The even-n Auslander–Reiten (n+2)-angle local-finiteness problem

The problem asks whether the odd-dimensional implication from local finiteness to Auslander–Reiten $(n+2)$-angles extends to even $n$. The general even-dimensional statement is presented as open, despite earlier papers asserting results that appear to cover all $n$.

Known results

• Zhou’s odd-$n$ implication is the established starting point for the even-dimensional question.
• A 2019 paper claims that every locally finite $(d+2)$-angulated category has Auslander–Reiten $(d+2)$-angles, for arbitrary $d$.
• A 2021 paper claims the broader statement for locally finite $n$-exangulated categories and records the $(n+2)$-angulated specialization.

August 2026 partial extension and earlier general claims

A development reported on August 27, 2026 extends Zhou’s implication to even $n$ under a sufficient condition and verifies that condition for broad examples; it explicitly leaves the general case open. The older general claims therefore amount to an unverified purported solution, not an accepted resolution.

Current status (as of August 2026): A conditional even-$n$ extension and broad examples are available, while the general statement remains open; older papers claim a full implication, but that claim is unverified.