Even though you're right in general, this is not the case here! We have proved that no smaller counter-example exists (again by using a computer search through optimized SAT-solving).
As for the question of why no smaller counter-examples, I'm afraid I don't have any nice answers and perhaps there simply isn't a nice answer. Let me explain what I mean. My advisor Marijn Heule finished the resolution of Keller's conjecture (a conjecture about how N-dimensional cubes work in the Euclidean N-dimensional space), and the final answer is that the conjecture fails for the first time in dimension 8. "Why 8?" Again it's the same situation: it seems that that's just the way math is, something in the way the definitions and the objects behave makes it so that is the smallest counterexample. "Why" is a hard question to answer...
As for the question of why no smaller counter-examples, I'm afraid I don't have any nice answers and perhaps there simply isn't a nice answer. Let me explain what I mean. My advisor Marijn Heule finished the resolution of Keller's conjecture (a conjecture about how N-dimensional cubes work in the Euclidean N-dimensional space), and the final answer is that the conjecture fails for the first time in dimension 8. "Why 8?" Again it's the same situation: it seems that that's just the way math is, something in the way the definitions and the objects behave makes it so that is the smallest counterexample. "Why" is a hard question to answer...