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Revision step 2 into the paper: section 9.1 (Tables 9b, 9c) as PREREG_controls prescribes. The analogy detector beats Lagouanelle-with-memory, Chandrupatla's band and no detector on corpus 4 (3656 vs 4488, 6236, 6165); with the multiplicity given, the classical transformation is 2.4-6.8x cheaper on every multiple or fractional root - stated as the price of estimating the order, in the abstract, 8.3, 8.6, the conclusion and as the first future-work question. verify_controls(): 26 checks. Server: verify 526 OK / 0 BAD, regression x2 1232 passed, PDF 30 pages, 0 errors, 0 undefined, 0 overfull.

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Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
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@@ -150,20 +150,27 @@ evaluations against Chandrupatla's 5717 and wins 177 problems to 28 ---
a corpus the author composed knowing where the method is strong: two of
its six families have multiple or fractional roots, and without them the
lead is 2403 against 3569, on the simple-root families alone 737 against
747. Over 855 problems in six corpora it is ahead of Brent, TOMS 748,
ITP and the recent ModAB everywhere and stays within its own bound of
twice bisection on every problem, a bound that Brent, TOMS 748 and
ModAB, which do not promise it, exceed on 118, 84 and 9 problems;
against Chandrupatla it is ahead in total on five corpora and level on
the sixth. The lead is large on multiple, fractional and nearly flat
roots and a matter of per cent on simple ones. Several of the resulting
mechanisms coincide with classical ones --- Lagouanelle's multiplicity
estimate, false position on \(|f|^{1/m}\) --- reached here from
similarity and analogy and recognised afterwards; what we found no
counterpart for is the decision layer, where every choice --- when to
give the model up, where to split, when to return --- is a judgment of
similarity or of analogy, and the similarity that sets the step, whose
form is decisive (replacing it costs 19\% to 425\%).
747. Against simpler detectors put in the place of the analogy ---
Lagouanelle's multiplicity estimate with the same memory, Chandrupatla's
acceptance band, or none --- it wins on that corpus in total and problem
by problem (3656 against 4488, 6236 and 6165); against the classical
transformation \(\operatorname{sign}(f)|f|^{1/m}\) with the multiplicity
given in advance it loses every multiple or fractional root, by a factor
of 2.4 to 6.8 --- the price of estimating the order. Over 855 problems
in six corpora it is ahead of Brent, TOMS 748, ITP and the recent ModAB
everywhere and stays within its own bound of twice bisection on every
problem, a bound that Brent, TOMS 748 and ModAB, which do not promise
it, exceed on 118, 84 and 9 problems; against Chandrupatla it is ahead
in total on five corpora and level on the sixth. The lead is large on
multiple, fractional and nearly flat roots and a matter of per cent on
simple ones. Several of the resulting mechanisms coincide with classical
ones --- Lagouanelle's multiplicity estimate, false position on
\(|f|^{1/m}\) --- reached here from similarity and analogy and
recognised afterwards; what we found no counterpart for is the decision
layer, where every choice --- when to give the model up, where to split,
when to return --- is a judgment of similarity or of analogy, and the
similarity that sets the step, whose form is decisive (replacing it
costs 19\% to 425\%).
\end{abstract}
\hypertarget{introduction}{%
@@ -1488,12 +1495,14 @@ which is exact for a pure power and becomes the midpoint as
\(\operatorname{sign}(f)|f|^{1/m}\) --- the transformation that turns a
root of multiplicity \(m\) into a simple one, classical since Schröder
{[}16{]} and Traub {[}17{]}; what is new here is only where \(m\) comes
from. The order read from a \emph{failed promise}, \((1-1/m)^m=r\), was
tried first and gave \(m=\infty\) in 163 of 174 entries, because the
base step is not a pure tangent; the order read from the \emph{analogy}
is the one that works. With it the flat families of the wide-bracket set
cost 743--847 evaluations against 1044 for pure bisection and 1068 for
Chandrupatla.
from. With the multiplicity given in advance the transformation is far
cheaper still (§9.1): the contribution is the estimate, and its price is
measured there. The order read from a \emph{failed promise},
\((1-1/m)^m=r\), was tried first and gave \(m=\infty\) in 163 of 174
entries, because the base step is not a pure tangent; the order read
from the \emph{analogy} is the one that works. With it the flat families
of the wide-bracket set cost 743--847 evaluations against 1044 for pure
bisection and 1068 for Chandrupatla.
