Zero-divisor methods for absolute concentration robustness

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.