Consider a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$, with notation and fixed versions as there. Fix $s\in[0,T]$ and let $\mathcal{H}_s$ denote the \reftext{def:independence-sigma-algebras-2026a}{$\sigma$-algebra generated} by the combined family of the random variables $\xi^{i}$ ($1\le i\le l$), $W^{j}_r$ ($1\le j\le m$, $0\le r\le s$), and $u^{j}_r$ ($1\le j\le\tilde l$, $0\le r\le s$). Since the values of $u$ up to time $s$ are $\mathcal{H}_s$-measurable, $\mathcal{G}_s\subseteq\mathcal{H}_s$.

\textbf{1. (Independence of later increments)} For every $n\ge1$ and all choices of component indices $j_1,\dots,j_n\in\{1,\dots,m\}$ and times $s\le u_p<v_p\le T$ ($1\le p\le n$), the $\sigma$-algebras

are independent.

\textbf{2. (Span structure of Wiener-integral increments)} Let $j\in\{1,\dots,m\}$, let $0\le s<t\le T$, let $f:[0,t]\to\mathbb{R}$ be \reftext{def:continuity-closed-interval-c54-2026b}{continuous}, and set

with \reftext{thm:vector-wiener-integral-gaussian-2026a}{Wiener integrals} (any fixed versions; for $s=0$ the subtracted term is $0$ by the convention of \ref{def:ito-integral-2026a}). Then $\mathbb{E}[I]=0$ with the \reftext{def:expectation-variance-2026a}{expectation}, and $I$ lies in the \reftext{lem:mean-square-span-closure-2026a}{closed mean-square span} of the family of increments $W^{j}_v-W^{j}_u$ with $s\le u<v\le t$.

\textbf{3. (Orthogonality)} With $I$ as in claim 2, for every \reftext{def:square-integrable-mean-square-2026a}{square-integrable} random variable $Q$ that is \reftext{def:almost-surely-2026a}{almost surely} equal to an $\mathcal{H}_s$-measurable random variable: $IQ$ is integrable, $\mathbb{E}[IQ]=0$, and, with the \reftext{def:covariance-square-integrable-2026a}{covariance}, $\operatorname{Cov}(I,Q)=0$.