When Labels Are Scarce: An Oscillatory State Space Model for Vibration Diagnosis

# When Labels Are Scarce: An Oscillatory State Space Model for Vibration Diagnosis
Source: [https://arxiv.org/html/2609.27411](https://arxiv.org/html/2609.27411)
###### Abstract

Machine fault diagnosis from vibration requires learning from scarce labelled fault recordings while meeting the computational constraints of edge devices for local inference\. We introduce DualRes, a compact oscillatory state\-space model that combines two complementary spectral views of vibration, capturing rapid changes and fine frequency structure\. Time\-aligned views are processed by selective oscillatory memory, which learns how long to retain temporal patterns\. The encoder contains 39,528 parameters\. We evaluate supervised learning across six bearing datasets and a gearbox benchmark, with an additional gearbox pilot\. Recording\-level splits and explicit accounting of labelled duration distinguish data efficiency from repeated exposure to correlated samples\. On the main gearbox benchmark, DualRes achieves state\-of\-the\-art performance among the nine evaluated methods at six of seven label budgets\. With about six labelled seconds per class, it improves macro\-F1 by 16\.1 percentage points over the next strongest comparator\. On the same benchmark, DualRes achieves a 1\.44\-fold recording\-level speedup and a 24\.8\-fold reduction in checkpoint storage relative to a selective state\-space baseline under matched hardware and runtime conditions\. Bearing results reveal task\-dependent trade\-offs\. These findings support oscillatory memory as a compact approach to vibration diagnosis under limited labelled exposure\.

## 1Introduction

Vibration sensors collect abundant operational data, but labelled recordings of representative faults remain scarce\. Windows extracted from the same recording share a component, mounting configuration and operating condition\. A useful classifier must learn from limited labelled exposure and remain effective when these factors change\. Deployment on resource\-constrained edge hardware imposes limits on model storage, memory and inference latency\.

Evaluation design is therefore central to the problem\. Randomly assigning windows from one recording to training and evaluation sets can place near\-identical signals in both; even recording\-disjoint splits may contain measurements from the same physical bearing\.[Vieira et al\. \(2026\)](https://arxiv.org/html/2609.27411#bib.bib17)document the effects of partitioning and bearing diversity\. We report the held\-out unit and unique labelled duration explicitly, and distinguish condition transfer from unseen\-component generalization\.

This raises two linked questions: which compact representation captures diagnostic vibration structure, and how should its data efficiency be evaluated? Model size alone answers neither question\. A compact classifier still needs to distinguish faults under changing conditions, and its execution cost depends on the signal processing and inference pipeline as well as its parameters\.

Mechanical vibration combines local excitations, resonant responses and temporal modulation\([Randall and Antoni, 2011](https://arxiv.org/html/2609.27411#bib.bib9)\)\. Short analysis windows localize changes; longer windows discriminate nearby frequencies\. We align both resolutions at common frame centers and model their evolution with selectively damped oscillatory memory\. We call this model DualRes, referring to its two aligned spectral resolutions\. Figure[1](https://arxiv.org/html/2609.27411#S1.F1)summarizes the design and its performance across labelled\-data budgets\.

![Refer to caption](https://arxiv.org/html/2609.27411v1/Arch_figures/Fig_top.png)Figure 1:Paper overview\. Left: representative sequence classifiers and DualRes’s aligned spectral views with oscillatory memory\. Right: MCC5 macro\-F1 across seven label budgets under the common supervised protocol\.Our contributions are threefold\.Architecture:a center\-aligned multi\-resolution classifier combines selective oscillatory memory and normalized state updates in a39,52839\{,\}528\-parameter encoder\.Evaluation:recording\-level splits and explicit duration accounting distinguish labelled exposure, recording coverage and operating\-condition shift\.Empirical analysis:comparisons, learning curves, component ablations and matched desktop measurements characterize the accuracy–cost trade\-off across bearing and gearbox tasks\.

## 2Related Work

#### Vibration and evaluation\.

Classical bearing analysis uses recurring excitations and resonances\([Randall and Antoni, 2011](https://arxiv.org/html/2609.27411#bib.bib9)\); public benchmarks enable data\-driven evaluation\([Lessmeier et al\., 2016](https://arxiv.org/html/2609.27411#bib.bib10);[Vieira et al\., 2026](https://arxiv.org/html/2609.27411#bib.bib17)\)\. DualRes learns from amplitudes produced by a fixed short\-time Fourier transform \(STFT\), without fault\-order templates or an assumption of speed invariance\. Unlike auxiliary\-task meta\-learning\([Snell et al\., 2017](https://arxiv.org/html/2609.27411#bib.bib7);[Finn et al\., 2017](https://arxiv.org/html/2609.27411#bib.bib8)\), our supervised setting restricts fitting to the stated labelled supports\.

#### Sequence representations\.

Structured and complex\-valued recurrent models, including S4, S4D, S5 and the linear recurrent unit \(LRU\), establish compact alternatives to attention\([Gu et al\., 2022a](https://arxiv.org/html/2609.27411#bib.bib6);[Gu et al\., 2022b](https://arxiv.org/html/2609.27411#bib.bib23);[Smith et al\., 2023](https://arxiv.org/html/2609.27411#bib.bib24);[Orvieto et al\., 2023](https://arxiv.org/html/2609.27411#bib.bib25)\); coRNN develops coupled oscillatory recurrence\([Rusch and Mishra, 2021](https://arxiv.org/html/2609.27411#bib.bib26)\)\. Mamba introduces selective updates, and Mamba\-2 relates state\-space models \(SSMs\) to structured attention\([Gu and Dao, 2024](https://arxiv.org/html/2609.27411#bib.bib1);[Dao and Gu, 2024](https://arxiv.org/html/2609.27411#bib.bib27)\)\. MambaSL adapts a single Mamba layer for time\-series classification, while TSCMamba combines wavelet and temporal views\([Jung and Kim, 2026](https://arxiv.org/html/2609.27411#bib.bib18);[Ahamed and Cheng, 2025](https://arxiv.org/html/2609.27411#bib.bib19)\)\. Our contribution is the specific combination of aligned spectral resolutions and normalized selective oscillatory updates, rather than complex states or time–frequency fusion alone\.

Attention models provide different ways to organize a sequence: PatchTST operates on patches, iTransformer on variates, and Crossformer on cross\-dimension structure\([Nie et al\., 2023](https://arxiv.org/html/2609.27411#bib.bib2);[Liu et al\., 2024](https://arxiv.org/html/2609.27411#bib.bib32);[Zhang and Yan, 2023](https://arxiv.org/html/2609.27411#bib.bib34)\); Medformer and ModernTCN provide multi\-granularity attention and convolutional alternatives\([Wang et al\., 2024](https://arxiv.org/html/2609.27411#bib.bib20);[Luo and Wang, 2024](https://arxiv.org/html/2609.27411#bib.bib21)\)\. ROCKET and MiniRocket classify sequences using convolutional feature transforms\([Dempster et al\., 2020](https://arxiv.org/html/2609.27411#bib.bib33);[Dempster et al\., 2021](https://arxiv.org/html/2609.27411#bib.bib22)\)\. The comparison spans these distinct model families, following classification benchmarking practice\([Middlehurst et al\., 2024](https://arxiv.org/html/2609.27411#bib.bib29)\)\. Appendix[D](https://arxiv.org/html/2609.27411#A4)documents their classification adaptations\.

#### Selection and deployment\.

Repeated evaluation can bias model selection even without within\-run leakage\([Cawley and Talbot, 2010](https://arxiv.org/html/2609.27411#bib.bib31)\)\. Research on resource\-constrained inference distinguishes parameter counts, activation memory and measured execution cost\([Lin et al\., 2020](https://arxiv.org/html/2609.27411#bib.bib3);[Cai et al\., 2020](https://arxiv.org/html/2609.27411#bib.bib4);[Banbury et al\., 2021](https://arxiv.org/html/2609.27411#bib.bib5)\); we therefore measure complete inference pipelines under matched conditions\.

## 3Multi\-Resolution Oscillatory State Space Model

DualRes separates local spectral analysis from temporal memory \(Figure[2](https://arxiv.org/html/2609.27411#S3.F2)\)\. It uses vibration and the acquisition sampling rate, with no operating\-condition inputs\. Appendix[A](https://arxiv.org/html/2609.27411#A1)provides full parameterizations, initialization and implementation details\.

![Refer to caption](https://arxiv.org/html/2609.27411v1/Arch_figures/Fig2.png)Figure 2:Overall architecture: two center\-aligned magnitude STFTs feed learned projections, an oscillatory selective scan, a gated residual readout and temporal mean pooling\. The classifier is trained directly on labelled vibration\.### 3\.1Aligned spectral representation

The frontend represents the same signal at two resolutions: one localizes short\-lived changes, while the other distinguishes more closely spaced frequencies\. Aligning their frame centers lets the memory process both views as one observation at each time step\. A windowx∈ℝLx\\in\\mathbb\{R\}^\{L\}is resampled tofs=64f\_\{s\}=64kHz, centered and divided by its root\-mean\-square \(RMS\) amplitude, usingL=32768L=32768\(0\.5120\.512s\)\. Normalization removes absolute amplitude while retaining within\-window variation\. LetX\(N\)j,:X^\{\(N\)\}\_\{j,:\}be its one\-sided, unnormalized STFT with periodic Hann window, no padding and hopH=128H=128\. We useNs=256N\_\{s\}=256andNl=1024N\_\{l\}=1024, providing44and1616ms supports with250250and62\.562\.5Hz bin spacing\([Harris, 1978](https://arxiv.org/html/2609.27411#bib.bib30)\)\.

