Effective Dimension Ratios under Symmetry Augmentation

We ask how data group symmetry $G$ acts on the effective dimension $d_{\mathrm{eff}} := (\mathrm{tr}\,\Sigma)^2 / \mathrm{tr}(\Sigma^2)$ of learned representations in deep networks. Under Reynolds decomposition $\Sigma = \Sigma_{\mathrm{inv}} \oplus \Sigma_{\mathrm{var}}$ and an isotypic isotropy assumption, $d_{\mathrm{eff}}$ admits a closed form in the symmetry index $S := \mathrm{tr}(P_G \Sigma P_G)/\mathrm{tr}(\Sigma)$: $$d_{\mathrm{eff}} = D \cdot \psi(S, |G|), \quad \psi(S, |G|) := \frac{1}{|G|\bigl(S^2 + (1-S)^2/(|G|-1)\bigr)};$$ however, this absolute prediction breaks down in practice because trained representations are strongly anisotropic. Our main result (Lemma 1') shows that under a mild conditional-invariance assumption on the task-induced spike, the **ratio** of effective dimensions across two augmentation conditions is universally given by the participation-ratio prefactor ratio: $$\frac{d_{\mathrm{eff}}(p{=}1)}{d_{\mathrm{eff}}(p{=}0)} = \frac{\psi(S(1), |G|)}{\psi(S(0), |G|)} \cdot (1 + O(\Delta\rho)).$$ On CIFAR-10 with three groups ($Z_2$, $D_2$, $D_4$) and a sweep of $20$ width--data cells, the prediction holds with median error $9.2\%$, $10.0\%$, and $5.6\%$ respectively (max $15.4\%$). This ratio is a representation-level signature of symmetry that holds independently of whether augmentation is helpful or harmful at the loss level.