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The boundedness and persistence conjectures for weakly reversible mass-action systems

A reaction network has complexes y∈Z≥0dy\in\mathbb Z_{\ge0}^d and reactions y→y′y\to y'; it is weakly reversible if every reaction has a directed return path. With positive rate constants, mass-action kinetics is x˙=∑y→y′κy→y′xy(y′−y)\dot x=\sum_{y\to y'}\kappa_{y\to y'}x^y(y'-y) on the positive orthant. Feinberg (1987) formulated the expectation that a positive trajectory of a weakly reversible system cannot converge to a point with a zero concentration. Anderson (2011) separated the boundedness conjecture (every positive trajectory is bounded) from the persistence conjecture (no bounded positive trajectory has lim inf⁡txi(t)=0\liminf_t x_i(t)=0). Known cases included one linkage class (Anderson; Gopalkrishnan-Miller-Shiu), two species (Craciun-Nazarov-Pantea) and two-dimensional stoichiometric subspace (Pantea). For weakly reversible mass-action systems with positive constant rates, is every positive trajectory global, bounded and persistent?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Chemical reaction network theory: mass-action kinetics
Posed by
Martin Feinberg, Chem. Eng. Sci. 42 (1987), Remark 6.1.E; split into the boundedness and persistence conjectures by David F. Anderson, J. Math. Chem. 49 (2011), Section 1.1
Year posed
1987
Years open
39y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
35 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for any finite weakly reversible network with positive constant rates and any positive x0x^0, the solution exists for all t≥0t\ge0 and some ε∈(0,1)\varepsilon\in(0,1), depending on the network, the rates and x0x^0, gives ε≤xi(t)≤ε−1\varepsilon\le x_i(t)\le\varepsilon^{-1} for all t≥0t\ge0 and all ii. The proof builds a compact convex forward-invariant polytope around each initial point from a finite concave piecewise-affine function. A remark derives convergence for complex-balanced systems, which the paper treats as known (citing Craciun's 2026 revision), not as new. The bounds are not uniform over a compatibility class (the permanence entry), and time-dependent rates are not treated.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The companion permanence manuscript (October 5) builds directly on this one; the README notes that some outputs build on earlier model results.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source; the proof was not refereed. Lean: the challenge MassAction (OAI.Problem326.global_bounded_persistent_solution, solution module OAI/Analysis/MassAction/Main.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Statement read here: for d>0d>0, a finite network of natural-number complexes with weak reversibility as a transitive closure, all rates positive and any positive initial state, there is ε∈(0,1)\varepsilon\in(0,1) such that a solution defined for all t≥0t\ge0 exists and every such solution stays in [ε,ε−1][\varepsilon,\varepsilon^{-1}] in every coordinate. That is the headline claim (solutions are functions on all of R\mathbb R with the derivative required for t≥0t\ge0, a harmless encoding choice). Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.

Sources

Changelog1 change

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