Row-balanced recurrent initialization

Integrate this at the recurrent-weight initialization and optionally as a training regularizer in a vanilla tanh RNN or gated-free state-space model. Choose hidden width $N$, gain $g$, timestep $\Delta t$, and a base matrix $J^0$ with iid Gaussian entries of standard deviation $g/\sqrt{N}$. For each row compute $\text{row\_mean}[i]=\text{mean}(J^0[i,:])$, then set $J^{(b)}=J^0-b\,\text{row\_mean}[:,\text{None}]$. Rescale $J^{(b)}$ so its empirical element variance equals $g^2/N$, preserving the comparison with the unbalanced baseline. Use the discrete update $x_{t+1}=\rho x_t+(1-\rho)J^{(b)}\phi(x_t)+Bu_t$, where $\rho=1-\Delta t$. During training, either freeze this structure or add $\lambda_{\mathrm{balance}}\mathcal{L}_{\mathrm{balance}}$ to the task loss, computing the penalty on the actual recurrent matrix after every optimizer step. Track row-sum RMS, hidden-state norm, gradient norm, task loss, hidden covariance rank, and a finite-time Lyapunov estimate. Estimate the latter by evolving two nearby states and repeatedly renormalizing their separation: $\hat\lambda=T^{-1}\sum_t\log(\|\delta_t\|/\|\delta_{t-1}\|)$. The row-centering and gain-preservation rules come from the mathematical construction; the regularization coefficient and balance schedule are empirical. Start with sequential MNIST or a synthetic delayed-copy task, $N=256$, and compare $b\in\{0,0.25,0.5,0.75,1\}$ for tanh, sigmoid, softsign, and ReLU at matched parameter count and initialization variance. Success means fewer exploding-gradient events, faster loss descent, or longer memory at equal compute. Test the predicted odd-function caveat directly by comparing tanh with sigmoid: balance should have a weaker qualitative effect for tanh than for a non-odd saturating activation.