Croppingκ=\(Nl−Ns\)/\(2​H\)=3\\kappa=\(N\_\{l\}\-N\_\{s\}\)/\(2H\)=3frames from each end of the short sequence aligns its centers with those of the long sequence:

\(j\+κ\)​H\+Ns/2=j​H\+Nl/2=τj,0≤j<T\.\(j\+\\kappa\)H\+N\_\{s\}/2=jH\+N\_\{l\}/2=\\tau\_\{j\},\\qquad 0\\leq j<T\.\(1\)This givesT=249T=249aligned pairs at token ratefz=fs/H=500f\_\{z\}=f\_\{s\}/H=500Hz\. Definesj=log\(1\+\|Xj\+κ,:\(Ns\)\|\)s\_\{j\}=\\log\(1\+\|X^\{\(N\_\{s\}\)\}\_\{j\+\\kappa,:\}\|\)andlj=log\(1\+\|Xj,:\(Nl\)\|\)l\_\{j\}=\\log\(1\+\|X^\{\(N\_\{l\}\)\}\_\{j,:\}\|\)\. Affine projections form

ej=\[Wssj\+as∥Wllj\+al\]∈ℝD,D=64,e\_\{j\}=\[W\_\{s\}s\_\{j\}\+a\_\{s\}\\;\\\|\\;W\_\{l\}l\_\{j\}\+a\_\{l\}\]\\in\\mathbb\{R\}^\{D\},\\qquad D=64,\(2\)withD/2D/2channels per branch\. All frequency bins are retained, but magnitudes discard STFT phase\. Figure[7](https://arxiv.org/html/2609.27411#A1.F7)illustrates the alignment\.

![Refer to caption](https://arxiv.org/html/2609.27411v1/Arch_figures/Fig4.png)Figure 3:Oscillatory memory rotates and damps the previous state, adds a normalized write, and reads the real projection\. The lower panel shows its equivalent real\-valued scan implementation; the illustrated spiral is schematic\.
### 3\.2Selective oscillatory memory

State rotation represents oscillatory dynamics between frames, while input\-dependent damping controls memory retention\. The input write is weighted by one minus the retention coefficient, coupling new information to state decay\. Letvt=LayerNorm⁡\(et\)v\_\{t\}=\\operatorname\{LayerNorm\}\(e\_\{t\}\)\. Learned affine maps produce inpututu\_\{t\}, positive damping incrementsδt,d\\delta\_\{t,d\}, and bounded complex write/read vectorsbt,k,ct,kb\_\{t,k\},c\_\{t,k\}\. Each channelddmaintainsK=8K=8complex states, initialized at zero:

qt,d,k=αt,d​e−i​θk​qt−1,d,k\+\(1−αt,d\)​bt,k​ut,d,αt,d=e−δt,d\.q\_\{t,d,k\}=\\alpha\_\{t,d\}e^\{\-i\\theta\_\{k\}\}q\_\{t\-1,d,k\}\+\(1\-\\alpha\_\{t,d\}\)b\_\{t,k\}u\_\{t,d\},\\qquad\\alpha\_\{t,d\}=e^\{\-\\delta\_\{t,d\}\}\.\(3\)Hereθk=2​π​fk/fz\\theta\_\{k\}=2\\pi f\_\{k\}/f\_\{z\}, with learned frequenciesfk∈\(0,fz/2\)f\_\{k\}\\in\(0,f\_\{z\}/2\)shared across channels\. These describe dynamics on the*spectral\-token clock*, not raw carrier frequencies\. The read/write vectors are shared across channels; states and damping are channel\-specific\. Figure[3](https://arxiv.org/html/2609.27411#S3.F3)illustrates the mechanism\.

As damping vanishes, the normalized write also diminishes\. Bounded writes yield bounded states by induction \(Appendix[A](https://arxiv.org/html/2609.27411#A1)\)\.

The real readout and gated residual are

yt,d\\displaystyle y\_\{t,d\}=∑k=1KRe⁡\(ct,k¯​qt,d,k\),\\displaystyle=\\sum\_\{k=1\}^\{K\}\\operatorname\{Re\}\(\\overline\{c\_\{t,k\}\}q\_\{t,d,k\}\),ot\\displaystyle o\_\{t\}=et\+\(Wm​yt\+am\)⊙SiLU⁡\(Wg​yt\+ag\)\.\\displaystyle=e\_\{t\}\+\(W\_\{m\}y\_\{t\}\+a\_\{m\}\)\\odot\\operatorname\{SiLU\}\(W\_\{g\}y\_\{t\}\+a\_\{g\}\)\.\(4\)The gate transforms the memory readout; the residual preserves fused input tokens\. The official real selective\-scan kernel\([Gu and Dao, 2024](https://arxiv.org/html/2609.27411#bib.bib1);[Gu and Dao, 2026](https://arxiv.org/html/2609.27411#bib.bib16)\)implements Eq\.[3](https://arxiv.org/html/2609.27411#S3.E3)via rotating coordinates, without an additional standard Mamba block\.

### 3\.3Learning and cost

Temporal mean pooling giveso¯=T−1​∑tot\\bar\{o\}=T^\{\-1\}\\sum\_\{t\}o\_\{t\}and logitsℓ=Wh​Dropout0\.1⁡\(o¯\)\+ah\\ell=W\_\{h\}\\operatorname\{Dropout\}\_\{0\.1\}\(\\bar\{o\}\)\+a\_\{h\}\. All parameters are trained jointly from labelled support𝒮\\mathcal\{S\}using

ℒ=−\|𝒮\|−1∑\(x,y\)∈𝒮logsoftmax\(ℓ\(x\)\)y\.\\mathcal\{L\}=\-\|\\mathcal\{S\}\|^\{\-1\}\\sum\_\{\(x,y\)\\in\\mathcal\{S\}\}\\log\\operatorname\{softmax\}\(\\ell\(x\)\)\_\{y\}\.\(5\)Training also applies positive multiplicative damping jitter \(σ=0\.10\\sigma=0\.10\), shared across time within each window/channel; jitter and dropout are disabled at inference\. The encoder has39,52839\{,\}528parameters and the head adds65​C65CforCCclasses\. The scan costsO⁡\(T​D​K\)O\(TDK\); fixed\-resolution frontend and pointwise operations also scale linearly in sequence length\. Measured latency and memory are reported separately from these analytical costs\.

## 4Benchmark Design and Evaluation Protocol

We evaluate how diagnostic performance changes with the amount and distribution of labelled vibration\. The source support is the labelled data available for fitting; evaluation recordings are kept separate before any windows are extracted\.

#### Study roadmap\.

We compare nine architectures under common label budgets, then repeat selected comparisons across three seeds\. Allocation controls separate recording diversity from duration, and a second gearbox dataset tests the selected training recipe\. Component ablations and matched desktop profiling assess architectural contributions and execution costs\. Appendix[E](https://arxiv.org/html/2609.27411#A5)details replication coverage\.

#### Tasks and split units\.

We use six public bearing datasets to test changes in component identity, operating conditions, geometry and speed profiles: Paderborn, Case Western Reserve University \(CWRU\), KAIST, UORED\-VAFCLS \(abbreviated UORED below\), HUST and Ottawa\([Lessmeier et al\., 2016](https://arxiv.org/html/2609.27411#bib.bib10);[Case Western Reserve University, 2026](https://arxiv.org/html/2609.27411#bib.bib11);[Jung et al\., 2023](https://arxiv.org/html/2609.27411#bib.bib12);[Sehri and Dumond, 2023](https://arxiv.org/html/2609.27411#bib.bib13);[Nguyen Duc Thuan and Hoang Si Hong, 2023](https://arxiv.org/html/2609.27411#bib.bib14);[Huang and Baddour, 2019](https://arxiv.org/html/2609.27411#bib.bib15)\)\. The eight\-class MCC5\-THU gearbox benchmark \(MCC5 below\) varies motion mode and fault severity\([Chen et al\., 2024](https://arxiv.org/html/2609.27411#bib.bib28)\)\. A separate two\-model pilot uses the labelled release of the 2009 Prognostics and Health Management gearbox challenge \(PHM2009\) for 14\-way configuration recognition\([PHM Society, 2009b](https://arxiv.org/html/2609.27411#bib.bib35);[PHM Society, 2009a](https://arxiv.org/html/2609.27411#bib.bib36)\)\. Paderborn/UORED hold out bearing identities; CWRU/KAIST hold out conditions; HUST holds out geometries; Ottawa holds out speed profiles\. MCC5 has two directions: low\-severity torque circulation to speed circulation \(M1\), and high\-severity speed circulation to torque circulation \(M2\)\. Target pools include available severities, so motion and severity both vary\. The dataset inventory, manifests and detailed split rules are in Appendices[B](https://arxiv.org/html/2609.27411#A2)and[C](https://arxiv.org/html/2609.27411#A3)\.

