Provable Edge-of-Stability for Adam on a One-Dimensional Quadratic

# Provable Edge-of-Stability for Adam on a One-Dimensional Quadratic
Source: [https://arxiv.org/html/2608.20638](https://arxiv.org/html/2608.20638)
Yiman FongThanks:School of Engineering and Applied Sciences, Harvard University\. Email:yiman\_fong@seas\.harvard\.edu, fangyimin05@gmail\.comHeng YangThanks:School of Engineering and Applied Sciences, Harvard University\. Email:hankyang@seas\.harvard\.edu

###### Abstract

The edge\-of\-stability \(EoS\) phenomenon of Adam has been widely observed, while its underlying dynamical mechanism is not yet fully understood\. We study uncorrected Adam on a one\-dimensional quadratic, a clean setting where constant curvature isolates the optimizer\-induced dynamics behind the EoS\. We characterize the resulting dynamics across the parameter space\. In broad regimes, we prove that Adam exhibits a restoring tendency toward its frozen stability threshold2​\(1\+β1\)/\[η⁡\(1−β1\)\]2\(1\+\\beta\_\{1\}\)/\[\\eta\(1\-\\beta\_\{1\}\)\]\. We also identify settings in which this edge\-seeking mechanism breaks down, including strictly subcritical periodic orbits and specially tuned trajectories that converge to the optimum while remaining uniformly supercritical\. These results give a concrete dynamical explanation for Adam’s EoS in a setting free of evolving loss geometry, while also exposing its limitations\.

## 1Introduction

Adam\([18](https://arxiv.org/html/2608.20638#bib.bib17)\)is among the most widely used optimizers in modern deep learning\. When neural networks are trained with first\-order methods such as gradient descent or Adam, a striking empirical regularity emerges—the*edge of stability*\(EoS\): the sharpness \(the largest eigenvalue of the Hessian of the loss function\) rises toward the stability threshold and then oscillates around it \(Figure[1](https://arxiv.org/html/2608.20638#S1.F1), left\)\. For gradient descent, this behavior has been observed extensively in neural\-network training, where the sharpness approaches the classical threshold2/η2/\\etafor learning rateη\\eta\([11](https://arxiv.org/html/2608.20638#bib.bib1)\)\. For adaptive methods,[12](https://arxiv.org/html/2608.20638#bib.bib2)observed a similar phenomenon for Adam: the*preconditioned sharpness*𝖲t\\mathsf\{S\}\_\{t\}—the largest eigenvalue of the Hessian after Adam’s coordinatewise rescaling—rises toward and then stays near the stability threshold of the corresponding frozen dynamics\. These results establish the EoS as a robust but so far largely empirical phenomenon and a strong departure from classical optimization theory\. While EoS attracts extensive theoretical investigation on gradient descent, the mechanism of EoS for Adam remains mysterious\. This leaves a basic question unresolved for Adam:*why*does this happen?

Figure 1:Edge\-of\-stability behavior and its negative\-feedback mechanism\. Left: Adam’s sharpness oscillates around its frozen stability threshold in both neural\-network training and the one\-dimensional quadratic studied here\. Right: above the edge, expansion raisesvtv\_\{t\}and pusheswtw\_\{t\}down; below the edge, contraction lowersvtv\_\{t\}and pusheswtw\_\{t\}up\. Experimental details are given in Appendix[H](https://arxiv.org/html/2608.20638#A8)\.In this work, we provide theoretical evidence that, in contrast to gradient descent, Adam’s adaptive nature by itself induces EoS\. More specifically, we work in the simplest possible setting, the one\-dimensional quadratic

f⁡\(x\)=12​x2,f\(x\)=\\tfrac\{1\}\{2\}x^\{2\},\(1\)and prove that edge\-of\-stability behavior actually occurs, identifying the mechanism that produces it, illustrated by the negative\-feedback loop in Figure[1](https://arxiv.org/html/2608.20638#S1.F1)\(right\)\. On this objective, uncorrected Adam iterates

mt\+1=β1​mt\+\(1−β1\)​xt,vt\+1=β2​vt\+\(1−β2\)​xt2,xt\+1=xt−ηvt\+1\+ε​mt\+1,m\_\{t\+1\}=\\beta\_\{1\}m\_\{t\}\+\(1\-\\beta\_\{1\}\)x\_\{t\},\\qquad v\_\{t\+1\}=\\beta\_\{2\}v\_\{t\}\+\(1\-\\beta\_\{2\}\)x\_\{t\}^\{2\},\\qquad x\_\{t\+1\}=x\_\{t\}\-\\frac\{\\eta\}\{\\sqrt\{v\_\{t\+1\}\}\+\\varepsilon\}m\_\{t\+1\},\(2\)with momentum and second\-moment parameters0≤β1<10\\leq\\beta\_\{1\}<1and0≤β2<10\\leq\\beta\_\{2\}<1, learning rateη\>0\\eta\>0, stabilizerε\>0\\varepsilon\>0, and initial state\(x0,m0,v0\)\(x\_\{0\},m\_\{0\},v\_\{0\}\)\. We use the normalized sharpnesswt=c​η​𝖲tw\_\{t\}=c\\eta\\,\\mathsf\{S\}\_\{t\}withc=\(1−β1\)/\(1\+β1\)c=\(1\-\\beta\_\{1\}\)/\(1\+\\beta\_\{1\}\), for which the frozen stability threshold becomes the parameter\-free boundaryw=2w=2\. The key mechanism is a negative\-feedback loop driven by Adam’s second\-moment adaptation: above the edge the iterates expand,vtv\_\{t\}inflates, the effective step size shrinks, andwtw\_\{t\}is pushed back down; below the edge the iterates contract,vtv\_\{t\}decays, andwtw\_\{t\}is pushed back up\. In broad regimes we prove that this feedback forceswtw\_\{t\}to keep fluctuating around22: it cannot persistently stay well above or well below the edge\. This built\-in self\-stabilization of the effective step size may also be part of why Adam performs so well in practice\.

Turning this mechanism into proofs requires arguments on the two sides of the edge; Figure[2](https://arxiv.org/html/2608.20638#S1.F2)summarizes the resulting proof structure \(Section[3](https://arxiv.org/html/2608.20638#S3)\)\. In the subcritical regime \(left\), an adaptive Lyapunov function certifies contraction up to an explicit cutoffW¯<2\\overline\{W\}<2, which weakens the second\-moment forcing and pusheswtw\_\{t\}upward toward the edge\. This mechanism is not universal: in the complementary parameter regime, under certain conditions, strictly subcritical periodic orbits can arise whenε=0\\varepsilon=0\. In the supercritical regime \(right\), the sign geometry separates two distinct behaviors\. Aligned trajectories expand and are forced out of every fixed supercritical band in finite time, whereas exceptional persistently misaligned trajectories can contract to the origin while remaining uniformly supercritical\.

Taken together, these results prove that Adam’s adaptive state can generate the negative feedback toward the frozen stability boundary illustrated in Figure[1](https://arxiv.org/html/2608.20638#S1.F1), even when the loss curvature is completely fixed\. At the same time, positive momentum introduces precise ways in which this edge\-seeking mechanism can fail, illustrating the additional complexity of the general setting\. Thus, our analysis identifies both an optimizer\-induced mechanism for the EoS and its limitations already in the simplest quadratic setting\. Extending this exact discrete analysis to higher dimensions is an important direction for future work\.

Subcritical phase \(wt<2w\_\{t\}<2\)β2=β12\\beta\_\{2\}=\\beta\_\{1\}^\{2\}β1\\beta\_\{1\}β2\\beta\_\{2\}\(0\.9,0\.999\)\(0\.9,\\,0\.999\)w=2w=2W¯\\overline\{W\}contraction pusheswtw\_\{t\}up toW¯\\overline\{W\}\(Prop\.[3\.1](https://arxiv.org/html/2608.20638#S3.Thmtheorem1), Cor\.[3\.2](https://arxiv.org/html/2608.20638#S3.Thmtheorem2)\)four\-cycles exist:\(Prop\.[3\.3](https://arxiv.org/html/2608.20638#S3.Thmtheorem3)\)Supercritical phase \(wt\>2w\_\{t\}\>2\)sign ofxt​\(mt\+c​xt\)x\_\{t\}\(m\_\{t\}\{\+\}cx\_\{t\}\)alignedxt​\(mt\+c​xt\)\>0x\_\{t\}\(m\_\{t\}\{\+\}cx\_\{t\}\)\>0misalignedxt​\(mt\+c​xt\)<0x\_\{t\}\(m\_\{t\}\{\+\}cx\_\{t\}\)<0w=2w=2finite exit back to the edge\(Lem\.[3\.4](https://arxiv.org/html/2608.20638#S3.Thmtheorem4), Cor\.[3\.5](https://arxiv.org/html/2608.20638#S3.Thmtheorem5)\)\|xt\|\|x\_\{t\}\|decays exponentially,converging while supercritical\(Prop\.[3\.6](https://arxiv.org/html/2608.20638#S3.Thmtheorem6); Fig\.[4](https://arxiv.org/html/2608.20638#A6.F4)\)generic perturbations re\-align

Figure 2:Proof roadmap forβ1\>0\\beta\_\{1\}\>0\. Left: below the edge, an adaptive Lyapunov function certifies contraction up to a cutoffW¯<2\\overline\{W\}<2whenβ2\>β12\\beta\_\{2\}\>\\beta\_\{1\}^\{2\}\(Proposition[3\.1](https://arxiv.org/html/2608.20638#S3.Thmtheorem1), Corollary[3\.2](https://arxiv.org/html/2608.20638#S3.Thmtheorem2)\), while strictly subcritical four\-cycles can arise for certain complementary parameter choices whenε=0\\varepsilon=0\(Proposition[3\.3](https://arxiv.org/html/2608.20638#S3.Thmtheorem3)\)\. Right: above the edge, the sign ofxt​htx\_\{t\}h\_\{t\}separates aligned trajectories, which exit every fixed supercritical band in finite time, from exceptional persistently misaligned trajectories that converge while remaining uniformly supercritical \(Lemma[3\.4](https://arxiv.org/html/2608.20638#S3.Thmtheorem4), Corollary[3\.5](https://arxiv.org/html/2608.20638#S3.Thmtheorem5), Proposition[3\.6](https://arxiv.org/html/2608.20638#S3.Thmtheorem6)\)\.
## 2Mathematical formulation and the case ofβ1=0\\beta\_\{1\}=0

### 2\.1The stability threshold

To see where the threshold comes from, freeze the second moment in \([2](https://arxiv.org/html/2608.20638#S1.E2)\) at a constant valuev≥0v\\geq 0\. Define

c:=1−β11\+β1,wt:=c​ηvt\+ε,wmax:=c​ηε\.c:=\\frac\{1\-\\beta\_\{1\}\}\{1\+\\beta\_\{1\}\},\\qquad w\_\{t\}:=\\frac\{c\\eta\}\{\\sqrt\{v\_\{t\}\}\+\\varepsilon\},\\qquad w\_\{\\max\}:=\\frac\{c\\eta\}\{\\varepsilon\}\.\(3\)Then0<wt≤wmax0<w\_\{t\}\\leq w\_\{\\max\}, and stability of the frozen update is decided by this scalar alone:

###### Lemma 2\.1\(Frozen stability in one dimension\)\.

Fixv≥0v\\geq 0and let𝒜v\\mathcal\{A\}\_\{v\}denote the map\(xt,mt\)↦\(xt\+1,mt\+1\)\(x\_\{t\},m\_\{t\}\)\\mapsto\(x\_\{t\+1\},m\_\{t\+1\}\)obtained from \([2](https://arxiv.org/html/2608.20638#S1.E2)\) by freezingvt\+1v\_\{t\+1\}atvv\. Then𝒜v\\mathcal\{A\}\_\{v\}is linear, and its spectral radius \(the largest modulus of its eigenvalues\) is smaller than11if and only if

w:=c​ηv\+ε<2\.w:=\\frac\{c\\eta\}\{\\sqrt\{v\}\+\\varepsilon\}<2\.

The lemma is the one\-dimensional case of Lemma[B\.2](https://arxiv.org/html/2608.20638#A2.Thmtheorem2)in Appendix[B](https://arxiv.org/html/2608.20638#A2), which treats general objectives in any dimension\. Thusw=2w=2is the frozen stability boundary, free of all parameters\. In practiceε\\varepsilonis tiny \(e\.g\.10−810^\{\-8\}\), sowmax\>2w\_\{\\max\}\>2in essentially every realistic configuration; this is the regime we study, and the globally subcritical casewmax<2w\_\{\\max\}<2is treated in Appendix[G](https://arxiv.org/html/2608.20638#A7)\.

Note that Schur stability of the frozen iteration is only a pointwise criterion:vtv\_\{t\}, and hencewtw\_\{t\}, evolves along the trajectory, so it does not by itself determine the stability of the adaptive dynamics—yet our results show that this local boundary is precisely the level around whichwtw\_\{t\}oscillates\. In the general setting of neural\-network training, the frozen operator, the preconditioned sharpness𝖲t\\mathsf\{S\}\_\{t\}, and its threshold𝖲⋆\\mathsf\{S\}^\{\\star\}are defined analogously in Appendix[B](https://arxiv.org/html/2608.20638#A2)\.

### 2\.2Illustrative example:β1=0\\beta\_\{1\}=0case

As an illustrative example, in this section we focus on the caseβ1=0\\beta\_\{1\}=0, in which Adam reduces to RMSProp\([26](https://arxiv.org/html/2608.20638#bib.bib30)\)\. This is the simpler case because the momentum variable disappears from the position recursion andc=1c=1\. Adam then reduces to

vt\+1=β2​vt\+\(1−β2\)​xt2,xt\+1=\(1−wt\+1\)​xt,v\_\{t\+1\}=\\beta\_\{2\}v\_\{t\}\+\(1\-\\beta\_\{2\}\)x\_\{t\}^\{2\},\\qquad x\_\{t\+1\}=\(1\-w\_\{t\+1\}\)x\_\{t\},\(4\)where we recallwt=ηvt\+εw\_\{t\}=\\frac\{\\eta\}\{\\sqrt\{v\_\{t\}\}\+\\varepsilon\}\.[5](https://arxiv.org/html/2608.20638#bib.bib23)studied a related threshold\-crossing mechanism for a momentum\-free one\-dimensional quadratic under additional conditions\. Appendix[D](https://arxiv.org/html/2608.20638#A4)strengthens this intuition through a two\-sided finite\-passage result: for everyw¯<2<w¯\\underline\{w\}<2<\\overline\{w\}, a trajectory cannot remain indefinitely beloww¯\\underline\{w\}or abovew¯\\overline\{w\}\. Since these levels may be chosen arbitrarily close to22, the result formalizes the restoring behavior toward the edge\. Ifwtw\_\{t\}remains uniformly below22, the position contracts, reducing the forcing ofvtv\_\{t\}and pushingwtw\_\{t\}upward; if it remains uniformly above22, the reverse mechanism pusheswtw\_\{t\}downward\. This is the feedback illustrated in Figure[1](https://arxiv.org/html/2608.20638#S1.F1)\. The positive\-momentum analysis in Section[3](https://arxiv.org/html/2608.20638#S3)follows the same high\-level strategy, but requires an adaptive Lyapunov argument below the edge and sign geometry above it\.

## 3Analysis of Adam with general parameters

The main focus of this paper is the positive\-momentum regime, corresponding to the practical choice\(β1,β2\)=\(0\.9,0\.999\)\(\\beta\_\{1\},\\beta\_\{2\}\)=\(0\.9,0\.999\)of Adam\. In this regime, we identify subtler EoS behavior of Adam, which requires more careful analysis\. The frozen threshold is stillw=2w=2, but momentum changes what happens near it\. In the subcritical regime, we need to control the joint evolution of position and momentum; in the supercritical regime, the key distinction is whether the two are aligned or misaligned\. All proofs for this section are deferred to Appendix[E](https://arxiv.org/html/2608.20638#A5)\(subcritical results\) and Appendix[F](https://arxiv.org/html/2608.20638#A6)\(supercritical results\)\.

##### Normalized dynamics and phase decomposition

Introduce the normalized momentum coordinate

ht:=c−1​mt\+xt,zt:=\(xtht\)\.h\_\{t\}:=c^\{\-1\}m\_\{t\}\+x\_\{t\},\\qquad z\_\{t\}:=\\binom\{x\_\{t\}\}\{h\_\{t\}\}\.\(5\)Then, the update rule on\(zt\)\(z\_\{t\}\)can be written as

zt\+1=A⁡\(wt\+1\)​zt,A⁡\(w\):=\(1−w−β1​w2−wβ1​\(1−w\)\)\.z\_\{t\+1\}=A\(w\_\{t\+1\}\)z\_\{t\},\\qquad A\(w\):=\\begin\{pmatrix\}1\-w&\-\\beta\_\{1\}w\\\\ 2\-w&\\beta\_\{1\}\(1\-w\)\\end\{pmatrix\}\.\(6\)SinceA⁡\(w\)A\(w\)is similar to the frozen map of Lemma[2\.1](https://arxiv.org/html/2608.20638#S2.Thmtheorem1), its spectral radius satisfiesρ⁡\(A⁡\(w\)\)<1\\rho\(A\(w\)\)<1exactly when0<w<20<w<2\. The matrices encountered along an Adam trajectory, however, vary withwtw\_\{t\}and need not commute\. Stability of every individual matrix is therefore not sufficient to control their product\. Our subcritical argument uses the exact restriction imposed on successive values ofwtw\_\{t\}by the second\-moment recursion, whereas the supercritical argument uses the sign geometry of\(xt,ht\)\(x\_\{t\},h\_\{t\}\)\.

### 3\.1Subcritical analysis

In the subcritical regime, decay of\(xt,mt\)\(x\_\{t\},m\_\{t\}\)reduces the forcing in the second\-moment recursion\. The resulting decrease ofvtv\_\{t\}raiseswtw\_\{t\}and therefore moves the trajectory toward the frozen boundary\. The principal difficulty is that the family\{A⁡\(w\):0<w<2\}\\\{A\(w\):0<w<2\\\}does not admit a common Euclidean contraction estimate\. We instead use the Lyapunov functional

Ψt:=xt2\+β1​wt2−wt⋅ht2\.\\displaystyle\\Psi\_\{t\}:=x\_\{t\}^\{2\}\+\\frac\{\\beta\_\{1\}w\_\{t\}\}\{2\-w\_\{t\}\}\\cdot h\_\{t\}^\{2\}\.\(7\)The coefficient ofht2h\_\{t\}^\{2\}is chosen as a function of the currentwtw\_\{t\}\. The exact one\-step quadratic inequality, together with the admissible change inwtw\_\{t\}, gives the following cutoff:

W¯:=2​\(β2−β1\)β2​\(1−β1\)−2​β1​\(1−β2\)/wmax\.\\overline\{W\}:=\\frac\{2\(\\sqrt\{\\beta\_\{2\}\}\-\\beta\_\{1\}\)\}\{\\sqrt\{\\beta\_\{2\}\}\(1\-\\beta\_\{1\}\)\-2\\beta\_\{1\}\(1\-\\sqrt\{\\beta\_\{2\}\}\)/w\_\{\\max\}\}\.\(8\)
###### Proposition 3\.1\.

Assumewmax\>2w\_\{\\max\}\>2andβ2\>β12\\beta\_\{2\}\>\\beta\_\{1\}^\{2\}\. Then0<W¯<20<\\overline\{W\}<2; ifΨt\>0\\Psi\_\{t\}\>0andwt,wt\+1<W¯w\_\{t\},w\_\{t\+1\}<\\overline\{W\}, thenΨt\+1<Ψt\\Psi\_\{t\+1\}<\\Psi\_\{t\}\. More quantitatively, for every0<a≤w<W¯0<a\\leq w<\\overline\{W\}, ifa≤wt,wt\+1≤wa\\leq w\_\{t\},w\_\{t\+1\}\\leq wthenΨt\+1≤\(1−δ⁡\(a,w\)\)​Ψt\\Psi\_\{t\+1\}\\leq\\bigl\(1\-\\delta\(a,w\)\\bigr\)\\Psi\_\{t\}, where the explicit quantityδ⁡\(a,w\)∈\(0,1\)\\delta\(a,w\)\\in\(0,1\)is defined in Appendix[E](https://arxiv.org/html/2608.20638#A5)\.

