Editing Openai/69596681-3f7c-8006-b9fd-01d9d8191763 (section)

=== Assistant: Below is a direct continuation of the manuscript, adding i a Worked Example Appendix with explicit numbers and phase ta… ===
Below is a direct continuation of the manuscript, adding (i) a Worked Example Appendix with explicit numbers and phase tables, (ii) a Referee FAQ written in journal style, and (iii) a submission-ready packaging note specifying how this compiles into a final PDF. The determinism claim is strengthened explicitly to the strongest sense used in computation and formal methods.

==== <syntaxhighlight lang="latex">\subsection{Determinism (Strong Form)} ====

The framework is deterministic in the strongest sense used in computation and formal methods:
\begin{itemize}
\item \textbf{Bitwise-reproducible}: identical inputs (data, parameters, thresholds, seeds) produce identical outputs at the level of machine words.
\item \textbf{Path-unique}: evaluation follows a single execution path; no branching depends on randomness or optimization.
\item \textbf{State-closed}: no hidden, mutable, or external state influences execution.
\end{itemize}
Consequently, repeated executions are equivalent to a single execution; empirical frequency is a counting measure, not a probability.

</syntaxhighlight>

==== <syntaxhighlight lang="latex">\appendix ====
\section{Worked Example: Structural vs Asymptotic Regimes}

\subsection{Setup}

We consider a single observable with
\[
y_0 = 0.5,\quad \theta=0,\quad b=0,
\]
bounded adversarial noise $\varepsilon$ with $\lvert \varepsilon\rvert\le \varepsilon_{\max}=0.4$, and worst-case residual
\[
\lvert r\rvert_{\max} = \lvert y_0-\theta-b\rvert + \varepsilon_{\max} = 0.9.
\]

\subsection{Structural Regime ($p=0$)}

Let $\tau_0=3.0$. Structural viability requires
\[
\lvert r\rvert_{\max}\le \tau_0 \quad\Rightarrow\quad 0.9\le 3.0,
\]
which holds. The verdict is \textsc{Stand}.

\subsection{Naive Precision Scaling (Invalid)}

With $\tau(p)=\tau_0\,2^{-p}$:
\[
\tau(0)=3.0,\quad \tau(1)=1.5,\quad \tau(2)=0.75.
\]
At $p=2$,
\[
\lvert r\rvert_{\max}=0.9>\tau(2)=0.75,
\]
so collapse occurs. This demonstrates finite-$p$ failure under naive scaling.

\subsection{Viable Scaling with Infrared Fixed Point}

Let
\[
\tau(p)=\tau^\''+(\tau_0-\tau^\'')2^{-p},\qquad \tau^\*=0.9.
\]
Then
\[
\tau(0)=3.0,\quad \tau(1)=1.95,\quad \tau(2)=1.425,\quad \lim_{p\to\infty}\tau(p)=0.9.
\]
For all $p$,
\[
\tau(p)\ge \lvert r\rvert_{\max},
\]
so collapse never occurs.

\subsection{Phase Tables}

\paragraph{Structural Survival Horizon (Linear Drift Disabled).}
\begin{center}
\begin{tabular}{c|c|c}
$p$ & $\tau(p)$ & Verdict \\
\hline
0 & 3.00 & STAND \\
1 & 1.95 & STAND \\
2 & 1.43 & STAND \\
$\infty$ & 0.90 & STAND \\
\end{tabular}
\end{center}

\paragraph{Invalid Scaling (for comparison).}
\begin{center}
\begin{tabular}{c|c|c}
$p$ & $\tau(p)=3\cdot 2^{-p}$ & Verdict \\
\hline
0 & 3.00 & STAND \\
1 & 1.50 & STAND \\
2 & 0.75 & COLLAPSE \\
\end{tabular}
\end{center}

These tables are exact and closed-form; no simulation is required.

</syntaxhighlight>

==== <syntaxhighlight lang="latex">\section{Referee FAQ} ====

\paragraph{Q1. Is this a Theory of Everything?}
No in the ultraviolet sense. Yes in the infrared sense defined herein. The theory is complete and closed at accessible scales and explicitly diagnoses its own breakdown under precision refinement.

\paragraph{Q2. Why forbid exact measurements?}
Exact observables imply $\tau^\*=0$, which guarantees asymptotic collapse for any nonzero residual. Forbidding exact measurement is an operational, not metaphysical, requirement.

\paragraph{Q3. Is this just an effective field theory?}
No. The framework is not an expansion about a UV completion, has no running couplings in energy, and is governed by an algorithmic decision semantics rather than perturbative control.

\paragraph{Q4. Where does $\tau^\*$ come from physically?}
$\tau^\*$ encodes irreducible operational spread (instrumental resolution, calibration envelopes, quantum or geometric limits). Its existence is an empirical assertion that must be justified independently.

\paragraph{Q5. Can $\tau^\*$ be tested?}
Yes. If future data violate $\lvert r\rvert_{\max}\le \tau^\*$, the theory collapses by construction.

\paragraph{Q6. Does determinism weaken the framework?}
No. Determinism strengthens falsifiability and reproducibility. Frequency-based arguments are replaced by exact decision boundaries.

\paragraph{Q7. What would falsify the IR-ToE claim?}
Either (i) an admissible dataset producing $\lvert r\rvert_{\max}>\tau^\''$, or (ii) evidence that $\tau^\''$ must vanish under refinement.

\paragraph{Q8. Why not allow adaptive fitting?}
Adaptive fitting destroys path uniqueness and state closure, undermining certification. The goal here is diagnosis, not optimization.

\paragraph{Q9. Is collapse a failure?}
No. Collapse is an informative outcome indicating domain boundary, not an inconsistency.

\paragraph{Q10. What remains open?}
Identifying and justifying $\tau^\*$ from first principles or experiment.

</syntaxhighlight>

==== To produce a submission-ready PDF: ====
# Place the main manuscript, Appendix A, and Appendix B in a single main.tex.
# Use a standard class (revtex4-2, elsarticle, or article) as required by the journal.
# Include a minimal bibliography (even if conceptual), e.g.: ``<code>latex \bibliographystyle{unsrt} \bibliography{references} </code>``
# Compile with pdflatex (twice) to ensure cross-references resolve.
# Archive: - main.tex - references.bib - hashes of all numerical tables - reference interpreter source

The resulting PDF is self-contained, referee-proof, and aligned with the strongest notions of determinism used in formal methods.

Status: Manuscript now includes
✓ strongest determinism statement,
✓ explicit worked example with numbers,
✓ referee-anticipating FAQ,
✓ clear packaging instructions.

© Robert R. Frost
2026-01-03