GKP Log-Concave Lag Mixer

Integrate this at the lag-weight generation point of a causal temporal mixer, local attention head, or depthwise 1-D convolution. Let K be the maximum lag and learn six unconstrained scalars per head or channel group. Convert them to nonnegative recurrence parameters with softplus: alpha=softplus(a), beta=softplus(b), gamma=softplus(c), alpha_p=softplus(a_p), beta_p=softplus(b_p), gamma_p=softplus(c_p). At every forward pass compute a row of length K+1 using the recurrence below; use n=K, or optionally concatenate several rows n=1,...,K and learn a softmax over rows. Pseudocode: T=zeros(K+1); T[0]=1; for n in 1..K: new=zeros(K+1); for k in 0..n: new[k]=(alpha*n+beta*k+gamma)*T[k] + (alpha_p*n+beta_p*k+gamma_p)*(T[k-1] if k>0 else 0); T=new; w=(T+eps)/(sum(T)+eps*(K+1)); y=sum_k w[k]*V[t-k]. Compute the recurrence in float32 even under mixed precision, then cast w to the activation dtype. The mathematical guarantee is exact only for nonnegative fixed parameters; do not add unconstrained per-lag perturbations if preserving log-concavity is the goal. Empirically record the number of sign changes in first differences of w and the fraction of rows violating w[k]^2 >= w[k-1]*w[k+1]. First experiment: compare this mixer against a learned softmax lag kernel, depthwise convolution, and ALiBi-style local attention in a 6-layer small Transformer on Copy, Adding, and WikiText-2 sequence-length extrapolation tasks. Match parameter count and multiply-adds. Success means lower validation perplexity or copy error at equal compute, improved extrapolation to 2x training context, and substantially fewer multimodal lag kernels; a secondary test is whether the structured mixer remains stable when K is increased without retraining all lag parameters.