diff --git a/apply_article_table10_split.py b/apply_article_table10_split.py
new file mode 100644
index 0000000..9a6fee2
--- /dev/null
+++ b/apply_article_table10_split.py
@@ -0,0 +1,50 @@
+"""Split Table 10 of article/sim_numerics.md in two - totals (with bound violations) and the
+per-problem counts - so that it fits a printed page.  The numbers are unchanged.
+
+    python -u apply_article_table10_split.py
+"""
+from __future__ import annotations
+
+PATH = "article/sim_numerics.md"
+
+OLD_START = "**Table 10.** A′ against the library rivals on all six corpora (uniform criterion)."
+OLD_END = "| all 855 | **13032** | 16481 | 16657 | 24565 | 23649 | | |"
+
+NEW = """**Table 10.** A′ against the library rivals on all six corpora (uniform criterion): total evaluations; in brackets the runs that break the $2\\times$-bisection bound.
+
+| corpus | A′ | Chandrupatla | ModAB | Brent | TOMS 748 |
+|----------------------|------------|--------------|------------|------------|------------|
+| APS (154) | **2259** (0) | 2595 (0) | 2726 (0) | 2707 (0) | 2636 (0) |
+| Chandrupatla (45) | **1074** (0) | 1505 (0) | 1681 (0) | 3238 (18) | 3088 (15) |
+| new corpus 1 (206) | **3290** (0) | 3716 (0) | 3809 (1) | 6395 (38) | 5746 (22) |
+| Roots.jl (78) | **1371** (0) | 1566 (0) | 1446 (0) | 2455 (11) | 2339 (9) |
+| applied + GSL (132) | 1382 (0) | 1382 (0) | 1460 (0) | 1503 (1) | 1756 (1) |
+| **corpus 4 (240), the one test** | **3656** (0) | 5717 (0) | 5535 (8) | 8267 (50) | 8084 (37) |
+| all 855 | **13032** | 16481 | 16657 | 24565 | 23649 |
+
+**Table 10b.** The same runs problem by problem: A′'s wins/ties/losses against Chandrupatla and against ModAB (exact two-sided sign test).
+
+| corpus | A′ vs Chandrupatla | A′ vs ModAB |
+|----------------------|------------------------------|--------------|
+| APS (154) | 103/20/31, $p=3\\cdot10^{-10}$ | 118/10/26 |
+| Chandrupatla (45) | 31/4/10, $p=0.001$ | 34/4/7 |
+| new corpus 1 (206) | 108/33/65, $p=0.001$ | 125/26/55 |
+| Roots.jl (78) | 44/14/20, $p=0.004$ | 47/11/20 |
+| applied + GSL (132) | 48/52/32, $p=0.09$ | 76/35/21 |
+| **corpus 4 (240), the one test** | **177/35/28, $p=10^{-27}$** | **183/23/34** |"""
+
+
+def main():
+    with open(PATH, encoding="utf8") as fh:
+        text = fh.read()
+    if text.count(OLD_START) != 1 or text.count(OLD_END) != 1:
+        raise SystemExit("Table 10 anchors not found exactly once")
+    i, j = text.index(OLD_START), text.index(OLD_END) + len(OLD_END)
+    text = text[:i] + NEW + text[j:]
+    with open(PATH, "w", encoding="utf8") as fh:
+        fh.write(text)
+    print("Table 10 split")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/article/sim_numerics.md b/article/sim_numerics.md
index 5fb398b..c96eeeb 100644
--- a/article/sim_numerics.md
+++ b/article/sim_numerics.md
@@ -135,15 +135,15 @@ Every quantity in it is either an evaluation of $f$, a divided difference of eva
 
 An interpolation-only rule has no worst-case bound. The obvious safeguards *replace* the computed step with a midpoint whenever some progress test fails, and all of them are ruinous here, for a reason specific to a fast method: a step that collapses the *error* need not halve the *bracket*, so a bracket-based test fires on exactly the steps that were best. With the three-point scale, replacing on that test drops the observed order from 1.94 to 1.00 and the Alefeld–Potra–Shi total from 2806 to 7260 — worse than bisection.
 
-The remedy is to keep as much of the computed step as the budget allows. Let $n_\max=2\lceil\log_2(w_0/\varepsilon_x)\rceil$ be twice the bisection step count for the initial width $w_0$ and $x$-tolerance $\varepsilon_x$, and let $\epsilon=\varepsilon_x/2$. At step $j$, with the bracket $[\ell,h]$, define the allowance and the projection
+The remedy is to keep as much of the computed step as the budget allows. Let $n_{\max}=2\lceil\log_2(w_0/\varepsilon_x)\rceil$ be twice the bisection step count for the initial width $w_0$ and $x$-tolerance $\varepsilon_x$, and let $\epsilon=\varepsilon_x/2$. At step $j$, with the bracket $[\ell,h]$, define the allowance and the projection
 
-$$\alpha_j = 2^{\,n_\max-j}\epsilon, \qquad x \;=\; \operatorname{clamp}\bigl(x_\mathrm{sim},\; h-\alpha_j,\; \ell+\alpha_j\bigr). \tag{3}$$
+$$\alpha_j = 2^{\,n_{\max}-j}\epsilon, \qquad x \;=\; \operatorname{clamp}\bigl(x_\mathrm{sim},\; h-\alpha_j,\; \ell+\alpha_j\bigr). \tag{3}$$
 
-The next bracket is at most $\max(x-\ell,\,h-x)$, so (3) gives $w_{j+1}\le\alpha_j$ directly, and induction bounds the run by $n_\max$ steps in exact arithmetic. In floating point the bracket can land exactly on the boundary, and the termination test then asks for one more; the guarantee stated and verified is therefore $2\lceil\log_2(w_0/\varepsilon_x)\rceil+3$ evaluations.
+The next bracket is at most $\max(x-\ell,\,h-x)$, so (3) gives $w_{j+1}\le\alpha_j$ directly, and induction bounds the run by $n_{\max}$ steps in exact arithmetic. In floating point the bracket can land exactly on the boundary, and the termination test then asks for one more; the guarantee stated and verified is therefore $2\lceil\log_2(w_0/\varepsilon_x)\rceil+3$ evaluations.
 
 **The multiplier is the method's one free constant.** An earlier version of this section argued that it is "the factor already claimed in the bound". That is circular — the bound is twice bisection because two was chosen — and measurement settles it (`measure_budget_factor.py`, Linux):
 
-**Table 1a.** The scheduled projection at budget multiplier $c$, i.e. $n_\max=\lceil c\cdot\log_2(w_0/\varepsilon_x)\rceil$. Total evaluations, failures charged.
+**Table 1a.** The scheduled projection at budget multiplier $c$, i.e. $n_{\max}=\lceil c\cdot\log_2(w_0/\varepsilon_x)\rceil$. Total evaluations, failures charged.
 
 | $c$ | APS-154 | CHA-45 | both |
 |---|---|---|---|
@@ -532,17 +532,28 @@ ModAB, the most recent bracketing method we know of, is run from its authors' pa
 
 **The history is part of the result.** The variant carried to corpora 1–3 was A′ without the tolerance step and the overshoot branch. It transferred to corpus 1 by every pre-registered criterion (total 10% below Chandrupatla, per problem 105/34/67, $p=0.005$), did *not* transfer to Roots.jl (total 6.5% below, but per problem 38/10/30, $p=0.4$) and lost on the applied corpus (14% above). The losses there were traced to two gaps of the base step — no closing of the bracket once the root is reached, and no recognition of roots of order below one — which became the last two readings of §8.4; they were developed on the seen corpora and tested once, on corpus 4.
 
-**Table 10.** A′ against the library rivals on all six corpora (uniform criterion). Totals; in brackets the runs that break the $2\times$-bisection bound. Per problem against Chandrupatla: wins/ties/losses.
+**Table 10.** A′ against the library rivals on all six corpora (uniform criterion): total evaluations; in brackets the runs that break the $2\times$-bisection bound.
 
-| corpus | A′ | Chandrupatla | ModAB | Brent | TOMS 748 | A′ vs Chandrupatla | A′ vs ModAB |
-|---|---|---|---|---|---|---|---|
-| APS (154) | **2259** (0) | 2595 (0) | 2726 (0) | 2707 (0) | 2636 (0) | 103/20/31, $p=3\cdot10^{-10}$ | 118/10/26 |
-| Chandrupatla (45) | **1074** (0) | 1505 (0) | 1681 (0) | 3238 (18) | 3088 (15) | 31/4/10, $p=0.001$ | 34/4/7 |
-| new corpus 1 (206) | **3290** (0) | 3716 (0) | 3809 (1) | 6395 (38) | 5746 (22) | 108/33/65, $p=0.001$ | 125/26/55 |
-| Roots.jl (78) | **1371** (0) | 1566 (0) | 1446 (0) | 2455 (11) | 2339 (9) | 44/14/20, $p=0.004$ | 47/11/20 |
-| applied + GSL (132) | 1382 (0) | 1382 (0) | 1460 (0) | 1503 (1) | 1756 (1) | 48/52/32, $p=0.09$ | 76/35/21 |
-| **corpus 4 (240), the one test** | **3656** (0) | 5717 (0) | 5535 (8) | 8267 (50) | 8084 (37) | **177/35/28, $p=10^{-27}$** | **183/23/34** |
-| all 855 | **13032** | 16481 | 16657 | 24565 | 23649 | | |
+| corpus | A′ | Chandrupatla | ModAB | Brent | TOMS 748 |
+|----------------------|------------|--------------|------------|------------|------------|
+| APS (154) | **2259** (0) | 2595 (0) | 2726 (0) | 2707 (0) | 2636 (0) |
+| Chandrupatla (45) | **1074** (0) | 1505 (0) | 1681 (0) | 3238 (18) | 3088 (15) |
+| new corpus 1 (206) | **3290** (0) | 3716 (0) | 3809 (1) | 6395 (38) | 5746 (22) |
+| Roots.jl (78) | **1371** (0) | 1566 (0) | 1446 (0) | 2455 (11) | 2339 (9) |
+| applied + GSL (132) | 1382 (0) | 1382 (0) | 1460 (0) | 1503 (1) | 1756 (1) |
+| **corpus 4 (240), the one test** | **3656** (0) | 5717 (0) | 5535 (8) | 8267 (50) | 8084 (37) |
+| all 855 | **13032** | 16481 | 16657 | 24565 | 23649 |
+
+**Table 10b.** The same runs problem by problem: A′'s wins/ties/losses against Chandrupatla and against ModAB (exact two-sided sign test).
+
+| corpus | A′ vs Chandrupatla | A′ vs ModAB |
+|----------------------|------------------------------|--------------|
+| APS (154) | 103/20/31, $p=3\cdot10^{-10}$ | 118/10/26 |
+| Chandrupatla (45) | 31/4/10, $p=0.001$ | 34/4/7 |
+| new corpus 1 (206) | 108/33/65, $p=0.001$ | 125/26/55 |
+| Roots.jl (78) | 44/14/20, $p=0.004$ | 47/11/20 |
+| applied + GSL (132) | 48/52/32, $p=0.09$ | 76/35/21 |
+| **corpus 4 (240), the one test** | **177/35/28, $p=10^{-27}$** | **183/23/34** |
 
 Only the last line of corpora is evidence for A′ in the strict sense; the first five were seen while it was developed and are reported as a description. On corpus 4 A′ is 36% below Chandrupatla and wins 177 problems to 28. Without the two classes aimed at by the last two readings it is 2403 against 3569, per problem 107/31/22 ($p=10^{-14}$); on the simple-root classes alone, 737 against 747 (−1.3%), per problem 43/22/15 ($p=3\cdot10^{-4}$).
 
diff --git a/article/sim_numerics.pdf b/article/sim_numerics.pdf
deleted file mode 100644
index 165ee44..0000000
Binary files a/article/sim_numerics.pdf and /dev/null differ
diff --git a/article/sim_numerics.tex b/article/sim_numerics.tex
index fbd09f2..526cdc3 100644
--- a/article/sim_numerics.tex
+++ b/article/sim_numerics.tex
@@ -1,987 +1,2285 @@
-%% =====================================================================
-%%  STALE.  This file lags behind article/sim_numerics.md, which is the
-%%  current text.  Do not build a PDF from it and do not send it out.
-%%
-%%  The .md has taken the first pass of revision corrections; this file
-%%  still contains statements that pass showed to be wrong:
-%%    - the proposed similarity called "ratio" (it is the truncated
-%%      linear / tent form; the ratio form is a different row)
-%%    - Proposition 1 stated for similarity rules in general (it holds
-%%      for the ratio form scaled by the far end, and for no other of
-%%      the five tested)
-%%    - Table 1's c = 0 rows given a "measured rate" (there is no rate;
-%%      the behaviour is sublinear, e_n ~ C/n with a predicted C)
-%%    - the three-point order given as 1.84 (it is 2)
-%%    - section 5.2 evaluation counts not reconciled with section 5
-%%
-%%  It will be regenerated from the .md once the text settles.
-%% =====================================================================
-\documentclass[11pt,a4paper]{article}
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{amsmath,amssymb,amsthm}
-\usepackage{booktabs}
+%% GENERATED from article/sim_numerics.md by build_tex.py - do not edit by hand.
+%% The .md is the canonical text; every number in it is checked by verify_article.py.
+% Options for packages loaded elsewhere
+\PassOptionsToPackage{unicode}{hyperref}
+\PassOptionsToPackage{hyphens}{url}
+%
+\documentclass[
+  11pt,
+  a4paper]{article}
+\usepackage{amsmath,amssymb}
+\usepackage{iftex}
+\ifPDFTeX
+  \usepackage[T1]{fontenc}
+  \usepackage[utf8]{inputenc}
+  \usepackage{textcomp} % provide euro and other symbols
+\else % if luatex or xetex
+  \usepackage{unicode-math} % this also loads fontspec
+  \defaultfontfeatures{Scale=MatchLowercase}
+  \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
+\fi
+\usepackage{lmodern}
+\ifPDFTeX\else
+  % xetex/luatex font selection
+\fi
+% Use upquote if available, for straight quotes in verbatim environments
+\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
+\IfFileExists{microtype.sty}{% use microtype if available
+  \usepackage[]{microtype}
+  \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
+}{}
+\makeatletter
+\@ifundefined{KOMAClassName}{% if non-KOMA class
+  \IfFileExists{parskip.sty}{%
+    \usepackage{parskip}
+  }{% else
+    \setlength{\parindent}{0pt}
+    \setlength{\parskip}{6pt plus 2pt minus 1pt}}
+}{% if KOMA class
+  \KOMAoptions{parskip=half}}
+\makeatother
+\usepackage{xcolor}
+\usepackage{longtable,booktabs,array}
+\usepackage{calc} % for calculating minipage widths
+% Correct order of tables after \paragraph or \subparagraph
+\usepackage{etoolbox}
+\makeatletter
+\patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{}
+\makeatother
+% Allow footnotes in longtable head/foot
+\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}}
+\makesavenoteenv{longtable}
+\setlength{\emergencystretch}{3em} % prevent overfull lines
+\providecommand{\tightlist}{%
+  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
+\usepackage{amsmath,amssymb}
+\usepackage{textcomp}
+\DeclareUnicodeCharacter{2032}{\ensuremath{'}}
+\DeclareUnicodeCharacter{2212}{\ensuremath{-}}
+\DeclareUnicodeCharacter{00D7}{\ensuremath{\times}}
+\DeclareUnicodeCharacter{00B5}{\textmu}
+\DeclareUnicodeCharacter{2192}{\ensuremath{\rightarrow}}
+\DeclareUnicodeCharacter{0394}{\ensuremath{\Delta}}
+\DeclareUnicodeCharacter{2082}{\textsubscript{2}}
 \usepackage[margin=2.5cm]{geometry}
-\usepackage{hyperref}
+\usepackage[htt]{hyphenat}
+\setlength{\emergencystretch}{3em}
+\usepackage{etoolbox}
+\AtBeginEnvironment{longtable}{\small}
+\ifLuaTeX
+  \usepackage{selnolig}  % disable illegal ligatures
+\fi
+\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
+\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
+\urlstyle{same}
+\hypersetup{
+  pdftitle={Similarity as a Step Rule: Parameter-Free Root Finding Derived from a Similarity Function},
+  pdfauthor={V. Sitnikov},
+  hidelinks,
+  pdfcreator={LaTeX via pandoc}}
 
-\newtheorem{proposition}{Proposition}
-\newtheorem{corollary}{Corollary}
-\theoremstyle{definition}
-\newtheorem{definition}{Definition}
-
-\title{Similarity as a Step Rule:\\
-Parameter-Free Root Finding Derived from a Similarity Function}
+\title{Similarity as a Step Rule: Parameter-Free Root Finding Derived
+from a Similarity Function}
 \author{V. Sitnikov}
 \date{}
 
 \begin{document}
 \maketitle
-
 \begin{abstract}
 Bisection places the next iterate at the midpoint because it uses no
 information about the function's value. We replace that rule with a
-\emph{similarity} rule: the step is proportional to how \emph{dissimilar} the
-current outcome $f(x)$ is to the expectation $0$, measured by a similarity
-function $S\colon\mathbb{R}\to[0,1]$ with $S(0)=1$. We derive the contraction
-factor of the resulting iteration in closed form and confirm it numerically to
-four digits. Three consequences follow and are all verified. First, the
-two-point normalisation of $S$ over the bracket reproduces the false-position
-formula identically, so a similarity rule calibrated by the bracket is not a
-new method. Second, any similarity function fixed in advance yields at best
-linear convergence at a rate that is a property of the problem rather than of
-the method; on eight standard problems, five to eight of them fail. Third, the
-rate vanishes exactly when the similarity's scale equals the local scale of the
-value axis; reading that scale off the data --- a divided difference, not a
-fitted coefficient --- turns the step into $|f|/|f'|$ and the convergence
-superlinear.
-
-The derivation fixes the method completely, before any benchmark is consulted:
-a similarity whose approach to $1$ is linear, the finest available data-derived
-scale, and a safeguard that projects the step onto the admissible interval
-rather than replacing it. That configuration carries a proven worst case of
-twice bisection plus one step and contains no tuning constant of any kind. On
-the $154$ problems of Alefeld, Potra and Shi it needs $2806$ evaluations of $f$
-against $2820$ for TOMS\,748 --- a margin of $0.5\%$, which is a tie --- and
-$2890$ for Brent, whom it beats on $102$ problems to $40$ head to head. On the
-$45$ problems of Chandrupatla it beats Brent $30$ to $4$ and TOMS\,748 $31$ to
-$8$, and loses to ITP. The result is parity with methods built on tuned
-constants, obtained without any.
-The ranking of the nine similarity functions across the whole grid matches the
-derived law without exception.
-Reading the same similarity in one more place --- which three points the local
-model is built on, selected by the shape of their configuration rather than by
-recency --- lowers the Alefeld--Potra--Shi count to $2527$, below every method
-measured here including Chandrupatla's $2785$, with the guarantee intact and
-still no tuning constant; reading it instead on the choice of far end gives
-$5511$ over both sets. Reading it in the decision to trust the model or halve
-lowers the canonical count further, to $2523$. Where the similarity is read
-turns out to matter more than which similarity it is. The gap that remains on
-the wide-bracket set is taken apart on the same evaluations: a third of it is
-the order of the model --- a convex two-point step cannot carry the negative
-weight of quadratic interpolation --- and two thirds is not a model at all but
-the decision when to give the model up, which Chandrupatla takes by bisecting
-on up to all of his steps and which no reading measured here takes correctly
-for ours.
+\emph{similarity} rule: the step is proportional to how
+\emph{dissimilar} the current outcome \(f(x)\) is to the expectation
+\(0\), measured by a similarity function \(S:\mathbb{R}\to[0,1]\) with
+\(S(0)=1\). We derive the contraction factor of the resulting iteration
+in closed form; it predicts a linear rate to four digits when \(1-S\)
+has a non-zero slope at the origin, and a sublinear \(e_n\sim C/n\) with
+a predicted \(C\) when it does not, both confirmed. Three consequences
+follow. First, with the \emph{ratio} similarity scaled by the far end
+--- and with that combination only, of the five tested --- the rule
+reproduces the false-position formula identically; so does the truncated
+linear similarity scaled by a divided difference over the \emph{bracket
+ends}, which shows that the boundary of novelty runs through which
+points the difference is taken over. Second, any similarity fixed in
+advance with the scale also fixed yields at best linear convergence at a
+rate that is a property of the problem rather than of the method; on the
+eight standard problems of §2.2, five to eight of them fail. Third, the
+rate vanishes exactly when the scale equals the local scale of the value
+axis; reading that scale off the data --- a divided difference, not a
+fitted coefficient --- turns the step into \(|f|/|f'|\) and the
+convergence superlinear.\\
+The derivation fixes a \emph{base rule} before any benchmark is
+consulted: a similarity whose approach to 1 is linear, the finest
+available data-derived scale, and a safeguard that projects the step
+onto the admissible interval rather than replacing it. It is,
+explicitly, a Newton step with a three-point divided-difference
+derivative, clipped to the bracket and projected onto the bisection
+budget. The similarity sets that step, and its form is decisive: with
+the tent the step is Newton's, with any of the other eight functions the
+same method costs from 12\% to four times more. It carries a proven
+worst case of twice bisection plus one step and exactly one free
+constant, which we declare rather than argue away: the multiplier of
+that budget. On the 154 problems of Alefeld, Potra and Shi it needs 2806
+evaluations of \(f\) against 2820 for TOMS 748, 2889 for Brent and 2787
+for Chandrupatla's method --- level in total --- and beats Chandrupatla
+problem by problem 87 to 42 (exact sign test \(p=9\cdot10^{-5}\)). On
+the 45 wide-bracket problems of Chandrupatla it beats Brent and TOMS 748
+and loses to ITP and to Chandrupatla (3220 against 2896 and 2441). For
+every data-derived scale, the ranking of the nine similarity functions
+matches the derived law without exception.\\
+The wide-bracket gap turns out to be not the order of the model but a
+decision --- when to give the model up --- and the decision is a
+similarity after all, of an unfamiliar kind: the similarity of a run to
+its own past. A pure power is self-similar under zoom, so the
+dimensionless configuration of the last three points recurs, while at a
+simple root it tends to the linear prototype; the analogous pair yields
+the local order, \(m=\ln L_f/\ln L_x\), and an exponential, self-similar
+by translation, is its limit \(m=\infty\). Entering a halving regime
+when that analogy holds with \(m\ge2\) --- the condition under which a
+tangent step loses to a halving --- splitting at the point the order
+predicts, and leaving when the split looks linear, together with four
+smaller readings of the same kind and one borrowed element, gives the
+method proposed here, A′. On the two sets it needs 2438 and 2005
+evaluations against Chandrupatla's 2787 and 2441 and beats him problem
+by problem on both, 103 to 30 and 29 to 7. The chain of decisions that
+produced it was selected on those sets; tested once, after its code was
+frozen, on 240 problems from six new families, it needs 3656 evaluations
+against Chandrupatla's 5717 and wins 177 problems to 28. Over 855
+problems in six corpora it is ahead of Brent, TOMS 748 and the recent
+ModAB everywhere and keeps its bound where they break theirs 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 12\% to 315\%).
 \end{abstract}
 
