HEXAGONbeta

Unistationarity of Type II_l autocatalytic cores

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 unistationarity of reversible Type II_l autocatalytic cores with l at least three under strictly positive degradation. The source-minimal wiring forces source exhaustion and unit return tails, allowing comparison of positive stationary states through current ratios, chain reduction and a cyclic determinant argument. The proof includes separated stems and the exceptional coincident three-fork branch, with all internal chain species retained. Exact six- and seven-species reversible mass-action examples show that the minimality assumptions cannot be dropped: both admit two distinct positive stationary states with strictly positive rates and losses. The principal uniqueness theorem has a complete registered Lean formalization. The manuscript distinguishes that theorem, broader historical formal evidence and conventional exact calculations; the supplied claim map records the boundaries of each.

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. M. Banaji and G. Craciun, Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems , Adv. Appl. Math. 44 (2010), 168--184. https://doi.org/10.1016/j.aam.2009.07.003DOI
  2. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences , Classics in Applied Mathematics 9, SIAM, Philadelphia, 1994.
  3. A. Blokhuis, D. Lacoste and P. Nghe, Universal motifs and the diversity of autocatalytic systems , Proc. Natl. Acad. Sci. USA 117 (2020), 25230--25236. https://doi.org/10.1073/pnas.2013527117DOI
  4. G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: I. The injectivity property , SIAM J. Appl. Math. 65 (2005), 1526--1546. https://doi.org/10.1137/S0036139904440278DOI
  5. L. de Moura and S. Ullrich, The Lean 4 theorem prover and programming language , in: Automated Deduction -- CADE 28, Lecture Notes in Computer Science 12699, Springer, 2021, pp. 625--635. https://doi.org/10.1007/978-3-030-79876-5_37DOI
  6. M. Feinberg, Foundations of Chemical Reaction Network Theory , Applied Mathematical Sciences 202, Springer, Cham, 2019.
  7. E. Feliu and C. Wiuf, Preclusion of switch behavior in networks with mass-action kinetics , Appl. Math. Comput. 219 (2012), 1449--1467. https://doi.org/10.1016/j.amc.2012.07.048DOI
  8. The mathlib Community, The Lean mathematical library , in: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), ACM, 2020, pp. 367--381. https://doi.org/10.1145/3372885.3373824DOI
  9. S. Müller, E. Feliu, G. Regensburger, C. Conradi, A. Shiu and A. Dickenstein, Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry , Found. Comput. Math. 16 (2016), 69--97. https://doi.org/10.1007/s10208-014-9239-3DOI
  10. P. Nandan, P. Nghe and J. Unterberger, Autocatalytic cores in the diluted regime: classification and properties , J. Math. Biol. 92 (2026), article 36. https://doi.org/10.1007/s00285-026-02357-7; arXiv:2507.15546.arXivDOI
Show all 11 referencesShow fewer references
  1. J. Unterberger and P. Nghe, Stoechiometric and dynamical autocatalysis for diluted chemical reaction networks , J. Math. Biol. 85 (2022), article 26. https://doi.org/10.1007/s00285-022-01798-0DOI

Version history

  1. v1Submitted by Michael MislanInitial depositCurrentOct 21, 2026