Abstract
Anderson and Kim conjectured that weak reversibility implies positive recurrence for stochastic mass-action systems. We prove this for bimolecular networks with one linkage class, removing the earlier requirement that every species occur in a pure unary or pure-double complex. Under weak reversibility, the states reachable from each initial population form a closed communicating class. Under the additional one-linkage and bimolecular hypotheses, for every positive rate vector each nonabsorbing reachable class is positive recurrent, whereas an absorbing singleton carries its point-mass law; consequently every reachable class has a unique stationary probability distribution. The closure statement follows by lifting directed return paths of complexes to population states. For recurrence, the proof marks the target of the most recently fired channel and applies a log-factorial potential after subtracting that target. Multiple linkage classes and molecularity above two remain open.
1. The marked-target mechanism
The embedded jump chain is augmented by the target complex \(t\) of the actual labelled reaction channel that just fired. If the post-jump population is \(x\), then the residual population \(x-t\) is nonnegative. For a next channel with source \(s\) and target \(u\), the residual log-factorial potential \(V(x,t)=\sum_i\log((x_i-t_i)!)\) has the exact increment
\[V(x-s+u,u)-V(x,t)=\log\frac{(x)_t}{(x)_s}.\]
Here \((x)_y\) is the stochastic falling-factorial monomial. Following the carried target, \(s=t\), therefore has exactly zero reward. Finite target-following paths move this information to a useful terminal complex. A scalar-envelope induction propagates the negative drift caused by a rare terminal source through the entire episode, even when intermediate propensities live on very different scales.
Every divergent residual sequence is compactified by normalized logarithms. Molecularity at most two gives an exhaustive alternative: either an enabled higher-weight source sits above a lower terminal complex, forcing the terminal source probability to vanish, or a reaction-wise linear invariant rules out the proposed divergence within the fixed communicating class. The argument then builds a finite nonempty exceptional set and a finite trace chain. Nonexplosion is proved at the population level from recurrent visits to a population state and the associated holding times.
The log-factorial growth is a discrete, target-shifted analogue of the classical pseudo-Helmholtz/Horn–Jackson entropy family. The classical entropy tradition motivates the growth scale; the exact subtraction of the reaction target and the resulting falling-factorial identity are the mechanism used here.
2. Theorem and stationary consequence
Closure lemma. For every finite weakly reversible network, the population states reachable from each initial population form a closed communicating class. This statement needs neither one linkage class nor bimolecularity.
Theorem. If the network also has one linkage class and molecularity at most two, then for every positive rate vector each nonabsorbing reachable class is nonexplosive and positive recurrent. An absorbing singleton carries its point-mass stationary law.
Consequence. Every initial population has a unique stationary probability law on its reachable class.
The lifted return-path lemma supplies closure from time zero. The theorem accommodates coordinate faces, siphons, parity restrictions, conservation relations, and other lattice constraints without assuming irreducibility on all of \(\mathbb N_0^d\).
3. Mathematical position and prior work
Anderson and Kim formulated the stochastic weak-reversibility positive-recurrence conjecture in 2018. Anderson, Cappelletti, and Kim proved the bimolecular one-linkage case in 2020 under the additional condition
\[\{S_i,2S_i\}\cap\mathcal C\ne\varnothing\quad\text{for every species }S_i.\]
That condition enters their final tier comparison at a disabled mixed top complex. The present marked-target construction avoids that boundary-availability step and removes the pure-species condition, so the 2020 theorem is a special case.
| Feature | Anderson–Cappelletti–Kim (2020) | Version 1.2.4 |
|---|---|---|
| Weakly reversible, one linkage class | Required | Required |
| Maximum molecularity | Two | Two |
| Positive rate vector | Arbitrary | Arbitrary |
| Every species in \(S_i\) or \(2S_i\) | Required | Removed |
| Closed boundary and lattice classes | Covered | Covered class-wise |
| Multiple linkage classes | Not covered | Not claimed |
| Molecularity above two | Not covered | Not claimed |
A complementary theorem for two-species weakly reversible systems has been publicly announced in conference talks since 2022, including the 10 June Geneva program and a 22 June Cornell recording. The current project page describes the proof as complete and sketches its approach, while listing the five-author manuscript as in preparation; no corresponding public manuscript was located in a 22 August 2026 check. Its public scope permits broader reaction-graph structure but fixes two species. This paper permits arbitrarily many species but assumes one linkage class and molecularity at most two. The claims are therefore complementary, and neither is presented as superseding the other. The May 2026 revision of Chuang Xu's On the Regulary of Reaction Systems [title as published] proves the relevant nonexplosion result and still describes bimolecular positive recurrence as open.
