Duality-Calibrated Jacobian Spectrum

1. Integration point: apply the method to the recurrent transition matrix of an RNN or SSM, or to minibatch Jacobians of an implicit layer. Start with state widths 64 to 256, where dense eigendecomposition is affordable. 2. Pseudocode: for each batch, compute \(J_t\) by automatic differentiation; for small states, calculate all eigenvalues \(z_i\); estimate the two-dimensional spectral density with a Gaussian kernel; locally unfold eigenvalue coordinates using that density; bin pair distances to obtain \(g_\theta(r)\); evaluate the selected class curve \(g_c(r)\); add the displayed correlation and stability penalties to the task loss; then backpropagate. For larger states, estimate leading eigenvalues with Arnoldi and use randomized Jacobian-vector products. 3. The paper supplies the symmetry classes and universal target statistics; the finite-width density, eigenvalues, and pair histogram are estimated empirically. 4. First experiment: use a small long-horizon sequence-prediction task and compare a vanilla RNN, spectral-radius regularization alone, and this method at equal parameter count and optimizer budget. Ablate the pair-statistics term while retaining the radius margin. Measure rollout loss, transient amplification, and distance to the target correlation curve. 5. Pre-register the signature: increasing \(\lambda_g\) must reduce correlation error monotonically, and the result must remain similar for widths 64, 128, and 256. At matched measured \(\rho(J)\), transient amplification should be lower than with radius-only control. Reject the transfer if pair-correlation error fails to decrease by 30%, long-horizon error fails to improve by 10% over the radius-only baseline, or the inferred class statistics vary substantially with width.