Projective stationary-energy initialization

(1) Integration point: use a recurrent cell with hidden state h=(h^(1),h^(2)), where each block has width W. This modifies only hidden-state initialization and optionally a slow normalization layer; it does not change the recurrent update. (2) Calibration pseudocode: run a baseline model for N short trajectories; for every hidden state compute x_t=(||h_t^(1)||^2,||h_t^(2)||^2,<h_t^(1),h_t^(2)>)/(||h_t||^2+1e-8). Fit a linear map Mhat by least squares from x_t to x_{t+1}; start from the empirical mean x and iterate x <- Mhat x / ((Mhat x)_1+(Mhat x)_2) for 20 to 100 steps. Use the converged x=(a,b,c) as the target block energies and correlation. (3) To sample an initial state, draw z_1,z_2 independently from N(0,I_W), set h^(1)=sqrt(a) z_1, set r=c/sqrt(ab+1e-8), and set h^(2)=sqrt(b)(r z_1+sqrt(max(0,1-r^2)) z_2); normalize the concatenated state. The projective update F and the moment formulas come from the paper. Mhat, the fixed point, and any exponential moving average are estimated empirically. (4) First experiment: compare zero, iid Gaussian, orthogonal, and projective-energy initialization in a two-block linear RNN and GRU on sequential MNIST and the adding problem. Measure loss during the first 20 sequence steps, time until block energies reach their stationary values, gradient norms, and final validation accuracy. The variance prediction C/W can be checked across widths W in {32,64,128,256}. Success is a shorter warm-up transient, lower early loss, and better long-sequence accuracy without extra parameters or per-step computation.