\hypertarget{five-more-readings}{%
\subsubsection{8.4 Five more readings}\label{five-more-readings}}
@@ -1761,7 +1770,9 @@ decisive. The methods for multiple roots we found are single-point
iterations without a bracket, with the multiplicity assumed, estimated
from derivatives, or removed by a transformation ({[}17{]}; Roots.jl's
\texttt{Order1B} and \texttt{Order2B} {[}12{]}); Chandrupatla takes the
decision to halve by an empirical acceptance band {[}3{]}.
decision to halve by an empirical acceptance band {[}3{]}. §9.1 puts
this claim against the detectors a reader would try first, and against
the transformation with the multiplicity known in advance.
\hypertarget{out-of-sample-evidence}{%
\subsection{9. Out-of-sample evidence}\label{out-of-sample-evidence}}
@@ -1972,6 +1983,135 @@ with a simple root in total (433 against 419, though ahead 20 to 12 in
problems), and the smooth equations of the applied corpus, where the two
are level.
\hypertarget{controls-simpler-detectors-and-the-multiplicity-known-in-advance}{%
\subsubsection{9.1 Controls: simpler detectors, and the multiplicity
known in
advance}\label{controls-simpler-detectors-and-the-multiplicity-known-in-advance}}
Two controls ask whether the gain of §8 comes from the recognition by
analogy or only from the split (7) with any reasonable detector. Both
were pre-registered, with what the paper would say under every outcome,
before the run (\texttt{sweep\_controls.py}).
\textbf{Other detectors in A′.} A′ is run with only its detector
swapped; the memory, the split, the exit and the five readings of §8.4
are unchanged. (a) \emph{Lagouanelle's estimate} \(m_L=1/(1-\theta)\),
\(\theta=ff''/f'^2\) from the parabola through the last three points
(\(m_L=\infty\) for \(\theta\ge1\)), with the same memory --- once seen,
held until the exit, a hysteresis --- entering at \(m_L\ge2\) and at
\(0<m_L\le\tfrac23\) and splitting at \(m_L\). (b) \emph{Chandrupatla's
acceptance band}, entering when the band rejects the current triple and
splitting at the current \(m_L\). (c) \emph{No detector}: the regime is
entered at every unkept promise.
\textbf{Table 9b.} A′ against its detector swapped: totals on the two
canonical sets, on corpus 4 and over the six corpora of §9 (uniform
criterion), and A′'s wins/ties/losses on corpus 4 with the exact sign
test. No variant breaks the bound.
\begin{longtable}[]{@{}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.3291}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.0759}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.0759}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.1013}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.1139}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.1392}}
>{\raggedright\arraybackslash}p{(\columnwidth - 12\tabcolsep) * \real{0.1646}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
detector
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
APS
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
CHA
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
corpus 4
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
six corpora
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
A′ on corpus 4
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\(p\)
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\textbf{analogy (A′)} & 2259 & \textbf{1074} & \textbf{3656} &
\textbf{13032} & --- & \\
Lagouanelle, with memory & \textbf{2235} & 1437 & 4488 & 15171 &
84/121/35 & \(8\cdot10^{-6}\) \\
Chandrupatla's band & 2682 & 2526 & 6236 & 20668 & 110/96/34 &
\(1.5\cdot10^{-10}\) \\
none & 2693 & 2526 & 6165 & 20749 & 113/87/40 & \(3\cdot10^{-9}\) \\
\end{longtable}
On corpus 4, the one test, A′ beats all three in total and problem by
problem, as the pre-registered criterion required. The competent
alternative is Lagouanelle's estimate: it is slightly better on the
canonical set (2235 against 2259), and on corpus 4 A′'s lead over it
lies in the multiple-root and fractional-power families (1163 against
1739, 924 against 1207); it is behind on \(\log(1+q^2)\) of a cubic (503
against 477) and level on the rest. The band and the regime without a
detector are worse than no regime at all. The decision layer by analogy
therefore earns its place against the detectors a reader would try
first.
\textbf{The multiplicity known in advance.} The second control asks how
good the classical transformation is when its multiplicity is given.
