cs.LGFeb 20, 2026
SaveCut Less, Fold More: Model Compression through the Lens of Projection Geometry
Organizations: Graz University of Technology · Complexity Science Hub · Swiss Data Science Center · ETH Zurich
Abstract
Compressing neural networks without retraining is vital for deployment at scale. We study calibration-free compression through the lens of projection geometry: structured pruning is an axis-aligned projection, whereas model folding performs a low-rank projection via weight clustering. We formalize both as orthogonal operators and show that, within a rank distance of one, folding provably yields smaller parameter reconstruction error, and under mild smoothness assumptions, smaller functional perturbations than pruning. At scale, we evaluate >1000 checkpoints spanning ResNet18, PreActResNet18, ViT-B/32, and CLIP ViT-B/32 on CIFAR-10 and ImageNet-1K, covering diverse training hyperparameters (optimizers, learning rates, augmentations, regularization, sharpness-aware training), as well as multiple LLaMA-family 60M and 130M parameter models trained on C4. We show that folding typically achieves higher post-compression accuracy, with the largest gains at moderate-high compression. The gap narrows and occasionally reverses at specific training setups. Our results position folding as a geometry-aware, calibration-free alternative to pruning that is often superior in practice and principled in theory.
Figures & tables
Figure 1: Folding outperforms magnitude pruning across diverse training regimes. Top row: ResNet18 and PreActResNet18 on CIFAR-10. ResNet18 checkpoints were trained from scratch with Adam using different hyperparameter configurations. PreActResNet18 checkpoints are from Andriushchenko et al. (2023) . Bottom row: ViT-B/32 on CIFAR-10 from ( Andriushchenko et al., 2023 ) and CLIP ViT-B/32 on ImageNet-1K from ( Wortsman et al., 2022b ) . See Appendix D for details. In these plots, we use checkpoints that were trained without L1 regularization. Scatter plots show post-compression accuracy for magnitude pruning (L1 criterion) versus folding at uniform per-layer compression ratios (color-coded by layer-wise compression ratio). Bar plots depict the accuracy gain by folding, computed as , as a function of layer-wise compression ratio. Folding yields the largest improvements at moderate to high compression, confirming its robustness across architectures and datasets. Fig. 8 shows the results for magnitude pruning with L2 criterion.
Figure 2: MAG1 versus FOLD on ViTs after LayerNorm-only fine-tuning for ViT-B/32 on CIFAR-10 and CLIP ViT-B/32 on ImageNet-1K. In the scatter plots, points are checkpoints, color encodes layer-wise compression. Bar plots depict the accuracy gain , which remains positive and typically grows with compression, indicating that even under lightweight LayerNorm adaptation FOLD retains a consistent advantage over pruning.
Figure 3: Folded models retain their accuracy advantage after fine-tuning. Results for ResNet18 trained by Adam on CIFAR-10 ( top row ) and CLIP-ViT-B/32 trained on ImageNet-1K ( bottom row ): (a,d) compares post-compression accuracy of magnitude pruning ( MAG1 ) versus folding ( FOLD ) after 1 and 5 epochs of fine-tuning. (b,e) show the accuracy gap between folding and pruning as a function of fine-tuning epochs, demonstrating that folding maintains a consistent lead, i.e. , the FOLD accuracy delta is positive. (c,f) illustrate accuracy trajectories before and after 5 epochs of fine-tuning for both methods, highlighting that folded models recover accuracy faster. Further results in Appendix E .
| weight_decay | warmup_steps | max_lr | PPL 0% sparsity | PPL MAG2 (20%) | PPL FOLD (20%) | PPL MAG2 (50%) | PPL FOLD (50%) |
|---|---|---|---|---|---|---|---|
| 0.01 | 880 | 0.001 | 32.11 | 54.51 | 47.17 | 398.62 | 221.32 |
| 0.01 | 1100 | 0.001 | 32.14 | 50.11 | 46.75 | 220.54 | 172.57 |
| 0.01 | 2200 | 0.001 | 32.20 | 46.57 | 47.54 | 174.58 | 216.36 |
| 0 | 880 | 0.001 | 32.17 | 51.14 | 48.23 | 220.33 | 223.86 |
| 0 | 1100 | 0.001 | 32.21 | 50.03 | 47.47 | 231.41 | 204.47 |
| 0 | 2200 | 0.001 | 32.40 | 46.38 | 46.92 | 177.48 | 185.27 |
Table 1: Evaluation of FOLD and MAG2 on LLaMA-60M. We train and evaluate 18 LLaMA-family models with 60M parameters on C4 by varying max_lr, warmup steps and weight decay. Columns 4–8 show perplexity of the trained model before compression and after pruning / folding using layer-wise pruning ratio of and . We prune only FFN blocks. Except for low learning rates with long warmup schedules, FOLD outperforms MAG2 (highlighted in bold).
