the wide-bracket gap is NOT closed, and now it has an address: the STEP RULE, not the guarantee and not the reading places. fun1 is one function at growing widths — 7/8, 12/12, 19/15, 59/24, 63/44 — so we tie when narrow and lose as it widens, while our observed order is the HIGHER one (3.63 vs 2.02); 59 on a 2e4 bracket is what bisection needs, i.e. one halving per evaluation against their 2.3. With the safeguard removed entirely we still spend 46/57/61/68 against bisection 57/77/45/45 and Chandrupatla 24/44 — worse than bisection on two of them. |f|/|f'| is an absolute length and cannot march across decades; written into the paper as the stated limit. Also fixed a tracer bug that had best_of double-counting evaluations (2993 vs 1487)

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

article/sim_numerics.tex

@@ -609,12 +609,41 @@ 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 \emph{worse} than bisection ($2136$); only the dissimilarity
-reading ($1740$) is clearly ahead. Every statement in this subsection is
+\subsection{Where the wide-bracket limit actually is}
+\label{sec:widelimit}
+
+The remaining gap looks diffuse --- per family $+57$, $+54$, $+46$, $+43$, $+35$
+and smaller --- but it has one address. The \texttt{fun1} family is the same
+function with brackets of growing width, and our cost against Chandrupatla's
+across them is $7/8$, $12/12$, $19/15$, $59/24$, $63/44$: we tie or win while the
+bracket is narrow and lose progressively as it widens. The endgame is not the
+cause --- our observed order on those very problems is the higher one ($3.63$
+against $2.02$ on the $2\times10^4$-wide bracket). The cause is the rate of
+progress far from the root: $59$ evaluations on a $2\times10^4$-wide bracket is
+what bisection itself needs ($\lceil\log_2(2\times10^{16})\rceil\approx54$), i.e.
+there we advance by one halving per evaluation while Chandrupatla advances by
+about $2.3$.
+
+It is not the guarantee that caps us. With the safeguard removed entirely the
+same problems cost $46$, $57$, $61$ and $68$ against bisection's $57$, $77$, $45$
+and $45$ --- better on the wide brackets, \emph{worse than bisection} on
+\texttt{fun3.1} and \texttt{fun5.1} --- and still far from Chandrupatla's $24$ and
+$44$. The cap is the step rule itself: $|f|/|f'|$ is an absolute length, and an
+absolute length cannot march across decades. That is the same wall the fixed
+fraction (\texttt{cbrt}) and the logarithmic change of variable each hit from a
+different side, both buying the wide brackets at the price of the canonical set.
+
+So the limit is stated exactly: the placement of the similarity reading improves
+the method up to the point where the \emph{form} of the derived step becomes the
+binding constraint, and on brackets spanning many decades it does. Removing that
+constraint means replacing the step the contraction law derives, which is a
+different method rather than a better reading of this one.
+
+On that set our shape reading ($2159$ re-scored) is \emph{worse} than bisection
+($2136$); only the dissimilarity reading ($1740$) is clearly ahead, which is the
+same statement seen from the other end: where the step cannot march, the reading
+that helps least is the one that only improves the local model. Every statement
+in these two subsections 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.