#### Label budgets\.

MCC5 usesr∈\{1,2,3,4,6\}r\\in\\\{1,2,3,4,6\\\}recordings/class and four non\-overlapping0\.5120\.512s windows/record:2\.0482\.048–12\.28812\.288labelled seconds/class\. The primary budget is3×43\\times 4windows \(6\.1446\.144s/class\)\. A matched\-duration pilot compares1×121\\times 12,3×43\\times 4and6×26\\times 2to study recording diversity\. The extension retains six recordings and increases windows/record to eight or twelve \(24\.57624\.576/36\.86436\.864s/class\)\. These nested supports separate the fixed\-pool duration study from joint diversity/exposure scaling\. The bearing protocol uses three overlapping windows covering1\.0241\.024unique seconds/class\.

PHM2009 reuses the primary support budget and training duration; Appendix[B](https://arxiv.org/html/2609.27411#A2)specifies its repeat\-separated folds\.

#### Fitting and evaluation\.

Methods receive identical supports and vibration channels within each comparison\. All models are fitted directly on labelled source support without pretraining\. Neural MCC5/bearing runs use300300/100100Adam updates, respectively; classical methods retain their own fitting procedures\. Appendix[C\.3](https://arxiv.org/html/2609.27411#A3.SS3)gives optimizer settings and gradient accumulation details\. More support increases computation even at fixed update count\. Terminal checkpoints avoid target\-based early stopping; repeated development on these datasets still limits claims of independent confirmation\([Cawley and Talbot, 2010](https://arxiv.org/html/2609.27411#bib.bib31)\)\. The primary metric is macro\-F1, the unweighted mean of per\-class F1 scores; accuracy is secondary\. Both use recording\-level predictions and are averaged equally across folds\. Neural models average window probabilities; MiniRocket averages decision scores\.

#### Comparisons and costs\.

Appendix[D](https://arxiv.org/html/2609.27411#A4)records classification adaptations of the released implementations\. Efficiency reporting separates parameters, storage, training time, memory and inference latency; Appendix[I\.2](https://arxiv.org/html/2609.27411#A9.SS2)defines the measured workloads and hardware scope\.

## 5Results and Analysis

### 5\.1Predictive performance under limited labels

#### Bearing diagnosis\.

At 1\.024 labelled seconds/class, DualRes exceeds MambaSL’s mean macro\-F1 on Paderborn, KAIST and UORED, ties on Ottawa, and trails on CWRU and HUST \(Table[1](https://arxiv.org/html/2609.27411#S5.T1)\)\. The largest positive difference is 7\.71 percentage points on KAIST\. UORED improves by 6\.24 points but exhibits substantial seed variability\. HUST remains difficult: MiniRocket performs best under its geometry shift\.

Table 1:Bearing macro\-F1 \(%\)\. Mean±\\pmsample SD over three seeds, each averaged over both frozen folds\. Labelled exposure: 1\.024 unique seconds/class; 100 supervised updates\. Bold: best three\-seed mean within each dataset\.iTransformer is additionally replicated on HUST \(Table[15](https://arxiv.org/html/2609.27411#A5.T15)\); other iTransformer datasets have one seed and are not included in this three\-seed table\.

#### Gearbox diagnosis across motion modes\.

DualRes leads the nine\-method MCC5 comparison at six of seven label budgets; MiniRocket leads at 2\.048 seconds/class \(Table[2](https://arxiv.org/html/2609.27411#S5.T2)\)\. In the replicated comparison, DualRes exceeds the stronger of MambaSL and MiniRocket by 16\.1 points at the primary budget and 11\.9 at the largest\. Appendix[E](https://arxiv.org/html/2609.27411#A5)gives accuracy, per\-fold results and repetitions\.

Table 2:MCC5 macro\-F1 \(%\): two\-fold means, seed 41\. Column headers are unique labelled seconds/class\. Neural methods use 300 terminal updates; MiniRocket uses its source\-fitted ridge pipeline\. Bold: best completed value per budget\.First five columns: 1/2/3/4/6 recordings×\\times4 windows\. Last two: 6 recordings×\\times8/12 windows\.

#### A separate gearbox pilot\.

The unchanged architecture and primary training recipe yield a 24\.91\-point macro\-F1 advantage over MambaSL on PHM2009, with gains in both folds \(Table[3](https://arxiv.org/html/2609.27411#S5.T3)\)\.

Table 3:PHM2009 configuration\-recognition pilot \(%\)\. Seed 41, 3 records×\\times4 disjoint windows/class, 300 updates; 140 evaluation recordings per fold\. Same architecture and recipe; no PHM model selection\.

### 5\.2How should a limited label budget be allocated?

Figure 4:Three different data\-efficiency questions on MCC5\. \(a\) Increase recordings and duration jointly\. \(b\) Increase duration in the same six\-record pool\. \(c\) Redistribute the same 12 windows across recordings\. Panels \(a,b\): markers show three\-seed means and shaded bands show one sample SD; Panel \(c\): seed\-41 bars\. Every value averages both folds\.DualRes improves from 22\.29% to 51\.12% mean macro\-F1 as coverage grows from one to six recordings/class \(Figure[4](https://arxiv.org/html/2609.27411#S5.F4)a\)\. This changes duration, diversity and computation together\. At fixed six\-record coverage, doubling duration gives little improvement, while the largest budget reaches 54\.74%\. More exposure is useful, but its effect is not monotonic\.

The fixed\-exposure control separates allocation from quantity\. Both methods peak at the3×43\\times 4allocation \(Figure[4](https://arxiv.org/html/2609.27411#S5.F4)c\)\. Thus balanced within\-record coverage and between\-record diversity outperform either extreme in this control\.

### 5\.3What do the components contribute?

Table 4:Component ablations: recording macro\-F1 \(%\), seed 41, both folds\. Bearing support 1\.024 s/class; MCC5 support 6\.144 s/class\. Bold: highest score in each row; all interventions are retained\.Matched controls remove a projected resolution, oscillator rotation or damping jitter while retaining the backbone and training recipe \(Table[4](https://arxiv.org/html/2609.27411#S5.T4)\)\. Removing rotation reduces MCC5 macro\-F1 by 8\.73 points and UORED by 27\.18 points, but slightly improves Paderborn\. The full frontend improves over either single\-resolution control on CWRU, UORED and HUST; the long\-only control is better on KAIST and MCC5\. Jitter has no consistent advantage\. Appendix[H](https://arxiv.org/html/2609.27411#A8)details the controls and their effects across datasets\.

### 5\.4How do the models optimize under the same label budget?

Figure 5:Training cross\-entropy for eight neural methods, seed 41, without smoothing, at 12\.288 labelled seconds/class \(six recordings, four windows each\)\. Table[2](https://arxiv.org/html/2609.27411#S5.T2)reports held\-out performance\.Several comparators reach near\-zero training loss before DualRes yet attain lower macro\-F1 on held\-out data \(Figure[5](https://arxiv.org/html/2609.27411#S5.F5); Table[2](https://arxiv.org/html/2609.27411#S5.T2)\)\. Lower training loss therefore does not necessarily imply better generalization\. Appendix[G\.2](https://arxiv.org/html/2609.27411#A7.SS2)documents source\-only duration selection\.

### 5\.5Accuracy, storage and measured execution cost

Figure 6:MCC5 primary setting, seed 41: predictive quality versus \(a\) registered parameters and \(b\) matched batch\-one window latency\. Dashed segments join the nondominated observed neural models\. Latency includes model\-specific preprocessing and transfers on one RTX A4000, FP32 with TF32 off\. MiniRocket’s CPU costs are reported separately\.DualRes has 40,048 parameters including the gearbox classification head, versus MambaSL’s 15,592\. Its checkpoint nevertheless requires 24\.8\-fold less storage \(0\.164 versus 4\.065 MiB\), since checkpoint size also depends on saved buffers\.

On MCC5, DualRes takes 1\.75 ms/window versus 2\.00 ms for MambaSL, and 196\.65 versus 283\.85 ms per native recording \(1\.44\-fold speedup\)\. Some attention models are faster but have lower macro\-F1 \(Figure[6](https://arxiv.org/html/2609.27411#S5.F6)\); Appendix[I\.2](https://arxiv.org/html/2609.27411#A9.SS2)gives all seven datasets’ measurements\.

Table 5:Seven\-dataset window costs: median \[range\] of dataset latency medians and maximum peak allocated CUDA memory\. RTX A4000, batch one, FP32\. Bold: lowest summary per metric\. Appendix[I\.2](https://arxiv.org/html/2609.27411#A9.SS2)gives individual datasets and CPU MiniRocket\.Table[5](https://arxiv.org/html/2609.27411#S5.T5)summarizes costs across all seven datasets\.

## 6Conclusion and Future Work

We presented DualRes, a compact vibration classifier that combines aligned multi\-resolution spectra with selective oscillatory memory\. The two spectral views retain complementary temporal and frequency structure, while learned rotation and damping model their evolution in a 39,528\-parameter encoder\. Across six bearing datasets and the main gearbox benchmark, the results characterize the accuracy–cost trade\-offs of learning from limited labelled recordings\. DualRes leads the nine\-method gearbox comparison at six of seven exposure budgets; bearing performance varies with the task and evaluation split\. The accompanying benchmark combines recording\-level separation, explicit labelled\-duration accounting, controlled support allocation and matched inference measurements\. These studies examine both the amount of labelled data and its allocation across recording conditions, alongside the execution costs of each model\.