-\section{Introduction}
+\hypertarget{introduction}{%
+\subsection{1. Introduction}\label{introduction}}
 
-A bracketing root finder must decide, at every step, where inside the bracket
-to evaluate next. Bisection answers ``in the middle'', which is the correct
-answer when nothing is known. Every improvement since has answered ``where a
-local model of $f$ says the root is'': a secant in false position, an inverse
-quadratic in Brent~\cite{brent}, a cubic in TOMS\,748~\cite{aps}, an
-interpolate--truncate--project cycle in ITP~\cite{itp}. Each of those answers
-carries constants: Illinois halves the stale ordinate, Brent accepts an
-interpolation subject to three empirical inequalities, ITP is governed by
-$\kappa_1,\kappa_2,n_0$.
+A bracketing root finder must decide, at every step, where inside the
+bracket to evaluate next. Bisection answers ``in the middle'', which is
+the correct answer when nothing is known. Every improvement since has
+answered ``where a local model of \(f\) says the root is'': a secant in
+false position, an inverse quadratic in Brent {[}2{]}, a cubic in TOMS
+748 {[}1{]}, an interpolate--truncate--project cycle in ITP {[}4{]}, an
+Anderson--Björck step behind a linearity check in ModAB {[}10, 20{]}.
+Each of those answers carries constants: Illinois halves the stale
+ordinate, Brent accepts an interpolation subject to three empirical
+inequalities, ITP is governed by \(\kappa_1,\kappa_2,n_0\).
 
-This paper asks a different question. Suppose the only thing available is a
-\emph{similarity function} --- a map $S$ from an outcome to $[0,1]$, where $1$
-means ``identical to what was wanted'' and $0$ means ``not similar at all'',
-in the sense used for parameter-free control in~\cite{sim}. Can a root finder
-be built out of $S$ alone, and how does it compare with methods built out of
-polynomial models and tuned constants?
+This paper asks a different question. Suppose the only thing available
+is a \emph{similarity function} --- a map \(S\) from an outcome to
+\([0,1]\), where 1 means ``identical to what was wanted'' and 0 means
+``not similar at all'', in the sense used for parameter-free control in
+{[}6{]}. Can a root finder be built out of \(S\) alone, and how does it
+compare with methods built out of polynomial models and tuned constants?
 
-The answer is sharp in both directions, and the negative half is as informative
-as the positive half.
+The answer is sharp in both directions, and the negative half is as
+informative as the positive half.
 
-\section{The similarity step rule}
+The paper proceeds in two stages. Sections 2--7 derive a base rule from
+the similarity alone, measure it, and locate what it lacks: not a better
+model but a decision, when to stop trusting the model it has. Section 8
+supplies that decision with a similarity of the run to its own past, and
+Section 9 tests it on four further corpora: three of them exposed two
+last gaps of the base step and were used to close them, and the fourth
+was fixed after the method's code was frozen and run once.
 
-\begin{definition}[Similarity step]
-Let $[a,b]$ bracket a root, $x$ be the current iterate and $x_{\mathrm{far}}$
-the opposite end. Let $S\colon\mathbb{R}\to[0,1]$ with $S(0)=1$ be a similarity
-function and $\sigma>0$ a scale. The next iterate is
-\begin{equation}
-\label{eq:step}
-x' \;=\; x \;+\; \bigl(1 - S(f(x)/\sigma)\bigr)\,(x_{\mathrm{far}} - x).
-\end{equation}
-\end{definition}
+\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
 
-The rule is exactly the informal statement ``step further when the outcome is
-dissimilar to the expectation, and less far when it is similar''. Bisection is
-the degenerate member $S\equiv\tfrac12$: a similarity function that carries no
-information about $f$.
+\hypertarget{the-similarity-step-rule}{%
+\subsection{2. The similarity step
+rule}\label{the-similarity-step-rule}}
 
-\subsection{The two-point normalisation is false position}
+\textbf{Definition 1 (Similarity step).} Let \([a,b]\) bracket a root,
+\(x\) be the current iterate and \(x_\mathrm{far}\) the opposite end.
+Let \(S:\mathbb{R}\to[0,1]\) with \(S(0)=1\) be a similarity function
+and \(\sigma>0\) a scale. The next iterate is
 
-The most obvious way to make~\eqref{eq:step} scale-free is to normalise the
-similarity by the two ends of the bracket, $S(a)=|f(b)|/(|f(a)|+|f(b)|)$ and
-$S(b)=1-S(a)$, and to place the next iterate at the similarity-weighted average
-of the ends.
+\[x' \;=\; x \;+\; \bigl(1 - S(f(x)/\sigma)\bigr)\,(x_\mathrm{far} - x). \tag{1}\]
 
-\begin{proposition}
-\label{prop:falsi}
-$S(a)\,a + S(b)\,b \;=\; \dfrac{a f(b) - b f(a)}{f(b)-f(a)}$.
-\end{proposition}
+The rule is exactly the informal statement ``step further when the
+outcome is dissimilar to the expectation, and less far when it is
+similar''. Bisection is the degenerate member \(S\equiv\tfrac12\): a
+similarity function that carries no information about \(f\).
 
-That is the false-position formula. The identity is elementary but decisive: a
-similarity rule whose scale is taken from the bracket is not a new method, and
-it inherits false position's stalling failure. In our measurements plain false
-position solves $127$ of the $154$ Alefeld--Potra--Shi problems, against $153$
-for bisection.
+\hypertarget{the-two-point-normalisation-is-false-position}{%
+\subsubsection{2.1 The two-point normalisation is false
+position}\label{the-two-point-normalisation-is-false-position}}
 
-\subsection{The contraction law}
+The most obvious way to make (1) scale-free is to normalise the
+similarity by the two ends of the bracket,
+\(S(a)=|f(b)|/(|f(a)|+|f(b)|)\) and \(S(b)=1-S(a)\), and to place the
+next iterate at the similarity-weighted average of the ends.
 
-Write $e_n = x_n - r$ for the error, $D = x_{\mathrm{far}} - r$, and let $c$ be
-the slope of $1-S$ at the origin, $1-S(z) = c\,z + O(z^2)$.
+\textbf{Proposition 1.} For the ratio similarity \(S(z)=1/(1+|z|)\) with
+\(\sigma=|f(x_\mathrm{far})|\), equivalently for the endpoint weights
+above, \[S(a)\,a + S(b)\,b \;=\; \frac{a f(b) - b f(a)}{f(b)-f(a)}.\]
 
-\begin{proposition}[Contraction law]
-\label{prop:law}
-For a simple root and $e_n$ small, rule~\eqref{eq:step} satisfies
-\begin{equation}
-\label{eq:law}
-e_{n+1} \;=\; e_n\left(1 - \frac{c\,|f'(r)|\,D}{\sigma}\right) + O(e_n^2),
-\qquad
-\text{rate} = \left|1 - \frac{c\,|f'(r)|\,D}{\sigma}\right| .
-\end{equation}
-\end{proposition}
+That is the false-position formula. Two things must be said about how
+far it reaches, because the obvious generalisation is false.
 
-\begin{proof}[Derivation]
-$|f(x_n)| = |f'(r)|\,|e_n| + O(e_n^2)$, hence
-$1-S(f(x_n)/\sigma) = c\,|f'(r)|\,|e_n|/\sigma + O(e_n^2)$, and
-$x_{\mathrm{far}}-x_n = D - e_n$. Substituting into~\eqref{eq:step} and
-discarding second-order terms gives~\eqref{eq:law}.
-\end{proof}
+It is specific to that similarity. On \(f=x^2-2\) over \([1,4]\), where
+false position gives 1.200000, the same construction gives 1.206812 for
+\(e^{-|z|}\), 1.015228 for \(1/(1+z^2)\), 1.007624 for
+\(1/\sqrt{1+z^2}\) and 1.214286 for the truncated linear form: one of
+five reproduces it. So ``a similarity rule scaled by the bracket is not
+a new method'' would be too broad a conclusion, and this paper's own
+Table 3 refutes it --- bracket-scaled configurations take first and
+fourth place on Chandrupatla and one of them solves 41 of 45, where
+false position solves 6.
 
-Table~\ref{tab:law-check} checks~\eqref{eq:law} against measurement on
-$f(x)=x^2-2$ over $[0,2]$ with $\sigma=|f(0)|$, for which $|f'(r)|=2\sqrt2$,
-$D=2-\sqrt2$ and $\sigma=2$.
+\textbf{Proposition 1'.} It is, however, specific to the \emph{bracket
+ends} rather than to the ratio form. Take the truncated linear
+similarity \(S(z)=\max(0,1-|z|)\) and the divided difference over the
+two ends, \(\sigma=|f[a,b]|\cdot|b-a|=|f(b)-f(a)|=|f(a)|+|f(b)|\). Then
+\(1-S=|f(a)|/(|f(a)|+|f(b)|)\), which is false position again, and the
+agreement is exact to the last bit.
 
-\begin{table}[h]
-\centering
-\caption{The contraction law against measurement. $c$ is the slope of $1-S$ at
-the origin. Similarities that approach $1$ quadratically have $c=0$ and are
-predicted to stagnate.}
-\label{tab:law-check}
-\begin{tabular}{lrrr}
-\toprule
-$S(z)$ & $c$ & predicted rate & measured rate\\
-\midrule
-$1/(1+|z|)$        & $1$ & $0.1716$ & $0.1716$\\
-$e^{-|z|}$         & $1$ & $0.1716$ & $0.1716$\\
-$1/(1+z^2)$        & $0$ & $1.0000$ & $0.9357$\\
-$e^{-z^2}$         & $0$ & $1.0000$ & $0.9444$\\
-$1/\sqrt{1+z^2}$   & $0$ & $1.0000$ & $0.9327$\\
-$\max(0,1-|z|)$    & $1$ & $0.1716$ & $1.0000$ (saturates)\\
-\bottomrule
-\end{tabular}
-\end{table}
+Together the two say where the boundary of novelty runs, and it is not
+where one would guess: it runs through the words \emph{most recent} in
+the definition of \(\hat f'\) in §3. A divided difference over the
+bracket ends is three centuries old; the same difference over the most
+recent iterates is the secant. In our measurements plain false position
+solves 127 of the 154 Alefeld--Potra--Shi problems, against 153 for
+bisection.
 
-\begin{corollary}[A chosen similarity is not enough]
-\label{cor:chosen}
-For fixed $S$ and fixed $\sigma$ the rate in~\eqref{eq:law} depends on
-$|f'(r)|$, $D$ and $\sigma$, none of which is under the method's control.
-Convergence is at best linear and occurs only when the product lands in
-$(0,2)$.
-\end{corollary}
+\hypertarget{the-contraction-law}{%
+\subsubsection{2.2 The contraction law}\label{the-contraction-law}}
 
-Empirically, with $\sigma$ taken as $|f|$ at the starting point, five to eight
-of eight standard problems fail outright for the six similarity functions
-tested.
+Write \(e_n = x_n - r\) for the error, \(D = x_\mathrm{far} - r\), and
+let \(c\) be the slope of \(1-S\) at the origin,
+\(1-S(z) = c\,z + O(z^2)\).
 
-\begin{corollary}[Faster is not better]
-\label{cor:faster}
-Increasing $c$ moves the rate away from $1$ in either direction. Similarities
-with unbounded slope --- $1-\sqrt{|z|}$ has $c\approx3.2\cdot10^{3}$ at the
-resolution used, $1-|z|^{1/3}$ has $c\approx4.6\cdot10^{4}$ --- overshoot as
-readily as flat ones stagnate, and fail on five and six of the eight problems
-respectively.
-\end{corollary}
+\textbf{Proposition 2 (Contraction law).} For a simple root and \(e_n\)
+small, rule (1) satisfies
 
-\begin{corollary}[The scale must come from the data]
-\label{cor:derived}
-The rate is zero exactly when $c\,|f'(r)|\,D/\sigma = 1$, i.e.
-$\sigma = c\,|f'(r)|\,D$. Substituting this $\sigma$ back into~\eqref{eq:step}
-with $c=1$ gives $x' - x = |f(x)|/|f'|$, the Newton step. The similarity rule
-is therefore superlinear precisely when its scale is the \emph{local scale of
-the value axis}, and that scale is not available a priori: it must be read off
-the evaluations.
-\end{corollary}
+\[e_{n+1} \;=\; e_n\left(1 - \frac{c\,|f'(r)|\,D}{\sigma}\right) + O(e_n^2), \qquad \text{rate} = \left|1 - \frac{c\,|f'(r)|\,D}{\sigma}\right|. \tag{2}\]
 
-\section{The method}
-\label{sec:method}
+\emph{Derivation.} \(|f(x_n)| = |f'(r)|\,|e_n| + O(e_n^2)\), hence
+\(1-S(f(x_n)/\sigma) = c\,|f'(r)|\,|e_n|/\sigma + O(e_n^2)\), and
+\(x_\mathrm{far}-x_n = D - e_n\). Substituting into (1) and discarding
+second-order terms gives (2). \(\square\)
 
-Corollaries~\ref{cor:chosen}--\ref{cor:derived} fix every choice in the method
-before any benchmark is consulted. We record them here as a prediction, and
-Section~\ref{sec:grid} tests it against the full grid.
+Table 1 checks (2) against measurement on \(f(x)=x^2-2\) over \([0,2]\)
+with \(\sigma=|f(0)|\), for which \(|f'(r)|=2\sqrt2\), \(D=2-\sqrt2\)
+and \(\sigma=2\).
 
-\paragraph{Similarity: any with $c\neq0$.} The law makes the shape of $S$
-almost irrelevant and its slope at the origin decisive. We take the plainest
-member, the ratio form $S(z)=\max(0,1-|z|)$.
+\textbf{Table 1.} The contraction law against measurement, all rows at
+the \emph{chosen} scale \(\sigma=|f(x_0)|\). \(c\) is the slope of
+\(1-S\) at the origin and \(c_2\) the coefficient of \(z^2\) when
+\(c=0\).
 
-\paragraph{Scale: the finest data-derived one.} We set
-$\sigma = |\hat f'|\cdot|x_{\mathrm{far}}-x|$ with $\hat f'$ a divided
-difference of the most recent evaluations. A divided difference is a
-computation, not a fitted coefficient: nothing is chosen, regressed or tuned.
-The two-point difference has slope error $O(h)$ and gives the secant order
-$\varphi\approx1.618$; the three-point difference --- the derivative at $x_n$
-of the parabola through the last three evaluations,
-$\hat f' = f[x_{n-1},x_n] + f[x_{n-2},x_{n-1},x_n](x_n-x_{n-1})$ --- has error
-$O(h^2)$ and lifts it. Measured medians are $1.584$ and $1.944$, against
-$1.809$ for Brent and $2.654$ for TOMS\,748.
+A similarity with \(c=0\) does not converge at a rate at all. Expanding
+\(1-S=c_2z^2+O(z^4)\) in the same way as (2) gives
+\(e_{n+1}=e_n-c_2|f'(r)|^2|D|e_n^2/\sigma^2+O(e_n^3)\), hence
 
-\subsection{The safeguard: project, do not replace}
-\label{sec:guard}
+\[e_n \;\simeq\; \frac{\sigma^2}{c_2\,|f'(r)|^2\,|D|}\cdot\frac1n, \tag{2a}\]
 
