Revisiting the Platonic Representation Hypothesis: An Aristotelian View — part 3/8
Cleaned-text code points [31277, 46820). Bibliography omitted here; source.md includes it.
For bounded similarities with maximum s max s_{\max} (often s max = 1 s_{\max}=1 ), we report a max-preserving calibrated aggregate T cal = max ( T obs − τ α agg s max − τ α agg , 0 ) . T_{\mathrm{cal}}=\max\!\left(\frac{T_{\mathrm{obs}}-\tau_{\alpha}^{\mathrm{agg}}}{s_{\max}-\tau_{\alpha}^{\mathrm{agg}}},\,0\right). (17) This score satisfies T cal = 0 T_{\mathrm{cal}}=0 when T obs ≤ τ α agg T_{\mathrm{obs}}\leq\tau_{\alpha}^{\mathrm{agg}} and T cal = 1 T_{\mathrm{cal}}=1 when T obs = s max T_{\mathrm{obs}}=s_{\max} . As above, T cal T_{\mathrm{cal}} depends on α \alpha via τ α agg \tau_{\alpha}^{\mathrm{agg}} ; we therefore report both T cal T_{\mathrm{cal}} (magnitude above null) and p agg p_{\mathrm{agg}} (evidence against null), applying multiplicity correction ( Holm, 1979 ; Benjamini and Hochberg, 1995 ) when many model pairs are evaluated.
### 5.3 Summary
To compute a calibrated similarity score: (i) fix a significance level α \alpha ( e.g. , α = 0.05 \alpha=0.05 ); (ii) generate K K null scores by permuting sample correspondences; (iii) compute critical value τ \tau as the ⌈ ( 1 − α ) ( K + 1 ) ⌉ \lceil(1-\alpha)(K+1)\rceil -th order statistic of the combined set (observed + null scores); (iv) return calibrated score, either s cal s_{\mathrm{cal}} or T cal T_{\mathrm{cal}} .
Use scalar calibration ( Section 5.1 ) when comparing a single pair of representations. Use aggregation-aware calibration ( Section 5.2 ) when reporting a summary statistic ( e.g. , maximum) over multiple layer pairs. Appendix D provides pseudocode for both procedures.
## 6 Experiments
We quantify the effects of the width and depth confounders in controlled synthetic experiments and show that our calibration framework effectively removes them. We then revisit the Platonic Representation Hypothesis using our calibration framework, assessing which convergence trends remain robust after controlling for these confounding factors.
### 6.1 Null-calibration removes width confounder
We validate that our calibration eliminates width-related inflation of similarity across metrics, regimes, and noise distributions, without metric-specific derivations.
We design controlled synthetic experiments as follows. Under H 0 H_{0} , we draw 𝐗 , 𝐘 ∈ ℝ n × d \mathbf{X},\mathbf{Y}\in\mathbb{R}^{n\times d} independently from Gaussian and heavy-tailed (Student- t t , Laplace) distributions. We vary the number of samples n ∈ { 128,256,512 , 1024 , 2048 , 4096 } n\in\{128,256,512,1024,2048,4096\} and the dimension d ∈ { 128,256,512 , 1024 , 2048 } d\in\{128,256,512,1024,2048\} . Under H 1 H_{1} , we inject a shared low-rank signal component and vary the signal-to-noise ratio. We evaluate representative metrics spanning three families. For spectral similarity, we use linear and RBF CKA , as well as CCA /SVCCA/PWCCA; for neighborhood similarity, we use mKNN (with k = 10 k=10 ); and for geometric similarities, we use RSA and Procrustes. Figure 3 reports a subset of these metrics for readability; additional metrics are reported in Section E.7 . For calibration, we use K = 200 K=200 permutations with α = 0.05 \alpha=0.05 .