When1−β2≪1−β11\-\\beta\_\{2\}\\ll 1\-\\beta\_\{1\}andε/\(c​η\)=1/wmax\\varepsilon/\(c\\eta\)=1/w\_\{\\max\}is negligible, the certified contraction region reaches close to the frozen boundary: forβ1=0\.9\\beta\_\{1\}=0\.9andβ2=0\.999\\beta\_\{2\}=0\.999the cutoff isW¯≈1\.991\\overline\{W\}\\approx 1\.991\. An expansion ofW¯\\overline\{W\}asβ2→1\\beta\_\{2\}\\to 1is given in \([56](https://arxiv.org/html/2608.20638#A5.E56)\) of Appendix[E](https://arxiv.org/html/2608.20638#A5)\.

BecauseΨt\\Psi\_\{t\}keeps decreasing, the trajectory cannot stay below any fixed levelw<W¯w<\\overline\{W\}forever\.

###### Corollary 3\.2\(Finite passage toward the subcritical cutoff\)\.

Assume the hypotheses of Proposition[3\.1](https://arxiv.org/html/2608.20638#S3.Thmtheorem1)\. Then, for anyw<W¯w<\\overline\{W\}andT∈ℕ0T\\in\\mathbb\{N\}\_\{0\}such thatwT≤ww\_\{T\}\\leq w, there existsτ<∞\\tau<\\inftysuch thatwT\+τ\>ww\_\{T\+\\tau\}\>w\.

A quantitative version of Corollary[3\.2](https://arxiv.org/html/2608.20638#S3.Thmtheorem2)is stated in Corollary[E\.2](https://arxiv.org/html/2608.20638#A5.Thmtheorem2)\.

#### 3\.1\.1Existence of subcritical cycles

Proposition[3\.1](https://arxiv.org/html/2608.20638#S3.Thmtheorem1)gives a sufficient contraction region forβ2\>β12\\beta\_\{2\}\>\\beta\_\{1\}^\{2\}which is indeed the regime of the standard parameter choice\(β1,β2\)=\(0\.9,0\.999\)\(\\beta\_\{1\},\\beta\_\{2\}\)=\(0\.9,0\.999\)\. In the complementary parameter range, strictly subcritical cycling can occur\.

###### Proposition 3\.3\(Existence of strictly subcritical four\-cycles\)\.

In the scale\-free caseε=0\\varepsilon=0, suppose that

2−1≤β1<1,0≤β2≤β12\.\\sqrt\{2\}\-1\\leq\\beta\_\{1\}<1,\\qquad 0\\leq\\beta\_\{2\}\\leq\\beta\_\{1\}^\{2\}\.\(9\)Then, for everyη\>0\\eta\>0, there exists an initial state\(x0,m0,v0\)\(x\_\{0\},m\_\{0\},v\_\{0\}\)whose Adam orbit has prime period four and satisfieswt<2w\_\{t\}<2for everyt≥0t\\geq 0\.

Figure[3](https://arxiv.org/html/2608.20638#A5.F3)in Appendix[E](https://arxiv.org/html/2608.20638#A5)displays the construction at a concrete point of the parameter region \([9](https://arxiv.org/html/2608.20638#S3.E9)\)\.

### 3\.2Supercritical analysis

In the supercritical regionw\>2w\>2, the sign ofxt​htx\_\{t\}h\_\{t\}separates two sharply different behaviors\. Whenxtx\_\{t\}andhth\_\{t\}are aligned, the next step preserves alignment and expands the position\. This expansion feeds the second\-moment recursion, increasesvtv\_\{t\}, and therefore pusheswtw\_\{t\}back toward the edge\. When they are misaligned, continued misalignment instead forces the position to contract even though the frozen matrix is unstable\. The following lemma records the one\-step geometry\.

###### Lemma 3\.4\(Supercritical sign geometry\)\.

Suppose thatwt\+1\>2w\_\{t\+1\}\>2\.

1. \(i\)Ifxt​ht\>0x\_\{t\}h\_\{t\}\>0, thenxt\+1​ht\+1\>0x\_\{t\+1\}h\_\{t\+1\}\>0and \|xt\+1\|\>\(wt\+1−1\)​\|xt\|\.\\left\\lvert x\_\{t\+1\}\\right\\rvert\>\(w\_\{t\+1\}\-1\)\\left\\lvert x\_\{t\}\\right\\rvert\.\(10\)
2. \(ii\)Ifxt​ht<0x\_\{t\}h\_\{t\}<0andxt\+1​ht\+1<0x\_\{t\+1\}h\_\{t\+1\}<0, then \|xt\+1\|<\|xt\|wt\+1−1\.\\left\\lvert x\_\{t\+1\}\\right\\rvert<\\frac\{\\left\\lvert x\_\{t\}\\right\\rvert\}\{w\_\{t\+1\}\-1\}\.\(11\)

In particular, an aligned trajectory that remains above a fixed margin2\+δ2\+\\deltaexpands at least at rate1\+δ1\+\\delta\. Such expansion cannot persist indefinitely: it drives the second moment so high thatwtw\_\{t\}can no longer stay above2\+δ2\+\\delta\. This yields a quantitative exit from every fixed band above the edge\.

###### Corollary 3\.5\(Finite exit from a uniformly supercritical band\)\.

Suppose thatxT​hT\>0x\_\{T\}h\_\{T\}\>0andwT≥2\+δw\_\{T\}\\geq 2\+\\deltafor someδ∈\(0,1\]\\delta\\in\(0,1\], and define

τ:=min⁡\{n∈ℕ≥1:wT\+n<2\+δ\}\.\\tau:=\\min\\\{n\\in\\mathbb\{N\}\_\{\\geq 1\}:w\_\{T\+n\}<2\+\\delta\\\}\.\(12\)Then

τ≤1\+δ−1​log⁡\(1\+δ​\(c​η\)2\(1−β2\)​xT2\)\.\\displaystyle\\tau\\leq 1\+\\delta^\{\-1\}\\log\\\!\\left\(1\+\\frac\{\\delta\(c\\eta\)^\{2\}\}\{\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\}\\right\)\.\(13\)

Consequently, if an aligned trajectory remains supercritical for all future times, thenlim inft→∞wt=2\\liminf\_\{t\\to\\infty\}w\_\{t\}=2\.

##### The unstable convergence stage

Part \(ii\) of Lemma[3\.4](https://arxiv.org/html/2608.20638#S3.Thmtheorem4)raises the question of whether misalignment must eventually end\. The answer is no: for every initialv0v\_\{0\}withw0\>2w\_\{0\}\>2, one can choose the initial position and momentum so that the trajectory remains misaligned forever\. It then converges to the origin along an exceptional contracting trajectory even though its frozen dynamics remain unstable\.

###### Proposition 3\.6\(Persistent supercritical misalignment\)\.

Suppose that0<β1<10<\\beta\_\{1\}<1andwmax\>2w\_\{\\max\}\>2\. For everyv0\>0v\_\{0\}\>0satisfyingw0=c​η/\(v0\+ε\)\>2w\_\{0\}=c\\eta/\(\\sqrt\{v\_\{0\}\}\+\\varepsilon\)\>2, there existx0≠0x\_\{0\}\\neq 0andm0∈ℝm\_\{0\}\\in\\mathbb\{R\}such that

xt​\(mt\+c​xt\)<0for every​t≥0\.x\_\{t\}\(m\_\{t\}\+cx\_\{t\}\)<0\\qquad\\text\{for every \}t\\geq 0\.\(14\)Along this trajectory,

wt≥w0,\|xt\|≤\(w0−1\)−t​\|x0\|,t≥0\.w\_\{t\}\\geq w\_\{0\},\\qquad\\left\\lvert x\_\{t\}\\right\\rvert\\leq\(w\_\{0\}\-1\)^\{\-t\}\\left\\lvert x\_\{0\}\\right\\rvert,\\qquad t\\geq 0\.\(15\)In particular,\(xt,mt,vt\)→\(0,0,0\)\(x\_\{t\},m\_\{t\},v\_\{t\}\)\\to\(0,0,0\)andwt→wmaxw\_\{t\}\\to w\_\{\\max\}, although the trajectory remains uniformly supercritical for all time\.

## 4Conclusion

We studied the exact discrete dynamics of uncorrected Adam on a one\-dimensional quadratic\. Our analysis identifies a restoring mechanism toward the frozen stability boundary\. With momentum, we establish this behavior up to an explicit subcritical cutoff close to the threshold under standard parameter regimes\. We also show its limitations through strictly subcritical periodic orbits and persistently supercritical yet convergent trajectories\. These results demonstrate that Adam can generate edge\-of\-stability behavior intrinsically, while momentum introduces dynamical mechanisms that can prevent universal convergence to the edge\. Extending this analysis to higher\-dimensional settings is an important direction for future work\.

## References

- A\. Agarwala, F\. Pedregosa, and J\. PenningtonSecond\-order regression models exhibit progressive sharpening to the edge of stability\.InProceedings of the 40th International Conference on Machine Learning,Vol\.202,pp\. 169–195\.Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Ahnet al\.\(2023\)K\. Ahn, S\. Bubeck, S\. Chewi, Y\. T\. Lee, F\. Suarez, and Y\. ZhangLearning threshold neurons via the “edge of stability”\.External Links:2212\.07469,[Link](https://arxiv.org/abs/2212.07469)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Aroraet al\.\(2022\)S\. Arora, Z\. Li, and A\. PanigrahiUnderstanding gradient descent on the edge of stability in deep learning\.InProceedings of the 39th International Conference on Machine Learning,K\. Chaudhuri, S\. Jegelka, L\. Song, C\. Szepesvari, G\. Niu, and S\. Sabato \(Eds\.\),Proceedings of Machine Learning Research, Vol\.162,pp\. 948–1024\.External Links:[Link](https://proceedings.mlr.press/v162/arora22a.html)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Baiet al\.\(2026a\)Z\. Bai, J\. Zhao, Z\. Zhou, Z\. J\. Xu, and Y\. ZhangTowards understanding adam convergence on highly degenerate polynomials\.External Links:2603\.09581,[Link](https://arxiv.org/abs/2603.09581)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1)\.
- Baiet al\.\(2026b\)Z\. Bai, Z\. Zhou, J\. Zhao, X\. Li, Z\. Li, F\. Xiong, H\. Yang, Y\. Zhang, and Z\. J\. XuAdaptive preconditioners trigger loss spikes in adam\.External Links:2506\.04805,[Link](https://arxiv.org/abs/2506.04805)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px3.p1.1),[§2\.2](https://arxiv.org/html/2608.20638#S2.SS2.p1.2)\.
- Barakat and Bianchi \(2020\)A\. Barakat and P\. BianchiConvergence and dynamical behavior of the adam algorithm for non\-convex stochastic optimization\.External Links:1810\.02263,[Link](https://arxiv.org/abs/1810.02263)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1)\.
- Bock and Weiß \(2021\)S\. Bock and M\. G\. WeißLocal convergence of adaptive gradient descent optimizers\.External Links:2102\.09804,[Link](https://arxiv.org/abs/2102.09804)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px3.p1.1)\.
- Bock and Weiß \(2019\)S\. Bock and M\. WeißNon\-convergence and limit cycles in the adam optimizer\.InArtificial Neural Networks and Machine Learning – ICANN 2019: Deep Learning,pp\. 232–243\.External Links:ISBN 9783030304843,ISSN 1611\-3349,[Link](http://dx.doi.org/10.1007/978-3-030-30484-3_20),[Document](https://dx.doi.org/10.1007/978-3-030-30484-3%5F20)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px3.p1.1)\.
- Chen and Bruna \(2023\)L\. Chen and J\. BrunaBeyond the edge of stability via two\-step gradient updates\.InProceedings of the 40th International Conference on Machine Learning,A\. Krause, E\. Brunskill, K\. Cho, B\. Engelhardt, S\. Sabato, and J\. Scarlett \(Eds\.\),Proceedings of Machine Learning Research, Vol\.202,pp\. 4330–4391\.External Links:[Link](https://proceedings.mlr.press/v202/chen23b.html)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Chenet al\.\(2019\)X\. Chen, S\. Liu, R\. Sun, and M\. HongOn the convergence of a class of adam\-type algorithms for non\-convex optimization\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=H1x-x309tm)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1)\.
- Cohenet al\.\(2021\)J\. Cohen, S\. Kaur, Y\. Li, J\. Z\. Kolter, and A\. TalwalkarGradient descent on neural networks typically occurs at the edge of stability\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=jh-rTtvkGeM)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.20638#S1.p1.1)\.
- Cohenet al\.\(2022\)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\.External Links:2207\.14484,[Link](https://arxiv.org/abs/2207.14484)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1),[Appendix H](https://arxiv.org/html/2608.20638#A8.p1.1),[§1](https://arxiv.org/html/2608.20638#S1.p1.1)\.
- da Silva and Gazeau \(2020\)A\. B\. da Silva and M\. GazeauA general system of differential equations to model first\-order adaptive algorithms\.Journal of Machine Learning Research21\(129\),pp\. 1–42\.External Links:[Link](http://jmlr.org/papers/v21/18-808.html)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.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\.InThe Eleventh International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=nhKHA59gXz)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Dereichet al\.\(2025\)S\. Dereich, R\. Graeber, A\. Jentzen, and A\. RiekertAsymptotic stability properties and a priori bounds for adam and other gradient descent optimization methods\.External Links:2509\.10476,[Link](https://arxiv.org/abs/2509.10476)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1),[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px3.p1.1)\.
- Duchiet al\.\(2011\)J\. Duchi, E\. Hazan, and Y\. SingerAdaptive subgradient methods for online learning and stochastic optimization\.Journal of Machine Learning Research12\(61\),pp\. 2121–2159\.External Links:[Link](http://jmlr.org/papers/v12/duchi11a.html)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1)\.
- Kalraet al\.\(2025\)D\. S\. Kalra, T\. He, and M\. BarkeshliUniversal sharpness dynamics in neural network training: fixed point analysis, edge of stability, and route to chaos\.External Links:2311\.02076,[Link](https://arxiv.org/abs/2311.02076)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Kingma and Ba \(2015\)D\. P\. Kingma and J\. BaAdam: a method for stochastic optimization\.InInternational Conference on Learning Representations,Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1),[§1](https://arxiv.org/html/2608.20638#S1.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/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Liet al\.\(2025\)X\. Li, H\. Wen, and K\. LyuAdam reduces a unique form of sharpness: theoretical insights near the minimizer manifold\.External Links:2511\.02773,[Link](https://arxiv.org/abs/2511.02773)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1)\.
- Liet al\.\(2022\)Z\. Li, Z\. Wang, and J\. LiAnalyzing sharpness along gd trajectory: progressive sharpening and edge of stability\.External Links:2207\.12678,[Link](https://arxiv.org/abs/2207.12678)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Mulayoff and Stich \(2026\)R\. Mulayoff and S\. U\. StichOn the stability of nonlinear dynamics in gd and sgd: beyond quadratic potentials\.InProceedings of Thirty Ninth Conference on Learning Theory,S\. Hanneke and T\. Lattimore \(Eds\.\),Proceedings of Machine Learning Research, Vol\.336,pp\. 5210–5243\.External Links:[Link](https://proceedings.mlr.press/v336/mulayoff26a.html)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Reddiet al\.\(2018\)S\. J\. Reddi, S\. Kale, and S\. KumarOn the convergence of adam and beyond\.InInternational Conference on Learning Representations,External Links:[Link](https://arxiv.org/abs/1904.09237)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1)\.
- Regis and Chewi \(2026\)E\. Regis and S\. ChewiA rod flow model for adam at the edge of stability\.External Links:2605\.06821,[Link](https://arxiv.org/abs/2605.06821)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px3.p1.1)\.
- Song and Yun \(2023\)M\. Song and C\. YunTrajectory alignment: understanding the edge of stability phenomenon via bifurcation theory\.External Links:2307\.04204,[Link](https://arxiv.org/abs/2307.04204)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.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 course lecture, Neural Networks for Machine LearningExternal Links:[Link](https://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf)Cited by:[§2\.2](https://arxiv.org/html/2608.20638#S2.SS2.p1.1)\.
- Wuet al\.\(2018\)L\. Wu C\. Maet al\.How sgd selects the global minima in over\-parameterized learning: a dynamical stability perspective\.Advances in Neural Information Processing Systems31\.Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.
- Xieet al\.\(2026\)Z\. Xie, Y\. Fong, H\. Yuan, and Z\. WenAccelerating optimization via differentiable stopping time\.Advances in Neural Information Processing Systems38,pp\. 72185–72212\.Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1)\.
- Zhanget al\.\(2023\)Y\. Zhang, C\. Chen, N\. Shi, R\. Sun, and Z\. LuoAdam can converge without any modification on update rules\.External Links:2208\.09632,[Link](https://arxiv.org/abs/2208.09632)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px2.p1.1)\.
- Zhuet al\.\(2023\)X\. Zhu, Z\. Wang, X\. Wang, M\. Zhou, and R\. GeUnderstanding edge\-of\-stability training dynamics with a minimalist example\.InThe Eleventh International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=p7EagBsMAEO)Cited by:[Appendix A](https://arxiv.org/html/2608.20638#A1.SS0.SSS0.Px1.p1.1)\.

## Appendix ARelated Work

##### Edge of stability\.

The edge\-of\-stability \(EoS\) phenomenon was systematically documented by[11](https://arxiv.org/html/2608.20638#bib.bib1), with early precursors in the catapult phase\[[19](https://arxiv.org/html/2608.20638#bib.bib26)\]; the role of dynamical stability in selecting minima was highlighted by[27](https://arxiv.org/html/2608.20638#bib.bib27)\. Subsequent work has studied its underlying mechanisms, including implicit regularization\[[3](https://arxiv.org/html/2608.20638#bib.bib4)\], self\-stabilization through higher\-order geometry\[[14](https://arxiv.org/html/2608.20638#bib.bib3)\], progressive sharpening\[[21](https://arxiv.org/html/2608.20638#bib.bib5)\], and EoS behavior in simplified models\[[1](https://arxiv.org/html/2608.20638#bib.bib8),[2](https://arxiv.org/html/2608.20638#bib.bib9),[30](https://arxiv.org/html/2608.20638#bib.bib11),[9](https://arxiv.org/html/2608.20638#bib.bib10)\]\. Other works connect EoS to bifurcation and nonlinear oscillatory dynamics\[[25](https://arxiv.org/html/2608.20638#bib.bib16),[22](https://arxiv.org/html/2608.20638#bib.bib14),[17](https://arxiv.org/html/2608.20638#bib.bib15)\]\. These results primarily concern gradient descent\. For adaptive methods,[12](https://arxiv.org/html/2608.20638#bib.bib2)identified an analogous adaptive EoS characterized by the preconditioned Hessian\. Our work studies this phenomenon for the exact discrete Adam dynamics on a quadratic objective, where the underlying curvature is fixed and the effective sharpness evolves solely through adaptive preconditioning\.

