The Sign Code: The Hidden Binary Nature of Deep Networks

Niclas Göring; Shuofeng Zhang; Roi Holtzman; Yoonsoo Nam; Ard A. Louis

cs.LGOct 16, 2026

The Sign Code: The Hidden Binary Nature of Deep Networks

Authors: Niclas Göring, Shuofeng Zhang, Roi Holtzman, Yoonsoo Nam, Ard A. Louis

Organizations: Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, UK · Kim Jaechul Graduate School of AI, KAIST, Seoul, South Korea

Abstract

Every neuron in a trained network outputs a number that can be separated into its sign (positive or negative) and its magnitude (absolute size). We call the set of all neuron signs the sign code. This raises a key question: does classification depend more on the sign code or on the magnitudes? For overparameterized vision classifiers with sign-preserving activations, the answer is overwhelmingly the sign code. A $k$ -nearest neighbor classifier operating only on this binary sign code achieves accuracy comparable to the full model across different architectures, activation functions, and datasets. Moreover, replacing all magnitudes in every layer with random noise while keeping the original signs barely changes residual network performance. In contrast, flipping just 20% of the signs reduces accuracy to chance. Models generalize well when same-class inputs share similar sign codes. When this overlap is disrupted by a regularizer, training accuracy remains high, but test performance collapses. A margin decomposition helps explain these results: the sign-only contribution to each logit grows as $\sqrt{d}$ in network width, dominating the magnitude residual. The natural unit of analysis for these networks may not be through their continuous weight space but instead through their discrete sign codes.

Figures & tables

Figure 1: a) Every neuron in a feedforward network computes a pre-activation $z_{j}^{(\ell)}$ , which the activation function maps to $h_{j}^{(\ell)}=\varphi(z_{j}^{(\ell)})$ . b) For any sign-preserving activation, this output decomposes as $h_{j}=S_{j}\odot M_{j}$ : a binary sign $S_{j}\in\{-1,+1\}$ encoding which side of the neuron’s hyperplane the input falls on, and a non-negative magnitude $M_{j}=|h_{j}|$ encoding how far . We test the causal role of each component with two channel-isolating interventions: randomly flipping signs while preserving magnitudes, or replacing magnitudes while enforcing the sign code. c) Flipping just 20% of sign bits collapses accuracy to chance ( red ); replacing every magnitude in every layer preserves far more classification performance ( blue ), and is nearly lossless for residual networks. The forward computation is dominated by binary routing decisions rather than raw activation strengths.

Figure 2: a) The sign gap

\Delta_{\mathrm{sgn}}

(Eq. 6 ) as a function of network depth and width for three architectures (FFNN, ResNet, ViT) with SiLU on CIFAR-10. The gap shrinks monotonically with increasing depth and width, with a clear phase boundary (yellow lines) beyond which

\Delta_{\mathrm{sgn}}<0.03

. b) Sign-

k

NN accuracy versus full model test accuracy across standard architectures (rows: FFNN, ResNet-18, ViT), activations (columns: ReLU, SiLU, GELU, Tanh), and datasets (colours: MNIST, CIFAR-10, CIFAR-100). The dashed line is

y{=}x

. Training and model details for all figures are in Appendix E .

Figure 3: Left (2

\times

2): Accuracy under channel destruction. Red : sign flip (sign code routing destroyed, magnitudes intact). Blue : magnitude replacement with random

|\mathcal{N}(0,1)|

draws (magnitudes destroyed, sign code routing intact). Solid: FFNN; dashed: ResNet; mean

\pm 1

s.d. over 5 draws. Right: Intervention formulas and summary of what each intervention removes. Routing destruction collapses accuracy to chance by

\alpha=0.2

across all conditions; projected replacement of every magnitude preserves far more classification performance, and is nearly lossless for residual networks.

Figure 4: a) Train LOO purity (

k{=}5

) versus test accuracy across models (grey: FFNN on SVHN, widths 4–1024, depths 1–12, SiLU) and

\mathcal{L}_{\mathrm{rep}}

-trained models (coloured). High neighbourhood purity tracks generalization (Spearman

\rho=0.834

);

\mathcal{L}_{\mathrm{rep}}

models fall on the same trend line. b) Training dynamics under

\mathcal{L}_{\mathrm{rep}}

(SiLU, width 1024, depth 8, SVHN). Train accuracy reaches

97\%

; test accuracy drops to

{\sim}40\%

. Dashed: Test accuracy of the normally trained baseline. c) Summary across activations (depth 8, width 1024, SVHN). Repulsion sharply degrades test accuracy and LOO purity for ReLU, SiLU, and GELU while training accuracy remains

{\geq}97\%

. Tanh is partially robust (

\delta\approx 0

; Appendix H ).

Figure 5: (FFNN, SiLU, CIFAR-10; widths 4–4096, depths 1–12, mean over 3 seeds). The yellow line marks the empirical gate-domination region where both the sign gap and sign-only disagreement are small; above it, $\Delta_{\mathrm{sgn}}<0.03$ and $\varepsilon_{1}<0.1$ (a) Sign code kNN gap $\Delta_{\mathrm{sgn}}$ : the sign code matches model test error (gap ${<}0.03$ ) throughout this region. (b) Median $Z_{1}$ : increases with both width ( $\sqrt{d}$ -averaging) and depth (stronger vote bias $m_{c}$ ). (c) Disagreement rate $\varepsilon_{1}$ : mirrors $Z_{1}$ with the same diagonal phase boundary. (d) $Z_{1}$ predicts $\varepsilon_{1}$ : the disagreement rate decreases monotonically with $Z_{1}$ across the plotted configurations, independent of how each model achieves its $Z_{1}$ value.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Neural collapse diagnostics across the width–depth grid (FFNN, SiLU, SVHN, 80 epochs). Top row: NC1 ( $S_{W}/S_{B}$ , log scale), NC2 (ETF deviation), NC3 (weight–mean cosine alignment). Bottom row: NC4 (nearest-class-mean test accuracy), sign purity, and NC summary curves across widths for selected depths. NC1 remains above $1$ and NC3 below $0.4$ across most configurations, confirming that the models have not undergone neural collapse. NCM test accuracy (NC4) peaks at ${\sim}70\%$ , well below full model accuracy. Despite this, sign purity is high wherever the network is sufficiently wide and deep—class-discriminative sign code structure emerges independently of, and prior to, neural collapse.