-An interpolation-only rule has no worst-case bound. The obvious safeguards
-\emph{replace} the computed step with a midpoint whenever some progress test
-fails, and all of them are ruinous here, for a reason that is specific to a
-fast method: a step that collapses the \emph{error} need not halve the
-\emph{bracket}, so a bracket-based test fires on exactly the steps that were
-best. With the three-point scale, replacing on that test drops the observed
-order from $1.94$ to $1.00$ and the Alefeld--Potra--Shi total from $2806$ to
-$7260$ --- worse than bisection.
+sublinear rather than stagnant. Reporting a ``measured rate'' for these
+rows would report the value of \(e_{n+1}/e_n\) at whatever \(n\) the
+window happened to end --- it creeps to 1 monotonically (0.925 at
+\(n=10\), 0.991 at \(n=100\), 0.9998 at \(n=4000\)) --- so the
+prediction tested is the coefficient in (2a) instead.
+
+\begin{longtable}[]{@{}llll@{}}
+\toprule\noalign{}
+\(S(z)\) & \(c\) & predicted & measured \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+\(1/(1+\lvert z\rvert)\) & 1 & rate 0.1716 & 0.1716 \\
+\(e^{-\lvert z\rvert}\) & 1 & rate 0.1716 & 0.1716 \\
+\(1/(1+z^2)\) & 0, \(c_2=1\) & \(n\,e_n\to0.8536\) & 0.8503 \\
+\(e^{-z^2}\) & 0, \(c_2=1\) & \(n\,e_n\to0.8536\) & 0.8498 \\
+\(1/\sqrt{1+z^2}\) & 0, \(c_2=\tfrac12\) & \(n\,e_n\to1.7071\) &
+1.6988 \\
+\(\max(0,1-\lvert z\rvert)\) & 1 & rate 0.1716 & captured, see below \\
+\end{longtable}
+
+The last row is not a slow rate but an immediate failure, and it matters
+because that similarity is the one this paper goes on to propose. At
+\(x_0=0\) the residual is \(z=-1\), so \(S=\max(0,1-1)=0\), the step is
+the whole bracket, the iterate lands exactly on \(x_\mathrm{far}\) and
+never leaves: the error is 0.5858 after one step and after two hundred.
+\textbf{The failure is a property of the chosen scale, not of the
+similarity.} Under the derived scale of §3 the residual \(|f|/\sigma\)
+stays below 1 --- on this problem the clip does not engage on a single
+one of the seven steps the method takes --- the truncation never fires,
+and the same function is the best of the nine (Table 4). This is the
+paper's own thesis, that the content sits in the scale and not in the
+shape of \(S\), showing up as a discontinuity in the behaviour of one
+function.
+
+\textbf{Corollary 1 (A chosen similarity is not enough).} For fixed
+\(S\) and fixed \(\sigma\) the rate in (2) depends on \(|f'(r)|\), \(D\)
+and \(\sigma\), none of which is under the method's control. Convergence
+is at best linear and occurs only when the product lands in \((0,2)\).
+
+Empirically, with \(\sigma\) taken as \(|f|\) at the starting point,
+five to eight of the eight standard problems fail outright for the six
+similarity functions tested. Those eight, used throughout as a quick set
+and never as evidence on their own, are: \(x^2-2\) on \([0,2]\);
+\(\cos x-x\) on \([0,1]\); \(e^x-2\) on \([0,2]\); \(x^3-x-2\) on
+\([1,2]\); \(\sin x-\tfrac12\) on \([0,1.5]\); \(x^{10}-1\) on
+\([0,1.3]\), which stalls false position; \(e^{-x}(x-1)\) on
+\([-1,40]\), whose right end is flat to \(10^{-16}\); and \(\arctan x\)
+on \([-1,3]\).
+
+\textbf{Corollary 2 (Missing the calibration in either direction
+costs).} The rate degrades as the product \(c\,|f'(r)|\,|D|/\sigma\)
+departs from 1, upwards or downwards. We claim no symmetry between the
+two directions, and the measurements do not support one: similarities
+with unbounded slope --- \(1-\sqrt{|z|}\) has \(c\approx3.2\cdot10^{3}\)
+at the resolution used, \(1-|z|^{1/3}\) has \(c\approx4.6\cdot10^{4}\)
+--- fail on five and six of the eight problems with a chosen scale, yet
+over the full benchmark they finish at 10497 and 11941, ahead of all
+three \(c=0\) similarities at 14314, 15102 and 16394, and one of them
+heads Table 3. Overshooting is the milder failure.
+
+\textbf{Corollary 3 (The scale must come from the data).} The rate is
+zero exactly when \(c\,|f'(r)|\,D/\sigma = 1\),
+i.e.~\(\sigma = c\,|f'(r)|\,D\). Substituting this \(\sigma\) back into
+(1) with \(c=1\) gives \(x' - x = |f(x)|/|f'|\), the Newton step. The
+similarity rule is therefore superlinear precisely when its scale is the
+\emph{local scale of the value axis}, and that scale is not available a
+priori: it must be read off the evaluations.
+
+\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
+
+\hypertarget{the-method}{%
+\subsection{3. The method}\label{the-method}}
+
+Corollaries 1--3 fix every choice in the method before any benchmark is
+consulted. We record them here as a prediction, and Section 5.3 tests it
+against the full grid. One number in the safeguard is not fixed by them,
+and §3.1 declares it as the method's one free constant.
+
+\textbf{Similarity: any with \(c\neq0\).} The law makes the shape of
+\(S\) almost irrelevant and its slope at the origin decisive. We take
+the plainest member, the \textbf{truncated linear} or tent form
+\(S(z)=\max(0,1-|z|)\). It is not the ratio form \(1/(1+|z|)\), which is
+a separate row of every table here; the two are distinguished throughout
+because Proposition 1 applies to the ratio form and Table 1's last row
+to the tent.
+
+\textbf{Scale: the finest data-derived one.} We set
+\(\sigma = |\hat f'|\cdot|x_\mathrm{far}-x|\) with \(\hat f'\) a divided
+difference of the \textbf{most recent evaluations} --- not of the
+bracket ends, which by Proposition 1' would give false position. A
+divided difference is a computation, not a fitted coefficient: nothing
+is chosen, regressed or tuned. The two-point difference has slope error
+\(O(h)\) and gives the secant order \(\varphi\approx1.618\); the
+three-point difference --- the derivative at \(x_n\) of the parabola
+through the last three evaluations,
+\(\hat f' = f[x_{n-1},x_n] + f[x_{n-2},x_{n-1},x_n](x_n-x_{n-1})\) ---
+has error \(O(h^2)\). A Newton step whose derivative carries an
+\(O(h^2)\) error satisfies
+\(e' = e-f/\hat f' = e-(e-ce^2+\dots)(1+O(e^2)) = ce^2+O(e^3)\), so the
+predicted order is \textbf{2} --- not Muller's 1.84, which belongs to a
+method using the parabola's \emph{root} rather than its
+\emph{derivative}. Measured medians are 1.584 for the two-point scale
+and 2.000 for the three-point, against 1.809 for Brent and 2.654 for
+TOMS 748.
+
+\textbf{What the rule reduces to.} With \(S(z)=\max(0,1-|z|)\) one has
+\(1-S=\min(1,|z|)\), and (1) becomes
+
+\[x' \;=\; x + \operatorname{sign}(x_\mathrm{far}-x)\cdot\min\!\Bigl(\frac{|f(x)|}{|\hat f'|},\;|x_\mathrm{far}-x|\Bigr), \tag{4}\]
+
+a Newton step with a three-point divided-difference derivative, clipped
+to the bracket and then projected as in §3.1. That the tent turns the
+similarity step into Newton's is a property of the similarity chosen,
+not a sign that the similarity has dropped out: which \(S\) stands in
+the step decides the step, and it is measured, not assumed --- Table 4
+ranks the nine functions over the whole grid, the readings of §7 pay
+most with the tent in the step and some of them lose with other
+functions (§7), and in the proposed method replacing the tent by any
+other function costs 12\% to 315\% (Table 9a).
+
+We state the reduction ourselves rather than leave it to be discovered,
+because it locates the contribution precisely. The base rule is not a
+new family of methods. It is (i) a derivation that fixes every choice
+--- which similarity, which scale, which safeguard --- before any
+benchmark is consulted, and then predicts the ranking of all nine
+similarity functions correctly; (ii) a safeguard of ITP's projection
+type with ITP's truncation stage, and hence all three of its constants,
+removed; (iii) the negative results of §10; (iv) the measurements. We
+call it the \textbf{base rule}; the method this paper proposes builds on
+it in Section 8.
+
+\textbf{Algorithm.} Given \([a,b]\) with \(f(a)f(b)<0\) and tolerance
+\(\varepsilon_x\):
+
+\begin{verbatim}
+evaluate fa = f(a), fb = f(b)
+A, B <- a, b                          the proven bracket
+x  <- the endpoint with the smaller |f|;  xp <- the other;  xp2 <- none
+n_max <- 2 * ceil(log2(|b-a| / xtol));   allow <- 2^n_max * xtol/2
+while |B - A| > xtol:
+    far   <- the end of [A,B] further from x
+    f'    <- derivative at x of the parabola through (x,xp,xp2)
+             (two-point difference while xp2 does not yet exist)
+    sigma <- |f'| * |far - x|
+    if sigma is 0 or not finite:  xn <- midpoint(A,B)
+    else:
+        xn <- x + min(1, |f(x)|/sigma) * (far - x)         # the step, eq (4)
+        xn <- clamp(xn, hi - allow, lo + allow)            # projection, eq (3)
+    if xn outside (A,B) or already evaluated:  xn <- midpoint(A,B)
+    evaluate fn = f(xn);   allow <- allow / 2
+    xp2, xp, x <- xp, x, xn
+    update the bracket: whichever of A,B keeps the sign change
+return midpoint(A,B)
+\end{verbatim}
+
+Every quantity in it is either an evaluation of \(f\), a divided
+difference of evaluations, or the halving that defines bisection ---
+with one exception, the factor 2 in \texttt{n\_max}, which §3.1 declares
+as the method's one free constant. The constant-free variant of §3.1
+replaces the two \texttt{allow} lines by the earned credit of (3').
+
+\hypertarget{the-safeguard-project-do-not-replace}{%
+\subsubsection{3.1 The safeguard: project, do not
+replace}\label{the-safeguard-project-do-not-replace}}
+
+An interpolation-only rule has no worst-case bound. The obvious
+safeguards \emph{replace} the computed step with a midpoint whenever
+some progress test fails, and all of them are ruinous here, for a reason
+specific to a fast method: a step that collapses the \emph{error} need
+not halve the \emph{bracket}, so a bracket-based test fires on exactly
+the steps that were best. With the three-point scale, replacing on that
+test drops the observed order from 1.94 to 1.00 and the
+Alefeld--Potra--Shi total from 2806 to 7260 --- worse than bisection.
 
 The remedy is to keep as much of the computed step as the budget allows.
-Let $n_{\max}=2\lceil\log_2(w_0/\varepsilon_x)\rceil$ be twice the bisection
-step count for the initial width $w_0$ and $x$-tolerance $\varepsilon_x$, and
-let $\epsilon=\varepsilon_x/2$. At step $j$, with the bracket $[\ell,h]$,
-define the allowance and the projection
-\begin{equation}
-\label{eq:proj}
-\alpha_j = 2^{\,n_{\max}-j}\epsilon,
-\qquad
-x \;=\; \operatorname{clamp}\bigl(x_{\mathrm{sim}},\; h-\alpha_j,\; \ell+\alpha_j\bigr).
-\end{equation}
-The next bracket is at most $\max(x-\ell,\,h-x)$, so~\eqref{eq:proj} gives
-$w_{j+1}\le\alpha_j$ directly, and induction bounds the run by $n_{\max}$ steps
-in exact arithmetic. In floating point the bracket can land exactly on the
-boundary, and the termination test then asks for one more; the guarantee
-stated and verified is therefore
-$2\lceil\log_2(w_0/\varepsilon_x)\rceil+3$ evaluations.
+Let \(n_{\max}=2\lceil\log_2(w_0/\varepsilon_x)\rceil\) be twice the
+bisection step count for the initial width \(w_0\) and \(x\)-tolerance
+\(\varepsilon_x\), and let \(\epsilon=\varepsilon_x/2\). At step \(j\),
+with the bracket \([\ell,h]\), define the allowance and the projection
+
+\[\alpha_j = 2^{\,n_{\max}-j}\epsilon, \qquad x \;=\; \operatorname{clamp}\bigl(x_\mathrm{sim},\; h-\alpha_j,\; \ell+\alpha_j\bigr). \tag{3}\]
+
+The next bracket is at most \(\max(x-\ell,\,h-x)\), so (3) gives
+\(w_{j+1}\le\alpha_j\) directly, and induction bounds the run by
+\(n_{\max}\) steps in exact arithmetic. In floating point the bracket
+can land exactly on the boundary, and the termination test then asks for
+one more; the guarantee stated and verified is therefore
+\(2\lceil\log_2(w_0/\varepsilon_x)\rceil+3\) evaluations.
+
+\textbf{The multiplier is the method's one free constant.} An earlier
+version of this section argued that it is ``the factor already claimed
+in the bound''. That is circular --- the bound is twice bisection
+because two was chosen --- and measurement settles it
+(\texttt{measure\_budget\_factor.py}, Linux):
+
+\textbf{Table 1a.} The scheduled projection at budget multiplier \(c\),
+i.e.~\(n_{\max}=\lceil c\cdot\log_2(w_0/\varepsilon_x)\rceil\). Total
+evaluations, failures charged.
+
+\begin{longtable}[]{@{}llll@{}}
+\toprule\noalign{}
+\(c\) & APS-154 & CHA-45 & both \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+1 & 4817 & 2997 & 7814 \\
+1.25 & 2806 & 2752 & 5558 \\
+1.5 & 2806 & 2889 & 5695 \\
+2 & 2806 & 3220 & 6026 \\
+3 & 2806 & 3756 & 6562 \\
+4 & 2806 & 4006 & 6812 \\
+\end{longtable}
+
+On the canonical set the projection never binds once \(c\ge1.25\), so
+the total does not move. On the wide-bracket set it moves monotonically
+with \(c\), and 2 is not even the best value. Nor is zero slack
+(\(c=1\)) the degenerate case it was claimed to be, ``a worst case of
+exactly bisection'' under which ``the guard behaves as bisection'': it
+costs 4817 and 2997, far from bisection's 7530 and 3090, and on the
+wide-bracket set it beats \(c=2\). The multiplicative family has no
+member that the bound singles out, so its multiplier is a genuinely free
+constant.
+
+\textbf{The budget earned rather than scheduled: a constant-free
+variant.} The constant can be removed. Let \(s_j=\log_2(w_j/w_{j+1})\)
+be the shrink a step delivers in bits --- bisection delivers exactly 1
+--- and keep a credit
+
+\[C \leftarrow \max(0,\;C+s_j-1),\qquad \alpha_j = w_j\,2^{\,C-1},\qquad x=\operatorname{clamp}\bigl(x_\mathrm{sim},\,h-\alpha_j,\,\ell+\alpha_j\bigr), \tag{3'}\]
+
+leaving the step alone while \(C\ge1\). A step may under-deliver only by
+what earlier steps banked, and since \(C\) never goes negative the
+shrinks over \(j\) steps sum to at least \(j-C_0\): the run costs at
+most bisection plus \(\lceil C_0\rceil\) steps. The endowment \(C_0\) is
+therefore the promise, not a setting. \(C_0=0\) makes the allowance
+exactly half the bracket on the first step, which forces the midpoint
+and makes the method bisection; every \(C_0\in(0,1]\) promises the same
+single extra step; and among those, 1 is the most permissive and ---
+measured --- the cheapest:
+
+\textbf{Table 1b.} The earned projection at endowment \(C_0\). Total
+evaluations, failures charged.
+
+\begin{longtable}[]{@{}lllll@{}}
+\toprule\noalign{}
+\(C_0\) & promise & APS-154 & CHA-45 & both \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+0.1 & bisection + 1 step & 5344 & 3051 & 8395 \\
+0.5 & bisection + 1 step & 4656 & 3002 & 7658 \\
+\textbf{1} & \textbf{bisection + 1 step} & \textbf{3568} & \textbf{3020}
+& \textbf{6588} \\
+2 & bisection + 2 steps & 3692 & 2712 & 6404 \\
+5 & bisection + 5 steps & 3006 & 2615 & 5621 \\
+\end{longtable}
+
+A looser promise buys evaluations, as it must; within one promise the
+cheapest endowment is determined. The constant-free variant is therefore
+the earned projection with \(C_0=1\), whose guarantee ---
+\(\lceil\log_2(w_0/\varepsilon_x)\rceil+4\) evaluations, bisection plus
+one step with one more for the floating-point boundary --- is verified
+on all 199 problems.
+
+We nonetheless propose the scheduled projection with \(c=2\) and declare
+its multiplier. On the canonical set it is 762 evaluations cheaper than
+the earned variant (2806 against 3568), and there its total does not
+depend on the multiplier at all; on the wide-bracket set the earned
+variant is the cheaper (3020 against 3220). The two are the ends of a
+trade stated in numbers: one declared constant and the better canonical
+result, or no constant, a tighter bound and the better wide-bracket
+result. The value 2 was fixed before any benchmark was consulted; what
+changed after the question whether it was load-bearing is only that
+it is now declared as a constant rather than argued to be none.
+
+Two remarks apply to both forms.
 
-Three remarks on why this introduces nothing.
 \begin{itemize}
-\item \textbf{The doubling is not a tuning constant.} It is the factor already
-claimed in the bound. Zero slack --- $n_{\max}=\lceil\log_2(w_0/\varepsilon_x)\rceil$,
-a worst case of exactly bisection --- is degenerate and was measured as such: a
-worst case of exactly bisection demands a halving at \emph{every} step, so the
-projection has no freedom, the radius on a unit bracket is $0.05$ from the
-first step, and the guard behaves as bisection.
-\item \textbf{This is ITP's projection stage without ITP's truncation stage.}
-All three of ITP's constants live in the truncation, whose purpose is to repair
-regula falsi; the similarity step needs no repair, so the truncation and every
-constant with it are dropped.
-\item \textbf{Writing the clamp in the form~\eqref{eq:proj} rather than as a
-radius $\alpha_j-w_j/2$ about the midpoint is not cosmetic.} That radius is a
-difference of two nearly equal numbers; the cancellation made the computed
-radius too large by a relative $8.9\cdot10^{-5}$, which broke the invariant on
-eleven Chandrupatla problems.
+\tightlist
+\item
+  \textbf{This is ITP's projection stage without ITP's truncation
+  stage.} All three of ITP's constants live in the truncation, whose
+  purpose is to repair regula falsi; the similarity step needs no
+  repair, so the truncation and every constant with it are dropped.
+\item
+  \textbf{Writing the clamp in the form (3) rather than as a radius
+  \(\alpha_j-w_j/2\) about the midpoint is not cosmetic.} That radius is
+  a difference of two nearly equal numbers; the cancellation made the
+  computed radius too large by a relative \(8.9\cdot10^{-5}\), which
+  broke the invariant on eleven Chandrupatla problems.
 \end{itemize}
 
-\section{Experimental protocol}
+\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
 
-\paragraph{Problem sets.} The $154$ problems of Alefeld, Potra and Shi~\cite{aps}
-and the $45$ of Chandrupatla~\cite{chandrupatla}, taken programmatically from
+\hypertarget{experimental-protocol}{%
+\subsection{4. Experimental protocol}\label{experimental-protocol}}
+
+\textbf{Problem sets.} The 154 problems of Alefeld, Potra and Shi
+{[}1{]} and the 45 of Chandrupatla {[}3{]}, taken programmatically from
 SciPy's \texttt{scipy.optimize.\_tstutils} so that no bracket or root is
 transcribed by hand.
 
-\paragraph{Cost model.} Evaluations of $f$. A run counts as solved when it
-reports convergence and lands within $10^{-6}$ of the published root;
-$x$-tolerance $10^{-12}$; budget $200$ evaluations. \emph{Failures are charged
-the full budget}, so no method can buy a low total by giving up early, and
-totals are reported over all problems rather than over the subset a method
-happened to solve. Rankings are by failures first, then by total.
+\textbf{Cost model.} Evaluations of \(f\). A run counts as solved when
+it reports convergence and lands within \(10^{-6}\) of the published
+root; \(x\)-tolerance \(10^{-12}\); budget 200 evaluations.
+\emph{Failures are charged the full budget}, so no method can buy a low
+total by giving up early, and totals are reported over all problems
+rather than over the subset a method happened to solve. Rankings are by
+failures first, then by total.
 
-\paragraph{Fairness to the competitors.} Three deliberate concessions.
-(i)~ITP is natively written against a half-width $\epsilon$ and terminates on
-$b-a\le2\epsilon$; feeding the common tolerance in as $\epsilon$ would let it
-stop at twice its competitors' bracket width, so $\epsilon=\varepsilon_x/2$ is
-used. (ii)~ITP is reported both at the authors' recommended constants and as a
-\emph{per-problem oracle}, the best of a $(\kappa_1,n_0)$ grid chosen
-separately for every problem --- not an achievable method, included so that the
-state of the art cannot be said to have been handicapped. (iii)~Our
-implementation of ITP is validated against the method's own theorem, that it
-never needs more than $n_{1/2}+n_0$ iterations, on all $154$ problems for
-$n_0\in\{0,1,2,5\}$.
+\textbf{Fairness to the competitors.} Three deliberate concessions.
 
