The permanence conjecture for weakly reversible mass-action systems with fixed rates

For a weakly reversible reaction network with mass-action kinetics $\dot x=\sum_{y\to y'}k_{y\to y'}x^y(y'-y)$ and fixed positive rates, let $S$ be the span of the reaction vectors $y'-y$ and $P=(c+S)\cap\mathbb R^d_{>0}$ a positive stoichiometric class, which may be unbounded. The system is permanent on $P$ if every solution starting in $P$ is global and positive and one compact $K\subset P$ eventually contains each of them, the entry time but not $K$ depending on the initial point. This is stronger than boundedness plus persistence of each trajectory. Known before: permanence for strongly endotactic networks, including weakly reversible ones with one linkage class (Gopalkrishnan-Miller-Shiu; Boros-Hofbauer), and for two species (Craciun-Nazarov-Pantea). Is every weakly reversible mass-action system with fixed positive rates permanent on every positive stoichiometric class?

Result

Proved(see note)

Status

Candidate (review pending)

AI contribution

AI-discovered

Method

Argument

Field

Chemical reaction network theory: mass-action kinetics

Posed by

Permanence conjecture of chemical reaction network theory; the manuscript names it without an originating source and takes the classwise definition from Boros and Hofbauer (2020, Definition 4.1)

Year posed

—

Years open

—

Solved

2026-10-05

Model

Unreleased internal OpenAI model

Vendor

OpenAI

Collaborators

—

Verification

Unreviewed

Publication

Announced

Collection

OpenAI math release (October 2026), version adc7f12

Significance

32 / 100*

Disclosed cost

—

Wikipedia

No dedicated article