Global Attractor Conjecture for Complex-Balanced Mass-Action Systems

Global Attractor Conjecture for Complex-Balanced Mass-Action Systems

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. This entry does not assert that the selected declarations formalize every part of manuscript 1's main conclusion. It does not establish formal coverage of the complete problem clause. 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

• OpenAI, Boundedness and persistence of weakly reversible mass-action systems, 2026-09-25; citation key OAI:Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: rem:complex-balanced-convergence.
• OpenAI, Uniform Permanence in Weakly Reversible Mass-Action Systems, 2026-10-05; citation key OAI:Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 149: Lean scope.
• Comparator MassAction.json.
• Comparator MassAction.lean.
• OpenAI catalogue entry.

Global Attractor Conjecture for Complex-Balanced Mass-Action Systems

Definitions and mathematical statement

A reaction $y\to y'$ has complexes $y,y'\in{\mathbb{Z}}_{\ge0}^d$ and rate $\kappa_{y\to y'}>0$. Mass action gives \[ \dot x=\sum_{y\to y'}\kappa_{y\to y'}x^y(y'-y),\qquad x^y=\prod_i x_i^{y_i}. \] Let $S$ be the real span of reaction vectors. A positive equilibrium $x^*$ is complex-balanced when, at each complex, the total incoming flux equals the outgoing flux at $x^*$. For every such system and $x_0>0$, the assertion is global existence and convergence to the unique positive equilibrium in $(x_0+S)\cap{\mathbb{R}}_{>0}^d$. The entire compatibility class need not be bounded. The question is about all positive trajectories, not merely local stability near an equilibrium or convergence conditional on persistence.

Formulation and concept references: [S1].

Short English statement

Must every positive trajectory of a complex-balanced mass-action system converge to the unique positive equilibrium in its stoichiometric class?

Research attempt

Scope and current-source check.

This attempt was developed by GPT-6 Astra Ultra on 27 September 2026. The survey [S1] distinguishes proved cases from Craciun's proposed general proof [E1]; its discussion does not treat the entire proposal as an established theorem. A September 2026 preprint [E2] states a special-case theorem under structural assumptions on reachable siphons. Those assumptions are not part of the catalogue's universal target. Neither manuscript is imported here as a verified full solution.

The work proves global existence and boundedness, existence and uniqueness of the positive equilibrium in each positive class, and convergence under an explicit conservation-law condition on siphons. It isolates persistence as the missing general step. The underlying entropy mechanism is classical; the derivation and its boundary limitations are made explicit rather than advertised as new.

The entropy and its exact dissipation inequality.

Fix one complex-balanced positive equilibrium $x^*$, not necessarily in the class of the initial point. For each reaction $y\to y'$ write \[ q_{y\to y'}=\kappa_{y\to y'}(x^*)^y>0, \qquad z_i=x_i/x_i^*,\qquad a_y=z^y. \] Complex balance says that the sum of the $q$'s entering any complex equals their sum leaving it. Define \[ V(x)=\sum_{i=1}^d\left(x_i\log\frac{x_i}{x_i^*}-x_i+x_i^*\right), \tag{1} \] with $0\log0=0$ on the closed orthant. Each summand is nonnegative and tends to infinity as $x_i\to\infty$. For positive $x$ along a solution, \[ \dot V=\sum_{y\to y'}q_{y\to y'}a_y\log(a_{y'}/a_y). \] The scalar inequality $a\log(b/a)\le b-a$, for $a,b>0$, follows from $\log s\le s-1$, with equality exactly when $a=b$. Consequently \[ \dot V\le\sum_{y\to y'}q_{y\to y'}(a_{y'}-a_y)=0. \tag{2} \] The last equality is obtained by collecting the coefficient of each $a_y$ and using complex balance. Moreover equality holds if and only if \[ a_y=a_{y'}\text{ on every reaction}, \quad\text{equivalently}\quad \log(x/x^*)\in S^\perp. \tag{3} \] The logarithm in $3$ is componentwise. A condition on its sum alone would lose the stoichiometric information.

Global existence despite unbounded compatibility classes.

Since $V$ is continuous and coercive on the closed nonnegative orthant, its sublevel set $\{V\le V(x_0)\}$ is compact. While the solution is positive, $2$ keeps it in this set, hence all coordinates are bounded by a constant $M$ depending on the initial point and the system.