Exact source comparisons and the fresh submission audit are included with the tagged Version 1.2.4 package.
4. Systems-biology interpretation
Stochastic mass-action chains are standard chemical-master-equation models for random molecule counts in low-copy-number intracellular networks, including signalling, enzymatic, gene-regulatory, and synthetic biochemical systems. For a network satisfying the theorem's exact structural assumptions, the result supplies a stationary probability law within each closed class and therefore long-run state frequencies and expectations of bounded observables in that class.
Finite means, variances, correlations, and other unbounded molecule-count moments require separate integrability; positive recurrence alone does not provide them. Nor does the theorem assert bounded sample paths or claim that all biochemical networks satisfy the one-linkage and bimolecular assumptions.
5. Verification, audits, and reproducibility
The universal theorem is analytic. The deterministic package is designed to falsify exact identities, exercise boundary cases, independently validate finite classifications, and detect software regressions; it is not a finite proof of recurrence.
- 57 mathematical/verifier tests plus seven release-safety and provenance tests. Across Python 3.11–3.14, the reproducer generates the canonical report twice and requires both outputs to be byte-identical to all committed copies; separate tests reject unsafe archive paths, unignored symbolic links, and absent, wrong, or lightweight release tags.
- 3,318 exact factorial identities and 172 exact entropy signatures. The source-probability rewrite is checked by exact rational prime signatures rather than floating-point comparison.
- 36 scalar-envelope checks. These cover both branches, their boundary, and pointwise monotonicity used in the backward induction.
- 98,261 exhaustive three-species top-complex cases. Each classification has an independently validated witness; atlas SHA-256:
6bc68fa9ffa0643e3ad4356b02d40839bb8cee28ed0fac026eb6b65881cedf27. - 5,000 fixed-seed four-species cases. Independently witness-validated stress tests; SHA-256:
4974d6213318ea627ae4dcec955d9f4d4c192a3c38a96c66504501cff63ea2d1. - Exact state-cycle and Anderson–Cappelletti–Kim calibrations. These exercise 58 lifted-edge witnesses, 26 finite reachability pairs, 24 boundary-example identities, rate degeneration, random-time Foster summation, regenerative occupation, absorbing singletons, parallel channels, equal displacements, finite classes, and boundary behavior.
The canonical verification report, unchanged Version 1.2.0 standalone verifier, reviewer checklist, adversarial audit, rate-degeneration calculation, complete manifest, canonical PDF builder, deterministic archive builder, and executable clean-checkout replay are all included in the tagged package.
The downloadable Version 1.2.4 package is the curated preservation object. Its complete durable-file manifest is available directly. All public download hashes, including the report, four PDFs, manifest, citation file, and archive, are collected in SHA256SUMS.txt.
6. Exact scope and quantitative limitations
The positive-recurrence result does not cover multiple linkage classes or complexes of molecularity greater than two. It supplies the regenerative occupation representation, but does not prove a closed-form or product-form stationary distribution, finite unbounded molecule-count moments, tail bounds, mixing rates, a spectral gap, exponential ergodicity, bounded sample paths, or uniform bounds on transient excursions.
The proof's exceptional set is finite for each fixed positive rate vector, but no useful general bound for its location or diameter is obtained. On the cycle \(0\to A\to A+B\to0\), the exact target-following drift from the audited family is
\[D_0(m,A)=-\frac{\kappa_2}{\kappa_1+\kappa_2}\log m+O\!\left(\frac{\log m}{m}\right).\]
The negative coefficient can approach zero through positive rate ratios. Consequently, this proof cannot give a rate-independent bound on the exceptional set based only on the numbers of species and complexes. Effective rate-dependent recurrence, tail, excursion, and mixing bounds remain open.
