Oracle symmetry-level selection

Add a symmetry-selection wrapper around a sparse functional network. Define candidate orbit sizes Q = {1, 2, 4, 8, ...} and train a shared-weight orbit-averaged model for each candidate, or evaluate several q values using nested prefixes of one maximum orbit. On a held-out validation set V, compute the empirical bias proxy A_hat_q as the average transformed-input loss minus the original-input loss, using the displayed formula. Also compute a consistency diagnostic D_q = (|V|q)^(-1) sum_{x in V} sum_j ||f_theta(g_j applied to x) - f_theta(x)||^2, which helps distinguish task noise from genuine symmetry violation. Estimate beta by fitting the slope of log validation error versus log m on a small pilot experiment; for the first implementation, set beta_hat = 1 and compare sensitivity to this choice. Estimate h from the median spacing of observed locations or from the bandwidth of a local interpolation layer, set q_sharp = floor(c_K/h), and score each q with (n m^2 q)^(-beta_hat/(beta_hat+1)) + lambda A_hat_q. Select the smallest q within one validation standard error of the minimum to avoid noisy over-selection. Alternatively use the simplified rule: select the largest q with A_hat_q <= tau/n and q <= q_sharp, calibrating tau on a development split. The mathematical contribution is the explicit decreasing variance proxy, increasing bias term, 1/n stopping floor, and saturation cap; lambda, tau, beta_hat, and h are estimated engineering choices. Start with synthetic periodic regression where a controllable asymmetric perturbation is added to otherwise rotation-invariant curves. Compare q = 1, fixed maximum q, and selected q. Success is lower test error than both fixed baselines under mixed symmetry, with selected q increasing as data become sparser or noisier and decreasing as asymmetry increases.