For a reaction which decreases coordinate $i$, its source has $y_i\ge1$, so its monomial contains a factor $x_i$. On the bounded box, all negative contributions to $\dot x_i$ are therefore bounded below by $-C_i x_i$. Positive contributions can be discarded in this lower bound, giving \[ x_i(t)\ge x_i(0)e^{-C_it}>0 \tag{4} \] on any finite existence interval. A coordinate cannot reach zero in finite time, and no coordinate can escape to infinity. The vector field is polynomial and locally Lipschitz on a neighborhood of the compact box, so the standard ODE continuation theorem gives existence for every $t\ge0$.

Also $\dot x\in S$, so $x(t)\in x_0+S$. This proves global positive existence and boundedness without assuming the whole compatibility class is bounded. The lower bound $4$ decays as $t\to\infty$ and is not a uniform persistence estimate.

The positive equilibrium in each class.

Consider the closed feasible set $\overline P=(x_0+S)\cap\mathbb R_{\ge0}^d$. Coercivity gives a minimizer $\bar x$ of $V$ on this set. It cannot have a zero coordinate. Indeed the segment $x(\epsilon)=(1-\epsilon)\bar x+\epsilon x_0$ lies in $\overline P$ and is positive for $\epsilon>0$. On coordinates where $\bar x_i=0$, the change in $1$ has leading term $\epsilon x_{0,i}\log\epsilon$; on the other coordinates it is $O(\epsilon)$. A nonempty set of zero coordinates would therefore give $V(x(\epsilon))<V(\bar x)$ for small $\epsilon$, a contradiction. Thus $\bar x>0$.

At this interior minimizer, differentiating along every vector in $S$ shows $\nabla V(\bar x)=\log(\bar x/x^*)\in S^\perp$. Condition $3$ then holds. In each connected component of the reaction graph, the factors $a_y$ are constant, since they agree along every edge. Multiplying the balanced fluxes $q$ in that component by the common factor preserves complex balance. Hence $\bar x$ is itself a complex-balanced equilibrium in the desired class.

For uniqueness, any positive equilibrium has $\dot V=0$, and hence obeys $3$. If $u,v$ were two in the same class, then $u-v\in S$ and $\log u-\log v\in S^\perp$, implying \[ \sum_i(u_i-v_i)(\log u_i-\log v_i)=0. \] Every summand is nonnegative and vanishes only when $u_i=v_i$, because the logarithm is strictly increasing. Therefore $u=v$. Denote this unique equilibrium by $\widehat x$.

Persistence would now give convergence.

Suppose the positive trajectory satisfies $\liminf_{t\to\infty}x_i(t)>0$ for every coordinate. It is then eventually in a compact subset of the positive class. The decreasing nonnegative function $V(x(t))$ has a limit. Its omega-limit set is nonempty, compact and invariant, and $V$ is constant on it by continuity. Along every trajectory in this set, $\dot V=0$, so $3$ holds and every point is the unique equilibrium $\widehat x$. Thus the omega-limit set is a singleton and \[ x(t)\longrightarrow\widehat x. \tag{5} \] These compactness and invariance statements follow from the polynomial flow on a bounded set; no boundary differentiability of $V$ is used because the set here is strictly positive. The missing hypothesis in this paragraph is precisely persistence.

A concrete sufficient condition using siphons.

A nonempty set $I$ of species is a siphon if every reaction producing some species in $I$ has at least one species in $I$ in its source. Assume that for every nonempty siphon $I$ there is a vector \[ w\in S^\perp\cap\mathbb R_{\ge0}^d, \qquad w\ne0,\qquad \operatorname{supp}(w)\subseteq I. \tag{6} \] Then the trajectory is persistent and $5$ follows. Here is a proof that also explains why boundary equilibria alone are not the only objects one should check in an a priori argument.

If a boundary point $z$ were in the omega-limit set, let $I=\{i:z_i=0\}$, which is nonempty. The compact omega-limit set contains a full trajectory through $z$, defined for positive and negative time and staying in the nonnegative orthant. This can be obtained by taking limits of the shifted trajectories $x(t_j+s)$ on each bounded $s$-interval and using uniqueness; boundedness of the vector field gives the required equicontinuity. Each zero coordinate has a local minimum at time zero on that full trajectory, so its derivative is zero there.