-\paragraph{Reproducibility.} All numbers come from a single Linux run. The test
-functions call \texttt{libm}, whose last digit differs between platforms; the
-same SciPy version scores $2819$ for TOMS\,748 on Windows and $2820$ on Linux
-for the same set. Every number stated in this paper is checked against the run
-data by a script that is part of the test suite.
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  ITP is natively written against a half-width \(\epsilon\) and
+  terminates on \(b-a\le2\epsilon\); feeding the common tolerance in as
+  \(\epsilon\) would let it stop at twice its competitors' bracket
+  width, so \(\epsilon=\varepsilon_x/2\) is used.
+\item
+  ITP is reported both at the authors' recommended constants and as a
+  \emph{per-problem oracle}, the best of a \((\kappa_1,n_0)\) grid
+  chosen separately for every problem --- not an achievable method,
+  included so that the state of the art cannot be said to have been
+  handicapped.
+\item
+  Our implementation of ITP is validated against the method's own
+  theorem, that it never needs more than \(n_{1/2}+n_0\) iterations, on
+  all 154 problems for \(n_0\in\{0,1,2,5\}\).
+\item
+  Where SciPy ships the method, the competitor \textbf{is the library}:
+  Chandrupatla is \texttt{scipy.optimize.elementwise.find\_root}, Brent
+  is \texttt{brentq}, TOMS 748 is \texttt{toms748}, each run with the
+  same stopping rule as every other method --- a bracket no wider than
+  \(\varepsilon_x\), the relative term at machine precision, no stopping
+  on \(|f|\) --- and each evaluation counted by the same wrapper around
+  \(f\). An earlier version gave each competitor the better of the
+  library's implementation and ours, problem by problem. That concession
+  backfired in the competitor's favour: our Chandrupatla stopped on a
+  bracket twice as wide as everybody else's --- the factor this protocol
+  corrects for ITP in item 1 and had not corrected there --- and the
+  per-problem minimum let the looser run win. With the tolerance
+  corrected our implementation and the library agree on all 199 problems
+  exactly, and Chandrupatla's totals move from 2785 to 2787 on the
+  canonical set and from 2423 to 2441 on the wide-bracket set. For Brent
+  the library is slightly dearer than our implementation (7004 against
+  6991 over both sets); we use the library anyway, because it is the
+  implementation a reader can check. ITP has no library implementation,
+  which is why it is checked against its own theorem instead; bisection
+  and the false-position family are not in SciPy either and are ours.
+\end{enumerate}
 
-\section{Results}
+\textbf{Reproducibility.} All numbers come from a single Linux run. The
+test functions call \texttt{libm}, whose last digit differs between
+platforms; the same SciPy version scores 2819 for TOMS 748 on Windows
+and 2820 on Linux for the same set. Every number stated in this paper is
+checked against the run data by a script that is part of the test suite.
 
-The grid is nine similarity functions $\times$ four scales $\times$ six
-safeguards $=216$ configurations, plus ten baselines. The configuration
-proposed in Section~3 --- ratio similarity, three-point scale, projection
-safeguard --- is one entry of that grid, fixed by the theory rather than chosen
-from the ranking.
+\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
 
-\begin{table}[h]
-\centering
-\caption{Alefeld--Potra--Shi, $154$ problems, out of $226$ configurations.}
-\label{tab:aps}
-\begin{tabular}{rlccrr}
-\toprule
-\# & configuration & tuned & solved & total evals & $>2\times$\\
-\midrule
-1 & \textbf{proposed: ratio, 3-point, projection} & \textbf{no} & 153/154 & \textbf{2806} & 0\\
-3 & TOMS\,748 & yes & 153/154 & 2820 & 0\\
-4 & Brent & yes & 153/154 & 2890 & 0\\
-11 & ITP, per-problem oracle & yes & 153/154 & 3260 & 0\\
-18 & Ridders & no & 153/154 & 3313 & 0\\
-31 & ITP, recommended constants & yes & 153/154 & 3957 & 0\\
-61 & Illinois & yes & 153/154 & 7045 & 0\\
-65 & bisection & no & 153/154 & 7530 & 0\\
-206 & false position & no & 127/154 & 10560 & --\\
-\bottomrule
-\end{tabular}
-\end{table}
+\hypertarget{results}{%
+\subsection{5. Results}\label{results}}
 
-On the problems both solve, the proposed configuration beats Brent
-$102/11/40$ (wins/ties/losses), TOMS\,748 $86/16/51$, ITP at its recommended
-constants $132/3/18$, the per-problem ITP oracle $121/4/28$, and bisection
-$152/1/0$.
+The grid is nine similarity functions × four scales × seven safeguards =
+252 configurations, plus twelve baselines: 264 in all. The base rule of
+Section 3 --- tent similarity, three-point scale, projection safeguard
+with its multiplier declared --- is one entry of that grid, fixed by the
+theory rather than chosen from the ranking; the constant-free earned
+variant of §3.1 is shown beside it. Chandrupatla's own method and
+Dekker's are among the baselines; an earlier version of these tables was
+drawn up before either was added.
 
-\begin{table}[h]
-\centering
-\caption{Chandrupatla, $45$ problems, brackets up to $10^{12}$ wide.}
-\label{tab:cha}
-\begin{tabular}{rlccr}
-\toprule
-\# & configuration & tuned & solved & total evals\\
-\midrule
-1 & similarity, $1-\sqrt{|z|}$, bracket scale, projection & no & \textbf{41/45} & 4516\\
-2 & ITP, per-problem oracle & yes & 40/45 & 2611\\
-3 & similarity, $e^{-|z|}$, bracket scale, both guards & no & 40/45 & 2733\\
-5 & ITP, recommended constants & yes & 40/45 & 2857\\
-6 & bisection & no & 40/45 & 3090\\
-10 & \textbf{proposed: ratio, 3-point, projection} & \textbf{no} & 40/45 & \textbf{3220}\\
-127 & TOMS\,748 & yes & 40/45 & 4000\\
-189 & Brent & yes & 38/45 & 4101\\
-\bottomrule
-\end{tabular}
-\end{table}
+\textbf{Table 2.} Alefeld--Potra--Shi, 154 problems, out of 264
+configurations. Places by failures first, then total.
 
-The set's nominal winner is a similarity configuration that solves one problem
-more than anything else, $41$ of $45$, at a worse total; ranking by failures
-first puts it on top, and we report it as such rather than by the total that
-would flatter our proposal. The proposed configuration is tenth by total and
-beats Brent $30/4/4$ and TOMS\,748 $31/1/8$, while losing to ITP $17/0/23$ and
-to the ITP oracle $16/0/24$.
+\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
+\#
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+configuration
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+tuned
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+solved
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+total evals
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+\(>2\times\)
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+1 & Chandrupatla & no & 153/154 & 2787 & 0 \\
+2 & the same rule with no safeguard at all & no & 153/154 & 2806 & 0 \\
+\textbf{3} & \textbf{base rule: tent, 3-point, projection (\(c=2\))} &
+\textbf{no} & 153/154 & \textbf{2806} & 0 \\
+4 & TOMS 748 & yes & 153/154 & 2820 & 0 \\
+5 & Brent & yes & 153/154 & 2889 & 0 \\
+12 & Dekker & no & 153/154 & 3158 & 0 \\
+13 & ITP, per-problem oracle & yes & 153/154 & 3260 & 0 \\
+20 & Ridders & no & 153/154 & 3313 & 0 \\
+29 & constant-free variant: earned projection & no & 153/154 & 3568 &
+0 \\
+34 & ITP, recommended constants & yes & 153/154 & 3957 & 0 \\
+73 & Illinois & yes & 153/154 & 7045 & 0 \\
+79 & bisection & no & 153/154 & 7530 & 0 \\
+244 & false position & no & 127/154 & 10560 & --- \\
+\end{longtable}
 
-\subsection{What the guarantee costs}
+The base rule and the same rule without any safeguard tie at 2806: the
+projection never binds on this set. By total, Chandrupatla's method is
+first with 2787, 0.7\% fewer --- the same order of margin as the 0.5\%
+by which the base rule leads TOMS 748, which we read as a tie. Problem
+by problem the picture is the other way round: on the problems both
+solve the base rule beats Chandrupatla 87/24/42 (wins/ties/losses; exact
+two-sided sign test \(p=9\cdot10^{-5}\)), TOMS 748 86/16/51, Brent
+102/11/40, ITP at its recommended constants 132/3/18, the per-problem
+ITP oracle 121/4/28, and bisection 152/1/0. It wins twice as many
+problems as it loses against Chandrupatla and trails him in total
+because its losses are the larger ones. The constant-free variant goes
+79/3/71 against Chandrupatla (\(p=0.57\), a tie in problems), 77/9/67
+against TOMS 748 and 78/5/70 against Brent, and loses on the total to
+all three.
 
-Nothing on one set, and less than nothing on the other. Against the identical
-rule with no safeguard at all:
+\textbf{Table 3.} Chandrupatla, 45 problems, brackets up to \(10^{12}\)
+wide, out of 264 configurations.
 
-\begin{center}
-\begin{tabular}{lrrr}
-\toprule
-& solved & total evals & runs over the $2\times$ bound\\
-\midrule
-APS, no safeguard        & 153/154 & 2806 & 0\\
-APS, projection          & 153/154 & 2806 & 0\\
-Chandrupatla, no safeguard & 35/45 & 4104 & 10\\
-Chandrupatla, projection   & \textbf{40/45} & \textbf{3220} & \textbf{0}\\
-\bottomrule
-\end{tabular}
-\end{center}
+\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
+\#
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+configuration
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+tuned
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+solved
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+total evals
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+\(>2\times\)
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+1 & similarity, \(1-\sqrt{\lvert z\rvert}\), bracket scale, projection &
+no & \textbf{41/45} & 4516 & 0 \\
+2 & Chandrupatla & no & 40/45 & 2441 & 0 \\
+3 & ITP, per-problem oracle & yes & 40/45 & 2647 & 0 \\
+4 & similarity, \(e^{-\lvert z\rvert}\), bracket scale, both guards & no
+& 40/45 & 2736 & 0 \\
+6 & ITP, recommended constants & yes & 40/45 & 2896 & 0 \\
+10 & constant-free variant: earned projection & no & 40/45 & 3020 & 0 \\
+16 & bisection & no & 40/45 & 3090 & 0 \\
+\textbf{45} & \textbf{base rule: tent, 3-point, projection (\(c=2\))} &
+\textbf{no} & 40/45 & \textbf{3220} & 0 \\
+159 & TOMS 748 & yes & 40/45 & 4000 & 15 \\
+227 & Brent & yes & 38/45 & 4115 & 18 \\
+228 & the same rule with no safeguard at all & no & 35/45 & 4104 & 10 \\
+\end{longtable}
 
-On the canonical set the projection never binds and the totals are identical to
-the last evaluation. On the wide-bracket set it solves five more problems for
-$21\%$ fewer evaluations. This is the opposite of the usual trade, and it is
-the point of projecting rather than replacing: the two safeguards that replace
-cost $7260$ and $7239$ on Alefeld--Potra--Shi, against $2806$.
+The set's nominal winner is a similarity configuration that solves one
+problem more than anything else, 41 of 45, at a worse total; ranking by
+failures first puts it on top, and we report it as such rather than by
+the total that would flatter our proposal. Here Chandrupatla's method is
+genuinely the stronger: it beats the base rule 26 problems to 9
+(\(p=0.006\)) and by 3220 against 2441 in total. The proposed rule beats
+Brent 30/4/4 and TOMS 748 31/1/8 --- both of which exceed the
+\(2\times\) bound on 18 and 15 problems --- and loses to ITP 17/0/23 and
+to the ITP oracle 16/0/24. The wide-bracket set is where the multiplier
+matters (Table 1a) and where the constant-free variant is the cheaper
+form, 3020 against 3220. Over both sets the base rule needs 6026 and the
+constant-free variant 6588, against 5228 for Chandrupatla, 5907 for the
+ITP oracle, 6820 for TOMS 748 and 7004 for Brent.
 
-\subsection{Wall-clock time, and why it is not the cost model}
+\hypertarget{what-the-guarantee-costs}{%
+\subsubsection{5.1 What the guarantee
+costs}\label{what-the-guarantee-costs}}
 
-Evaluations of $f$ are the cost model because in any real problem $f$ is the
-expensive part. Wall-clock time measures something narrower --- the method's
-own arithmetic per step --- and it is implementation- and language-bound. We
-report it because a reader will ask, and because what it shows is instructive.
+It depends on which bound is bought. Against the identical rule with no
+safeguard at all:
 
-On the $199$ problems as they stand, every $f$ being a formula of a few
-operations, the proposed rule is the \emph{slowest} of the pure-Python methods
-compared, at $32.8$\,ms against $18.4$ for Brent, $21.1$ for ITP, $14.2$ for
-Ridders and $22.9$ for bisection --- while using the fewest evaluations of all
-of them, $4991$ against $5932$, $5675$, $6421$ and $9474$. Bisection, needing
-nearly twice the evaluations, finishes in two thirds of our time.
+\begin{longtable}[]{@{}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}@{}}
+\toprule\noalign{}
+\begin{minipage}[b]{\linewidth}\raggedright
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+solved
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+total evals
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+runs over the \(2\times\) bound
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+APS, no safeguard & 153/154 & 2806 & 0 \\
+APS, projection --- base rule (\(2\times\) bisection) & 153/154 & 2806 &
+0 \\
+APS, earned projection --- constant-free (bisection + 1 step) & 153/154
+& 3568 & 0 \\
+Chandrupatla, no safeguard & 35/45 & 4104 & 10 \\
+Chandrupatla, projection --- base rule & 40/45 & 3220 & 0 \\
+Chandrupatla, earned projection --- constant-free & \textbf{40/45} &
+\textbf{3020} & \textbf{0} \\
+\end{longtable}
 
-That inversion is the point. Making $f$ progressively more expensive by a
-controlled amount of arithmetic and timing the sweep directly --- nothing
-subtracted, nothing modelled --- the proposed rule overtakes ITP once one call
-to $f$ costs about $9\,\mu$s and Brent at about $20\,\mu$s, of the order of a
-hundred arithmetic operations on this machine. Below that the ranking is
-decided by per-step overhead; above it, by evaluations.
+The projection of the base rule never binds on the canonical set, so its
+totals are identical to the last evaluation; on the wide-bracket set it
+solves five more problems for 21\% fewer evaluations. The earned
+projection promises twice as much --- a worst case of bisection plus one
+step instead of twice bisection --- and pays for it on the canonical
+set, 762 evaluations or 27\%, while on the wide-bracket set it is the
+cheapest of the three. Both are the opposite of the usual trade on at
+least one set, and that is the point of projecting rather than
+replacing: the two safeguards that replace cost 7260 and 7239 on
+Alefeld--Potra--Shi, against 2806.
 
-Three caveats, all limitations of the measurement rather than of the methods.
-Timings vary by tens of percent between runs even on an idle machine. An
-attempt to separate our per-step cost into ``algorithm'' and ``Python call
-overhead'', by timing a flattened rewrite of the same rule, produced a variant
-that diverged from the original on $42$ of the $199$ problems and was
-discarded rather than quoted; the question stays open. And an earlier version
-of this measurement estimated a per-evaluation overhead by subtracting the
-measured cost of $f$, which returned \emph{negative} overheads for four of six
-methods --- an impossibility that condemned the estimator, and the reason
-nothing is subtracted here.
+\hypertarget{wall-clock-time-and-why-it-is-not-the-cost-model}{%
+\subsubsection{5.2 Wall-clock time, and why it is not the cost
+model}\label{wall-clock-time-and-why-it-is-not-the-cost-model}}
 
-\subsection{The law across the whole grid}
-\label{sec:grid}
+Evaluations of \(f\) are the cost model because in any real problem
+\(f\) is the expensive part. Wall-clock time measures something narrower
+--- the method's own arithmetic per step --- and it is implementation-
+and language-bound. We report it because a reader will ask, and because
+what it shows is instructive.
 
-Corollaries~\ref{cor:chosen}--\ref{cor:derived} predict an ordering of the nine
-similarity functions and of the four scales. Table~\ref{tab:grid} reports it.
-The prediction holds without exception for every data-derived scale: the three
-similarities with $c=1$ occupy the top three places and the three with $c=0$
-the bottom three. For the chosen scale the two groups are indistinguishable
-($16102$ against $15345$), which is consistent --- with a chosen scale nothing
-converges well and the ordering carries no information.
+On the 199 problems as they stand, every \(f\) being a formula of a few
+operations, the base rule is the \textbf{slowest} of the pure-Python
+methods compared, at 32.8 ms against 18.4 for Brent, 21.1 for ITP, 14.2
+for Ridders and 22.9 for bisection --- while using the fewest
+evaluations of all of them, 4991 against 5932, 5675, 6421 and 9474.
+Bisection, needing nearly twice the evaluations, finishes in two thirds
+of our time.
 
-\begin{table}[h]
-\centering
-\caption{Mean total evaluations over both sets, step guard, grouped by the
-slope $c$ of $1-S$ at the origin.}
-\label{tab:grid}
-\begin{tabular}{lrr}
-\toprule
-scale & $c=1$ (linear approach) & $c=0$ (quadratic approach)\\
-\midrule
-3-point derived  & \textbf{6809} & 22294\\
-2-point derived  & 6906 & 22167\\
-bracket          & 10263 & 19137\\
-start (chosen)   & 16102 & 15345\\
-\bottomrule
-\end{tabular}
-\end{table}
+Those evaluation counts are \emph{raw}: they are what the timing loop
+actually spent, and a failed run contributes only the evaluations it
+made before the budget ran out. They are therefore not the totals of §5,
+where a failure is charged the full 200-evaluation budget so that no
+method can buy a low score by giving up --- 4991 raw against 6026
+charged for the base rule, 5932 against 6991 for Brent (our pure-Python
+implementation, which is what is timed; the tables use the library's
+\texttt{brentq}, 7004), 9474 against 10620 for bisection. Wall-clock
+time can only be compared against what was really executed, so the raw
+counts are the right companions to the milliseconds; every comparison in
+§5 uses the charged ones. Both are stated to avoid the appearance of two
+incompatible sets of numbers for the same methods.
 
-Ranked individually and best over scales, the similarities come out $1-|z|$
-($6496$), $e^{-|z|}$ ($6822$), $1/(1+|z|)$ ($7038$), $(1+|z|)^{-2}$ ($9502$),
-$1-\sqrt{|z|}$ ($10497$), $1-|z|^{1/3}$ ($11941$), $e^{-z^2}$ ($14314$),
-$1/(1+z^2)$ ($15102$), $1/\sqrt{1+z^2}$ ($16394$); the scales come out 3-point
-($6496$), 2-point ($6628$), bracket ($9081$), start ($14540$). Both orderings
-are those the law predicts, and both were predicted before the grid was run.
+That inversion is the point. Making \(f\) progressively more expensive
+by a controlled amount of arithmetic and timing the sweep directly ---
+nothing subtracted, nothing modelled --- the base rule overtakes ITP
+once one call to \(f\) costs about 9 µs and Brent at about 20 µs, which
+on this machine is of the order of a hundred arithmetic operations.
+Below that the ranking is decided by per-step overhead; above it, by
+evaluations.
 
-\section{Where the evaluations go}
+Three caveats, all of them limitations of the measurement rather than of
+the methods. Timings vary by tens of percent between runs even on an
+idle machine. An attempt to separate our per-step cost into
+``algorithm'' and ``Python call overhead'', by timing a flattened
+rewrite of the same rule, produced a variant that diverged from the
+original on 42 of the 199 problems and was therefore discarded rather
+than quoted; the question stays open. And an earlier version of this
+measurement estimated a per-evaluation overhead by subtracting the
+measured cost of \(f\), which returned \emph{negative} overheads for
+four of six methods --- an impossibility that condemned the estimator,
+and the reason nothing is subtracted here.
+
+\hypertarget{the-law-across-the-whole-grid}{%
+\subsubsection{5.3 The law across the whole
+grid}\label{the-law-across-the-whole-grid}}
+
+Corollaries 1--3 predict an ordering of the nine similarity functions
+and of the four scales. Table 4 reports it. The prediction holds without
+exception for every data-derived scale: the three similarities with
+\(c=1\) occupy the top three places and the three with \(c=0\) the
+bottom three. For the chosen scale the two groups are indistinguishable
+(16102 against 15345), which is consistent --- with a chosen scale
+nothing converges well and the ordering carries no information.
+
+\textbf{Table 4.} Mean total evaluations over both sets, step guard,
+grouped by the slope \(c\) of \(1-S\) at the origin.
+
+\begin{longtable}[]{@{}lll@{}}
+\toprule\noalign{}
+scale & \(c=1\) (linear approach) & \(c=0\) (quadratic approach) \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+3-point derived & \textbf{6893} & 22315 \\
+2-point derived & 6981 & 22206 \\
+bracket & 10279 & 19127 \\
+start (chosen) & 16102 & 15345 \\
+\end{longtable}
+
+Ranked individually and best over scales, the similarities come out
+\(1-|z|\) (6560), \(e^{-|z|}\) (6895), \(1/(1+|z|)\) (7118),
+\((1+|z|)^{-2}\) (9578), \(1-\sqrt{|z|}\) (10546), \(1-|z|^{1/3}\)
+(12002), \(e^{-z^2}\) (14307), \(1/(1+z^2)\) (15102), \(1/\sqrt{1+z^2}\)
+(16394); the scales come out 3-point (6560), 2-point (6702), bracket
+(9107), start (14540). Both orderings are those the law predicts, and
+both were predicted before the grid was run.
+
+\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
+
+\hypertarget{where-the-evaluations-go}{%
+\subsection{6. Where the evaluations
+go}\label{where-the-evaluations-go}}
 
 Splitting each run at the geometric mean of the initial width and the
-attainable tolerance --- the log-midpoint of the journey, at which bisection by
-construction has spent half its evaluations --- separates the \emph{approach}
-from the \emph{endgame}. On Alefeld--Potra--Shi the two-point rule spends
-$2235$ on the approach and $668$ on the endgame against Brent's $2175$ and
-$515$: its deficit is almost entirely the endgame, i.e.\ the order, which is
-what motivated the three-point scale. On Chandrupatla it spends $1068$ and
-$1161$ against Brent's $1343$ and $1358$, winning both phases.
+attainable tolerance --- the log-midpoint of the journey, at which
+bisection by construction has spent half its evaluations --- separates
+the \emph{approach} from the \emph{endgame}. On Alefeld--Potra--Shi the
+two-point rule spends 2235 on the approach and 668 on the endgame
+against Brent's 2175 and 515: its deficit is almost entirely the
+endgame, i.e.~the order, which is what motivated the three-point scale.
+On Chandrupatla it spends 1068 and 1161 against Brent's 1343 and 1358,
+winning both phases.
 
