\section{Introduction}\label{sec:introduction}
Permanence asks for an eventual compact region inside a positive stoichiometric class. For mass-action systems this question separates the direction of the reaction vectors from the sizes and time dependence of their coefficients. Craciun, Nazarov and Pantea formulated the general endotactic permanence problem; Johnston, Pantea and Donnell state it as Conjecture~1.3 and give the ordinary-endotactic witness condition in Definition~3.1~\cite{CraciunNazarovPantea2013,JohnstonPanteaDonnell2016}. Strong endotacticity and weak reversibility provide important special cases~\cite{GopalkrishnanMillerShiu2014,BorosHofbauer2020,CraciunDeshpande2020}.

This paper develops the qualitative theorem and its reverse-rate consequences in one account. Part~\ref{part:uniform} proves permanence for ordinary endotactic networks under arbitrary measurable histories within a fixed positive rate box. The compact convex target is chosen before the initial compact set and history; the absorption deadline is chosen before the initial point and history. Existence and uniqueness are conclusions. The proof is for nonnegative real complexes. A registered formal statement also covers arbitrary signed real complexes on the positive orthant.

Part~\ref{part:reverse} asks what survives quantitatively when some coefficients approach zero. It treats autonomous weakly reversible networks with natural-number complexes and rates $k_e=\bar k_e\varepsilon^{a_e}$, with prefactors in a fixed positive box. For each fixed $\varepsilon>0$ the rates remain positive; the quantitative theorem tracks the deterioration of concentration bounds as $\varepsilon\downarrow0$. This is a parameter family, rather than a history whose coefficients vanish in time. Uniform permanence over one fixed rate box does not supply a bound uniform over these degenerating boxes.

\paragraph{The common geometric input and the new estimates.}
Both arguments use the attributed finite affine construction of OpenAI~\cite{OpenAI2026Boundedness,OpenAI2026Uniform,OpenAI2026Formal}. It supplies a concave, piecewise-affine barrier $F_h(x)=\min_{\ell\in I}\{r_\ell\cdot x+b_\ell(h)\}$. The offsets are chosen so that a slope active at $x=h^q$ approximates the logarithmic concentration profile $q=\log x/\log h$, even for a discontinuous pointwise tolerance. Thus the directions of the affine pieces encode which concentrations are near the small- or large-coordinate faces of a positive box. For ordinary endotacticity, a single forward witness below all adverse sources gives a rate-box drift margin after the projection lengths are retained. For weak reversibility, a return-path estimate across source-level cuts exposes the ratio of the largest and smallest rates. Convex plateaux and dyadic scale descent then supply absorption.

\paragraph{What the quantitative examples establish.}
The measurable-rate example in Section~\ref{u:sec:example} is neither weakly reversible nor strongly endotactic; its staged scalar clocks give a history-uniform operating deadline $347/16$. The reverse-rate example in Section~\ref{q:sec:coupled-proof} is a different, weakly reversible three-species network. Its sharp concentration scales and recovery upper bound require a declared prepared basin. An inventory preserved by the fast reactions supplies necessary recovery times from arbitrary positive seeds, but does not prove that every seed reaches the sharp corridor. The stochastic comparison supplies a lower bound on recovery duration, not a success probability or a stochastic upper deadline.

\paragraph{Evidence and organization.}
The conventional proofs are given here in full at their stated scope. Section~\ref{sec:formal} identifies the two registered formalization records, their selected statements and the remaining conventional or numerical components. The registered interfaces support the real-complex permanence theorem and a state-dependent inventory comparison. The stochastic and optimization arguments retain their conventional proofs, and declared numerical trajectories provide illustrations. Source and proof snapshots, historical receipts and computational files are preserved separately from the manuscript-only TeX archive. No proof development or numerical campaign was rerun to prepare this edition.

Part~\ref{part:uniform} contains the model, drift argument, solution construction and measurable-rate example. Part~\ref{part:reverse} gives the rate-spread theorem, the coupled-network certificate, recovery clocks, stochastic comparison and dimensional limitations. The final evidence section and Appendix~\ref{q:app:certificate} make the computational and formal boundaries explicit. Throughout, $E$ indexes reactions, $V$ denotes the comparison directions, $q$ the logarithmic exponent vector, and $\ell\in I$ an affine label. We use $P_c$ for a positive stoichiometric class, $C_h$ for an affine plateau, $D_*$ for the largest source degree and $h_0$ for a dyadic base. The scalar $p=uh$ is reserved for the rate parameters of the second example; the two examples have distinct reaction lists and assumptions. Part~\ref{part:reverse} reuses the geometric and absorption machinery proved in Part~\ref{part:uniform}.