##### Adam\.

AdaGrad\[[16](https://arxiv.org/html/2608.20638#bib.bib18)\]introduced coordinatewise adaptive learning rates based on accumulated gradients, and[18](https://arxiv.org/html/2608.20638#bib.bib17)later introduced Adam by combining adaptive second\-moment scaling with momentum\. Despite its empirical success, Adam can fail to converge\[[23](https://arxiv.org/html/2608.20638#bib.bib7)\], motivating convergence analyses under additional assumptions\[[10](https://arxiv.org/html/2608.20638#bib.bib21),[29](https://arxiv.org/html/2608.20638#bib.bib28),[15](https://arxiv.org/html/2608.20638#bib.bib29)\]\. Adam has also been studied through continuous\-time dynamical models\[[13](https://arxiv.org/html/2608.20638#bib.bib19),[6](https://arxiv.org/html/2608.20638#bib.bib13)\]\. A separate line of work studies how to tune Adam’s hyperparameters, for instance adapting its learning rate online\[[28](https://arxiv.org/html/2608.20638#bib.bib25)\]\. More recent work has investigated Adam’s behavior on degenerate objectives and its implicit effect on sharpness\[[4](https://arxiv.org/html/2608.20638#bib.bib20),[20](https://arxiv.org/html/2608.20638#bib.bib12)\]\. However, comparatively little work characterizes how Adam’s adaptive preconditioner drives its sharpness relative to a finite\-step stability threshold\.

##### Adam at the edge of stability\.

The works most closely related to ours study Adam directly through stability and dynamical perspectives\.[5](https://arxiv.org/html/2608.20638#bib.bib23)explain loss spikes through the evolution of the adaptive preconditioner, with their theoretical analysis focusing on a one\-dimensional quadratic setting withβ1=0\\beta\_\{1\}=0\.[24](https://arxiv.org/html/2608.20638#bib.bib22)instead develop a continuous\-time model for Adam in the EoS regime\. From a discrete dynamical perspective,[8](https://arxiv.org/html/2608.20638#bib.bib6)showed that Adam can admit non\-convergent limit cycles, including quadratic examples, while[7](https://arxiv.org/html/2608.20638#bib.bib24)analyzed local convergence through linear stability near fixed points\. Recently,[15](https://arxiv.org/html/2608.20638#bib.bib29)established a priori bounds and asymptotic stability properties for Adam on strongly convex quadratic objectives\. In contrast, we study the exact finite\-step dynamics across both subcritical and supercritical regimes, including finite threshold passage, periodic orbits, and trajectories whose behavior cannot be inferred from frozen stability alone\.

## Appendix BThe general Adam iteration and the frozen stability threshold

For a twice continuously differentiable objectivef:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}, uncorrected Adam maintains a momentum estimatemtm\_\{t\}and a second\-moment estimatevtv\_\{t\}and iterates

mt\+1\\displaystyle m\_\{t\+1\}=β1mt\+\(1−β1\)∇f\(xt\),\\displaystyle=\\beta\_\{1\}m\_\{t\}\+\(1\-\\beta\_\{1\}\)\\nabla f\(x\_\{t\}\),\(16\)vt\+1\\displaystyle v\_\{t\+1\}=β2​vt\+\(1−β2\)​\(∇f​\(xt\)⊙∇f​\(xt\)\),\\displaystyle=\\beta\_\{2\}v\_\{t\}\+\(1\-\\beta\_\{2\}\)\\bigl\(\\nabla f\(x\_\{t\}\)\\odot\\nabla f\(x\_\{t\}\)\\bigr\),xt\+1\\displaystyle x\_\{t\+1\}=xt−η​\(diag⁡\(vt\+1\)\+ε​I\)−1​mt\+1,\\displaystyle=x\_\{t\}\-\\eta\\bigl\(\\operatorname\{diag\}\(\\sqrt\{v\_\{t\+1\}\}\)\+\\varepsilon I\\bigr\)^\{\-1\}m\_\{t\+1\},with0≤β1<10\\leq\\beta\_\{1\}<1,0≤β2<10\\leq\\beta\_\{2\}<1,η\>0\\eta\>0,ε\>0\\varepsilon\>0, and initial state\(x0,m0,v0\)∈ℝd×ℝd×\[0,∞\)d\(x\_\{0\},m\_\{0\},v\_\{0\}\)\\in\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\\times\[0,\\infty\)^\{d\}; here⊙\\odotand the square root ofvtv\_\{t\}are taken coordinatewise\. The one\-dimensional iteration \([2](https://arxiv.org/html/2608.20638#S1.E2)\) of the main text is the special casef⁡\(x\)=12​x2f\(x\)=\\tfrac\{1\}\{2\}x^\{2\}\.

A useful EoS perspective is to freeze the second\-moment variable and study the resulting operator on position and momentum:

𝒜v:\(xm\)⟼\(x−η\(diag\(v\)\+εI\)−1\(β1m\+\(1−β1\)∇f\(x\)\)β1m\+\(1−β1\)∇f\(x\)\)\.\\mathcal\{A\}\_\{v\}:\\binom\{x\}\{m\}\\longmapsto\\binom\{x\-\\eta\\bigl\(\\operatorname\{diag\}\(\\sqrt\{v\}\)\+\\varepsilon I\\bigr\)^\{\-1\}\\bigl\(\\beta\_\{1\}m\+\(1\-\\beta\_\{1\}\)\\nabla f\(x\)\\bigr\)\}\{\\beta\_\{1\}m\+\(1\-\\beta\_\{1\}\)\\nabla f\(x\)\}\.\(17\)The following preconditioned Hessian and its largest eigenvalue identify the stability boundary of the frozen linearization\.

###### Definition B\.1\(Preconditioned sharpness\)\.

Define

H~t:=\(diag⁡\(vt\)\+ε​I\)−1​∇2f​\(xt\),𝖲t:=λmax​\(H~t\)\.\\widetilde\{H\}\_\{t\}:=\\bigl\(\\operatorname\{diag\}\(\\sqrt\{v\_\{t\}\}\)\+\\varepsilon I\\bigr\)^\{\-1\}\\nabla^\{2\}f\(x\_\{t\}\),\\qquad\\mathsf\{S\}\_\{t\}:=\\lambda\_\{\\max\}\(\\widetilde\{H\}\_\{t\}\)\.\(18\)

###### Lemma B\.2\(Frozen stability\)\.

Suppose that∇2f​\(xt\)≻0\\nabla^\{2\}f\(x\_\{t\}\)\\succ 0\. The linearization of𝒜vt\\mathcal\{A\}\_\{v\_\{t\}\}is Schur stable if and only if

𝖲t<𝖲⋆,𝖲⋆:=2​\(1\+β1\)η⁡\(1−β1\)\.\\mathsf\{S\}\_\{t\}<\\mathsf\{S\}^\{\\star\},\\qquad\\mathsf\{S\}^\{\\star\}:=\\frac\{2\(1\+\\beta\_\{1\}\)\}\{\\eta\(1\-\\beta\_\{1\}\)\}\.\(19\)

###### Proof\.

Recall that when∇2f​\(xt\)≻0\\nabla^\{2\}f\(x\_\{t\}\)\\succ 0, the matrixH~t\\widetilde\{H\}\_\{t\}of \([18](https://arxiv.org/html/2608.20638#A2.E18)\) is similar to a symmetric positive\-definite matrix, so its eigenvalues are real and positive\. Fixttand writeD=diag⁡\(vt\)\+ε​ID=\\operatorname\{diag\}\(\\sqrt\{v\_\{t\}\}\)\+\\varepsilon IandH=∇2f​\(xt\)H=\\nabla^\{2\}f\(x\_\{t\}\)\. In the coordinatesy=D1/2​Δ​xy=D^\{1/2\}\\Delta xandp=D−1/2Δmp=D^\{\-1/2\}\\Delta m, the linearization of𝒜vt\\mathcal\{A\}\_\{v\_\{t\}\}is

p\+=β1p\+\(1−β1\)D−1/2HD−1/2y,y\+=y−ηp\+\.p^\{\+\}=\\beta\_\{1\}p\+\(1\-\\beta\_\{1\}\)D^\{\-1/2\}HD^\{\-1/2\}y,\\qquad y^\{\+\}=y\-\\eta p^\{\+\}\.\(20\)The middle matrix is symmetric positive definite and has the same eigenvalues asD−1​H=H~tD^\{\-1\}H=\\widetilde\{H\}\_\{t\}\. Along an eigenvector with eigenvalueλ\>0\\lambda\>0, the characteristic polynomial is

z2−\(1\+β1−η⁡\(1−β1\)​λ\)​z\+β1\.z^\{2\}\-\\bigl\(1\+\\beta\_\{1\}\-\\eta\(1\-\\beta\_\{1\}\)\\lambda\\bigr\)z\+\\beta\_\{1\}\.\(21\)The quadratic Jury criterion gives Schur stability exactly when0<η⁡\(1−β1\)​λ<2​\(1\+β1\)0<\\eta\(1\-\\beta\_\{1\}\)\\lambda<2\(1\+\\beta\_\{1\}\)\. This holds for every mode if and only ifλmax​\(H~t\)<2​\(1\+β1\)/\(η⁡\(1−β1\)\)\\lambda\_\{\\max\}\(\\widetilde\{H\}\_\{t\}\)<2\(1\+\\beta\_\{1\}\)/\(\\eta\(1\-\\beta\_\{1\}\)\), as claimed\. ∎

## Appendix CNotation retained for Section[2\.2](https://arxiv.org/html/2608.20638#S2.SS2)

This section only records the quantities used in the statements of Section[2\.2](https://arxiv.org/html/2608.20638#S2.SS2); no proof for that section is included\. Forn∈ℕ≥1n\\in\\mathbb\{N\}\_\{\\geq 1\}andr,s≥0r,s\\geq 0, let

Gn​\(r,s\):=\{\(rn−sn\)/\(r−s\),r≠s,n​rn−1,r=s\.G\_\{n\}\(r,s\):=\\begin\{cases\}\(r^\{n\}\-s^\{n\}\)/\(r\-s\),&r\\neq s,\\\\ nr^\{n\-1\},&r=s\.\\end\{cases\}\(22\)Also set

σw^​\(T\)\\displaystyle\\sigma\_\{\\widehat\{w\}\}\(T\):=inf\{n∈ℕ0:wT\+n≥w^\},\\displaystyle:=\\inf\\\{n\\in\\mathbb\{N\}\_\{0\}:w\_\{T\+n\}\\geq\\widehat\{w\}\\\},\(23\)Aδ\\displaystyle A\_\{\\delta\}:=\(1\+δ\)2,\\displaystyle:=\(1\+\\delta\)^\{2\},τδ​\(T\)\\displaystyle\\tau\_\{\\delta\}\(T\):=inf\{n∈ℕ≥1:wT\+n<2\+δ\},\\displaystyle:=\\inf\\\{n\\in\\mathbb\{N\}\_\{\\geq 1\}:w\_\{T\+n\}<2\+\\delta\\\},Ln​\(T,a\)\\displaystyle L\_\{n\}\(T,a\):=β2n​vT\+\(1−β2\)​a2​Gn​\(Aδ,β2\),\\displaystyle:=\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)a^\{2\}G\_\{n\}\(A\_\{\\delta\},\\beta\_\{2\}\),withinf∅=\+∞\\inf\\varnothing=\+\\infty\.

## Appendix DProofs for theβ1=0\\beta\_\{1\}=0case

Here we prove the results announced in Section[2\.2](https://arxiv.org/html/2608.20638#S2.SS2): for theβ1=0\\beta\_\{1\}=0dynamics, we bound the precise number of steps needed to approach the threshold from either side\.

##### Subcritical stage

IfwT≥w^w\_\{T\}\\geq\\widehat\{w\}, then by definitionσw^​\(T\)=0\\sigma\_\{\\widehat\{w\}\}\(T\)=0\. We therefore consider the nontrivial casewT<w^w\_\{T\}<\\widehat\{w\}\.

###### Theorem D\.1\(Finite passage to a strict subcritical target\)\.

Assumeβ1=0\\beta\_\{1\}=0,wmax\>2w\_\{\\max\}\>2, and

wT<w^<2\.w\_\{T\}<\\widehat\{w\}<2\.Define

MT\\displaystyle M\_\{T\}:=max⁡\{vT,xT2\},\\displaystyle:=\\max\\\{v\_\{T\},x\_\{T\}^\{2\}\\\},w¯T\\displaystyle\\underline\{w\}\_\{T\}:=ηMT\+ε,\\displaystyle:=\\frac\{\\eta\}\{\\sqrt\{M\_\{T\}\}\+\\varepsilon\},\(24\)ρT\\displaystyle\\rho\_\{T\}:=max⁡\{\|1−w¯T\|,\|1−w^\|\},\\displaystyle:=\\max\\\{\\left\\lvert 1\-\\underline\{w\}\_\{T\}\\right\\rvert,\\left\\lvert 1\-\\widehat\{w\}\\right\\rvert\\\},rT\\displaystyle r\_\{T\}:=ρT2,\\displaystyle:=\\rho\_\{T\}^\{2\},\(25\)and

Un\(T\):=β2n​vT\+\(1−β2\)​xT2​Gn​\(rT,β2\),n∈ℕ≥1\.U\_\{n\}^\{\(T\)\}:=\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}G\_\{n\}\(r\_\{T\},\\beta\_\{2\}\),\\qquad n\\in\\mathbb\{N\}\_\{\\geq 1\}\.\(26\)Then

NTimp:=min⁡\{n∈ℕ≥1:Un\(T\)≤\(ηw^−ε\)2\}<∞,N\_\{T\}^\{\\mathrm\{imp\}\}:=\\min\\left\\\{n\\in\\mathbb\{N\}\_\{\\geq 1\}:U\_\{n\}^\{\(T\)\}\\leq\\left\(\\frac\{\\eta\}\{\\widehat\{w\}\}\-\\varepsilon\\right\)^\{2\}\\right\\\}<\\infty,\(27\)and

1≤σw^​\(T\)≤NTimp\.1\\leq\\sigma\_\{\\widehat\{w\}\}\(T\)\\leq N\_\{T\}^\{\\mathrm\{imp\}\}\.\(28\)
Moreover, introduce

Cβ,T\\displaystyle C\_\{\\beta,T\}:=vT\+1−β2β2−rT​xT2\\displaystyle:=v\_\{T\}\+\\frac\{1\-\\beta\_\{2\}\}\{\\beta\_\{2\}\-r\_\{T\}\}x\_\{T\}^\{2\}if​β2\>rT,\\displaystyle\\text\{if \}\\beta\_\{2\}\>r\_\{T\},Cr,T\\displaystyle C\_\{r,T\}:=vT\+1−β2rT−β2​xT2\\displaystyle:=v\_\{T\}\+\\frac\{1\-\\beta\_\{2\}\}\{r\_\{T\}\-\\beta\_\{2\}\}x\_\{T\}^\{2\}if​β2<rT,\\displaystyle\\text\{if \}\\beta\_\{2\}<r\_\{T\},C=,T\\displaystyle C\_\{=,T\}:=vT\+1−rT\(1−rT\)2​xT2\\displaystyle:=v\_\{T\}\+\\frac\{1\-r\_\{T\}\}\{\(1\-\\sqrt\{r\_\{T\}\}\)^\{2\}\}x\_\{T\}^\{2\}if​β2=rT\.\\displaystyle\\text\{if \}\\beta\_\{2\}=r\_\{T\}\.The implicit bound satisfies

NTimp≤N¯T:=\{⌈log⁡\(Cβ,T\(η/w^−ε\)2\)log⁡\(1/β2\)⌉,β2\>rT,⌈log⁡\(Cr,T\(η/w^−ε\)2\)log⁡\(1/rT\)⌉,β2<rT,1\+⌈log⁡\(C=,T\(η/w^−ε\)2\)log⁡\(1/rT\)⌉,β2=rT\.N\_\{T\}^\{\\mathrm\{imp\}\}\\leq\\overline\{N\}\_\{T\}:=\\begin\{cases\}\\displaystyle\\left\\lceil\\frac\{\\log\\\!\\left\(\\dfrac\{C\_\{\\beta,T\}\}\{\\left\(\\eta/\\widehat\{w\}\-\\varepsilon\\right\)^\{2\}\}\\right\)\}\{\\log\(1/\\beta\_\{2\}\)\}\\right\\rceil,&\\beta\_\{2\}\>r\_\{T\},\\\\\[14\.22636pt\] \\displaystyle\\left\\lceil\\frac\{\\log\\\!\\left\(\\dfrac\{C\_\{r,T\}\}\{\\left\(\\eta/\\widehat\{w\}\-\\varepsilon\\right\)^\{2\}\}\\right\)\}\{\\log\(1/r\_\{T\}\)\}\\right\\rceil,&\\beta\_\{2\}<r\_\{T\},\\\\\[14\.22636pt\] \\displaystyle 1\+\\left\\lceil\\frac\{\\log\\\!\\left\(\\dfrac\{C\_\{=,T\}\}\{\\left\(\\eta/\\widehat\{w\}\-\\varepsilon\\right\)^\{2\}\}\\right\)\}\{\\log\(1/\\sqrt\{r\_\{T\}\}\)\}\\right\\rceil,&\\beta\_\{2\}=r\_\{T\}\.\\end\{cases\}\(29\)

###### Proof\.

Becauseβ1=0\\beta\_\{1\}=0, we have

wt=ηvt\+ε,wmax=ηε\.w\_\{t\}=\\frac\{\\eta\}\{\\sqrt\{v\_\{t\}\}\+\\varepsilon\},\\qquad w\_\{\\max\}=\\frac\{\\eta\}\{\\varepsilon\}\.Sincew^<2<wmax\\widehat\{w\}<2<w\_\{\\max\},

ηw^−ε\>0\.\\frac\{\\eta\}\{\\widehat\{w\}\}\-\\varepsilon\>0\.The assumptionwT<w^w\_\{T\}<\\widehat\{w\}therefore gives

vT\>\(ηw^−ε\)2\.v\_\{T\}\>\\left\(\\frac\{\\eta\}\{\\widehat\{w\}\}\-\\varepsilon\\right\)^\{2\}\.In particular,MT\>0M\_\{T\}\>0, and sinceMT≥vTM\_\{T\}\\geq v\_\{T\},

0<w¯T≤wT<w^<2\.0<\\underline\{w\}\_\{T\}\\leq w\_\{T\}<\\widehat\{w\}<2\.Hence

0<ρT<1,0<rT<1\.0<\\rho\_\{T\}<1,\\qquad 0<r\_\{T\}<1\.Also, sincewT<w^w\_\{T\}<\\widehat\{w\}, we haveσw^​\(T\)≥1\\sigma\_\{\\widehat\{w\}\}\(T\)\\geq 1\.

