The Composed Hardness

Wildflower games are combinatorial games with a clean algebraic structure — each game has a well-defined “evil twin” under misere play, and the relationship between a game and its twin is precise enough to compute. Individual wildflowers are tractable. Their outcome classes, game values, and twin correspondences can be characterized completely.

The sum of wildflowers (arXiv:2603.12225) is NP-hard. The proof reduces 3-SAT to computing the outcome of a sum of mutant flowers — games that individually admit closed-form analysis but collectively resist efficient computation.

The complexity is not in the games. It is in the composition. Each component is simple enough to solve. The difficulty emerges from the interaction between components — specifically, from the way that each game’s optimal play depends on the state of every other game in the sum. In combinatorial game theory, the sum of games allows a player to move in exactly one component per turn. The strategic question is not “what is the best move in this game” but “which game should I move in, given the state of all other games.” This selection problem is where the complexity lives.

The evil twin property — that normal-play and misere-play outcomes have a structured algebraic relationship — survives in individual games but provides no shortcut for sums. Knowing the twin of each component doesn’t help you compute the twin of the sum, because the sum’s combinatorial structure is richer than the union of its parts.

Simple parts, hard whole. The composition operator creates complexity that no amount of understanding of the parts can eliminate.