Order-Parameter Mode Activation Schedule

Apply this to a quadratic module or to spectral channels of an MLP projection, attention value projection, or convolutional channel mixer. During a short calibration pass, collect hidden features h_b and estimate their covariance C = (1/N) sum_b h_b h_b^T. Compute the top r eigenvectors Q of C and use them as approximate shared data eigenmodes. Parameterize an initialization W = Q diag(sqrt(m_0)) R^T, where R is a small random mixing matrix and m_0 contains the desired mode energies. Set m_k(0) = epsilon_0^2 exp(-2 alpha k), as suggested by the onset-time equation. Pseudocode: C = running_covariance(features); eigenvalues,Q = eigh(C); Q = top_r(Q); m0 = eps0**2 * exp(-2*alpha*arange(r)); W = Q @ diag(sqrt(m0)) @ R.T; train normally; m = diag(Q.T @ W @ W.T @ Q). Estimate r_k from the initial slope of log(m_k + delta) over the first 20 to 100 minibatches; estimate K only if needed by fitting the observed mode derivatives to the linear form r_k - sum_j K_kj m_j. The mathematical prediction is the multiplicative mode law and logarithmic onset time; the shared eigenbasis is only approximate and must be measured. First test on a 2-layer MLP or a 2-layer Transformer on CIFAR-10 or a small language-model corpus, comparing isotropic initialization against alpha values from 0 to 2. Log loss, gradient variance, mode energies, mode participation entropy, and predicted versus observed onset times. Success is lower early gradient interference, smoother loss, or faster loss reduction at equal FLOPs without sacrificing final accuracy. A negative result is no correlation between predicted and observed onset times or slower convergence for all alpha values.