Whitfield touched the projection controls. The extra number disappeared. "Very observant," he said in a voice so thin the compliment barely survived. "Now perhaps we can finally examine your actual work." Elijah nodded, feeling a little steadier, and fitted his flash drive into the presentation console. For a second he feared that it would not recognize the files, but his first slide appeared, showing a hand-drawn tiled rectangle surrounded by four colored pencils.
He began at the beginning because Okonkwo had taught him that simple definitions were a form of kindness. Two regions were neighbors if they shared an actual boundary segment, not merely a point. A proper coloring used different colors for neighboring regions. The classical four-color theorem gave an existence result for planar maps, but Hartwell's periodic-extension question imposed extra requirements on how a coloring behaved when a repeating patch was continued in different directions.
"A wall of tiles can repeat even if the colors painted on it don't repeat the same way," Elijah explained. "That's the first thing I had to learn. When I say a coloring is compatible with the repeating patch, I mean the edge rules match wherever two copies of the patch are joined. The real puzzle is to find a repeating recipe, not just to color a picture that happens to be big."
Ruiz nodded as he spoke. His language was simple, but it avoided a common error: he did not claim that ordinary infinite planar graphs were uncolorable, or that adding periodicity was necessary for all four-colorings to exist. Instead, he identified a specialized constructive problem with additional boundary conditions. Whitfield knew that distinction very well, and it made him more attentive than he wanted anyone to notice.
Elijah switched slides. A colored diagram filled the screen, with little arrows at the edges of each repeating unit. "I tried doing every possible coloring," he said, "but there are too many, and most are the same after you rename the colors. Then I started writing down what I need to know at a boundary instead of every color inside a square. If the boundary information works when I go across and when I go up, it should work when I reach the same corner from both directions."
He called that the corner check. His notebook contained a proposed method for arranging the admissible boundary states into a finite transition structure and checking a consistency condition for a particular class of periodic networks. The claim was narrower than a sweeping solution to all imaginable periodic maps. It concerned a family whose local patches met explicit compatibility assumptions, which were listed on the final page of his manuscript. The importance lay in whether those assumptions were sufficient to make his construction general across that family.
Whitfield lifted his hand. "You're confusing a sufficient condition with a necessary one," he said. "You've shown some repeating configurations behave well. That does not mean nonperiodic configurations cannot be colored." His tone made the distinction sound fatal. A few audience members nodded before Elijah answered.