 Two natural explanations of the Chandrupatla advantage were tested and
-\emph{refuted}. Brent discards $53.3\%$ of its interpolations to its safeguard
-on the set it wins and only $34.0\%$ on the set it loses, so safeguard rate is
-not the mechanism; and on Chandrupatla Brent is the bolder of the two ---
-$56.3\%$ of its probes land in the outer tenths of the bracket against our
-$41.8\%$ --- so step length is not the mechanism either.
+\emph{refuted}. Brent discards 53.3\% of its interpolations to its
+safeguard on the set it wins and only 34.0\% on the set it loses, so
+safeguard rate is not the mechanism; and on Chandrupatla Brent is the
+bolder of the two --- 56.3\% of its probes land in the outer tenths of
+the bracket against our 41.8\% --- so step length is not the mechanism
+either.
 
-The mechanism is step \emph{quality}. Reconstructing each method's bracket from
-the signs it observed, the median probe removes $0.500$ of the bracket for
-every method on Alefeld--Potra--Shi, but on Chandrupatla it removes $0.275$ for
-the similarity rule against $0.058$ for Brent and $0.068$ for TOMS\,748 ---
-and that comparison already flatters them, since $69$ and $141$ of their probes
-respectively fall outside the proven bracket and are excluded, against none of
-ours. High-order interpolation is a wager that $f$ is locally polynomial. When
-the wager pays, order $1.84$ or $2.65$ wins the endgame; when it fails, the
-probes are wasted. The similarity rule makes the weakest available assumption
---- a single local slope --- and degrades gracefully.
+The mechanism is step \emph{quality}. Reconstructing each method's
+bracket from the signs it observed, the median probe removes 0.500 of
+the bracket for every method on Alefeld--Potra--Shi, but on Chandrupatla
+it removes 0.275 for the similarity rule against 0.058 for Brent and
+0.068 for TOMS 748 --- and that comparison already flatters them, since
+69 and 141 of their probes respectively fall outside the proven bracket
+and are excluded, against none of ours. High-order interpolation is a
+wager that \(f\) is locally polynomial. When the wager pays, order 1.84
+or 2.65 wins the endgame; when it fails, the probes are wasted. The
+similarity rule makes the weakest available assumption --- a single
+local slope --- and degrades gracefully.
 
-\section{Where the similarity is read}
-\label{sec:placement}
+\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
 
-Everything above reads the similarity in one place: the \emph{length} of the
-step, $(1-S(z))$ times the distance to the far end. But a bracketing iteration
-takes several decisions per step, and the similarity was not consulted in any
-of the others: which end counts as far (decided by distance), which points the
-local slope is estimated from (decided by recency), and whether the step is
-worth taking at all (decided after the fact, by the safeguard). Each of those
-is a place where a similarity can be read, and the placement turns out to
-matter more than the form of $S$.
+\hypertarget{where-the-similarity-is-read}{%
+\subsection{7. Where the similarity is
+read}\label{where-the-similarity-is-read}}
 
-Three readings were added, each parameter-free and each measured on the same
-protocol; every number below is re-checked against its sweep file by
-\texttt{verify\_article.py}, and every configuration keeps the
-$2\lceil\log_2(w_0/\varepsilon_x)\rceil+3$ guarantee ($0$ violations) and solves
-$193$ of the $199$ problems.
+Everything above reads the similarity in one place: the \emph{length} of
+the step, \((1-S(z))\) times the distance to the far end. But a
+bracketing iteration takes several decisions per step, and the
+similarity was not consulted in any of the others: which end counts as
+far (decided by distance), which points the local slope is estimated
+from (decided by recency), and whether the step is worth taking at all
+(decided after the fact, by the safeguard). Each of those is a place
+where a similarity can be read, and the placement turns out to matter
+more than the form of \(S\).
+
+Three readings were added to the base rule of §3 (tent similarity,
+three-point scale), each parameter-free and each measured on the
+protocol of §4; every number below is re-checked against its own sweep
+file by \texttt{verify\_article.py}, and every configuration in Table 5
+keeps the \(2\lceil\log_2(w_0/\varepsilon_x)\rceil+3\) guarantee (0
+violations) and solves 193 of the 199 problems.
+
+Where a safeguard other than the projection of §3.1 appears, it is one
+of the grid's. \emph{safe} is the projection preceded by a step-length
+test --- a step not at most half the step taken two evaluations earlier
+is replaced by a halving --- with the projection still carrying the
+bound. \emph{credit} replaces the scheduled allowance of (3) by one
+earned step by step: a surplus is banked whenever a step shrinks the
+bracket by more than half and drawn on when it shrinks it by less, which
+bounds the run by bisection plus one step. \emph{bracket} and
+\emph{both} are the replacing safeguards of §3.1, which keep a bound at
+the price of the order; \emph{step} alone and \emph{none} carry no
+bound.
 
 \begin{enumerate}
-\item \textbf{Trust before the step.} The step trusts a local slope, so the
-question of trust is whether that slope has settled: if two consecutive
-estimates have opposite signs the local model is not established, and the
-evaluation is spent halving instead. A sign is compared, not a magnitude.
-\item \textbf{Which points the model uses.} For a linear $f$ the ratio of
-positions $\xi=(x-u)/(v-u)$ and the ratio of values
-$\varphi=(f(x)-f(u))/(f(v)-f(u))$ coincide, so $|\varphi-\xi|$ measures how
-unlike a linear configuration a triple is. Among the declared candidate triples
-the one minimising it is \emph{selected} --- an argmin over what is already in
-hand, with no threshold.
-\item \textbf{Which end is far.} Read by dissimilarity instead of distance: the
-end whose $|f|$ is larger is the one less similar to the target.
+\def\labelenumi{\arabic{enumi}.}
+\tightlist
+\item
+  \textbf{Trust before the step.} The step trusts a local slope, so the
+  question of trust is whether that slope has settled: if two
+  consecutive estimates have opposite signs the local model is not
+  established, and the evaluation is spent halving instead. A sign is
+  compared, not a magnitude.
+\item
+  \textbf{Which points the model uses.} For a linear \(f\) the ratio of
+  positions \(\xi=(x-u)/(v-u)\) and the ratio of values
+  \(\varphi=(f(x)-f(u))/(f(v)-f(u))\) coincide, so \(|\varphi-\xi|\)
+  measures how unlike a linear configuration a triple is. Among the
+  declared candidate triples the one minimising it is \emph{selected}
+  --- an argmin over what is already in hand, with no threshold.
+\item
+  \textbf{Which end is far.} Read by dissimilarity instead of distance:
+  the end whose \(|f|\) is larger is the one less similar to the target.
 \end{enumerate}
 
-\begin{table}[h]
-\centering
-\begin{tabular}{lrrr}
-\hline
-configuration (all with the guarantee, $193/199$) & APS-154 & CHA-45 & both \\
-\hline
-proposed method of \S\ref{sec:method} (credit) & $3568$ & $3020$ & $6588$ \\
-projection variant & $2806$ & $3220$ & $6026$ \\
-\hline
-(1) trust before the step & $2848$ & $3074$ & $5922$ \\
-(1)+(2) points chosen by shape & $\mathbf{2527}$ & $3108$ & $5635$ \\
-(1)+(3) far end by dissimilarity & $2726$ & $\mathbf{2785}$ & $\mathbf{5511}$ \\
-(1)+(2)+(3) & $2884$ & $2859$ & $5743$ \\
-\hline
-Chandrupatla & $2785$ & $2423$ & $5208$ \\
-TOMS\,748 & $2820$ & $4000$ & $6820$ \\
-ITP (oracle over its constants) & $3260$ & $2611$ & $5871$ \\
-\hline
-\end{tabular}
-\caption{Reading the similarity in one more place is worth more than changing
-$S$. Evaluations of $f$; failures charged the full budget.}
-\end{table}
+\textbf{Table 5.} Reading the similarity in one more place is worth more
+than changing \(S\). Evaluations of \(f\); failures charged the full
+budget.
 
-Two results here are worth stating plainly. First, the canonical set improves
-from $2806$ to $\mathbf{2527}$, which is below every method measured, including
-Chandrupatla's $2785$, and it is obtained with the worst-case guarantee intact
-and still without a tuning constant. Second, on the two sets together the
-$5511$ of reading~(3) passes ITP's oracle ($5871$) but not Chandrupatla
-($5208$): the remaining gap is entirely on the wide-bracket families, where
-Chandrupatla needs $2423$ against our $2683$.
+\begin{longtable}[]{@{}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}@{}}
+\toprule\noalign{}
+\begin{minipage}[b]{\linewidth}\raggedright
+configuration (all with the guarantee, 193/199)
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+APS-154
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+CHA-45
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+both
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+base rule: tent, 3-point, projection \(c=2\) (§3) & 2806 & 3220 &
+6026 \\
+constant-free variant: earned projection (§3.1) & 3568 & 3020 & 6588 \\
+(1) trust before the step, projection & 2848 & 3114 & 5962 \\
+(1)+(2) points chosen by shape, \emph{safe} & \textbf{2527} & 3126 &
+5653 \\
+(1)+(3) far end by dissimilarity, projection & 2726 & \textbf{2799} &
+\textbf{5525} \\
+(1)+(2)+(3), \emph{safe} & 2884 & 2871 & 5755 \\
+Chandrupatla & 2787 & 2441 & 5228 \\
+TOMS 748 & 2820 & 4000 & 6820 \\
+ITP, per-problem oracle & 3260 & 2647 & 5907 \\
+\end{longtable}
 
-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.
+Two results here are worth stating plainly. First, the canonical set
+improves from 2806 to \textbf{2527}, which is below every method
+measured, including Chandrupatla's 2787, with the worst-case guarantee
+intact. It hardly feels the declared multiplier of §3.1: 2527 for every
+multiplier from 1.5 to 4, 2541 at 1.25
+(\texttt{measure\_records\_vs\_multiplier.py}). Second, on the two sets
+together the 5525 of reading (3) passes ITP's oracle (5907) but not
+Chandrupatla (5228): the remaining gap is entirely on the wide-bracket
+families, where Chandrupatla needs 2441 against our 2695 (reading (3)
+under \emph{safe}).
 
-\subsection{The similarity function belongs to the place, not to the method}
-\label{sec:perplace}
+The readings do not compose. Taken together, (2) and (3) give 2884 and
+2871 --- worse than either alone on the canonical set, and on the
+wide-bracket set worse than (3) alone. Two selections that are each an
+improvement interfere, which is a reason to measure composition rather
+than assume it.
 
-Everything above varies \emph{where} the similarity is read while keeping one
-function $S$ for the whole method. That is itself an assumption, and it is
-measurable. Proposition~\ref{prop:falsi} says the two-point normalisation is
-false position; what it does not say is \emph{why} false position stalls. Write
-$|f|\sim K e^m$ near the current point, $e=|x-r|$. Then the residual ratio
-against the far end needs no slope at all,
-\begin{equation}
-\label{eq:orderz}
-z \;=\; \frac{|f(x)|}{|f(x_{\mathrm{far}})|} \;=\; \Bigl(\frac{e}{E}\Bigr)^{m},
-\qquad\text{so}\qquad z^{1/m} \;=\; \frac{e}{E},
-\end{equation}
-and under $S=1/(1+z)$ the step fraction is $z/(1+z)$, which is exactly inverse
-linear interpolation. False position therefore reads a dimensionless ratio
-\emph{at order one} on an approach of order $m$: its three-century-old stall is
-a mismatch between the similarity function and the place it is read in. The
-exponent $1/m$ is not a constant --- $m=1/(1-ff''/f'^2)$ comes from three
-evaluations already paid for, and is $1$ whenever the power model does not
-apply.
+\textbf{Which similarity stands in the step decides whether a reading
+pays.} The readings above, and the decision \emph{defect} of §7.2,
+compare quantities directly --- an ordinal judgment of which is more
+similar, in which every decreasing \(S\) would decide alike. Their gains
+nevertheless depend on the \(S\) that sets the step
+(\texttt{audit\_similarity\_places.py}, on the same sweeps): on the
+canonical set the triple chosen by shape gains 493 evaluations with the
+tent in the step and loses 250 with \(1/\sqrt{1+z^2}\); \emph{defect}
+gains 283 with the tent and loses 215 with \(1-|z|^{1/3}\). The records
+of this section are the pairing of the tent step with these judgments,
+not the judgments alone.
 
-Measured over the $252$ cells of the grid, the correction behaves exactly as
-that reading predicts and not at all as a universal improvement. Paired inside
-each (function, safeguard) cell of the far reading it is cheaper in $29$ of
-$60$ cells --- indistinguishable from a coin. But in the one pairing where the
-far reading \emph{is} exact inverse interpolation, $S=1/(1+z)$, it is cheaper
-in all seven safeguards:
+Under the constant-free earned projection the same readings gain far
+less. Reading (1) gives 3507 and 2997, (1)+(2) 3755 and 3021, (1)+(3)
+3857 and 3002, all three together 4354 and 3037, against 3568 and 3020
+for the constant-free variant itself: the earned budget binds early on
+the canonical problems, and what the readings buy under the projection
+is mostly room that the earned budget does not grant.
 
-\begin{table}[h]
-\centering
-\begin{tabular}{lrrrr}
-\toprule
-safeguard & APS $m{=}1$ & APS measured $m$ & CHA $m{=}1$ & CHA measured $m$\\
-\midrule
-both    & 4522 & \textbf{3675} & 3387 & \textbf{3216}\\
-bracket & 4606 & \textbf{3542} & 3571 & \textbf{3303}\\
-credit  & 5190 & \textbf{4686} & 3013 & \textbf{2884}\\
-project & 6932 & \textbf{6364} & 4794 & \textbf{4559}\\
-safe    & 5987 & \textbf{5229} & 4108 & \textbf{3923}\\
-step    & 6435 & \textbf{5551} & 4257 & \textbf{4022}\\
-none    & 10676 & \textbf{10043} & 8265 & \textbf{7794}\\
-\bottomrule
-\end{tabular}
-\caption{The measured order in the step place, on the far reading. Cheaper in
-$7$ of $7$ safeguards for the similarity that matches the place, and in $29$ of
-$60$ cells over all nine functions.}
-\end{table}
+\hypertarget{the-similarity-function-belongs-to-the-place-not-to-the-method}{%
+\subsubsection{7.1 The similarity function belongs to the place, not to
+the
+method}\label{the-similarity-function-belongs-to-the-place-not-to-the-method}}
 
-Two limits must be stated with it. The estimator is \emph{biased low}: on a
-pure cube, with no noise at all, it returns $2.830$ rather than $3$, because a
-parabola through three points carries only the second difference. The bias is
-towards $m=1$, that is towards the uncorrected reading, which is the harmless
-direction, but it is a bias. And the correction does not rescue the bracket
-reading as a method: the best cell it reaches on the wide-bracket set is
-$2884$, against $2683$ for reading~(3) above and $2423$ for Chandrupatla. In
-the \emph{slope} reading the same exponent is harmful, $2806\to2854$, which is
-the same statement read from the other side --- a function that does not belong
+Everything above varies \emph{where} the similarity is read while
+keeping one function \(S\) for the whole method. That is itself an
+assumption, and it is measurable. Proposition 1 says that the ratio
+similarity scaled by the far end is false position; what it does not say
+is \emph{why} false position stalls. Write \(|f|\sim K e^m\) near the
+current point, \(e=|x-r|\). Then the residual ratio against the far end
+needs no slope at all,
+
+\[z \;=\; \frac{|f(x)|}{|f(x_\mathrm{far})|} \;=\; \Bigl(\frac{e}{E}\Bigr)^{m}, \qquad\text{so}\qquad z^{1/m} \;=\; \frac{e}{E}, \tag{5}\]
+
+and under the ratio similarity \(S=1/(1+|z|)\) the step fraction is
+\(z/(1+z)\), which is exactly inverse linear interpolation. False
+position therefore reads a dimensionless ratio \emph{at order one} on an
+approach of order \(m\): its three-century-old stall is a mismatch
+between the similarity function and the place it is read in. The
+exponent \(1/m\) is not a constant --- \(m=1/(1-ff''/f'^2)\),
+Lagouanelle's estimate of the multiplicity {[}13{]} with the derivatives
+replaced by divided differences, comes from three evaluations already
+paid for, as in item 5 of §10, and is 1 whenever the power model does
+not apply.
+
+Measured over the 252 cells of the grid, the correction behaves exactly
+as that reading predicts and not at all as a universal improvement.
+Paired inside each (function, safeguard) cell of the far-end reading it
+is cheaper in 29 of 60 cells --- indistinguishable from a coin. But in
+the one pairing where that reading \emph{is} exact inverse
+interpolation, the ratio similarity, it is cheaper under all seven
+safeguards:
+
+\textbf{Table 6.} The measured order in the step place, on the far-end
+reading with the ratio similarity \(1/(1+|z|)\). Cheaper in 7 of 7
+safeguards; in 29 of 60 cells over all nine functions.
+
+\begin{longtable}[]{@{}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}@{}}
+\toprule\noalign{}
+\begin{minipage}[b]{\linewidth}\raggedright
+safeguard
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+APS, \(m=1\)
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+APS, measured \(m\)
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+CHA, \(m=1\)
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+CHA, measured \(m\)
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+both & 4522 & \textbf{3675} & 3406 & \textbf{3241} \\
+bracket & 4606 & \textbf{3542} & 3602 & \textbf{3359} \\
+credit & 5190 & \textbf{4686} & 3013 & \textbf{2884} \\
+projection & 6932 & \textbf{6364} & 4789 & \textbf{4554} \\
+safe & 5987 & \textbf{5229} & 4134 & \textbf{3954} \\
+step & 6435 & \textbf{5551} & 4287 & \textbf{4068} \\
+none & 10676 & \textbf{10043} & 8265 & \textbf{7794} \\
+\end{longtable}
+
+Two limits must be stated with it. The estimator is \emph{biased low}:
+on a pure cube, with no noise at all, it returns 2.830 rather than 3,
+because a parabola through three points carries only the second
+difference. The bias is towards \(m=1\), that is towards the uncorrected
+reading, which is the harmless direction, but it is a bias. And the
+correction does not rescue the far-end reading as a method: the best
+cell it reaches on the wide-bracket set is 2884, against 2695 for
+reading (3) above and 2441 for Chandrupatla. In the \emph{slope} reading
+of §3 the same exponent is harmful, \(2806\to2854\), which is the same
+statement read from the other side --- a function that does not belong
 to a place does not become right by being applied there.
 
-\subsection{The decision place, and the one test that settles the gap}
-\label{sec:decision}
+\hypertarget{the-decision-place}{%
+\subsubsection{7.2 The decision place}\label{the-decision-place}}
 
-Three places have now been measured: the step length, the far end, and the
-points the model is built on. The fourth is the \emph{decision} --- whether the
-local model deserves this evaluation or halving is the better use of it.
-Chandrupatla's method is strong exactly there: it refuses its model on most
-steps by the acceptance band $1-\sqrt{1-\xi}<\varphi<\sqrt{\xi}$, where
-$\xi=(x-u)/(v-u)$ and $\varphi=(f(x)-f(u))/(f(v)-f(u))$ are ratios of positions
-and of values, equal for a linear $f$. Our method has no similarity in this
-place at all: a sign gate and a projection that repairs \emph{after} the step.
+Three places have now been measured: the step length, the far end, and
+the points the model is built on. The fourth is the \emph{decision} ---
+whether the local model deserves this evaluation or halving is the
+better use of it. Chandrupatla's method is strong exactly there: it
+refuses its model on most steps by the acceptance band
+\(1-\sqrt{1-\xi}<\varphi<\sqrt{\xi}\), with \(\xi\) and \(\varphi\) the
+ratios of positions and of values defined above. The rule of §3 has no
+similarity in this place at all: a sign gate and a projection that
+repairs \emph{after} the step.
 