Fixn∈ℕ≥1n\\in\\mathbb\{N\}\_\{\\geq 1\}\. Ifσw^​\(T\)<n\\sigma\_\{\\widehat\{w\}\}\(T\)<n, the target has already been reached before timeT\+nT\+n, so no further estimate is needed\. Suppose instead that

σw^​\(T\)≥n\.\\sigma\_\{\\widehat\{w\}\}\(T\)\\geq n\.Then

wT\+j<w^,j=0,…,n−1\.w\_\{T\+j\}<\\widehat\{w\},\\qquad j=0,\\ldots,n\-1\.We claim that throughout this stopped trajectory,

vT\+j≤MT,\|xT\+j\|≤ρTj\|xT\|,j=0,…,n−1\.v\_\{T\+j\}\\leq M\_\{T\},\\qquad\\left\\lvert x\_\{T\+j\}\\right\\rvert\\leq\\rho\_\{T\}^\{j\}\\left\\lvert x\_\{T\}\\right\\rvert,\\qquad j=0,\\ldots,n\-1\.\(30\)
We prove the two estimates simultaneously by induction\. They are immediate atj=0j=0\. Suppose they hold up to somej−1j\-1with1≤j≤n−11\\leq j\\leq n\-1\. SincewT\+j<w^<2w\_\{T\+j\}<\\widehat\{w\}<2,

\|xT\+j\|=\|1−wT\+j\|​\|xT\+j−1\|<\|xT\+j−1\|,\\left\\lvert x\_\{T\+j\}\\right\\rvert=\\left\\lvert 1\-w\_\{T\+j\}\\right\\rvert\\left\\lvert x\_\{T\+j\-1\}\\right\\rvert<\\left\\lvert x\_\{T\+j\-1\}\\right\\rvert,and hence\|xT\+j−1\|≤\|xT\|\\left\\lvert x\_\{T\+j\-1\}\\right\\rvert\\leq\\left\\lvert x\_\{T\}\\right\\rvert\. Using

vT\+j=β2​vT\+j−1\+\(1−β2\)​xT\+j−12,v\_\{T\+j\}=\\beta\_\{2\}v\_\{T\+j\-1\}\+\(1\-\\beta\_\{2\}\)x\_\{T\+j\-1\}^\{2\},together withvT\+j−1≤MTv\_\{T\+j\-1\}\\leq M\_\{T\}andxT\+j−12≤xT2≤MTx\_\{T\+j\-1\}^\{2\}\\leq x\_\{T\}^\{2\}\\leq M\_\{T\}, gives

vT\+j≤MT\.v\_\{T\+j\}\\leq M\_\{T\}\.Consequently,

w¯T=ηMT\+ε≤wT\+j<w^\.\\underline\{w\}\_\{T\}=\\frac\{\\eta\}\{\\sqrt\{M\_\{T\}\}\+\\varepsilon\}\\leq w\_\{T\+j\}<\\widehat\{w\}\.Sincew↦\|1−w\|w\\mapsto\\left\\lvert 1\-w\\right\\rvertis convex, its maximum on\[w¯T,w^\]\[\\underline\{w\}\_\{T\},\\widehat\{w\}\]is attained at one of the endpoints\. Therefore

\|1−wT\+j\|≤max⁡\{\|1−w¯T\|,\|1−w^\|\}=ρT,\\left\\lvert 1\-w\_\{T\+j\}\\right\\rvert\\leq\\max\\\{\\left\\lvert 1\-\\underline\{w\}\_\{T\}\\right\\rvert,\\left\\lvert 1\-\\widehat\{w\}\\right\\rvert\\\}=\\rho\_\{T\},and thus

\|xT\+j\|≤ρT​\|xT\+j−1\|≤ρTj​\|xT\|\.\\left\\lvert x\_\{T\+j\}\\right\\rvert\\leq\\rho\_\{T\}\\left\\lvert x\_\{T\+j\-1\}\\right\\rvert\\leq\\rho\_\{T\}^\{j\}\\left\\lvert x\_\{T\}\\right\\rvert\.This proves \([30](https://arxiv.org/html/2608.20638#A4.E30)\)\.

Unrolling the second\-moment recursion overnnsteps and using \([30](https://arxiv.org/html/2608.20638#A4.E30)\) gives

vT\+n\\displaystyle v\_\{T\+n\}=β2n​vT\+\(1−β2\)​∑j=0n−1β2n−1−j​xT\+j2\\displaystyle=\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)\\sum\_\{j=0\}^\{n\-1\}\\beta\_\{2\}^\{n\-1\-j\}x\_\{T\+j\}^\{2\}≤β2n​vT\+\(1−β2\)​xT2​∑j=0n−1β2n−1−j​rTj\\displaystyle\\leq\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\\sum\_\{j=0\}^\{n\-1\}\\beta\_\{2\}^\{n\-1\-j\}r\_\{T\}^\{j\}=Un\(T\)\.\\displaystyle=U\_\{n\}^\{\(T\)\}\.Sinceβ2∈\[0,1\)\\beta\_\{2\}\\in\[0,1\)andrT∈\(0,1\)r\_\{T\}\\in\(0,1\),Un\(T\)→0U\_\{n\}^\{\(T\)\}\\to 0\. Indeed, whenβ2≠rT\\beta\_\{2\}\\neq r\_\{T\}, this follows directly from the closed form ofGn​\(rT,β2\)G\_\{n\}\(r\_\{T\},\\beta\_\{2\}\), while forβ2=rT\\beta\_\{2\}=r\_\{T\},Gn​\(rT,rT\)=n​rTn−1→0G\_\{n\}\(r\_\{T\},r\_\{T\}\)=nr\_\{T\}^\{n\-1\}\\to 0\. Because

\(ηw^−ε\)2\>0,\\left\(\\frac\{\\eta\}\{\\widehat\{w\}\}\-\\varepsilon\\right\)^\{2\}\>0,there exists a finitennfor which

Un\(T\)≤\(ηw^−ε\)2\.U\_\{n\}^\{\(T\)\}\\leq\\left\(\\frac\{\\eta\}\{\\widehat\{w\}\}\-\\varepsilon\\right\)^\{2\}\.HenceNTimp<∞N\_\{T\}^\{\\mathrm\{imp\}\}<\\infty\.

Now taken=NTimpn=N\_\{T\}^\{\\mathrm\{imp\}\}\. Ifσw^​\(T\)<n\\sigma\_\{\\widehat\{w\}\}\(T\)<n, then certainlyσw^​\(T\)≤NTimp\\sigma\_\{\\widehat\{w\}\}\(T\)\\leq N\_\{T\}^\{\\mathrm\{imp\}\}\. Otherwise the stopped estimate applies and yields

vT\+n≤Un\(T\)≤\(ηw^−ε\)2\.v\_\{T\+n\}\\leq U\_\{n\}^\{\(T\)\}\\leq\\left\(\\frac\{\\eta\}\{\\widehat\{w\}\}\-\\varepsilon\\right\)^\{2\}\.Sinceη/w^−ε\>0\\eta/\\widehat\{w\}\-\\varepsilon\>0, this implies

vT\+n\+ε≤ηw^,\\sqrt\{v\_\{T\+n\}\}\+\\varepsilon\\leq\\frac\{\\eta\}\{\\widehat\{w\}\},and therefore

wT\+n=ηvT\+n\+ε≥w^\.w\_\{T\+n\}=\\frac\{\\eta\}\{\\sqrt\{v\_\{T\+n\}\}\+\\varepsilon\}\\geq\\widehat\{w\}\.Thus

1≤σw^​\(T\)≤NTimp\.1\\leq\\sigma\_\{\\widehat\{w\}\}\(T\)\\leq N\_\{T\}^\{\\mathrm\{imp\}\}\.
It remains to prove the explicit bounds\.

Ifβ2\>rT\\beta\_\{2\}\>r\_\{T\}, then

Gn​\(rT,β2\)=β2n−rTnβ2−rT≤β2nβ2−rT,G\_\{n\}\(r\_\{T\},\\beta\_\{2\}\)=\\frac\{\\beta\_\{2\}^\{n\}\-r\_\{T\}^\{n\}\}\{\\beta\_\{2\}\-r\_\{T\}\}\\leq\\frac\{\\beta\_\{2\}^\{n\}\}\{\\beta\_\{2\}\-r\_\{T\}\},so

Un\(T\)≤Cβ,T​β2n\.U\_\{n\}^\{\(T\)\}\\leq C\_\{\\beta,T\}\\beta\_\{2\}^\{n\}\.Hence

n≥log⁡\(Cβ,T\(η/w^−ε\)2\)log⁡\(1/β2\)n\\geq\\frac\{\\log\\\!\\left\(\\dfrac\{C\_\{\\beta,T\}\}\{\(\\eta/\\widehat\{w\}\-\\varepsilon\)^\{2\}\}\\right\)\}\{\\log\(1/\\beta\_\{2\}\)\}is sufficient to ensure

Un\(T\)≤\(ηw^−ε\)2,U\_\{n\}^\{\(T\)\}\\leq\\left\(\\frac\{\\eta\}\{\\widehat\{w\}\}\-\\varepsilon\\right\)^\{2\},which gives the first case of \([29](https://arxiv.org/html/2608.20638#A4.E29)\)\.

Ifβ2<rT\\beta\_\{2\}<r\_\{T\}, then

Gn​\(rT,β2\)=rTn−β2nrT−β2≤rTnrT−β2,β2n≤rTn\.G\_\{n\}\(r\_\{T\},\\beta\_\{2\}\)=\\frac\{r\_\{T\}^\{n\}\-\\beta\_\{2\}^\{n\}\}\{r\_\{T\}\-\\beta\_\{2\}\}\\leq\\frac\{r\_\{T\}^\{n\}\}\{r\_\{T\}\-\\beta\_\{2\}\},\\qquad\\beta\_\{2\}^\{n\}\\leq r\_\{T\}^\{n\}\.Consequently,

Un\(T\)≤Cr,T​rTn\.U\_\{n\}^\{\(T\)\}\\leq C\_\{r,T\}r\_\{T\}^\{n\}\.Thus

n≥log⁡\(Cr,T\(η/w^−ε\)2\)log⁡\(1/rT\)n\\geq\\frac\{\\log\\\!\\left\(\\dfrac\{C\_\{r,T\}\}\{\(\\eta/\\widehat\{w\}\-\\varepsilon\)^\{2\}\}\\right\)\}\{\\log\(1/r\_\{T\}\)\}is sufficient, giving the second case\.

Finally, supposeβ2=rT\\beta\_\{2\}=r\_\{T\}\. Then

Gn​\(rT,rT\)=n​rTn−1\.G\_\{n\}\(r\_\{T\},r\_\{T\}\)=nr\_\{T\}^\{n\-1\}\.For everyn≥1n\\geq 1,

n​rTn−1\\displaystyle nr\_\{T\}^\{n\-1\}=\(rT\)n−1​\(n​\(rT\)n−1\)\\displaystyle=\(\\sqrt\{r\_\{T\}\}\)^\{\\,n\-1\}\\left\(n\(\\sqrt\{r\_\{T\}\}\)^\{\\,n\-1\}\\right\)≤\(rT\)n−1\(1−rT\)2,\\displaystyle\\leq\\frac\{\(\\sqrt\{r\_\{T\}\}\)^\{\\,n\-1\}\}\{\(1\-\\sqrt\{r\_\{T\}\}\)^\{2\}\},because

n​\(rT\)n−1≤∑k=1∞k​\(rT\)k−1=1\(1−rT\)2\.n\(\\sqrt\{r\_\{T\}\}\)^\{\\,n\-1\}\\leq\\sum\_\{k=1\}^\{\\infty\}k\(\\sqrt\{r\_\{T\}\}\)^\{\\,k\-1\}=\\frac\{1\}\{\(1\-\\sqrt\{r\_\{T\}\}\)^\{2\}\}\.Moreover,

rTn=\(rT\)2​n≤\(rT\)n−1\.r\_\{T\}^\{n\}=\(\\sqrt\{r\_\{T\}\}\)^\{2n\}\\leq\(\\sqrt\{r\_\{T\}\}\)^\{\\,n\-1\}\.Therefore

Un\(T\)≤C=,T​\(rT\)n−1\.U\_\{n\}^\{\(T\)\}\\leq C\_\{=,T\}\(\\sqrt\{r\_\{T\}\}\)^\{\\,n\-1\}\.Hence it is sufficient that

n−1≥log⁡\(C=,T\(η/w^−ε\)2\)log⁡\(1/rT\),n\-1\\geq\\frac\{\\log\\\!\\left\(\\dfrac\{C\_\{=,T\}\}\{\(\\eta/\\widehat\{w\}\-\\varepsilon\)^\{2\}\}\\right\)\}\{\\log\(1/\\sqrt\{r\_\{T\}\}\)\},which gives the final case of \([29](https://arxiv.org/html/2608.20638#A4.E29)\)\. ∎

##### Supercritical stage

We next show that a nonzero trajectory cannot remain indefinitely in a uniformly supercritical band\. The argument is the reverse of the subcritical mechanism: as long aswtw\_\{t\}remains above2\+δ2\+\\delta, the position expands geometrically, which forces the second moment upward and eventually makes such a large value ofwtw\_\{t\}impossible\.

###### Theorem D\.2\(Finite supercritical exit forβ1=0\\beta\_\{1\}=0\)\.

LetT∈ℕ0T\\in\\mathbb\{N\}\_\{0\}and letδ\>0\\delta\>0satisfy

2\+δ≤wmax\.2\+\\delta\\leq w\_\{\\max\}\.Assume

β1=0,wT≥2\+δ,xT≠0\.\\beta\_\{1\}=0,\\qquad w\_\{T\}\\geq 2\+\\delta,\\qquad x\_\{T\}\\neq 0\.Define

Nδ,Timp:=min⁡\{n∈ℕ≥1:Ln​\(T,\|xT\|\)\>\(η2\+δ−ε\)2\}\.N\_\{\\delta,T\}^\{\\mathrm\{imp\}\}:=\\min\\left\\\{n\\in\\mathbb\{N\}\_\{\\geq 1\}:L\_\{n\}\(T,\\left\\lvert x\_\{T\}\\right\\rvert\)\>\\left\(\\frac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\right\)^\{2\}\\right\\\}\.\(31\)Then this minimum is finite and

τδ​\(T\)≤Nδ,Timp≤Nδ,Texp<∞,\\tau\_\{\\delta\}\(T\)\\leq N\_\{\\delta,T\}^\{\\mathrm\{imp\}\}\\leq N\_\{\\delta,T\}^\{\\mathrm\{exp\}\}<\\infty,\(32\)where

Nδ,Texp:=1\+⌊log⁡\(1\+\(Aδ−β2\)​\(η2\+δ−ε\)2\(1−β2\)​xT2\)log⁡Aδ⌋\.N\_\{\\delta,T\}^\{\\mathrm\{exp\}\}:=1\+\\left\\lfloor\\frac\{\\log\\\!\\left\(1\+\\frac\{\(A\_\{\\delta\}\-\\beta\_\{2\}\)\\left\(\\dfrac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\right\)^\{2\}\}\{\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\}\\right\)\}\{\\log A\_\{\\delta\}\}\\right\\rfloor\.\(33\)

###### Proof\.

Becauseβ1=0\\beta\_\{1\}=0,

wt=ηvt\+ε\.w\_\{t\}=\\frac\{\\eta\}\{\\sqrt\{v\_\{t\}\}\+\\varepsilon\}\.The assumption

2\+δ≤wmax=ηε2\+\\delta\\leq w\_\{\\max\}=\\frac\{\\eta\}\{\\varepsilon\}implies

η2\+δ−ε≥0\.\\frac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\geq 0\.Consequently,

wt≥2\+δ⟺vt≤\(η2\+δ−ε\)2\.w\_\{t\}\\geq 2\+\\delta\\quad\\Longleftrightarrow\\quad v\_\{t\}\\leq\\left\(\\frac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\right\)^\{2\}\.\(34\)
Fixn∈ℕ≥1n\\in\\mathbb\{N\}\_\{\\geq 1\}\. Ifτδ​\(T\)≤n\\tau\_\{\\delta\}\(T\)\\leq n, then the trajectory has already left the prescribed supercritical band by timeT\+nT\+n, so there is nothing to prove\. Suppose instead that

τδ​\(T\)\>n\.\\tau\_\{\\delta\}\(T\)\>n\.By the definition ofτδ​\(T\)\\tau\_\{\\delta\}\(T\),

wT\+k≥2\+δ,k=1,…,n\.w\_\{T\+k\}\\geq 2\+\\delta,\\qquad k=1,\\ldots,n\.Using the zero\-momentum position recursion

xt\+1=\(1−wt\+1\)​xt,x\_\{t\+1\}=\(1\-w\_\{t\+1\}\)x\_\{t\},we obtain, for everyk=1,…,nk=1,\\ldots,n,

\|xT\+k\|\\displaystyle\\left\\lvert x\_\{T\+k\}\\right\\rvert=\|1−wT\+k\|​\|xT\+k−1\|\\displaystyle=\\left\\lvert 1\-w\_\{T\+k\}\\right\\rvert\\left\\lvert x\_\{T\+k\-1\}\\right\\rvert=\(wT\+k−1\)​\|xT\+k−1\|\\displaystyle=\(w\_\{T\+k\}\-1\)\\left\\lvert x\_\{T\+k\-1\}\\right\\rvert≥\(1\+δ\)​\|xT\+k−1\|\.\\displaystyle\\geq\(1\+\\delta\)\\left\\lvert x\_\{T\+k\-1\}\\right\\rvert\.Iterating gives

\|xT\+j\|≥\(1\+δ\)j\|xT\|,j=0,…,n,\\left\\lvert x\_\{T\+j\}\\right\\rvert\\geq\(1\+\\delta\)^\{j\}\\left\\lvert x\_\{T\}\\right\\rvert,\\qquad j=0,\\ldots,n,and hence

xT\+j2≥AδjxT2,j=0,…,n,x\_\{T\+j\}^\{2\}\\geq A\_\{\\delta\}^\{j\}x\_\{T\}^\{2\},\\qquad j=0,\\ldots,n,\(35\)whereAδ=\(1\+δ\)2A\_\{\\delta\}=\(1\+\\delta\)^\{2\}\.

Unrolling the second\-moment recursion overnnsteps yields

vT\+n\\displaystyle v\_\{T\+n\}=β2n​vT\+\(1−β2\)​∑j=0n−1β2n−1−j​xT\+j2\\displaystyle=\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)\\sum\_\{j=0\}^\{n\-1\}\\beta\_\{2\}^\{n\-1\-j\}x\_\{T\+j\}^\{2\}≥β2n​vT\+\(1−β2\)​xT2​∑j=0n−1β2n−1−j​Aδj\\displaystyle\\geq\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\\sum\_\{j=0\}^\{n\-1\}\\beta\_\{2\}^\{n\-1\-j\}A\_\{\\delta\}^\{j\}=β2n​vT\+\(1−β2\)​xT2​Gn​\(Aδ,β2\)\\displaystyle=\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}G\_\{n\}\(A\_\{\\delta\},\\beta\_\{2\}\)=Ln​\(T,\|xT\|\)\.\\displaystyle=L\_\{n\}\(T,\\left\\lvert x\_\{T\}\\right\\rvert\)\.
On the other hand,τδ​\(T\)\>n\\tau\_\{\\delta\}\(T\)\>nimplieswT\+n≥2\+δw\_\{T\+n\}\\geq 2\+\\delta\. Therefore, by \([34](https://arxiv.org/html/2608.20638#A4.E34)\),

vT\+n≤\(η2\+δ−ε\)2\.v\_\{T\+n\}\\leq\\left\(\\frac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\right\)^\{2\}\.Hence survival in the supercritical band through timeT\+nT\+nnecessarily implies

Ln​\(T,\|xT\|\)≤\(η2\+δ−ε\)2\.L\_\{n\}\(T,\\left\\lvert x\_\{T\}\\right\\rvert\)\\leq\\left\(\\frac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\right\)^\{2\}\.\(36\)Thus, whenever

Ln​\(T,\|xT\|\)\>\(η2\+δ−ε\)2,L\_\{n\}\(T,\\left\\lvert x\_\{T\}\\right\\rvert\)\>\\left\(\\frac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\right\)^\{2\},the trajectory must have exited the band no later than timeT\+nT\+n\.