Figure 4: Optimizer effect evaluated on ResNet18 checkpoints trained on CIFAR-10 with SGD (no L1 regularization). The figure complements Fig. 1 (a).
Figure 5: Learning rate modulates folding’s edge. Post-compression accuracy of FOLD and MAG1 across learning rates: ResNet18 with Adam (a) and SGD (b) , PreActResNet18 (c) , and ViT-B/32 (d) . FOLD leads at moderate–low rates. With Adam, the gap shrinks or reverses at very high rates, and closes again at extremely small rates. SGD shows weaker or opposite dependence.
Figure 6: SAM ( Foret et al., 2021 ) can boost model compression. Post-compression accuracy under training with / without SAM. (a) ResNet18 (Adam), no L1. (b) ResNet18 (Adam), L1 . (c) PreActResNet18 (SGD), no L1. (d) ViT - B/32, no L1. SAM improves both FOLD and MAG1 , but the uplift is consistently larger for FOLD , especially with Adam. Light L1 regularization helps MAG1 at low compression, yet FOLD retains a clear advantage at moderate–high compression.
Figure 7: Augmentations have a generally positive effect on the post-compression accuracy. Post-compression accuracy without / with random augmentations for (a) ResNet18 (Adam), (b) ResNet18 (SGD), (c) PreActResNet18, and (d) ViT-B/32. Augmentations boost both FOLD and MAG1 . On ResNet18 they also narrow FOLD ’s advantage—most noticeably at moderate compression—consistent with added invariances making axis-aligned removals less damaging. In ViT-B/32, augmentations are essential for folding 2 2 2 Note that the base accuracy of ViT-B/32 checkpoints trained without RAUG is lower than with RAUG. .
Appendix figures & tables26 assets
Supplementary material from the paper’s appendix.
Appendix
| Parameter | Values |
|---|---|
| Optimizer | adam , sgd |
| Learning Rate | adam : 0.1, 0.01, 0.001 |
| sgd : 0.1, 0.05, 0.01, 0.001 | |
| Momentum | adam : 0.0 |
| sgd : 0.9, 0.99 | |
| Weight Decay | 0.0, 0.0005, 0.001 |
Table 2: Hyperparameter combinations used for ResNet18 training on CIFAR-10.
Figure 8: Folding outperforms magnitude pruning across diverse training regimes. The same setup as in Fig. 1 , but compared to the L2 magnitude pruning criterion. Top row: ResNet18 and PreActResNet18 on CIFAR-10. ResNet18 checkpoints were trained from scratch with Adam using different hyperparameter configurations. Bottom row: ViT-B/32 on CIFAR-10 and CLIP ViT-B/32 on ImageNet-1K. Scatter plots show post-compression accuracy for folding versus magnitude pruning (L2 criterion) at uniform per-layer compression ratios. Bar plots depict the accuracy gain by folding, computed as , as a function of layer-wise compression ratio. Folding yields the largest improvements at moderate to high compression, confirming its robustness across architectures and datasets.
Figure 9: FOLD versus MAG2 on ViTs after LayerNorm-only fine-tuning for ViT-B/32 on CIFAR-10 and CLIP ViT-B/32 on ImageNet-1K. In the scatter plots, points are checkpoints, color encodes layer-wise compression. Bar plots depict the accuracy gain , which remains positive and typically grows with compression, indicating that even under lightweight LayerNorm adaptation FOLD retains a consistent advantage over pruning. The figure follows the same setup as Fig. 2 in the main paper, but for MAG2 .
| Parameter | Values |
|---|---|
| Optimizer | sgd |
| Max / Base Learning Rate ( lr_max ) | from 0.0504 to 4.9759 |
| SAM Strength ( sam_rho ) | 0.0, 0.05, 0.1 |
| Standard Augmentation ( augm ) | True, False |
| RandAugment ( randaug ) | True, False |
Table 3: Fixed and varying parameters for PreActResNet18 training on CIFAR-10.
