Vacancy-preserving collision-free router

Integrate this as a token-to-slot router before a sparse MoE block, slot-attention block, external memory read, or one-to-one cross-attention head. Use a group of m tokens and n>=m+1 slots, with logits z_{i,j} produced by a learned router. Maintain a hard assignment and one vacant slot. At each routing update, greedily choose a target assignment by sorting logits while respecting unused columns. Construct a path from the current assignment to the target: whenever row i must move to a currently occupied target column, first move the row occupying that column into the vacancy, then move row i into the newly vacant column. During a move from column j to vacant column k, use lambda_i=(1-alpha)e_j+alpha e_k for alpha increasing from zero to one; all other rows remain one-hot. Evaluate the sparse module using these weights, or use the interpolation as a short continuation over several training steps. For soft supports, store each row's weights as barycentric coordinates in Delta(S_i), and only interpolate states whose combined support has no repeated column. The exact math is row normalization, column exclusion, and affine feasibility; target selection and move schedules are empirical algorithms. First test m=8,n=9 on a synthetic permutation task, then compare against softmax top-1 routing and Sinkhorn routing in a small MoE classifier or set transformer. Record collision rate, dropped-token rate, routing churn, gradient variance, throughput, and task loss. Success requires zero collisions, smoother loss under token perturbations, and equal accuracy with less overflow or routing overhead than the baseline.