Future work will investigate adaptive spectral resolution for changing signal characteristics and streaming oscillatory inference for continuous monitoring\. Additional machine types and independently collected operating conditions will test how broadly the learned representations generalize\.

## Reproducibility, Ethics and AI\-Use Statements

Appendix[J](https://arxiv.org/html/2609.27411#A10)documents the evidence artifacts and reproduction entry points\. Dataset redistribution remains governed by each source’s terms\. The experiments use machine signals rather than personal data; deployment would require application\-specific failure analysis\. AI tools assisted code development, analysis and manuscript preparation\. The authors are responsible for verifying the implementation, numerical evidence, citations and final claims\.

## References

- Ahamed and Cheng \(2025\)M\. A\. Ahamed and Q\. ChengTSCMamba: mamba meets multi\-view learning for time series classification\.Information Fusion120,pp\. 103079\.External Links:[Document](https://dx.doi.org/10.1016/j.inffus.2025.103079),[Link](https://doi.org/10.1016/j.inffus.2025.103079)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p1.1)\.
- Banburyet al\.\(2021\)C\. Banbury, V\. J\. Reddi, P\. Torelli, J\. Holleman, N\. Jeffries, C\. Kiraly, P\. Montino, D\. Kanter, S\. Ahmed, D\. Pau, U\. Thakker, A\. Torrini, P\. Warden, J\. Cordaro, G\. Di Guglielmo, J\. Duarte, S\. Gibellini, V\. Parekh, H\. Tran, N\. Tran, W\. Niu, and X\. XuMLPerf Tiny benchmark\.InProceedings of the Neural Information Processing Systems Track on Datasets and Benchmarks,Vol\.1\.External Links:[Link](https://datasets-benchmarks-proceedings.neurips.cc/paper_files/paper/2021/hash/da4fb5c6e93e74d3df8527599fa62642-Abstract-round1.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px3.p1.1)\.
- Caiet al\.\(2020\)H\. Cai, C\. Gan, L\. Zhu, and S\. HanTinyTL: reduce memory, not parameters for efficient on\-device learning\.InAdvances in Neural Information Processing Systems,Vol\.33\.External Links:[Link](https://proceedings.neurips.cc/paper/2020/hash/81f7acabd411274fcf65ce2070ed568a-Abstract.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px3.p1.1)\.
- Case Western Reserve University \(2026\)Case Western Reserve UniversityBearing data center: download a data file\.Note:Accessed September 2026; 48 kHz drive\-end releaseExternal Links:[Link](https://engineering.case.edu/bearingdatacenter/download-data-file)Cited by:[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px2.p1.1)\.
- Cawley and Talbot \(2010\)G\. C\. Cawley and N\. L\. C\. TalbotOn over\-fitting in model selection and subsequent selection bias in performance evaluation\.Journal of Machine Learning Research11,pp\. 2079–2107\.External Links:[Link](https://jmlr.org/papers/v11/cawley10a.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px3.p1.1),[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px4.p1.1)\.
- Chenet al\.\(2024\)S\. Chen, Z\. Liu, X\. He, D\. Zou, and D\. ZhouMulti\-mode fault diagnosis datasets of gearbox under variable working conditions\.Data in Brief54,pp\. 110453\.External Links:[Document](https://dx.doi.org/10.1016/j.dib.2024.110453),[Link](https://pmc.ncbi.nlm.nih.gov/articles/PMC11070670/)Cited by:[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px2.p1.1)\.
- Dao and Gu \(2024\)T\. Dao and A\. GuTransformers are SSMs: generalized models and efficient algorithms through structured state space duality\.InProceedings of the 41st International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.235,pp\. 10041–10071\.External Links:[Link](https://proceedings.mlr.press/v235/dao24a.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p1.1)\.
- Dempsteret al\.\(2020\)A\. Dempster, F\. Petitjean, and G\. I\. WebbROCKET: exceptionally fast and accurate time series classification using random convolutional kernels\.Data Mining and Knowledge Discovery\.External Links:[Document](https://dx.doi.org/10.1007/s10618-020-00701-z),[Link](https://doi.org/10.1007/s10618-020-00701-z)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p2.1)\.
- Dempsteret al\.\(2021\)A\. Dempster, D\. F\. Schmidt, and G\. I\. WebbMINIROCKET: a very fast \(almost\) deterministic transform for time series classification\.InACM SIGKDD Conference on Knowledge Discovery and Data Mining,External Links:[Document](https://dx.doi.org/10.1145/3447548.3467231),[Link](https://doi.org/10.1145/3447548.3467231)Cited by:[§D\.2](https://arxiv.org/html/2609.27411#A4.SS2.p1.1),[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p2.1)\.
- Finnet al\.\(2017\)C\. Finn, P\. Abbeel, and S\. LevineModel\-agnostic meta\-learning for fast adaptation of deep networks\.InProceedings of the 34th International Conference on Machine Learning,Vol\.70,pp\. 1126–1135\.External Links:[Link](https://proceedings.mlr.press/v70/finn17a.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px1.p1.1)\.
- Gu and Dao \(2024\)A\. Gu and T\. DaoMamba: linear\-time sequence modeling with selective state spaces\.InConference on Language Modeling,External Links:[Link](https://openreview.net/forum?id=tEYskw1VY2)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p1.1),[§3\.2](https://arxiv.org/html/2609.27411#S3.SS2.p3.2)\.
- Gu and Dao \(2026\)A\. Gu and T\. DaoOfficial Mamba implementation\.Note:Software requirements and implementation; accessed September 2026External Links:[Link](https://github.com/state-spaces/mamba)Cited by:[§3\.2](https://arxiv.org/html/2609.27411#S3.SS2.p3.2)\.
- Guet al\.\(2022a\)A\. Gu, K\. Goel, and C\. RéEfficiently modeling long sequences with structured state spaces\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=uYLFoz1vlAC)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p1.1)\.
- Guet al\.\(2022b\)A\. Gu, A\. Gupta, K\. Goel, and C\. RéOn the parameterization and initialization of diagonal state space models\.InAdvances in Neural Information Processing Systems,External Links:[Link](https://proceedings.neurips.cc/paper_files/paper/2022/hash/e9a32fade47b906de908431991440f7c-Abstract-Conference.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p1.1)\.
- Harris \(1978\)F\. J\. HarrisOn the use of windows for harmonic analysis with the discrete fourier transform\.Proceedings of the IEEE66\(1\),pp\. 51–83\.External Links:[Link](https://web.mit.edu/xiphmont/Public/windows.pdf)Cited by:[§3\.1](https://arxiv.org/html/2609.27411#S3.SS1.p1.1)\.
- Huang and Baddour \(2019\)H\. Huang and N\. BaddourBearing vibration data under time\-varying rotational speed conditions\.Note:Mendeley Data, version 2External Links:[Document](https://dx.doi.org/10.17632/v43hmbwxpm.2),[Link](https://data.mendeley.com/datasets/v43hmbwxpm/2)Cited by:[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px2.p1.1)\.
- Junget al\.\(2023\)W\. Jung, S\. Kim, S\. Yun, J\. Bae, and Y\. ParkVibration, acoustic, temperature, and motor current dataset of rotating machine under varying load conditions for fault diagnosis\.Note:Mendeley Data, version 6External Links:[Document](https://dx.doi.org/10.17632/ztmf3m7h5x.6),[Link](https://data.mendeley.com/datasets/ztmf3m7h5x/6)Cited by:[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px2.p1.1)\.
- Jung and Kim \(2026\)Y\. Jung and L\. KimMambaSL: exploring single\-layer mamba for time series classification\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=YDl4vqQqGP)Cited by:[§D\.2](https://arxiv.org/html/2609.27411#A4.SS2.p1.1),[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p1.1)\.
- Lessmeieret al\.\(2016\)C\. Lessmeier, J\. K\. Kimotho, D\. Zimmer, and W\. SextroCondition monitoring of bearing damage in electromechanical drive systems by using motor current signals of electric motors: a benchmark data set for data\-driven classification\.InPHM Society European Conference,Vol\.3\.External Links:[Document](https://dx.doi.org/10.36001/phme.2016.v3i1.1577)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px1.p1.1),[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px2.p1.1)\.
- Linet al\.\(2020\)J\. Lin, W\. Chen, Y\. Lin, J\. Cohn, C\. Gan, and S\. HanMCUNet: tiny deep learning on IoT devices\.InAdvances in Neural Information Processing Systems,Vol\.33\.External Links:[Link](https://proceedings.neurips.cc/paper/2020/hash/86c51678350f656dcc7f490a43946ee5-Abstract.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px3.p1.1)\.
- Liuet al\.\(2024\)Y\. Liu, T\. Hu, H\. Zhang, H\. Wu, S\. Wang, L\. Ma, and M\. LongiTransformer: inverted transformers are effective for time series forecasting\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=JePfAI8fah)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p2.1)\.
- Luo and Wang \(2024\)D\. Luo and X\. WangModernTCN: a modern pure convolution structure for general time series analysis\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=vpJMJerXHU)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p2.1)\.
- Middlehurstet al\.\(2024\)M\. Middlehurst, P\. Schäfer, and A\. BagnallBake off redux: a review and experimental evaluation of recent time series classification algorithms\.Data Mining and Knowledge Discovery\.External Links:[Link](https://doi.org/10.1007/s10618-024-01022-1)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p2.1)\.
- Nguyen Duc Thuan and Hoang Si Hong \(2023\)Nguyen Duc Thuan and Hoang Si HongHUST bearing: a practical dataset for ball bearing fault diagnosis\.BMC Research Notes16\.External Links:[Document](https://dx.doi.org/10.1186/s13104-023-06400-4)Cited by:[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px2.p1.1)\.
- Nieet al\.\(2023\)Y\. Nie, N\. H\. Nguyen, P\. Sinthong, and J\. KalagnanamA time series is worth 64 words: long\-term forecasting with transformers\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=Jbdc0vTOcol)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p2.1)\.
- Orvietoet al\.\(2023\)A\. Orvieto, S\. L\. Smith, A\. Gu, A\. Fernando, C\. Gulcehre, R\. Pascanu, and S\. DeResurrecting recurrent neural networks for long sequences\.InProceedings of the 40th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.202,pp\. 26670–26698\.External Links:[Link](https://proceedings.mlr.press/v202/orvieto23a.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p1.1)\.
- PHM Society \(2009a\)PHM SocietyPHM09 data challenge: apparatus\.Note:Accessed September 2026External Links:[Link](https://phmsociety.org/data-analysis-competition/apparatus/)Cited by:[§B\.2](https://arxiv.org/html/2609.27411#A2.SS2.SSS0.Px2.p1.1),[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px2.p1.1)\.
- PHM Society \(2009b\)PHM SocietyPublic data sets: 2009 phm challenge competition data set\.Note:Labelled expanded release; accessed September 2026External Links:[Link](https://phmsociety.org/public-data-sets/)Cited by:[§B\.2](https://arxiv.org/html/2609.27411#A2.SS2.SSS0.Px2.p1.1),[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px2.p1.1)\.
- Randall and Antoni \(2011\)R\. B\. Randall and J\. AntoniRolling element bearing diagnostics—a tutorial\.Mechanical Systems and Signal Processing25,pp\. 485–520\.External Links:[Document](https://dx.doi.org/10.1016/j.ymssp.2010.07.017)Cited by:[§1](https://arxiv.org/html/2609.27411#S1.p4.1),[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px1.p1.1)\.
- Rusch and Mishra \(2021\)T\. K\. Rusch and S\. MishraCoupled oscillatory recurrent neural network \(coRNN\): an accurate and \(gradient\) stable architecture for learning long time dependencies\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=F3s69XzWOia)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p1.1)\.
- Sehri and Dumond \(2023\)M\. Sehri and P\. DumondUniversity of ottawa rolling\-element dataset—vibration and acoustic faults under constant load and speed conditions \(UORED\-VAFCLS\)\.Note:Mendeley Data, version 5External Links:[Document](https://dx.doi.org/10.17632/y2px5tg92h.5),[Link](https://data.mendeley.com/datasets/y2px5tg92h/5)Cited by:[§4](https://arxiv.org/html/2609.27411#S4.SS0.SSS0.Px2.p1.1)\.
- Smithet al\.\(2023\)J\. T\. H\. Smith, A\. Warrington, and S\. W\. LindermanSimplified state space layers for sequence modeling\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=Ai8Hw3AXqks)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p1.1)\.
- Snellet al\.\(2017\)J\. Snell, K\. Swersky, and R\. S\. ZemelPrototypical networks for few\-shot learning\.InAdvances in Neural Information Processing Systems,Vol\.30\.External Links:[Link](https://proceedings.neurips.cc/paper/2017/hash/cb8da6767461f2812ae4290eac7cbc42-Abstract.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px1.p1.1)\.
- Vieiraet al\.\(2026\)J\. P\. Vieira, V\. A\. Bauler, R\. K\. Rosa, and D\. SilvaTowards a more realistic evaluation of machine learning models for bearing fault diagnosis\.Mechanical Systems and Signal Processing258,pp\. 114640\.External Links:[Document](https://dx.doi.org/10.1016/j.ymssp.2026.114640),[Link](https://doi.org/10.1016/j.ymssp.2026.114640)Cited by:[§1](https://arxiv.org/html/2609.27411#S1.p2.1),[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px1.p1.1)\.
- Wanget al\.\(2024\)Y\. Wang, N\. Huang, T\. Li, Y\. Yan, and X\. ZhangMedformer: a multi\-granularity patching transformer for medical time\-series classification\.InAdvances in Neural Information Processing Systems,External Links:[Link](https://proceedings.neurips.cc/paper_files/paper/2024/hash/3fe2a777282299ecb4f9e7ebb531f0ab-Abstract-Conference.html)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p2.1)\.
- Zhang and Yan \(2023\)Y\. Zhang and J\. YanCrossformer: transformer utilizing cross\-dimension dependency for multivariate time series forecasting\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=vSVLM2j9eie)Cited by:[§2](https://arxiv.org/html/2609.27411#S2.SS0.SSS0.Px2.p2.1)\.