-Three readings of that place were measured, each on quantities already in hand
-and each without a constant. Two are ours: refuse when the configuration is
-less like a linear one than the iterate is off-centre,
-$|\varphi-\xi|>\min(\xi,1-\xi)$ (``defect''); and refuse when the
-configuration's similarity to linear falls below the iterate's similarity to
-the root, $S(|\varphi-\xi|)<S(z)$, with the \emph{same} $S$ in both places
-(``simpair''). The third is \emph{Chandrupatla's own band}, placed inside our
-step rule. It is not a proposal; it is an instrument, and it answers the only
-question that matters here --- whether the remaining gap lives in this place.
+Three readings of that place were measured, each on quantities already
+in hand and each without a constant. Two are ours: refuse when the
+configuration is less like a linear one than the iterate is off-centre,
+\(|\varphi-\xi|>\min(\xi,1-\xi)\) (``defect''); and refuse when the
+configuration's similarity to linear falls below the iterate's
+similarity to the root, \(S(|\varphi-\xi|)<S(z)\), with the \emph{same}
+\(S\) in both places (``simpair''). The third is \emph{Chandrupatla's
+own band}, placed inside our step rule. It is not a proposal; it is an
+instrument, and it answers whether the remaining gap lives in this
+place.
 
-In total, no. With Chandrupatla's band in our step, the wide-bracket set costs
-$3066$ under the projection guard and $2899$ under credit, against his $2423$,
-while the canonical set rises from $2806$ to $5432$. The band is not idle ---
-it refuses $63.6\%$ of steps --- so this is a decision that was taken and did
-not pay \emph{in total}. Per family it did pay, and where: it closes almost the
-whole gap on the flat families and gives back more than that on the families
-where our step is good. \S\ref{sec:widelimit} takes this apart; the short form is
-that the decision place does hold most of the gap, but no reading measured so far
-makes that decision correctly for \emph{our} step.
+In total, no. With Chandrupatla's band in our step, the wide-bracket set
+costs 3059 under the projection and 2892 under \emph{credit}, against
+his 2441, while the canonical set rises from 2806 to 5432. The band is
+not idle --- it refuses 63.5\% of steps --- so this is a decision that
+was taken and did not pay \emph{in total}. Per family it did pay, and
+where: it closes almost the whole gap on the flat families and gives
+back more than that on the families where our step is good. §7.4 takes
+this apart; the short form is that the decision place does hold most of
+the gap, but no reading measured so far makes that decision correctly
+for \emph{our} step.
 
-Our own readings do not close it either (best $2868$ on the wide-bracket set),
-but ``defect'' sets a new best on the canonical set: $\mathbf{2523}$ at
-$153/154$, below the $2527$ of reading~(2) and below every method measured,
-with the guarantee intact and no constant. It refuses only $7.0\%$ of steps,
-the fewest of the three. Over both sets it costs $5733$ against the $5511$ of
-reading~(3), because it gives back on the wide brackets ($3210$) what it gains
-on the canonical problems --- so it is a second point on the same trade, not a
-replacement.
+Our own readings do not close it either (best 2897 on the wide-bracket
+set), but ``defect'' sets a new best on the canonical set: \textbf{2523}
+at 153/154 under the projection of the base rule, below the 2527 of
+reading (2) and below every method measured, with the guarantee intact
+--- and, unlike 2527, the same for every multiplier from 1.25 to 4.
+Under the earned projection it gives 3545 and 2949 (6494 against 6588),
+the best of the readings of this section without a constant; the flat
+regime of §8 does better still. It refuses only 6.6\% of steps, the
+fewest of the three. Over both sets it costs 5733 against the 5525 of
+reading (3), because it gives back on the wide brackets (3210) what it
+gains on the canonical problems --- so it is a second point on the same
+trade, not a replacement.
 
-\subsection{Six problems of the two sets are scored unsolved for every method}
-\label{sec:fun7}
+\hypertarget{six-problems-of-the-two-sets-are-scored-unsolved-for-every-method}{%
+\subsubsection{7.3 Six problems of the two sets are scored unsolved for
+every
+method}\label{six-problems-of-the-two-sets-are-scored-unsolved-for-every-method}}
 
-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.
+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: 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.
+The consequence is arithmetic. Each such problem charges the full budget
+of 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}
+\textbf{Table 7.} 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:
+235 evaluations, or 16\%, not 10\%.
 
-\subsection{Where the wide-bracket limit actually is}
-\label{sec:widelimit}
+\begin{longtable}[]{@{}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}@{}}
+\toprule\noalign{}
+\begin{minipage}[b]{\linewidth}\raggedright
+method
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+APS as scored
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+APS re-scored
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+CHA as scored
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+CHA re-scored
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+ours, (1)+(2) points by shape & \textbf{2527} & \textbf{2337} & 3108 &
+2159 \\
+ours, base rule (projection) & 2806 & 2634 & 3220 & 2357 \\
+Chandrupatla & 2787 & 2595 & \textbf{2441} & \textbf{1505} \\
+TOMS 748 & 2820 & 2636 & 4000 & 3088 \\
+Brent & 2889 & 2707 & 4115 & 3238 \\
+ITP & 3957 & 3767 & 2896 & 1907 \\
+bisection & 7530 & 7338 & 3090 & 2136 \\
+\end{longtable}
 
-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 $2.3$.
+\hypertarget{where-the-wide-bracket-limit-actually-is}{%
+\subsubsection{7.4 Where the wide-bracket limit actually
+is}\label{where-the-wide-bracket-limit-actually-is}}
 
-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 remaining gap looks diffuse --- per family +57, +49, +42, +39, +30
+and smaller --- but part of 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 about one halving per evaluation while Chandrupatla advances by about
+2.3.
 
-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 $87.5\%$.
+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.
 
-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.
+Nor is it that the step fails to be relative to the bracket. With the
+tent similarity \(1-S=\min(1,|f|/(|\hat f'|D))\), so by (4) the step is
+exactly \(\min(D,|f|/|\hat f'|)\): 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\%.
 
-That mechanism is right for \texttt{fun1} and \emph{wrong as a statement about the
-whole gap}, which an earlier version of this section made. Replaying our own
-trajectories on the $33$ problems where we lose, and placing both models on the
-\emph{same} evaluations --- one extra evaluation each, from the same state ---
-splits the $262$ evaluations of the gap in two. On \texttt{fun1} ($58$ of them)
-inverse quadratic interpolation from our own three points removes far more of
-the bracket than our step did: $10.37$ halvings against $0.62$ on the
-$2\times10^4$-wide bracket. There the gap is the order of the model, and the
-reason is structural: our step $x+(1-S)(x_{\mathrm{far}}-x)=S\,x+(1-S)\,x_{\mathrm{far}}$
-is a convex combination of two points, while the quadratic interpolant's
-Lagrange weights at $0$ always include exactly one \emph{negative} weight when
-the triple brackets the root (verified on all $428$ non-degenerate states; in
-the other five the negative weight underflowed to $-0$). A similarity in $[0,1]$
-cannot produce a negative weight --- Proposition~\ref{prop:falsi} extended from
-two points to three.
+What differs on \texttt{fun1} is how much each probe removes. Ours
+shrinks the bracket by 0.93 halvings per evaluation there,
+Chandrupatla's by 2.30. 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; a model that carries the curvature --- inverse quadratic
+interpolation --- lands far closer.
 
-But $178$ of the $262$ --- two thirds --- sit on \texttt{fun3}--\texttt{fun6},
-which are pure powers of order $3$, $5$, $9$ and $19$, and there \emph{no}
-interpolating model works: our step removes $0.41$--$0.81$ halvings of the
-bracket per evaluation, inverse quadratic interpolation from the same points only
-$0.01$--$0.17$. Chandrupatla does not win those families with his model --- his
-traced run bisects $93$, $96$, $100$ and $100\%$ of his steps. We lose them
-because our step is slower than bisection there and our guarantee, at most twice
-bisection, licenses exactly that. His band inside our step confirms it family by
-family: under the projection guard it takes \texttt{fun4}, \texttt{fun5},
-\texttt{fun6} from $573$, $456$, $456$ to $296$, $239$, $235$ --- against his own
-$293$, $228$, $228$ --- and costs $+164$, $+111$, $+170$, $+144$ on \texttt{fun1},
-\texttt{fun2}, \texttt{fun8}, \texttt{fun9}, where it refuses steps that were good.
+That mechanism is right for \texttt{fun1} and \emph{wrong as a statement
+about the whole gap}. Replaying our own trajectories on the 33 problems
+where we lose, and placing both models on the \emph{same} evaluations
+--- one extra evaluation each, from the same state --- splits the 256
+evaluations of the gap in two. On \texttt{fun1} (58 of them, about a
+quarter) inverse quadratic interpolation from our own three points
+removes far more of the bracket than our step did: 10.37 halvings
+against 0.62 on the \(2\times10^4\)-wide bracket. There the gap is the
+order of the model, and the reason is structural: our step
+\(x+(1-S)(x_\mathrm{far}-x)=S\,x+(1-S)\,x_\mathrm{far}\) is a convex
+combination of two points, while the quadratic interpolant's Lagrange
+weights at 0 always include exactly one \emph{negative} weight when the
+triple brackets the root (verified on all 428 non-degenerate states; in
+the other three the negative weight underflowed to \(-0\)). A similarity
+in \([0,1]\) cannot produce a negative weight --- Proposition 1 extended
+from two points to three.
 
-So most of what remains is not the order of the model but the decision \emph{when
-to give the model up}, and it is open because every detector measured so far
-confounds ``the model is useless here'' with something else: the band with ``the
-configuration is not shaped for inverse quadratic interpolation'', bracket
-progress with ``the step has not moved the bracket yet'' ($4449$ on the canonical
-set), and the local order with nothing at all --- $m\ge2$ is exactly the
-condition under which a tangent step loses to a halving, since it lands at
-$e(1-1/m)$, and it is clean ($0.2\%$ of canonical states) but blind: it reports
-$m\ge2$ on only $5.8\%$ of states of families whose order is by construction $3$
-to $19$. For the \texttt{fun1} part, three parameter-free ways round the model
-order have 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.
+But 172 of the 256 --- two thirds --- sit on
+\texttt{fun3}--\texttt{fun6}, which are pure powers of order 3, 5, 9 and
+19, and there \emph{no} interpolating model works: our step removes
+0.41--0.75 halvings of the bracket per evaluation, inverse quadratic
+interpolation from the same points only 0.01--0.17. Chandrupatla does
+not win those families with his model --- his traced run bisects 92, 96,
+100 and 100\% of his steps. We lose them because our step is slower than
+bisection there and our guarantee, at most twice bisection, licenses
+exactly that. His band inside our step confirms it family by family:
+under the projection it takes \texttt{fun4}, \texttt{fun5},
+\texttt{fun6} from 573, 456, 456 to 296, 239, 228 --- against his own
+297, 233, 233 --- and costs +164, +111, +170, +144 on \texttt{fun1},
+\texttt{fun2}, \texttt{fun8}, \texttt{fun9}, where it refuses steps that
+were good.
 
-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.
+So most of what remains is not the order of the model but the decision
+\emph{when to give the model up}, and it is open because every detector
+measured so far confounds ``the model is useless here'' with something
+else: the band with ``the configuration is not shaped for inverse
+quadratic interpolation'', bracket progress with ``the step has not
+moved the bracket yet'' (4449 on the canonical set, §10), and the local
+order with nothing at all --- \(m\ge2\) is exactly the condition under
+which a tangent step loses to a halving, since it lands at \(e(1-1/m)\),
+and it is clean (0.2\% of canonical states) but blind: it reports
+\(m\ge2\) on none of the states of families whose order is by
+construction 3 to 19. For the \texttt{fun1} part, three parameter-free
+ways round the model order have 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 (the cube-root similarity), a logarithmic
+change of variable, and the curvature-corrected scale of §10 item 5. The
+fourth way is to model the curvature directly, which is inverse
+quadratic interpolation --- a different method rather than another
+reading of this one. Section 8 shows that the decision is reachable
+without a new model: by a similarity of the run to its own past.
 
