantity in terms of a quadratic load𝒖⊤​𝑯¯η,𝒖​𝒖\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}and occasionally interprets this as an approximation of the second Taylor term𝑯¯η,𝒖≃𝑯\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\\simeq\{\\bm\{H\}\}\. However, this simplification is only for our conceptual understanding of the quantity\. It does*not*affect the exactness of our realized EoSζ\\zeta\. This section provides a more detailed analysis to support this simplification\. Higher\-order geometry does not explain*why*the edge is occupied\. It determines*how*an occupied quadratic load is realized as a finite step, and whether that realization rescues or amplifies a one\-step crossing ofζ=1\\zeta=1\.

### D\.1Exact nonlinear realization

Fix a step𝜽\+:=𝜽−η​𝒖\{\\bm\{\\theta\}\}^\{\+\}:=\{\\bm\{\\theta\}\}\-\\eta\{\\bm\{u\}\}withA:=𝒈⊤​𝒖\>0A:=\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\>0andH𝒖:=𝒖⊤​𝑯​𝒖\>0H\_\{\\bm\{u\}\}:=\{\\bm\{u\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{u\}\}\>0\. The quadratic and exact secant expenditures are defined as

B2:=η2​H𝒖,B:=A\+ℒ⁡\(𝜽\+\)−ℒ⁡\(𝜽\)η=η2​𝒖⊤​𝑯¯η,𝒖​𝒖,B\_\{2\}\\;:=\\;\\frac\{\\eta\}\{2\}\\,H\_\{\\bm\{u\}\},\\qquad B\\;:=\\;A\+\\frac\{\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}^\{\+\}\)\-\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)\}\{\\eta\}\\;=\\;\\frac\{\\eta\}\{2\}\\,\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\},\(51\)the second equality being \([13](https://arxiv.org/html/2608.18415#S4.E13)\)\. We also define the corresponding loadsζ2:=B2/A\\zeta\_\{2\}:=B\_\{2\}/Aandζ:=B/A\\zeta:=B/A\. Let the*nonlinear realization factor*be the ratio of the two,

h:=ζζ2=BB2,h\\;:=\\;\\frac\{\\zeta\}\{\\zeta\_\{2\}\}\\;=\\;\\frac\{B\}\{B\_\{2\}\},\(52\)so that, identically,

ζ=ζ2​h\.\\zeta\\;=\\;\\zeta\_\{2\}\\,h\.\(53\)Note that the spatial factorς\\varsigmaof \([♡‣4](https://arxiv.org/html/2608.18415#S4.Ex4)\) is built from the*secant*Hessian, hence already containshh\. Write

ς2:=𝒖⊤​𝑯​𝒖S​‖𝒖‖𝑷2\\varsigma\_\{2\}\\;:=\\;\\frac\{\{\\bm\{u\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{u\}\}\}\{S\\\|\{\\bm\{u\}\}\\\|\_\{\{\\bm\{P\}\}\}^\{2\}\}\(54\)for the quadratic occupancy of the same ray\. Thenς=ς2​h\\varsigma=\\varsigma\_\{2\}h, and the main\-text factorization splits as

ζ2=χ​ς2τ,ζ=χ​ς2τ​h\.\\zeta\_\{2\}\\;=\\;\\chi\\,\\frac\{\\varsigma\_\{2\}\}\{\\tau\},\\qquad\\zeta\\;=\\;\\chi\\,\\frac\{\\varsigma\_\{2\}\}\{\\tau\}\\,h\.\(55\)Thus we can further interpret that \(1\)χ\\chiis*available load*, \(2\)ς2/τ\\varsigma\_\{2\}/\\tauoccupies a*quadratic\-only load*along𝒖\{\\bm\{u\}\}, and \(3\)hh*realizes*that load as a finite step\. We need not apply the correction factorhhto the main\-textς\\varsigmaagain, which is already exact asς=ς2​h\\varsigma=\\varsigma\_\{2\}h\. Multiplying the main\-textς\\varsigmaby a furtherhhwould double\-count the same correction\.

Since we are dealing with directional derivatives along the update direction𝒖\{\\bm\{u\}\}actuated by the optimizer, the factorhhis a Taylor series of one*scalar*, not a collection of tensors\. WriteT𝒖:=∇3ℒ​\[𝒖\]3T\_\{\\bm\{u\}\}:=\\nabla^\{3\}\{\\mathcal\{L\}\}\[\{\\bm\{u\}\}\]^\{3\},F𝒖:=∇4ℒ​\[𝒖\]4F\_\{\\bm\{u\}\}:=\\nabla^\{4\}\{\\mathcal\{L\}\}\[\{\\bm\{u\}\}\]^\{4\},P𝒖:=∇5ℒ​\[𝒖\]5P\_\{\\bm\{u\}\}:=\\nabla^\{5\}\{\\mathcal\{L\}\}\[\{\\bm\{u\}\}\]^\{5\}, and