7. Version history
Version 1.2.4 is the current preprint-submission candidate. It distinguishes the regenerative stationary-law representation from unavailable closed and product forms, points directly to the corollary's regenerative proof, records the zero initial expected reward in the Anderson–Cappelletti–Kim episode, defines the active complex set explicitly, and qualifies the deterministic motivation by strictly positive initial conditions. It also requires the literal annotated release tag before a complete replay can pass and documents the exact scope of non-load-bearing computational checks. The theorem and standalone verifier are unchanged.
Version 1.2.3 identified the exact Anderson–Cappelletti–Kim entropy function, recorded the tier-method lineage, made the properness selection literal, and presented nonexplosion before the physical-time return estimate. The theorem and standalone verifier were unchanged.
Version 1.2.2 stated both facets of the live ConStRAINeD status—a complete-proof description and an in-preparation manuscript listing—and added an exact human upload checklist for the current bioRxiv workflow. The theorem, proof, and standalone verifier were unchanged.
Version 1.2.1 corrected published-version locators, reduced notation collisions, clarified the top-complex trichotomy and three Markov-chain interfaces, and made the PDF-facing version/status text durable without changing the theorem.
Version 1.2 hardened the abstract quantifiers, Anderson–Cappelletti–Kim boundary display, regenerative occupation interface, disclosure and metadata; added a standalone supplementary-note PDF; and made the PDFs, regular wheel, manifest, archive, and tagged-release replay fully specified and deterministic.
Version 1.1 proved the stronger state-space formulation used here: under weak reversibility, every population reachability set is already a closed communicating class. It also corrected the fixed-population rate-limit sentence and added the exact Anderson–Cappelletti–Kim Example 4.1 comparison.
Version 1.0 is a curated publication candidate with a revised title and exposition, exact scope and systems-biology framing, an expanded literature audit, a targeted proof-interface replay, the quantitative rate-degeneration limitation, 45 substantive tests, independently validated atlases, three deterministic manuscript builds, and a complete first-contact archive. It intentionally omits abandoned discovery material from the new archive; that history remains preserved in Git and with Version 0.3.
Version 0.3 preserved the marked-target proof after an adversarial reconstruction and repaired the proof interfaces, verifier, manifests, bibliography, disclosure, and reproducibility record. Its manuscript, journal-layout PDF, and tagged package remain available.
Version 0.2 supplied the first independent Gates A1–A12 reconstruction. Its byte-identical archival files remain available as paper-v0.2.pdf and jap-submission-v0.2.pdf. The unversioned aliases paper.pdf and jap-submission.pdf now follow the current Version 1.2.4 release.
AI-assistance and verification disclosure
Generative-AI systems were used materially from 5–22 August 2026: OpenAI ChatGPT with GPT-5.6 Pro; Anthropic Claude and Claude Code with Opus 5; and OpenAI Codex desktop with a GPT-5-family coding/research deployment whose exact backend identifier was not exposed. Uses included mathematical exploration, counterexample search, adversarial proof review, exact verification, literature and policy checking, drafting, revision, and reproducibility validation.
The author directed the research, checked retained outputs, determined the released claims, curated the package, and assumes responsibility. AI output was treated as working material rather than scholarly authority or independent validation. No AI system is an author, and no independent expert human validation is claimed. The full tool-by-tool statement accompanies the tagged package.
Suggested citation
Alec Kriebel, “Positive Recurrence for Single-Linkage Bimolecular Weakly Reversible Stochastic Reaction Networks,” Version 1.2.4, unrefereed manuscript, 22 August 2026. Permanent project page. Tagged source: bimolecular-positive-recurrence-v1.2.4. Machine-readable citation metadata: CITATION-v1.2.4.cff. No DOI has been assigned.
Contact
Alec Kriebel
Independent Researcher
Technical correspondence: me@aleckriebel.com