"I didn't say they can't," Elijah replied. "That's not my claim. I said the ordinary coloring problem and the compatible repeating-recipe problem are different questions. I'm only trying to prove the recipe works for the family I defined."
Whitfield stood. "Then let's test the limits of your reasoning. Suppose I give you a construction with a nonperiodic arrangement of regions. Does your method have anything to say about it?"
Elijah shook his head. "Not unless it satisfies my conditions, sir." Whitfield seemed annoyed by the careful answer because it left him no easy contradiction to announce.
He walked to the digital board and began drawing an intricate network. The lines spread across the screen, crossing near clusters of numbered vertices and looping around a complicated corner. He used colors to indicate an attempted assignment, moving quickly and speaking confidently as he worked. "Here we have an arrangement that doesn't repeat in the way your examples do," he said. "And we can still color it, which means your insistence on periodicity adds nothing."
Elijah watched the drawing evolve. He was not certain how to respond to the philosophical point, but his attention kept returning to a tight cluster at the top right. Two blue regions seemed to share a short edge. He blinked and adjusted his glasses, afraid that the projection had played another trick on him. Then Whitfield stepped away and declared the example finished.
"Dr. Whitfield," Elijah said, "could you zoom in on the area near regions forty-seven and fifty-two?"
Whitfield sighed. "What now?"
"I think those two are both blue. I can't see clearly from here, but if they're neighbors, this coloring doesn't work."
The request brought the auditorium to a near silence. Whitfield enlarged the corner, and an unmistakable blue line touched another blue region along a shared boundary segment. Ruiz reached for a pen. Brooks leaned across the table to examine the connection. A few mathematicians in the audience stood for a better look.
"He's right," Brooks said at last. "The coloring shown on the screen violates its own adjacency rule."
Whitfield stared at the display. "I drew it too quickly," he said. "A minor drafting error."
"I do that," Elijah answered, relieved that nobody was asking him to explain a graph he had never studied. "That's why I write the region numbers beside the colors. I caught the same mistake in my first notebook." His attempt to be friendly drew a burst of laughter that was not aimed at him. Some people smiled because he had treated the great professor as a fellow human capable of mistakes, a kindness Whitfield had not offered him.
Ruiz raised a hand. "Lawrence, whether this example can be repaired is beside the point. Elijah has been careful about the scope of his claim, and so far the objections haven't addressed the lemma he's actually proposing. I would like to hear the rest." Her voice was formal, but the reprimand was unmistakable. Whitfield returned to his chair, his shoulders rigid.