Degree-Calibrated Stable Residual Flow

Use this in a neural ODE, continuous-depth ResNet, or recurrent state-space block. Set z_star to zero initially and parameterize the hidden dynamics as f_phi(z,t,x) = -a r^m z + P_perp(z) h_phi(z,t,x). For an Euler block, update z_next = z + delta_t f_phi(z,t,x); for longer trajectories use an adaptive ODE solver. At sampled hidden states, compute r and the radial derivative z transpose f_phi(z), then add L_rad to the task loss. A hard alternative is to rescale any proposed update whose empirical radial derivative exceeds the allowed value. Compute the radial quantities exactly on minibatch states; estimate global validity by sampling states, inputs, and small perturbations because the penalty is not a global certificate. The paper supplies the degree-to-decay comparison; the tangent learned field and sampled penalty are the proposed neural adaptation. First test m equal to 0, 1, and 2 against an unconstrained residual MLP and a spectrally normalized residual block on a synthetic long-horizon sequence task and sequential MNIST. After the input is removed, measure hidden-state norms for 100 integration steps and plot r on log-log axes. The prediction is exponential decay for m equal to zero and a log-log slope near -1/m for r when m is positive, with no asymptotic growth provided the radial inequality holds. Increase delta_t to identify the numerical instability boundary and report whether the observed decay law persists below that boundary.