Under H 0 H_{0} , uncalibrated scores are systematically inflated, while our calibrated scores stay at zero across settings ( Figure 3 ). This confirms that the similarity scores of wider models can arise purely from high-dimensional finite-sample effects, and our calibration removes this spurious baseline. Importantly, the magnitude of the null baseline is metric-dependent, consistent with our theory: CKA’s baseline scales as 𝒪 ( d / n ) \mathcal{O}(d/n) ( Proposition 4.1 ), while mKNN ’s baseline scales as 𝒪 ( k / n ) \mathcal{O}(k/n) ( Proposition 4.2 ). Intuitively, mKNN compares local neighborhood overlap at a fixed k k , thus only comparing relationships instead of local distances, making its null baseline insensitive to representation width d d , which explains the order-of-magnitude gap observed in raw scores. The same pattern holds for heavy-tailed noise ( Section E.7 ).
The same width inflation arises under a genuine shared signal, and calibration corrects it there as well, as we show analytically ( Proposition C.5 ) and on real networks at fixed n n ( Section E.5 ).
Next, we verify the statistical guarantees of our empirical null calibration. For Type-I error control, rejection rates stay at or below the nominal α = 0.05 \alpha=0.05 across ( n , d / n ) (n,d/n) configurations ( Figure 4 a). Crucially, our calibration does not sacrifice sensitivity to real alignment: detection rates increase rapidly with signal strength ( Figure 4 b). Overall, our calibration preserves signal structure: in the high-signal regime, raw and calibrated scores show the same pattern, while in the low-signal regime, calibration correctly gates scores to zero ( Section E.2 ).
Furthermore, we verify that our empirical calibration closely matches existing analytical bias corrections for CKA ( Murphy et al., 2024 ) , recovering the width correction without metric-specific derivation ( Section E.4 ).
Additionally, we perform ablations on different noise distributions used in the synthetic experiments ( Section E.1 ), different calibration approaches ( Section E.3 ), and an ablation on the influence of the number of permutations K K used for calibration ( Section E.6 ).
### 6.2 Null-calibration removes depth confounder
We validate that our aggregation-aware null-calibration eliminates the depth confounder. To build a controlled synthetic setting, we construct two synthetic models, A A and B B , each with L L layers. Under H 0 H_{0} , we sample layer representations { 𝐗 ℓ } ℓ = 1 L \{\mathbf{X}_{\ell}\}_{\ell=1}^{L} and { 𝐘 ℓ ′ } ℓ ′ = 1 L \{\mathbf{Y}_{\ell^{\prime}}\}_{\ell^{\prime}=1}^{L} , where each 𝐗 ℓ , 𝐘 ℓ ′ ∈ ℝ n × d \mathbf{X}_{\ell},\mathbf{Y}_{\ell^{\prime}}\in\mathbb{R}^{n\times d} has i.i.d. 𝒩 ( 0 , 1 ) \mathcal{N}(0,1) entries (independent across layers and between models), using d / n = 8 d/n=8 to match the upper range of the Platonic Representation Hypothesis setting. We then compute the layerwise similarity matrix S ℓ , ℓ ′ = CKA ( 𝐗 ℓ , 𝐘 ℓ ′ ) S_{\ell,\ell^{\prime}}=\operatorname{CKA}(\mathbf{X}_{\ell},\mathbf{Y}_{\ell^{\prime}}) and summarize it with standard aggregates.
The uncalibrated max-aggregated scores inflate with layer count even under H 0 H_{0} ( Figure 6 ): raw max-scores are systematically higher at L = 128 L=128 than at L = 2 L=2 , despite no true signal. Our aggregation-aware calibration eliminates this bias: calibrated aggregates remain stable regardless of depth. We further show that naively calibrating each scalar comparison still leads to inflation, highlighting the importance of calibrating the final statistic. Furthermore, since deeper models tend to be wider as well, raw comparisons are doubly confounded.
### 6.3 Revisiting the Platonic Representation Hypothesis
A central claim behind the Platonic Representation Hypothesis is that, as models become more capable, their representations begin to converge across modalities. We revisit this claim through our calibration framework to determine whether the observed alignment reflects genuine shared representation structure or instead arises from width and depth confounders.