Library Chandrupatla and Brent are run on
\(g=\operatorname{sign}(f)\,|f|^{1/m^*}\), with \(m^*\) the order read
at the known root --- the slope of \(\ln|f|\) against \(\ln|x-r|\) on
both sides at \(h=10^{-5}\), rounded to 0.01, and 1 where either side
shows no zero (a jump). It is not an achievable method but an upper
bound. On the multiple and fractional roots, \(m^*\ge2\) or
\(m^*\le\tfrac23\) --- the branches of §8.1 where a tangent step loses
to a halving:
\textbf{Table 9c.} Multiple and fractional roots with the multiplicity
given: totals, failures charged. A′ wins none of the 148 problems.
\begin{longtable}[]{@{}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
corpus
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
problems
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
A′
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Chandrupatla on \(g\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Brent on \(g\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Chandrupatla on \(f\)
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Chandrupatla & 20 & 654 & 96 & 66 & 1068 \\
new corpus 1 & 40 & 1229 & 358 & 337 & 1792 \\
corpus 4 & 88 & 2159 & 901 & 844 & 4151 \\
\end{longtable}
With the multiplicity known the transformation is 2.4 to 6.8 times
cheaper than A′. A′ is far ahead of Chandrupatla on \(f\) itself, which
does not know the multiplicity either. On the remaining problems the
transformed and untransformed runs differ little and A′ is close to
them: ahead on the canonical sets (2259 against 2640, 420 against 437
for Chandrupatla on \(g\)), behind on corpora 1 and 4 (2061 against
1924, 1497 against 1427). So on multiple roots the contribution of A′ is
the estimate of the order without knowing it, and the gap to Table 9c is
the price of that estimate --- large, and the clearest direction for
improving the method.
\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
\hypertarget{negative-results}{%
@@ -2183,11 +2323,15 @@ the local order, the order decides between the model and a halving by
bisection's own \(\tfrac12\), and it says where to split. The method
built on it, A′, is ahead of Chandrupatla on both canonical sets in
total and problem by problem, and on a corpus fixed after its code was
frozen it needs 36\% fewer evaluations and wins 177 problems to 28. The
claim the measurements support is a lead over Brent, TOMS 748, ITP and
ModAB everywhere, within a bound of twice bisection that Brent, TOMS 748
and ModAB do not promise and exceed, and over Chandrupatla a lead that
is large where roots are multiple, fractional or nearly flat and small
frozen it needs 36\% fewer evaluations and wins 177 problems to 28. Its
decision layer beats Lagouanelle's estimate, Chandrupatla's band and no
detector put in its place; with the multiplicity known in advance the
classical transformation is 2.4 to 6.8 times cheaper on multiple roots,
which measures what estimating the order costs. The claim the
measurements support is a lead over Brent, TOMS 748, ITP and ModAB
everywhere, within a bound of twice bisection that Brent, TOMS 748 and
ModAB do not promise and exceed, and over Chandrupatla a lead that is
large where roots are multiple, fractional or nearly flat and small
where they are simple --- not superiority by the same margin everywhere.
\texttt{fun9}, near-cancelling terms and smooth applied equations are
where it is behind or level.
@@ -2201,13 +2345,19 @@ hand is measured rather than asserted.
\hypertarget{future-work}{%
\subsubsection{Future work}\label{future-work}}
Two questions remain, both narrower than the one this section used to
Three questions remain, all narrower than the one this section used to
pose.
The first is the steep flank. Where the loss remains --- \texttt{fun9},
The first is the price of estimating the order. With the multiplicity
given, the transformation of §8.3 is 2.4 to 6.8 times cheaper than A′ on
multiple and fractional roots (Table 9c): the order read from the
analogous pair, or its use in the split, loses most of what the
transformation could deliver.
The second is the steep flank. Where the loss remains --- \texttt{fun9},
the near-cancelling class, 4 − tan x at its pole --- the base step
approaches along a flank far steeper than its model, and the half
contradiction of §8.4 removes only part of the cost. The second is the
contradiction of §8.4 removes only part of the cost. The third is the
one geometric choice of §8.2, the Euclidean metric of the
\((\xi,\varphi)\) plane: a metric derived from the configuration's own
sensitivity would remove it.