It remains to show that such annnexists\. Since

Aδ=\(1\+δ\)2\>1\>β2A\_\{\\delta\}=\(1\+\\delta\)^\{2\}\>1\>\\beta\_\{2\}andxT≠0x\_\{T\}\\neq 0, we have

Ln​\(T,\|xT\|\)\\displaystyle L\_\{n\}\(T,\\left\\lvert x\_\{T\}\\right\\rvert\)=β2n​vT\+\(1−β2\)​xT2​Gn​\(Aδ,β2\)\\displaystyle=\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}G\_\{n\}\(A\_\{\\delta\},\\beta\_\{2\}\)≥\(1−β2\)​xT2Aδ−β2​\(Aδn−β2n\)\.\\displaystyle\\geq\\frac\{\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\}\{A\_\{\\delta\}\-\\beta\_\{2\}\}\\left\(A\_\{\\delta\}^\{n\}\-\\beta\_\{2\}^\{n\}\\right\)\.BecauseAδ\>1A\_\{\\delta\}\>1andβ2<1\\beta\_\{2\}<1,Ln​\(T,\|xT\|\)→∞L\_\{n\}\(T,\\left\\lvert x\_\{T\}\\right\\rvert\)\\to\\infty\. Therefore,

Nδ,Timp<∞\.N\_\{\\delta,T\}^\{\\mathrm\{imp\}\}<\\infty\.The survival implication above gives

τδ​\(T\)≤Nδ,Timp\.\\tau\_\{\\delta\}\(T\)\\leq N\_\{\\delta,T\}^\{\\mathrm\{imp\}\}\.
It remains to derive the explicit bound\. Since0≤β2<10\\leq\\beta\_\{2\}<1, one hasβ2n≤1\\beta\_\{2\}^\{n\}\\leq 1, and hence

Gn​\(Aδ,β2\)\\displaystyle G\_\{n\}\(A\_\{\\delta\},\\beta\_\{2\}\)=Aδn−β2nAδ−β2\\displaystyle=\\frac\{A\_\{\\delta\}^\{n\}\-\\beta\_\{2\}^\{n\}\}\{A\_\{\\delta\}\-\\beta\_\{2\}\}≥Aδn−1Aδ−β2\.\\displaystyle\\geq\\frac\{A\_\{\\delta\}^\{n\}\-1\}\{A\_\{\\delta\}\-\\beta\_\{2\}\}\.Thus

Ln​\(T,\|xT\|\)≥\(1−β2\)​xT2Aδ−β2​\(Aδn−1\)\.L\_\{n\}\(T,\\left\\lvert x\_\{T\}\\right\\rvert\)\\geq\\frac\{\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\}\{A\_\{\\delta\}\-\\beta\_\{2\}\}\\left\(A\_\{\\delta\}^\{n\}\-1\\right\)\.\(37\)Therefore it is sufficient that

Aδn\>1\+\(Aδ−β2\)​\(η2\+δ−ε\)2\(1−β2\)​xT2\.A\_\{\\delta\}^\{n\}\>1\+\\frac\{\(A\_\{\\delta\}\-\\beta\_\{2\}\)\\left\(\\dfrac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\right\)^\{2\}\}\{\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\}\.By the definition ofNδ,TexpN\_\{\\delta,T\}^\{\\mathrm\{exp\}\},

AδNδ,Texp\>1\+\(Aδ−β2\)​\(η2\+δ−ε\)2\(1−β2\)​xT2\.A\_\{\\delta\}^\{N\_\{\\delta,T\}^\{\\mathrm\{exp\}\}\}\>1\+\\frac\{\(A\_\{\\delta\}\-\\beta\_\{2\}\)\\left\(\\dfrac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\right\)^\{2\}\}\{\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\}\.Combining this with \([37](https://arxiv.org/html/2608.20638#A4.E37)\) yields

LNδ,Texp​\(T,\|xT\|\)\>\(η2\+δ−ε\)2\.L\_\{N\_\{\\delta,T\}^\{\\mathrm\{exp\}\}\}\(T,\\left\\lvert x\_\{T\}\\right\\rvert\)\>\\left\(\\frac\{\\eta\}\{2\+\\delta\}\-\\varepsilon\\right\)^\{2\}\.Hence

Nδ,Timp≤Nδ,Texp,N\_\{\\delta,T\}^\{\\mathrm\{imp\}\}\\leq N\_\{\\delta,T\}^\{\\mathrm\{exp\}\},which completes the proof\. ∎

## Appendix EProofs for general parameters in the subcritical stage

### E\.1Proof of Proposition[3\.1](https://arxiv.org/html/2608.20638#S3.Thmtheorem1)

###### Proof\.

Let

θ⁡\(y\):=y2−y\.\\theta\(y\):=\\frac\{y\}\{2\-y\}\.\(38\)Recall that in the present section0<β1<10<\\beta\_\{1\}<1\.

We first record the restriction that the second\-moment recursion places on the pair\(wt,wt\+1\)\(w\_\{t\},w\_\{t\+1\}\)\. Since

vt=c​η​\(1wt−1wmax\),\\sqrt\{v\_\{t\}\}=c\\eta\\left\(\\frac\{1\}\{w\_\{t\}\}\-\\frac\{1\}\{w\_\{\\max\}\}\\right\),its exact normalized form is

\(1wt\+1−1wmax\)2=β2​\(1wt−1wmax\)2\+\(1−β2\)​\(xtc​η\)2\.\\left\(\\frac\{1\}\{w\_\{t\+1\}\}\-\\frac\{1\}\{w\_\{\\max\}\}\\right\)^\{2\}=\\beta\_\{2\}\\left\(\\frac\{1\}\{w\_\{t\}\}\-\\frac\{1\}\{w\_\{\\max\}\}\\right\)^\{2\}\+\(1\-\\beta\_\{2\}\)\\left\(\\frac\{x\_\{t\}\}\{c\\eta\}\\right\)^\{2\}\.\(39\)Taking nonnegative square roots gives

1wt\+1−1wmax≥β2​\(1wt−1wmax\),\\frac\{1\}\{w\_\{t\+1\}\}\-\\frac\{1\}\{w\_\{\\max\}\}\\geq\\sqrt\{\\beta\_\{2\}\}\\left\(\\frac\{1\}\{w\_\{t\}\}\-\\frac\{1\}\{w\_\{\\max\}\}\\right\),and hence

wt\+1≤wtβ2\+\(1−β2\)​wt/wmax,w\_\{t\+1\}\\leq\\frac\{w\_\{t\}\}\{\\sqrt\{\\beta\_\{2\}\}\+\(1\-\\sqrt\{\\beta\_\{2\}\}\)w\_\{t\}/w\_\{\\max\}\},or equivalently,

wt≥β2​wt\+11−\(1−β2\)​wt\+1/wmax\.w\_\{t\}\\geq\\frac\{\\sqrt\{\\beta\_\{2\}\}\\,w\_\{t\+1\}\}\{1\-\(1\-\\sqrt\{\\beta\_\{2\}\}\)w\_\{t\+1\}/w\_\{\\max\}\}\.\(40\)Equality holds exactly whenxt=0x\_\{t\}=0\.

We next solve the one\-step quadratic inequality\. Fory,Y\>0y,Y\>0, put

Py:=diag⁡\(1,y\),D⁡\(y,Y,wt\+1\):=Py−A​\(wt\+1\)⊤​PY​A​\(wt\+1\)\.P\_\{y\}:=\\operatorname\{diag\}\(1,y\),\\qquad D\(y,Y;w\_\{t\+1\}\):=P\_\{y\}\-A\(w\_\{t\+1\}\)^\{\\top\}P\_\{Y\}A\(w\_\{t\+1\}\)\.\(41\)Direct multiplication gives the exact identities

D11​\(y,Y,wt\+1\)=\(2−wt\+1\)​\(wt\+1−\(2−wt\+1\)​Y\),D\_\{11\}\(y,Y;w\_\{t\+1\}\)=\(2\-w\_\{t\+1\}\)\\bigl\(w\_\{t\+1\}\-\(2\-w\_\{t\+1\}\)Y\\bigr\),and

detD⁡\(y,Y,wt\+1\)=\(wt\+1−\(2−wt\+1\)​Y\)​\(\(2−wt\+1\)​y−β12​wt\+1\)\.\\det D\(y,Y;w\_\{t\+1\}\)=\\bigl\(w\_\{t\+1\}\-\(2\-w\_\{t\+1\}\)Y\\bigr\)\\bigl\(\(2\-w\_\{t\+1\}\)y\-\\beta\_\{1\}^\{2\}w\_\{t\+1\}\\bigr\)\.\(42\)Because0<wt\+1<20<w\_\{t\+1\}<2, Sylvester’s criterion shows that

A​\(wt\+1\)⊤​PY​A​\(wt\+1\)≺PyA\(w\_\{t\+1\}\)^\{\\top\}P\_\{Y\}A\(w\_\{t\+1\}\)\\prec P\_\{y\}if and only if

Y<θ⁡\(wt\+1\),y\>β12​θ​\(wt\+1\)\.Y<\\theta\(w\_\{t\+1\}\),\\qquad y\>\\beta\_\{1\}^\{2\}\\theta\(w\_\{t\+1\}\)\.\(43\)
For the Lyapunov function in the proposition, the source and target weights are

y=β1​θ​\(wt\),Y=β1​θ​\(wt\+1\)\.y=\\beta\_\{1\}\\theta\(w\_\{t\}\),\\qquad Y=\\beta\_\{1\}\\theta\(w\_\{t\+1\}\)\.The first inequality in \([43](https://arxiv.org/html/2608.20638#A5.E43)\) is automatic becauseβ1<1\\beta\_\{1\}<1, while the second becomes

θ⁡\(wt\)\>β1​θ​\(wt\+1\)\.\\theta\(w\_\{t\}\)\>\\beta\_\{1\}\\theta\(w\_\{t\+1\}\)\.\(44\)
Sinceθ\\thetais strictly increasing on\(0,2\)\(0,2\), \([40](https://arxiv.org/html/2608.20638#A5.E40)\) implies

θ⁡\(wt\)≥θ⁡\(β2​wt\+11−\(1−β2\)​wt\+1/wmax\)\.\\theta\(w\_\{t\}\)\\geq\\theta\\left\(\\frac\{\\sqrt\{\\beta\_\{2\}\}\\,w\_\{t\+1\}\}\{1\-\(1\-\\sqrt\{\\beta\_\{2\}\}\)w\_\{t\+1\}/w\_\{\\max\}\}\\right\)\.Therefore, after direct simplification,

θ⁡\(wt\+1\)θ⁡\(wt\)≤2−wt\+1​\(β2\+2​\(1−β2\)/wmax\)β2​\(2−wt\+1\)\.\\frac\{\\theta\(w\_\{t\+1\}\)\}\{\\theta\(w\_\{t\}\)\}\\leq\\frac\{2\-w\_\{t\+1\}\\left\(\\sqrt\{\\beta\_\{2\}\}\+2\(1\-\\sqrt\{\\beta\_\{2\}\}\)/w\_\{\\max\}\\right\)\}\{\\sqrt\{\\beta\_\{2\}\}\(2\-w\_\{t\+1\}\)\}\.\(45\)The right\-hand side is increasing inwt\+1w\_\{t\+1\}, since

β2\+2​\(1−β2\)wmax<1\\sqrt\{\\beta\_\{2\}\}\+\\frac\{2\(1\-\\sqrt\{\\beta\_\{2\}\}\)\}\{w\_\{\\max\}\}<1whenwmax\>2w\_\{\\max\}\>2\. Thus, if

wt,wt\+1≤w<2,w\_\{t\},w\_\{t\+1\}\\leq w<2,then

θ⁡\(wt\+1\)θ⁡\(wt\)≤2−w⁡\(β2\+2​\(1−β2\)/wmax\)β2​\(2−w\)\.\\frac\{\\theta\(w\_\{t\+1\}\)\}\{\\theta\(w\_\{t\}\)\}\\leq\\frac\{2\-w\\left\(\\sqrt\{\\beta\_\{2\}\}\+2\(1\-\\sqrt\{\\beta\_\{2\}\}\)/w\_\{\\max\}\\right\)\}\{\\sqrt\{\\beta\_\{2\}\}\(2\-w\)\}\.\(46\)The product of the last expression withβ1\\beta\_\{1\}is strictly below one exactly when

w<2​\(β2−β1\)β2​\(1−β1\)−2​β1​\(1−β2\)/wmax=:W\.w<\\frac\{2\(\\sqrt\{\\beta\_\{2\}\}\-\\beta\_\{1\}\)\}\{\\sqrt\{\\beta\_\{2\}\}\(1\-\\beta\_\{1\}\)\-2\\beta\_\{1\}\(1\-\\sqrt\{\\beta\_\{2\}\}\)/w\_\{\\max\}\}=:W\.\(47\)Sinceβ2\>β1\\sqrt\{\\beta\_\{2\}\}\>\\beta\_\{1\}andwmax\>2w\_\{\\max\}\>2,

β2​\(1−β1\)−2​β1​\(1−β2\)wmax−\(β2−β1\)\\displaystyle\\sqrt\{\\beta\_\{2\}\}\(1\-\\beta\_\{1\}\)\-\\frac\{2\\beta\_\{1\}\(1\-\\sqrt\{\\beta\_\{2\}\}\)\}\{w\_\{\\max\}\}\-\(\\sqrt\{\\beta\_\{2\}\}\-\\beta\_\{1\}\)=β1​\(1−β2\)​\(1−2wmax\)\>0\.\\displaystyle=\\beta\_\{1\}\(1\-\\sqrt\{\\beta\_\{2\}\}\)\\left\(1\-\\frac\{2\}\{w\_\{\\max\}\}\\right\)\>0\.Hence the denominator in \([47](https://arxiv.org/html/2608.20638#A5.E47)\) is positive and strictly larger thanβ2−β1\\sqrt\{\\beta\_\{2\}\}\-\\beta\_\{1\}, and therefore

Equations \([43](https://arxiv.org/html/2608.20638#A5.E43)\)–\([47](https://arxiv.org/html/2608.20638#A5.E47)\) prove the strict decrease for every nonzeroztz\_\{t\}\.

It remains to make the contraction factor explicit\. Set

P⁡\(wt\):=diag⁡\(1,β1​θ​\(wt\)\),P\(w\_\{t\}\):=\\operatorname\{diag\}\\bigl\(1,\\beta\_\{1\}\\theta\(w\_\{t\}\)\\bigr\),and

D:=P⁡\(wt\)−A​\(wt\+1\)⊤​P​\(wt\+1\)​A​\(wt\+1\)\.D:=P\(w\_\{t\}\)\-A\(w\_\{t\+1\}\)^\{\\top\}P\(w\_\{t\+1\}\)A\(w\_\{t\+1\}\)\.\(48\)Specializing \([42](https://arxiv.org/html/2608.20638#A5.E42)\), or equivalently taking a Schur complement, gives

D11=\(1−β1\)​wt\+1​\(2−wt\+1\),D\_\{11\}=\(1\-\\beta\_\{1\}\)w\_\{t\+1\}\(2\-w\_\{t\+1\}\),and

D22−D122D11=β1​\(θ⁡\(wt\)−β1​θ​\(wt\+1\)\)\.D\_\{22\}\-\\frac\{D\_\{12\}^\{2\}\}\{D\_\{11\}\}=\\beta\_\{1\}\\left\(\\theta\(w\_\{t\}\)\-\\beta\_\{1\}\\theta\(w\_\{t\+1\}\)\\right\)\.\(49\)If

a≤wt,wt\+1≤w<W,a\\leq w\_\{t\},w\_\{t\+1\}\\leq w<W,then

D11≥\(1−β1\)​min⁡\{a⁡\(2−a\),w⁡\(2−w\)\},D\_\{11\}\\geq\(1\-\\beta\_\{1\}\)\\min\\\{a\(2\-a\),\\,w\(2\-w\)\\\},and, using \([46](https://arxiv.org/html/2608.20638#A5.E46)\),

D22−D122D11≥\\displaystyle D\_\{22\}\-\\frac\{D\_\{12\}^\{2\}\}\{D\_\{11\}\}\\geq\{\}β1​a2−a​\[1−β1​2−w⁡\(β2\+2​\(1−β2\)/wmax\)β2​\(2−w\)\]\.\\displaystyle\\frac\{\\beta\_\{1\}a\}\{2\-a\}\\Bigg\[1\-\\beta\_\{1\}\\frac\{2\-w\\left\(\\sqrt\{\\beta\_\{2\}\}\+2\(1\-\\sqrt\{\\beta\_\{2\}\}\)/w\_\{\\max\}\\right\)\}\{\\sqrt\{\\beta\_\{2\}\}\(2\-w\)\}\\Bigg\]\.\(50\)The bracket is positive by \([47](https://arxiv.org/html/2608.20638#A5.E47)\)\.

For convenience, define

K⁡\(w\):=2−w⁡\(β2\+2​\(1−β2\)/wmax\)β2​\(2−w\),K\(w\):=\\frac\{2\-w\\left\(\\sqrt\{\\beta\_\{2\}\}\+2\(1\-\\sqrt\{\\beta\_\{2\}\}\)/w\_\{\\max\}\\right\)\}\{\\sqrt\{\\beta\_\{2\}\}\(2\-w\)\},and

δ⁡\(a,w\):=\(1−β1\)​min⁡\{a⁡\(2−a\),w⁡\(2−w\)\}​β1​a2−a​\(1−β1​K​\(w\)\)2​\(1\+β1​w2−w\)​max⁡\{1,β1​w2−w\}\.\\delta\(a,w\):=\\frac\{\(1\-\\beta\_\{1\}\)\\min\\\{a\(2\-a\),\\,w\(2\-w\)\\\}\\displaystyle\\frac\{\\beta\_\{1\}a\}\{2\-a\}\\bigl\(1\-\\beta\_\{1\}K\(w\)\\bigr\)\}\{2\\left\(1\+\\frac\{\\beta\_\{1\}w\}\{2\-w\}\\right\)\\max\\left\\\{1,\\frac\{\\beta\_\{1\}w\}\{2\-w\}\\right\\\}\}\.\(51\)All factors in this expression are strictly positive whenever

so

Moreover,

P⁡\(wt\)−D=A​\(wt\+1\)⊤​P​\(wt\+1\)​A​\(wt\+1\)≻0,P\(w\_\{t\}\)\-D=A\(w\_\{t\+1\}\)^\{\\top\}P\(w\_\{t\+1\}\)A\(w\_\{t\+1\}\)\\succ 0,becauseP⁡\(wt\+1\)≻0P\(w\_\{t\+1\}\)\\succ 0and

detA⁡\(wt\+1\)=β1\>0\.\\det A\(w\_\{t\+1\}\)=\\beta\_\{1\}\>0\.Thus

0≺D≺P⁡\(wt\)\.0\\prec D\\prec P\(w\_\{t\}\)\.Sincewt≤ww\_\{t\}\\leq wandθ\\thetais increasing,

tr⁡D<tr⁡P⁡\(wt\)≤1\+β1​w2−w\.\\operatorname\{tr\}D<\\operatorname\{tr\}P\(w\_\{t\}\)\\leq 1\+\\frac\{\\beta\_\{1\}w\}\{2\-w\}\.\(52\)
Also, by \([49](https://arxiv.org/html/2608.20638#A5.E49)\)–\([50](https://arxiv.org/html/2608.20638#A5.E50)\),

detD\\displaystyle\\det D=D11​\(D22−D122D11\)\\displaystyle=D\_\{11\}\\left\(D\_\{22\}\-\\frac\{D\_\{12\}^\{2\}\}\{D\_\{11\}\}\\right\)\(53\)≥\(1−β1\)​min⁡\{a⁡\(2−a\),w⁡\(2−w\)\}​β1​a2−a​\(1−β1​K​\(w\)\)\.\\displaystyle\\geq\(1\-\\beta\_\{1\}\)\\min\\\{a\(2\-a\),\\,w\(2\-w\)\\\}\\frac\{\\beta\_\{1\}a\}\{2\-a\}\\bigl\(1\-\\beta\_\{1\}K\(w\)\\bigr\)\.
For a positive\-definite2×22\\times 2matrix,

λmin​\(D\)≥detDtr⁡D\.\\lambda\_\{\\min\}\(D\)\\geq\\frac\{\\det D\}\{\\operatorname\{tr\}D\}\.Hence, by \([51](https://arxiv.org/html/2608.20638#A5.E51)\)–\([53](https://arxiv.org/html/2608.20638#A5.E53)\),

λmin​\(D\)≥2​δ​\(a,w\)​max⁡\{1,β1​w2−w\}\.\\lambda\_\{\\min\}\(D\)\\geq 2\\delta\(a,w\)\\max\\left\\\{1,\\frac\{\\beta\_\{1\}w\}\{2\-w\}\\right\\\}\.On the other hand,

P⁡\(wt\)⪯max⁡\{1,β1​w2−w\}​I\.P\(w\_\{t\}\)\\preceq\\max\\left\\\{1,\\frac\{\\beta\_\{1\}w\}\{2\-w\}\\right\\\}I\.Consequently,

D⪰2​δ​\(a,w\)​max⁡\{1,β1​w2−w\}​I⪰2​δ​\(a,w\)​P​\(wt\)⪰δ⁡\(a,w\)​P​\(wt\)\.D\\succeq 2\\delta\(a,w\)\\max\\left\\\{1,\\frac\{\\beta\_\{1\}w\}\{2\-w\}\\right\\\}I\\succeq 2\\delta\(a,w\)P\(w\_\{t\}\)\\succeq\\delta\(a,w\)P\(w\_\{t\}\)\.\(54\)SinceD≺P⁡\(wt\)D\\prec P\(w\_\{t\}\), the inequality above implies

2​δ​\(a,w\)<1,2\\delta\(a,w\)<1,and hence, in particular,

0<δ⁡\(a,w\)<1\.0<\\delta\(a,w\)<1\.
Finally,

Ψt\+1\\displaystyle\\Psi\_\{t\+1\}=zt⊤​A​\(wt\+1\)⊤​P​\(wt\+1\)​A​\(wt\+1\)​zt\\displaystyle=z\_\{t\}^\{\\top\}A\(w\_\{t\+1\}\)^\{\\top\}P\(w\_\{t\+1\}\)A\(w\_\{t\+1\}\)z\_\{t\}\(55\)=Ψt−zt⊤​D​zt\\displaystyle=\\Psi\_\{t\}\-z\_\{t\}^\{\\top\}Dz\_\{t\}≤\(1−δ⁡\(a,w\)\)​Ψt\.\\displaystyle\\leq\\bigl\(1\-\\delta\(a,w\)\\bigr\)\\Psi\_\{t\}\.This proves the quantitative contraction estimate\.