Supplementary Material

## Appendix AMathematical and Implementation Details

### A\.1Tensor shapes and parameter accounting

Table 6:Per\-window tensor shapes; batch axis omitted\.Eight complex states correspond to 16 real coordinates/channel\. Shared read/write maps account for 2,080 parameters; they are not independent64×864\\times 8projections for every channel\.

### A\.2Normalization and complete spectral construction

Letx∈ℝLx\\in\\mathbb\{R\}^\{L\}be a univariate vibration window after resampling tofs=64,000f\_\{s\}=64\{,\}000Hz\. We useL=32,768L=32\{,\}768, corresponding to0\.5120\.512s\. Each window is centered and scaled by its own root\-mean\-square amplitude:

x~n=xn−μxmax⁡\(rx,10−10\),μx=1L​∑nxn,rx=1L​∑n\(xn−μx\)2\.\\widetilde\{x\}\_\{n\}=\\frac\{x\_\{n\}\-\\mu\_\{x\}\}\{\\max\(r\_\{x\},10^\{\-10\}\)\},\\qquad\\mu\_\{x\}=\\frac\{1\}\{L\}\\sum\_\{n\}x\_\{n\},\\quad r\_\{x\}=\\sqrt\{\\frac\{1\}\{L\}\\sum\_\{n\}\(x\_\{n\}\-\\mu\_\{x\}\)^\{2\}\}\.\(6\)For analysis lengthN∈\{256,1024\}N\\in\\\{256,1024\\\}and common hopH=128H=128, we compute

Xj,k\(N\)=∑m=0N−1x~j​H\+mwN\[m\]e−2πikm/N,0≤k≤N/2,X^\{\(N\)\}\_\{j,k\}=\\sum\_\{m=0\}^\{N\-1\}\\widetilde\{x\}\_\{jH\+m\}w\_\{N\}\[m\]e^\{\-2\\pi ikm/N\},\\qquad 0\\leq k\\leq N/2,\(7\)wherewN​\[m\]=12​\(1−cos⁡\(2​π​m/N\)\)w\_\{N\}\[m\]=\\tfrac\{1\}\{2\}\(1\-\\cos\(2\\pi m/N\)\)is a periodic Hann window\. WriteNs=256N\_\{s\}=256andNl=1024N\_\{l\}=1024for the short and long analysis lengths, andFs=Ns/2\+1F\_\{s\}=N\_\{s\}/2\+1andFl=Nl/2\+1F\_\{l\}=N\_\{l\}/2\+1for their one\-sided frequency\-bin counts\. The diagrams show batch sizeBBexplicitly; the equations omit that independent batch dimension\.

The short transform yields255255frames and the long transform249249\. Define the frame offsetκ=\(Nl−Ns\)/\(2​H\)=3\\kappa=\(N\_\{l\}\-N\_\{s\}\)/\(2H\)=3and the shared centerτj=j​H\+Nl/2\\tau\_\{j\}=jH\+N\_\{l\}/2\. Retaining short framesj\+κj\+\\kappagives the shared center in Eq\.[1](https://arxiv.org/html/2609.27411#S3.E1)\. HereT=⌊\(L−Nl\)/H⌋\+1=249T=\\lfloor\(L\-N\_\{l\}\)/H\\rfloor\+1=249\. Exact alignment by an integer\-index crop requires integralκ\\kappa, as satisfied by our settings\. The resulting token rate isfz=fs/H=500f\_\{z\}=f\_\{s\}/H=500Hz\. Figure[7](https://arxiv.org/html/2609.27411#A1.F7)illustrates this alignment and the subsequent branch projections\.

