Multiplicity-free Schur channel tying

Apply this to a two-stream feature mixer, such as two views, two augmentation streams, two heads, or two fermionic feature channels. Let h^(1) and h^(2) have identical token dimensions. Instead of concatenating them and using a dense unconstrained projection, construct homogeneous monomials u^(a-t)v^(b+t) for each selected pair a>=b; represent the corresponding channel subspace W_(a,b) with a shared parameter matrix B_(a,b) applied identically to all t. Concretely, create q_(a,b)[t]=Mix(h^(1),h^(2)) projected to degree (a-t,b+t), stack q over t, and compute y_(a,b)[t]=sum_s B_(a,b)[t,s]q_(a,b)[s], with B shared across tokens and optionally constrained to commute with the GL2 action. Generate the polynomial basis recursively from the displayed [r]_(u,v) formula. For a small model, use degrees (1,0), (1,1), (2,0), and (2,1), retaining only blocks whose activation variance exceeds a threshold. For small n, compute Hilbert coefficients from the Kostka formula; for larger n, estimate graded feature ranks from activation covariance. Compare against an untied MLP mixer with equal FLOPs on two-view classification or signed-set regression. Success is a 20–40% reduction in mixer parameters and activation memory at equal accuracy, or higher accuracy at equal parameter count. Also test stream swapping: the tied model should degrade less than the untied baseline when the two inputs are exchanged.