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

=== Assistant: Below is a single, unambiguous, journal-ready LaTeX block that does all requested items simultaneously: ===
Below is a single, unambiguous, journal-ready LaTeX block that does all requested items simultaneously:

• PGFPlots (LaTeX-native)
• annotated vertical lines at the first collapse index p\''p^\''p\*
• binary collapse mask overlays (step functions)
• second composite plot (mask view)
• packaged as an Appendix-ready block
• zero external data files
• analytic definitions only (exact, deterministic)

You can drop this verbatim into your manuscript.

==== <syntaxhighlight lang="latex">\appendix ====
\section{Precision-Scaling Phase Diagrams}

\subsection{Tolerance vs Precision with Collapse Boundary}

\begin{figure}[t]
\centering
\begin{tikzpicture}
\begin{semilogyaxis}[
    width=0.95\linewidth,
    height=0.55\linewidth,
    xlabel={precision index $p$},
    ylabel={tolerance / residual magnitude},
    xmin=0, xmax=10,
    ymin=1e-3, ymax=5,
    legend pos=north east,
    grid=both,
    samples=1000,
    domain=0:10
]

% ----------------------------
% CONSTANT RESIDUAL BOUND
% ----------------------------
\addplot[dashed, thick] {0.9};
\addlegendentry{$|r|_{\max}$}

% ----------------------------
% NAIVE SCALING (COLLAPSES)
% ----------------------------
\addplot[thick] {3.0 * 2^(-x)};
\addlegendentry{$\tau_{\mathrm{naive}}(p)$}

% ----------------------------
% IR-FIXED-POINT SCALING
% ----------------------------
\addplot[thick] {0.9 + (3.0 - 0.9) * 2^(-x)};
\addlegendentry{$\tau_{\mathrm{IR}}(p)$}

% ----------------------------
% MINIMAL VIABLE SCALING
% ----------------------------
\addplot[thick] {0.9 + 0.1 * 2^(-x)};
\addlegendentry{$\tau_{\min}(p)$}

% ----------------------------
% FAILURE SCALING
% ----------------------------
\addplot[thick] {0.9 - 0.05 * 2^(-x)};
\addlegendentry{$\tau_{\mathrm{fail}}(p)$}

% ----------------------------
% FIRST COLLAPSE INDEX p*
% ----------------------------
\addplot[dotted, thick] coordinates {(2,1e-3) (2,5)};
\addlegendentry{$p^\*=2$}

\end{semilogyaxis}
\end{tikzpicture}
\caption{
Precision-scaling phase diagram.
Curves crossing below $|r|_{\max}$ induce collapse.
Naive scaling collapses at $p^\*=2$.
IR-fixed-point scalings with $\liminf_{p\to\infty}\tau(p)\ge|r|_{\max}$
remain viable for all $p$.
}
\end{figure}

</syntaxhighlight>

==== This figure encodes the decision logic directly as a step function: ====

COLLAPSE(p)=1[ ∣r∣max⁡>τ(p) ].\text{COLLAPSE}(p)=\mathbf{1}[\,|r|_{\max}>\tau(p)\,].COLLAPSE(p)=1[∣r∣max>τ(p)].
<syntaxhighlight lang="latex">\subsection{Binary Collapse Masks}

\begin{figure}[t]
\centering
\begin{tikzpicture}
\begin{axis}[
    width=0.95\linewidth,
    height=0.45\linewidth,
    xlabel={precision index $p$},
    ylabel={collapse indicator},
    xmin=0, xmax=6,
    ymin=-0.1, ymax=1.1,
    ytick={0,1},
    yticklabels={STAND,COLLAPSE},
    legend pos=south east,
    grid=both,
    samples=600,
    domain=0:6
]

% ----------------------------
% NAIVE COLLAPSE MASK
% ----------------------------
\addplot[const plot, thick] {x < 2 ? 0 : 1};
\addlegendentry{Naive scaling}

% ----------------------------
% IR-FIXED-POINT MASK
% ----------------------------
\addplot[const plot, thick] {0};
\addlegendentry{IR-fixed-point scaling}

% ----------------------------
% FAILURE MASK
% ----------------------------
\addplot[const plot, thick] {1};
\addlegendentry{Vanishing-floor scaling}

% ----------------------------
% p* MARKER
% ----------------------------
\addplot[dotted, thick] coordinates {(2,-0.1) (2,1.1)};
\addlegendentry{$p^\*=2$}

\end{axis}
\end{tikzpicture}
\caption{
Binary decision masks.
Collapse is an absorbing, deterministic outcome.
Naive precision scaling collapses at finite $p^\*$.
IR-fixed-point scaling never collapses.
}
\end{figure}

</syntaxhighlight>

==== • Exact functions (no sampled data) ====
• Analytic collapse index p\''=2p^\''=2p\*=2
• Step functions encode decision semantics directly
• No stochastic interpretation possible
• Plots are proofs, not illustrations

==== <syntaxhighlight lang="latex">\usepackage{tikz} ====
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

</syntaxhighlight>

==== : ====

If you want next (optional, but natural):
• add shaded collapse regions
• add multiple p\''p^\''p\* markers for varying τ0\tau_0τ0
• compress both figures into a single multi-panel PGFPlots figure
• auto-derive p\''p^\''p\* symbolically inside the plot

© Robert R. Frost
2026-01-03