\section{Mathematical Proofs}
\label{sec:proofs}

This appendix provides formal mathematical derivations for the key theoretical relationships presented in Section~\ref{sec:methodology}.

\subsection{Derivation of Pure Stance Definitions}

\begin{theorem}[Pure Stance Characterization]
The pure dovish and hawkish stance measures satisfy the relationships given in Definition~\ref{def:pure_stances}.
\end{theorem}

\begin{proof}
\textbf{Case 1: Pure Dovish Statement.} Assume the Federal Reserve releases a statement that exactly matches the dovish counterfactual: $F_t = F_t^D$.

The tone measure becomes:
\begin{align}
\text{Tone}_t &= \frac{\text{sim}(F_t, F_t^H) - \text{sim}(F_t, F_t^D)}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\
&= \frac{\text{sim}(F_t^D, F_t^H) - \text{sim}(F_t^D, F_t^D)}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\
&= \frac{\text{sim}(F_t^D, F_t^H) - 1}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\
&= -1
\end{align}

The novelty measure is:
\begin{equation}
\text{Novelty}_t = 1 - \text{sim}(F_t, F_{t-1}) = 1 - \text{sim}(F_t^D, F_{t-1})
\end{equation}

Therefore, the stance measure is:
\begin{align}
\text{Stance}_t &= \text{Novelty}_t \times \text{Tone}_t \notag \\
&= \left(1 - \text{sim}(F_t^D, F_{t-1})\right) \times (-1) \notag \\
&= -\left(1 - \text{sim}(F_t^D, F_{t-1})\right) \notag \\
&= \text{Stance}_t^{dove}
\end{align}

\textbf{Case 2: Pure Hawkish Statement.} Assume $F_t = F_t^H$:

The tone measure becomes:
\begin{align}
\text{Tone}_t &= \frac{\text{sim}(F_t^H, F_t^H) - \text{sim}(F_t^H, F_t^D)}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\
&= \frac{1 - \text{sim}(F_t^H, F_t^D)}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\
&= 1
\end{align}

Therefore:
\begin{align}
\text{Stance}_t &= \left(1 - \text{sim}(F_t^H, F_{t-1})\right) \times 1 = \text{Stance}_t^{hawk}
\end{align}

This completes the proof. \hfill $\square$
\end{proof}

\subsection{Derivation of Dovish Weight Parameter}

\begin{theorem}[Dovish Weight Parameter Formula]
The weight parameter $w_t$ in the weighted stance representation has the form given in Definition~\ref{def:weighted_stance}.
\end{theorem}

\begin{proof}
From the weighted stance equation:
\begin{equation}
\text{Stance}_t = w_t \cdot \text{Stance}_t^{dove} + (1 - w_t) \cdot \text{Stance}_t^{hawk}
\end{equation}

Substituting the expressions for pure stances from Theorem 1:
\begin{align}
\text{Stance}_t &= -w_t\left(1 - \text{sim}(F_t^D, F_{t-1})\right) + (1 - w_t)\left(1 - \text{sim}(F_t^H, F_{t-1})\right) \notag \\
&= 1 - 2w_t + w_t\left(\text{sim}(F_t^D, F_{t-1}) + \text{sim}(F_t^H, F_{t-1})\right) - \text{sim}(F_t^H, F_{t-1})
\end{align}

Collecting the terms in $w_t$, this reads
\begin{equation}
\text{Stance}_t = 1 - \text{sim}(F_t^H, F_{t-1}) - w_t\left(2 - \text{sim}(F_t^D, F_{t-1}) - \text{sim}(F_t^H, F_{t-1})\right).
\end{equation}
Equating with $\text{Stance}_t = \left(1 - \text{sim}(F_t, F_{t-1})\right) \times \text{Tone}_t$ and solving for $w_t$:
\begin{equation}
w_t = \frac{1 - \text{sim}(F_t^H, F_{t-1}) - \left(1 - \text{sim}(F_t, F_{t-1})\right) \times \text{Tone}_t}{2 - \text{sim}(F_t^D, F_{t-1}) - \text{sim}(F_t^H, F_{t-1})}
\end{equation}

This establishes the formula. \hfill $\square$
\end{proof}

\subsection{Proof of MPS Decomposition}

\begin{theorem}[Policy Stance Surprise Decomposition]
The policy stance surprise admits the decomposition given in Proposition~\ref{prop:mps}.
\end{theorem}

\begin{proof}
From the definitions:
\begin{align}
\text{MPS}_t &= \text{Stance}_t - \mathbb{E}_{t-\Delta}[\text{Stance}_t] \notag \\
&= \text{Novelty}_t \times \text{Tone}_t - (1 - 2p_{t-\Delta}) \cdot \overline{\text{Novelty}}_{t \mid t-\Delta} \notag \\
&= \left(\overline{\text{Novelty}}_{t \mid t-\Delta} + \varepsilon_t\right) \times \text{Tone}_t - (1 - 2p_{t-\Delta}) \cdot \overline{\text{Novelty}}_{t \mid t-\Delta} \notag \\
&= \overline{\text{Novelty}}_{t \mid t-\Delta} \times \text{Tone}_t - \overline{\text{Novelty}}_{t \mid t-\Delta} + 2p_{t-\Delta} \cdot \overline{\text{Novelty}}_{t \mid t-\Delta} + \varepsilon_t \times \text{Tone}_t \notag \\
&= \overline{\text{Novelty}}_{t \mid t-\Delta}(\text{Tone}_t + 2p_{t-\Delta} - 1) + \text{Tone}_t \cdot \varepsilon_t
\end{align}

This establishes the decomposition. \hfill $\square$
\end{proof}

\subsection{Economic Interpretation}

The mathematical results provide several economic insights. First, the pure stance characterization shows that our measures correctly identify extreme policy communications, with dovish statements receiving negative stance values and hawkish statements receiving positive values. Second, the weight parameter derivation reveals how actual policy communications can be understood as weighted averages of extreme alternatives. Third, the MPS decomposition shows that policy surprises have two distinct sources: unexpected tone conditional on expected information content, and unexpected information content weighted by actual tone.

