Brown-measure stability initialization

Integrate this at initialization and as an optional diagnostic for the recurrent matrix in an RNN, S4-like state-space model, or linear residual block. Factor the transition as J=A B, where A is the task-specific deterministic matrix or a learned diagonal/low-rank operator and B=U diag(s) V^T with U and V sampled as Haar-orthogonal matrices generated by QR decomposition of Gaussian matrices; include a scalar alpha multiplying B. For a calibration width, use K=6 to 10 and estimate m_k=(1/N)tr[(JJ^T)^k] with Hutchinson probes. For each random Rademacher vector v, set q_0=v and repeatedly apply q_{j+1}=JJ^Tq_j; estimate m_k by the average of v^Tq_k over probes. Fit the truncated moment polynomial M(x), numerically invert M on z in [-1,0], and solve S(F(r)-1)=1/r^2 over a radius grid. Use one-dimensional bisection on alpha so the predicted outer radius r_out is 0.95 to 0.99 for a plain RNN, or 1+epsilon for a residual or discretized state-space transition. Optionally require a predicted inner radius above 0.2 to avoid rapid mode collapse. The paper-derived quantities are the radial law and S-transform relation; moment truncation, Hutchinson estimation, and bisection are engineering approximations. Since finite-width and non-normal effects can invalidate the asymptotic proxy, also estimate the actual spectral radius with Arnoldi and the largest singular value with power iteration; if prediction error exceeds 10%, use empirical rescaling. First test a width-256 tanh RNN on copy-memory and sequential-MNIST, comparing standard orthogonal initialization, scaled Gaussian initialization, and the proposed initialization at equal parameter count. Measure gradient-norm variance versus sequence length, final accuracy, loss descent at equal FLOPs, and the actual eigenvalue cloud. Success is longer copy-memory retention and lower gradient variance without an increased largest Jacobian eigenvalue.