The phase-shift formula for multi-block box-ball systems

Let a box-ball system have $n$ blocks, with $Q_k$ balls in the $k$-th block, and suppose the blocks are separated sufficiently so that the asymptotic soliton structure can be identified by ordering $(Q_k)_{k=1}^n$. After sufficiently many time steps, once all collisions have finished, the phase-shift conjecture. The total phase shift experienced by the $k$-th block is

$$2\sum_{\substack{j>k\Q_j<Q_k}} \min(Q_j,Q_k)-2\sum_{\substack{j<k\Q_j>Q_k}} \min(Q_j,Q_k).$$

In particular, if $Q_1>Q_2>\cdots>Q_n$, then the phase shift experienced by the $k$-th block is

$$2\sum_{j>k} \min(Q_j,Q_k)-2\sum_{j<k} \min(Q_j,Q_k).$$

This extends the two-block phase-shift calculation and proposes the expected cumulative contribution of every collision; its general validity remains open in the source.