Measuring the Symmetry--Data Exchange Rate

• URL: http://arxiv.org/abs/2606.01090v1
• Date: Sun, 31 May 2026 08:17:40 GMT
• Title: Measuring the Symmetry--Data Exchange Rate
• Abstract summary: We show that a wrong-group control with identical orbit size and matched compute is worse than no constraint.<n>Relative exchange rate beta_diff = 1.28 is consistent in sign and order of magnitude with the theoretical bootstrap 1.0.<n>The methodological contributions -- the relative-rate estimator that cancels the shared-difficulty confound, the wrong-group control, and a pre-specified failure taxonomy -- transfer to any inductive bias whose strength can be parameterised.
• Score: 0.2538209532048867
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Equivariance theory predicts that an architectural symmetry prior reduces sample complexity by a factor of |G|; this is widely cited but rarely measured as a scaling law with controls that separate the prior from its confounds. On a controlled C_n-symmetric task, we report three findings. First, a wrong-group control with identical orbit size and matched compute is worse than no constraint (joint pairwise CI [+0.79, +3.26] excludes zero, robust across estimators); misaligned constraint is actively harmful, not merely unhelpful. Second, an augmentation baseline equipped with test-time orbit averaging matches the equivariant model exactly -- bit-identical per-epoch validation curves across matched cells -- so the architecture-vs-augmentation gap is conditional on asymmetric test-time computation, not unconditional. Third, the relative exchange rate beta_diff = 1.28 is consistent in sign and order of magnitude with the theoretical 1.0 (single-level CI [+0.92, +2.05]); the more conservative two-level bootstrap (seeds x group sizes) widens this to [-0.63, +1.72], including zero, and a finer-N replication on a sqrt(2)-spaced grid is inconclusive (point estimate -0.82). The methodological contributions -- the relative-rate estimator that cancels the shared-difficulty confound, the wrong-group control, and a pre-specified failure taxonomy -- transfer to any inductive bias whose strength can be parameterised. Honest scoping: the primary estimator beta_diff was adopted post-hoc after the initial analysis revealed a positive-slope identifiability problem; the design was never externally pre-registered; and the headline number rests on an OLS slope over seven group sizes on a coarse N grid. This is an exploratory study, not a confirmatory measurement; the wrong-group result is the cleanest finding and the one we report with the most confidence. A registered replication on fresh seeds is future work.

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