q-Boson Spectral Initialization

Integrate this at initialization of every wide MLP or projection matrix W in a transformer block, before training starts. Choose a truncation size L such as 32 or 64 and q in {0.5,0.8,0.95}. Build the symmetric tridiagonal T_L with off-diagonals sqrt((1-q^r)/(1-q)), compute its eigenpairs using a symmetric eigensolver, and form the nonnegative singular-value list s_j=alpha*abs(lambda_j). For an m-by-d layer with rank r=min(m,d), draw independent Haar orthogonal matrices U_m and V_d, select r quadrature nodes by sampling indices j with probabilities p_j=(U_T[0,j])^2, and set W=U_m diag(s_{a_1},...,s_{a_r}) V_d^T. A simpler deterministic option repeats the L-node list to fill rank r. Normalize alpha using the displayed second-moment formula, then apply ordinary residual-branch scaling separately. The paper supplies the Jacobi recurrence and moment construction; eigendecomposition, node sampling, and matrix assembly are standard numerical operations. Measure actual layer singular spectra, activation variance across depth, gradient norms, and largest singular values during the first 1,000 updates. First test a 12-layer width-256 MLP on CIFAR-10 against Xavier, orthogonal, and truncated-normal initialization at equal parameter count and optimizer settings. Success means lower activation and gradient variance, faster loss decrease, or stable training at a depth or learning rate where Xavier becomes unstable.