![Refer to caption](https://arxiv.org/html/2609.27411v1/Arch_figures/Fig3.png)Figure 7:Dual\-resolution alignment\. Short and long Hann windows have different supports but paired frames share the centerτj\\tau\_\{j\}\. Cropping the short sequence byκ=\(Nl−Ns\)/\(2​H\)\\kappa=\(N\_\{l\}\-N\_\{s\}\)/\(2H\)givesTTaligned frames\. Independent projections map theFsF\_\{s\}andFlF\_\{l\}magnitude bins intoD/2D/2channels before concatenation;BBdenotes batch size\.
### A\.3Coefficient parameterization and initialization

For tokentt, letvt=LayerNorm⁡\(et\)v\_\{t\}=\\operatorname\{LayerNorm\}\(e\_\{t\}\)\. An affine map producesut∈ℝDu\_\{t\}\\in\\mathbb\{R\}^\{D\}\. Each channelddmaintainsK=8K=8complex statesqt,d,kq\_\{t,d,k\}, initialized at zero\. Frequencies are shared across channels:

fk\\displaystyle f\_\{k\}=fz2​sigmoid⁡\(ρk\),\\displaystyle=\\frac\{f\_\{z\}\}\{2\}\\operatorname\{sigmoid\}\(\\rho\_\{k\}\),θk\\displaystyle\\theta\_\{k\}=2​π​fk/fz,\\displaystyle=2\\pi f\_\{k\}/f\_\{z\},\(8\)δt,d\\displaystyle\\delta\_\{t,d\}=exp⁡\(λd\+tanh⁡\(\[Wγ​vt\+aγ\]d\)\)fz,\\displaystyle=\\frac\{\\exp\(\\lambda\_\{d\}\+\\tanh\(\[W\_\{\\gamma\}v\_\{t\}\+a\_\{\\gamma\}\]\_\{d\}\)\)\}\{f\_\{z\}\},αt,d\\displaystyle\\alpha\_\{t,d\}=exp⁡\(−δt,d\)\.\\displaystyle=\\exp\(\-\\delta\_\{t,d\}\)\.\(9\)On the spectral\-token clock defined in Section[3](https://arxiv.org/html/2609.27411#S3), initial frequencies are log\-spaced from1010to200200Hz\. Baseline damping ratesexp⁡\(λd\)\\exp\(\\lambda\_\{d\}\)are initialized from half\-lives log\-spaced between11and6464ms, and the input\-dependent damping projection starts at zero\.

Separate affine maps fromvtv\_\{t\}to2​K2Kreal coordinates parameterize

bt,k\\displaystyle b\_\{t,k\}=tanh⁡\(βt,kR\)\+i​tanh⁡\(βt,kI\),\\displaystyle=\\tanh\(\\beta^\{\\mathrm\{R\}\}\_\{t,k\}\)\+i\\tanh\(\\beta^\{\\mathrm\{I\}\}\_\{t,k\}\),\(10\)ct,k\\displaystyle c\_\{t,k\}=\(tanh⁡\(χt,kR\)\+i​tanh⁡\(χt,kI\)\)/K\.\\displaystyle=\\big\(\\tanh\(\\chi^\{\\mathrm\{R\}\}\_\{t,k\}\)\+i\\tanh\(\\chi^\{\\mathrm\{I\}\}\_\{t,k\}\)\\big\)/\\sqrt\{K\}\.\(11\)

### A\.4Bounded\-state argument

For bounded writes\|bt,k​ut,d\|≤M\|b\_\{t,k\}u\_\{t,d\}\|\\leq M, the triangle inequality gives

\|qt,d,k\|≤αt,d​\|qt−1,d,k\|\+\(1−αt,d\)​M≤max⁡\{\|q0,d,k\|,M\}\.\|q\_\{t,d,k\}\|\\leq\\alpha\_\{t,d\}\|q\_\{t\-1,d,k\}\|\+\(1\-\\alpha\_\{t,d\}\)M\\leq\\max\\\{\|q\_\{0,d,k\}\|,M\\\}\.\(12\)The second inequality follows by induction\. This state bound assumes bounded writes; it does not constrain every learned projection or imply generalization\. Equation[3](https://arxiv.org/html/2609.27411#S3.E3)defines a discrete update rather than an exact discretization of a general forced continuous oscillator\.

### A\.5Exact damping regularization and cost

During training only, damping increments receive multiplicative jitter:

δt,d′=δt,d​exp⁡\(ϵd−σ2/2\),ϵd∼𝒩⁡\(0,σ2\),σ=0\.10\.\\delta^\{\\prime\}\_\{t,d\}=\\delta\_\{t,d\}\\exp\(\\epsilon\_\{d\}\-\\sigma^\{2\}/2\),\\qquad\\epsilon\_\{d\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}\),\\quad\\sigma=0\.10\.\(13\)Noise is independent across windows and channels but shared across token positions within a window\. Its positive multiplier has mean one and preserves positive damping; it does not preserve the expected decay coefficient itself\. The perturbedδ′\\delta^\{\\prime\}defines both decay and normalized writing\.

For fixed resolutions, STFT cost isO⁡\(T⁡\(256​log⁡256\+1024​log⁡1024\)\)O\(T\(256\\log 256\+1024\\log 1024\)\); projection and mixing costs areO⁡\(T⁡\(32​\(129\+513\)\+D2\+D​K\)\)O\(T\(32\(129\+513\)\+D^\{2\}\+DK\)\), and the state scan costsO⁡\(T​D​K\)O\(TDK\)\. All scale linearly with sequence length at fixed widths and analysis windows\. The recurrent state containsD​KDKcomplex values; measured activation memory is discussed in Appendix[I\.2](https://arxiv.org/html/2609.27411#A9.SS2)\.

### A\.6Efficient implementation in rotating coordinates

The complex recurrence can be evaluated using a real selective\-scan kernel\. Factoring the known rotation out of the state leaves a real decay and moves the phase factors into the read and write vectors\. Writeat,d=exp⁡\(−δt,d\)a\_\{t,d\}=\\exp\(\-\\delta\_\{t,d\}\)and let the recurrence be Eq\.[3](https://arxiv.org/html/2609.27411#S3.E3)\. The substitutionrt,d,k=ei​t​θk​qt,d,kr\_\{t,d,k\}=e^\{it\\theta\_\{k\}\}q\_\{t,d,k\}yields

rt=at​rt−1\+\(1−at\)​ei​t​θ​bt​ut\.r\_\{t\}=a\_\{t\}r\_\{t\-1\}\+\(1\-a\_\{t\}\)e^\{it\\theta\}b\_\{t\}u\_\{t\}\.\(14\)Rotating the read vector by the same phase leavesRe⁡\(c¯t​qt\)\\operatorname\{Re\}\(\\bar\{c\}\_\{t\}q\_\{t\}\)unchanged\. The official real selective scan usesA=−1A=\-1,Δ=δ\\Delta=\\deltaand a write termΔ​B​u\\Delta Bu\. Scaling its input by

g⁡\(δ\)=−expm1⁡\(−δ\)δ,limδ→0g⁡\(δ\)=1,g\(\\delta\)=\\frac\{\-\\operatorname\{expm1\}\(\-\\delta\)\}\{\\delta\},\\qquad\\lim\_\{\\delta\\to 0\}g\(\\delta\)=1,\(15\)reproduces the normalized write without a complex CUDA kernel\. This is an algebraic representation of the chosen discrete recurrence, not a claim that an unmodified Mamba block implements the same model\. In the homogeneous system,\|qt\|=exp\(−∑j≤tδj\)\|q0\|\|q\_\{t\}\|=\\exp\(\-\\sum\_\{j\\leq t\}\\delta\_\{j\}\)\|q\_\{0\}\|\. Asymptotic decay requires the cumulative damping to diverge; positivity alone does not supply a uniform decay rate over every possible input\. Neither state boundedness nor asymptotic decay ensures bounded optimization gradients\.

### A\.7Gradient paths and memory use

Gradients pass through both spectral projections, input\-dependent coefficients, the scan, output gate and classifier\. The FFT itself has no trained weights\. The full\-window scan also stores sequence activations, so its memory demand exceeds the recurrent\-state footprint\.

## Appendix BDatasets and Task Definitions

Table[7](https://arxiv.org/html/2609.27411#A2.T7)consolidates the retained labels, acquisition rates and evaluation split for each dataset\. All tasks use one vibration channel; metadata defines recording roles but is not an input feature\. The retained label space may be smaller than the original release\.

Table 7:Public datasets and evaluated tasks\. Rates refer to the selected native signal, before conversion to 64 kHz\. IR/OR/RE denote inner\-race, outer\-race and rolling\-element faults\. Each task has two frozen folds\.### B\.1Bearing task boundaries

#### Paderborn\.

Combined\-fault bearings are outside the retained task\. Reserved confirmation bearingsK001,KA30,KI21andKB23are rejected before signal indexing\. Both artificial and real damage occur in the release; the manifests identify the participating bearings\.

#### CWRU and KAIST\.

These folds test operating\-condition changes, not independent bearing populations\. The fixed manifests and signal\-channel choices are common to every method\.

#### UORED\-VAFCLS\.

All available stages of one bearing remain in one split\. Manufacturer–fault, load–fault and stage\-dependent acquisition correlations remain limitations\. Centering removes constant DC offsets, not manufacturer or gain effects\. The scalar speed marker is not a tachometer trace and is not used by the model\.

#### HUST and Ottawa\.

In HUST, bearing identity changes together with geometry, so the split cannot isolate their effects\. Ottawa tests speed\-profile changes on the recorded apparatus; near\-perfect scores do not establish independent\-machine transfer\.

### B\.2Gearbox recording roles

#### MCC5\-THU\.

Version 2 contains 240 recordings\. We use onlygearbox\_vibration\_x\. Health and missing teeth each contribute 12 recordings; every other label contributes 36\. M1 trains on low\-severity torque circulation and evaluates all speed\-circulation recordings; M2 trains on high\-severity speed circulation and evaluates all torque\-circulation recordings\. Health and missing teeth have no severity suffix\. Each evaluation fold has 120 recordings; motion and severity composition change together\.