We follow the experimental protocol of Huh et al. (2024) using n = 1024 n=1024 image–text pairs (WIT; Srinivasan et al. (2021) ) and embeddings from three language model families (BLOOM, OpenLLaMA, LLaMA) and five vision model families (ImageNet-21K, MAE, DINOv2, CLIP, CLIP-finetuned) across multiple scales. As in Huh et al. (2024) , we extract per-layer representations using the class token for vision models and mean pooling over tokens for language models. This yields 204 vision–language model pairs spanning d / n ∈ [ 0.19 , 8 ] d/n\in[0.19,8] . For each pair, we compute layer-wise similarity and report the maximum across layers, as in the original work. We evaluate both global spectral metrics ( CKA linear/RBF) and local neighborhood metrics ( mKNN , cycle- k k NN, CKNNA). Following Huh et al. (2024) , we evaluate mKNN , cycle- k k NN, and CKNNA with k = 10 k=10 . We further apply Benjamini-Hochberg FDR correction ( Benjamini and Hochberg, 1995 ) to control for multiple comparisons across model pairs.
For the global similarity, we find that uncalibrated CKA scores increase with model scale (dotted lines in Figure 5 a), reproducing the trend interpreted as evidence of cross-modal convergence ( Huh et al., 2024 ) . However, this trend disappears after our calibration (solid lines): calibrated CKA shows no systematic increase with model size. This indicates that global convergence in uncalibrated CKA is largely attributable to width and depth confounders rather than a genuine increase in representational similarity. The same holds for the other global metrics, including shape- and geometry-based ones: SVCCA , the RV coefficient, and Procrustes distance all lose their apparent scaling trend after calibration ( Section E.8 ).
In contrast, for the local similarity, evidence of cross-modal convergence remains strong for neighborhood-based metrics even under our calibration ( Figure 5 b). The same qualitative conclusion holds for other neighborhood-based measures (cycle- k k NN and CKNNA; Section E.8 ) and different choices of α \alpha ( Section E.11 ). Further analysis ( Section E.10 ) reveals that models converge in local neighborhood structure: models increasingly agree on which points are neighbors, but do not agree on the pairwise distances, since CKA-RBF with a small bandwidth shows no alignment after calibration.
CKA and mKNN capture different invariances, so their raw magnitudes are not directly comparable. We therefore compare, for each metric separately, how strongly its score tracks language-model capability before and after calibration (its Pearson correlation with the model ranking). After calibration this correlation collapses for global metrics (for linear CKA it falls from 0.86 0.86 to 0.45 0.45 , and for Procrustes distance from 0.89 0.89 to 0.39 0.39 ) but is essentially unchanged for local ones ( mKNN stays near 0.85 0.85 and CKNNA near 0.87 0.87 ). Section E.8 reports all metrics.
To test whether these findings generalize beyond images and text, we extend our analysis to video–language alignment following Zhu et al. (2026) . We compare video encoders (VideoMAE small/base/large/huge, fine-tuned on Kinetics) against the same language model families. Consistent with our previous findings, the global similarity ( CKA ) shows no trend with model capacity ( Figure 7 ). In contrast, for local similarity ( mKNN ), a clear scaling trend emerges with VideoMAE-Large/Huge, whereas smaller video encoders appear to act as a bottleneck, limiting alignment regardless of language model size. This confirms that local neighborhood convergence extends to video–language alignment, provided that representations are sufficiently powerful. Section E.9 further compares non-finetuned VideoMAE versions and a variety of image models at the frame level on the same dataset, showing the same trend.
Taken together, these results suggest a refined version of the Platonic Representation Hypothesis. After calibration, we find little evidence that representations converge in global spectral structure as models scale, at least under the considered setting. What reliably persists is local geometric alignment: different models preserve similar neighborhood relationships among inputs. We therefore propose the alternative Aristotelian Representation Hypothesis : As models become capable, their representations converge to shared local neighborhood relationships .
