Structure is in the zoom: probing neural symmetry through dimensionality scaling

How much of a neural population's measured dimensionality reflects its representational structure, and how much is generated by the sampling process itself? We decompose the scaling exponent into a matched-null floor, the component retained under label-blind subsampling, and an axis-resolved structural shift $\delta$. On mouse-V1 calcium-imaging data, a random-stimulus axis produces pure floor ($\delta \approx 0$, 18 recordings); a direction-aligned axis yields positive shifts ($+0.20$ to $+0.46$ in all eight grating recordings) that vanish under label permutation. The successful axis follows the code's approximate $O(2)$ symmetry: the signal correlation decomposes into sectors dominated by the quadrupole (orientation, $\ell{=}2$) over the dipole (direction, $\ell{=}1$) by $2.4\times$, and the harmonic coefficients rationalize the ordering of every ladder we tested, including one that reversed a registered prediction. The result replicates in a second laboratory: across 167 Neuropixels populations (32 mice, 10 visual areas), the shift is positive in 158 and correlates specifically with orientation-tuning strength ($r = +0.41$) rather than direction selectivity. A calibrated generative model and a model-free subspace measurement show that the shift depends jointly on between-class harmonic structure and class-dependent within-class covariance. On ground truth, Ising and nematic lattices and a rotation-equivariant network, the shift reads exactly the structure each system is known to contain, and the network's measured harmonics prospectively predict its ladder ordering. With a matched floor and a declared probe axis, dimensionality scaling becomes an axis-resolved measurement of neural and artificial representations rather than a single global summary; when the axis follows a known symmetry, the shift resolves its sector content.