θ3:=η​T𝒖3​H𝒖,θ4:=η2​F𝒖12​H𝒖,θ5:=η3​P𝒖60​H𝒖\.\\theta\_\{3\}:=\\frac\{\\eta\\,T\_\{\\bm\{u\}\}\}\{3H\_\{\\bm\{u\}\}\},\\qquad\\theta\_\{4\}:=\\frac\{\\eta^\{2\}F\_\{\\bm\{u\}\}\}\{12H\_\{\\bm\{u\}\}\},\\qquad\\theta\_\{5\}:=\\frac\{\\eta^\{3\}P\_\{\\bm\{u\}\}\}\{60H\_\{\\bm\{u\}\}\}\.\(56\)
###### Lemma D\.1\(Secant Hessian expansion\)\.

Letℒ\{\\mathcal\{L\}\}beCm\+2C^\{m\+2\}on a neighbourhood of the executed segment\{𝛉−s​η​𝐮:s∈\[0,1\]\}\\\{\{\\bm\{\\theta\}\}\-s\\eta\{\\bm\{u\}\}:s\\in\[0,1\]\\\}\. \(If the activation is piecewise linear, this requires that the segment not cross a kink\.\) Then the weighted secant Hessian \([10](https://arxiv.org/html/2608.18415#S4.E10)\) satisfies

𝑯¯η,𝒖=2​∑k=0m\(−η\)k\(k\+2\)\!​∇k\+2ℒ​\(𝜽\)​\[𝒖\]k\+O⁡\(ηm\+1\),\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\\;=\\;2\\sum\_\{k=0\}^\{m\}\\frac\{\(\-\\eta\)^\{k\}\}\{\(k\+2\)\!\}\\,\\nabla^\{k\+2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)\[\{\\bm\{u\}\}\]^\{k\}\\;\+\\;O\(\\eta^\{m\+1\}\),\(57\)where∇jℒ​\[𝐮\]k\\nabla^\{j\}\{\\mathcal\{L\}\}\[\{\\bm\{u\}\}\]^\{k\}denotes thejj\-th derivative tensor contractedkktimes with𝐮\{\\bm\{u\}\}\.

###### Proof\.

Expand∇2ℒ​\(𝜽−s​η​𝒖\)=∑k≥0\(−s​η\)kk\!​∇k\+2ℒ​\(𝜽\)​\[𝒖\]k\\nabla^\{2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-s\\eta\{\\bm\{u\}\}\)=\\sum\_\{k\\geq 0\}\\frac\{\(\-s\\eta\)^\{k\}\}\{k\!\}\\nabla^\{k\+2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)\[\{\\bm\{u\}\}\]^\{k\}and integrate against the kernel2​\(1−s\)2\(1\-s\)using2​∫01\(1−s\)​sk​𝑑s=2\(k\+1\)​\(k\+2\)2\\int\_\{0\}^\{1\}\(1\-s\)s^\{k\}\\,ds=\\frac\{2\}\{\(k\+1\)\(k\+2\)\}\. The coefficient of∇k\+2ℒ​\[𝒖\]k\\nabla^\{k\+2\}\{\\mathcal\{L\}\}\[\{\\bm\{u\}\}\]^\{k\}is therefore2​\(−η\)kk\!​\(k\+1\)​\(k\+2\)=2​\(−η\)k\(k\+2\)\!\\frac\{2\(\-\\eta\)^\{k\}\}\{k\!\\,\(k\+1\)\(k\+2\)\}=\\frac\{2\(\-\\eta\)^\{k\}\}\{\(k\+2\)\!\}\. ∎

###### Proposition D\.2\(Impedance series and its loss\-only form\)\.

WithA≠0A\\neq 0,

ζ=∑j≥2\(−1\)j​ηj−1j\!​A​∇jℒ​\(𝜽\)​\[𝒖\]j=ζ2​\(1−θ3\+θ4−θ5\+O⁡\(η4\)\),\\zeta\\;=\\;\\sum\_\{j\\geq 2\}\\frac\{\(\-1\)^\{j\}\\eta^\{j\-1\}\}\{j\!\\,A\}\\,\\nabla^\{j\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)\[\{\\bm\{u\}\}\]^\{j\}\\;=\\;\\zeta\_\{2\}\\bigl\(1\-\\theta\_\{3\}\+\\theta\_\{4\}\-\\theta\_\{5\}\+O\(\\eta^\{4\}\)\\bigr\),\(58\)and thereforeh=1−θ3\+θ4−θ5\+O⁡\(η4\)h=1\-\\theta\_\{3\}\+\\theta\_\{4\}\-\\theta\_\{5\}\+O\(\\eta^\{4\}\)\. Moreover, withΔ​ℒ:=ℒ⁡\(𝛉−η​𝐮\)−ℒ⁡\(𝛉\)\\Delta\{\\mathcal\{L\}\}:=\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-\\eta\{\\bm\{u\}\}\)\-\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\),

