Step three: bridge the two paths. Show that the standard approach and the boundary approach converge to the same answer when, and only when, the assumption is removed.
Step four: simplify the final expression. One line, four terms, reduced to its absolute cleanest form.
On the left side of the board: Caldwell's solution, half a page of steps, an assumption buried in the middle, and a wrong answer at the boundary.
On the right side: Miles' solution, four lines, no assumptions, correct in every case.
He capped the marker, placed it on the tray, turned around, and looked at Caldwell. He didn't say a word.
The auditorium went silent. Not gradually—instantly, like someone had pressed mute on three hundred people at once. The air conditioning hummed. A chair creaked somewhere in the back. That was it.
Dr. Nina Ashford, third row center, was leaning forward. Her eyes moved across the board: left side, right side, left side again. She was checking, verifying, running the logic in her head the way only someone with a PhD in mathematics can.
Then she nodded slowly, once.
Vice Principal Ellison, standing by the side door, uncrossed his arms. He took a half step forward. His face was unreadable, but his posture said everything: he was paying attention now.
Daphne Sullivan, in the student section, closed her eyes. When she opened them, they were wet. She pressed her lips together and clapped once against her knee, then again—quiet, almost private.
A student in the middle section started clapping. Then another. Then five more. Then the parents, then the board members. It built like a wave—not an explosion, but a wave, starting small and growing with every second until three hundred people were on their feet, the sound filling Founders Hall from the stage to the back wall.
Miles stood on the stage. He didn't pump his fist. Didn't bow. Didn't smile. He just stood there, the marker tray behind him, two solutions on the board, and three hundred people standing.
Dr. Ashford rose from her seat. She didn't wait for the applause to die; her voice cut through it, clear, measured, and loud enough for the entire hall: