The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal

Fix parameters as in The Ising Population Data (i) and adopt the Ising population data with those parameters, together with The Ising Population Model Instantiates the Data of the Fluctuation Theory, whose claims identify $\beta$, its aggregate state drift $b$, the extensions $(U,V,\bar{\beta})$ and $(U_{c},\bar{L},\bar{G})$, the extended aggregate state drift $\bar{b}$, and the partial derivatives of $\bar{L}$. Adopt the coordinate and partial-derivative notation $\partial_{i}$ of the extension definitions, as in that lemma. Write $\mathcal{U}_{\mathcal{A}}$ for the set of $\mathcal{A}$-valued controls on $[0,\theta]$ for a horizon $\theta>0$, $F^{[\theta]}$ for the mean-field cost and $J^{*[\theta]}_{x}$, $\mathcal{M}^{*[\theta]}_{x}$ for the optimal mean-field value and the set of optimal mean-field controls from $x\in\Delta^{2}$, all formed with the data $(\beta_{0},\beta_{1})$, $(L,G)$ and the horizon $\theta$; the superscript $[\theta]$ is dropped when $\theta=T$.

Define the equilibrium triple(i) of these data to be the three constant maps $S,A,P:[0,T]\to\mathbb{R}^{2}$ given by

$$S_{t}=\Bigl(\tfrac{1}{2},\tfrac{1}{2}\Bigr),\qquad A_{t}=(1,1),\qquad P_{t}=(0,0)\qquad(t\in[0,T]) :$$

an evenly split population, held there by unit transition rates, with vanishing co-state. Then the following hold.

1. (A trajectory pair.)(ii) $b(S_{t},A_{t})=0$ for every $t\in[0,T]$, and $(S,A)$ is a mean-field trajectory pair for $\beta$ with horizon $T$. Moreover $S$ takes values in $\Delta^{2}$, $S_{0}=\bigl(\tfrac{1}{2},\tfrac{1}{2}\bigr)$, and $S^{\gamma}_{t}=S^{\gamma}_{0}+\int_{[0,t]}b^{\gamma}(S_{s},A_{s})\,ds$ for every $t\in[0,T]$ and $\gamma\in\{1,2\}$, so that $S$ satisfies, with $x_{0}=S_{0}$, the standing hypothesis on the map written $S^{*}$ in The Block Cascade of Anchored Good-Set Clocks: Adapted Good Sets, Matched Escape Bounds, and the Energy Ledger.

2. (A stationary mean-field triple.)(iii) All four first-order partial derivatives of $\bar{L}$ vanish at $(S_{t},A_{t})$ for every $t\in[0,T]$. The map $P$ is a stationary co-state for $\beta$, $(L,G)$ and the chosen extensions, so that $(S,A,P)$ is a stationary mean-field triple for them; and the constant $C_{P}=0$ satisfies $\sum_{\delta=1}^{2}|P^{\delta}_{t}|\le C_{P}$ for every $t\in[0,T]$.

3. (Hypothesis (U).)(iv) The mean-field Hamiltonian along the triple, in the sense of Quadratic Growth of the Mean-Field Hamiltonian in the Control along a Stationary Mean-Field Triple, is

$$\mathcal{H}_{t}(a)=\frac{1}{2\chi}\bigl(\phi(a^{1})+\phi(a^{2})\bigr)\qquad(t\in[0,T],\ a\in V),$$

and for every $t\in[0,T]$ the point $A_{t}=(1,1)$ is the unique minimiser of $\mathcal{H}_{t}$ on $\mathcal{A}$, with $\mathcal{H}_{t}(A_{t})=0$; that is, hypothesis (U) of that lemma holds.

4. (Optimality of the equilibrium control.)(v) Let $\theta>0$ be a real number. Then $J^{*[\theta]}_{x}\ge0$ for every $x\in\Delta^{2}$, and $J^{*[\theta]}_{(1/2,1/2)}=0$, this value being attained by the class in $\mathcal{U}_{\mathcal{A}}$ of the constant map with value $(1,1)$ on $[0,\theta]$. In particular, with $\theta=T$, the class $[A]$ lies in $\mathcal{M}^{*}_{S_{0}}$.

5. (Hypothesis (TG).)(vi) Hypothesis (TG) of Quadratic Expansion Bounds for the Mean-Field To-Go Value along a Stationary Mean-Field Triple holds for these data and the triple $(S,A,P)$, with the constants $\varepsilon_{tg}=1$ and $C_{tg}=0$.