ζ−1=Δ​ℒη​A=ℒ⁡\(𝜽−η​𝒖\)−ℒ⁡\(𝜽\)η​𝒈⊤​𝒖,\\zeta\-1\\;=\\;\\frac\{\\Delta\{\\mathcal\{L\}\}\}\{\\eta A\}\\;=\\;\\frac\{\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-\\eta\{\\bm\{u\}\}\)\-\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)\}\{\\eta\\,\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\},\(59\)so \([58](https://arxiv.org/html/2608.18415#A4.E58)\) expands a quantity that is exactly the one\-step loss change normalized by the learning power\.

###### Proof\.

The first equality follows from Lemma[D\.1](https://arxiv.org/html/2608.18415#A4.Thmtheorem1)andζ=η2​A​𝒖⊤​𝑯¯η,𝒖​𝒖\\zeta=\\frac\{\\eta\}\{2A\}\{\\bm\{u\}\}^\{\\top\}\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\{\\bm\{u\}\}\. The factorization is \([56](https://arxiv.org/html/2608.18415#A4.E56)\) collected term by term\. For \([59](https://arxiv.org/html/2608.18415#A4.E59)\), expandΔ​ℒ=∑j≥1\(−η\)jj\!​∇jℒ​\[𝒖\]j=−η​A\+∑j≥2\(−η\)jj\!​∇jℒ​\[𝒖\]j\\Delta\{\\mathcal\{L\}\}=\\sum\_\{j\\geq 1\}\\frac\{\(\-\\eta\)^\{j\}\}\{j\!\}\\nabla^\{j\}\{\\mathcal\{L\}\}\[\{\\bm\{u\}\}\]^\{j\}=\-\\eta A\+\\sum\_\{j\\geq 2\}\\frac\{\(\-\\eta\)^\{j\}\}\{j\!\}\\nabla^\{j\}\{\\mathcal\{L\}\}\[\{\\bm\{u\}\}\]^\{j\}and divide byη​A\\eta A\. Equivalently, it is \([13](https://arxiv.org/html/2608.18415#S4.E13)\) restated\. ∎

Equation \([59](https://arxiv.org/html/2608.18415#A4.E59)\) is Hessian\-free, which was mentioned in the main text as well\. Only two forward passes and the inner productAAare needed, which are already computed by the optimizer\. Only the quadratic loadζ2\\zeta\_\{2\}requires a Hessian\-vector product\. The series \([58](https://arxiv.org/html/2608.18415#A4.E58)\) is simply the Taylor expansion of the exact ratiohh, not an independent mechanism\. The following corollary upper bounds the approximation error when we treat the secant Hessian as the true Hessian\.

###### Corollary D\.3\(Approximation of the edge of stability\)\.

If the Hessian𝐇=∇𝛉2​ℒ​\(𝛉\)\{\\bm\{H\}\}=\\nabla\_\{\{\\bm\{\\theta\}\}\}^\{2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\)isL𝐇L\_\{\\bm\{H\}\}\-Lipschitz on the segment\{𝛉−s​η​𝐮:s∈\[0,1\]\}\\\{\{\\bm\{\\theta\}\}\-s\\eta\{\\bm\{u\}\}:s\\in\[0,1\]\\\}, then

\|𝒖⊤​\(𝑯¯η,𝒖−𝑯\)​𝒖\|≤L𝑯​η​‖𝒖‖33\.\\left\|\{\\bm\{u\}\}^\{\\top\}\(\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\-\{\\bm\{H\}\}\)\{\\bm\{u\}\}\\right\|\\;\\leq\\;\\frac\{L\_\{\\bm\{H\}\}\\eta\\\|\{\\bm\{u\}\}\\\|^\{3\}\}\{3\}\.\(60\)Writing

ζ¯:=η​𝒖⊤​𝑯​𝒖2​𝒈⊤​𝒖\\bar\{\\zeta\}\\;:=\\;\\frac\{\\eta\{\\bm\{u\}\}^\{\\top\}\{\\bm\{H\}\}\{\\bm\{u\}\}\}\{2\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\}\(61\)for the quadratic estimator \(i\.e\.ζ2\\zeta\_\{2\}of \([58](https://arxiv.org/html/2608.18415#A4.E58)\)\), the approximation error is bounded by

\|ζ−ζ¯\|≤L𝑯​η2​‖𝒖‖36​\|𝒈⊤​𝒖\|\.\\left\|\\zeta\-\\bar\{\\zeta\}\\right\|\\;\\leq\\;\\frac\{L\_\{\\bm\{H\}\}\\eta^\{2\}\\\|\{\\bm\{u\}\}\\\|^\{3\}\}\{6\\,\\lvert\{\\bm\{g\}\}^\{\\top\}\{\\bm\{u\}\}\\rvert\}\.\(62\)