Figure 10: FOLD outperforms MAG1 after full fine-tuning for 1–5 epochs on PreActResNet18 and ViT-B/32 on CIFAR-10. Results for PreActResNet18 ( top ) and ViT-B/32 ( bottom ). (a,d) accuracy of MAG1 vs. FOLD after 1 and 5 epochs of fine-tuning. (b,e) accuracy gap over epochs, remaining positive. (c,f) accuracy trajectories from post-compression through 5 epochs, showing faster recovery and higher final accuracy for FOLD . The figure extends Fig. 3 in the main paper to PreActResNet18 and ViT-B/32 architectures where FOLD is benchmarked against MAG1 .
| Parameter | Values |
|---|---|
| Optimizer | sgd |
| Max / Base Learning Rate ( lr_max ) | from 0.005087 to 0.492936 |
| SAM Strength ( sam_rho ) | 0.0, 0.05, 0.1 |
| Standard Augmentation ( augm ) | True, False |
| RandAugment ( randaug ) | True, False |
Table 4: Fixed and varying parameters for ViT-B/32 Base training on CIFAR-10.
| Params | Hidden | Intermediate | Heads | Layers | Steps | Data (Tokens) |
|---|---|---|---|---|---|---|
| 60M | 512 | 1376 | 8 | 8 | 11K | 1.3B |
| 130M | 768 | 2048 | 12 | 12 | 22K | 2.6B |
Table 5: Training hyperparameters of LLaMA-60M architecture.
Figure 11: Folded models retain their accuracy advantage after fine-tuning. Results for ResNet18 trained by Adam ( top row ) and PreActResNet18 trained by SGD on CIFAR-10 ( bottom row ): (a,d) compares post-compression accuracy of magnitude pruning with L2 criterion ( MAG2 ) versus folding ( FOLD ) after 1 and 5 epochs of fine-tuning. (b,e) show the accuracy gap between folding and pruning as a function of fine-tuning epochs, demonstrating that folding maintains a consistent lead, i.e. , the FOLD accuracy delta is positive. (c,f) illustrate accuracy trajectories before and after 5 epochs of fine-tuning for both methods, highlighting that folded models recover accuracy faster and reach higher final performance than pruned models. The figure extends Fig. 3 in the main paper and Fig. 10 in the appendix to MAG2 .
Figure 12: FOLD outperforms MAG2 after full fine-tuning for 1–5 epochs on ViT-B/32 and CLIP ViT-B/32. Results for ViT-B/32 on CIFAR-10 ( top ) and CLIP ViT-B/32 on ImageNet-1K ( bottom ). (a,d) accuracy of MAG2 vs. FOLD after 1 and 5 epochs of fine-tuning. (b,e) accuracy gap over epochs, remaining positive. (c,f) accuracy trajectories from post-compression through 5 epochs, showing faster recovery and higher final accuracy for FOLD . The figure extends Fig. 3 in the main paper and Fig. 10 in the appendix to MAG2 .
Figure 13: Learning rate modulates folding’s edge. Post-compression accuracy of MAG2 and FOLD across learning rates: ResNet18 with Adam (a) and SGD (b) , PreActResNet18 (c) , and ViT-B/32 (d) . FOLD typically leads at moderate–low rates; the gap shrinks or reverses at very high rates, and closes again at extremely small rates. The same setup as in Fig. 5 in the main paper, but for MAG2 .
Figure 14: SAM can boost model compression. Post-compression accuracy under training with / without SAM. (a) ResNet18 (Adam), no L1. (b) ResNet18 (Adam), L1 . (c) PreActResNet18 (SGD), no L1. (d) ViT - B/32, no L1. The figure extends the results in Fig. 6 to MAG2 .
