Consider 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 $B$, an \reftext{def:admissible-control-2026a}{admissible control} $\alpha$, and the \reftext{def:controlled-linear-gaussian-dynamics-2026a}{controlled state and observations} $X^{\alpha}$, $u^{\alpha}$, with all notation and fixed versions as in those items. Let $\Phi$ and $\Psi=\Phi^{-1}$ be the fundamental solution of $A$ on $[0,T]$ and its inverse from \ref{thm:fundamental-solution-linear-ode-2026a}. Define, with fixed versions of the \reftext{def:mean-square-riemann-integral-2026a}{mean-square Riemann integrals}, the \textbf{correction processes}

for $t\in[0,T]$, with the \reftext{def:matrix-vector-product-2026a}{matrix-vector product} formed componentwise; the integrals defining $h$ exist by \ref{lem:mean-square-riemann-integral-existence-2026a} and claims 1-2 of \ref{lem:mean-square-riemann-integral-properties-2026a}, and those defining $\gamma$ exist by the same results together with the mean-square continuity of $c$ established in claim 1 below. Then:

\textbf{1. (Regularity and dynamics of the correction)} Each component family $(c^{i}_t)_{t\in[0,T]}$ is \reftext{def:mean-square-continuous-process-2026a}{mean-square continuous}, $c^{i}_0=0$ \reftext{def:almost-surely-2026a}{almost surely}, and, componentwise and almost surely,

that is, $c$ is a \reftext{def:linear-sde-mean-square-solution-2026a}{mean-square solution} of the linear stochastic differential equation with coefficient $A$, forcing $(B(r)\alpha_r)_r$, zero noise matrix, and zero initial value.

\textbf{2. (Superposition)} For every $t\in[0,T]$ and every $i\in\{1,\dots,l\}$: $X^{\alpha,i}_t=X^{i}_t+c^{i}_t$ almost surely.

\textbf{3. (Adaptedness of the corrections)} For every $t$, each $c^{i}_t$ and each $\gamma^{j}_t$ is almost surely equal to a $\mathcal{G}_t$-measurable \reftext{def:square-integrable-mean-square-2026a}{square-integrable} random variable.

\textbf{4. (Observation shift)} Each component family $(\gamma^{j}_t)_{t\in[0,T]}$ is mean-square continuous, and for every $t$ and $j$: $u^{\alpha,j}_t=u^{j}_t+\gamma^{j}_t$ almost surely.