Somewhere between Phases 4 and 5, Richard holed up at the Schloss during Mud Month—the weeks following the end of the ski season—and wrote out some ideas that had been brewing in his mind since the deepest and most lugubrious weeks of Phase 3. Reading them, Corvallis had identified this as an "inflection point," which was another of those terms that meant nothing to Richard but that was—to judge from the vigorous shifts in body language it elicited in meetings—of infinite significance to math geeks. As far as Richard could make out, it denoted the hardly-obvious-at-the-time moment when, seen later in retrospect, everything had changed.