Result 149, Dynamical systems and ergodic theory

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.

Lean formalization Proof

The bigger picture

Why it matters

In mass-action models, reaction speeds are determined by reactant concentrations and rate constants. The manuscript claims that a reversible-path structure in the reaction network prevents both eventual species loss and unlimited concentration growth, with bounds shared across compatible starting states.

What changes?

The claim covers every finite weakly reversible network with fixed positive rate constants. Weak reversibility means each reaction has a return path through other reactions. A positive stoichiometric compatibility class consists of positive concentration vectors differing by linear combinations of reaction changes. Each such class, even an unbounded one, reportedly has a common compact, convex absorbing set: a bounded, closed region that every positive trajectory eventually enters and never leaves, where all concentrations have positive lower and finite upper bounds.

What does that help mathematicians do?

The key strengthening is uniformity within a class, rather than bounds chosen separately for each initial condition. The absorbing set depends only on the network, rates, and class, although entry time may depend on the starting state. Researchers could therefore confine long-term behavior to one region separated from zero concentrations, even when the class itself permits arbitrarily large concentrations. This does not establish convergence to equilibrium.

Are there practical applications?

The immediate value is foundational for reaction-network dynamics. The claimed result supplies a shared bounded region for studying long-term behavior in mass-action systems satisfying its assumptions. It is not a numerical recipe for computing concentration limits or waiting times, and it does not by itself establish the behavior of systems with changing rates or different kinetic laws.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Uniform Permanence in Weakly Reversible Mass-Action Systems

October 5, 2026 13 pages

We prove the permanence conjecture for finite weakly reversible mass-action systems with fixed positive reaction rates. Every positive stoichiometric compatibility class admits one compact convex forward-invariant set that every positive trajectory in that class enters in finite time, even when the class is unbounded. The set depends only on the network, the rates, and the class; the entry time may depend on the initial state. Thus, after its entry time, every concentration satisfies the same positive lower and finite upper bounds throughout that class.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026,
  author = {{OpenAI}},
  title = {{Uniform Permanence in Weakly Reversible Mass-Action Systems}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf}{OAI:Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026}},
  year = {2026}
}

Boundedness and persistence of weakly reversible mass-action systems

September 25, 2026 16 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/149.md.

Classwise permanence for weakly reversible mass-action systems

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The boundedness and persistence conjectures for mass-action systems ask whether positive concentrations remain finite and separated from zero. The formalization proves this for every finite weakly reversible reaction network with positive constant reaction rates and every strictly positive initial state. A global forward solution exists, and one ε∈(0,1)\varepsilon\in(0,1) bounds every concentration of every global forward solution between ε\varepsilon and ε−1\varepsilon^{-1} for all nonnegative times. The bound may depend on the network, rates, and initial state.

Comparator links

Result Comparator statement
Global boundedness and persistence for weakly reversible mass-action systems MassAction.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.