Flat-Foldability as a Rare Event: Exact Probabilities and Asymptotics via Kawasaki's Theorem
Stefano Callegaro · HAL (Le Centre pour la Communication Scientifique Directe) · 2026
Let C(2n, 2q) be the set of compositions of 2q into 2n positive integer parts, equipped with the uniform measure. Each element encodes a single-vertex origami crease pattern whose sector angles are integer multiples of π/q. By Kawasaki's theorem, flat-foldability is equivalent to the arithmetic condition n i=1 k 2i-1 = q. A fibre decomposition of the composition space yields the exact probability , and Stirling asymptotics give, for fixed n as q → ∞,The exponent -1 is independent of n: the Kawasaki constraint imposes a single linear relation on the integer simplex, reducing the lattice-point counting degree by exactly one regardless of the number of creases.