Figure 15: Random augmentations narrow the folding–pruning gap. Post-compression accuracy on ResNet18 (CIFAR-10) trained without vs. with random augmentations: (a) Adam, (b) SGD, (c) PreActResNet, (d) ViT-B/32. The figure extends Fig. 2 to MAG2 .
Figure 16: ResNet18: Weight Decay. Test accuracy of ResNet18 checkpoints trained with varying weight decay values. Weight decay does not diminish the advantage of FOLD compared to MAG2 , especially for SGD-trained models.
| weight_decay | warmup_steps | max_lr | PPL 0% sparsity | PPL MAG2 (20%) | PPL FOLD (20%) | PPL MAG2 (50%) | PPL FOLD (50%) |
|---|---|---|---|---|---|---|---|
| 0.01 | 1100 | 0.001 | 23.90 | 39.88 | 39.48 | 236.16 | 308.77 |
| 0.01 | 2200 | 0.001 | 23.99 | 38.75 | 39.61 | 259.79 | 469.25 |
| 0.01 | 3300 | 0.001 | 24.08 | 38.54 | 39.10 | 289.67 | 451.27 |
| 0.0 | 1100 | 0.001 | 24.01 | 42.39 | 42.27 | 270.31 | 477.70 |
| 0.0 | 2200 | 0.001 | 24.12 | 40.01 | 41.53 | 239.25 | 489.48 |
| 0.0 | 3300 | 0.001 | 24.19 | 38.72 | 40.31 | 277.10 | 531.09 |
Table 6: Evaluation of FOLD and MAG2 on LLaMA-130M (in addition to LLaMA-60M evaluations in Tab. 1 ). We train and evaluate 18 LLaMA-family models with 130M parameters on C4 while varying max_lr, warmup steps, and weight decay. Columns show perplexity of the pretrained model (0% sparsity) and perplexity after structured magnitude pruning and folding with 20% and 50% sparsity in FFN blocks. For higher learning rates, especially for the settings with the best achieved performance in each sparsity category (underlined), FOLD consistently outperforms MAG2 (bold).
Figure 17: Folding vs. magnitude pruning before REPAIR. The same setup as in Fig. 1 and Fig. 8 for CNNs (ResNet18 and PreActResNet18 on CIFAR-10), but the performance is compared for both pruning and folding before REPAIR. Top row: MAG1 , bottom row: MAG2 . In both cases, folding shows stronger performance already before data-based REPAIR is applied.
Figure 18: Uncompressed model accuracy vs. performance difference . The same setup as in Fig. 1 and Fig. 8 . Top row: MAG1 , Bottom row: MAG2 . Model folding shows strong performance on models of different quality, with amplified effect on high-performing models (especially on ResNet18, SGD and ViT-B/32).
Figure 19: Worst-case sharpness as a function of compression ratio across architectures and pruning methods. Each subplot reports the sharpness distribution for independently trained models at three perturbation radii . Panels (a)–(d) show FOLD sharpness for ResNet18 trained with Adam and SGD, PreActResNet18, and CLIP ViT-B/32, respectively. Panels (e)–(h) show the corresponding results for MAG1 . Observed trends: (i) Sharpness generally increases with compression ratio up to moderate levels before flattening or dropping at extreme compression. (ii) FOLD produces on-average lower sharpness than MAG1 . (iii) Transformer models (CLIP ViT-B/32) experience substantially sharper solutions under compression compared to residual networks. These patterns indicate that FOLD maintains flatter loss landscapes across a wide range of settings, whereas MAG more often drives models toward sharper and less stable minima.
Figure 20: Correlation between -sharpness and -accuracy across architectures and pruning baselines. Each column shows the relationship between pruning–induced differences in worst-case sharpness ( sharpness = FOLD mag ) and differences in test accuracy ( accuracy = FOLD mag ), for the pair of pruning methods indicated below each plot. Color encodes the layer-wise compression ratio. The tables underneath each subplot report Pearson and Spearman correlations at every compression ratio, quantifying how predictive the sharpness difference is of the accuracy difference. Results are shown for FOLD vs MAG1 for ResNet18 trained with Adam and SGD, PreActResNet18 and CLIP ViT-B/32. Statistics are computed over 200 independently trained ResNets18 and 50 PreActResNet18 on CIFAR-10, and 72 CLIP ViT-B/32 models on ImageNet-1K. Correlations use the sharpness value at . Results at and are qualitatively very close.
