For five minutes the room listened without interruption. Elijah described how he had first noticed the corner problem, how his early attempts failed, and how converting the boundary choices into states made the reasoning easier to inspect. He spoke slowly, sometimes looking down to recover a sentence he had rehearsed. When he mentioned a false lemma from his first notebook, he even smiled. "I liked this one," he told the panel, "but it was wrong, so I kept it to remind myself not to use it again."
At the sixth minute Whitfield raised his hand. Park acknowledged him with a warning glance. "The presenter has described his problem as potentially ill-posed in earlier notes," Whitfield began. "Elijah, can you explain what you mean by that using Hadamard's criteria for a well-posed problem?" Elijah stared at him. He had heard the name during practice but did not know the formal criteria, and the sudden switch from coloring constructions to terminology made his thoughts scatter.
"I don't know that definition," he said. "I meant that the old versions of Hartwell's statement didn't always say which boundary conditions they required, so different people could be answering different questions."
Whitfield turned slightly toward the audience. "There is our difficulty. A promising intuitive observation should not be confused with full professional preparation." Ruiz raised a hand before he could continue.
"Lawrence, Hadamard's terminology is not necessary to interpret the claim before us. Elijah has explained the ambiguity he means, and he has supplied formal definitions in his manuscript. We should examine those."
Whitfield tightened his lips but remained seated. Elijah tried to continue, and Park gave him a few seconds to locate his place. Then the professor raised another question, this time holding a printed email. He explained that Tanaka in Kyoto had reviewed part of the argument overnight and identified what might be an unproved extension step. With Park's permission, the text of the technical question appeared on the screen, stripped of the commentary Whitfield had prepared around it.
Elijah recognized the passage immediately. His face warmed because the displayed line was one he had already underlined the night before. Tanaka's note asked whether a property preserved across one admissible patch must also hold across an arbitrary infinite extension. Without a domain restriction, the inference could be circular. Whitfield folded his hands as though the answer were obvious and waited for Elijah to admit defeat.
"Can I see my earlier definition?" Elijah asked.
Park nodded to the projection operator, who returned to the requested page. Elijah found the line restricting the argument to specified admissible periodic extensions and explained that he was not quantifying over arbitrary extensions. Tanaka's objection was valuable, he said, because the sentence Whitfield had projected was imprecise if read alone. He would rewrite it to refer explicitly to the earlier domain definition, but that correction did not introduce the circular step Tanaka was worried about.