Chapter 12

The room goes silent.Dr. Brooks speaks carefully. "Lawrence, you sent Elijah's proof to an external reviewer without permission?""I sent a potential solution to an unsolved problem to a colleague. That's standard peer review."Dr. Park turns to Elijah. "Do you understand the objection?"Elijah's mouth is dry. "He's saying I skipped a step.""Not skipped—assumed." Whitfield walks toward the board. "You assumed something that requires proof. Your argument is circular: you used your conclusion to prove your conclusion. In mathematics, circular reasoning is death."Elijah stares at the screen: line 127, his own handwriting projected ten feet high. He thought he had checked everything.In Roxbury, Dr. Okonkwo grips the armrests, her knuckles turning white. On stage, Elijah's hands start to shake.Elijah's voice comes out steadier than he feels. "Can I see the line he's talking about?"They project his notebook: page 8, line 127. His own handwriting fills the screen. He reads it once, twice, his mind racing through logic he built four months ago.The silence stretches: ten seconds, twenty. Then Elijah looks up."Dr. Tanaka is right. That line is wrong."The audience gasps. In Roxbury, someone starts crying. Whitfield tries not to smile. "Well, there we have it.""But he's only right because he's reading it wrong."The room freezes."Line 127 says bipartite structure holds under infinite extension. Dr. Tanaka thinks I'm claiming it holds for all infinite extensions—that would be circular." Elijah walks to the board, picks up the stylus with shaking hands. "But line 119 already defines that I'm only talking about periodic extensions. The bipartite property doesn't need separate proof; it's inherited from the periodicity constraint."Dr. Brooks checks the notebook. "He's right. Line 119 limits the domain. Tanaka's objection doesn't apply."Whitfield doesn't back down. "That's semantic at best. The notation is ambiguous.""The notation is clear if you read the proof in order," Dr. Ruiz says.

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