Spectral region of four-cycle stochastic matrices
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Which complex numbers can be eigenvalues of the specified row-stochastic 4×4 matrices supported on a four-cycle? For nonreal a+ib, the target region includes 0≤a≤1, a+|b|≤1 and (b²+a²+a)²+2a²−b²≥0, with realization and boundary questions.
Why it matters and what counts as progress
Why it matters: A complete spectral region requires realization as well as necessary inequalities, giving more than a one-sided conjectured bound.
What counts as progress: Prove both necessity and realization for the region and its boundary arcs, including the Ran–Teng conjectured inequality.
How the problem developed
- 2024 · Conjecture [1]
Ran and Teng Conjectured the exact nonreal eigenvalue region for the specified four-cycle row-stochastic matrix family.
- 21 Feb 2026 · Theorem report [2]
Brecht Verbeken, Brando Vagenende, Marie-Anne Guerry, Andres Algaba and Vincent Ginis Reported a necessary-and-sufficient spectral-region characterization with explicit boundary attainment, documenting an LLM-assisted proof and human verification.
- 7 May 2026 · Theorem [1]
Andres Algaba, Brando Vagenende, Brecht Verbeken and Marie-Anne Guerry Published a self-contained characterization of the real and nonreal spectra of four-cycle row-stochastic matrices, including converse realization of the interior region.
AI contribution
Early Evidence of Vibe-Proving with Consumer LLMs: A Case Study on Spectral Region Characterization with ChatGPT-5.2 (Thinking)
For the specified row-stochastic 4×4 matrices supported on a 4-cycle, every nonreal eigenvalue a+ib has 0≤a≤1, a+|b|≤1 and (b²+a²+a)²+2a²−b²≥0. Conversely, upper-half-plane points with b>0, 0≤a≤1, a+b<1 and the polynomial nonnegative occur; the stated boundary arcs are attained. This includes Ran and Teng’s Conjecture 20.
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AI-assisted proof characterizes a four-cycle matrix spectrum