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Optimal affinity of autocatalytic networks beyond gross stoichiometry

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.OC — Optimization and Control

Contributed by Michael Mislan ↗

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

An explicit reversible autocatalytic network has a unique global maximum of controlled production whose affinity exponential is strictly below its gross stoichiometric seed ratio. This refutes the proposed general lower bound. We identify a response-profile capacity that retains the missing kinetic information, analyze a literal recycling family approaching the source boundary and give a zero-diagonal counterexample. Under the stated rooted accessibility and response hypotheses, a local mass-action realization theorem makes the capacity sharp; an explicit power family supplies global maxima, while interacting cores delimit broader claims. The registered Lean project verifies the original refutation and checks the explicit witness against the literature conventions. Broader historical formal evidence and conventional arguments are distinguished in the manuscript and supporting claim map.

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. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences , Classics in Applied Mathematics 9, SIAM, Philadelphia, 1994.
  2. 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
  3. 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
  4. A. Despons, Y. De Decker and D. Lacoste, Structural constraints limit the regime of optimal flux in autocatalytic reaction networks , Commun. Phys. 7 (2024), article 224. https://doi.org/10.1038/s42005-024-01704-8; arXiv:2306.02366.arXivDOI
  5. M. Feinberg, Foundations of Chemical Reaction Network Theory , Applied Mathematical Sciences 202, Springer, Cham, 2019.
  6. M. Idel, A review of matrix scaling and Sinkhorn's normal form for matrices and positive maps , arXiv:1609.06349 (2016).arXiv
  7. J. G. Kemeny and J. L. Snell, Finite Markov Chains , Undergraduate Texts in Mathematics, Springer, New York, 1976.
  8. T. Kosc, D. Kuperberg, E. Rajon and S. Charlat, Thermodynamic consistency of autocatalytic cycles , Proc. Natl. Acad. Sci. USA 122 (2025), e2421274122. https://doi.org/10.1073/pnas.2421274122DOI
  9. 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
  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 17 referencesShow fewer references
  1. M. Polettini and M. Esposito, Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws , J. Chem. Phys. 141 (2014), 024117. https://doi.org/10.1063/1.4886396DOI
  2. R. Rao and M. Esposito, Nonequilibrium thermodynamics of chemical reaction networks: wisdom from stochastic thermodynamics , Phys. Rev. X 6 (2016), 041064. https://doi.org/10.1103/PhysRevX.6.041064DOI
  3. J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems , Rev. Mod. Phys. 48 (1976), 571--585. https://doi.org/10.1103/RevModPhys.48.571DOI
  4. E. Seneta, Non-negative Matrices and Markov Chains , Springer Series in Statistics, Springer, New York, revised printing, 2006.
  5. R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices , Ann. Math. Statist. 35 (1964), 876--879. https://doi.org/10.1214/aoms/1177703591DOI
  6. R. Sinkhorn and P. Knopp, Concerning nonnegative matrices and doubly stochastic matrices , Pacific J. Math. 21 (1967), 343--348.
  7. 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