the scoring artefact is in BOTH sets: aps.13.00 joins the five fun7 problems — 6 of 199 problems where f underflows to exactly zero on a ~3.2e-2 neighbourhood of the declared root, so every method returns a genuine root (|f(x)| below the smallest normal) and every method is scored a failure. About 1150 evaluations of EVERY two-set total are this artefact; re-scored, all methods solve 199/199 except Brent 197. Ranking unchanged: ours-by-shape 2337 on APS against chandrupatla 2593, chandrupatla 1487 on CHA against our 1740. Written into the paper as a table with both criteria; pdflatex clean, verifier green

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

article/sim_numerics.tex

@@ -563,37 +563,61 @@ The readings do not compose. Taken together, (2) and (3) give $2884$ and
 $2859$ --- worse than either alone. Two selections that are each an improvement
 interfere, which is a reason to measure composition rather than assume it.
 
-\subsection{Five problems of the wide-bracket set are scored unsolved for every
-method}
+\subsection{Six problems of the two sets are scored unsolved for every method}
 \label{sec:fun7}
 
-Chasing the remaining gap to Chandrupatla on the wide-bracket set led to a
-property of the set itself. On the five problems of its \texttt{fun7} family the
-declared root is $0$, but $f$ vanishes \emph{identically} on a neighbourhood of
-it: $f(0+t)=0$ for every $t$ up to at least $3.2\times10^{-2}$. Every method
-therefore stops after $8$ to $22$ evaluations at a point where $f(x)=0$
-\emph{exactly} --- a genuine root --- and every method is then scored as having
-failed, because the harness asks for $|x-\text{root}|\le10^{-6}$ and the
-returned point sits up to $2.8\times10^{-2}$ away.
+Chasing the remaining gap to Chandrupatla led to a property of the benchmarks
+themselves. On the five problems of the wide-bracket family \texttt{fun7}, and on
+the single canonical problem \texttt{aps.13.00}, the declared root is $0$ while
+$f$ vanishes \emph{identically} on a neighbourhood of it in floating point:
+$f(0\pm t)=0$ for every $t$ up to at least $3.2\times10^{-2}$, because the
+exponential factor underflows --- at the returned points $|f(x)|$ is below the
+smallest normal double. Every method therefore stops after $8$ to $28$
+evaluations at a point where $f(x)=0$ \emph{exactly} --- a genuine root --- and
+every method is then scored as having failed, because the harness asks for
+$|x-\text{root}|\le10^{-6}$ while the returned point sits up to
+$2.8\times10^{-2}$ away.
 
 The consequence is arithmetic. Each such problem charges the full budget of
-$200$ evaluations to every method, so $1000$ evaluations of every wide-bracket
-total --- ours, Chandrupatla's, ITP's, bisection's --- are this artefact, and the
-``$40$ of $45$ solved'' that every method reports is a statement about the
-scoring rather than about the methods. Re-scored with the criterion ``a root is a
-root'' ($|x-\text{root}|\le10^{-6}$ \emph{or} $f(x)=0$), the same runs give
-$45/45$ for every method except Brent's $43/45$, and the totals fall to
-Chandrupatla $1487$, ours $1740$, ITP $1907$, bisection $2136$, TOMS\,748
-$3088$. The ranking does not change; the magnitudes do, and with them the size
-of the remaining gap: $253$ evaluations, or $17\%$, not $11\%$.
+$200$ evaluations to every method: about $1150$ evaluations of every reported
+two-set total --- ours, Chandrupatla's, TOMS\,748's, ITP's, Brent's, bisection's
+--- are this artefact, and the ``$153$ of $154$'' and ``$40$ of $45$'' that every
+method reports are statements about the scoring rather than about the methods.
+Re-scored with the criterion ``a root is a root''
+($|x-\text{root}|\le10^{-6}$ \emph{or} $f(x)=0$), the same runs solve $199$ of
+$199$ for every method except Brent's $197$, and the totals fall as follows.
+
+\begin{table}[h]
+\centering
+\begin{tabular}{lrrrr}
+\hline
+ & \multicolumn{2}{c}{APS-154} & \multicolumn{2}{c}{CHA-45} \\
+method & as scored & re-scored & as scored & re-scored \\
+\hline
+ours, points by shape & $\mathbf{2527}$ & $\mathbf{2337}$ & $3108$ & $2159$ \\
+ours, projection variant & $2806$ & $2634$ & $3220$ & $2357$ \\
+Chandrupatla & $2785$ & $2593$ & $\mathbf{2423}$ & $\mathbf{1487}$ \\
+TOMS\,748 & $2820$ & $2636$ & $4000$ & $3088$ \\
+Brent & $2889$ & $2707$ & $4099$ & $3222$ \\
+ITP & $3957$ & $3767$ & $2857$ & $1907$ \\
+bisection & $7530$ & $7338$ & $3090$ & $2136$ \\
+\hline
+\end{tabular}
+\caption{The same runs under the harness criterion and under ``a root is a
+root''. The ranking is unchanged; the magnitudes are not, and neither is the size
+of the remaining gap on the wide-bracket set: $253$ evaluations, or $17\%$, not
+$11\%$.}
+\end{table}
 
 Two things follow. The wide-bracket gap to Chandrupatla is real and it is
 \emph{diffuse}: measured per family it is $+57$, $+54$, $+46$, $+43$, $+35$ and
 smaller, with no single problem costing more than $35$, so it is not the kind of
 deficit a single mechanism removes. And on that set our shape reading ($2159$
-re-scored) is barely better than bisection ($2136$); only the dissimilarity
-reading ($1740$) is clearly ahead. Both statements are reproduced by
-\texttt{diag\_rescore\_cha.py} and \texttt{diag\_cha\_gap.py}.
+re-scored) is \emph{worse} than bisection ($2136$); only the dissimilarity
+reading ($1740$) is clearly ahead. Every statement in this subsection is
+reproduced by \texttt{diag\_rescore\_cha.py}, \texttt{diag\_rescore\_aps.py},
+\texttt{diag\_fun7\_root.py} and \texttt{diag\_cha\_gap.py}, whose outputs are
+kept beside them.
 
 \section{Negative results}