%% Vol III, Chapter 2 — Mathematical Companion
%% SOURCE: docs/working/papers/SSRN_companion/Math_Companion_SSRN.tex (full LaTeX source)
%%         docs/working/papers/SSRN_companion/figures/ (7 figure PDFs)

\chapter{Mathematical Companion}
\label{vol3:chap:math}

%% TODO(bob): this chapter is a formal reference for the mathematical and
%% engineering structures described in Math_Companion_SSRN.tex.
%% The full source is available at:
%%   docs/working/papers/SSRN_companion/Math_Companion_SSRN.tex
%%
%% Option A: include it directly:
%%   \input{../../working/papers/SSRN_companion/Math_Companion_SSRN.tex}
%%   (note: preamble collision likely — requires refactoring the companion source)
%%
%% Option B: promote key sections by hand.
%%
%% Recommend Option B for maintainability. Dictate which sections to promote.

\section{Q48.16 Fixed-Point Arithmetic}
\label{vol3:sec:math:q4816}

%% SOURCE: Math_Companion_SSRN.tex (Q48.16 section)
%% TODO(bob): promote the Q48.16 definition, operations, and precision analysis.

\section{Feedback Loop Equations}
\label{vol3:sec:math:loops}

%% SOURCE: Math_Companion_SSRN.tex (feedback loop mathematical formulations)
%% TODO(bob): promote the equations for all 7 loops.

\section{Figures}
\label{vol3:sec:math:figures}

%% SOURCE: docs/working/papers/SSRN_companion/figures/ (7 PDFs)
%% TODO(bob): decide which figures to include in this volume.
%% Available figures:
%%   fig1_snake_trajectory.pdf      — phase-space snake trajectory
%%   fig2_K_vs_window.pdf           — K parameter vs window width
%%   fig3_FFT_spectrum.pdf          — FFT spectrum of adaptation signal
%%   fig4_golden_ratio_penalties.pdf — penalty function shape
%%   fig5_bimodal_distributions.pdf — heat distribution bimodality
%%   fig6_performance_vs_k.pdf      — performance vs K sweep
%%   fig7_performance_by_regime.pdf — performance by operating regime

\begin{figure}[ht]
  \centering
  %% TODO(bob): uncomment and adjust path when figures are accessible from build:
  %% \includegraphics[width=0.8\linewidth]{../../working/papers/SSRN_companion/figures/fig1_snake_trajectory.pdf}
  \caption{Phase-space trajectory (snake plot).}
  \label{fig:snake}
\end{figure}