Sicherman dice have non-standard faces: one shows {1, 3, 4, 5, 6, 8}, the other {1, 2, 2, 3, 3, 4}. Roll them together and the sum distribution is identical to standard dice. Different mechanism, same statistics. They were invented in 1977 as a curiosity.
Tamuz and Sandomirskiy (Mathematische Annalen, 2025) used Sicherman dice — and their infinite family of generalizations — to prove that the Boltzmann distribution is the only probability law that correctly describes independent systems without creating false connections between unrelated choices. Every alternative theory, when tested against the full space of possible dice, produces inconsistencies: it would say that your preference for one die face changes the probability of an outcome on the other die, even when the dice are physically independent.
The proof strategy is worth noting. You can't test uniqueness against one example, or ten, or a thousand. Any finite set of dice might be consistent with many distributions. The proof works because polynomials — the mathematical objects representing dice — form an infinite-dimensional space, and only the Boltzmann distribution maintains independence across all of it. The crazy dice aren't just a pedagogical illustration. They're an infinite family of stress tests, and only one law survives them all.
The same distribution appears in economics as the multinomial logit model, where it describes consumer choice. There, the requirement is that irrelevant alternatives don't affect the choice between relevant ones — the economic version of physical independence. The convergence isn't a coincidence. It's the same mathematical constraint wearing different labels.
The deeper point: the Boltzmann distribution was discovered empirically, justified thermodynamically, and derived from maximum entropy arguments. Now it has a uniqueness proof from a direction nobody expected — the combinatorics of non-standard dice. The law isn't just the best description of independent systems. It's the only one that doesn't break independence. There is no room for an alternative.