When both sides value the case very differently, how does a mediator actually close that gap?

Slowly, confidentially, and with reality rather than pressure. Big gaps usually come from each side privately believing its best-case scenario will hold — the plaintiff assuming full liability and a runaway verdict, the defense assuming contributory negligence or a weak expert wins the day. In caucus I test both of those assumptions against what a jury in that venue is actually likely to do, what each contested ruling would cost if it goes the wrong way, and what another year of litigation and fees does to the net recovery on each side. I don't tell anyone what to accept. What I do is narrow the range by replacing optimism with risk-adjusted numbers, find the terms beyond money that can bridge the last distance, and keep both sides at the table long enough to get there. Most cases that feel hopeless at 9 a.m. have a workable shape by late afternoon once the real risk is on the table.