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.

翻译：等变性理论预测，架构对称性先验可将样本复杂度降低|G|倍；这一结论被广泛引用，但鲜少在控制混杂因素的缩放律中通过分离先验效应进行测量。在受控的C_n对称任务中，我们报告三项发现。第一，在轨道大小相等且计算量匹配条件下，错误群组控制的性能低于无约束基线（联合成对CI [+0.79, +3.26]排除了零，在不同估计器间表现稳健）；错位约束不仅无益，反而具有实际危害。第二，配备测试时轨道平均的数据增强基线可与等变模型精确匹配——在匹配单元上逐周期验证曲线达到比特级一致——因此架构与增强方法的差距条件性地取决于非对称测试时计算，而非无条件存在。第三，相对交换率beta_diff = 1.28在符号和量级上与理论值1.0一致（单层CI [+0.92, +2.05]）；更保守的双层自举法（种子×群组大小）将区间扩大至[-0.63, +1.72]（包含零），而采用sqrt(2)间距精细N网格的复现结果不具决定性（点估计-0.82）。方法论贡献——可消解共享难度混杂效应的相对交换率估计器、错误群组控制方法及预定义失败分类体系——可迁移至任何强度可参数化的归纳偏置。诚实声明：初始分析揭示正斜率可辨识性问题后，主估计量beta_diff系事后采用；设计从未经过外部预注册；核心数值基于粗粒度N网格上七个群组大小的OLS斜率。本研究属探索性分析而非验证性测量；错误群组结果最为清晰，也是我们报告时最具信心的发现。基于新种子的注册复现留待后续工作。