Stable equilibrium requires an attained global entropy maximum

Let $\mathcal{F}$ be the explicitly declared feasible endpoint set of an isolated compound with the chosen independent separated macroscopic subsystem split and any explicitly modeled named auxiliaries, as defined in Isolated compound and feasible equilibrium set. For this route, adopt the physical criterion that any endpoint tuple the model classifies as a stable equilibrium under these fixed totals, retained partitions, and permitted exchanges satisfies

$S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x^{\mathrm{e}\mathrm{q}})={\max }_{x\in \mathcal{F}}{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}(x),S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x):={\sum }_{i=1}^m{S}_{c_i}(x_i)+{\sum }_{j=1}^k{S}_j^{\mathrm{a}\mathrm{u}\mathrm{x}}(y_j),$

Here $\max$ means that $x^{\mathrm{e}\mathrm{q}}\in \mathcal{F}$ and $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x^{\mathrm{e}\mathrm{q}})\ge S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x)$ for every $x\in \mathcal{F}$, using maximum in the sense of Maximum and minimum of a set. Here $x=(\mathbf{x},\mathbf{y})$ is a complete endpoint tuple from Isolated compound and feasible equilibrium set; $k=0$ gives the original thermodynamic-only formula. The finite sums are as in Finite sums and finite products, by recursion. The macroscopic summands are those of the separated constituents under Concave macroscopic fundamental entropy, and any auxiliary summands are the explicitly adopted endpoint entropy maps of the same complete compound model. This route assumes that these contributions add for the complete admitted split. A constant auxiliary entropy may be dropped as an additive constant when comparing maxima, whereas variable auxiliary entropy must be retained. Whenever the stable-equilibrium label is used, attainment is assumed: at least one maximizing endpoint lies in $\mathcal{F}$. If $\mathcal{F}$ is empty or the entropy supremum is not attained, this item supplies no equilibrium state.

“Global” refers to every endpoint in this exact $\mathcal{F}$, not just nearby states, states reachable by a restricted short-time dynamics, or states on a selected branch. A local maximum that is not global, or a long-lived preparation trapped by kinetic barriers, is not declared stable equilibrium by this postulate. No uniqueness or relaxation dynamics is assumed.