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Uniform Permanence in Weakly Reversible Mass-Action Systems
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Abstract
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.
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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 its companion 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). The scope note lists only the companion paper as accompanying and formalizes its pointwise statement: global existence and an ε bound depending on the network, the rates and the initial state. The classwise uniformity of this paper's Theorem 1.1 is not in the stated scope. [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]
- Michael Mislan, palomar-quantitative-permanence (Lean repository) (GitHub; Palomar record PALOMAR-2026-10-11-000020, 2026-10-11). Lean statements for quantitative permanence and recovery bounds with vanishing reverse rates; its README says the general construction builds on an attributed affine permanence development from this preprint. [AI-generated per source]
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)
- Polly Y. Yu, Gheorghe Craciun, Mathematical Analysis of Chemical Reaction Systems (arXiv:1805.10371, 2018). This survey states the Permanence Conjecture as a globally attracting compact set in every stoichiometric compatibility class, the formulation that Theorem 1.1 addresses.
- 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]
- RMJ / VideoToBlog, Uniformní permanence ve slabě reverzibilních reakčních systémech: jedna bezpečná oblast pro všechny kladné začátky (undated). This Czech exposition explains the October manuscript's common absorbing set, initial-state-dependent entry times and proof outline using a one-species reversible example.
Elsewhere
- Hexagon: Endotactic permanence and quantitative recovery with vanishing reverse rates (AI-generated; human understanding: some)
- Palomar: Quantitative permanence and recovery with vanishing reverse rates (Lean formalization (adapts))
- VibeMathed: Problem record
- Emergent Mind: family 149
Independent sites; their content is not reviewed here.
Cite
@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}
}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:
- Boundedness and persistence of weakly reversible mass-action systems · September 25, 2026