\documentclass[11pt,a4paper]{article}
\usepackage{amsmath,amssymb,physics}
\usepackage{geometry}
\usepackage{hyperref}
\usepackage{xcolor}
\usepackage{booktabs}

\geometry{margin=2.5cm}
\hypersetup{colorlinks=true,linkcolor=blue!60!black}

\title{\textbf{Research Flow Notebook}\\[0.5em]
\large Topic: \texttt{demo-topic} \quad Lane: \texttt{formal\_theory}}
\author{AITP v2 Protocol}
\date{2026-04-20}

\begin{document}
\maketitle

\begin{abstract}
This flow notebook archives the complete research trajectory for topic
\texttt{demo-topic}, from question framing through derivation to
validated claim. It serves as the mandatory pre-paper artifact required
before L5 writing.
\end{abstract}

\tableofcontents
\newpage

% ============================================================
\section{Research Question}
% ============================================================

\textbf{Bounded Question:} What quantity is bounded here?

\medskip
\noindent\textbf{Scope:} One model, one regime.

\noindent\textbf{Target Quantities:} Gap and symmetry sector.

\subsection{Non-Success Conditions}
\begin{itemize}
  \item No convergence in the numerical solution after $10^4$ iterations.
  \item Dimensional analysis yields inconsistent units.
\end{itemize}

\subsection{Uncertainty Markers}
\begin{itemize}
  \item Sign convention ambiguity in the Green's function.
  \item Unknown validity of perturbative expansion beyond second order.
\end{itemize}

% ============================================================
\section{Conventions And Regime}
% ============================================================

\subsection{Notation Choices}
\begin{tabular}{lll}
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Source} \\
\midrule
$G(\mathbf{r}, \mathbf{r}')$ & Green's function & source-a \\
$\Sigma$ & Self-energy & source-a \\
$E_{\text{gap}}$ & Band gap & convention \\
\bottomrule
\end{tabular}

\medskip
\noindent\textbf{Units:} Natural units ($\hbar = 1$).

\noindent\textbf{Sign Convention:} $(-,+,+,+)$ metric signature.

% ============================================================
\section{Derivation Route}
% ============================================================

\subsection{Ideation}
The core idea is to verify whether the AITP v2 protocol produces
correct provenance chains from framing through validation.

\textbf{Motivation:} Without a complete provenance chain, claims
cannot be audited or reproduced.

\subsection{Planning}
\begin{enumerate}
  \item Frame the research question with explicit scope boundaries.
  \item Walk through all five L3 subplanes in sequence.
  \item Validate the distilled claim against physics checks.
  \item Render the flow notebook as the archive checkpoint.
\end{enumerate}

\textbf{Derivation Route:} Step-by-step verification of each protocol
stage gate.

\subsection{Analysis}
Applied unit tests and manual inspection to verify:
\begin{itemize}
  \item Each L3 subplane artifact has required frontmatter fields.
  \item Each subplane artifact contains required headings.
  \item Transitions follow the allowed directed graph.
\end{itemize}

Key equation from analysis:
\begin{equation}
  \Sigma(\mathbf{r}, \mathbf{r}'; \omega)
  = i \int \frac{d\omega'}{2\pi} \,
    G(\mathbf{r}, \mathbf{r}'; \omega + \omega') \,
    W(\mathbf{r}, \mathbf{r}'; \omega')
\end{equation}

\subsection{Result Integration}
All stages connected properly. The provenance chain from L1 framing
through L3 derivation to L4 validation is complete and consistent.

\textbf{Findings:}
\begin{itemize}
  \item All 74 unit tests pass across 6 waves.
  \item Flow notebook TeX renders correctly with all required sections.
  \item Promotion to global L2 succeeds with correct trust classification.
\end{itemize}

\subsection{Distillation}
\textbf{Distilled Claim:} The AITP v2 protocol produces correct
provenance chains from question framing through validated claim
promotion.

\textbf{Evidence Summary:} Tests pass, flow TeX renders correctly,
trust classification is $\text{basis} = \text{validated}$, $\text{scope}
= \text{bounded\_reusable}$.

% ============================================================
\section{Validation And Checks}
% ============================================================

\subsection{Validation Contract}
\begin{tabular}{ll}
\toprule
\textbf{Field} & \textbf{Value} \\
\midrule
Candidate ID & \texttt{cand-1} \\
Mandatory checks & dimensional\_consistency, limiting\_case\_check \\
\bottomrule
\end{tabular}

\subsection{Review: \texttt{cand-1}}
\begin{tabular}{ll}
\toprule
\textbf{Field} & \textbf{Value} \\
\midrule
Outcome & \textcolor{green!60!black}{\textbf{pass}} \\
Notes & All physics checks passed. \\
Dimensional consistency & \textcolor{green!60!black}{\checkmark} \\
Limiting case check & \textcolor{green!60!black}{\checkmark} \\
\bottomrule
\end{tabular}

% ============================================================
\section{Current Claim Boundary}
% ============================================================

\textbf{Claim:} The AITP v2 protocol produces correct provenance chains
from question framing through validated claim promotion.

\medskip
\textbf{Trust Classification:}
\begin{tabular}{ll}
\toprule
\textbf{Dimension} & \textbf{Value} \\
\midrule
Trust basis & validated \\
Trust scope & bounded\_reusable \\
Version & 1 \\
\bottomrule
\end{tabular}

\medskip
\textbf{Assumptions:}
\begin{enumerate}
  \item The topic follows the \texttt{formal\_theory} lane.
  \item All L1 artifacts are filled before advancing to L3.
  \item At least one candidate passes L4 review before promotion.
\end{enumerate}

% ============================================================
\section{Failures And Open Problems}
% ============================================================

\subsection{Known Limitations}
\begin{itemize}
  \item The flow notebook currently embeds raw Markdown in TeX; a
        proper renderer should convert headings to \LaTeX\ sections.
  \item The \texttt{contradiction\_register} requires manual population;
        it does not auto-detect contradictions from source analysis.
\end{itemize}

\subsection{Open Problems}
\begin{itemize}
  \item Extending the protocol to support multi-model comparison topics.
  \item Integration with Lean 4 for formal verification of mathematical claims.
  \item Automatic detection of scope creep during L3 derivation.
\end{itemize}

\subsection{Negative Results}
\begin{itemize}
  \item Attempting to skip subplanes (e.g., jumping from ideation to
        distillation) is correctly blocked by the protocol gates.
\end{itemize}

\end{document}
