Spectral-gap adaptive Sinkhorn attention

Integrate this into the Sinkhorn normalization branch of an attention layer, immediately after attention logits S are converted to a positive kernel A=exp((S-c)/tau), where c is the per-row maximum used for numerical stability. Assume square sequence attention first, with p=q=1/L for sequence length L; for rectangular cross-attention use uniform marginals with equal total mass. Initialize u=v=ones and perform alternating updates in log-space: log u <- log p - log(A v) and log v <- log q - log(A^T u), using logsumexp rather than explicit division. After every two updates, construct the row and column marginal residuals r_k; do not materialize extra matrices beyond the current normalized attention X. Compute the residual ratio rho_hat from consecutive residual norms, smooth it with rho_ema <- 0.8 rho_ema + 0.2 rho_hat, and estimate the remaining iteration count from j_pred=ceil(log(eps/||r_k||)/log(rho_ema)). Continue until either ||r_k||<=eps or the configured maximum iteration count is reached. Require at least two iterations before early stopping, clamp rho_ema to [0.05,0.999], and reject a prediction if the residual increased. The exact mathematical quantity is rho=1-sigma_2; sigma_2 can optionally be estimated on a small subsampled kernel by 5-10 power iterations on B=P^{-1/2}XQ^{-1}X^T P^{-1/2}, while the MVP should use residual ratios as a cheaper proxy. First test on a small 4-8 layer Transformer trained on WikiText-2 or a synthetic assignment task, comparing fixed 10/20 Sinkhorn iterations against adaptive stopping at equal output tolerance. Measure average iterations, normalization FLOPs, marginal residual, training loss, validation perplexity, and gradient-norm variance. Success means at least 25% fewer average iterations at the same residual tolerance with no degradation in perplexity or assignment accuracy; stability is indicated by fewer residual spikes and NaNs at low attention temperature.