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Math-2431-A1 · revision r1 · on Math-2431

OpenAI claim summary for Math-2431

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 149, 2026-09-25): All trajectories are bounded and persistent individually, but the bound may depend on the initial point; classwise uniform eventual bounds required for permanence are explicitly not asserted.

Manuscript 2 (family 149, 2026-10-05): The theorem supplies persistence and one compact absorbing set common to all positive initial states within each fixed stoichiometric class and rate choice, exactly the standard classwise meaning of permanence.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Persistence and Permanence of Weakly Reversible Reaction Networks
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Persistence and Permanence of Weakly Reversible Reaction Networks

OpenAI's claim

Manuscript 1 (family 149, 2026-09-25): Every positive trajectory of a finite weakly reversible constant-rate mass-action system is claimed to exist forever and keep all concentrations between positive finite bounds, even in unbounded compatibility classes. Manuscript 2 (family 149, 2026-10-05): Each positive stoichiometric class of a weakly reversible constant-rate mass-action system is claimed to have one compact convex forward-invariant set eventually entered by every positive trajectory.

Mathematical statement of the claimed result

Mathematical claim 1.
Take a finite reaction network $\mathcal C\subset\mathbb Z_{\ge0}^d$, $\mathcal R\subset\mathcal C\times\mathcal C$, with positive constant rates $\kappa_{y\to y'}$. Assume weak reversibility: every directed reaction lies on a directed cycle. Its mass-action system is \[ \dot x=\sum_{y\to y'}\kappa_{y\to y'}x^y(y'-y),\qquad x^y=\prod_i x_i^{y_i}. \] OpenAI claims every positive initial point has a global positive solution and a trajectory-dependent $\varepsilon\in(0,1)$ with $\varepsilon\le x_i(t)\le\varepsilon^{-1}$ for all $t\ge0$ and all species $i$. With the extra complex-balance hypothesis (at some positive equilibrium, total inflow equals total outflow at each complex), it records convergence within each positive stoichiometric class to its sole positive equilibrium $x^0+\operatorname{span}\{y'-y\}$ intersected with the positive orthant. The convergence conclusion is not asserted from weak reversibility alone.
Mathematical claim 2.
For a finite weakly reversible mass-action network with fixed positive rates, define its stoichiometric subspace $S=\operatorname{span}\{y'-y\}$ and positive class $P=(x^0+S)\cap\mathbb R_{>0}^d$. OpenAI claims a nonempty compact convex forward-invariant $K_P\subset P$ eventually containing every trajectory from $P$. Each such solution is global and positive, and \[ \exists\varepsilon_P\in(0,1)\ \forall x^0\in P\ \exists T(x^0)<\infty:\quad \varepsilon_P\le x_i(t)\le\varepsilon_P^{-1}\quad(t\ge T(x^0)). \] Weak reversibility means every directed reaction lies on a directed cycle, and the dynamics are $\dot x=\sum\kappa_{y\to y'}x^y(y'-y)$. The bounds depend on the network, rates and class, rather than the initial point; only the entry time may depend on that point.

Outline of the argument

Manuscript 1 (family 149, 2026-09-25): An affine approximation lemma controls exponent pairings separately rather than relying on a single large concentration product. It builds a compact convex polytope inside the positive orthant around each initial condition, with all boundary reaction directions pointing inward; invariance supplies boundedness, persistence, and global existence.

Manuscript 2 (family 149, 2026-10-05): The affine trapping construction is strengthened by strict increase outside a selected trapping set. Connecting such sets across scales removes their dependence on the initial point, giving a common absorbing region and hence concentration bounds uniform over the whole fixed class after trajectory-dependent entry times.

Formal verification

For family 149, the scope page documents the following selections. The selected concentration bounds hold for all forward times with one positive lower bound depending on the initial state, network, and fixed rates. Classwise eventual bounds uniform over initial states are outside this selection. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Manuscript 2 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause. Comparator files: MassAction.lean. Declarations: OAI.Problem326.global_bounded_persistent_solution.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources