ChatGPT 5.6 Sol Finds Gap Results for Finite Posets

In a preprint first uploaded on August 13, 2026, Alireza Haqi proves an old conjectural relation between the width of a finite poset and gaps in its elements' expected ranks under a uniformly random linear extension. Writing $\operatorname{gap}(P)$ for the largest spacing, including the endpoint spacings, and $w(P)$ for the width, the paper establishes the explicit bound $\operatorname{gap}(P)\leq2w(P)-1$.

The paper also gives two counterexample constructions to stronger expectations. For every prescribed $L$, it constructs a width-two poset in which every maximal chain has an expected-rank gap of at least $L$. It further constructs posets $P_r$ with $\operatorname{gap}(P_r)\geq(3/2)^r$ while the entropy per selected element of every induced relative order remains below three bits, answering a gap-entropy question negatively.

Haqi states that ChatGPT 5.6 Sol found the key ideas and theorem, while he wrote the manuscript with assistance from Codex. The preprint also credits Jeff Kahn and Max Aires with identifying an application of the main gap-width estimate.