###### Proof\.

By \([10](https://arxiv.org/html/2608.18415#S4.E10)\),𝒖⊤​\(𝑯¯η,𝒖−𝑯\)​𝒖=2​∫01\(1−s\)​𝒖⊤​\(∇2ℒ​\(𝜽−s​η​𝒖\)−𝑯\)​𝒖​𝑑s\{\\bm\{u\}\}^\{\\top\}\(\\bar\{\{\\bm\{H\}\}\}\_\{\\eta,\{\\bm\{u\}\}\}\-\{\\bm\{H\}\}\)\{\\bm\{u\}\}=2\\int\_\{0\}^\{1\}\(1\-s\)\\,\{\\bm\{u\}\}^\{\\top\}\\bigl\(\\nabla^\{2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-s\\eta\{\\bm\{u\}\}\)\-\{\\bm\{H\}\}\\bigr\)\{\\bm\{u\}\}\\,ds\. Lipschitz continuity of𝑯\{\\bm\{H\}\}bounds the integrand byL𝑯​\(s​η​‖𝒖‖\)​‖𝒖‖2L\_\{\\bm\{H\}\}\(s\\eta\\\|\{\\bm\{u\}\}\\\|\)\\,\\\|\{\\bm\{u\}\}\\\|^\{2\}, so the integral is at most2​L𝑯​η​‖𝒖‖3​∫01\(1−s\)​s​𝑑s=L𝑯​η​‖𝒖‖3/32L\_\{\\bm\{H\}\}\\eta\\\|\{\\bm\{u\}\}\\\|^\{3\}\\int\_\{0\}^\{1\}\(1\-s\)s\\,ds=L\_\{\\bm\{H\}\}\\eta\\\|\{\\bm\{u\}\}\\\|^\{3\}/3\. Dividing by2​\|A\|/η2\\lvert A\\rvert/\\etayields \([62](https://arxiv.org/html/2608.18415#A4.E62)\)\. This is them=0m=0remainder of Lemma[D\.1](https://arxiv.org/html/2608.18415#A4.Thmtheorem1)under a Lipschitz rather than aC3C^\{3\}hypothesis\. ∎

All higher\-order content inζ\\zetais this one scalar\. It is the ratio of the averaged directional curvature over the step to the instantaneous directional curvature at its start\. The coefficients−θ3\+θ4−⋯\-\\theta\_\{3\}\+\\theta\_\{4\}\-\\cdotsare its Taylor coefficients\. This scalar can act in either side of the edge\.

Let𝒖^:=𝒖/‖𝒖‖\\hat\{\\bm\{u\}\}:=\{\\bm\{u\}\}/\\\|\{\\bm\{u\}\}\\\|, letr:=η​‖𝒖‖r:=\\eta\\\|\{\\bm\{u\}\}\\\|be the step length, and define the*directional curvature profile*

ψ⁡\(s\):=𝒖^⊤​∇2ℒ​\(𝜽−s​𝒖^\)​𝒖^,g𝒖:=𝒈⊤​𝒖^\>0\.\\psi\(s\)\\;:=\\;\\hat\{\\bm\{u\}\}^\{\\top\}\\nabla^\{2\}\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-s\\hat\{\\bm\{u\}\}\)\\,\\hat\{\\bm\{u\}\},\\qquad g\_\{\\bm\{u\}\}\\;:=\\;\{\\bm\{g\}\}^\{\\top\}\\hat\{\\bm\{u\}\}\\;\>\\;0\.\(63\)
###### Proposition D\.4\(Profile form\)\.

The impedance is the\(r−s\)\(r\-s\)\-weighted mean of the directional curvature over the executed step,

ζ=1r​g𝒖​∫0r\(r−s\)​ψ​\(s\)​𝑑s=r​ψ¯r2​g𝒖,ψ¯r:=2r2​∫0r\(r−s\)​ψ​\(s\)​𝑑s,\\zeta\\;=\\;\\frac\{1\}\{r\\,g\_\{\\bm\{u\}\}\}\\int\_\{0\}^\{r\}\(r\-s\)\\,\\psi\(s\)\\,ds\\;=\\;\\frac\{r\\,\\bar\{\\psi\}\_\{r\}\}\{2\\,g\_\{\\bm\{u\}\}\},\\qquad\\bar\{\\psi\}\_\{r\}:=\\frac\{2\}\{r^\{2\}\}\\int\_\{0\}^\{r\}\(r\-s\)\\,\\psi\(s\)\\,ds,\(64\)and therefore, withζ2=r​ψ​\(0\)/\(2​g𝐮\)\\zeta\_\{2\}=r\\psi\(0\)/\(2g\_\{\\bm\{u\}\}\),

h=ζζ2=ψ¯rψ⁡\(0\)\.h\\;=\\;\\frac\{\\zeta\}\{\\zeta\_\{2\}\}\\;=\\;\\frac\{\\bar\{\\psi\}\_\{r\}\}\{\\psi\(0\)\}\.\(65\)

