Oliu–Barton–Vigeral conjecture on absorbing-game values

Oliu-Barton and Vigeral conjectured that every algebraic number of degree $m$ is the value of a rational $m \times m$ absorbing game. Before the new preprint, the conjecture was open for $m \ge 3$.

Known results

• Degree $1$: trivial.
• Degree $2$: every algebraic number is realized by a rational $2 \times 2$ absorbing game (Oliu-Barton and Vigeral, 2023).
• Degree $3$: a rational $3 \times 3$ example with an irrational value was constructed (Oliu-Barton, 2023).
• For every $m$, some rational $m \times m$ game has a value of algebraic degree $m$; this did not establish realization of every number.

September 2026 claimed proof

Ali Asadi and Krishnendu Chatterjee’s preprint Values of Absorbing Recursive Games Express All Real Algebraic Numbers claims the full realization theorem, via a stronger strictly absorbing recursive construction. The result is presented as an unrefereed preprint, so the claimed resolution remains unconfirmed.

Current status (as of September 2026): The conjecture has a claimed complete proof for all degrees, but that proof is unverified; the previously established cases include degrees $1$ and $2$, with earlier degree-$3$ examples.