-\section{Negative results}
+On the re-scored wide-bracket set of Table 7 our shape reading (2159) 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 §§7.3--7.4 is reproduced by
+\texttt{diag\_rescore\_cha.py}, \texttt{diag\_rescore\_aps.py},
+\texttt{diag\_fun7\_root.py}, \texttt{diag\_cha\_gap.py},
+\texttt{diag\_model\_order.py}, \texttt{diag\_model\_order\_checks.py},
+\texttt{diag\_flat\_detectable.py} and
+\texttt{diag\_band\_by\_family.py}, whose outputs are kept beside them.
+
+\hypertarget{the-flat-regime-recognised-by-analogy}{%
+\subsection{8. The flat regime, recognised by
+analogy}\label{the-flat-regime-recognised-by-analogy}}
+
+§7.4 left one decision open: \emph{when to give the model up}. On the
+pure powers \texttt{fun3}--\texttt{fun6} the right move is to halve,
+Chandrupatla halves on up to all of his steps, and every reading of a
+single state measured so far confuses ``the model is useless here'' with
+something else. This section closes that decision, and the reading that
+closes it is not a similarity of the current state to a prototype but a
+similarity of the run to \emph{its own past} --- an analogy in Pólya's
+sense {[}8{]}, two situations agreeing in more than one relation. Every
+rule below was pre-registered before its code and its run, with a stop
+rule; the rejected ones are listed in §10 with their numbers.
+
+\hypertarget{the-flag-a-promise-that-was-not-kept}{%
+\subsubsection{8.1 The flag: a promise that was not
+kept}\label{the-flag-a-promise-that-was-not-kept}}
+
+A tangent step on an approach of local order \(m\) lands at \(e(1-1/m)\)
+and leaves the residual \(r=|f_\text{new}/f_\text{old}|=(1-1/m)^m\) on
+the same side of the root. It beats a halving exactly when
+\(|1-1/m|<\tfrac12\), that is for \(\tfrac23<m<2\). At the upper end
+\(r=(1-\tfrac12)^2=\tfrac14\). So a model step that did not cross the
+root and left \(r\ge\tfrac14\) is a \emph{promise not kept}: the model
+is in the regime where a halving would have done better. The constant is
+bisection's \(\tfrac12\), carried through the formula. The relation
+between the order and the contraction, \(e(1-1/m)\), is the one behind
+the classical estimates of multiplicity from successive Newton iterates
+{[}14, 15{]}; here it is not used to estimate \(m\) but to set the point
+at which a model step has lost to a halving.
+
+The flag sees the regime --- on the flat families it rises on most model
+steps --- but acted upon step by step it does not pay. Replacing the one
+flagged step by a halving costs the canonical set 2806 → 3029, because a
+simple root far from convergence also breaks its promises (the flag
+rises at least once on 142 of the 154 canonical problems). Every
+refinement of \emph{when} to halve that reads the moment alone --- twice
+in a row, the promise together with a creep in one direction, a halving
+that sticks once begun, a judgment of the evaluated point against the
+line through the bracket --- trades one set for the other (§10, items
+12--14).
+
+\hypertarget{the-analogy-similarity-of-a-run-to-its-own-past}{%
+\subsubsection{8.2 The analogy: similarity of a run to its own
+past}\label{the-analogy-similarity-of-a-run-to-its-own-past}}
+
+What distinguishes a pure power from a simple root is not the shape of
+one configuration --- \texttt{fun1} on a wide bracket looks as
+non-linear as \(x^9\) --- but what happens under zoom. A pure power
+\(f=c(x-r)^m\) is similar to itself: zooming about the root by \(L\)
+scales \(x-r\) by \(L\) and \(f\) by \(L^m\), so the \emph{dimensionless
+configuration} of the last three points,
+
+\[\xi=\frac{x_k-x_{k-1}}{x_{k-2}-x_{k-1}},\qquad \varphi=\frac{f_k-f_{k-1}}{f_{k-2}-f_{k-1}},\]
+
+recurs as the iteration proceeds. At a simple root, zooming in makes
+\(f\) look like its tangent, and the configuration tends to the
+\emph{linear prototype} \(\varphi=\xi\) instead. Two prototypes, then:
+the line, and the run's own past. The judgment is ordinal --- is the
+current \((\xi,\varphi)\) nearer to some earlier configuration of this
+run than to the line? --- with the plane's Euclidean distance and the
+perpendicular to the line, the one geometric choice in this section,
+declared as such.
+
+Nearness of configurations is one relation. An analogy in the full sense
+needs a second, and the pair supplies it: if the analogous pair shrank
+\(x\) by \(L_x\) and \(f\) by \(L_f\), the power law requires
+\(L_f=L_x^{\,m}\), so the order is \emph{read from the analogy itself},
+
+\[m=\frac{\ln L_f}{\ln L_x}.\tag{6}\]
+
+Formula (6) is the log--log slope by which any power-law exponent is
+estimated, and has the shape of the computational order of convergence
+and its approximation from differences of iterates {[}18, 19{]}. What is
+not classical is \emph{which} two states it compares: not consecutive
+iterates, but a pair the similarity of their configurations has found.
+Under an exact zoom a pure power of order \(m\) reads exactly \(m\);
+along a real run the analogous pairs are only nearly homothetic and the
+reading scatters about the order, but the regime asks only whether it
+reaches 2, the order from which a tangent step loses to a halving (§8.1)
+--- no new constant. On the traces of the base rule, at the moments the
+flag rises, the three conditions together hold on 3 of the 154 canonical
+problems, on all 20 flat problems, and on none of \texttt{fun2},
+\texttt{fun8}, \texttt{fun9}.
+
+\textbf{The analogy belongs to the run, the flag to the moment.} In the
+cycle the base rule runs on a pure power --- undershoot, undershoot,
+crossing, a nearly empty step --- the flag rises in the phases whose
+configuration does \emph{not} recur (the empty step grows emptier every
+cycle), while the recurring phases have the flag down. Requiring both in
+one state delayed the entry on the flat problems to their thirteenth
+state on average. Once established, the analogy is therefore remembered
+for the run, and the regime is entered at the next unkept promise
+(eighth state on average).
+
+\textbf{The translation limit.} The exponential is self-similar by
+\emph{translation}, \(e^{x-1}=e^x/e\): along its flank the step does not
+shrink while \(f\) does, so (6) is undefined or negative. That is the
+limit \(m\to\infty\) of (6), steeper than any power, and it is read as
+\(m=\infty\): when an analogous pair shows \(L_f<1\le L_x\). A high
+power far from its root is locally an exponential, \(x^m=e^{m\ln x}\),
+and it was there, not on the flank of \texttt{fun9} it was built for,
+that the limit paid.
+
+\textbf{Exit.} Inside the regime the step splits the bracket (§8.3). At
+every split the evaluated \(f\) is judged against two predictions: the
+line through the bracket ends, \(F_A(1-t)+F_Bt\) at the split fraction
+\(t\), and 0. Nearer the line, the regime is left and the remembered
+analogy is cleared. A sticky halving without this exit cost the
+canonical set 6897; with it, 3125 --- the exit is what makes a false
+entry cheap.
+
+\hypertarget{where-to-split-the-point-from-the-order}{%
+\subsubsection{8.3 Where to split: the point from the
+order}\label{where-to-split-the-point-from-the-order}}
+
+Halving is the answer when nothing is known about the shape; inside the
+regime the order is known. For \(|f|\propto|x-r|^m\) the ends of the
+bracket lie at distances in the ratio \(|F_A|^{1/m}:|F_B|^{1/m}\) from
+the root, so the regime splits at
+
+\[t=\frac{|F_A|^{1/m}}{|F_A|^{1/m}+|F_B|^{1/m}},\tag{7}\]
+
+which is exact for a pure power and becomes the midpoint as
+\(m\to\infty\). It is false position applied to
+\(\operatorname{sign}(f)|f|^{1/m}\) --- 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.
+
+\hypertarget{five-more-readings}{%
+\subsubsection{8.4 Five more readings}\label{five-more-readings}}
+
+\textbf{The half the step promises.} \(S(z)\ge\tfrac12\) means a step of
+at most half the way to the far end: the model promises the root in the
+half of the bracket \emph{near} \(x\). The line through the bracket ends
+puts it nearer the end with the smaller \(|f|\). When the far end has
+the smaller \(|f|\) the two disagree about the half, and the evaluation
+halves. On a steep flank --- a slope dominated by
+\(f(81)=1.5\cdot10^{35}\) on \texttt{fun9.5} --- this is the false
+promise ``the root is here'', and removing it alone takes the canonical
+set from 2806 to 2564. It is a relative of Brent's acceptance condition
+{[}2{]}, which admits an interpolant only between the best point and
+\((3a+b)/4\): both refuse a model that points towards the worse end of
+the bracket.
+
+\textbf{The sign contradiction.} The bracket says which way \(f\) must
+go between its ends; the last two points say which way it goes here.
+Opposite signs --- the local picture contradicts the bracket --- and the
+evaluation halves. It is the cause of the \texttt{fun9} well: a
+near-cancelling end reads the root as close while \(|f|\) doubles for
+seventeen steps.
+
+\textbf{The repeated step pattern.} Two successive moves the same way,
+of similar length, mean the iterate creeps along a geometric series
+whose remainder is \(\text{step}\cdot s/(1-s)\); the step is lengthened
+by \(1/(1-s)\), with \(s\) the similarity of successive step lengths,
+but only when the pattern of successive length ratios repeats with a
+period read from the data (the lag of highest mean similarity) and that
+similarity exceeds \(\tfrac12\).
+
+\textbf{The overshoot branch.} §8.1's inequality has a second branch,
+\(m<\tfrac23\): a root of order below one, where the tangent step
+overshoots and the iterate oscillates about the root
+(\(\sqrt{|x|}\operatorname{sign}x\): 84 evaluations of the base rule).
+The flag for it is the unkept promise \emph{with} a sign change,
+\(f_\text{new}/f_\text{old}\le-\tfrac14\), and an analogous pair reading
+\(0<m\le\tfrac23\); the split (7) is again exact for a pure power. The
+exit judgment of §8.2 separates a flat root from a simple one; at a root
+of order below one \(f\) is far from 0 on both sides and the exit fires
+falsely, so the memory of this branch is \emph{not} cleared by it;
+clearing it was measured on the development corpora and was worse
+(recorded in the pre-registration amendment of this branch).
+
+\textbf{Borrowed: the tolerance step.} Once the root is reached from one
+side the bracket's far end can stay far: on \(\sin\) over \([3,4]\) the
+base rule reaches \(|f|\approx10^{-16}\) in 7 evaluations and then
+halves the remaining \(1.3\cdot10^{-7}\) down to the tolerance in 17
+more. Dekker {[}9{]} and Brent {[}2{]} step at least the tolerance; we
+do the same, \(\varepsilon_x/2\) towards the far end, when the model's
+step is shorter --- but only after the model has \emph{kept} its
+previous promise (\(r<\tfrac14\)), because at the start of a run a slope
+dominated by a huge far value promises the root at an end. This element
+is not ours and is named as such.
+
+\hypertarget{the-proposed-method}{%
+\subsubsection{8.5 The proposed method}\label{the-proposed-method}}
+
+The method this paper proposes, \textbf{A′}, is the base rule of §3 ---
+tent similarity, three-point scale, projection with the declared
+multiplier \(c=2\) --- with the regime of §8.2--8.3 (the analogy
+remembered, its translation limit, the split (7), the exit) and the five
+readings of §8.4. Its constants are bisection's \(\tfrac12\) and what
+follows from it: \(\tfrac14=(1-\tfrac12)^2\) and the two branches
+\(m\ge2\), \(m\le\tfrac23\) of \(|1-1/m|>\tfrac12\). Three choices are
+declared rather than derived: the multiplier \(c\) of the projection
+(§3.1), the Euclidean geometry of the \((\xi,\varphi)\) plane, and the
+pattern test of the step lengthening (lags from 2 to half the history,
+threshold \(\tfrac12\)). The projection carries the guarantee,
+\(2\lceil\log_2(w_0/\varepsilon_x)\rceil+3\) evaluations; it is verified
+with no violation on every corpus of this paper and under every
+safeguard of the grid.
+
+\textbf{Table 8.} A′ on the protocol of §4 (published roots, failures
+charged 200). Per problem against each rival: wins/ties/losses, exact
+two-sided sign test.
+
+\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
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+APS-154
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+CHA-45
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+both
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+vs A′ per problem, APS
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+vs A′ per problem, CHA
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+\textbf{A′ (proposed)} & \textbf{2438} & \textbf{2005} & \textbf{4443} &
+--- & --- \\
+A′, earned budget (no free constant) & 3608 & 2807 & 6415 & & \\
+base rule (§3) & 2806 & 3220 & 6026 & & \\
+Chandrupatla & 2787 & 2441 & 5228 & A′ 103/20/30, \(p=1.5\cdot10^{-10}\)
+& A′ 29/4/7, \(p=3\cdot10^{-4}\) \\
+TOMS 748 & 2820 & 4000 & 6820 & A′ 127/6/20 & A′ 38/2/0 \\
+Brent & 2889 & 4115 & 7004 & A′ 118/17/18 & A′ 33/3/2 \\
+ITP, per-problem oracle & 3260 & 2647 & 5907 & A′ 126/7/20 & A′
+35/1/4 \\
+ModAB {[}10, 20{]} & 2914 & 2436 & 5350 & A′ 118/11/25 & A′ 32/9/4 \\
+\end{longtable}
+
+A′ is 12.5\% below Chandrupatla on the canonical set and 17.9\% below on
+the wide-bracket set, and beats every rival problem by problem on both.
+On the wide-bracket families it is ahead on seven of nine, level on
+\texttt{fun7} (the scoring artefact of §7.3) and behind on
+\texttt{fun9}, 65 against 55. The earned-budget form keeps its
+constant-free guarantee of bisection plus one step and pays for it as in
+§3.1.
+
+\textbf{Table 9.} Leave-one-out ablation of A′ on all six corpora of §9
+(uniform criterion, a sign change within \(10^{-6}\)). Δ = total without
+the component − total with it; positive means the component earns its
+place.
+
+\begin{longtable}[]{@{}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 14\tabcolsep) * \real{0.1250}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 14\tabcolsep) * \real{0.1250}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 14\tabcolsep) * \real{0.1250}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 14\tabcolsep) * \real{0.1250}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 14\tabcolsep) * \real{0.1250}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 14\tabcolsep) * \real{0.1250}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 14\tabcolsep) * \real{0.1250}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 14\tabcolsep) * \real{0.1250}}@{}}
+\toprule\noalign{}
+\begin{minipage}[b]{\linewidth}\raggedright
+component removed
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+APS
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+CHA
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+new-1
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+Roots.jl
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+applied
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+fourth
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+sum
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+the whole regime & −7 & +1013 & +1660 & +324 & +77 & +2269 &
+\textbf{+5336} \\
+the split (7) (midpoint instead) & −5 & +479 & +667 & +272 & +30 & +1672
+& \textbf{+3115} \\
+overshoot branch & −6 & 0 & −7 & +16 & +63 & +893 & +959 \\
+tolerance step (borrowed) & +130 & +35 & +97 & +78 & +135 & −2 & +473 \\
+the half the step promises & +249 & +65 & +82 & −16 & −5 & +1 & +376 \\
+translation limit & +31 & +143 & +100 & −21 & −2 & +87 & +338 \\
+repeated step pattern & −22 & −6 & +123 & −30 & −10 & +34 & +89 \\
+sign contradiction & +11 & −12 & +29 & 0 & +1 & −1 & +28 \\
+\end{longtable}
+
+Every component is positive in sum; the regime and the split carry the
+method. Two are marginal and we say so: the step pattern earns on two
+corpora of six and loses on four, and the sign contradiction is within
+noise (+28 over 855 problems). Removing both would change A′'s total by
+under one per cent (117 of 13032 evaluations); they are kept because the
+pre-registered rule removes only a component that earns nowhere.
+
+\textbf{Table 9a.} The similarity in the step of A′: the same method
+with each of the nine similarity functions of §5.3 in place of the tent,
+everything else unchanged. Totals on the protocol of §4 and over the six
+corpora of §9 (uniform criterion); the guarantee holds with every
+function.
+
+\begin{longtable}[]{@{}lll@{}}
+\toprule\noalign{}
+\(S\) in the step & APS + CHA & six corpora (855) \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+\textbf{tent \(1-|z|\) (A′)} & \textbf{4443} & \textbf{13032} \\
+\(e^{-|z|}\) & 4988 & 14142 \\
+\(1/(1+|z|)\) & 5486 & 14843 \\
+\((1+|z|)^{-2}\) & 8360 & 28500 \\
+\(1-\sqrt{|z|}\) & 12718 & 45552 \\
+\(1-|z|^{1/3}\) & 17207 & 63031 \\
+\(e^{-z^2}\) & 18410 & 67701 \\
+\(1/\sqrt{1+z^2}\) & 18412 & 67854 \\
+\(1/(1+z^2)\) & 18422 & 67394 \\
+\end{longtable}
+
+The similarity in the step earns its place in the proposed method as it
+did in the base rule: the nearest other function costs 12\% more on the
+canonical protocol and 9\% more over the six corpora, the functions that
+approach 1 quadratically cost four to five times more, and the ranking
+is the one the contraction law of §2.2 predicts, the three functions
+with a linear approach first and the three with a quadratic approach
+last.
+
+\hypertarget{what-is-known-and-what-is-new}{%
+\subsubsection{8.6 What is known, and what is
+new}\label{what-is-known-and-what-is-new}}
+
+Several mechanisms of A′ coincide with classical ones, and the
+coincidence is worth stating, because they were not taken from the
+literature but reached from the principles of the paper --- similarity
+to a prototype and analogy as similarity of relations --- and recognised
+afterwards:
+
+\begin{longtable}[]{@{}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}}@{}}
+\toprule\noalign{}
+\begin{minipage}[b]{\linewidth}\raggedright
+in A′
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+classical counterpart
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+the three-point order estimate of §7.1 & Lagouanelle's multiplicity
+estimate \(m=f'^2/(f'^2-ff'')\) {[}13{]} \\
+the contraction \(e(1-1/m)\) behind the flag of §8.1 & multiplicity
+estimates from successive Newton iterates {[}14, 15{]} \\
+the order (6) of an analogous pair & the log--log slope of a power law;
+the (approximate) computational order of convergence {[}18, 19{]} \\
+the split (7) & false position on \(\operatorname{sign}(f)|f|^{1/m}\)
+{[}16, 17{]} \\
+the half the step promises & a relative of Brent's acceptance condition
+{[}2{]} \\
+the tolerance step & Dekker {[}9{]}, Brent {[}2{]} --- borrowed and
+named as such \\
+the exit judgment of §8.2 (the value at a split against the line) &
+ModAB's linearity check at the midpoint {[}10, 20{]},
+\(|y_m-y_3|<k\,(|y_3|+|y_m|)\) with \(k\) from a symmetry factor; ours
+compares two predictions, the line and 0, without a factor \\
+\end{longtable}
+
+What we found no counterpart for, in the literature we could reach, is
+the part that carries the gain in Table 9: the \emph{decision} layer. A
+bracketing method that recognises the regime in which its model has lost
+by the similarity of the run to its own past --- the recurrence of a
+dimensionless configuration against the linear prototype --- reads the
+order from the analogous pair itself, distinguishes similarity by
+scaling from similarity by translation, and leaves the regime by a
+similarity judgment at a point the model did not choose; and a step
+whose length is set by a similarity function whose form, measured, is
+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{]}.
+
+\hypertarget{out-of-sample-evidence}{%
+\subsection{9. Out-of-sample evidence}\label{out-of-sample-evidence}}
+
+Every decision above was taken on the two sets of §4. Pre-registration
+protects each decision but not the chain: the final configuration was
+selected on the same 199 problems. Four further corpora were therefore
+used, each fixed and committed before any method was run on it, each
+with its criteria and stop rule written down first.
 
 \begin{enumerate}
-\item \textbf{Placement beats form, and the wrong placement is expensive.} The
-same ``refuse before the step'' idea, with refusal decided by \emph{promised
-bracket progress} instead of by the shape of the model, costs $4449$ on the
-canonical set against $2527$ --- a difference of $1922$ evaluations for one
-choice of criterion. A secant step routinely collapses the error while barely
-moving the proven bracket, so bracket progress rejects steps that were
-excellent; this is the same mechanism that makes the bracket guard cost $7260$.
-\item \textbf{Making the scale independent of the bracket does not help either.}
-The natural objection to the derived scale is that it defeats the very principle
-of a similarity step: with $\sigma=|f'|D$ the normalised residual is
-$|f|/(|f'|D)$, which on the $2\times10^4$-wide bracket of $x^3-2x-5$ at
-$x=10^4$ equals $0.167$, so $S=0.833$ --- ``similar'' --- and the step is small
-although the root is some five thousand step-lengths away. The similarity is
-reading what fraction of the \emph{bracket} the Newton step is, not how far the
-iterate is from the root. Selecting the scale between the two quantities already
-computed, $\sigma=\min(|f'|D,\max(|f(A)|,|f(B)|))$ --- no new constant, a
-\texttt{min} over what is in hand --- does give a full-bracket step far away and
-Newton's step near the root, and on eight hand-picked wide problems it improves
-$403$ evaluations to $390$. Over the whole grid it is worse on both sets:
-$3288$ on the canonical set against $2726$, and $3111$ on the wide-bracket set
-against $2683$. Its apparent extra solves ($194$ instead of $193$) are two
-\texttt{fun7} problems where it lands inside the flat zero region close enough to
-satisfy the harness --- the artefact of \S\ref{sec:fun7}, not a capability.
-\item \textbf{Reading the similarity at the measured order does not cure false
-position.} Since false position's stall is a similarity read at order one
-(\S\ref{sec:perplace}), the obvious hope is that reading it at the measured
-order is the parameter-free version of the classical fix. It is not. Unguarded,
-the correction takes the bracket reading from $127/154$ and $10676$ to $128/154$
-and $10043$ --- one problem and six per cent --- where Illinois, at the price of
-one constant (halve the stale ordinate), reaches $153/154$ and $7045$. A sixth
-of the gap is not the gap. The correction is real and it is in the right place;
-it is simply much smaller than a constant that was chosen for exactly this
-failure. \emph{Our} $10676$ and the $10560$ of Table~\ref{tab:aps} are not
-the same algorithm --- our iterate starts at the midpoint of the bracket, plain
-false position at an end --- and the two must not be differenced.
-\item \textbf{A logarithmic change of variable does not buy the wide brackets.}
-Working in $u=\operatorname{sign}(x-c)\log(1+|x-c|/\varepsilon_x)$ makes the
-step multiplicative, which looked like the right diagnosis of the flat families
---- for the same similarity, scale and safeguard CHA improves from $3643$ to
-$3474$ --- but the canonical set gets worse, $5822$ to $6940$, against the $2806$
-of the projection variant. An earlier version compared the $3474$ with $4516$,
-the total of a different configuration. A march needs a \emph{ratio} of distances while an identity on narrow
-brackets needs an \emph{absolute} unit, and no parameter-free unit provides
-both.
-\item \textbf{The prototype-aggregation form of the method loses.} A richer
-construction --- prototypes over \emph{pairs} of evaluated points, softmax
-weights over the whole memory, similarity on both the value and the abscissa
-axis --- leads on eight hand-picked problems but loses decisively on the real
-set: $0$ wins, $6$ ties and $147$ losses against Brent. The simplest form of
-the idea won.
-\item \textbf{Every safeguard that replaces the step destroys the order.}
-Bracket-based and step-based replacement both drop the three-point rule from
-order $1.94$ to $1.00$ and cost $7260$ and $7239$ evaluations on
-Alefeld--Potra--Shi. Only projection preserves both. We record this because it
-is the trap the construction naturally falls into.
-\item \textbf{Zero slack in the projection is degenerate.} A worst case of
-exactly bisection demands a halving at every step and leaves the projection no
-freedom at all.
-\item \textbf{Regimes exist, at the level of a function family, and cannot be
-exploited by a switch.} The cube-root similarity beats the proposed rule on
-all five members of Chandrupatla's \texttt{fun4} --- $32/32/40/47/59$ against
-$89/93/113/125/153$, a factor of $2.6$ --- and consistency across a family at
-different parameters is the signature of structure rather than luck. But the
-same similarity over the whole Chandrupatla set costs $4099$ against $3220$,
-so a switch chosen per benchmark is worse than no switch at all: $6905$
-against $6026$ over both sets. The scale axis offers only $5.3\%$ headroom
-even to an oracle handed the best scale per problem, two thirds of it inside a
-single family, and the winners are not separated by anything the method can
-see before it commits --- the proposed scale wins on bracket widths from
-$10^0$ to $10^{12}$.
-
-\item \textbf{The phase detector: derivation sound, estimator not.} Redoing
-the contraction law without the assumption $|f|\approx|f'(r)|e$ --- taking
-$|f|\approx K e^m$ instead --- produces the march exponent on its own,
-$p = L/(L-\ln m)$ with $L=\ln(e/D)$, giving $p=1$ at a simple root and dropping
-to $0.50$ and $0.23$ as the approach flattens, which is the neighbourhood of
-the cube root that wins those families. Substituted back it collapses to the
-multiplier $m$, measurable as $m = 1/(1-\theta)$ with $\theta = f f''/(f')^2$
-from three divided differences and no constant. It preserves the order
-($2.001$) and improves the eight standard problems ($82$ against $84$) --- and
-loses on the full benchmark, $6304$ against $6026$. The estimator is the
-failure: on Alefeld--Potra--Shi, where almost every root is simple and the
-truth is $m=1$, it reports $m>1$ on $27.9\%$ of steps with excursions to
-$m=34.8$, against $35.8\%$ on the flat families. $27.9$ against $35.8$ is no
-separation. A cruder variant, correcting by the fraction of its own promise
-the previous step kept, fails everything: the order collapses to $0.904$ and
-the standard problems double to $161$.
-
-\item \textbf{One problem defeats every method}, \texttt{aps.13.00},
-$f(x)=x e^{-1/x^2}$, all of whose derivatives vanish at the root and which
-underflows to exactly zero throughout $|x|<0.086$; every method returns a
-machine zero at distance up to $0.086$ from the root.
-\item \textbf{The symmetric ratio similarity degenerates.} For
-$S(x,y)=\min(|x|,|y|)/\max(|x|,|y|)$ with $y$ the target $0$, $S\equiv0$; the
-only usable reading is $1-|f|/\sigma$, which saturates and stagnates
-(Table~\ref{tab:law-check}, last row).
+\def\labelenumi{\arabic{enumi}.}
+\tightlist
+\item
+  \textbf{New corpus 1} --- 206 problems: SciPy's \texttt{original} set
+  (6) and five generated classes of 40 (seed 20260921): smooth simple
+  roots, transcendental simple roots, multiple roots \(|x-r|^p\) with
+  \(p\sim U(2,20)\), exponential flanks, and near-cancelling terms of
+  the \texttt{fun9} kind.
+\item
+  \textbf{Roots.jl / ModAB} --- the published bracketing benchmark of
+  the Julia package Roots.jl {[}12{]}, taken from the ModAB benchmark of
+  Ganchovski and Traykov {[}10{]}: 91 problems, of which 13 coincide
+  with APS or Chandrupatla functions (a seven-point identity test) and
+  are excluded, leaving 78, many of them discontinuous or pathological.
+\item
+  \textbf{Applied and GSL} --- the 10 bracketed problems of the GNU
+  Scientific Library's root-finding test suite {[}11{]} and 122
+  equations from physics and engineering on grids fixed beforehand:
+  Kepler's equation (eccentricity up to 0.999), Colebrook--White,
+  Lambert's W, Planck/Wien, van der Waals for CO₂ (including
+  near-critical), internal rate of return, the finite square well.
+\item
+  \textbf{Corpus 4} --- 240 problems in six new families (seed
+  20260922): random polynomials with a simple root, polynomials with a
+  multiple root, fractional powers \(|x-r|^p\) with
+  \(p\sim U(0.15,0.95)\), sums of exponentials and sines, compositions
+  \(\tanh\), \(\arctan\), \(\log(1+q^2)\) of a cubic, and functions odd
+  about the root. The code of A′ was committed \emph{before} this corpus
+  was run, and nothing was changed after.
 \end{enumerate}
 