Finally,

β2=1−1−β22\+O⁡\(\(1−β2\)2\),\\sqrt\{\\beta\_\{2\}\}=1\-\\frac\{1\-\\beta\_\{2\}\}\{2\}\+O\\bigl\(\(1\-\\beta\_\{2\}\)^\{2\}\\bigr\),and substituting this expansion into the definition ofWWgives \([56](https://arxiv.org/html/2608.20638#A5.E56)\)\. ∎

### E\.2Proof of Corollary[3\.2](https://arxiv.org/html/2608.20638#S3.Thmtheorem2)

###### Proof\.

Suppose, to the contrary, thatwT\+n≤ww\_\{T\+n\}\\leq wfor everyn≥0n\\geq 0, and set

M:=max⁡\{vT,ΨT\},a:=c​ηM\+ε\.M:=\\max\\\{v\_\{T\},\\Psi\_\{T\}\\\},\\qquad a:=\\frac\{c\\eta\}\{\\sqrt\{M\}\+\\varepsilon\}\.\(57\)Then0<a≤wT≤w0<a\\leq w\_\{T\}\\leq w\. Proposition[3\.1](https://arxiv.org/html/2608.20638#S3.Thmtheorem1)and the factxt2≤Ψtx\_\{t\}^\{2\}\\leq\\Psi\_\{t\}show inductively that

vT\+n≤M,ΨT\+n≤\(1−δ⁡\(a,w\)\)n​ΨT\.v\_\{T\+n\}\\leq M,\\qquad\\Psi\_\{T\+n\}\\leq\\bigl\(1\-\\delta\(a,w\)\\bigr\)^\{n\}\\Psi\_\{T\}\.\(58\)Unrolling the second\-moment recursion and using the second inequality gives

vT\+n≤β2n​vT\+\(1−β2\)​ΨT​∑j=0n−1β2n−1−j​\(1−δ⁡\(a,w\)\)j\.v\_\{T\+n\}\\leq\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)\\Psi\_\{T\}\\sum\_\{j=0\}^\{n\-1\}\\beta\_\{2\}^\{n\-1\-j\}\\bigl\(1\-\\delta\(a,w\)\\bigr\)^\{j\}\.\(59\)The right\-hand side tends to zero\. HencewT\+n→wmax\>2\>ww\_\{T\+n\}\\to w\_\{\\max\}\>2\>w, a contradiction\. ∎

The following is the quantitative version referenced after Corollary[3\.2](https://arxiv.org/html/2608.20638#S3.Thmtheorem2)\.

###### Corollary E\.2\(Quantitative passage toward the subcritical cutoff\)\.

Assume the hypotheses of Proposition[3\.1](https://arxiv.org/html/2608.20638#S3.Thmtheorem1)andwmax\>4w\_\{\\max\}\>4\. Fix0<δ<W¯0<\\delta<\\overline\{W\}andT∈ℕ0T\\in\\mathbb\{N\}\_\{0\}, and let

τ:=min⁡\{n∈ℕ0:wT\+n\>W¯−δ\}\.\\tau:=\\min\\\{n\\in\\mathbb\{N\}\_\{0\}:w\_\{T\+n\}\>\\overline\{W\}\-\\delta\\\}\.\(60\)IfwT\>W¯−δw\_\{T\}\>\\overline\{W\}\-\\delta, thenτ=0\\tau=0\. Otherwise, set

q:=1−δ⁡\(c​ηmax⁡\{vT,ΨT\}\+c​η/4,W¯−δ\)\.q:=1\-\\delta\\\!\\left\(\\frac\{c\\eta\}\{\\sqrt\{\\max\\\{v\_\{T\},\\Psi\_\{T\}\\\}\}\+c\\eta/4\},\\overline\{W\}\-\\delta\\right\)\.\(61\)Then

τ≤1\+⌈21−max⁡\{β2,q\}​log⁡\[16c2​η2​\(vT\+2​\(1−β2\)1−max⁡\{β2,q\}​ΨT\)\]⌉\.\\displaystyle\\tau\\leq 1\+\\Bigg\\lceil\\frac\{2\}\{1\-\\max\\\{\\beta\_\{2\},q\\\}\}\\log\\Bigg\[\\frac\{16\}\{c^\{2\}\\eta^\{2\}\}\\left\(v\_\{T\}\+\\frac\{2\(1\-\\beta\_\{2\}\)\}\{1\-\\max\\\{\\beta\_\{2\},q\\\}\}\\Psi\_\{T\}\\right\)\\Bigg\]\\Bigg\\rceil\.\(62\)

###### Proof\.

Only the casewT≤W¯−δw\_\{T\}\\leq\\overline\{W\}\-\\deltaneeds consideration\. Putλ=max⁡\{β2,q\}\\lambda=\\max\\\{\\beta\_\{2\},q\\\}andγ=\(1\+λ\)/2\\gamma=\(1\+\\lambda\)/2\. Sincewmax\>4w\_\{\\max\}\>4, one hasε<c​η/4\\varepsilon<c\\eta/4\. Until timeτ\\tau, the argument in \([58](https://arxiv.org/html/2608.20638#A5.E58)\) therefore applies with the lower endpoint used in \([61](https://arxiv.org/html/2608.20638#A5.E61)\), and gives

vT\+n≤β2n​vT\+\(1−β2\)​ΨT​∑j=0n−1β2n−1−j​qj\.v\_\{T\+n\}\\leq\\beta\_\{2\}^\{n\}v\_\{T\}\+\(1\-\\beta\_\{2\}\)\\Psi\_\{T\}\\sum\_\{j=0\}^\{n\-1\}\\beta\_\{2\}^\{n\-1\-j\}q^\{j\}\.\(63\)The sum is at mostn​λn−1n\\lambda^\{n\-1\}and hence at most2​γn/\(1−λ\)2\\gamma^\{n\}/\(1\-\\lambda\)\. Thus the right\-hand side is bounded by

γn​\(vT\+2​\(1−β2\)1−λ​ΨT\)\.\\gamma^\{n\}\\left\(v\_\{T\}\+\\frac\{2\(1\-\\beta\_\{2\}\)\}\{1\-\\lambda\}\\Psi\_\{T\}\\right\)\.\(64\)Moreover, ifvT\+n≤\(c​η/4\)2v\_\{T\+n\}\\leq\(c\\eta/4\)^\{2\}then, sinceε<c​η/4\\varepsilon<c\\eta/4andW¯−δ<2\\overline\{W\}\-\\delta<2, one haswT\+n=c​η/\(vT\+n\+ε\)\>2\>W¯−δw\_\{T\+n\}=c\\eta/\(\\sqrt\{v\_\{T\+n\}\}\+\\varepsilon\)\>2\>\\overline\{W\}\-\\delta\. Finally,log⁡\(1/γ\)≥1−γ=\(1−λ\)/2\\log\(1/\\gamma\)\\geq 1\-\\gamma=\(1\-\\lambda\)/2\. Takingnnequal to the ceiling in \([62](https://arxiv.org/html/2608.20638#A5.E62)\) makes \([64](https://arxiv.org/html/2608.20638#A5.E64)\) no larger thanc2​η2/16c^\{2\}\\eta^\{2\}/16, so the target has been crossed by that time\. The extra11in \([62](https://arxiv.org/html/2608.20638#A5.E62)\) is harmless\. ∎

### E\.3Proof of Proposition[3\.3](https://arxiv.org/html/2608.20638#S3.Thmtheorem3)

###### Proof\.

Fors∈\(0,1\)s\\in\(0,1\)define, only within this proof,

R⁡\(s\)\\displaystyle R\(s\):=\[\(1−β1​s\)​\(1\+s\)\(1−s\)​\(β1\+s\)\]2,\\displaystyle:=\\left\[\\frac\{\(1\-\\beta\_\{1\}s\)\(1\+s\)\}\{\(1\-s\)\(\\beta\_\{1\}\+s\)\}\\right\]^\{2\},\(65\)G⁡\(s\)\\displaystyle G\(s\):=1−R⁡\(s\)​s2R⁡\(s\)−s2\.\\displaystyle:=\\frac\{1\-R\(s\)s^\{2\}\}\{R\(s\)\-s^\{2\}\}\.The ratio inside the square is larger than one\. Lets0∈\(0,1\)s\_\{0\}\\in\(0,1\)be the first solution of

s​\(1−β1​s\)​\(1\+s\)\(1−s\)​\(β1\+s\)=1\.s\\frac\{\(1\-\\beta\_\{1\}s\)\(1\+s\)\}\{\(1\-s\)\(\\beta\_\{1\}\+s\)\}=1\.\(66\)Such a solution exists because the left\-hand side is zero ats=0s=0and tends to\+∞\+\\inftyass→1−s\\to 1^\{\-\}\. On\[0,s0\]\[0,s\_\{0\}\]the denominator inGGis positive, and

G⁡\(0\)=β12,G⁡\(s0\)=0\.G\(0\)=\\beta\_\{1\}^\{2\},\\qquad G\(s\_\{0\}\)=0\.\(67\)Thus everyβ2∈\[0,β12\)\\beta\_\{2\}\\in\[0,\\beta\_\{1\}^\{2\}\)equalsG⁡\(s\)G\(s\)for somes∈\(0,s0\]s\\in\(0,s\_\{0\}\]\. Whenβ2=β12\\beta\_\{2\}=\\beta\_\{1\}^\{2\}, direct differentiation gives

G′​\(0\)=2​β1​\(1−β1\)2\>0;G^\{\\prime\}\(0\)=2\\beta\_\{1\}\(1\-\\beta\_\{1\}\)^\{2\}\>0;sinceG⁡\(s0\)=0G\(s\_\{0\}\)=0, there is again a solution withs∈\(0,s0\)s\\in\(0,s\_\{0\}\)\. We henceforth fix such a positivess\.

Set

A\\displaystyle A:=1\+β2​s21\+β2,\\displaystyle:=\\sqrt\{\\frac\{1\+\\beta\_\{2\}s^\{2\}\}\{1\+\\beta\_\{2\}\}\},B\\displaystyle B:=s2\+β21\+β2,\\displaystyle:=\\sqrt\{\\frac\{s^\{2\}\+\\beta\_\{2\}\}\{1\+\\beta\_\{2\}\}\},\(68\)f\\displaystyle f:=1−β1​s1−s,\\displaystyle:=\\frac\{1\-\\beta\_\{1\}s\}\{1\-s\},g\\displaystyle g:=β1\+s1\+s,\\displaystyle:=\\frac\{\\beta\_\{1\}\+s\}\{1\+s\},k\\displaystyle k:=1−β11\+β12\.\\displaystyle:=\\frac\{1\-\\beta\_\{1\}\}\{1\+\\beta\_\{1\}^\{2\}\}\.The equationβ2=G⁡\(s\)\\beta\_\{2\}=G\(s\)is equivalent to

fA=gB\.\\frac\{f\}\{A\}=\\frac\{g\}\{B\}\.\(69\)For the prescribed learning rateη\>0\\eta\>0, define

a:=η​k​fA=η​k​gB\>0a:=\\eta k\\frac\{f\}\{A\}=\\eta k\\frac\{g\}\{B\}\>0\(70\)and initialize

x0=a,m0=−k​a​\(β1\+s\),v0=a2​B2\.x\_\{0\}=a,\\qquad m\_\{0\}=\-ka\(\\beta\_\{1\}\+s\),\\qquad v\_\{0\}=a^\{2\}B^\{2\}\.\(71\)The momentum and second\-moment recursions give

tmod40123xtas​a−a−s​amt−k​a​\(β1\+s\)k​a​\(1−β1​s\)k​a​\(β1\+s\)−k​a​\(1−β1​s\)vta2​B2a2​A2a2​B2a2​A2\.\\begin\{array\}\[\]\{c\|cccc\}t\\bmod 4&0&1&2&3\\\\ \\hline\\cr x\_\{t\}&a&sa&\-a&\-sa\\\\ m\_\{t\}&\-ka\(\\beta\_\{1\}\+s\)&ka\(1\-\\beta\_\{1\}s\)&ka\(\\beta\_\{1\}\+s\)&\-ka\(1\-\\beta\_\{1\}s\)\\\\ v\_\{t\}&a^\{2\}B^\{2\}&a^\{2\}A^\{2\}&a^\{2\}B^\{2\}&a^\{2\}A^\{2\}\.\\end\{array\}\(72\)Indeed,a​A=η​k​faA=\\eta kfanda​B=η​k​gaB=\\eta kgverify the first two position updates whenε=0\\varepsilon=0; the remaining two follow by half\-turn symmetry\. This also shows directly why the construction works for everyη\>0\\eta\>0: changingη\\etamerely rescalesaa,mtm\_\{t\}, andvt\\sqrt\{v\_\{t\}\}by the same factor\.

The two values ofwtw\_\{t\}are

1\+β121\+β1​1−s1−β1​s,1\+β121\+β1​1\+sβ1\+s\.\\frac\{1\+\\beta\_\{1\}^\{2\}\}\{1\+\\beta\_\{1\}\}\\frac\{1\-s\}\{1\-\\beta\_\{1\}s\},\\qquad\\frac\{1\+\\beta\_\{1\}^\{2\}\}\{1\+\\beta\_\{1\}\}\\frac\{1\+s\}\{\\beta\_\{1\}\+s\}\.\(73\)The first is always below22, while the second is below22exactly when

s\>1−2​β1−β121\+2​β1−β12\.s\>\\frac\{1\-2\\beta\_\{1\}\-\\beta\_\{1\}^\{2\}\}\{1\+2\\beta\_\{1\}\-\\beta\_\{1\}^\{2\}\}\.\(74\)Sinceβ1≥2−1\\beta\_\{1\}\\geq\\sqrt\{2\}\-1, the right\-hand side is nonpositive\. Therefore the orbit is strictly subcritical\. The four position values are distinct, so its prime period is four\. ∎

Figure 3:An exact four\-cycle of Proposition[3\.3](https://arxiv.org/html/2608.20638#S3.Thmtheorem3)at\(β1,β2\)=\(12,15\)\(\\beta\_\{1\},\\beta\_\{2\}\)=\(\\tfrac\{1\}\{2\},\\,\\frac\{1\}\{5\}\),ε=0\\varepsilon=0, initialized at the point \([71](https://arxiv.org/html/2608.20638#A5.E71)\) withs≈0\.2587s\\approx 0\.2587anda≈0\.5112​ηa\\approx 0\.5112\\,\\eta\. Left: the orbit onf⁡\(x\)=12​x2f\(x\)=\\tfrac\{1\}\{2\}x^\{2\}\. Right:wtw\_\{t\}alternates between≈1\.38\\approx 1\.38and≈0\.71\\approx 0\.71, beloww=2w=2\.

## Appendix FProofs for the supercritical results

### F\.1Proof of Lemma[3\.4](https://arxiv.org/html/2608.20638#S3.Thmtheorem4)

###### Proof\.

Ifxtx\_\{t\}andhth\_\{t\}have the same sign, the two rows ofA⁡\(wt\+1\)A\(w\_\{t\+1\}\)show that both signs reverse whenwt\+1\>2w\_\{t\+1\}\>2, and

\|xt\+1\|=\(wt\+1−1\)​\|xt\|\+β1​wt\+1​\|ht\|\>\(wt\+1−1\)​\|xt\|\.\\left\\lvert x\_\{t\+1\}\\right\\rvert=\(w\_\{t\+1\}\-1\)\\left\\lvert x\_\{t\}\\right\\rvert\+\\beta\_\{1\}w\_\{t\+1\}\\left\\lvert h\_\{t\}\\right\\rvert\>\(w\_\{t\+1\}\-1\)\\left\\lvert x\_\{t\}\\right\\rvert\.\(75\)
Ifxt​ht<0x\_\{t\}h\_\{t\}<0, writeht=−r​xth\_\{t\}=\-rx\_\{t\}withr\>0r\>0\. The next state is also misaligned exactly when

wt\+1−2β1​\(wt\+1−1\)<r<wt\+1−1β1​wt\+1\.\\frac\{w\_\{t\+1\}\-2\}\{\\beta\_\{1\}\(w\_\{t\+1\}\-1\)\}<r<\\frac\{w\_\{t\+1\}\-1\}\{\\beta\_\{1\}w\_\{t\+1\}\}\.\(76\)Within this interval,

\|xt\+1\|\\displaystyle\\left\\lvert x\_\{t\+1\}\\right\\rvert=\(wt\+1−1−β1​wt\+1​r\)​\|xt\|\\displaystyle=\\bigl\(w\_\{t\+1\}\-1\-\\beta\_\{1\}w\_\{t\+1\}r\\bigr\)\\left\\lvert x\_\{t\}\\right\\rvert\(77\)<\(wt\+1−1−wt\+1​\(wt\+1−2\)wt\+1−1\)​\|xt\|=\|xt\|wt\+1−1\.\\displaystyle<\\left\(w\_\{t\+1\}\-1\-\\frac\{w\_\{t\+1\}\(w\_\{t\+1\}\-2\)\}\{w\_\{t\+1\}\-1\}\\right\)\\left\\lvert x\_\{t\}\\right\\rvert=\\frac\{\\left\\lvert x\_\{t\}\\right\\rvert\}\{w\_\{t\+1\}\-1\}\.This proves both assertions\. ∎

### F\.2Proof of Corollary[3\.5](https://arxiv.org/html/2608.20638#S3.Thmtheorem5)

###### Proof\.

WhilewT\+j≥2\+δw\_\{T\+j\}\\geq 2\+\\delta, Lemma[3\.4](https://arxiv.org/html/2608.20638#S3.Thmtheorem4)\(i\) preserves alignment\. The following stopped estimate is obtained by substitutingwt\+1=c​η/\(vt\+1\+ε\)w\_\{t\+1\}=c\\eta/\(\\sqrt\{v\_\{t\+1\}\}\+\\varepsilon\)in \([75](https://arxiv.org/html/2608.20638#A6.E75)\) and usingvt\+1=β2​vt\+\(1−β2\)​xt2v\_\{t\+1\}=\\beta\_\{2\}v\_\{t\}\+\(1\-\\beta\_\{2\}\)x\_\{t\}^\{2\}: for every1≤n<τ1\\leq n<\\tau,

\(1\+δ\)2​n−1<δ​\(c​η\)2\(1−β2\)​xT2\.\(1\+\\delta\)^\{2n\}\-1<\\frac\{\\delta\(c\\eta\)^\{2\}\}\{\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\}\.\(78\)This is a direct induction onnn; after clearing the positive denominators, the induction step is precisely the sum of the nonnegative momentum term in \([75](https://arxiv.org/html/2608.20638#A6.E75)\) and the nonnegative termβ2​vt\\beta\_\{2\}v\_\{t\}in the second\-moment update\.