#### PHM2009\.

The labelled mirror contains 280 recordings: five speeds, two loads and two repeats per case\. P1 uses repeat 1 at 30/35/40 Hz and high load for support and evaluates repeat 2; P2 uses repeat 2 at 50/45/40 Hz and low load for support and evaluates repeat 1\. Four disjoint windows per support recording start at 1 s, giving 6\.144 s/class\. Both models use seed 41 and 300 terminal updates\. This expanded labelled release differs from the original unlabeled challenge pool and its fault type/location/magnitude output\([PHM Society, 2009b](https://arxiv.org/html/2609.27411#bib.bib35);[PHM Society, 2009a](https://arxiv.org/html/2609.27411#bib.bib36)\)\. We use the first vibration column\.

### B\.3Acquisition compatibility and provenance

Native rates determine polyphase resampling to 64 kHz \(MCC5:5/15/1; PHM2009:24/2524/25\)\. Conversion cannot restore information beyond the native Nyquist limit\. Per\-window normalization follows Eq\.[6](https://arxiv.org/html/2609.27411#A1.E6)and removes absolute amplitude\. Acquisition manifests retain original identities, release paths, versions and available license metadata; raw redistribution remains governed by each release’s terms\. Fold sizes and support accounting are specified below\.

## Appendix CFull Benchmark Specification

### C\.1Manifests, split construction and support selection

The bearing experiments use fixed recording manifests\. The MCC5 scaling experiment usesROLES\.json, which enumerates the source record indices at each level and the entire opposite\-mode evaluation set\. Roles are assigned before windows are extracted\. No evaluation record enters the labelled support or the fitted classical feature scaler\. For M1 the nested source\-condition order is 1000 rpm/10 Nm, 3000/20, 2000/10, 1000/20, 3000/10 and 2000/20\. For M2 it is 20 Nm/3000 rpm, 10/1000, 10/2000, 20/1000, 10/3000 and 20/2000\. The firstnnentries are retained for every class\.

### C\.2Window placement and exposure accounting

MCC5 support windows begin at resampled index15​fs\+j​L15f\_\{s\}\+jL, forj=0,…,W−1j=0,\\ldots,W\-1\. Adjacent support windows are disjoint\. Withnnrecordings per class,C=8C=8classes andL/fs=0\.512L/f\_\{s\}=0\.512s,

Nsupport=C​n​W,Eclass=0\.512​n​W,Etotal=C​Eclass\.N\_\{\\mathrm\{support\}\}=CnW,\\qquad E\_\{\\mathrm\{class\}\}=0\.512nW,\\qquad E\_\{\\mathrm\{total\}\}=CE\_\{\\mathrm\{class\}\}\.\(16\)
Table 8:MCC5 exposure accounting; no overlap correction is needed for these support windows\.The bearing protocol uses 50% overlap*within*a labelled support recording\. Fornwn\_\{w\}windows its unique duration is0\.512\+0\.256​\(nw−1\)0\.512\+0\.256\(n\_\{w\}\-1\)seconds/class when one support recording is used per class\. Thus three windows cover 1\.024 unique seconds\. Within\-recording overlap does not cross split boundaries\.

### C\.3Training and checkpoint rules

Neural models use Adam with initial learning rate5×10−45\\times 10^\{\-4\}, decay0\.990\.99per update and microbatches of four windows\. Updateeetherefore uses learning rate5×10−4​0\.99e−15\\times 10^\{\-4\}0\.99^\{e\-1\}\. A microbatch ofbbwindows contributes\(b/N\)​ℒb\(b/N\)\\mathcal\{L\}\_\{b\}to the accumulated gradient, yielding the support\-average objective before one optimizer step\. Batch\-dependent layers still see microbatch statistics; accumulation is not equivalent to a single full\-batch forward pass\. Evaluation uses terminal checkpoints from completed fits\. Seed repetitions vary model initialization while keeping the development folds and evaluation domains fixed\.

### C\.4Recording\-level scoring

The aggregation in Section[4](https://arxiv.org/html/2609.27411#S4)assigns one predicted label per recording; windows are not independent evaluation replicates\. WithCCfixed classes, the primary metric is

macroF1=1C​∑c=1C2​T​Pc2​T​Pc\+FPc\+FNc,\\mathrm\{macroF1\}=\\frac\{1\}\{C\}\\sum\_\{c=1\}^\{C\}\\frac\{2\\mathrm\{TP\}\_\{c\}\}\{2\\mathrm\{TP\}\_\{c\}\+\\mathrm\{FP\}\_\{c\}\+\\mathrm\{FN\}\_\{c\}\},\(17\)with undefined class ratios set to zero\. Source balance does not imply evaluation balance: MCC5 has six health and six missing\-teeth recordings per evaluation fold, versus 18 for each other label\.

Figure 8:Three support\-allocation experiments\. Each cell denotes a disjoint 0\.512s window; each row/color within a group is a different recording\. Only source recordings supply these windows\. The diagrams distinguish adding data from reallocating a fixed duration\.

## Appendix DComparator Configurations and Adaptations

### D\.1Shared recipe and released classification paths

All compared neural models receive the same 32768\-sample support windows and label budget under the MCC5 recipe above\. The comparison is of released classification architectures under a fixed scarce\-label recipe, not a reproduction of each paper’s dataset\-specific tuning grid\. Internal normalization and augmentation follow each documented implementation\.

Table 9:MCC5 comparator settings\. FFN denotes the feed\-forward network width\. Full configurations accompany the saved fits\.
### D\.2Method\-specific interpretation

MambaSL uses a fixed released long\-sequence configuration across tasks\([Jung and Kim, 2026](https://arxiv.org/html/2609.27411#bib.bib18)\)\. TSCMamba uses the released scalar Morlet scale64, with the resulting CWT resized to64×6464\\times 64\. Its raw\-view orientation is repaired to match the released linear projection, and continuous wavelet transform \(CWT\) extrema are fitted on source support only\. The isolated scikit\-image dependency is 0\.24 rather than the release’s 0\.22 for NumPy compatibility\. MiniRocket’s source features are centered and divided by the featurewise centeredL2L\_\{2\}norm, with a unit replacement for near\-zero scales\. Its ridge coefficient grid has ten log\-spaced values from10−310^\{\-3\}to10310^\{3\}\. MiniRocket follows its fixed\-transform classifier pipeline\([Dempster et al\., 2021](https://arxiv.org/html/2609.27411#bib.bib22)\)\. Ridge fitting produces no iterative neural loss curve; we report its fitted coefficients and storage separately from neural parameter counts\.

## Appendix EExpanded Predictive Results and Replication

### E\.1Bearing tasks and variation across folds

Table 10:Bearing accuracy \(%\)\. Mean±\\pmsample SD over three seeds, each averaged over both frozen folds\. Labelled exposure: 1\.024 unique seconds/class; 100 supervised updates\. Bold: best three\-seed mean within each dataset\.iTransformer is additionally replicated on HUST \(Table[15](https://arxiv.org/html/2609.27411#A5.T15)\); other iTransformer datasets have one seed and are not included in this three\-seed table\.

Figure 9:All six bearing tasks at 1\.024 unique labelled seconds/class\. Grouped bars show macro\-F1 means over three fold\-averaged seeds; error bars show one sample standard deviation\. All panels share the same scale\. HUST iTransformer is reported separately in Table[15](https://arxiv.org/html/2609.27411#A5.T15)\.Accuracy complements macro\-F1 when evaluation classes have unequal frequencies\.

Table 11:Bearing macro\-F1 \(%\), seed 41, at 1\.024 unique seconds/class\. Both folds use 100 supervised updates for neural models; MiniRocket fits its ridge pipeline\. Fold 1/2 follow each frozen manifest\. Bold ranks the displayed two\-fold means within each dataset\. This table exposes fold variation beneath the three\-seed summaries\.The seed\-41 table reports the variation between folds underlying the three\-seed summary\.

### E\.2Gearbox performance across label budgets

Table 12:MCC5 accuracy \(%\): two\-fold means, seed 41\. Column headers are unique labelled seconds/class\. Neural methods use 300 terminal updates; MiniRocket uses its source\-fitted ridge pipeline\. Bold: best completed value per budget\.First five columns: 1/2/3/4/6 recordings×\\times4 windows\. Last two: 6 recordings×\\times8/12 windows\.

Figure 10:Broad MCC5 comparison at all seven label budgets, seed 41\. Macro\-F1 and accuracy are shown on a shared scale\. Each bar averages both folds\. The first five budgets increase the number of recordings, whereas the last two increase duration within the six\-record pool\.Table 13:MCC5 macro\-F1 \(%\) by fold, seed41\. All nine evaluated methods; bold is highest in each column\.The per\-fold table reports both directions of motion\-mode transfer at the primary and six\-record settings\.

### E\.3Three\-seed replication coverage

Table 14:MCC5 replication macro\-F1 \(%\)\. Mean±\\pmsample SD across fold\-averaged seeds 41/42/43\. Bold: best completed mean per row\.The MCC5 replication compares DualRes, MiniRocket and MambaSL across all seven budgets, with seeds 41, 42 and 43 and both folds per seed\. Within this nine\-method scope, these are the three highest seed\-41 means across the two extended budgets\. All three methods also have complete bearing repetitions\. HUST additionally replicates iTransformer, which ranks third in the broad seed\-41 comparison; Table[15](https://arxiv.org/html/2609.27411#A5.T15)reports its full seed variation\. The seed\-41 fit is shared with the broad comparison at each matching configuration\.

