What was already known
Reading had proved equality for most irreducible finite Coxeter types, leaving exceptional types including H₄ and E₆ unresolved. No simplicial-arrangement example above rank was known in the cited baseline.
AI involvement
Poliakova credits ChatGPT 5.6 Sol Ultra with finding the small obstruction subgraphs after she proposed searching for counterexamples. She supplied the final proof and explanation.
The author expressly assigns the central finite obstruction discovery to ChatGPT, while providing the conjectural direction and final written argument herself.
Importance
The two examples answer the general rank-equality question negatively, including a crystallographic case.
Limitations
The paper gives incomparable pairs and noncolorability arguments but does not document a completed check of every finite comparison. It does not claim exact dimensions or settle Segovia’s extremal question.
Technical verification
The manuscript supplies explicit words, comparison rules and short noncolorability proofs, but its statement that readers can check them by hand is an invitation, not documentation of a completed check. No machine check is cited.
Human understanding
Poliakova explains how incomparable pairs form an incompatibility graph, why its chromatic number lower-bounds order dimension, and how the H4 and E6 subgraphs force the claimed inequalities.
Correction and publication history
Initial publication of this result.
Read this published revision