###### Proof\.

Letϕ⁡\(s\):=ℒ⁡\(𝜽−s​𝒖^\)\\phi\(s\):=\{\\mathcal\{L\}\}\(\{\\bm\{\\theta\}\}\-s\\hat\{\\bm\{u\}\}\), soϕ′​\(0\)=−g𝒖\\phi^\{\\prime\}\(0\)=\-g\_\{\\bm\{u\}\}andϕ′′​\(s\)=ψ​\(s\)\\phi^\{\\prime\\prime\}\(s\)=\\psi\(s\)\. Taylor’s theorem with integral remainder givesϕ⁡\(r\)−ϕ⁡\(0\)=−r​g𝒖\+∫0r\(r−s\)​ψ​\(s\)​𝑑s\\phi\(r\)\-\\phi\(0\)=\-rg\_\{\\bm\{u\}\}\+\\int\_\{0\}^\{r\}\(r\-s\)\\psi\(s\)\\,ds, andη​A=r​g𝒖\\eta A=rg\_\{\\bm\{u\}\}, so by \([59](https://arxiv.org/html/2608.18415#A4.E59)\)ζ=1\+ϕ⁡\(r\)−ϕ⁡\(0\)r​g𝒖\\zeta=1\+\\frac\{\\phi\(r\)\-\\phi\(0\)\}\{rg\_\{\\bm\{u\}\}\}, which is \([64](https://arxiv.org/html/2608.18415#A4.E64)\)\. ∎

###### Corollary D\.5\(Monotone softening and stiffening\)\.

Ifψ\\psiis nonincreasing on\[0,r\]\[0,r\]thenh≤1h\\leq 1, and ifψ\\psiis nondecreasing thenh≥1h\\geq 1\. Both inequalities are strict if the monotonicity is strict on a set of positive measure\. More quantitatively, ifψ⁡\(s\)≤\(1−μ\)​ψ​\(0\)\\psi\(s\)\\leq\(1\-\\mu\)\\psi\(0\)fors≥ϱ​rs\\geq\\varrho rwithμ∈\(0,1\)\\mu\\in\(0,1\)andϱ∈\(0,1\)\\varrho\\in\(0,1\), then

h≤1−μ​\(1−ϱ\)2\.h\\;\\leq\\;1\-\\mu\(1\-\\varrho\)^\{2\}\.\(66\)

###### Proof\.

The kernel2​\(r−s\)/r22\(r\-s\)/r^\{2\}is a probability density on\[0,r\]\[0,r\], soψ¯r\\bar\{\\psi\}\_\{r\}is an average ofψ\\psiand \([65](https://arxiv.org/html/2608.18415#A4.E65)\) gives the two monotone statements\. For the bound,∫ϱ​rr2​\(r−s\)r2​𝑑s=\(1−ϱ\)2\\int\_\{\\varrho r\}^\{r\}\\frac\{2\(r\-s\)\}\{r^\{2\}\}\\,ds=\(1\-\\varrho\)^\{2\}, and on that sub\-intervalψ\\psiis below\(1−μ\)​ψ​\(0\)\(1\-\\mu\)\\psi\(0\)while elsewhere it is at mostψ⁡\(0\)\\psi\(0\)\. ∎

Thus we can callh<1h<1*directional softening*andh\>1h\>1*directional stiffening*\.

### D\.2Entry, return, and maintenance

Becauseζ=ζ2​h\\zeta=\\zeta\_\{2\}h, a step lies below, on, or above the realized edge according ashhlies below, on, or above1/ζ21/\\zeta\_\{2\}:

ζ<1⟺h<1ζ2,ζ=1⟺h=1ζ2,ζ\>1⟺h\>1ζ2\.\\zeta<1\\;\\;\\Longleftrightarrow\\;\\;h<\\frac\{1\}\{\\zeta\_\{2\}\},\\qquad\\zeta=1\\;\\;\\Longleftrightarrow\\;\\;h=\\frac\{1\}\{\\zeta\_\{2\}\},\\qquad\\zeta\>1\\;\\;\\Longleftrightarrow\\;\\;h\>\\frac\{1\}\{\\zeta\_\{2\}\}\.\(67\)These are simple identities\. When we are discussing the higher\-order termhh, we are mainly interested in the condition whether this term rescues or destabilizes a one\-step crossing ofζ=1\\zeta=1, i\.e\., whetherhhis below, on, or above1/ζ21/\\zeta\_\{2\}\.

###### Theorem D\.6\(Nonlinear rescue and destabilization\)\.

AssumeA\>0A\>0andH𝐮\>0H\_\{\\bm\{u\}\}\>0\.

