\section{Numerical diagnostics and reproducibility}\label{numerics:section}
\input{figures/numerical_data}
The computations in this section are diagnostics, not inputs to a proof.
All plotted coordinates are extracted from saved result files. The source
archive contains the CSV files and their SHA--256 provenance manifest.
Sampling errors and numerical discretization errors are different quantities;
we report both where relevant. A finite arithmetic census has no sampling
standard error, but still has numerical evaluation and missed-root risks.

\subsection{The Gaussian crossing calculation}
For the Gaussian process with mean and covariance corresponding to
\(\theta=1\), the two-point crossing probability can be integrated directly.
An independent conditional Monte Carlo calculation used one million independent
standard normal draws, with seed \(20261014\). Figure~\ref{numerics:gaussian} compares the two
calculations. The same samples were used at different mesh widths, so the
displayed estimates are correlated across widths. At \(h=0.005\), quadrature
gives \(0.13651707788667264\); the conditional Monte Carlo estimate is
\(0.1364148646262289\), with standard error
\(0.00010298793221565028\). Comparing 64 and 128 quadrature nodes gives a
small numerical discrepancy, not an interval-arithmetic certificate.

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{semilogxaxis}[width=.88\textwidth,height=5.8cm,
 xlabel={Mesh width $h$},ylabel={Crossing probability divided by $h$},
 legend style={font=\small,at={(.02,.03)},anchor=south west},
 scaled y ticks=false,yticklabel style={/pgf/number format/fixed,/pgf/number format/precision=4},
 grid=major]
\addplot[BMblue,thick,mark=*,error bars/.cd,y dir=both,y explicit]
 table[x=h,y=mc,y error=twose,col sep=comma]{\BMgaussianmesh};
\addlegendentry{Conditional MC, $n=10^6$, $\pm2$ SE}
\addplot[BMorange,thick,dashed,mark=square*]
 table[x=h,y=quadrature,col sep=comma]{\BMgaussianmesh};
\addlegendentry{Deterministic quadrature}
\addplot[BMgray,densely dotted,thick]
 table[x=h,y=rice,col sep=comma]{\BMgaussianmesh};
\addlegendentry{Rice intensity $\cBM$}
\end{semilogxaxis}
\end{tikzpicture}
\caption{A check of the Gaussian two-point crossing calculation. This is
Gaussian data, not actual arithmetic or an Euler-product simulation. The
finite-mesh bias and Monte Carlo standard error are both visible.}
\label{numerics:gaussian}
\end{figure}

\subsection{Local quadratic distributions and finite root counts}
The quadratic computation evaluated all \(60\,786\) nonprincipal fundamental
discriminants with \(|D|\le10^5\). For the local-distribution comparison we
conditioned on \(\chi_D(2)=1\), giving \(10\,125\) positive and \(10\,136\)
negative discriminants. The random comparison used \(20\,000\) independent
Euler samples, primes up to \(10^6\), and seed \(68120261014\), with the same
condition at 2. No Gaussian tail was added. Figure~\ref{numerics:cdf} uses
\(\delta=1/2\); the omitted random prime tail has estimated standard
deviation \(0.00385\), which is not a certified error bound. For the random
empirical CDF, the pointwise sampling standard error is
\(\sqrt{\widehat F(1-\widehat F)/20000}\le0.00354\).

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{axis}[width=.88\textwidth,height=5.5cm,xmin=-3,xmax=2,ymin=0,ymax=1,
 xlabel={$x$: value of $2\delta(-L'/L)(1/2+\delta)$},
 ylabel={Empirical cumulative probability},grid=major,
 legend style={font=\small,at={(.03,.97)},anchor=north west}]
\addplot[BMblue,thick] table[x=x,y=positive,col sep=comma]{\BMquadraticcdf};
\addlegendentry{$D>0$, $n=10\,125$}
\addplot[BMorange,thick,dashed] table[x=x,y=negative,col sep=comma]{\BMquadraticcdf};
\addlegendentry{$D<0$, $n=10\,136$}
\addplot[BMgray,thick,densely dotted] table[x=x,y=random,col sep=comma]{\BMquadraticcdf};
\addlegendentry{Finite Euler model, $n=20\,000$}
\end{axis}
\end{tikzpicture}
\caption{Local distributions at $\delta=1/2$, conditioned on
$\chi_D(2)=X(2)=1$. These finite-conductor distributions need not be
Gaussian. Sampling uncertainty applies only to the random curve; its
pointwise standard error is at most $0.00354$. The comparison tests a local
statistic and does not test the real-zero asymptotic.}
\label{numerics:cdf}
\end{figure}

The empirical root counts in Figure~\ref{numerics:counts} agreed on meshes
with 64, 128, and 512 subdivisions and in two winding checks. Selected
values were also recomputed at higher precision. These agreements do not
certify all roots or their multiplicities. There are \(30\,394\) positive
and \(30\,392\) negative discriminants. Their recorded mean counts are
\(0.27406725\) and \(0.31248355\), respectively. The upper bound
\(\log\log(10^5)\) is only \(2.44347036\), so these data cannot distinguish
the asymptotic constant from bounded and preasymptotic effects.

