Zero-divisor methods for absolute concentration robustness: completeness, positive geometry, and quantitative certificates

A mass-action system has absolute concentration robustness (ACR) in a species whose concentration is the same at every positive steady state. García Puente, Gross, Harrington, Johnston, Meshkat, Pérez Millán and Shiu proposed to detect ACR values as the positive numbers $\alpha$ for which $x_i-\alpha$ lies in, or is a zero divisor of, the steady-state ideal, and to list them from the leading coefficients of a Gröbner basis. We settle two questions they left open. For networks with at most bimolecular complexes, uniqueness of such a value is neither necessary nor sufficient for ACR. Their candidate algorithm is incomplete for an elimination order that their definition admits, and we prove that it is complete for block orders, which establishes their conjecture in that form. We then show that one positive steady state at which the Jacobian has rank equal to the stoichiometric rank forces every ACR value, global or local, to be a zero-divisor value, with a multiplier that does not vanish at that state. The second half of the paper makes such identities quantitative: a lower bound on the multiplier converts a polynomial identity into a bound on $|x_i-\alpha|$ in terms of steady-state residuals and signed load terms, and without such a bound a small residual can indicate extinction instead of accuracy. For a loaded reactor we obtain the exact positive operating region, a strictly smaller conditioning region, local stability throughout, and an explicit rational Lyapunov ellipsoid on which the multiplier bound holds for all time. For the EnvZ/OmpR model every positive steady state is regular, steady states are classified by total amounts, and uniform multiplier bounds hold over parameter boxes. Finally, replacing a reaction that has more than two product molecules by a chain of private intermediates preserves the steady-state quotient ring, the positive steady states and all ACR values, while adding explicit residual, storage and leakage terms to a certificate. The algebraic results and the identities and inequalities behind the quantitative ones are compiled in Lean 4 with Mathlib; the geometric and dynamical arguments are conventional proofs and are marked as such.