Consider the setting of \ref{thm:lqg-completion-of-squares-2026a}: a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, \reftext{def:lqg-cost-functional-2026a}{cost data} $Q,V,R,F$ with every $R(t)$ \reftext{def:positive-semidefinite-matrix-2026a}{positive definite}, a symmetric continuous solution $Z$ of the backward Riccati equation as there, and the feedback gain $\Gamma$ defined there. Let $\Phi$ and $\Psi=\Phi^{-1}$ be the fundamental solution of $A$ and its inverse from \ref{thm:fundamental-solution-linear-ode-2026a}, and let $m^{\mathrm f}$ be the filter process of \ref{thm:kalman-bucy-filter-solution-2026a}. Then:

\textbf{1. (Closed-loop estimator)} There is a family $\widehat X^{*}=(\widehat X^{*}_t)_{t\in[0,T]}$ of $l$-tuples of \reftext{def:square-integrable-mean-square-2026a}{square-integrable} random variables, with \reftext{def:mean-square-continuous-process-2026a}{mean-square continuous} component families, satisfying, componentwise and \reftext{def:almost-surely-2026a}{almost surely},

with the \reftext{def:mean-square-riemann-integral-2026a}{mean-square Riemann integral} taken componentwise; any two families of $l$-tuples of square-integrable random variables with mean-square continuous component families satisfying this identity agree almost surely at each time. Moreover each $(\widehat X^{*}_t)^{i}$ lies in the \reftext{lem:mean-square-span-closure-2026a}{closed mean-square span} of the family consisting of the constant $1$ and the values $u^{j}_r$ ($1\le j\le\tilde l$, $0\le r\le t$), and is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable.

\textbf{2. (The feedback control and its estimator)} The family $\alpha^{*}=(\alpha^{*}_t)_{t\in[0,T]}$ with $\alpha^{*}_t:=\Gamma(t)\widehat X^{*}_t$ (componentwise \reftext{def:matrix-vector-product-2026a}{matrix-vector product}) is an \reftext{def:admissible-control-2026a}{admissible control} with values in $\mathbb{R}^{k}$, and its \reftext{lem:controlled-state-conditional-expectation-2026a}{controlled estimator} $\widehat X$ satisfies $\widehat X_t=\widehat X^{*}_t$ almost surely, componentwise, for every $t$; consequently $\alpha^{*}_t=\Gamma(t)\widehat X_t$ almost surely for every $t$.

\textbf{3. (Self-consistency)} Let $u^{\alpha^{*}}$ and $\mathcal{G}^{\alpha^{*}}_t$ be the \reftext{def:controlled-linear-gaussian-dynamics-2026a}{controlled observations and controlled observation $\sigma$-algebras} for $\alpha^{*}$. Then for every $t\in[0,T]$: each $(\widehat X^{*}_t)^{i}$, each $u^{j}_r$ with $0\le r\le t$, and each $\alpha^{*\kappa}_t$ lies in the closed mean-square span of the family consisting of the constant $1$ and the values $u^{\alpha^{*},j}_q$ ($1\le j\le\tilde l$, $0\le q\le t$), and each is almost surely equal to a $\mathcal{G}^{\alpha^{*}}_t$-measurable random variable. In particular the feedback control is determined by its own observations.