AI-assisted classification of field-dependent path Euler characteristic.

Fu and Ivanov showed that the path Euler characteristic of a digraph can depend on the coefficient field. In their example, the rational and mod-2 path Euler characteristics are different. This behaviour cannot occur for ordinary topological spaces, so it reveals a genuinely non-classical feature of path homology.

Field dependence is related to thick non-bipartite classes in the graph of short moves. However, this does not yet give a full classification: one must understand when such classes occur, when they are minimal, and when their contributions to the Euler characteristic do or do not cancel across dimensions.

The aim of this problem is to use AI to organise this phenomenon systematically. For finite digraphs, we want to search for examples where the path Euler characteristic over the rationals differs from the path Euler characteristic over the field with two elements.

Problem: Which directed graphs have field-dependent path Euler characteristic, and what directed structures are responsible?