%% Vol I, Chapter 4 — Formal Verification
%% SOURCE: proof/ directory (24 .thy files)
%%         .claude/CLAUDE.md (Formal Verification section)
%%         docs/working/architecture/03-architecture/word-acl/DESIGN.md (ACL proofs)

\chapter{Formal Verification}
\label{vol1:chap:proofs}

%% TODO(bob): introductory paragraph — Isabelle/HOL proof-checked verification
%% of all 7 feedback loops, word categories, and the ACL system.
%% 24 theory files total (19 core + 5 ACL).

\section{Proof Infrastructure}
\label{vol1:sec:proofs:infra}

%% TODO(bob): describe the proof/ directory ROOT file, Isabelle session,
%% and how to run: isabelle build -D proof/
%% Note that the tool is Isabelle/HOL 2024 (or later); see DEVELOPER.md for setup.

\section{Base Definitions}
\label{vol1:sec:proofs:base}

%% SOURCE: proof/StarForth_Base.thy
%% TODO(bob): summarize the base type system, VM state type, and key
%% definitions exported to all downstream theories.

\section{Physics Loop Proofs}
\label{vol1:sec:proofs:loops}

%% SOURCE: proof/StarForth_Loop{1..7}_*.thy (7 files)
%% TODO(bob): one subsection per loop proving its stated property.
%% Template:

%% \subsection{Loop \#1: Execution Heat Monotonicity}
%% Source theory: \texttt{StarForth\_Loop1\_Heat.thy} \cite{james:2025:proof:loop1}
%% TODO(bob): state the theorem and its intuition in 2--3 sentences.

\subsection{Loop \#1: Execution Heat}
%% TODO(bob): theorem statement and 2-3 sentence intuition

\subsection{Loop \#2: Rolling Window Bounded Growth}
%% TODO(bob): theorem statement and 2-3 sentence intuition

\subsection{Loop \#3: Decay Convergence}
%% TODO(bob): theorem statement and 2-3 sentence intuition

\subsection{Loop \#4: Pipelining Metrics Safety}
%% TODO(bob): theorem statement and 2-3 sentence intuition

\subsection{Loop \#5: Window Width Inference Termination}
%% TODO(bob): theorem statement and 2-3 sentence intuition

\subsection{Loop \#6: Decay Slope Inference Soundness}
%% TODO(bob): theorem statement and 2-3 sentence intuition

\subsection{Loop \#7: Heartrate Adaptive Stability}
%% TODO(bob): theorem statement and 2-3 sentence intuition

\section{Word-Category Proofs}
\label{vol1:sec:proofs:words}

%% SOURCE: proof/StarForth_{Arithmetic,Stack,Logical,Memory,Return_Stack,Q48_16}_Words.thy
%% TODO(bob): table of word-category proofs with theorem names.

\section{Concurrency and Mutex}
\label{vol1:sec:proofs:concurrent}

%% SOURCE: proof/StarForth_Concurrent.thy, proof/StarForth_Mutex.thy
%% TODO(bob): describe the concurrency model and what mutual-exclusion
%% properties are proven. Key context: heartbeat thread vs. main thread race
%% identified in segfault-analysis.md; tuning_lock fix applied.

\section{Overall Correctness}
\label{vol1:sec:proofs:correctness}

%% SOURCE: proof/StarForth_Correctness.thy \cite{james:2025:proof:correctness}
%% TODO(bob): state the top-level correctness theorem binding all 24 files.

\section{ACL System Proofs}
\label{vol1:sec:proofs:acl}

%% SOURCE: proof/ACL_Pin_Monotone.thy, ACL_Inherit_Clears_Pin.thy,
%%         ACL_TTL_Bounded.thy, ACL_Emergency_Bypass.thy, ACL_No_Escalation.thy
%% TODO(bob): one paragraph per ACL theorem. Cite james:2025:proof:acl.
%% These 5 theorems cover: pin monotonicity, inheritance clears pin,
%% TTL bounded, emergency bypass correctness, no-escalation safety.