## 7 Conclusion
Representational similarity metrics are widely used to study learned features, but their interpretation is systematically distorted by two artifacts: width-dependent null baselines and depth-dependent selection inflation. We introduced a unified null-calibration framework that corrects both, turning similarity scores into effect sizes with principled zero points and valid p p -values. Applying our framework to the Platonic Representation Hypothesis reveals that previously reported global spectral convergence is largely confounded by width and depth, whereas local neighborhood alignment remains significant, motivating an Aristotelian Representation Hypothesis.
##### Relationship to the Platonic hypothesis.
The Aristotelian hypothesis refines rather than refutes the Platonic one. Global convergence implies local convergence, because matching the full geometry preserves neighborhoods, but the converse does not hold. The Platonic hypothesis therefore implies the Aristotelian one, which is a weaker criterion. Our experiments support this local form and, after calibration, find little evidence for the global one. Prior reports of convergence thus remain compatible with our analysis, and the Aristotelian hypothesis offers a more precise lens on what converges across models and modalities.
##### Limitations and outlook.
Our conclusions come with two caveats. First, representational similarity has no ground-truth scale, so we report the presence or absence of calibrated evidence for convergence rather than proving it. Second, our guarantees assume exchangeability, so grouped or clustered samples require restricted permutations that preserve their dependence structure. Why representations converge in local neighborhoods but not in global geometry remains the key open question.
## Acknowledgements
We thank Artyom Gadetsky, Siba Smarak Panigrahi, Debajyoti Dasgupta, David Frühbuss, Shin Matsushima, Rishubh Singh, Adriana Moreno Castan, and Gioele La Manno for their valuable suggestions, which helped improve the manuscript. We are especially grateful to Simone Lionetti for additional input and support. We gratefully acknowledge the support of the Swiss National Science Foundation (SNSF) starting grant TMSGI2_226252/1, SNSF grant IC00I0_231922, SNSF grant 10.004.411, and the Swiss AI Initiative Large Call #32. M.B. is a CIFAR Fellow in the Multiscale Human Program.
## Impact statement
This paper presents work whose goal is to advance the field of machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
## Appendix A Existing calibration approaches for representational similarity metrics
## Appendix B Metrics and score definitions
This appendix gives the definitions of the similarity metrics s ( 𝐗 , 𝐘 ) s(\mathbf{X},\mathbf{Y}) used throughout the paper. The main text focuses on the calibration procedure ( Sections 5.1 and 5.2 ). Here we provide concrete instantiations of the metrics referenced in Section 3 and Section 6 .
### B.1 Preprocessing and basic notation
Let 𝐗 ∈ ℝ n × d x \mathbf{X}\in\mathbb{R}^{n\times d_{x}} and 𝐘 ∈ ℝ n × d y \mathbf{Y}\in\mathbb{R}^{n\times d_{y}} denote row-aligned representations evaluated on the same n n inputs. We use the centering matrix 𝐇 = 𝐈 n − 1 n 𝟙 n 𝟙 n ⊤ , \mathbf{H}\;=\;\mathbf{I}_{n}-\frac{1}{n}\mathbbm{1}_{n}\mathbbm{1}_{n}^{\top}, (18) where 𝐈 n ∈ ℝ n × n \mathbf{I}_{n}\in\mathbb{R}^{n\times n} is the identity matrix and 𝟙 n ∈ ℝ n \mathbbm{1}_{n}\in\mathbb{R}^{n} is the all-ones vector. We define row-centered representations 𝐗 c = 𝐇𝐗 \mathbf{X}_{c}=\mathbf{H}\mathbf{X} and 𝐘 c = 𝐇𝐘 \mathbf{Y}_{c}=\mathbf{H}\mathbf{Y} . Unless stated otherwise, similarities are computed on centered representations.
### B.2 Raw similarity metrics