Proof of The Asymptotically Optimal Value of the Recentred N-Agent Cost of the Ising Population Model

Claim 1. We go through the common data and the standing hypotheses of The Fluctuation LQG Value Is the Asymptotically Optimal Value of the Recentred N-Agent Cost over Admissible Families of Observation-Driven Policies, and Is Attained by the Approximate Kalman Policies, under Deterministic Initial States in the order in which they are listed there.

The natural numbers are $l=2\ge2$, $m=2\ge1$ and $\tilde{l}=2\ge1$. By claims The Ising Population Model Instantiates the Data of the Fluctuation Theory (i) and The Ising Population Model Instantiates the Data of the Fluctuation Theory (ii), $(\beta_{0},\beta_{1})$ is an affine-controlled transition-rate family on $2$ states whose control set $\mathcal{A}$ is nonempty, convex and compact for the topology determined by the Euclidean distance, with control bound $\sup_{a\in\mathcal{A}}|a|=\sqrt{2}\,\bar{a}$, and the associated transition-rate family is $\beta$, with rate bound $B=\bar{a}$. By claim The Ising Population Model Instantiates the Data of the Fluctuation Theory (iv), $(U,V,\bar{\beta})$ is a twice continuously differentiable extension of $\beta$ with derivative bound $K=1$, and $\bar{b}$ is its extended aggregate state drift. By claim The Ising Population Model Instantiates the Data of the Fluctuation Theory (v), $\tilde{\beta}$ is an observation-rate family with $\tilde{l}=2$ channels and rate bound $\tilde{B}=q+q_{0}$, with aggregate observation drift $\tilde{b}$. The horizon is the real number $T>0$ of clause The Ising Population Data (i). By claims The Ising Population Model Instantiates the Data of the Fluctuation Theory (vii) and The Ising Population Model Instantiates the Data of the Fluctuation Theory (viii), $(L,G)$ is population cost data convex in the control on $\mathcal{A}$, with twice continuously differentiable extension $(U_{c},\bar{L},\bar{G})$. By claim The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal (iii), $(S,A,P)$ is a stationary mean-field triple for these data, with $S_{0}=\bigl(\tfrac12,\tfrac12\bigr)\in\Delta^{2}$ and $P_{0}=(0,0)$. By claim The Ising Population Model Instantiates the Data of the Fluctuation Theory (iii), $\Theta$ is the aggregate fluctuation covariance of $\beta$, and $\Theta^{\star}_{t}=\Theta(S_{t},A_{t})=vv^{\top}$ by claim The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity (iii). The matrices $E_{t}$, $\mathsf{B}_{t}$, $Q_{t}$, $V_{t}$, $R_{t}$, $\hat{F}$ are those computed in claim The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity (ii), defined there by the very formulas the theorem uses, and hypothesis (H2) holds with the family $Z$ by claim The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form (iii).

The standing hypotheses are supplied as follows: (A) by claim The Ising Population Model Instantiates the Data of the Fluctuation Theory (i); (U) by claim The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal (iv); (JC) with the constant $c_{J}>0$ by claim The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity (v); (H2) as just recorded; (LipC) by claim The Ising Population Model Instantiates the Data of the Fluctuation Theory (vii); the optimality $[A]\in\mathcal{M}^{*}_{S_{0}}$ by claim The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal (v); (TG) by claim The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal (vi); and the standing hypothesis on $S^{*}=S$ by claim The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal (ii). The three further hypotheses of the theorem hold as well: (OC) with $\underline{b}=q_{0}>0$ by claim The Ising Population Model Instantiates the Data of the Fluctuation Theory (v); (X$'$) with $\tilde{K}=0$ by claim The Ising Population Model Instantiates the Data of the Fluctuation Theory (vi); and (H5) with $\varrho_{\mathcal{A}}=\min\{1-\underline{a},\bar{a}-1\}>0$ by claim The Ising Population Model Instantiates the Data of the Fluctuation Theory (i), since $A_{t}=(1,1)$ for every $t$ and every $a$ within Euclidean distance $\varrho_{\mathcal{A}}$ of $(1,1)$ lies in $\mathcal{A}$. Together with the assumed Kalman family and (DK), every hypothesis of the theorem is met.

