O(n)-parameter transitive pairwise head

(1) Use this in a ranking, preference, retrieval, or set-comparison module whose current final output is an n by n score matrix. Replace its final pairwise layer with a linear or MLP head that outputs alpha of shape [batch,n-1]. Store a fixed tensor B of shape [n-1,n,n], normalized so (B[k]*B[k]).sum() is one. (2) Reconstruct with S=(alpha[..., :, None, None]*B).sum(dim=-3), set the diagonal to zero, and use sigmoid(S[i,j]) for pairwise probabilities or sort a utility vector recovered from the basis coordinates. The simplest equivalent implementation predicts utilities u of shape [batch,n] and computes S=u[..., :,None]-u[...,None,:]; the basis form is useful when the paper's sparse or windowed basis is desired. (3) The fixed basis, dimensionality reduction, and cycle guarantee come from the paper; alpha is learned by ordinary backpropagation. For variable-size sets, use a maximum-size basis with a validity mask, or generate an orthonormal centered basis for each n at runtime. (4) First run a synthetic n=32 Bradley-Terry benchmark and a MovieLens pairwise-preference benchmark. Compare the O(n^2) unconstrained pairwise head with the O(n) transitive head at matched hidden width and training steps. Measure parameter count, activation memory, forward latency, pairwise accuracy, ranking regret, and triangle-cycle count. The expected result is zero predicted cycles, substantially lower memory and latency, and comparable or better held-out ranking accuracy when labels contain contradictions. A negative result would be a large clean-data accuracy gap caused by the transitivity assumption.