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

=== Assistant: Below is a precise replacement that implements all three requests without ambiguity: ===
Below is a precise replacement that implements all three requests without ambiguity:
# Linear time decay → logarithmic experiment accumulation
# Discrete experiment layers (finite NNN, not continuous ttt)
# Optimal experiment stopping rule derived analytically from the 4D geometry

Everything is analytic, deterministic, and journal-uploadable as a standalone <code>.tikz</code>.

==== Replace linear decay with log-experiment accumulation: ====

Let n∈N0n\in\mathbb{N}_0n∈N0 be the experiment count. Define the effective tightening factor

κ(n)≡1−αlog⁡ ⁣(1+n/n0),0<α<1,  n0>0,\kappa(n) \equiv 1-\alpha \log\!\bigl(1+n/n_0\bigr),
\quad 0<\alpha<1,\; n_0>0,κ(n)≡1−αlog(1+n/n0),0<α<1,n0>0,
with the constraint κ(n)≥0\kappa(n)\ge 0κ(n)≥0.

Naive tolerance with accumulation

τnaive(p,τ0,n)=τ0 2−p κ(n).\tau_{\text{naive}}(p,\tau_0,n)=\tau_0\,2^{-p}\,\kappa(n).τnaive(p,τ0,n)=τ02−pκ(n).
IR-fixed-point tolerance (time-invariant)

τIR(p,τ0)=τ\''+(τ0−τ\'')2−p,τ\''≥∣r∣max⁡.\tau_{\text{IR}}(p,\tau_0)=\tau^\''+(\tau_0-\tau^\'')2^{-p},\quad \tau^\''\ge |r|_{\max}.τIR(p,τ0)=τ\''+(τ0−τ\'')2−p,τ\*≥∣r∣max.
Collapse condition

τ(p,τ0,n)<∣r∣max⁡.\tau(p,\tau_0,n)<|r|_{\max}.τ(p,τ0,n)<∣r∣max.

==== File: precision_phase_4D_logaccum_discrete.tikz ====

<syntaxhighlight lang="latex">%==========================================================
% Standalone TikZ / PGFPlots
% 4D Projection with log-experiment accumulation
% Discrete experiment layers
%==========================================================
\documentclass[tikz,border=2mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{tikzpicture}

% --------- PARAMETERS (FROZEN) ----------
\def\rmax{0.9}
\def\taustar{0.9}
\def\alpha{0.25}
\def\nzero{5} % n0 in log accumulation

% Discrete experiment layers
\def\nset{{0,2,5,10,20}}

\begin{axis}[
    view={135}{30},
    width=14cm,
    height=9cm,
    xlabel={precision index $p$},
    ylabel={initial tolerance $\tau_0$},
    zlabel={tolerance $\tau(p,\tau_0,n)$},
    xmin=0, xmax=6,
    ymin=1, ymax=6,
    zmin=1e-3, zmax=6,
    zmode=log,
    grid=both,
    legend style={at={(0.02,0.98)},anchor=north west}
]

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

% --------- DISCRETE NAIVE SURFACES (LOG ACCUMULATION) ----------
\foreach \n in \nset {
    % kappa(n) = 1 - alpha * ln(1+n/n0)
    \pgfmathsetmacro{\kappa}{max(0,1-\alpha*ln(1+\n/\nzero))}
    \addplot3[
        surf, opacity=0.85,
        domain=0:6, y domain=1:6,
        samples=25, samples y=25
    ] {y '' 2^(-x) '' \kappa};
    \addlegendentry{$\tau_{\mathrm{naive}}(n=\n)$}
}

% --------- IR FIXED-POINT SURFACE ----------
\addplot3[
    surf, opacity=0.35,
    domain=0:6, y domain=1:6,
    samples=25, samples y=25
]{\taustar + (y-\taustar)*2^(-x)};
\addlegendentry{$\tau_{\mathrm{IR}}$}

\end{axis}
\end{tikzpicture}
\end{document}

</syntaxhighlight>

What this shows: stacked 3D surfaces for discrete experiment counts n={0,2,5,10,20}n=\{0,2,5,10,20\}n={0,2,5,10,20}. As nnn increases, naive surfaces sink logarithmically; IR surface remains invariant.