Uniqueness of $Z$: apply The Control Riccati Equation with Cross Term: Global Existence and Uniqueness, Positive Semidefiniteness, and the Lyapunov Bound with $l=k=2$, horizon $T$, and the data $A(t)=E_{t}$, $B(t)=\mathsf{B}_{t}$, $Q(t)=Q_{t}$, $V(t)=\tfrac12V_{t}=0$, $R(t)=R_{t}$ and $F=\hat{F}=0$. All are constant except the null matrix $V$, hence continuous; $R_{t}=(4\chi)^{-1}I$ is symmetric positive definite; $Q_{t}=\tfrac12\psi\,vv^{\top}+\mu\,\mathsf{n}\mathsf{n}^{\top}$ is symmetric and positive semidefinite because $x\cdot Q_{t}x=\tfrac12\psi(v\cdot x)^{2}+\mu(\mathsf{n}\cdot x)^{2}\ge0$; $Q(t)-V(t)R(t)^{-1}V(t)^{\top}=Q_{t}$ is positive semidefinite; and $F=0$ is symmetric positive semidefinite. With $V(t)=0$ the corollary's equation is exactly the equation of (H2), so it has exactly one continuous solution, which by claim The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form (iii) is $Z$.

The theorem defines $\tilde{D}_{t}=\tilde{\mathcal{E}}_{t}^{\top}(\tilde{\Theta}^{\star}_{t})^{-1}\tilde{\mathcal{E}}_{t}$, $W_{t}=Z_{t}\mathsf{B}_{t}+\tfrac12V_{t}$ and $\Xi_{t}=W_{t}R_{t}^{-1}W_{t}^{\top}$, which are the matrices computed in claims The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity (iv) and The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form (iii); and it defines $\Pi$ as the solution of the Kalman covariance Riccati equation on $[0,T]$ with the data $\mathcal{E}$, $\Theta^{\star}$, $\tilde{D}$ and the zero matrix as initial value, which by claim The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form (iv) is $\Pi_{t}=p_{t}vv^{\top}$.

Finally, the theorem defines $V^{*}$ as the Lebesgue integral over $[0,T]$ of $t\mapsto\sum_{\gamma,\delta}Z^{\gamma\delta}_{t}\Theta^{\star\gamma\delta}_{t}+\sum_{\gamma,\delta}\Xi^{\gamma\delta}_{t}\Pi^{\gamma\delta}_{t}$, which by claim The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form (v) equals $4z_{t}+32\chi z_{t}^{2}p_{t}$, a continuous function of $t$ not involving $\mu$. This is the displayed formula for $V^{*}$.

For the bounds, claims The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form (i) and The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form (ii) give $0\le z_{t}<(\Delta-1)/(4\chi)$ and $0\le p_{t}<1/(\Gamma+2)$ for every $t\in[0,T]$. Hence the integrand is nonnegative and at most

$$4\cdot\frac{\Delta-1}{4\chi}+32\chi\cdot\frac{(\Delta-1)^{2}}{16\chi^{2}}\cdot\frac{1}{\Gamma+2}=\frac{\Delta-1}{\chi}+\frac{2(\Delta-1)^{2}}{\chi\,(\Gamma+2)}=\frac{\Delta-1}{\chi}\Bigl(1+\frac{2(\Delta-1)}{\Gamma+2}\Bigr),$$

and the stated bounds on $V^{*}$ follow by monotonicity of the integral, Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, the interval $[0,T]$ having measure $T$.

Claims 2, 3 and 4. By claim 1 every hypothesis of The Fluctuation LQG Value Is the Asymptotically Optimal Value of the Recentred N-Agent Cost over Admissible Families of Observation-Driven Policies, and Is Attained by the Approximate Kalman Policies, under Deterministic Initial States holds for the present data, triple, Riccati family and Kalman family, and the real number $V^{*}$ of that theorem is the one displayed in claim 1. Claim 2 is then conclusion 2 of the theorem, claim 3 is conclusion 3, and claim 4 is conclusion 4, each restated for these data.