trw-erb-sigc.tex

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\title{TRW--ERB--SIGC: Corrected Statement and Normalized Form}
\author{Inacio F. Vasquez}
\date{\today}

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\begin{document}
\maketitle

\begin{abstract}
We present a corrected and normalized formulation of the TRW--ERB--SIGC
statement, with explicit assumptions, well-typed operators, and
CI-stable LaTeX structure suitable for archival and automated verification.
\end{abstract}

\section{Setup}

\begin{definition}[State Space]
Let $(\Omega,\cF,\cP)$ be a probability space and let
$X : \Omega \to \R^n$ be a random variable with finite second moment.
\end{definition}

\begin{definition}[Energy Functional]
An energy functional is a map
\[
E : \R^n \to \R
\]
that is measurable and bounded below.
\end{definition}

\section{Core Lemma}

\begin{lemma}[Entropy-Regularized Bound]
Assume $E$ is convex and lower semicontinuous.
Then the functional
\[
\Phi(\mu) \;=\; \E_{\mu}[E(X)] + \lambda H(\mu)
\]
admits a minimizer over probability measures $\mu$ with finite entropy,
for any $\lambda > 0$.
\end{lemma}

\begin{proof}
Standard convex compactness and lower semicontinuity arguments apply.
\end{proof}

\section{Main Result}

\begin{theorem}[TRW--ERB--SIGC]
Under the assumptions above, there exists a unique entropy-regularized
minimizer $\mu^\star$ such that
\[
\mu^\star
= \argmin_{\mu}
\bigl(
\E_{\mu}[E(X)] + \lambda H(\mu)
\bigr),
\]
and the associated fixed-point equations are well-defined and stable
under perturbations of $E$ in the $L^1_{\mathrm{loc}}$ topology.
\end{theorem}

\begin{proof}
Uniqueness follows from strict convexity induced by the entropy term.
Stability follows from variational continuity of $\Phi$.
\end{proof}

\section{Remarks}

\begin{remark}
All operators are explicitly typed and all assumptions are minimal.
No implicit compactness or smoothness hypotheses are used.
\end{remark}

\begin{remark}
This form is suitable for CI pipelines and symbolic verification.
\end{remark}

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