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Boundedness and persistence of weakly reversible mass-action systems
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Abstract
We prove the boundedness and persistence conjectures for finite weakly reversible mass-action systems with positive constant reaction rates. For every positive initial condition, the solution exists for all forward time, and every concentration remains bounded above and bounded away from zero. The bounds may depend on the initial condition, and no boundedness assumption is imposed on its stoichiometric compatibility class.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Michael Mislan, Endotactic permanence and quantitative recovery with vanishing reverse rates (hexagon:2610.00321, October 2026). Cites this paper and the companion uniform-permanence paper and uses "the finite affine construction of OpenAI" to prove classwise permanence for ordinary endotactic networks with measurable rates in a fixed positive rate box; it also gives quantitative concentration bounds and recovery times for weakly reversible networks whose rates scale as powers of a small parameter. [AI-generated per source] [family-level]
Formalization
- OpenAI math release, Lean scope note Classwise permanence for weakly reversible mass-action systems (
lean/docs/149.md) (openai/math release, October 2026). Lists this paper as the accompanying paper. Its scope is Theorem 1.1: global existence and an ε bound that depends on the network, the rates and the initial state; the comparator statement is inlean/ComparatorChallenges/MassAction.lean. [family-level] - Michael Mislan, palomar-endotactic-permanence (Lean repository) (GitHub; registered as Palomar record PALOMAR-2026-10-09-000015 per hexagon:2610.00321, 2026). A Lean development of uniform permanence for endotactic networks with measurable rates; its README says the finite affine barrier construction is adapted from OpenAI's mass-action development. [AI-generated per source] [family-level]
Related prior literature mentioned by others
- Matthew D. Johnston, Casian Pantea, Pete Donnell, A computational approach to persistence, permanence, and endotacticity of biochemical reaction systems (arXiv:1412.4662, 2014; Journal of Mathematical Biology 72, 2016). Cited by hexagon:2610.00321 as the source of the endotactic permanence conjecture (its Conjecture 1.3), which that follow-up proves using the affine construction from this family. [family-level]
- Gheorghe Craciun, Abhishek Deshpande, Endotactic networks and toric differential inclusions (arXiv:1906.08384, 2019). Cited by hexagon:2610.00321 alongside Boros–Hofbauer as related work on endotactic networks. [family-level]
Possibly relevant literature (found by the agent, not checked)
- David Angeli, Patrick De Leenheer, Eduardo Sontag, A Petri Net approach to the study of persistence in chemical reaction networks (arXiv:q-bio/0608019, 2006; Math. Biosci. 2007). Gives Petri-net (siphon) sufficient conditions for persistence, an earlier, structurally different route to the persistence half of Theorem 1.1.
- Polly Y. Yu, Gheorghe Craciun, Mathematical Analysis of Chemical Reaction Systems (arXiv:1805.10371, 2018). A survey stating the persistence and permanence conjectures for weakly reversible mass-action systems, which the introduction's history section discusses.
- James D. Brunner and Gheorghe Craciun, Robust persistence and permanence of polynomial and power law dynamical systems (arXiv:1705.06785, 2017). Its two-dimensional tropically endotactic permanence theorem provides a comparison for the family's all-dimensional weakly reversible claims and treatment of fixed versus time-dependent parameters. [family-level]
Digests and expositions
- Latent Seminar, Weakly reversible mass-action systems: persistence and permanence (6 October 2026). Explains both manuscripts' affine barriers, cut-flux estimates and scale iteration, explicitly distinguishing their quantifiers and the release's formalization scope without claiming complete proof verification. [family-level]
Elsewhere
- Hexagon: Endotactic permanence and quantitative recovery with vanishing reverse rates (AI-generated; human understanding: some)
- VibeMathed: Problem record
- Emergent Mind: family 149
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026,
author = {{OpenAI}},
title = {{Boundedness and persistence of weakly reversible mass-action systems}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf}{OAI:Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/149.md. The Lean statements have not been checked by people against the paper's claims.
Family 149: Classwise permanence for weakly reversible mass-action systems
Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds.
Other papers in this family:
- Uniform Permanence in Weakly Reversible Mass-Action Systems · October 5, 2026