Orthogonal transitivity projection for pairwise logits

(1) Insert the layer after a network produces pairwise logits Z with shape [batch,n,n], and before Bradley-Terry, sigmoid, softmax, or ranking losses. First form S=(Z-Z.transpose(-1,-2))/2; optionally mask invalid pairs. Precompute basis matrices B[k] with shape [n-1,n,n], using the paper's orthogonal basis or an equivalent QR-orthogonalized basis spanning matrices u 1^T-1 u^T. (2) Compute denom[k]=(B[k]*B[k]).sum(), alpha[k]=(S*B[k]).sum((-1,-2))/denom[k], and S_hat=(alpha[...,k,None,None]*B[k]).sum(-3). Use loss=bce_with_logits(S_hat, pairwise_labels) and retain gradients through all operations. (3) The basis and projection coefficients are computed exactly from the linear-algebra formula; no spectrum or hyperparameter estimate is required. If ratios M are used instead of logits, compute L=log(clamp(M,eps)), project L, and return exp(L_hat). (4) Start with n=16 or n=32 synthetic Bradley-Terry data: draw true utilities u, label pairs from sigmoid(u_i-u_j), then flip 10%, 20%, and 30% of labels to inject inconsistent cycles. Compare an unconstrained pairwise head, projection only at inference, and projection during training. Report pairwise accuracy, ranking regret, held-out noisy-label loss, number of predicted 3-cycles, and gradient variance. The expected signal is fewer cycles and better noisy-data generalization at similar clean accuracy, with no measurable inference instability.