1. 1\.Ifζ2=1\+δ\\zeta\_\{2\}=1\+\\deltawithδ\>0\\delta\>0, the step is quadratically supercritical\. It is rescued toζ≤1\\zeta\\leq 1if and only if h≤11\+δi\.e\.1−h≥δ1\+δ\.h\\;\\leq\\;\\frac\{1\}\{1\+\\delta\}\\qquad\\text\{i\.e\.\}\\qquad 1\-h\\;\\geq\\;\\frac\{\\delta\}\{1\+\\delta\}\.\(68\)To the order retained in \([58](https://arxiv.org/html/2608.18415#A4.E58)\), the same comparison readsθ3−θ4\+O⁡\(η3\)≥δ/\(1\+δ\)\\theta\_\{3\}\-\\theta\_\{4\}\+O\(\\eta^\{3\}\)\\geq\\delta/\(1\+\\delta\), with remainder controlled by Corollary[D\.3](https://arxiv.org/html/2608.18415#A4.Thmtheorem3)when only the Lipschitz modulus is assumed\.
2. 2\.Ifζ2=1−δ\\zeta\_\{2\}=1\-\\deltawithδ∈\(0,1\)\\delta\\in\(0,1\), the step is quadratically subcritical\. Nonlinear stiffening pushes it above the edge if and only ifh\>1/\(1−δ\)h\>1/\(1\-\\delta\)\.
3. 3\.Ifψ\\psiis nonincreasing andζ2≤1\\zeta\_\{2\}\\leq 1, thenζ≤ζ2≤1\\zeta\\leq\\zeta\_\{2\}\\leq 1\. Ifψ\\psiis nondecreasing andζ2≥1\\zeta\_\{2\}\\geq 1, thenζ≥ζ2≥1\\zeta\\geq\\zeta\_\{2\}\\geq 1\.

###### Proof\.

Claims 1–2 are \([67](https://arxiv.org/html/2608.18415#A4.E67)\) rewritten inδ\\delta\. Claim 3 is Corollary[D\.5](https://arxiv.org/html/2608.18415#A4.Thmtheorem5)\. ∎

A quadratically supercritical step is therefore rescued precisely when the fractional drop in path\-averaged directional curvature exceeds the fractional linear excess\. A quadratically subcritical step is destabilized when stiffening consumes more than the quadratic margin\. We only need to consider two event indicators, both measurable from\(ζ2,ζ\)\(\\zeta\_\{2\},\\zeta\)alone:*nonlinear rescue*isζ2\>1\\zeta\_\{2\}\>1andζ≤1\\zeta\\leq 1;*nonlinear destabilization*isζ2≤1\\zeta\_\{2\}\\leq 1andζ\>1\\zeta\>1\. Anϵ\\epsilon\-band1−ϵ≤ζ≤1\+ϵ1\-\\epsilon\\leq\\zeta\\leq 1\+\\epsilonis equivalent to\(1−ϵ\)/ζ2≤h≤\(1\+ϵ\)/ζ2\(1\-\\epsilon\)/\\zeta\_\{2\}\\leq h\\leq\(1\+\\epsilon\)/\\zeta\_\{2\}\. That is a one\-step constraint onhh, not a statement that a period\-22orbit remains in1≤ζ<1\+ϵ1\\leq\\zeta<1\+\\epsilon\(a nontrivial two\-cycle withAt\>0A\_\{t\}\>0must change the sign ofζ−1\\zeta\-1\)\.

The factorization \([55](https://arxiv.org/html/2608.18415#A4.E55)\) also splits*changes*ofζ\\zeta\. On the positive domain,

log⁡ζ=log⁡χ\+log⁡ς2−log⁡τ\+log⁡h,\\log\\zeta\\;=\\;\\log\\chi\+\\log\\varsigma\_\{2\}\-\\log\\tau\+\\log h,\(69\)hence

Δ​log⁡ζ=Δ​log⁡χ\+Δ​log​ς2−Δ​log​τ\+Δ​log​h\.\\Delta\\log\\zeta\\;=\\;\\Delta\\log\\chi\+\\Delta\\log\\varsigma\_\{2\}\-\\Delta\\log\\tau\+\\Delta\\log h\.\(70\)Basically, this is the same as the one in the main text, but with one additional granularity, decomposing the spatial participationς\\varsigmainto the quadratic participationς2\\varsigma\_\{2\}and higher\-order maintenancehh\. Each summand is a channel of diagnostics, we may not treat them as causes while analyzing the dynamics\. Therefore, we can read the diagnostics by following interpretations:

Entry from below \(ζt<1\\zeta\_\{t\}<1andΔ​log⁡ζt\>0\\Delta\\log\\zeta\_\{t\}\>0\) isΔ​log⁡χ\+Δ​log​ς2\+Δ​log​h\>Δ​log​τ\\Delta\\log\\chi\+\\Delta\\log\\varsigma\_\{2\}\+\\Delta\\log h\>\\Delta\\log\\tau, return from above is the opposite inequalityΔ​log⁡χ\+Δ​log​ς2\+Δ​log​h<Δ​log​τ\\Delta\\log\\chi\+\\Delta\\log\\varsigma\_\{2\}\+\\Delta\\log h<\\Delta\\log\\tau, and maintenance nearζ≃1\\zeta\\simeq 1is approximate balance of the four increments\. These are measurable channel\-balance conditions\. Note that these are not an attractor theorem: the quantities do not assert that higher\-order terms should driveζ\\zetatoward11, nor thathhis the dominant channel\.

