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Zero-divisor methods for absolute concentration robustness

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

Contributed by Michael Mislan ↗

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

We study the relation between absolute concentration robustness and zero divisors in steady-state coordinate rings. Explicit positive rational, two-sided bimolecular examples disprove both necessity and sufficiency of the proposed unrestricted criterion. A separate example shows why elimination order matters without contradicting lexicographic or product-order statements. We prove completeness of a block-compatible algebraic interface under explicit basis, leading-coverage, nonvacuity and zero-divisor hypotheses. The manuscript develops complementary criteria from regular positive geometry and quantitative algebraic certificates, with examples from loaded reactors, EnvZ and product-release networks. Selected counterexamples, the conditional completeness theorem and exact certificate identities are formalized in Lean 4. Geometric and nonlinear dynamical arguments supplied conventionally are identified separately.

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

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Version history

  1. v1Submitted by Michael MislanInitial depositCurrentOct 21, 2026