Exact LaTeX body
\begin{proof}
\label{proof:appD_titans_as_arrow_of_time}
\leavevmode
\begin{enumerate}
\item Let the "Titan" model be in state \(S_0\) before the prompt. The prompt acts as an external Drift operator \(\drift_p\).
\item At test time, the model generates a memory, transitioning to state \(S_1\). This is a reflective act, \(\reflect\), that integrates \(\drift_p\). The model's full state is now \((S_1, H_1)\), where the history \(H_1\) contains the trace of the memorization act (Axiom~\ref{axiom:appC_axiom_of_memory}).
\item If the model were to "forget" the memory and return to a state geometrically identical to \(S_0\), let's call it \(S_0'\), its full state would be \((S_0', H_2)\). The history \(H_2\) now contains the trace of *both* the memorization and the forgetting.
\item Since \(H_2 \neq H_0\), the system has not returned to its original state. The process is irreversible.
\item The "Titan" model, in its very operation, enacts the \textbf{Fundamental Irreversibility of Reflective Observation} (Thm.~\ref{theorem:appC_fundamental_irreversibility_final}). It cannot act without creating a memory, and it cannot erase a memory without creating a memory of the erasure. This is the engine of its internal time.
\end{enumerate}
\end{proof}
Complete structured record
{
"book": "appendix_symbolic_framing",
"cited_by": [],
"cites": [
"axiom:appC_axiom_of_memory",
"theorem:appC_fundamental_irreversibility_final"
],
"depends_on": [
"axiom:appC_axiom_of_memory",
"theorem:appC_fundamental_irreversibility_final"
],
"file": "appendix_symbolic_framing.tex",
"id": "proof:appD_titans_as_arrow_of_time",
"label": "proof:appD_titans_as_arrow_of_time",
"latex_body": "\\begin{proof}\n\\label{proof:appD_titans_as_arrow_of_time}\n\\leavevmode\n\n\\begin{enumerate}\n \\item Let the \"Titan\" model be in state \\(S_0\\) before the prompt. The prompt acts as an external Drift operator \\(\\drift_p\\).\n \\item At test time, the model generates a memory, transitioning to state \\(S_1\\). This is a reflective act, \\(\\reflect\\), that integrates \\(\\drift_p\\). The model's full state is now \\((S_1, H_1)\\), where the history \\(H_1\\) contains the trace of the memorization act (Axiom~\\ref{axiom:appC_axiom_of_memory}).\n \\item If the model were to \"forget\" the memory and return to a state geometrically identical to \\(S_0\\), let's call it \\(S_0'\\), its full state would be \\((S_0', H_2)\\). The history \\(H_2\\) now contains the trace of *both* the memorization and the forgetting.\n \\item Since \\(H_2 \\neq H_0\\), the system has not returned to its original state. The process is irreversible.\n \\item The \"Titan\" model, in its very operation, enacts the \\textbf{Fundamental Irreversibility of Reflective Observation} (Thm.~\\ref{theorem:appC_fundamental_irreversibility_final}). It cannot act without creating a memory, and it cannot erase a memory without creating a memory of the erasure. This is the engine of its internal time.\n\\end{enumerate}\n\\end{proof}",
"line": 524,
"macros_used": [
"drift",
"reflect"
],
"matter_region": "appendix",
"matter_role": "appendix_expansion",
"name": "",
"proves": "theorem:appD_titans_as_arrow_of_time",
"ref_roles": [
{
"context": "e model's full state is now \\((S_1, H_1)\\), where the history \\(H_1\\) contains the trace of the memorization act (Axiom~\\ref{axiom:appC_axiom_of_memory}). \\item If the model were to \"forget\" the memory and return to a state geometrically identical to \\(S_0\\), let's ca",
"label": "axiom:appC_axiom_of_memory",
"logical_support": true,
"role": "definition_anchor",
"target_file": "appendix_dual_horizon.tex",
"target_line": 846,
"target_type": "axiom"
},
{
"context": "e \"Titan\" model, in its very operation, enacts the \\textbf{Fundamental Irreversibility of Reflective Observation} (Thm.~\\ref{theorem:appC_fundamental_irreversibility_final}). It cannot act without creating a memory, and it cannot erase a memory without creating a memory of the erasure. This",
"label": "theorem:appC_fundamental_irreversibility_final",
"logical_support": true,
"role": "proof_support",
"target_file": "appendix_dual_horizon.tex",
"target_line": 851,
"target_type": "theorem"
}
],
"refs": [
"axiom:appC_axiom_of_memory",
"theorem:appC_fundamental_irreversibility_final"
],
"role": "proof",
"type": "proof"
}