  We show that the Hausdorff distance between any forward and any backward
surgery paths in the sphere graph is at most 2. From this it follows that the
Hausdorff distance between any two surgery paths with the same initial sphere
system and same target sphere system is at most 4. Our proof relies on
understanding how surgeries affect the Guirardel core associated to sphere
systems. We show that applying a surgery is equivalent to performing a Rips
move on the Guirardel core.