Architecture	Tensor	Code type	Full $\mathbf{S}$	§ Section 3
FFNN	$\mathbf{h}^{(\ell)}=\varphi(\mathbf{z}^{(\ell)})$	Layer sign	✓	last only
FFNN	Output logits	—	$\times$	$\times$
ResNet	Stem ReLU output	Stem	✓	$\times$
	$\mathbf{a}=\mathrm{ReLU}(\mathrm{bn1}(\mathrm{conv1}(\cdot)))$	Int.	✓	$\times$
	$\mathbf{h}^{(\ell+1)}=\mathrm{ReLU}(\mathbf{b}+\mathrm{skip})$	Out.	✓	last only
	Avgpool / fc	—	$\times$	$\times$

Appendix

Table 1: Summary of sign code extraction across architectures. Int. = internal branch/MLP hidden code; Out. = block-output code; Attn. = post-attention residual stream code. “Full

\mathbf{S}

” indicates inclusion in the full-network sign code; indicates the code used for the last-layer sign-space

k

-NN experiment in Section 3 .

Figure 7: Empirical diagnostics for the gate-domination theorem conditions (FFNN, SiLU, SVHN; widths 4–1024, depths 1–12). Left: Positive vote bias. Each cell shows the fraction of test inputs for which $m_{c}(x)>0$ for all competitor classes. In the overparameterized regime (depth $\geq 4$ , width $\geq 64$ ), this fraction exceeds $0.80$ ; the sign-only and full classifiers agree on exactly these inputs. Centre: Residual scaling diagnostic. Each cell shows the median of the per-input ratio $R_{\hat{y},c}^{2}(x)/D_{c}(x)$ , where $D_{c}=\sum_{j}(w_{\hat{y},j}-w_{c,j})^{2}\sigma_{j}^{2}(S_{j})$ is the diagonal (uncorrelated) variance proxy. Under approximate conditional uncorrelatedness and a CLT approximation over $d$ neurons, this ratio is approximately $\chi^{2}(1)$ with median ${\approx}0.45$ . The observed values are consistent with this prediction in the transitional regime (depth ${\geq}3$ , width ${\geq}32$ ), supporting the residual-scaling condition. Right: Finite second moments. All cells show $1.0$ : every gauge value, residual, and logit is finite across the entire grid, as guaranteed by bounded SVHN inputs and finite network parameters.

		SiLU				Tanh
Width	Depth	$\bar{\delta}$	$\hat{\sigma}_{r}$	$\tilde{\gamma}$	$Z_{1}$	$\bar{\delta}$	$\hat{\sigma}_{r}$	$\tilde{\gamma}$	$Z_{1}$
64	6	0.94	8.39	1.46	${\sim}4$	0.007	0.84	0.86	${\sim}0.6$
64	12	0.93	6.53	1.44	${\sim}6$	0.003	0.89	1.00	${\sim}0.5$
256	6	0.86	3.68	1.40	${\sim}5$	0.000	0.87	1.26	${\sim}0.6$
256	12	0.71	1.82	2.53	${\sim}9$	0.001	0.75	1.17	${\sim}0.5$
1024	6	0.55	1.19	2.08	${\sim}7$	0.001	0.65	0.98	${\sim}0.5$

Appendix

Table 2: Magnitude asymmetry and gate-domination diagnostics for SiLU versus tanh on CIFAR-10 (FFNN, selected configurations, averaged over 2 seeds).

Figure 8: Activation asymmetry explains the $Z_{1}$ gap between SiLU and tanh. All panels show FFNN on CIFAR-10, widths 4–4096, depths 1–12, averaged over 2 seeds. Top row: the magnitude asymmetry $\bar{\delta}$ (Eq. 32 ) is uniformly large for SiLU and uniformly near zero for tanh. Middle row: gate-only margins $\tilde{\gamma}_{\mathrm{med}}$ (Eq. 36 ) are 2–3 $\times$ larger for SiLU, because the asymmetry offset in ( 35 ) amplifies the contribution of correctly-routed neurons. Bottom row: residual noise $\hat{\sigma}_{r}$ is comparable or larger for SiLU, confirming that the $Z_{1}$ gap is entirely due to the margin, not the noise.

Figure 9: Asymmetry predicts gate-only margin. Each point is one trained FFNN on CIFAR-10. SiLU models (orange) span

\bar{\delta}\in[0.3,0.95]

with margins up to

3.2

; tanh models (blue) cluster at

\bar{\delta}\approx 0

with margins below

1.6

Figure 10: Theoretical quantities across the depth-width grid for a ResNet on Cifar-10 with SiLU activation.

Configuration	Sign causal (§ 4 )	Sign-kNN tracks accuracy (§ 3.1 )	$Z_{1}$ certifies
FFNN + SiLU/GELU/ReLU	✓	✓	✓
FFNN + Tanh	✓	✓	× (small $\delta$ )
ResNet-18 + SiLU	✓	✓	× (skip connections)
ViT + GELU	✓	✓	not tested

Appendix

Table 3: Summary of gate-domination evidence across activations and architectures.

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