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

=== Assistant: Below is a standalone, journal-uploadable .tikz file that adds a 4D projection by encoding time / experiment count as a… ===
Below is a standalone, journal-uploadable <code>.tikz</code> file that adds a 4D projection by encoding time / experiment count as a parametric dimension (color + layered contours), while remaining pure PGFPlots, analytic, deterministic, and self-contained.

Design choice (explicit):
4D (p,τ0,τ,t)(p,\tau_0,\tau,t)(p,τ0,τ,t) is projected as:
* 3D surface: (p,τ0,τ(p,τ0))(p,\tau_0,\tau(p,\tau_0))(p,τ0,τ(p,τ0))
* 4th dimension ttt (time / experiment count) encoded as a continuous color parameter and time-indexed contour slices
* No animation, no external files, PRD-safe static figure

==== <syntaxhighlight lang="latex">%========================================================== ====
% Standalone TikZ / PGFPlots file
% 4D Projection: (p, tau0, tau, t)
%==========================================================
\documentclass[tikz,border=2mm]{standalone}

\usepackage{pgfplots}
\usepackage{pgfplotstable}
\pgfplotsset{compat=1.18}

\begin{document}

\begin{tikzpicture}

%==========================================================
% GLOBAL PARAMETERS (FROZEN)
%==========================================================
\def\rmax{0.9}        % worst-case residual bound
\def\taustar{0.9}     % IR fixed point
% Time / experiment index t \in [0,1]
% Interpretation: cumulative experiment count or refinement epoch

%==========================================================
% 4D PROJECTION PANEL
%==========================================================
\begin{axis}[
    view={135}{30},
    width=14cm,
    height=9cm,
    xlabel={precision index $p$},
    ylabel={initial tolerance $\tau_0$},
    zlabel={tolerance $\tau(p,\tau_0)$},
    xmin=0, xmax=6,
    ymin=1, ymax=6,
    zmin=1e-3, zmax=6,
    zmode=log,
    grid=both,
    colormap={time}{
        rgb255(0cm)=(33,102,172);   % early (blue)
        rgb255(1cm)=(247,247,247); % mid (white)
        rgb255(2cm)=(178,24,43)    % late (red)
    },
    colorbar,
    colorbar style={
        title={$t$ (experiment count / time)},
        ticks={0,0.5,1},
        ticklabels={early,mid,late}
    }
]

%==========================================================
% NAIVE SURFACE WITH TIME MODULATION (COLOR = t)
% tau_naive(p,tau0,t) = tau0 '' 2^{-p} '' (1 - 0.3 t)
%==========================================================
\addplot3[
    surf,
    shader=interp,
    domain=0:6,
    y domain=1:6,
    samples=35,
    samples y=35,
    point meta expr={t},
    variable=t,
    variable y=\tauzero
]
({x},{\tauzero},{\tauzero '' 2^(-x) '' (1 - 0.3*t)});

%==========================================================
% IR-FIXED-POINT SURFACE (TIME-INVARIANT)
%==========================================================
\addplot3[
    surf,
    opacity=0.35,
    domain=0:6,
    y domain=1:6,
    samples=30,
    samples y=30
]
{\taustar + (y - \taustar) * 2^(-x)};

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

%==========================================================
% TIME-SLICED p* ISOCURVES (CONTOUR STACK)
% p*(tau0,t) = log2(tau0(1-0.3t)/rmax)
%==========================================================
\foreach \tval in {0.0,0.5,1.0} {

    \addplot3[
        thick,
        contour gnuplot={
            levels={0},
            draw color=black
        }
    ]
    {ln((y''(1-0.3''\tval))/\rmax)/ln(2) - x};

}

\end{axis}

\end{tikzpicture}

\end{document}

</syntaxhighlight>

==== ### ====
* x=px = px=p : precision index
* y=τ0y = \tau_0y=τ0 : initial tolerance
* z=τ(p,τ0)z = \tau(p,\tau_0)z=τ(p,τ0) : tolerance flow
* Color = ttt : time / experiment count / refinement epoch

===== τnaive(p,τ0,t)=τ0 2−p (1−αt),α=0.3\tau_{\text{naive}}(p,\tau_0,t) =====
= \tau_0\,2^{-p}\,(1-\alpha t),\quad \alpha=0.3τnaive(p,τ0,t)=τ02−p(1−αt),α=0.3
* t=0t=0t=0: early experiments (blue)
* t=1t=1t=1: late experiments (red, tighter effective tolerance)
* Models cumulative tightening from repeated experiments or meta-analysis

IR surface is time-invariant → encodes structural robustness.

==== 1. 4D collapse surface exists - Collapse is a codimension-1 manifold in (p,τ0,t)(p,\tau_0,t)(p,τ0,t) ====
# Time accelerates collapse under naive scaling - p\''(τ0,t)p^\''(\tau_0,t)p\*(τ0,t) decreases monotonically with ttt
# IR-fixed-point surface is time-stable - No intersection with collapse plane for any ttt
# Certification horizon shrinks with experiment count p\''(t)=log⁡2 ⁣(τ0(1−αt)∣r∣max⁡)p^\''(t)=\log_2\!\left(\frac{\tau_0(1-\alpha t)}{|r|_{\max}}\right)p\*(t)=log2(∣r∣maxτ0(1−αt))

==== - This is a single-file <code>.tikz</code> → upload directly to PRD / APS ====
* No animation (APS-safe)
* No external data
* Analytic surfaces only
* Colorbar explicitly defined

==== • replace linear time decay with log-experiment accumulation ====
• add discrete experiment layers instead of continuous ttt
• derive optimal experiment stopping rule from 4D geometry

If you want any of those, say which one.

© Robert R. Frost
2026-01-03