Positive-Cycle Matching Repair

(1) Integrate this after the network computes a compatibility matrix S in R^(n x n) between n query or red embeddings and n candidate or blue embeddings, and before the downstream loss or routing operation. Obtain an initial bijection pi using greedy row-wise assignment, truncated Sinkhorn rounding, or the router's current top-1 decisions. (2) Construct a directed graph on red indices. For every ordered pair (i,j), compute w_ij=S[i,pi[j]]-S[i,pi[i]], with w_ii=0. Repeatedly search for a positive directed cycle. If C=(i_1,...,i_m) has G(C)=sum_r w_{i_r,i_{r+1}}>epsilon, replace pi[i_r] by pi[i_{r+1}] for every r. For an MVP, enumerate all 2-cycles and 3-cycles, costing O(n^3) per repair pass. For arbitrary cycles, run Bellman-Ford on edge weights -w; a negative cycle in -w is a positive-gain cycle in w. Stop after a fixed number of augmentations or when the largest detected gain is at most epsilon. (3) The exact mathematical quantities are each cycle gain and the guaranteed score increase after an accepted rotation. The initial matching, cycle-length limit, epsilon, and number of passes are empirical choices. During training, sample 16-64 cycles of lengths 2-6 and add lambda times L_cycle to the task loss; detach the discrete pi or use soft Sinkhorn probabilities for a differentiable approximation. (4) First test on a small two-tower retrieval model or a 4-layer Transformer with batches of 32-128 paired items. Compare greedy matching, 20-step Sinkhorn, Hungarian matching, and greedy-plus-cycle-repair on total assignment score, downstream accuracy, latency, and collision count. The expected signal is a consistent score increase over greedy or truncated Sinkhorn at equal inference cost, convergence toward Hungarian-quality assignments, and lower residual positive-cycle mass on validation batches.