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

=== Assistant: Below is a journal-ready PGFPlots extension that adds i the IR-fixed-point surface in 3D and ii a signed difference sur… ===
Below is a journal-ready PGFPlots extension that adds (i) the IR-fixed-point surface in 3D and (ii) a signed difference surface (naive − IR) for direct comparison. Everything is analytic, deterministic, and self-contained.

==== <syntaxhighlight lang="latex">\appendix ====
\section{Three-Dimensional Comparative Surfaces}

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

% =========================================================
% GLOBAL PARAMETERS (FROZEN)
% =========================================================
\def\rmax{0.9}        % worst-case residual bound
\def\taustar{0.9}     % IR fixed point (>= r_max)

% =========================================================
% PANEL (a): 3D TOLERANCE SURFACES (OVERLAID)
% =========================================================
\begin{axis}[
    view={135}{30},
    width=0.95\linewidth,
    height=0.55\linewidth,
    xlabel={precision index $p$},
    ylabel={initial tolerance $\tau_0$},
    zlabel={tolerance $\tau(p)$},
    xmin=0, xmax=6,
    ymin=1, ymax=6,
    zmin=1e-3, zmax=6,
    zmode=log,
    colormap/viridis,
    grid=both,
    legend style={at={(0.02,0.98)},anchor=north west}
]

% --- NAIVE SURFACE: τ_naive(p,τ0)=τ0 2^{-p}
\addplot3[
    surf,
    opacity=0.85,
    domain=0:6,
    y domain=1:6,
    samples=35,
    samples y=35
]
{y * 2^(-x)};
\addlegendentry{$\tau_{\mathrm{naive}}(p,\tau_0)$}

% --- IR-FIXED-POINT SURFACE: τ_IR(p,τ0)=τ''+(τ0-τ'')2^{-p}
\addplot3[
    surf,
    opacity=0.85,
    domain=0:6,
    y domain=1:6,
    samples=35,
    samples y=35
]
{\taustar + (y - \taustar) * 2^(-x)};
\addlegendentry{$\tau_{\mathrm{IR}}(p,\tau_0)$}

% --- COLLAPSE PLANE: τ=|r|max
\addplot3[
    surf,
    opacity=0.25,
    draw=none,
    domain=0:6,
    y domain=1:6
]
{\rmax};
\addlegendentry{$\tau=|r|_{\max}$}

\end{axis}

\end{tikzpicture}
\caption{
\textbf{3D tolerance surfaces under precision scaling.}
Naive scaling (solid) decays to zero and intersects the collapse plane at finite $p$.
IR-fixed-point scaling (solid) saturates at $\tau^\*\ge|r|_{\max}$ and remains viable.
}
\end{figure}

</syntaxhighlight>

==== This surface isolates where and by how much naive scaling violates IR viability. ====

<syntaxhighlight lang="latex">\section{Difference Surface Between Scaling Laws}

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

\begin{axis}[
    view={135}{30},
    width=0.95\linewidth,
    height=0.55\linewidth,
    xlabel={precision index $p$},
    ylabel={initial tolerance $\tau_0$},
    zlabel={$\Delta\tau(p)=\tau_{\mathrm{naive}}-\tau_{\mathrm{IR}}$},
    xmin=0, xmax=6,
    ymin=1, ymax=6,
    zmin=-6, zmax=6,
    colormap/viridis,
    grid=both,
    legend style={at={(0.02,0.98)},anchor=north west}
]

% --- DIFFERENCE SURFACE (ANALYTIC)
\addplot3[
    surf,
    domain=0:6,
    y domain=1:6,
    samples=40,
    samples y=40
]
{(y '' 2^(-x)) - (\taustar + (y - \taustar) '' 2^(-x))};
\addlegendentry{$\Delta\tau(p,\tau_0)$}

% --- ZERO PLANE (REFERENCE)
\addplot3[
    surf,
    opacity=0.25,
    draw=none,
    domain=0:6,
    y domain=1:6
]
{0};
\addlegendentry{$\Delta\tau=0$}

\end{axis}

\end{tikzpicture}
\caption{
\textbf{Difference surface between naive and IR-fixed-point scaling.}
Negative regions indicate where naive scaling undershoots the IR floor,
quantifying the structural deficit that induces collapse.
}
\end{figure}

</syntaxhighlight>

==== For reference, the plotted surfaces are: ====

τnaive(p,τ0)=τ0 2−p,τIR(p,τ0)=τ\''+(τ0−τ\'') 2−p,Δτ(p,τ0)=τnaive−τIR=−τ\*(1−2−p)≤0.\begin{aligned}
\tau_{\mathrm{naive}}(p,\tau_0) &= \tau_0\,2^{-p},\\
\tau_{\mathrm{IR}}(p,\tau_0) &= \tau^\'' + (\tau_0-\tau^\'')\,2^{-p},\\
\Delta\tau(p,\tau_0) &= \tau_{\mathrm{naive}} - \tau_{\mathrm{IR}}
= -\tau^\*\bigl(1-2^{-p}\bigr)\le 0.
\end{aligned}τnaive(p,τ0)τIR(p,τ0)Δτ(p,τ0)=τ02−p,=τ\''+(τ0−τ\'')2−p,=τnaive−τIR=−τ\*(1−2−p)≤0.
The sign of Δτ\Delta\tauΔτ is independent of τ0\tau_0τ0 and strictly negative for all p>0p>0p>0, proving that naive scaling is uniformly more fragile than IR-fixed-point scaling.

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

</syntaxhighlight>

===== • add a signed colormap highlighting Δτ<0\Delta\tau<0Δτ<0 =====
• overlay p\''p^\''p\* isocurves on the difference surface
• compress all 3D panels into a single multi-row figure suitable for PRD

© Robert R. Frost
2026-01-03