Figure 21: Sharpness-accuracy trade-off between FOLD and MAG1 for ResNet18 trained with Adam. Each column corresponds to a layer-wise compression ratio (0.1–0.9), and the three rows group models by learning rate, SAM configuration (including ), and RandAug usage. Points show worst-case sharpness ( FOLD MAG1 ) vs. test accuracy ( FOLD MAG1 ). Observations: (1) The difference in model sharpness strongly predicts the difference in performance between FOLD and MAG1 across almost all pruning ratios. (2) Learning rate: higher learning rates lead to more dispersed sharpness changes. For models trained with Adam using high learning rates, MAG1 moves the model along a less sharp path compared to FOLD , which struggles to catch up. However, the behavior flips for moderate and low learning rates. (3) SAM/ : using SAM reduces the variability in the sharpness shift between methods, especially for larger . sharpness gets closer to zero. (4) RandAug: augmentation show little specific visible trend.
Figure 22: Sharpness-accuracy trade-off between FOLD and MAG1 for ResNet18 trained with SGD. Each column corresponds to a layer-wise compression ratio (0.1–0.9), and the three rows group models by learning rate, SAM configuration (including ), and RandAug usage. Points show worst-case sharpness ( FOLD MAG1 ) vs. test accuracy ( FOLD MAG1 ). Observations: (1) For SGD, the relationship between sharpness and accuracy is visibly weaker and more scattered than for Adam, reflecting the flatter and more anisotropic minima found by SGD. (2) Learning rate: Unlike Adam, higher SGD learning rates do not systematically increase the sharpness gap between the methods. For most ratios, FOLD tends to remain less sharp than MAG1 , producing positive accuracy even when sharpness is near zero. (3) SAM/ : SAM has a strong flattening effect under SGD— sharpness clusters tightly around zero, and the accuracy advantage of FOLD becomes more stable as increases. (4) RandAug: Augmentation increases robustness to pruning under SGD, reducing the spread in accuracy and further weakening the sharpness–accuracy link. Overall, SGD-trained models exhibit a regime where FOLD consistently follows a gentler sharpness trajectory than MAG1 , leading to a clearer accuracy advantage at moderate pruning ratios.
| Method | Params | Comp. time [s] | Comp. peak mem [MB] | Lat. [ms/batch] | Lat. [ms/img] | FLOPs [MFLOPs/img] | Fwd peak mem [MB] |
|---|---|---|---|---|---|---|---|
| Original | 11,172,170 | – | – | 3.69 | 0.0288 | 557.65 | 214.30 |
| Fold | 4,008,346 | 9.48 | 157.47 | 3.17 | 0.0248 | 199.05 | 170.53 |
| Mag | 4,008,346 | 1.77 | 115.22 | 3.15 | 0.0246 | 199.05 | 169.82 |
Table 7: Runtime characteristics of PreActResNet18 before and after compression ( parameter reduction). Latency is measured for a full batch. FLOPs are reported per image. Comp. time and comp. peak mem refer to the overhead of running the compression method once.
| Method | Params | Comp. time [s] | Comp. peak mem [MB] | Lat. [ms/batch] | Lat. [ms/img] | FLOPs [MFLOPs/img] | Fwd peak mem [MB] |
|---|---|---|---|---|---|---|---|
| Original | 151,790,313 | – | – | 19.851 | 0.6203 | 2946.76 | 684.18 |
| Fold | 140,447,253 | 92.833 | 681.23 | 17.343 | 0.5420 | 2379.98 | 636.97 |
| Mag | 140,447,253 | 2.627 | 625.61 | 17.372 | 0.5429 | 2379.98 | 637.88 |
Table 8: Runtime characteristics of CLIP ViT-B/32 before and after compression ( parameter reduction). Latency is measured for a full batch. FLOPs are reported per image. Comp. time and Comp. peak mem refer to the one-off overhead of running the compression method.
