Minimum pushes in Sokoban
A push changes both crate access and future geometry, while an ordinary player step changes only position. Crate Bloom therefore ranks pushes first and moves second, and never calls a stored replay globally optimal without a suitable proof.
Pushes versus moves
Two solutions may use the same number of pushes but very different walking routes. Minimum pushes means no solution exists with fewer crate pushes. Minimum moves is a separate, stricter claim that also compares every player step.
The proof used here
The admission solver groups player positions that can reach one another without pushing, then explores every legal next push in breadth-first order. The first completed push-state depth proves the minimum push count within the fully explored admitted graph.
Why reference moves are labeled
The replay stored with a level is legal and reaches completion. Its move count is shown as a reference, not a global minimum, because collapsing a reachable player region intentionally does not prove the shortest walk among equal-push solutions.
How to improve
First match the verified push target. Then keep the same crate order and shorten walking by ending each push near the standing square for the next one. Compare like with like: assisted results remain playable but are marked Assisted.