CORRECTION, prompted by Valeri: the step IS a fraction of the bracket — with the ratio similarity 1-S = min(1,|f|/(|f'|D)) so step = min(D,|f|/|f'|) — and my 'absolute length cannot march' was wrong, as was the follow-up guess that we creep along one side: measured, 91.5% of our probes DO shrink the bracket, more often than Chandrupatla's 87.5%. What differs is how much each probe removes: 0.93 halvings per evaluation against 2.26. A linear model of the value axis undershoots on a wide bracket of a convex function, so the new bracket is about half; inverse quadratic interpolation lands near the root and removes four fifths. The limit is the ORDER OF THE MODEL the similarity is read against, and all three parameter-free ways round it are now measured failures (cbrt, log change of variable, derived3_m). Paper corrected

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

article/sim_numerics.tex

@@ -628,16 +628,35 @@ 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.
+$44$.
 
-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.
+Nor is it that the step fails to be relative to the bracket. With the ratio
+similarity $1-S=\min(1,|f|/(|f'|D))$, so the step is exactly
+$\min(D,|f|/|f'|)$: a \emph{fraction} of the distance to the far end, and the
+similarity is that fraction. Nor does the iterate creep along one side without
+moving the proven bracket: measured probe by probe, $91.5\%$ of our evaluations
+on the $2\times10^4$-wide bracket do shrink the bracket, slightly \emph{more}
+often than Chandrupatla's $87.5\%$.
+
+What differs is how much each probe removes. Ours shrinks the bracket by
+$0.93$ halvings per evaluation there, Chandrupatla's by $2.26$ --- a factor of
+about two against a factor of about five. On a wide bracket of a convex function
+the Newton step systematically undershoots, because a linear model of the value
+axis cannot see the curvature that separates the iterate from the root, so our
+probe lands well short and the new bracket is about half the old one; a model
+that carries the curvature --- inverse quadratic interpolation --- lands far
+closer and removes about four fifths.
+
+So the limit is stated exactly, and it is the \emph{order of the model the
+similarity is read against}, not the placement of the reading and not the
+guarantee. Three parameter-free ways round it have now been measured and all
+three fail in the same way, buying the wide brackets at the price of the
+canonical set: a fixed fraction of the bracket (\texttt{cbrt}), a logarithmic
+change of variable, and a curvature-corrected scale (\texttt{derived3\_m},
+whose derivation holds but whose estimator of the local order fires on $27.9\%$
+of steps where the truth is a simple root). The fourth way is to model the
+curvature directly, which is inverse quadratic interpolation --- a different
+method rather than another 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