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Spectrally Simple Spectra without Irreducible Nonnegative Realizations

A mix of human-written and AI-generated textHuman understanding: all partsmath.RA — Rings and Algebras

Contributed by Yair Lavi ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We give a counterexample to a conjecture of Johnson, Marijuán, and Pisonero: we show that the spectrum {1,1/2,±i/3}\{1,1/2,\pm i/\sqrt3\} is realizable and spectrally simple, but has no irreducible nonnegative realization. More generally, we prove their nonexistence conjecture for an order-four boundary family.

Tools used

OpenAI
GPT-5.6-sol
OpenAI
GPT-6 Astra
Anthropic
Claude FableVersion 5
Anthropic
Claude FableVersion 5.1

References

  1. C. R. Johnson, C. Marijuán, and M. Pisonero, The INIEP: Irreducible and Positive Realizations , preprint (2026). arXiv:2605.23552v1 (https://arxiv.org/abs/2605.23552v1).arXiv

Version history

  1. v1 · 5 pagesSubmitted by Yair LaviInitial depositCurrentNov 08, 2026