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{axis}[ybar,bar width=14pt,width=.82\textwidth,height=5.3cm,
 xlabel={Recorded number of real roots on $[1/2,1]$},
 ylabel={Proportion of the finite family},xtick={0,1,2},ymin=0,ymax=.85,
 enlarge x limits=.3,legend style={font=\small,at={(.97,.97)},anchor=north east}]
\addplot[fill=BMblue!65,draw=BMblue] table[x=roots,y=positive,col sep=comma]{\BMquadraticcounts};
\addlegendentry{$D>0$, $n=30\,394$}
\addplot[fill=BMorange!35,draw=BMorange,postaction={pattern=north east lines}]
 table[x=roots,y=negative,col sep=comma]{\BMquadraticcounts};
\addlegendentry{$D<0$, $n=30\,392$}
\end{axis}
\end{tikzpicture}
\caption{Empirical counts for $|D|\le10^5$. Counts at two roots are
3 and 43 in the positive and negative families and are barely visible at
this scale. Sampling SE is inapplicable to the census; missed-root and
finite-precision errors have not been interval-certified. The available
$\log\log$ range is too small to verify $\cBM$.}
\label{numerics:counts}
\end{figure}

\subsection{An actual modular twist}
For the normalized weight-12 form \(\Delta\), an independent Mellin-integral
calculation produced values of the primitive twists at small signed
fundamental discriminants. Figure~\ref{numerics:twist} shows \(D=17\), with
conductor \(17^2=289\). The saved refined computation used 70-digit
arithmetic, 20-point Gauss--Legendre panels of maximum width 0.2, and 128 real
subintervals. Comparison to 45 digits, 12-point panels of maximum width 0.3, and
64 subintervals gave differences below \(2.42\cdot10^{-27}\) for \(L\)
and \(6.53\cdot10^{-27}\) for \(L'\) on common points. Separate high-precision
Mellin evaluations checked selected points and bracket endpoints.

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{axis}[width=.88\textwidth,height=5.5cm,xmin=.5,xmax=1,
 xlabel={Real argument $\sigma$},ylabel={Value of the normalized twist},
 grid=major,legend style={font=\small,at={(.03,.97)},anchor=north west}]
\addplot[BMblue,thick] table[x=s,y=L,col sep=comma]{\BMdeltatwist};
\addlegendentry{$L(\sigma,\Delta\otimes\chi_{17})$}
\addplot[BMorange,thick,dashed] table[x=s,y=Lprime,col sep=comma]{\BMdeltatwist};
\addlegendentry{$L'(\sigma,\Delta\otimes\chi_{17})$}
\end{axis}
\end{tikzpicture}
\caption{An actual modular twist, sampled at 129 real arguments. The
derivative changes sign between $1/2$ and $1/2+1/128$. This is a numerical
bracket, not an interval certificate or a count of all roots. The calculation
is deterministic, so sampling SE is inapplicable; precision and quadrature
comparisons are given in the text.}
\label{numerics:twist}
\end{figure}

\subsection{The cost of eight fixed anchors}
Positive definiteness of the two fixed Gram matrices used in the proof is
analytic, but their conditioning explains the size of possible constants.
For the central nodes \(v_j=-\log(1+j/9)\) and bridge nodes
\(v_j=-1/8+j/36\), \(1\le j\le8\), the matrix is
\(\bigl(r(v_i-v_j)\bigr)\). The smallest eigenvalues are approximately
\(5.2463\cdot10^{-16}\) and \(5.8477\cdot10^{-22}\), respectively.
Figure~\ref{numerics:gram} uses the saved 260-digit calculation. Comparison
with 180 digits gave relative changes below \(10^{-159}\) in the smallest
eigenvalues. The central matrix also has an exact rational
\(LDL^{\mathsf T}\) decomposition with positive pivots. Ordinary binary64
arithmetic can incorrectly return a negative smallest bridge eigenvalue.
The bridge eigenvalue calculation itself is not an interval certificate.

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{semilogyaxis}[width=.85\textwidth,height=5.5cm,xlabel={Eigenvalue index, increasing},
 ylabel={Eigenvalue of the correlation matrix},xtick={1,...,8},grid=major,
 legend style={font=\small,at={(.03,.97)},anchor=north west}]
\addplot[BMblue,thick,mark=*] table[x=index,y=central,col sep=comma]{\BMeightanchorgram};
\addlegendentry{Central logarithmic nodes}
\addplot[BMorange,thick,dashed,mark=square*] table[x=index,y=bridge,col sep=comma]{\BMeightanchorgram};
\addlegendentry{Bridge linear nodes}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Eigenvalues of the two fixed $8\times8$ Gram matrices. This
deterministic calculation has no sampling SE. It illustrates large constants
in finite-dimensional small-ball estimates and cannot give an effective
conductor threshold for the arithmetic theorems.}
\label{numerics:gram}
\end{figure}

As a separate exact algebra check, enumeration of primitive cubic characters
up to conductor 5000 gave 1590 oriented characters and 795 fields. The
conductor-9 decomposition and mixed local words were checked with exact
arithmetic. Such finite checks detect normalization or conjugation errors;
they do not prove the asymptotic orthogonality estimate and do not evaluate
actual cubic $L$-function zeros.