PLMS Copula Augmentation Layer

Integrate this at the latent augmentation point of a VAE, contrastive encoder, or multimodal embedding model. Given a minibatch of latent vectors z in R^d, choose two coordinates or two examples, transform each selected coordinate to an empirical uniform value using U=rank(z)/(B+1), and sample a PLMS map alpha. Store a partition 0=a_0<...<a_N=1 and a sign s_i in {+1,-1} for each branch. For every scalar u, locate interval i and compute v=(u-a_{i-1})/(a_i-a_{i-1}) when s_i=+1, otherwise v=(a_i-u)/(a_i-a_{i-1}). Map back through the desired marginal inverse CDF, or simply use v as a latent coordinate. Pseudocode: u = rank(z_coord)/(B+1); i = bucketize(u,a); v = where(sign[i]>0,(u-a[i-1])/(a[i]-a[i-1]),(a[i]-u)/(a[i]-a[i-1])); z_aug = replace(z_coord,v). Estimate achieved dependence empirically with pairwise concordance, while the uniform-marginal guarantee and branch formulas come directly from the PLMS construction. Start with CIFAR-10 SimCLR using a 2-layer projection head and compare independent noise, Gaussian-copula noise, and PLMS coupling at equal augmentation count and FLOPs. Test partitions with two to eight branches and random sign patterns. Success is higher linear-probe accuracy or contrastive retrieval accuracy at equal training compute, with marginal histograms remaining uniform and measured tau matching the intended coupling.