Revisiting the Platonic Representation Hypothesis: An Aristotelian View — part 8/8

Cleaned-text code points [109382, 117409). Bibliography omitted here; source.md includes it.

The pattern mirrors the image–language findings. For spectral metrics, raw scores suggest alignment, whereas calibrated scores drop significantly, indicating that much of the apparent alignment is attributable to width and depth confounders. In contrast, neighborhood metrics retain significant alignment after calibration, confirming that video and language representations share local topological structure. This local alignment strengthens with the capability of the video encoder: both fine-tuning on Kinetics and increasing scale raise the calibrated neighborhood alignment, in line with the Aristotelian hypothesis that more capable representations converge more in local structure. Calibration removes the spectral inflation throughout.

### E.10 Characterizing the locality of cross-modal alignment

The main text establishes that local neighborhood metrics retain significant alignment after calibration, while global spectral metrics do not. We next characterize how local this alignment is. Both mKNN and CKA -RBF have hyperparameters that control their sensitivity to local versus global structure. By varying these parameters, we can characterize the scale at which cross-modal alignment emerges.

##### Experimental setup.

We vary two locality parameters: the neighborhood size k k in mKNN , testing k ∈ { 10 , 20 , 50 , 100 } k\in\{10,20,50,100\} where smaller values focus on immediate neighbors and larger values consider broader local structure, and the RBF kernel bandwidth σ \sigma in CKA -RBF, testing σ ∈ { 0.1 , 0.5 , 2.0 , 5.0 } \sigma\in\{0.1,0.5,2.0,5.0\} , which controls the length scale over which the kernel assigns significant weight.

##### RBF bandwidth.

The RBF (radial basis function) kernel is defined as k ( 𝐱 , 𝐲 ) = exp ( − ∥ 𝐱 − 𝐲 ∥ 2 / 2 σ 2 ) k(\mathbf{x},\mathbf{y})=\exp\left(-\|\mathbf{x}-\mathbf{y}\|^{2}/2\sigma^{2}\right) . The bandwidth σ \sigma determines the length scale of similarity. When σ \sigma is small ( e.g. , 0.1 0.1 ), the kernel is sharply peaked: only very close points contribute significantly to the Gram matrix, making the similarity measure sensitive to exact pairwise distances in the immediate neighborhood. When σ \sigma is large ( e.g. , 5.0 5.0 ), the kernel is broad: even moderately distant points contribute, and the similarity measure aggregates information over larger neighborhoods, becoming sensitive to coarser geometric structure.

##### Neighborhood size.

For mKNN , the parameter k k controls how many nearest neighbors are considered when measuring overlap. Small k k ( e.g. , 10 10 ) measures agreement on immediate neighbors, i.e. , the closest points to each sample, capturing fine-grained local topology. Large k k ( e.g. , 100 100 ) measures agreement on a broader neighborhood. With n = 1000 n=1000 samples and k = 100 k=100 , we ask whether the 10 % 10\% closest points agree across representations. Crucially, mKNN is a rank-based metric: it asks which points are neighbors (ordinal information), not how close they are (cardinal information).

##### mKNN across k k values.

Figure 26 shows the PRH alignment results for mKNN with varying k k . A consistent pattern emerges: all k k values show significant alignment after calibration, with calibrated scores remaining well above zero even at k = 100 k=100 . However, the scaling trend is most pronounced at small k k . For k = 10 k=10 , raw scores show a clear upward trend with model capacity that persists after calibration. At large k k , this trend flattens even in raw scores. For k = 100 k=100 , raw scores plateau for larger models, suggesting that broader neighborhood agreement is already saturated across model scales. This pattern indicates that scaling-driven improvement in alignment is concentrated at the finest topological level.

##### CKA-RBF across bandwidth values.

Figure 27 , and the accompanying p p -values in Figure 28 , show results for CKA-RBF with varying bandwidth σ \sigma , revealing a different pattern from mKNN. At σ = 0.1 \sigma=0.1 (very local), there is no significant alignment after calibration: raw scores are near 1.0 1.0 , reflecting the high similarity of any high-dimensional representations under a sharply peaked kernel. However, calibrated scores collapse to approximately zero with p p -values exceeding 0.05 0.05 for most model pairs, indicating that the observed similarity is indistinguishable from chance. At σ = 0.5 \sigma=0.5 , alignment emerges, but with a flattening trend after calibration. Calibrated scores initially rise with model scale, then plateau and slightly decline for the largest models. At σ = 2.0 \sigma=2.0 and σ = 5.0 \sigma=5.0 , significant alignment persists, but the calibrated trend also flattens, resembling the pattern observed for large- k k mKNN: alignment exists, but scaling-driven improvement disappears after calibration.

##### Topological versus metric alignment.

The contrasting behavior of mKNN and small- σ \sigma CKA-RBF reveals a fundamental distinction in what “local alignment” means. On one hand, mKNN measures topological alignment: do the representations agree on which points are neighbors? This captures ordinal information where the ranking of distances matters but not their absolute values. On the other hand, small- σ \sigma CKA-RBF measures metric alignment: do the representations agree on how close neighbors are? This captures cardinal information where exact distance values matter.

The fact that mKNN shows alignment at all k k values while small- σ \sigma CKA-RBF shows no alignment reveals that cross-modal representations agree on neighborhood identity (which points are close) but not on exact local distances (how close they are). This finding is consistent with the observation that different training objectives and architectures induce different distance scales in representation space while preserving the relative ordering of neighbors. The Aristotelian Representation Hypothesis should therefore be understood as convergence to shared topological structure rather than shared metric structure.

### E.11 Sensitivity to significance level α \alpha

The main text uses a significance level of α = 0.05 \alpha=0.05 throughout. To confirm that the PRH conclusions are not sensitive to this choice, we repeat the PRH evaluation from Section 6.3 with α ∈ { 0.01 , 0.05 , 0.10 } \alpha\in\{0.01,0.05,0.10\} for representative global (CKA linear, CKA RBF) and local ( mKNN with k = 10 k=10 ) metrics.

Figures 29 , 30 and 31 show that the conclusions are entirely invariant to the choice of α \alpha . For global metrics, calibrated scores show no convergence trend at any significance level. For local metrics, calibrated scores retain their alignment trend across all three α \alpha values. Stricter thresholds ( α = 0.01 \alpha=0.01 ) produce slightly lower calibrated scores, while more permissive thresholds ( α = 0.10 \alpha=0.10 ) produce slightly higher ones, but the qualitative pattern is unchanged. This confirms that our findings are not an artifact of a particular significance level.

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