Permanence of weakly reversible mass-action systems
Permanence of weakly reversible mass-action systems is the subject of two manuscripts dated September and October 2026 in OpenAI's public mathematics collection (github.com/openai/math). They concern the long-time behaviour of chemical reaction networks with mass-action kinetics. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original results
- Claim. For every finite weakly reversible mass-action system with fixed positive rate constants, every positive stoichiometric compatibility class, even an unbounded one, has one compact convex forward-invariant set that every positive trajectory in that class eventually enters. All positive trajectories in a class therefore eventually share positive lower and finite upper concentration bounds; the manuscript presents this as a proof of the permanence conjecture. Source: Uniform permanence in weakly reversible mass-action systems (5 October 2026). An earlier manuscript of the family treats boundedness and persistence: Boundedness and persistence of weakly reversible mass-action systems (25 September 2026).
- Formalization, as described by the repository. The scope note lean/docs/149.md states that the Lean formalization proves boundedness and persistence for every finite weakly reversible network with positive constant rates and every strictly positive initial state: a global forward solution exists, and one ε ∈ (0, 1) bounds every concentration between ε and 1/ε for all nonnegative times, with the bound allowed to depend on the network, the rates and the initial state.
- Earlier results, as summarized in the October manuscript. Anderson proved convergence to equilibrium for complex-balanced systems with one linkage class; Craciun, Nazarov and Pantea proved permanence for two-species weakly reversible systems as part of a theorem on endotactic networks; Pantea proved persistence of bounded weakly reversible trajectories when the stoichiometric subspace has dimension two; Gopalkrishnan, Miller and Shiu proved permanence for strongly endotactic networks, including every weakly reversible network with one linkage class, with another proof by Boros and Hofbauer. A revised manuscript of Craciun (23 September 2026) proves persistence for bounded positive trajectories of weakly reversible systems.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 14 October 2026) |
|---|---|---|---|---|
| 2026-10-09 | M. Mislan, mmislan/palomar-endotactic-permanence (GitHub; Palomar PALOMAR-2026-10-09-000015, version 1) | Extension (formal) | Uniform permanence for finite endotactic reaction networks with arbitrary real complexes and measurable rate coefficients bounded above and away from zero. The registry abstract states that this resolves the Endotactic Permanence Conjecture (Conjecture 1.3 of Johnston, Pantea and Donnell, arXiv:1412.4662), using an ordinary-endotactic drift estimate together with the finite affine barrier construction from OpenAI's work. | Registered in the Palomar registry on 9 October 2026. |
| 2026-10-11 | M. Mislan, mmislan/palomar-shared-resource-realizations (GitHub; Palomar PALOMAR-2026-10-11-000011, version 1) | Use (formal) | Mass-action realizations and dynamics of a family of two-species modules coupled through a shared resource and buffer; the registry record states that it builds on OpenAI's Lean mass-action development. | Registered in the Palomar registry on 11 October 2026. |
| 2026-10-11 | M. Mislan, mmislan/palomar-quantitative-permanence (GitHub; Palomar PALOMAR-2026-10-11-000020, version 1) | Extension (formal) | Quantitative permanence and recovery bounds for mass-action systems whose reverse rates vanish as powers of a parameter, with explicit bounds for a three-species network; the record cites the uniform-permanence manuscript and the family's Lean scope note as sources it adapts. | Registered in the Palomar registry on 11 October 2026. |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.