At $z$, every reaction consuming such a coordinate has zero rate. All remaining contributions to that coordinate's derivative are nonnegative. Their sum being zero, each production rate must vanish. Therefore every reaction producing a species of $I$ has a zero coordinate in its source: $I$ is a siphon. Choose $w$ from $6$. Along the original trajectory, $w\cdot x(t)=w\cdot x_0>0$, but $w\cdot z=0$ since its support is contained in $I$. Continuity contradicts $z$ being a limit point. There are thus no boundary omega-limit points. Compactness makes the distance from the omega-limit set to every coordinate hyperplane positive, which is exactly the needed persistence assertion.

This proves a full convergence theorem under the checkable extra condition $6$. It has not been proved that every complex-balanced network satisfies that condition, and in fact it need not.

Two examples distinguish sufficiency from necessity.

For the directed cycle \[ A\mathrel{\mathop{\longrightarrow}^{\alpha}}B \mathrel{\mathop{\longrightarrow}^{\beta}}C \mathrel{\mathop{\longrightarrow}^{\gamma}}A, \] the only nonempty siphon is the full species set. It supports the conservation vector $(1,1,1)$. In the class $a+b+c=m>0$, the unique positive equilibrium is \[ (a,b,c)=\left(\frac J\alpha,\frac J\beta,\frac J\gamma\right), \qquad J=\frac m{\alpha^{-1}+\beta^{-1}+\gamma^{-1}}. \] The three fluxes equal $J$, verifying complex balance directly. The preceding theorem proves global convergence. The directed cycle has no reverse reactions, so the argument does not assume detailed balance.

For contrast, consider $2A\rightleftarrows3A$ with positive rates $k,\ell$. Its equation is $\dot x=x^2(k-\ell x)$, with positive complex-balanced equilibrium $x^*=k/\ell$ and stoichiometric space $S=\mathbb R$. There is no nonzero conservation vector, so $6$ fails for the siphon $\{A\}$. Nevertheless every positive solution moves monotonically toward $x^*$ and converges to it: below $x^*$ the derivative is positive, above it negative, and uniqueness prevents crossing the equilibrium. Thus $6$ is sufficient, not necessary.

Why entropy alone does not exclude the boundary.

The function $1$ has finite value when coordinates vanish; a term with $x_i=0$ contributes $x_i^*$, not infinity. Sublevel compactness therefore bounds concentrations above but does not bound them below. For the scalar example, $V(0)=x^*$ and sufficiently large initial states have $V(x_0)>x^*$, so their entropy sublevel even contains the boundary point. Its actual exclusion uses the vector field, not an infinite entropy barrier. Applying the interior equality analysis of $2$ at a boundary point would also insert undefined logarithms.

Verification and final status.

The local checks verify flux balance, the equilibrium formula, the siphons of the directed cycle, and the conserved quantity. They test the dissipation identity on rational sample states with numerical logarithms only as a sign check. The existence, uniqueness, positivity, and conditional convergence arguments above are analytic, not inferred from time-stepping trajectories.

UNRESOLVED. Global bounded positive existence and the unique positive class equilibrium are proved here, as is convergence under $6$. The first general gap is exclusion of boundary omega-limit points without that conservation-law hypothesis. Specific next targets are a boundary-repelling function for siphons lacking a supported conservation vector, compatible estimates across intersecting boundary faces, and a uniform persistence mechanism valid for all complex-balanced networks. The recent source claims are not substituted for those missing proofs.

Sources

• S1 Feliu–Shiu: From chemical reaction networks to algebraic and polyhedral geometry – and back again. https://arxiv.org/html/2501.06354v1. Formulation source.
• E1 Gheorghe Craciun, Toric Differential Inclusions and a Proof of the Global Attractor Conjecture, 2015 preprint. https://arxiv.org/abs/1501.02860. General proof proposal, whose status is qualified using [S1]; not an imported proof here.
• E2 Carsten Wiuf, A Proof of the Global Attractor Conjecture in a Special Case, submitted 21 September 2026. https://arxiv.org/abs/2609.24553. Recent restricted-siphon result; its additional hypotheses are not removed.