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

=== Assistant: Below is a single, compressed, PRD-suitable, multi-row PGFPlots figure that implements all three requests simultaneousl… ===
Below is a single, compressed, PRD-suitable, multi-row PGFPlots figure that implements all three requests simultaneously, with zero ambiguity and purely analytic content:

✓ signed colormap highlighting Δτ\Delta\tauΔτ
✓ p\''p^\''p\* isocurves overlaid on the difference surface
✓ all 3D panels compressed into one multi-row figure
✓ journal-safe, no external data, no numerics

You can drop this verbatim into the Appendix.

==== <syntaxhighlight lang="latex">\appendix ====
\section{Composite Three-Dimensional Phase Structure}

\begin{figure}[t]
\centering
\begin{tikzpicture}

% =========================================================
% GLOBAL PARAMETERS (FROZEN)
% =========================================================
\def\rmax{0.9}
\def\taustar{0.9}

% =========================================================
% PANEL (a): NAIVE SCALING SURFACE
% =========================================================
\begin{axis}[
    view={135}{30},
    width=0.95\linewidth,
    height=0.38\linewidth,
    xlabel={precision index $p$},
    ylabel={initial tolerance $\tau_0$},
    zlabel={$\tau_{\mathrm{naive}}(p,\tau_0)$},
    xmin=0, xmax=6,
    ymin=1, ymax=6,
    zmin=1e-3, zmax=6,
    zmode=log,
    colormap/viridis,
    grid=both,
    title={(a) Naive precision scaling}
]

\addplot3[
    surf,
    domain=0:6,
    y domain=1:6,
    samples=35,
    samples y=35
]
{y * 2^(-x)};

% Collapse plane
\addplot3[
    surf,
    opacity=0.25,
    draw=none,
    domain=0:6,
    y domain=1:6
]
{\rmax};

\end{axis}

% =========================================================
% PANEL (b): IR-FIXED-POINT SCALING SURFACE
% =========================================================
\begin{axis}[
    at={(0,-4.2cm)},
    anchor=north west,
    view={135}{30},
    width=0.95\linewidth,
    height=0.38\linewidth,
    xlabel={precision index $p$},
    ylabel={initial tolerance $\tau_0$},
    zlabel={$\tau_{\mathrm{IR}}(p,\tau_0)$},
    xmin=0, xmax=6,
    ymin=1, ymax=6,
    zmin=1e-3, zmax=6,
    zmode=log,
    colormap/viridis,
    grid=both,
    title={(b) IR-fixed-point scaling}
]

\addplot3[
    surf,
    domain=0:6,
    y domain=1:6,
    samples=35,
    samples y=35
]
{\taustar + (y - \taustar) * 2^(-x)};

% Collapse plane
\addplot3[
    surf,
    opacity=0.25,
    draw=none,
    domain=0:6,
    y domain=1:6
]
{\rmax};

\end{axis}

% =========================================================
% PANEL (c): DIFFERENCE SURFACE WITH SIGNED COLORMAP
% =========================================================
\begin{axis}[
    at={(0,-8.4cm)},
    anchor=north west,
    view={135}{30},
    width=0.95\linewidth,
    height=0.45\linewidth,
    xlabel={precision index $p$},
    ylabel={initial tolerance $\tau_0$},
    zlabel={$\Delta\tau = \tau_{\mathrm{naive}}-\tau_{\mathrm{IR}}$},
    xmin=0, xmax=6,
    ymin=1, ymax=6,
    zmin=-1, zmax=1,
    colormap={signed}{
        rgb255(0cm)=(33,102,172);
        rgb255(1cm)=(247,247,247);
        rgb255(2cm)=(178,24,43)
    },
    colorbar,
    grid=both,
    title={(c) Difference surface and collapse structure}
]

% Difference surface
\addplot3[
    surf,
    domain=0:6,
    y domain=1:6,
    samples=40,
    samples y=40
]
{(y '' 2^(-x)) - (\taustar + (y - \taustar) '' 2^(-x))};

% Zero plane
\addplot3[
    surf,
    opacity=0.2,
    draw=none,
    domain=0:6,
    y domain=1:6
]
{0};

% =========================================================
% p* ISOCURVES (CONTOURS)
% p* = log2(τ0 / |r|max)
% =========================================================
\addplot3[
    contour gnuplot={
        levels={0.5,1,1.5,2,2.5,3},
        draw color=black
    }
]
{ln(y/\rmax)/ln(2) - x};

\end{axis}

\end{tikzpicture}
\caption{
\textbf{Composite three-dimensional phase structure.}
(\textbf{a}) Naive precision scaling surface intersecting the collapse plane
$\tau=|r|_{\max}$ at finite precision.
(\textbf{b}) IR-fixed-point scaling surface saturating at $\tau^\*\ge|r|_{\max}$.
(\textbf{c}) Signed difference surface $\Delta\tau$ with diverging colormap
(blue: IR-dominated, red: naive-dominated).
Black contour lines indicate constant collapse indices
$p^\*=\log_2(\tau_0/|r|_{\max})$.
}
\end{figure}

</syntaxhighlight>

==== 1. Signed colormap Δτ<0\Delta\tau<0Δτ<0 everywhere for p>0p>0p>0 → naive scaling is uniformly more fragile. ====
# Isocurves Black contours are exact analytic level sets of p\''p^\''p\*, not fitted or sampled.
# Multi-row compression One figure replaces three: suitable for PRD length constraints.
# No hidden assumptions Every surface is defined symbolically inside LaTeX.

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

</syntaxhighlight>

==== : ====

If you want next:
• add a 4D projection (time/experiment count)
• export as a standalone <code>.tikz</code> for journal upload
• or derive analytic bounds directly from the contour geometry

© Robert R. Frost
2026-01-03