HEXAGONbeta

Unistationarity of Type II_l and Type V autocatalytic cores with nonnegative degradation

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systems

Contributed by Michael Mislan ↗

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

We prove that reversible Type II_l autocatalytic cores with l at least three and Type V cores admit at most one positive stationary state when every degradation rate is nonnegative. The Type II result assumes source-minimality, while the Type V statement uses its literal normal form. Strictly positive reaction rate constants are retained throughout. A reduction to the fork skeleton, source-exhaustion identities and diagonal current criteria control both positive-loss and boundary cases, including internal chains. We also derive determinant estimates and local continuation results, with small-degradation existence and stability under the stated hypotheses. The main nonnegative-degradation uniqueness theorem is registered in Lean 4; the manuscript and supporting claim map distinguish the formal interface from additional conventional results.

Provenance statement

The text is primarily AI-generated. AI systems assisted the mathematical development, computations and Lean formalization under human direction; human understanding covers some parts of the work.

Formalizations

References

  1. Banaji, Murad; Craciun, Gheorghe. Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems. Advances in Applied Mathematics, vol. 44, no. 2, pp. 168–184. 2010. https://doi.org/10.1016/j.aam.2009.07.003link
  2. Berman, Abraham; Plemmons, Robert J. Nonnegative Matrices in the Mathematical Sciences. SIAM, vol. 9. 1994. https://doi.org/10.1137/1.9781611971262link
  3. Blokhuis, Alex; Lacoste, David; Nghe, Philippe. Universal motifs and the diversity of autocatalytic systems. Proceedings of the National Academy of Sciences of the USA, vol. 117, no. 41, pp. 25230–25236. 2020. https://doi.org/10.1073/pnas.2013527117link
  4. Conradi, Carsten; Feliu, Elisenda; Mincheva, Maya; Wiuf, Carsten. Identifying parameter regions for multistationarity. PLoS Computational Biology, vol. 13, no. 10, pp. e1005751. 2017. https://doi.org/10.1371/journal.pcbi.1005751link
  5. Craciun, Gheorghe; Feinberg, Martin. Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM Journal on Applied Mathematics, vol. 65, no. 5, pp. 1526–1546. 2005. https://doi.org/10.1137/S0036139904440278link
  6. Craciun, Gheorghe; Feinberg, Martin. Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph. SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1321–1338. 2006. https://doi.org/10.1137/050634177link
  7. de Moura, Leonardo; Ullrich, Sebastian. The Lean 4 theorem prover and programming language. Automated Deduction – CADE 28, vol. 12699, pp. 625–635. 2021. https://doi.org/10.1007/978-3-030-79876-5_37link
  8. Feinberg, Martin. Chemical reaction network structure and the stability of complex isothermal reactors–I. The deficiency zero and deficiency one theorems. Chemical Engineering Science, vol. 42, no. 10, pp. 2229–2268. 1987. https://doi.org/10.1016/0009-2509(87)80099-4link
  9. Feinberg, Martin. Foundations of Chemical Reaction Network Theory. Springer, vol. 202. 2019. https://doi.org/10.1007/978-3-030-03858-8link
  10. Feliu, Elisenda; Wiuf, Carsten. Preclusion of switch behavior in networks with mass-action kinetics. Applied Mathematics and Computation, vol. 219, no. 4, pp. 1449–1467. 2012. https://doi.org/10.1016/j.amc.2012.07.048link
Show all 21 referencesShow fewer references
  1. Hordijk, Wim; Steel, Mike. Detecting autocatalytic, self-sustaining sets in chemical reaction systems. Journal of Theoretical Biology, vol. 227, no. 4, pp. 451–461. 2004. https://doi.org/10.1016/j.jtbi.2003.11.020link
  2. Horn, Fritz; Jackson, Roy. General mass action kinetics. Archive for Rational Mechanics and Analysis, vol. 47, pp. 81–116. 1972. https://doi.org/10.1007/BF00251225link
  3. Horn, Roger A.; Johnson, Charles R. Matrix Analysis. Cambridge University Press. 2013
  4. Joshi, Badal; Shiu, Anne. A survey of methods for deciding whether a reaction network is multistationary. Mathematical Modelling of Natural Phenomena, vol. 10, no. 5, pp. 47–67. 2015. https://doi.org/10.1051/mmnp/201510504link
  5. Kauffman, Stuart A. Autocatalytic sets of proteins. Journal of Theoretical Biology, vol. 119, no. 1, pp. 1–24. 1986. https://doi.org/10.1016/S0022-5193(86)80047-9link
  6. Krantz, Steven G.; Parks, Harold R. The Implicit Function Theorem: History, Theory, and Applications. Birkhäuser. 2002. https://doi.org/10.1007/978-1-4612-0059-8link
  7. Mislan, Michael. Unistationarity of Type IIℓ_\ell autocatalytic cores. 2026. https://doi.org/10.5281/zenodo.23479694. Zenodo, version 1.0DOI
  8. Müller, Stefan; Feliu, Elisenda; Regensburger, Georg; Conradi, Carsten; Shiu, Anne; Dickenstein, Alicia. Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. Foundations of Computational Mathematics, vol. 16, no. 1, pp. 69–97. 2016. https://doi.org/10.1007/s10208-014-9239-3link
  9. Nandan, Praneet; Nghe, Philippe; Unterberger, Jérémie. Autocatalytic cores in the diluted regime: classification and properties. Journal of Mathematical Biology, vol. 92, pp. article 36. 2026. https://doi.org/10.1007/s00285-026-02357-7; arXiv:2507.15546arXiv
  10. The mathlib Community. The Lean mathematical library. Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), pp. 367–381. 2020. https://doi.org/10.1145/3372885.3373824link
  11. Unterberger, Jérémie; Nghe, Philippe. Stoechiometric and dynamical autocatalysis for diluted chemical reaction networks. Journal of Mathematical Biology, vol. 85, pp. article 26. 2022. https://doi.org/10.1007/s00285-022-01798-0link

Version history

  1. v1Submitted by Michael MislanInitial depositCurrentOct 21, 2026