Which piece to keep: limiting side counts for iterated random chord cutting of convex polygons

We study how the retained piece determines the long-run side count of a repeatedly cut convex polygon. Each cut joins independent uniform points on a uniformly chosen pair of distinct sides, and one piece is kept. If a piece is kept with probability proportional to its number of sides plus an offset, the stationary side count is shifted confluent hypergeometric, and its mean decreases strictly in the offset towards the uniform-retention value. Among all retention rules, including those using geometry and history, keeping the piece with more sides maximizes the expected side count at every step. Its limit is a quotient of two explicit, rapidly convergent series, and the generating function of its stationary law satisfies no algebraic Mahler equation. Every intermediate limit is attained by a randomized rule, and rules based on areas or centres converge geometrically. Whether the maximal limit has a closed form remains open. MSC 2020: 60D05 (primary); 60J10, 60J05, 52A22, 90C40, 33C15 (secondary). Files: the paper (PDF) and chord-cutting-side-counts-1.0.0-reproduction.zip, the programs and outputs that reproduce every number in the paper, including the maximal limit to 370 decimal places.