She wasn't nodding, wasn't waving, wasn't mouthing words of encouragement. She was just sitting there, looking at him, with a face that said one thing and one thing only: I've watched you carry this your whole life. I'm not going anywhere.
Miles looked at her for two seconds, maybe three. Then he turned back to the board and uncapped the marker.
He didn't find courage from anger. He didn't find it from pride. He found it from the quietest person in the room: a seventy-two-year-old woman in the back row wearing the nicest dress she owned, who never understood a single equation he wrote, but who understood him.
Miles put the marker to the board, and he didn't hesitate again.
Miles wrote the first line: clean, steady. The marker didn't shake. He underlined step three of Caldwell's solution, drew a small bracket around the assumption, and then next to it, in clear block letters, he wrote: "This assumes the function is always positive across the full domain."
He turned to the audience for one second—not for approval, just to make sure they were reading. They were, every single one of them.
Then he went back to the board. He constructed a boundary case: a specific set of values—simple numbers, nothing exotic—where the variable reaches zero. He plugged those values into Caldwell's step three and showed, line by line, what happens to the function at the boundary.
It breaks. Not approximately, not arguably; completely. Caldwell's optimal solution at the boundary produces a result that is mathematically worse than doing nothing at all. Her answer doesn't just become inaccurate; it becomes wrong.
He wrote Caldwell's final answer on the left side of the board. Then he wrote the correct answer for the boundary case on the right side. Two numbers, side by side, completely different. Undeniably, visually, obviously different. No one in the audience needed a math degree to see it: two numbers that didn't match. That was enough.
But Miles wasn't done.
He moved to a clean section of the board, and he solved the entire problem from scratch. Not Caldwell's way—his way. Modular arithmetic: the method Caldwell had called unclear, the method she deducted points for, the method she erased when Daphne used it ten minutes ago on this same stage.
His handwriting was steady. Each step flowed into the next like water finding its path downhill: no hesitation, no crossing out, no going back.
Step one: reframe the constraint. Instead of assuming the function was always positive, he transformed the variable so the boundary case was built into the structure from the start.
Step two: apply the modular framework. Reduce the problem space. Eliminate the noise.