Since2​log⁡\(1\+δ\)≥δ2\\log\(1\+\\delta\)\\geq\\deltafor0<δ≤10<\\delta\\leq 1, \([78](https://arxiv.org/html/2608.20638#A6.E78)\) implies

n<δ−1​log⁡\(1\+δ​\(c​η\)2\(1−β2\)​xT2\)\(1≤n<τ\)\.n<\\delta^\{\-1\}\\log\\\!\\left\(1\+\\frac\{\\delta\(c\\eta\)^\{2\}\}\{\(1\-\\beta\_\{2\}\)x\_\{T\}^\{2\}\}\\right\)\\qquad\(1\\leq n<\\tau\)\.\(79\)Taking the largest surviving integer gives \([13](https://arxiv.org/html/2608.20638#S3.E13)\)\. If an aligned trajectory remained supercritical and hadlim inftwt\>2\\liminf\_\{t\}w\_\{t\}\>2, it would eventually remain inwt≥2\+δw\_\{t\}\\geq 2\+\\deltafor someδ∈\(0,1\]\\delta\\in\(0,1\], contradicting the bound just proved\. Hence its lower limit is22\. ∎

### F\.3Proof of Proposition[3\.6](https://arxiv.org/html/2608.20638#S3.Thmtheorem6)

###### Proof\.

Fix the prescribedv0\>0v\_\{0\}\>0and choose any

0<\|x0\|≤v0\.0<\\left\\lvert x\_\{0\}\\right\\rvert\\leq\\sqrt\{v\_\{0\}\}\.\(80\)We will select the momentum by a one\-dimensional shooting argument\. For a temporary sloper0\>0r\_\{0\}\>0, let

h0=−r0​x0,m0=c⁡\(h0−x0\)=−c⁡\(1\+r0\)​x0\.h\_\{0\}=\-r\_\{0\}x\_\{0\},\\qquad m\_\{0\}=c\(h\_\{0\}\-x\_\{0\}\)=\-c\(1\+r\_\{0\}\)x\_\{0\}\.\(81\)Letσ=sign⁡\(x0\)\\sigma=\\operatorname\{sign\}\(x\_\{0\}\)and remove the alternating sign by setting

Xt:=σ​\(−1\)t​xt,Ht:=σ​\(−1\)t​ht\.X\_\{t\}:=\\sigma\(\-1\)^\{t\}x\_\{t\},\\qquad H\_\{t\}:=\\sigma\(\-1\)^\{t\}h\_\{t\}\.\(82\)ThusX0=\|x0\|\>0X\_\{0\}=\\left\\lvert x\_\{0\}\\right\\rvert\>0andH0=−r0​\|x0\|<0H\_\{0\}=\-r\_\{0\}\\left\\lvert x\_\{0\}\\right\\rvert<0\. From \([6](https://arxiv.org/html/2608.20638#S3.E6)\),

\(Xt\+1Ht\+1\)=\(wt\+1−1β1​wt\+1wt\+1−2β1​\(wt\+1−1\)\)​\(XtHt\)\.\\binom\{X\_\{t\+1\}\}\{H\_\{t\+1\}\}=\\begin\{pmatrix\}w\_\{t\+1\}\-1&\\beta\_\{1\}w\_\{t\+1\}\\\\ w\_\{t\+1\}\-2&\\beta\_\{1\}\(w\_\{t\+1\}\-1\)\\end\{pmatrix\}\\binom\{X\_\{t\}\}\{H\_\{t\}\}\.\(83\)
We first verify that every finite misaligned orbit stays in the prescribed supercritical box\. If misalignment survives from timettto timet\+1t\+1, Lemma[3\.4](https://arxiv.org/html/2608.20638#S3.Thmtheorem4)\(ii\) gives

\|xt\+1\|<\|xt\|wt\+1−1<\|xt\|\.\\left\\lvert x\_\{t\+1\}\\right\\rvert<\\frac\{\\left\\lvert x\_\{t\}\\right\\rvert\}\{w\_\{t\+1\}\-1\}<\\left\\lvert x\_\{t\}\\right\\rvert\.\(84\)Starting from \([80](https://arxiv.org/html/2608.20638#A6.E80)\), induction and the convex\-combination identityvt\+1=β2​vt\+\(1−β2\)​xt2v\_\{t\+1\}=\\beta\_\{2\}v\_\{t\}\+\(1\-\\beta\_\{2\}\)x\_\{t\}^\{2\}therefore give

xt2≤v0,vt≤v0x\_\{t\}^\{2\}\\leq v\_\{0\},\\qquad v\_\{t\}\\leq v\_\{0\}\(85\)at every surviving time\. Hence

wt≥c​ηv0\+ε=w0\>2\.w\_\{t\}\\geq\\frac\{c\\eta\}\{\\sqrt\{v\_\{0\}\}\+\\varepsilon\}=w\_\{0\}\>2\.\(86\)This closes the small\-box argument: the supercritical hypothesis needed in Lemma[3\.4](https://arxiv.org/html/2608.20638#S3.Thmtheorem4)\(ii\) is automatically preserved for as long as the trajectory remains misaligned\.

We now construct an orbit for which misalignment never ends\. On a finite surviving orbit define

rt:=−HtXt=−htxt\>0\.r\_\{t\}:=\-\\frac\{H\_\{t\}\}\{X\_\{t\}\}=\-\\frac\{h\_\{t\}\}\{x\_\{t\}\}\>0\.\(87\)Substitution ofHt=−rt​XtH\_\{t\}=\-r\_\{t\}X\_\{t\}in \([83](https://arxiv.org/html/2608.20638#A6.E83)\) shows that the next state is misaligned exactly when

ℓ⁡\(wt\+1\)<rt<q⁡\(wt\+1\),\\ell\(w\_\{t\+1\}\)<r\_\{t\}<q\(w\_\{t\+1\}\),\(88\)where, within this proof,

ℓ⁡\(w\):=w−2β1​\(w−1\),q⁡\(w\):=w−1β1​w\.\\ell\(w\):=\\frac\{w\-2\}\{\\beta\_\{1\}\(w\-1\)\},\\qquad q\(w\):=\\frac\{w\-1\}\{\\beta\_\{1\}w\}\.\(89\)When \([88](https://arxiv.org/html/2608.20638#A6.E88)\) holds, the next ratio is

rt\+1=Φwt\+1​\(rt\):=β1​\(wt\+1−1\)​rt−\(wt\+1−2\)wt\+1−1−β1​wt\+1​rt\.r\_\{t\+1\}=\\Phi\_\{w\_\{t\+1\}\}\(r\_\{t\}\):=\\frac\{\\beta\_\{1\}\(w\_\{t\+1\}\-1\)r\_\{t\}\-\(w\_\{t\+1\}\-2\)\}\{w\_\{t\+1\}\-1\-\\beta\_\{1\}w\_\{t\+1\}r\_\{t\}\}\.\(90\)For each fixedw\>2w\>2, this map is continuous and strictly increasing on\(ℓ⁡\(w\),q⁡\(w\)\)\(\\ell\(w\),q\(w\)\), and

limr→ℓ​\(w\)\+Φw​\(r\)=0,limr→q​\(w\)−Φw​\(r\)=\+∞\.\\lim\_\{r\\to\\ell\(w\)^\{\+\}\}\\Phi\_\{w\}\(r\)=0,\\qquad\\lim\_\{r\\to q\(w\)^\{\-\}\}\\Phi\_\{w\}\(r\)=\+\\infty\.\(91\)
We next build nested shooting intervals\. Sincev1v\_\{1\}depends onx0x\_\{0\}andv0v\_\{0\}but not onr0r\_\{0\}, the numberw1w\_\{1\}is fixed\. Define

I1:=\(ℓ⁡\(w1\),q⁡\(w1\)\)\.I\_\{1\}:=\(\\ell\(w\_\{1\}\),q\(w\_\{1\}\)\)\.\(92\)Everyr0∈I1r\_\{0\}\\in I\_\{1\}survives through time one, the mapr0↦r1r\_\{0\}\\mapsto r\_\{1\}is continuous, and \([91](https://arxiv.org/html/2608.20638#A6.E91)\) says that its range is\(0,\+∞\)\(0,\+\\infty\)\.

Suppose inductively thatIN=\(aN,bN\)I\_\{N\}=\(a\_\{N\},b\_\{N\}\)is a nonempty open interval such that everyr0∈INr\_\{0\}\\in I\_\{N\}survives through timeNN, the mapr0↦rN​\(r0\)r\_\{0\}\\mapsto r\_\{N\}\(r\_\{0\}\)is continuous, and

limr0→aN\+rN​\(r0\)=0,limr0→bN−rN​\(r0\)=\+∞\.\\lim\_\{r\_\{0\}\\to a\_\{N\}^\{\+\}\}r\_\{N\}\(r\_\{0\}\)=0,\\qquad\\lim\_\{r\_\{0\}\\to b\_\{N\}^\{\-\}\}r\_\{N\}\(r\_\{0\}\)=\+\\infty\.\(93\)The quantitieswN\+1​\(r0\)w\_\{N\+1\}\(r\_\{0\}\),ℓ⁡\(wN\+1​\(r0\)\)\\ell\(w\_\{N\+1\}\(r\_\{0\}\)\), andq​\(wN\+1​\(r0\)\)q\(w\_\{N\+1\}\(r\_\{0\}\)\)are continuous onINI\_\{N\}\. Furthermore, \([86](https://arxiv.org/html/2608.20638#A6.E86)\) implies the uniform bounds

ℓ⁡\(wN\+1\)≥w0−2β1​\(w0−1\)\>0,q⁡\(wN\+1\)<1β1\.\\ell\(w\_\{N\+1\}\)\\geq\\frac\{w\_\{0\}\-2\}\{\\beta\_\{1\}\(w\_\{0\}\-1\)\}\>0,\\qquad q\(w\_\{N\+1\}\)<\\frac\{1\}\{\\beta\_\{1\}\}\.\(94\)Thus the graph ofrNr\_\{N\}starts below the lower boundary in \([88](https://arxiv.org/html/2608.20638#A6.E88)\) and ends above its upper boundary\. Choose a componentIN\+1=\(aN\+1,bN\+1\)I\_\{N\+1\}=\(a\_\{N\+1\},b\_\{N\+1\}\)on which

ℓ⁡\(wN\+1\)<rN<q⁡\(wN\+1\)\\ell\(w\_\{N\+1\}\)<r\_\{N\}<q\(w\_\{N\+1\}\)\(95\)and whose left and right endpoints meet the lower and upper boundary, respectively\. Then

I¯N\+1⊂IN\.\\overline\{I\}\_\{N\+1\}\\subset I\_\{N\}\.\(96\)Applying \([91](https://arxiv.org/html/2608.20638#A6.E91)\) at the two new endpoints shows thatr0↦rN\+1r\_\{0\}\\mapsto r\_\{N\+1\}again has the endpoint behavior in \([93](https://arxiv.org/html/2608.20638#A6.E93)\)\. This completes the induction\.

The nonempty compact intervalsI¯N\\overline\{I\}\_\{N\}are nested, so choose

r0∗∈⋂N≥1I¯N\.r\_\{0\}^\{\*\}\\in\\bigcap\_\{N\\geq 1\}\\overline\{I\}\_\{N\}\.\(97\)BecauseI¯N\+1⊂IN\\overline\{I\}\_\{N\+1\}\\subset I\_\{N\}, the selected point actually belongs to every open survival intervalINI\_\{N\}\. The initialization \([81](https://arxiv.org/html/2608.20638#A6.E81)\) withr0=r0∗r\_\{0\}=r\_\{0\}^\{\*\}therefore remains strictly misaligned at every finite time\. Sincemt\+c​xt=c​htm\_\{t\}\+cx\_\{t\}=ch\_\{t\}, this givesxt​\(mt\+c​xt\)=c​xt​ht<0x\_\{t\}\(m\_\{t\}\+cx\_\{t\}\)=cx\_\{t\}h\_\{t\}<0for everyt≥0t\\geq 0\.

It remains only to record the convergence estimates\. Iterating Lemma[3\.4](https://arxiv.org/html/2608.20638#S3.Thmtheorem4)\(ii\) and usingwt≥w0w\_\{t\}\\geq w\_\{0\}gives

\|xt\|≤\|x0\|​∏j=1t1wj−1≤\(w0−1\)−t​\|x0\|\.\\left\\lvert x\_\{t\}\\right\\rvert\\leq\\left\\lvert x\_\{0\}\\right\\rvert\\prod\_\{j=1\}^\{t\}\\frac\{1\}\{w\_\{j\}\-1\}\\leq\(w\_\{0\}\-1\)^\{\-t\}\\left\\lvert x\_\{0\}\\right\\rvert\.\(98\)Alsort<q⁡\(wt\+1\)<1/β1r\_\{t\}<q\(w\_\{t\+1\}\)<1/\\beta\_\{1\}, andmt=c⁡\(ht−xt\)=−c⁡\(1\+rt\)​xtm\_\{t\}=c\(h\_\{t\}\-x\_\{t\}\)=\-c\(1\+r\_\{t\}\)x\_\{t\}, so

\|mt\|≤c⁡\(1\+1β1\)​\|xt\|→0\.\\left\\lvert m\_\{t\}\\right\\rvert\\leq c\\left\(1\+\\frac\{1\}\{\\beta\_\{1\}\}\\right\)\\left\\lvert x\_\{t\}\\right\\rvert\\to 0\.\(99\)Finally,

vt=β2t​v0\+\(1−β2\)​∑j=0t−1β2t−1−j​xj2→0,v\_\{t\}=\\beta\_\{2\}^\{t\}v\_\{0\}\+\(1\-\\beta\_\{2\}\)\\sum\_\{j=0\}^\{t\-1\}\\beta\_\{2\}^\{t\-1\-j\}x\_\{j\}^\{2\}\\to 0,\(100\)becausexj2x\_\{j\}^\{2\}decays geometrically\. Hence\(xt,mt,vt\)→\(0,0,0\)\(x\_\{t\},m\_\{t\},v\_\{t\}\)\\to\(0,0,0\)andwt→wmaxw\_\{t\}\\to w\_\{\\max\}\. ∎

### F\.4Numerical illustration of Proposition[3\.6](https://arxiv.org/html/2608.20638#S3.Thmtheorem6)

Take

β1=12,β2=0\.9,η=1,ε=112,\\beta\_\{1\}=\\tfrac\{1\}\{2\},\\qquad\\beta\_\{2\}=0\.9,\\qquad\\eta=1,\\qquad\\varepsilon=\\tfrac\{1\}\{12\},so thatc=13c=\\tfrac\{1\}\{3\}andwmax=4w\_\{\\max\}=4, and prescribev0=0\.0025v\_\{0\}=0\.0025andx0=0\.04≤v0x\_\{0\}=0\.04\\leq\\sqrt\{v\_\{0\}\}, hencew0=c​η/\(v0\+ε\)=52\>2w\_\{0\}=c\\eta/\(\\sqrt\{v\_\{0\}\}\+\\varepsilon\)=\\tfrac\{5\}\{2\}\>2\. Following the shooting argument of Appendix[F\.3](https://arxiv.org/html/2608.20638#A6.SS3), the momentum is initialized asm0=−c⁡\(1\+r0\)​x0m\_\{0\}=\-c\(1\+r\_\{0\}\)x\_\{0\}and the persistent sloper0∗∈⋂NI¯Nr\_\{0\}^\{\\ast\}\\in\\bigcap\_\{N\}\\overline\{I\}\_\{N\}is located by bisection onr0r\_\{0\}, running the recursion \([2](https://arxiv.org/html/2608.20638#S1.E2)\)\. With this initialization the orbit remains misaligned,xt​\(mt\+c​xt\)<0x\_\{t\}\(m\_\{t\}\+cx\_\{t\}\)<0, over the1212steps shown in Figure[4](https://arxiv.org/html/2608.20638#A6.F4)\(and beyond\):wt≥w0w\_\{t\}\\geq w\_\{0\}increases monotonically towardwmax=4w\_\{\\max\}=4, the envelope\|xt\|≤\(w0−1\)−t​\|x0\|\\left\\lvert x\_\{t\}\\right\\rvert\\leq\(w\_\{0\}\-1\)^\{\-t\}\\left\\lvert x\_\{0\}\\right\\rvertof \([15](https://arxiv.org/html/2608.20638#S3.E15)\) holds at every step, and already\|x12\|<10−9\\left\\lvert x\_\{12\}\\right\\rvert<10^\{\-9\}: the trajectory converges to the origin without ever leaving the supercritical region\.

Figure 4:The exceptional orbit of Proposition[3\.6](https://arxiv.org/html/2608.20638#S3.Thmtheorem6)forβ1=12\\beta\_\{1\}=\\frac\{1\}\{2\},β2=0\.9\\beta\_\{2\}=0\.9,η=1\\eta=1,ε=112\\varepsilon=\\frac\{1\}\{12\},v0=0\.0025v\_\{0\}=0\.0025,x0=0\.04x\_\{0\}=0\.04, withm0m\_\{0\}determined by the shooting argument of Appendix[F\.3](https://arxiv.org/html/2608.20638#A6.SS3)\. Left: the sign ofxtx\_\{t\}alternates at every step and\|xt\|\\left\\lvert x\_\{t\}\\right\\rvertdecays well inside the certified envelope\(w0−1\)−t​\|x0\|\(w\_\{0\}\-1\)^\{\-t\}\\left\\lvert x\_\{0\}\\right\\rvertof \([15](https://arxiv.org/html/2608.20638#S3.E15)\) \(dashed\)\. Right: along the same stepswtw\_\{t\}increases monotonically fromw0=52w\_\{0\}=\\tfrac\{5\}\{2\}towardwmax=4w\_\{\\max\}=4and never leaves the supercritical region\.

## Appendix GProofs for thewmax<2w\_\{\\max\}<2case

### G\.1Global convergence in the globally subcritical zero\-momentum regime

We record separately the complementary regimewmax<2w\_\{\\max\}<2forβ1=0\\beta\_\{1\}=0\. In this case the entire trajectory remains strictly below the frozen stability threshold, and the stopped\-trajectory argument used in Theorem[D\.1](https://arxiv.org/html/2608.20638#A4.Thmtheorem1)becomes a global contraction argument\.

###### Theorem G\.1\(Global exponential convergence forβ1=0\\beta\_\{1\}=0\)\.

Assume

β1=0,wmax=ηε<2\.\\beta\_\{1\}=0,\\qquad w\_\{\\max\}=\\frac\{\\eta\}\{\\varepsilon\}<2\.Then every trajectory of Adam converges to the origin\. More precisely, for every initial state\(x0,m0,v0\)∈ℝ2×\[0,∞\)\(x\_\{0\},m\_\{0\},v\_\{0\}\)\\in\\mathbb\{R\}^\{2\}\\times\[0,\\infty\), there exist constantsC\>0C\>0andγ∈\(0,1\)\\gamma\\in\(0,1\), depending on the initial state and the parameters, such that

\|xt\|\+\|mt\|\+vt≤C​γt,t∈ℕ0\.\\left\\lvert x\_\{t\}\\right\\rvert\+\\left\\lvert m\_\{t\}\\right\\rvert\+v\_\{t\}\\leq C\\gamma^\{t\},\\qquad t\\in\\mathbb\{N\}\_\{0\}\.In particular,\(xt,mt,vt\)→\(0,0,0\)\(x\_\{t\},m\_\{t\},v\_\{t\}\)\\to\(0,0,0\)\.

###### Proof\.

Sinceβ1=0\\beta\_\{1\}=0, one hasc=1c=1and

mt\+1=xt,vt\+1=β2​vt\+\(1−β2\)​xt2,xt\+1=\(1−wt\+1\)​xt,m\_\{t\+1\}=x\_\{t\},\\qquad v\_\{t\+1\}=\\beta\_\{2\}v\_\{t\}\+\(1\-\\beta\_\{2\}\)x\_\{t\}^\{2\},\\qquad x\_\{t\+1\}=\(1\-w\_\{t\+1\}\)x\_\{t\},where

wt=ηvt\+ε\.w\_\{t\}=\\frac\{\\eta\}\{\\sqrt\{v\_\{t\}\}\+\\varepsilon\}\.Becausevt≥0v\_\{t\}\\geq 0,

0<wt≤wmax<2for every​t≥0\.0<w\_\{t\}\\leq w\_\{\\max\}<2\\qquad\\text\{for every \}t\\geq 0\.Hence

\|xt\+1\|=\|1−wt\+1\|​\|xt\|≤\|xt\|,\\left\\lvert x\_\{t\+1\}\\right\\rvert=\\left\\lvert 1\-w\_\{t\+1\}\\right\\rvert\\,\\left\\lvert x\_\{t\}\\right\\rvert\\leq\\left\\lvert x\_\{t\}\\right\\rvert,so that\|xt\|≤\|x0\|\\left\\lvert x\_\{t\}\\right\\rvert\\leq\\left\\lvert x\_\{0\}\\right\\rvertfor everyt≥0t\\geq 0\.

SetM:=max⁡\{v0,x02\}M:=\\max\\\{v\_\{0\},x\_\{0\}^\{2\}\\\}\. We claim thatvt≤Mv\_\{t\}\\leq Mfor everyt≥0t\\geq 0\. Indeed, this is true att=0t=0, and ifvt≤Mv\_\{t\}\\leq M, then

vt\+1=β2​vt\+\(1−β2\)​xt2≤β2​M\+\(1−β2\)​M=M\.v\_\{t\+1\}=\\beta\_\{2\}v\_\{t\}\+\(1\-\\beta\_\{2\}\)x\_\{t\}^\{2\}\\leq\\beta\_\{2\}M\+\(1\-\\beta\_\{2\}\)M=M\.Therefore

w¯:=ηM\+ε≤wt≤wmax<2for every​t≥0\.\\underline\{w\}:=\\frac\{\\eta\}\{\\sqrt\{M\}\+\\varepsilon\}\\leq w\_\{t\}\\leq w\_\{\\max\}<2\\qquad\\text\{for every \}t\\geq 0\.Define

ρ:=max⁡\{\|1−w¯\|,\|1−wmax\|\}\.\\rho:=\\max\\bigl\\\{\\left\\lvert 1\-\\underline\{w\}\\right\\rvert,\\left\\lvert 1\-w\_\{\\max\}\\right\\rvert\\bigr\\\}\.Since0<w¯≤wmax<20<\\underline\{w\}\\leq w\_\{\\max\}<2, we have0≤ρ<10\\leq\\rho<1\. The convexity ofw↦\|1−w\|w\\mapsto\\left\\lvert 1\-w\\right\\rverttherefore gives\|1−wt\|≤ρ\\left\\lvert 1\-w\_\{t\}\\right\\rvert\\leq\\rhofor everyt≥0t\\geq 0\. Consequently,\|xt\+1\|≤ρ​\|xt\|\\left\\lvert x\_\{t\+1\}\\right\\rvert\\leq\\rho\\left\\lvert x\_\{t\}\\right\\rvert, and hence

\|xt\|≤ρt​\|x0\|\.\\left\\lvert x\_\{t\}\\right\\rvert\\leq\\rho^\{t\}\\left\\lvert x\_\{0\}\\right\\rvert\.\(101\)
It remains to control the second moment\. Unrolling its recursion yields

vt=β2t​v0\+\(1−β2\)​∑j=0t−1β2t−1−j​xj2\.v\_\{t\}=\\beta\_\{2\}^\{t\}v\_\{0\}\+\(1\-\\beta\_\{2\}\)\\sum\_\{j=0\}^\{t\-1\}\\beta\_\{2\}^\{t\-1\-j\}x\_\{j\}^\{2\}\.Using \([101](https://arxiv.org/html/2608.20638#A7.E101)\),

vt≤β2t​v0\+\(1−β2\)​x02​∑j=0t−1β2t−1−j​ρ2​j=β2t​v0\+\(1−β2\)​x02​Gt​\(ρ2,β2\)\.v\_\{t\}\\leq\\beta\_\{2\}^\{t\}v\_\{0\}\+\(1\-\\beta\_\{2\}\)x\_\{0\}^\{2\}\\sum\_\{j=0\}^\{t\-1\}\\beta\_\{2\}^\{t\-1\-j\}\\rho^\{2j\}=\\beta\_\{2\}^\{t\}v\_\{0\}\+\(1\-\\beta\_\{2\}\)x\_\{0\}^\{2\}G\_\{t\}\(\\rho^\{2\},\\beta\_\{2\}\)\.Sinceρ2<1\\rho^\{2\}<1andβ2<1\\beta\_\{2\}<1, the right\-hand side converges to zero\. Moreover, lettingλ:=max⁡\{ρ2,β2\}<1\\lambda:=\\max\\\{\\rho^\{2\},\\beta\_\{2\}\\\}<1and choosing anyγ∈\(λ,1\)\\gamma\\in\(\\lambda,1\), the geometric convolution satisfiesGt​\(ρ2,β2\)≤C1​γtG\_\{t\}\(\\rho^\{2\},\\beta\_\{2\}\)\\leq C\_\{1\}\\gamma^\{t\}for someC1\>0C\_\{1\}\>0\. Thusvt≤C2​γtv\_\{t\}\\leq C\_\{2\}\\gamma^\{t\}after enlargingC2C\_\{2\}if necessary\.

Finally,mt\+1=xtm\_\{t\+1\}=x\_\{t\}, so\|mt\+1\|≤ρt​\|x0\|\\left\\lvert m\_\{t\+1\}\\right\\rvert\\leq\\rho^\{t\}\\left\\lvert x\_\{0\}\\right\\rvert\. Combining the preceding estimates, and absorbing the finite initial values into the constant, gives

\|xt\|\+\|mt\|\+vt≤C​γ~t\\left\\lvert x\_\{t\}\\right\\rvert\+\\left\\lvert m\_\{t\}\\right\\rvert\+v\_\{t\}\\leq C\\widetilde\{\\gamma\}^\{t\}for someC\>0C\>0andγ~∈\(max⁡\{ρ,γ\},1\)\\widetilde\{\\gamma\}\\in\(\\max\\\{\\rho,\\gamma\\\},1\)\. Therefore\(xt,mt,vt\)→\(0,0,0\)\(x\_\{t\},m\_\{t\},v\_\{t\}\)\\to\(0,0,0\)exponentially\. ∎

### G\.2Global convergence in the globally subcritical regime with positive momentum

The next result removes the zero\-momentum restriction\.

###### Theorem G\.3\(Global convergence in the strictly subcritical regime\)\.

Assume

0<β1<1,0≤β2<1,β2\>β12,wmax<2\.0<\\beta\_\{1\}<1,\\qquad 0\\leq\\beta\_\{2\}<1,\\qquad\\sqrt\{\\beta\_\{2\}\}\>\\beta\_\{1\}^\{2\},\\qquad w\_\{\\max\}<2\.Then every trajectory of Adam onf⁡\(x\)=12​x2f\(x\)=\\tfrac\{1\}\{2\}x^\{2\}converges to the origin\. More precisely, for every initial state\(x0,m0,v0\)∈ℝ2×\[0,∞\)\(x\_\{0\},m\_\{0\},v\_\{0\}\)\\in\\mathbb\{R\}^\{2\}\\times\[0,\\infty\), there exist constantsC\>0C\>0andγ0∈\(0,1\)\\gamma\_\{0\}\\in\(0,1\), possibly depending on the initial state, such that

\|xt\|\+\|mt\|\+vt≤C​γ0t,t≥0\.\\left\\lvert x\_\{t\}\\right\\rvert\+\\left\\lvert m\_\{t\}\\right\\rvert\+v\_\{t\}\\leq C\\gamma\_\{0\}^\{t\},\\qquad t\\geq 0\.In particular,\(xt,mt,vt\)→\(0,0,0\)\(x\_\{t\},m\_\{t\},v\_\{t\}\)\\to\(0,0,0\)andwt→wmaxw\_\{t\}\\to w\_\{\\max\}\.

###### Proof\.

Recall

ξt:=mt\+c​xtc,zt:=\(xtξt\),\\xi\_\{t\}:=\\frac\{m\_\{t\}\+cx\_\{t\}\}\{c\},\\qquad z\_\{t\}:=\\binom\{x\_\{t\}\}\{\\xi\_\{t\}\},so thatzt\+1=A⁡\(wt\+1\)​ztz\_\{t\+1\}=A\(w\_\{t\+1\}\)z\_\{t\}\. Also recall

θ⁡\(w\):=w2−w,Φ⁡\(w\):=wβ2\+\(1−β2\)​w/wmax\.\\theta\(w\):=\\frac\{w\}\{2\-w\},\\qquad\\Phi\(w\):=\\frac\{w\}\{\\sqrt\{\\beta\_\{2\}\}\+\(1\-\\sqrt\{\\beta\_\{2\}\}\)w/w\_\{\\max\}\}\.Every Adam transition satisfies0<wt\+1≤Φ⁡\(wt\)0<w\_\{t\+1\}\\leq\\Phi\(w\_\{t\}\)\.

We first construct a quadratic weight valid on the entire interval\(0,wmax\]\(0,w\_\{\\max\}\]\. Define

R⁡\(w\):=β12​θ​\(Φ⁡\(w\)\)θ⁡\(w\)=β12​2−w2​β2−\[1−2​\(1−β2\)/wmax\]​w\.R\(w\):=\\frac\{\\beta\_\{1\}^\{2\}\\theta\(\\Phi\(w\)\)\}\{\\theta\(w\)\}=\\beta\_\{1\}^\{2\}\\frac\{2\-w\}\{2\\sqrt\{\\beta\_\{2\}\}\-\\bigl\[1\-2\(1\-\\sqrt\{\\beta\_\{2\}\}\)/w\_\{\\max\}\\bigr\]w\}\.A direct differentiation gives

R′​\(w\)=2​β12​\(1−β2\)​\(1−2/wmax\)\{2​β2−\[1−2​\(1−β2\)/wmax\]​w\}2\.R^\{\\prime\}\(w\)=\\frac\{2\\beta\_\{1\}^\{2\}\(1\-\\sqrt\{\\beta\_\{2\}\}\)\(1\-2/w\_\{\\max\}\)\}\{\\bigl\\\{2\\sqrt\{\\beta\_\{2\}\}\-\\bigl\[1\-2\(1\-\\sqrt\{\\beta\_\{2\}\}\)/w\_\{\\max\}\\bigr\]w\\bigr\\\}^\{2\}\}\.Sincewmax<2w\_\{\\max\}<2, one hasR′​\(w\)<0R^\{\\prime\}\(w\)<0\. Moreover,

limw↓0R⁡\(w\)=β12β2<1\\lim\_\{w\\downarrow 0\}R\(w\)=\\frac\{\\beta\_\{1\}^\{2\}\}\{\\sqrt\{\\beta\_\{2\}\}\}<1by the assumptionβ2\>β12\\sqrt\{\\beta\_\{2\}\}\>\\beta\_\{1\}^\{2\}\. Hence

β12​θ​\(Φ⁡\(w\)\)<θ⁡\(w\),0<w≤wmax\.\\beta\_\{1\}^\{2\}\\theta\(\\Phi\(w\)\)<\\theta\(w\),\\qquad 0<w\\leq w\_\{\\max\}\.Define

y⁡\(w\):=β1​θ⁡\(w\)​θ​\(Φ⁡\(w\)\)\.y\(w\):=\\beta\_\{1\}\\sqrt\{\\theta\(w\)\\theta\(\\Phi\(w\)\)\}\.Then

β12​θ​\(Φ⁡\(w\)\)<y⁡\(w\)<θ⁡\(w\),0<w≤wmax\.\\beta\_\{1\}^\{2\}\\theta\(\\Phi\(w\)\)<y\(w\)<\\theta\(w\),\\qquad 0<w\\leq w\_\{\\max\}\.For any valid transitionw↦uw\\mapsto u, monotonicity ofθ\\thetaandu≤Φ⁡\(w\)u\\leq\\Phi\(w\)give

y⁡\(w\)\>β12​θ​\(u\),y⁡\(u\)<θ⁡\(u\)\.y\(w\)\>\\beta\_\{1\}^\{2\}\\theta\(u\),\\qquad y\(u\)<\\theta\(u\)\.The exact one\-step diagonal quadratic criterion therefore yields

A​\(u\)⊤​P​\(y⁡\(u\)\)​A​\(u\)≺P⁡\(y⁡\(w\)\),P⁡\(y\):=diag⁡\(1,y\)\.A\(u\)^\{\\top\}P\(y\(u\)\)A\(u\)\\prec P\(y\(w\)\),\\qquad P\(y\):=\\operatorname\{diag\}\(1,y\)\.Set

Et:=zt⊤​P​\(y⁡\(wt\)\)​zt=xt2\+y⁡\(wt\)c2​\(mt\+c​xt\)2\.E\_\{t\}:=z\_\{t\}^\{\\top\}P\(y\(w\_\{t\}\)\)z\_\{t\}=x\_\{t\}^\{2\}\+\\frac\{y\(w\_\{t\}\)\}\{c^\{2\}\}\(m\_\{t\}\+cx\_\{t\}\)^\{2\}\.It follows thatEt\+1≤EtE\_\{t\+1\}\\leq E\_\{t\}, with strict inequality wheneverzt≠0z\_\{t\}\\neq 0\. In particular,xt2≤Et≤E0x\_\{t\}^\{2\}\\leq E\_\{t\}\\leq E\_\{0\}\.

LetM:=max⁡\{v0,E0\}M:=\\max\\\{v\_\{0\},E\_\{0\}\\\}\. The second\-moment recursion then gives inductivelyvt≤Mv\_\{t\}\\leq M, and hence

0<w¯:=c​ηM\+ε≤wt≤wmax<2\.0<\\underline\{w\}:=\\frac\{c\\eta\}\{\\sqrt\{M\}\+\\varepsilon\}\\leq w\_\{t\}\\leq w\_\{\\max\}<2\.Thus all transitions of the trajectory lie in the compact set

𝒦:=\{\(w,u\):w¯≤w,u≤wmax,u≤Φ\(w\)\}\.\\mathcal\{K\}:=\\\{\(w,u\):\\underline\{w\}\\leq w,u\\leq w\_\{\\max\},\\;u\\leq\\Phi\(w\)\\\}\.The strict quadratic inequality is continuous on𝒦\\mathcal\{K\}\. Consequently, there existsq∈\(0,1\)q\\in\(0,1\)such thatEt\+1≤q2​EtE\_\{t\+1\}\\leq q^\{2\}E\_\{t\}, and thereforeEt≤q2​t​E0E\_\{t\}\\leq q^\{2t\}E\_\{0\}\. Sinceyyhas a positive minimum on\[w¯,wmax\]\[\\underline\{w\},w\_\{\\max\}\], bothxtx\_\{t\}andξt\\xi\_\{t\}, and hencemtm\_\{t\}, decay geometrically\.

Finally,

vt=β2t​v0\+\(1−β2\)​∑j=0t−1β2t−1−j​xj2,v\_\{t\}=\\beta\_\{2\}^\{t\}v\_\{0\}\+\(1\-\\beta\_\{2\}\)\\sum\_\{j=0\}^\{t\-1\}\\beta\_\{2\}^\{t\-1\-j\}x\_\{j\}^\{2\},and the geometric bound onxj2x\_\{j\}^\{2\}implies geometric decay ofvtv\_\{t\}\. Thus there existC\>0C\>0andγ0∈\(0,1\)\\gamma\_\{0\}\\in\(0,1\)such that\|xt\|\+\|mt\|\+vt≤C​γ0t\\left\\lvert x\_\{t\}\\right\\rvert\+\\left\\lvert m\_\{t\}\\right\\rvert\+v\_\{t\}\\leq C\\gamma\_\{0\}^\{t\}\. Hence\(xt,mt,vt\)→\(0,0,0\)\(x\_\{t\},m\_\{t\},v\_\{t\}\)\\to\(0,0,0\)and, consequently,wt→wmaxw\_\{t\}\\to w\_\{\\max\}\. ∎

## Appendix HExperimental details for Figure[1](https://arxiv.org/html/2608.20638#S1.F1)

*Left, top\.*Standard \(bias\-corrected\) full\-batch Adam withβ1=0\.9\\beta\_\{1\}=0\.9,β2=0\.999\\beta\_\{2\}=0\.999,η=10−2\\eta=10^\{\-2\},ε=10−8\\varepsilon=10^\{\-8\}trains a two\-layer networkx↦W2⊤​tanh⁡\(W1​x\+b1\)\+b2x\\mapsto W\_\{2\}^\{\\top\}\\tanh\(W\_\{1\}x\+b\_\{1\}\)\+b\_\{2\}with1616hidden units on the squared loss for the regression tasky=sin⁡\(2​x\)y=\\sin\(2x\)over3232equispaced inputs in\[−2,2\]\[\-2,2\]\. Every4040steps the Hessian is computed by central finite differences and we plot the preconditioned sharpness𝖲t\\mathsf\{S\}\_\{t\}of \([18](https://arxiv.org/html/2608.20638#A2.E18)\), withvtv\_\{t\}replaced by its bias\-corrected estimate;𝖲t\\mathsf\{S\}\_\{t\}equilibrates near the frozen threshold𝖲⋆=2​\(1\+β1\)/\(η⁡\(1−β1\)\)=38/η\\mathsf\{S\}^\{\\star\}=2\(1\+\\beta\_\{1\}\)/\\bigl\(\\eta\(1\-\\beta\_\{1\}\)\\bigr\)=38/\\eta, the adaptive edge of stability of[12](https://arxiv.org/html/2608.20638#bib.bib2)\.

*Left, bottom\.*Uncorrected Adam \([2](https://arxiv.org/html/2608.20638#S1.E2)\) on the quadratic \([1](https://arxiv.org/html/2608.20638#S1.E1)\) with the same\(β1,β2\)\(\\beta\_\{1\},\\beta\_\{2\}\)andη=1\\eta=1,wmax=10w\_\{\\max\}=10\(equivalentlyε=c​η/10\\varepsilon=c\\eta/10\),x0=0\.1x\_\{0\}=0\.1,m0=0m\_\{0\}=0,v0=0\.01v\_\{0\}=0\.01; the normalized sharpnesswtw\_\{t\}oscillates around the parameter\-free boundaryw=2w=2\.

*Right\.*The negative\-feedback loop: supercritical steps expand the state and inflatevtv\_\{t\}, pushingwtw\_\{t\}down, while subcritical steps contract the state and deflatevtv\_\{t\}, pushingwtw\_\{t\}up\. Forβ1=0\\beta\_\{1\}=0, Appendix[D](https://arxiv.org/html/2608.20638#A4)gives explicit finite\-step bounds for both transitions\.