==== Solve τnaive(p,τ0,n)=∣r∣max⁡\tau_{\text{naive}}(p,\tau_0,n)=|r|_{\max}τnaive(p,τ0,n)=∣r∣max: ====

τ0 2−p κ(n)=∣r∣max⁡⇒p\*(τ0,n)=log⁡2 ⁣(τ0 κ(n)∣r∣max⁡).\tau_0\,2^{-p}\,\kappa(n)=|r|_{\max}
\quad\Rightarrow\quad
p^\''(\tau_0,n)=\log_2\!\left(\frac{\tau_0\,\kappa(n)}{|r|_{\max}}\right).τ02−pκ(n)=∣r∣max⇒p\''(τ0,n)=log2(∣r∣maxτ0κ(n)).
Existence condition

κ(n)>0  ⟺  n<nmax⁡wherenmax⁡=n0(e1/α−1).\kappa(n)>0 \iff n < n_{\max}
\quad\text{where}\quad
n_{\max}=n_0\bigl(e^{1/\alpha}-1\bigr).κ(n)>0⟺n<nmaxwherenmax=n0(e1/α−1).
Beyond nmax⁡n_{\max}nmax, collapse is immediate for any p>0p>0p>0.

==== Define the certification horizon PPP (maximum required precision). To avoid collapse up to PPP, require: ====

τ0 2−P κ(n)  ≥  ∣r∣max⁡.\tau_0\,2^{-P}\,\kappa(n)\;\ge\;|r|_{\max}.τ02−Pκ(n)≥∣r∣max.
Solve for the maximum admissible experiment count:

n  ≤  n0 ⁣[exp⁡ ⁣(1α(1−∣r∣max⁡τ02P))−1]\boxed{
n \;\le\; n_0\!\left[
\exp\!\left(\frac{1}{\alpha}\left(1-\frac{|r|_{\max}}{\tau_0}2^{P}\right)\right)-1
\right]
}n≤n0[exp(α1(1−τ0∣r∣max2P))−1]
This yields the optimal stopping rule:

: 

===== - Each additional experiment tightens tolerance logarithmically, not linearly. =====
* There is a finite experiment budget compatible with a given precision target.
* Exceeding it guarantees collapse regardless of further data quality.

==== For IR-fixed-point scaling: ====

inf⁡pτIR(p,τ0)=τ\''≥∣r∣max⁡⇒No stopping required.\inf_p \tau_{\mathrm{IR}}(p,\tau_0)=\tau^\''\ge |r|_{\max}
\quad\Rightarrow\quad
\boxed{\text{No stopping required.}}pinfτIR(p,τ0)=τ\*≥∣r∣max⇒No stopping required.
Experiments do not erode viability.

==== κ(n)=1−αlog⁡(1+n/n0),p\*(τ0,n)=log⁡2 ⁣(τ0κ(n)∣r∣max⁡),nmax⁡=n0(e1/α−1),nstop(P)=n0 ⁣[exp⁡ ⁣(1α(1−∣r∣max⁡τ02P))−1].\boxed{ ====
\begin{aligned}
\kappa(n)&=1-\alpha\log(1+n/n_0),\\
p^\*(\tau_0,n)&=\log_2\!\left(\frac{\tau_0\kappa(n)}{|r|_{\max}}\right),\\
n_{\max}&=n_0(e^{1/\alpha}-1),\\
n_{\text{stop}}(P)&=n_0\!\left[\exp\!\left(\frac{1}{\alpha}\left(1-\frac{|r|_{\max}}{\tau_0}2^{P}\right)\right)-1\right].
\end{aligned}
}κ(n)p\*(τ0,n)nmaxnstop(P)=1−αlog(1+n/n0),=log2(∣r∣maxτ0κ(n)),=n0(e1/α−1),=n0[exp(α1(1−τ0∣r∣max2P))−1].
These bounds are read directly from the 4D geometry; no simulations are used.

If you want next, I can:
* add isocurves of nstopn_{\text{stop}}nstop to the plot,
* include a decision-flow appendix (when to stop vs refine),
* or provide a formal theorem + proof block for the stopping rule.

© Robert R. Frost
2026-01-03