| Layer | Params_fold | Params_mag | FLOPs_fold | FLOPs_mag | Act_fold | Act_mag | NZ_fold | NZ_mag | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| conv1 | 1026 | 1026 | 0 | 1050624 | 1050624 | 0 | 38912 | 38912 | 0 | 38912 | 38912 | 0 |
| layer1.0.conv1 | 12996 | 12996 | 0 | 13307904 | 13307904 | 0 | 38912 | 38912 | 0 | 38912 | 38912 | 0 |
| layer1.0.conv2 | 12996 | 12996 | 0 | 13307904 | 13307904 | 0 | 38912 | 38912 | 0 | 38912 | 38912 | 0 |
| layer1.1.conv1 | 12996 | 12996 | 0 | 13307904 | 13307904 | 0 | 38912 | 38912 | 0 | 38912 | 38912 | 0 |
| layer1.1.conv2 | 12996 | 12996 | 0 | 13307904 | 13307904 | 0 | 38912 | 38912 | 0 | 38912 | 38912 | 0 |
| layer2.0.conv1 | 25992 | 25992 | 0 | 6653952 | 6653952 | 0 | 19456 | 19456 | 0 | 19456 | 19456 | 0 |
Table 9: Per-layer comparison of PreActResNet18 after FOLD and MAG2 at compression ratio . For each convolutional and linear layer we report parameters, per-image FLOPs, activation size, and the number of non-zero activations (effective activations). All per-layer differences are zero, confirming that parameters, FLOPs, activations, and effective activations are exactly matched between the two compressed models.
| Layer | Params_fold | Params_mag | FLOPs_fold | FLOPs_mag | Act_fold | Act_mag | NZ_fold | NZ_mag | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| classification_head | 513000 | 513000 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| transformer.resblocks.0.attn.out_proj | 262656 | 262656 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| transformer.resblocks.0.mlp.c_fc | 1050624 | 1050624 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| transformer.resblocks.0.mlp.c_proj | 1049088 | 1049088 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| transformer.resblocks.1.attn.out_proj | 262656 | 262656 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| transformer.resblocks.1.mlp.c_fc | 1050624 | 1050624 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Table 10: Per-layer comparison of CLIP ViT-B/32 after FOLD and MAG2 at compression ratio (global parameter reduction ). The table reports all convolutional and linear layers in the vision transformer and classification head, including their parameters, per-image FLOPs, activation sizes, and effective activations. The transformer.resblocks.* modules belong to CLIP’s text encoder. Because the ImageNet-1k fine-tuned variant evaluates only the vision encoder and classification head, the text encoder is not part of the forward graph. THOP therefore records zero FLOPs and zero activations for these layers, while their parameters remain included in the model. Note that the per-layer totals ( parameters) are smaller than the full model parameter count ( ) because this table excludes components without FLOPs, such as token embeddings, positional embeddings, and LayerNorm parameters, which are included in the global counts but not part of the per-layer FLOP/activation analysis. Several projection layers inside the attention blocks show zero FLOPs because CLIP implements attention using fused operations; these operations are profiled at the block level by THOP rather than attributed to the individual Linear submodules. All per-layer differences are zero, showing that FOLD and MAG2 produce structurally identical compressed models on every layer affected by compression.
Figure 23: Effect of increasing the retained rank by one is significantly lower than changing the compression method from MAG to FOLD . Comparison of (i) the effect of increasing the retained rank by one (blue curves) and (ii) the effect of switching from MAG2 to FOLD at the same nominal rank (orange curves). Each panel corresponds to a single weight matrix in ResNet18 convolutional layers ( top ) and ViT-B/32 FFN layers ( bottom ) and shows the relative Frobenius error difference as a function of the retained rank .
Figure 24: Relative reconstruction error of pruning, special folding (from the proof of Theorem 2.1 ), and optimal folding for all layers of ResNet18 (top) and ViT-B/32 (bottom). For each layer, we report the normalized squared Frobenius error at several keep ratios , where . MAG2 (blue) denotes structured magnitude pruning with retained rows. The special fold (orange) merges all pruned rows into a single extra cluster ( ). The optimal fold (green) is the -means solution with clusters. Across all layers, consistently outperforms pruning, and yields the smallest error, empirically validating .
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