And he found it.
Step three. Caldwell's solution assumed that the variable $x$ was always greater than zero across the entire domain. She never tested the boundary; she never checked what happens when $x$ equals zero. She just assumed it couldn't be and built her entire answer on that assumption.
But Miles knew better. He'd seen this exact trap before. Two years ago on an MIT open forum, a graduate student posted a similar problem. The thread argued about it for weeks. Miles spent three weeks in his grandmother's kitchen working through every edge case. He knew what happened at the boundary. When $x$ equals zero, the function behaves completely differently. Caldwell's optimal answer doesn't just become inaccurate; it becomes wrong. Flat-out, provably, mathematically wrong.
Miles opened his notebook. Right there, standing in front of the bulletin board, he wrote a clean correction on a single page.
First, he identified the hidden assumption: "Step three assumes $x$ is always positive. That's never proven. It's just assumed."
Second, he constructed a specific boundary case: a concrete set of values where $x$ equals zero. Simple numbers, easy to check, undeniable.
Third, he showed the consequence: plug those values into Caldwell's final answer, and the result was worse than a basic approach. Her optimal solution was actually inferior at the boundary.
Fourth, he solved it correctly using modular arithmetic, the same method she deducted eight points for. On the page, his solution was four lines: clean, elegant, and correct in every case, including the one Caldwell missed.
He tore the page from his notebook and pinned it right next to Caldwell's solution on the bulletin board. Then he picked up his bag and walked away. No announcement, no drama; just the math, side by side for anyone who cared to look.
By noon, they were looking.
It started with two students from the honors track. They stopped, read both solutions, and pulled out their calculators. Five minutes later, they called their friends over. By the end of lunch, there were fifteen students crowded around the bulletin board, checking, rechecking, and whispering.
One by one, the same realization hit: Miles was right. Caldwell was wrong.
Daphne Sullivan pushed through the crowd. She read Caldwell's solution, then read Miles' correction. Then she sat on the hallway bench and did the boundary calculation herself from scratch. Her face went through three stages: confusion, verification, and then something she clearly didn't enjoy feeling—the realization that she, Caldwell's best student, hadn't caught it either.
She looked up from her notes and said quietly to no one in particular, "He's right."