## Appendix EAdditional Experiments

This section presents the experimental setup and all experimental results\.

### E\.1Implementation Details

Table 5:Optimizer grids on used in the paper\. Learning rates are five geometrically spaced values\. The displayed endpoints are the grid minima and maxima\. “BC” is bias correction\.ppis the preconditioner exponent on the second moment\.wd\\mathrm\{wd\}is decoupled weight decay\.Familyη\\etagridβ1\\beta\_\{1\}/β\\betaβ2\\beta\_\{2\}ε\\varepsilonBC /ppwd\\mathrm\{wd\}Adam3×10−53\\times 10^\{\-5\}–10−310^\{\-3\}0\.90\.90\.9990\.99910−710^\{\-7\}yes /0\.50\.500AdamW3×10−53\\times 10^\{\-5\}–10−310^\{\-3\}0\.90\.90\.9990\.99910−710^\{\-7\}yes /0\.50\.50\.010\.01Adafactor10−510^\{\-5\}–10−310^\{\-3\}0\.90\.90\.80\.810−710^\{\-7\}no /0\.50\.500AMSGrad10−510^\{\-5\}–10−310^\{\-3\}0\.90\.90\.9990\.99910−710^\{\-7\}no /0\.50\.500PAdam10−310^\{\-3\}–10−110^\{\-1\}0\.90\.90\.9990\.99910−710^\{\-7\}no /0\.250\.2500NAdam10−510^\{\-5\}–10−310^\{\-3\}0\.90\.90\.9990\.99910−710^\{\-7\}no /0\.50\.500RMSProp10−410^\{\-4\}–10−310^\{\-3\}000\.9950\.99510−710^\{\-7\}no /0\.50\.500AdaGrad10−210^\{\-2\}–10−110^\{\-1\}001110−1010^\{\-10\}no /0\.50\.500GD0\.020\.02–0\.20\.2————00Heavy Ball0\.0020\.002–0\.020\.020\.90\.9———00Nesterov0\.0020\.002–0\.020\.020\.90\.9———00QHM0\.0593750\.059375–0\.593750\.593750\.90\.9——ν=0\.7\\nu=0\.700two\-pole0\.4/0\.70\.4/0\.70\.02570\.0257–0\.2570\.257———cascade00two\-pole0\.7/0\.850\.7/0\.850\.006810\.00681–0\.06810\.0681———cascade00#### Dataset\.

We use CIFAR\-10\([29](https://arxiv.org/html/2608.18415#bib.bib47)\)dataset, all training samples are used\. Every update uses one deterministic50,00050\{,\}000\-example full batch without shuffling\.

#### Architecture\.

The network flattens the3×32×323\\times 32\\times 32input, applies five fully connected hidden layers of width200200withtanh\\tanhactivations, and produces1010logits\. The objective is mean categorical cross\-entropy on the full training set\. This is exactly the same as the5×2005\\times 200architecture of[12](https://arxiv.org/html/2608.18415#bib.bib26), not the2×2002\\times 200network of[13](https://arxiv.org/html/2608.18415#bib.bib4)\.

#### Optimizers and their hyperparameters\.

Table[5](https://arxiv.org/html/2608.18415#A5.T5)displays the learning rate grids and optimizer hyperparameters\. AdamW weight decay is0\.010\.01and AdaGradε\\varepsilonis10−1010^\{\-10\}\. Adam uses bias correction\. Adafactor, AMSGrad, PAdam, NAdam, RMSProp and AdaGrad do not\. PAdam uses partial\-adaptivity exponent0\.250\.25\. QHM usesβ=0\.9\\beta=0\.9,ν=0\.7\\nu=0\.7\. The two\-pole filters are cascaded EMA poles at\(0\.4,0\.7\)\(0\.4,0\.7\)and\(0\.7,0\.85\)\(0\.7,0\.85\), i\.e\., Grokfast\-style\([30](https://arxiv.org/html/2608.18415#bib.bib37)\)cascades rather than parallel mixtures \(Appendix[C\.2](https://arxiv.org/html/2608.18415#A3.SS2)\)\.

#### Temporal filtersQQ\.

The1414optimizer names collapse to eight distinct causal filtersQQ, whose transfer functions are calculated in Table[4](https://arxiv.org/html/2608.18415#A3.T4)\. Heavy Ball and Nesterov are the classical \(not dc\-normalized\) recurrences, soQ⁡\(1\)=1/\(1−β\)=10Q\(1\)=1/\(1\-\\beta\)=10atβ=0\.9\\beta=0\.9\. Adam\-family first moments are dc\-normalized and share the EMA filter of the dc\-normalized Heavy Ball\. The two\-pole rows are the only filters whose first root\-locus exit is off Nyquist\. Refer to Figure[7](https://arxiv.org/html/2608.18415#A3.F7)for the full root locus\.

Table 6:Measured relative edgeχ=η​S​Γ/2\\chi=\\eta S\\Gamma/2offsets for different optimizers\. Each cell is the median ofχ\\chibetween first entry above0\.750\.75and last exit below0\.750\.75\(open windows remain open at the horizon\)\. A dash means the trajectory never exceeded0\.750\.75\. Extreme casesχ\>1\.2\\chi\>1\.2are highlighted in boldface\.

### E\.2Full Comparison of Worst\-Case EoSχ\\chiand Realized EoSζ\\zeta

This section shows a full comparison of worst\-case EoSχ\\chiand realized EoSζ\\zeta, extending Figure[1](https://arxiv.org/html/2608.18415#S1.F1)in the main text\. Figures[8](https://arxiv.org/html/2608.18415#A5.F8)and Figures[9](https://arxiv.org/html/2608.18415#A5.F9)show the results\. Table[6](https://arxiv.org/html/2608.18415#A5.T6)is the complete numerical results, measured as the median ofχ\\chion the closed interval from firstχ\>0\.75\\chi\>0\.75to lastχ<0\.75\\chi<0\.75, or to the terminal checkpoint if the trajectory never returns below0\.750\.75\. Two settings never exceed0\.750\.75\(Adafactorη=10−5\\eta=10^\{\-5\}, Heavy Ballη=0\.002\\eta=0\.002, two\-pole0\.7/0\.850\.7/0\.85atη=0\.00681\\eta=0\.00681\) and are recorded as dashes\. Besides the discussion in Section[3](https://arxiv.org/html/2608.18415#S3), Table[6](https://arxiv.org/html/2608.18415#A5.T6)also suggests that the offsets are not an artefact of decoupled weight decay\. Adam \(λ=0\\lambda=0\) and AdamW \(λ=0\.01\\lambda=0\.01\) share the same temporal filter, the sameη\\etagrid, and the same architecture; their edge\-conditioned family centers are0\.9600\.960and0\.9710\.971\.

Figure 8:Comparison of worst\-case EoSχ\\chiand realized EoSζ\\zetaon the primary CIFAR\-10 experiments\.Figure 9:Comparison of worst\-case EoSχ\\chiand realized EoSζ\\zetaon the primary CIFAR\-10 experiments\.
### E\.3Full Decomposition of Realized EoSζ\\zeta

This section shows a full decomposition of realized EoSζ\\zetafor each individual setting of \(optimizer, learning rate\), extending Figure[4](https://arxiv.org/html/2608.18415#S5.F4)in the main text\. Full results are visualized in Figures[10](https://arxiv.org/html/2608.18415#A5.F10)through Figure[13](https://arxiv.org/html/2608.18415#A5.F13)\.

Figure 10:Decomposition of the realized EoSζ\\zetainto factors in log\-scale, involving the \(relative\) edge of stabilityχ\\chi\(redarea and line\), the spatial participation factorς\\varsigma\(bluearea and line\), and the temporal calibration factorτ\\tau\(greyoverlay and black line\)\. Note thatlog⁡τ\\log\\tauoperates in the opposite direction by subtraction\.Figure 11:Decomposition of the realized EoSζ\\zetainto factors in log\-scale, involving the \(relative\) edge of stabilityχ\\chi\(redarea and line\), the spatial participation factorς\\varsigma\(bluearea and line\), and the temporal calibration factorτ\\tau\(greyoverlay and black line\)\. Note thatlog⁡τ\\log\\tauoperates in the opposite direction by subtraction\.Figure 12:Decomposition of the realized EoSζ\\zetainto factors in log\-scale, involving the \(relative\) edge of stabilityχ\\chi\(redarea and line\), the spatial participation factorς\\varsigma\(bluearea and line\), and the temporal calibration factorτ\\tau\(greyoverlay and black line\)\. Note thatlog⁡τ\\log\\tauoperates in the opposite direction by subtraction\.Figure 13:Decomposition of the realized EoSζ\\zetainto factors in log\-scale, involving the \(relative\) edge of stabilityχ\\chi\(redarea and line\), the spatial participation factorς\\varsigma\(bluearea and line\), and the temporal calibration factorτ\\tau\(greyoverlay and black line\)\. Note thatlog⁡τ\\log\\tauoperates in the opposite direction by subtraction\.