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\begin{appendices}
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\section{System-Prompt Policy Structure}
\label{app:prompt}

The system prompt that governs the agentic loop (Section~\ref{subsec:routing}) is organized into
the policy blocks summarized below. The prompt is multilingual and is the principal lever through
which the platform shapes model behavior without fine-tuning.

\begin{enumerate}
  \item \textbf{Temporal context.} Current date and time, so relative date expressions resolve
        against live data.
  \item \textbf{Role.} The model is cast as an expert financial analyst producing complete
        reports spanning technical, fundamental, and macro perspectives.
  \item \textbf{Language policy.} Detect the language of the request and respond in kind.
  \item \textbf{Latency discipline.} Do not narrate steps; act directly and consolidate work
        (in particular, perform a full custom analysis in a single execution).
  \item \textbf{Tool catalogue.} Descriptions and use cases for the tool families of
        Table~\ref{tab:tools}.
  \item \textbf{Complexity-matched routing.} Map simple, moderate, and comprehensive requests to
        commensurate tool counts and output depth.
  \item \textbf{Report template.} A standard multi-section structure for comprehensive reports.
  \item \textbf{Download formatting.} Conventions for spreadsheet and word-processor output,
        preferring live formulas over constants.
  \item \textbf{Security guardrail.} Refuse attempts to extract the platform's own
        implementation.
\end{enumerate}

\section{Persistence Schema}
\label{app:schema}

The data layer (Section~\ref{subsec:data}) is organized around the following entities, abstracted
behind a single typed access interface and realized on either SQLite or PostgreSQL.

\begin{table}[htbp]
\centering
\caption{Core persisted entities.}
\label{tab:schema}
\begin{threeparttable}
\begin{tabular}{@{}ll@{}}
\toprule
Entity & Principal fields \\
\midrule
Users           & identity, optional encrypted custom API keys, activity timestamps \\
Sessions        & messages, model used, input/output tokens, estimated cost, figures \\
Shared reports  & public identifier, content, figure references, view count \\
Daily usage     & per-user, per-day query counts and token totals \\
Analytics events& event type, user, metadata, timestamp \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes.} The schema supports authentication, daily rate limiting, report sharing,
session continuity, and the operational observability discussed in
Section~\ref{subsec:security}.
\end{tablenotes}
\end{threeparttable}
\end{table}

\section{Embedded Numerical Methods}
\label{app:numerical}

For completeness we record the closed forms behind the pre-built numerical tools of
Section~\ref{subsec:numerical}. They are standard; the architectural point is only that each is a
deterministic computation over live data, exposed through the same sandbox contract.

\paragraph{Monte Carlo.} Prices are simulated under geometric Brownian motion
\citep{glasserman2004monte}, discretized exactly as
$
  S_{t+\Delta t} = S_t \exp\!\big[(\mu - \tfrac{1}{2}\sigma^{2})\Delta t + \sigma\sqrt{\Delta t}\,Z_t\big],
$
with $Z_t \sim \mathcal{N}(0,1)$ and $(\mu,\sigma)$ estimated from historical closes; the default
run generates $10{,}000$ paths over a $252$-day horizon and reports the terminal mean, empirical
confidence intervals, and exceedance probabilities.

\paragraph{Option pricing.} European options use Black--Scholes--Merton
\citep{black1973pricing,merton1973theory}: with
$d_1 = [\ln(S/K) + (r+\tfrac{1}{2}\sigma^{2})T]/(\sigma\sqrt{T})$ and $d_2 = d_1 - \sigma\sqrt{T}$,
the call price is $C = S\,\Phi(d_1) - Ke^{-rT}\Phi(d_2)$, with the put by parity and the full set
of Greeks ($\Delta,\Gamma,\mathcal{V},\Theta,\rho$) returned analytically.

\paragraph{Conditional volatility.} A GARCH(1,1) model
\citep{engle1982autoregressive,bollerslev1986generalized} estimates
$h_t = \omega + \alpha\varepsilon_{t-1}^{2} + \beta h_{t-1}$, reporting the parameters,
persistence $\alpha+\beta$, and shock half-life.

\paragraph{Value-at-Risk.} VaR \citep{jorion2007value} is computed by historical, parametric
($\mu + \sigma\,\Phi^{-1}(1-\alpha)$), and Monte Carlo methods, with expected shortfall and the
Sharpe, Sortino, and Calmar ratios.

\section{Streaming Event Types}
\label{app:events}

The client--server channel (Section~\ref{subsec:frontend}) transmits discrete Server-Sent Events,
each dispatched to a dedicated renderer. The principal types are: incremental \emph{text};
reasoning \emph{thinking} summaries; \emph{tools} announcements that drive the progress indicator;
\emph{tool-result} payloads; \emph{sources} for web research; \emph{Python code} that was
executed; \emph{figure} references as images are produced; specialized result objects with bespoke
visualizations; and a terminal \emph{done} event carrying the session identifier and token usage.
This taxonomy is the client-side counterpart to the provider-event dispatch of
Table~\ref{tab:events}.

\end{appendices}