Structural Results for 4 x n Chomp: Unique Extension, Bimodal Asymptotic Structure, and Period-112 Geometry

Abstract

We present a complete computational tabulation of all 961,619,972 P-positions in 4xn Chomp for n <= 3000, obtained via a new O(n^4) shadow-array sieve that replaces the O(n^5) hash-set approach of prior work. Three structural results are reported. First, we prove the Unique Extension property: for any triple (a,b,c), there is at most one value of d such that (a,b,c,d) is a P-position. The proof is a short contradiction using the move structure of Chomp and generalizes immediately to all k-row Chomp. Second, the P-positions exhibit a persistent bimodal decomposition into two subfamilies, HIGH and LOW, separated by a clean gap in the per-a median of d/a that grows monotonically from 0.040 at n=500 to 0.062 at n=3000, with the HIGH subfamily maintaining a stable density of 56.2% throughout. The previously conjectured global limit d/a -> 2/9 is shown to be a mixture artifact. Third, within each family the two larger row-length ratios satisfy an exact quadratic relation at machine precision, and numerical evidence suggests d/a -> 1/4 in the HIGH family, though a power-law convergence fit gives an asymptote of approximately 0.248 with exponent alpha ~ 0.05, leaving the exact limit open. The LOW family limit L3 ~ 0.183 is not well approximated by 3/16; the best rational with denominator at most 2000 is 20/109. Code and the n <= 500 dataset are available at https://github.com/gargarnav/chomp-4xn-v2.