Balanced-Jordan Residual Mixer

Integrate this at the token-mixing operation of a residual Transformer block, or at the state-transition matrix of a linear/nonlinear SSM. For a sequence length or state width \(N\), choose \(k=\lfloor N/2\rfloor\) or \(k=\lceil N/2\rceil\), and construct a sparse row-sum-zero directed tridiagonal or banded Laplacian using a balanced chain-plus-core pattern: two sparse chains of lengths approximately \(k\) and \(N-k\) feed into a central coupling block. Use nonnegative edge weights and set each diagonal to the outgoing-weight sum so \(L{\bf 1}=0\). In the forward pass replace ordinary token mixing \(XW\) by \(X+\alpha MX\), where \(M=-L\), or equivalently use \(X'=(I-\alpha L)X\); constrain \(\alpha=\alpha_{\max}\sigma(\rho)\), with \(\alpha_{\max}=1.9/\lambda_{\max}(L)\). Pseudocode: initialize balanced \(L\); estimate \(\lambda_{\max}\) by five-step power iteration; set \(M=I-\alpha L\); for each layer compute \(Y=\operatorname{Norm}(MX)\), \(Z=X+gY\), then apply the usual MLP; optionally learn only edge weights and \(\rho\), retaining the sparsity mask. The paper supplies the Jordan-size and rank rationale; estimate \(\lambda_{\max}\), activation growth, and gradient amplification empirically. First test a six-layer, width-128 Transformer on Tiny Shakespeare or CIFAR-10 with sequence length \(N=32\), comparing this mixer against a random tridiagonal mixer, dense linear mixing, and identity residuals at equal FLOPs. Log \(\|M^t\|_2\), maximum activation norm, gradient-spike frequency, and validation loss versus steps. Success means lower early-training gradient and activation spikes, faster loss descent at equal FLOPs, and no degradation in final validation accuracy.