Table 15:Additional HUST finalist replication: iTransformer at 1\.024 labelled seconds/class, 100 terminal updates\. Each seed averages both frozen folds\. Its mean macro\-F1 is below MiniRocket \(22\.76%\) and MambaSL \(20\.63%\), and above DualRes \(14\.16%\); selection used the original seed\-41 ranking\.
### E\.4Generalizing the training recipe to another gearbox dataset

All weights are fitted on PHM source support; only the MCC5 training recipe is reused\. Table[3](https://arxiv.org/html/2609.27411#S5.T3)reports aggregate scores\. Figure[11](https://arxiv.org/html/2609.27411#A5.F11)instead identifies difficult case classes, using recall averaged over folds\. Each class has ten evaluation recordings per fold; Table[7](https://arxiv.org/html/2609.27411#A2.T7)defines H1–H6 and S1–S8\.

Figure 11:PHM class recall, seed 41, averaged over both repeat\-separated folds\. The dashed separator distinguishes helical and spur configurations\. This class\-level view complements the aggregate scores in Table[3](https://arxiv.org/html/2609.27411#S5.T3)\.

## Appendix FRecording Diversity versus Labelled Exposure

### F\.1Matched exposure control

The allocations in Figure[4](https://arxiv.org/html/2609.27411#S5.F4)c share both total support size and update count\. The table below reports their scores; the recording conditions are drawn from the same source pool\.

Table 16:Fixed 6\.144 seconds/class: macro\-F1 \(%\), two\-fold mean, seed 41\.The three\-record allocation exceeds the one\-record allocation for both methods in both folds, but increasing from three to six recordings does not consistently improve performance\.

## Appendix GTraining Dynamics and Duration Selection

### G\.1Optimization across architectures

Figure[5](https://arxiv.org/html/2609.27411#S5.F5)in the main paper reports the cross\-architecture trajectories\. Each history records the update number, support\-averaged loss and learning rate\. Architecture\-specific stochastic layers can affect the trajectories even though the eight neural methods share the cross\-entropy definition\.

### G\.2Source\-only selection of training duration

The separate duration pilot uses five source\-mode recordings/class for training and holds one operating condition out for validation: M1 holds2000rpm\_20Nm, M2 holds20Nm\-2000rpm\. Training uses three overlapping windows per recording \(5\.120 unique seconds/class\); this differs from the later non\-overlapping benchmark\. Checkpoints at 50/100/200/300 updates are assessed only on these source\-mode validation files\. Opposite\-motion recordings are never accessed by this study\. The highest mean validation macro\-F1 across both models and folds selects 300 updates, with lower log loss then earlier update as tie\-breakers\. This shared duration is subsequently fixed for all MCC5 neural comparators\.

Figure 12:Selecting training duration using source conditions\. \(a,b\) Unsmoothed training losses; \(c\) source\-validation recording macro\-F1 averaged over the two folds, seed41\. The shared 300\-update choice uses both methods, not target\-fold accuracy\.The histories record update indices without per\-update wall\-clock timestamps; training time is therefore reported separately from these learning curves\.

## Appendix HComponent Ablations

The ablation study comprises 28 control configurations and 56 fold fits\. Each is compared with the seed\-41 full model at the matching support budget\. The full table is in the main paper; Figure[13](https://arxiv.org/html/2609.27411#A8.F13)gives signed full\-minus\-control differences on every dataset\. These are descriptive one\-seed effects; they do not establish uncertainty across initializations\.

### H\.1Frontend resolution controls

Short\-only and long\-only zero the other projected branch before LayerNorm\. Both retain the 64\-channel backbone, 249 frame centers, state capacity and head\. Disabled projections are frozen, so trainable and registered counts differ\. These are controlled information removals, not optimized single\-scale competitors\.

### H\.2Memory and regularization controls

The no\-rotation control fixes oscillator frequencies at zero\. The no\-jitter control sets jitter strength to zero while retaining the random draw, preserving the sequence of dropout random numbers\. Pooling, initialization of retained weights and the remaining recipe stay fixed\.

Figure 13:Component contributions shown as full\-model minus control macro\-F1\. Positive bars favor the full model; negative bars favor the control\. All seven datasets use the same vertical scale, and zero is explicitly marked\.

## Appendix IDesktop Efficiency Measurements

### I\.1Cost definitions and available fit measurements

Registered parameters count all model parameters; active parameters count tensors receiving gradients in the training path\. Serialized bytes include persistent buffers \(such as positional tables\) and checkpoint\-format overhead, and need not equal four bytes per parameter\. GPU allocation and host\-memory measurements are defined below\.

Table 17:MCC5 training and storage costs at 6\.144 labelled seconds/class \(seed 41, both folds\)\. Fit time and peak allocated training memory were recorded on a shared host; they are not isolated inference benchmarks\. Bold: lowest observed neural value per column\.†\\daggerMiniRocket has 9,996 fixed\-transform features and 79,976 learned linear coefficients; CPU fitting is not a neural parameter comparison\. Storage includes persistent buffers and serialization overhead\. GPU allocation is a framework counter, not total device memory\. Matched desktop inference is reported separately\.

The training timer encloses the optimization loop, with CUDA synchronization at its boundaries and excludes evaluation\. Extended\-budget evaluation recovered from saved terminal checkpoints after CUDA errors; recovery runtime is not reported as full\-workflow training time\. Training on a shared host is identified separately from isolated inference measurements\.

### I\.2Matched desktop inference

We measure all 63 combinations of nine methods and seven datasets\. Each measurement uses a fresh process, four CPU/linear\-algebra threads and the same first source\-support recording within its dataset\. Neural paths use one RTX A4000, FP32, TF32 disabled; MiniRocket runs on CPU and is labelled separately\. The saved checkpoint is seed41/first fold at the primary support budget\. No training or other experiment runs concurrently with these measurements\.

The window workload maps a normalized CPU window to a CPU score, including transfers and method\-specific transforms\. It uses 20 warm\-ups and 120 timed repetitions\. Native\-record timing starts from the selected channel in host memory and includes resampling, normalization and all prescribed window predictions; it excludes disk parsing and model loading\. Both prepared\-record and native\-record workloads use two warm\-ups, ten repetitions and a common batch cap of four\. Bearing evaluation prefixes and the full MCC5 window count follow the original evaluation code\. Recording\-level timings across datasets therefore correspond to different workloads\.

Figure 14:Matched window latency for every profiled dataset and method\. Common logarithmic axes preserve large cost differences without dropping slow methods\. Hatching identifies MiniRocket’s CPU backend; the other paths include GPU inference and their required CPU work\.Table 18:Window median latency \(ms\)\. GPU paths use one RTX A4000; MiniRocket uses four CPU threads and forms a separate backend group\. Bold: lowest neural median in each dataset\. Record durations differ across datasets\.Table 19:Native\-record median latency \(ms\)\. GPU paths use one RTX A4000; MiniRocket uses four CPU threads and forms a separate backend group\. Bold: lowest neural median in each dataset\. Record durations differ across datasets\.Table 20:Absolute peak allocated CUDA memory \(MiB\), batch\-one window workload\. Includes loaded tensors and temporary allocations; excludes the CUDA context and driver memory\. Bold: smallest neural value per dataset\.Table 21:Profiled source workloads and DualRes tail latency\. The 95th\-percentile \(P95\) estimates use 120 window and 10 record repetitions\. RSS is the sampled absolute process resident memory and is not model\-only storage\.Memory is measured in a separate untimed pass\. CUDA allocated and reserved bytes are distinct allocator counters; neither measures the full device or operating\-system footprint\. Process resident set size \(RSS\) is sampled every 10 ms and may miss shorter peaks\. The exported ledger also contains prepared\-record timings, incremental memory, buffers, head/encoder counts, raw timing repetitions, input/checkpoint hashes and prediction agreement checks\. Zero incremental RSS does not mean zero memory demand: previously allocated memory can be reused\. Tail timings are descriptive estimates from these finite repetition counts\.

## Appendix JBenchmark Reproduction and Artifact Provenance

### J\.1Artifacts and implementation entry points

The benchmark rootresearch/mcc5\_all13\_scaling/providesrun\_neural,run\_tscmambaandrun\_minirocket;research/mcc5\_extended\_exposure/contains thewindows8andwindows12extensions\. From the repository root:

> python \-m research\.mcc5\_all13\_scaling\.run\_neural dualres \-\-gpu 0

The model isresearch/stft\_dualres\_ossm/model\.py;research/dualres\_seed41\_reference/is the architecture reference\. The package contains standalone DualRes code, comparator adapters, manifests and manuscript sources\. Remaining local paths require adjustment for portable use\.

### J\.2Seeds, runtime and evidence provenance

Broad MCC5 results use model seed 41 and support permutation seed 1041; replications add model seeds 42 and 43\. Encoder and classifier initializations both use the requested seed\. The runtime is WSL Ubuntu 22\.04, Python 3\.10, officialmamba\_ssmselective scan and RTX A4000 GPUs\. Per\-job plans record source hashes\.

Result\-source paths and SHA\-256 hashes are recorded inCOMPLETED\_EVIDENCE\.json,COMPLETED\_ADDITIONS\.jsonandAPPENDIX\_EVIDENCE\.json\. Tables and plots use these completed results; ablations use the matched campaign rather than architecture\-search trials\.