-\section{Conclusion}
+ModAB, the most recent bracketing method we know of, is run from its
+authors' package (\texttt{pymodab} 1.0.9, a wrapper of their C code)
+with the stopping rule of every other method --- a bracket no wider than
+\(\varepsilon_x\), the relative term at machine precision, no stopping
+on \(|f|\) --- and counted by the same wrapper around \(f\), whose count
+agrees with the package's own. Where roots are not known in closed form,
+and for uniformity on all six corpora, a run is solved when \(f\)
+changes sign within \(10^{-6}\) of its answer; the \(x\)-tolerance, the
+budget and the charge for a failure are those of §4.
+
+\textbf{The history is part of the result.} The variant carried to
+corpora 1--3 was A′ without the tolerance step and the overshoot branch.
+It transferred to corpus 1 by every pre-registered criterion (total 10\%
+below Chandrupatla, per problem 105/34/67, \(p=0.005\)), did \emph{not}
+transfer to Roots.jl (total 6.5\% below, but per problem 38/10/30,
+\(p=0.4\)) and lost on the applied corpus (14\% above). The losses there
+were traced to two gaps of the base step --- no closing of the bracket
+once the root is reached, and no recognition of roots of order below one
+--- which became the last two readings of §8.4; they were developed on
+the seen corpora and tested once, on corpus 4.
+
+\textbf{Table 10.} A′ against the library rivals on all six corpora
+(uniform criterion): total evaluations; in brackets the runs that break
+the \(2\times\)-bisection bound.
+
+\begin{longtable}[]{@{}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.2619}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1429}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1667}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1429}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1429}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 10\tabcolsep) * \real{0.1429}}@{}}
+\toprule\noalign{}
+\begin{minipage}[b]{\linewidth}\raggedright
+corpus
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+A′
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+Chandrupatla
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+ModAB
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+Brent
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+TOMS 748
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+APS (154) & \textbf{2259} (0) & 2595 (0) & 2726 (0) & 2707 (0) & 2636
+(0) \\
+Chandrupatla (45) & \textbf{1074} (0) & 1505 (0) & 1681 (0) & 3238 (18)
+& 3088 (15) \\
+new corpus 1 (206) & \textbf{3290} (0) & 3716 (0) & 3809 (1) & 6395 (38)
+& 5746 (22) \\
+Roots.jl (78) & \textbf{1371} (0) & 1566 (0) & 1446 (0) & 2455 (11) &
+2339 (9) \\
+applied + GSL (132) & 1382 (0) & 1382 (0) & 1460 (0) & 1503 (1) & 1756
+(1) \\
+\textbf{corpus 4 (240), the one test} & \textbf{3656} (0) & 5717 (0) &
+5535 (8) & 8267 (50) & 8084 (37) \\
+all 855 & \textbf{13032} & 16481 & 16657 & 24565 & 23649 \\
+\end{longtable}
+
+\textbf{Table 10b.} The same runs problem by problem: A′'s
+wins/ties/losses against Chandrupatla and against ModAB (exact two-sided
+sign test).
+
+\begin{longtable}[]{@{}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3333}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.4545}}
+  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.2121}}@{}}
+\toprule\noalign{}
+\begin{minipage}[b]{\linewidth}\raggedright
+corpus
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+A′ vs Chandrupatla
+\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
+A′ vs ModAB
+\end{minipage} \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+APS (154) & 103/20/31, \(p=3\cdot10^{-10}\) & 118/10/26 \\
+Chandrupatla (45) & 31/4/10, \(p=0.001\) & 34/4/7 \\
+new corpus 1 (206) & 108/33/65, \(p=0.001\) & 125/26/55 \\
+Roots.jl (78) & 44/14/20, \(p=0.004\) & 47/11/20 \\
+applied + GSL (132) & 48/52/32, \(p=0.09\) & 76/35/21 \\
+\textbf{corpus 4 (240), the one test} & \textbf{177/35/28,
+\(p=10^{-27}\)} & \textbf{183/23/34} \\
+\end{longtable}
+
+Only the last line of corpora is evidence for A′ in the strict sense;
+the first five were seen while it was developed and are reported as a
+description. On corpus 4 A′ is 36\% below Chandrupatla and wins 177
+problems to 28. Without the two classes aimed at by the last two
+readings it is 2403 against 3569, per problem 107/31/22
+(\(p=10^{-14}\)); on the simple-root classes alone, 737 against 747
+(−1.3\%), per problem 43/22/15 (\(p=3\cdot10^{-4}\)).
+
+\textbf{Table 11.} Corpus 4 by class: totals of A′ / Chandrupatla, per
+problem wins/ties/losses.
+
+\begin{longtable}[]{@{}llll@{}}
+\toprule\noalign{}
+class & A′ & Chandrupatla & per problem \\
+\midrule\noalign{}
+\endhead
+\bottomrule\noalign{}
+\endlastfoot
+random polynomial, simple root & 433 & 419 & 20/8/12 \\
+polynomial, multiple root & 1163 & 1805 & 39/1/0 \\
+fractional power, \(p<1\) & 924 & 1784 & 37/1/2 \\
+sums of exponentials and sines & 304 & 328 & 23/14/3 \\
+\(\tanh\), \(\arctan\), \(\log(1+q^2)\) of a cubic & 503 & 1017 &
+25/8/7 \\
+odd about the root & 329 & 364 & 33/3/4 \\
+\end{longtable}
+
+\textbf{What the evidence supports, and what it does not.} Against Brent
+and TOMS 748 A′ is ahead on every corpus, by 8\% to 67\% in total, and
+it keeps its bound on all 855 problems, which Brent breaks on 118 and
+TOMS 748 on 84. Against ModAB it is ahead on every corpus, in total and
+problem by problem (\(p\le0.0013\)), on the one test of corpus 4 by 3656
+to 5535 and 183 problems to 34 (\(p=7\cdot10^{-26}\)), and on ModAB's
+own benchmark as well --- all 91 problems of it, 1507 against 1588,
+56/13/22 (\(p=1.5\cdot10^{-4}\)); ModAB breaks the bound on 9 problems.
+Against Chandrupatla it is ahead in total on five corpora and level on
+the sixth, and ahead problem by problem on all six, significantly on
+five. The size of the lead is not uniform: it is large on multiple,
+fractional and nearly flat roots --- the regime §8 recognises --- and
+small on simple roots, where it is a matter of per cent. Where it is
+behind is named: \texttt{fun9} (65 against 55), the near-cancelling
+class of corpus 1 (491 against 391), random polynomials 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.
+
+\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
+
+\hypertarget{negative-results}{%
+\subsection{10. Negative results}\label{negative-results}}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  \textbf{The prototype-aggregation form of the method loses.} A richer
+  construction --- prototypes over \emph{pairs} of evaluated points,
+  softmax weights over the whole memory, similarity on both the value
+  and the abscissa axis --- leads on eight hand-picked problems but
+  loses decisively on the real set: 0 wins, 6 ties and 147 losses
+  against Brent. The simplest form of the idea won.
+\item
+  \textbf{Every safeguard that replaces the step destroys the order.}
+  Bracket-based and step-based replacement both drop the three-point
+  rule from order 1.94 to 1.00 and cost 7260 and 7239 evaluations on
+  Alefeld--Potra--Shi. Only projection preserves both. We record this
+  because it is the trap the construction naturally falls into.
+\item
+  \textbf{Zero slack in the projection is not the degenerate case it was
+  said to be.} An earlier version reported that a worst case of exactly
+  bisection leaves the projection no freedom and makes the guard behave
+  as bisection. Measured (Table 1a), multiplier 1 costs 4817 and 2997,
+  against bisection's 7530 and 3090 --- and on the wide-bracket set it
+  beats the multiplier of 2. It remains the most expensive multiplier on
+  the canonical set, which is all that survives of the claim.
+\item
+  \textbf{Regimes exist, at the level of a function family, and cannot
+  be exploited by a switch.} The cube-root similarity beats the base
+  rule on all five members of Chandrupatla's \texttt{fun4} ---
+  32/32/40/47/59 against 89/93/113/125/153, a factor of 2.6 --- and
+  consistency across a family at different parameters is the signature
+  of structure rather than luck. But the same similarity over the whole
+  Chandrupatla set costs 4109 against 3220, so a switch chosen per
+  benchmark is worse than no switch at all: 6915 against 6026 over both
+  sets. The scale axis offers only 5.3\% headroom even to an oracle
+  handed the best scale per problem, two thirds of it inside a single
+  problem family, and the winners are not separated by anything the
+  method can see before it commits --- the derived scale wins on bracket
+  widths from \(10^0\) to \(10^{12}\).
+\item
+  \textbf{The phase detector: derivation sound, estimator not.} Redoing
+  the contraction law without the assumption \(|f|\approx|f'(r)|e\) ---
+  taking \(|f|\approx K e^m\) instead --- does produce the march
+  exponent on its own, \(p = L/(L-\ln m)\) with \(L=\ln(e/D)\), giving
+  \(p=1\) at a simple root and dropping to 0.50 and 0.23 as the approach
+  flattens, which is the neighbourhood of the cube root that wins those
+  families. Substituted back it collapses to the multiplier \(m\),
+  measurable as \(m = 1/(1-\theta)\) with \(\theta = f f''/(f')^2\) from
+  three divided differences and no constant --- which is Lagouanelle's
+  multiplicity estimate {[}13{]}. It preserves the order (2.001) and
+  improves the eight standard problems (82 against 84) --- and loses on
+  the full benchmark, 6304 against 6026. The estimator is the failure:
+  on Alefeld--Potra--Shi, where almost every root is simple and the
+  truth is \(m=1\), it reports \(m>1\) on 27.9\% of steps with
+  excursions to \(m=34.8\), against 35.8\% on the flat families. 27.9
+  against 35.8 is no separation. A cruder variant, correcting by the
+  fraction of its own promise the previous step kept, fails everything:
+  the order collapses to 0.904 and the standard problems double to 161.
+\item
+  \textbf{One problem is scored unsolved for every method},
+  \texttt{aps.13.00}, \(f(x)=x e^{-1/x^2}\), all of whose derivatives
+  vanish at the root and which underflows to exactly zero throughout
+  \(|x|<0.086\). Every method returns a point where \(f(x)=0\) exactly
+  --- a genuine root --- which the harness scores as a failure because
+  it lies more than \(10^{-6}\) from the declared root. It is a property
+  of the scoring, not a failure of the methods; §7.3 finds five more
+  such problems in the wide-bracket set.
+\item
+  \textbf{The symmetric ratio similarity degenerates.} For
+  \(S(x,y)=\min(|x|,|y|)/\max(|x|,|y|)\) with \(y\) the target 0,
+  \(S\equiv0\); the only usable reading is \(1-|f|/\sigma\), which
+  saturates and stagnates (Table 1, last row).
+\item
+  \textbf{Placement beats form, and the wrong placement is expensive.}
+  The same ``refuse before the step'' idea as reading (1) of §7, with
+  refusal decided by \emph{promised bracket progress} instead of by the
+  shape of the model, costs 4449 on the canonical set against 2527 --- a
+  difference of 1922 evaluations for one choice of criterion. A secant
+  step routinely collapses the error while barely moving the proven
+  bracket, so bracket progress rejects steps that were excellent; this
+  is the same mechanism that makes the bracket guard of §3.1 cost 7260.
+\item
+  \textbf{Making the scale independent of the bracket does not help
+  either.} The natural objection to the derived scale is that it defeats
+  the very principle of a similarity step: with \(\sigma=|\hat f'|D\)
+  the normalised residual is \(|f|/(|\hat f'|D)\), which on the
+  \(2\times10^4\)-wide bracket of \(x^3-2x-5\) at \(x=10^4\) equals
+  0.167, so \(S=0.833\) --- ``similar'' --- and the step is small
+  although the root is some five thousand step-lengths away. The
+  similarity is reading what fraction of the \emph{bracket} the Newton
+  step is, not how far the iterate is from the root. Selecting the scale
+  between the two quantities already computed,
+  \(\sigma=\min(|\hat f'|D,\max(|f(A)|,|f(B)|))\) --- no new constant, a
+  minimum over what is in hand --- does give a full-bracket step far
+  away and Newton's step near the root, and on eight hand-picked wide
+  problems it improves 403 evaluations to 390. Over the whole grid it is
+  worse on both sets: 3288 on the canonical set against 2726, and 3111
+  on the wide-bracket set against 2695. Its apparent extra solves (194
+  instead of 193) are two \texttt{fun7} problems where it lands inside
+  the flat zero region close enough to satisfy the harness --- the
+  artefact of §7.3, not a capability.
+\item
+  \textbf{Reading the similarity at the measured order does not cure
+  false position.} Since false position's stall is a similarity read at
+  order one (§7.1), the obvious hope is that reading it at the measured
+  order is the parameter-free version of the classical fix. It is not.
+  Unguarded, the correction takes the far-end reading from 127/154 and
+  10676 to 128/154 and 10043 --- one problem and six per cent --- where
+  Illinois, at the price of one constant (halve the stale ordinate),
+  reaches 153/154 and 7045. A sixth of the gap is not the gap. The
+  correction is real and it is in the right place; it is simply much
+  smaller than a constant that was chosen for exactly this failure.
+  \emph{Our} 10676 and the 10560 of Table 2 are not the same algorithm
+  --- our iterate starts at the midpoint of the bracket, plain false
+  position at an end --- and the two must not be differenced.
+\item
+  \textbf{A logarithmic change of variable does not buy the wide
+  brackets.} Working in
+  \(u=\operatorname{sign}(x-c)\log(1+|x-c|/\varepsilon_x)\) makes the
+  step multiplicative, which looked like the right diagnosis of the flat
+  families --- for the same similarity, scale and safeguard
+  (\((1+|z|)^{-2}\), two-point, \emph{safe}) the wide-bracket set
+  improves from 3695 to 3541 --- but the canonical set gets worse, 5822
+  to 6940, against the 2806 of the base rule. An earlier version
+  compared the 3541 with 4516, which is the total of a different
+  configuration. A march needs a \emph{ratio} of distances while an
+  identity on narrow brackets needs an \emph{absolute} unit, and no
+  parameter-free unit provides both. §7.4 shows the diagnosis itself was
+  wrong: on the flat families the right move is to halve, not to march.
+\item
+  \textbf{Reading the unkept promise one step at a time does not pay.}
+  The flag of §8.1 sees the flat regime, but a halving in place of each
+  flagged step takes the canonical set from 2806 to 3029 and the
+  wide-bracket set only from 3220 to 3198: a simple root far from
+  convergence breaks its promises too. Requiring the promise to fail
+  twice in a row keeps the canonical set (2803) but fires one step late,
+  before the crossing, and replaces the most useful step of the cycle
+  (wide brackets 3341); requiring a creep in one direction as well gives
+  2878 and 3354.
+\item
+  \textbf{A halving that sticks once begun wins the flat families and
+  loses everything else.} Without an exit the flat families fall to 1057
+  --- pure bisection's level --- and the canonical set rises to 6897,
+  because a false entry now lasts to the end of the run. Splitting at
+  the order read from the failed promise, \((1-1/m)^m=r\), changes
+  nothing: that order is \(m=\infty\) in 163 of 174 entries (6781).
+\item
+  \textbf{The model's own point cannot tell a flat root from a simple
+  one.} Judging the evaluated point against the line through the bracket
+  ends and against 0 --- the exit judgment of §8.2 --- also at
+  \emph{entry} gives 3009 and 2486 against 3125 and 2503 for the exit
+  alone: the model aims at the root, so \(f\) is small there by
+  construction and looks flat at a simple root as readily as at a power.
+  The same judgment works at a split, which the model did not choose.
+\item
+  \textbf{The decision reading ``defect'' does not transfer.} Combined
+  with the regime it was the best configuration on the two development
+  sets (§7.2's canonical record 2523 with the flat families of §8), and
+  it passed every pre-registered criterion there. On new corpus 1 it did
+  not: per problem 95/33/78 against Chandrupatla (\(p=0.22\)), losing
+  the class of exponential flanks 9 to 26. It is therefore not part of
+  A′.
+\item
+  \textbf{A component rejected alone can pay in combination.}
+  Remembering the analogy for the run was rejected by its own
+  pre-registration (canonical set 2814 against 2806 without the regime),
+  and with the split of §8.3 it is the better form (with the step
+  pattern, wide brackets 2117 against 2221 for the analogy read in one
+  state). Composition was measured, not assumed, throughout --- which is
+  also why §7's readings (2) and (3), each an improvement, are not
+  combined.
+\end{enumerate}
+
+\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
+
+\hypertarget{conclusion}{%
+\subsection{11. Conclusion}\label{conclusion}}
 
 A root finder can be derived from a similarity function alone, and the
-derivation shows where the content of such a method actually sits. It is not in
-the shape of $S$ --- nine shapes were tried and their ranking is dictated by a
-single number, the slope of $1-S$ at the origin. It is not in making the
-similarity change faster --- that overshoots exactly as readily as flatness
-stagnates. It is in the \emph{scale} against which similarity is judged:
-chosen, the method is brittle and merely linear; taken from the bracket, it is
-false position and three centuries old; read off the data, it is superlinear.
-And the safeguard that makes such a method safe must project the step, not
-replace it: replacing costs the order, projecting costs nothing on one
-benchmark and improves the method on the other.
+derivation shows where the content of such a method actually sits. It is
+not in the shape of \(S\) --- nine shapes were tried and their ranking
+is dictated by a single number, the slope of \(1-S\) at the origin. It
+is not in making the similarity change faster --- that overshoots
+exactly as readily as flatness stagnates. It is in the \emph{scale}
+against which similarity is judged: chosen, the method is brittle and
+merely linear; taken from the bracket, it is false position and three
+centuries old; read off the data, it is superlinear. And the safeguard
+that makes such a method safe must project the step, not replace it:
+replacing costs the order, projecting costs nothing on one benchmark and
+improves the method on the other. The projection's budget multiplier is
+a free constant, and we declare it: it does not move the canonical
+result and it does move the wide-bracket one; projecting onto a budget
+the steps earn instead removes it, at a measured price.
 
-The resulting rule has no tuning constant and carries a worst case of twice
-bisection plus one step. On the canonical Alefeld--Potra--Shi set it is level
-with TOMS\,748 --- $2806$ evaluations against $2820$ is a margin of $0.5\%$ and
-should be read as a tie --- and ahead of Brent by $2.9\%$, which it also beats
-$102$ problems to $40$ head to head. The claim the measurements support is
-parity with methods that carry tuned constants, not superiority over them.
+The base rule has one declared constant and carries a worst case of
+twice bisection plus one step. On the canonical Alefeld--Potra--Shi set
+it is level with TOMS 748 and with Chandrupatla's method in total and
+ahead of both problem by problem; on the wide-bracket set Chandrupatla
+is genuinely stronger (2441 against 3220). §7.4 locates the cause: not
+the order of the model but the decision to give it up, which
+Chandrupatla takes by halving and which no reading of a single state
+takes correctly for our step.
 
-This is the same movement as in parameter-free similarity control~\cite{sim},
-where membership functions are eliminated by normalising the input space from
-the data rather than by choosing widths. Here the object normalised is the
-value axis, and the price of choosing it by hand is measured rather than
-asserted.
+That decision is a similarity too --- of the run to its own past. A pure
+power repeats its configuration under zoom, an exponential under
+translation, a simple root tends to the line; the analogous pair reads
+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 and ModAB
+everywhere, with the guarantee they do not keep, 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.
 
-\subsection*{Future work}
+This is the same movement as in parameter-free similarity control
+{[}6{]}, where membership functions are eliminated by normalising the
+input space from the data rather than by choosing widths. Here the
+object normalised is the value axis, and the price of choosing it by
+hand is measured rather than asserted.
 
-The diagnosis of \S\ref{sec:widelimit} leaves one question, and it is narrower
-than the one this section used to pose. An earlier version proposed a
-logarithmic change of variable to turn an additive step into a geometric
-march; that was measured and refuted (\S\ref{sec:widelimit} and the negative
-results), and the step turned out to be a fraction of the bracket already.
+\hypertarget{future-work}{%
+\subsubsection{Future work}\label{future-work}}
 
-What is open is a decision, not a model. On two thirds of the remaining gap
-the right move is to halve, and Chandrupatla halves; our guarantee permits a
-step that is slower than halving, and nothing we have measured tells our step
-when it is. The condition itself is known exactly --- a tangent step on an
-approach of local order $m$ lands at $e(1-1/m)$ and loses to a halving when
-$m\ge2$ --- so the constant is bisection's $\tfrac12$ and not a choice. What is
-missing is a reading of $m$ that sees it: the three-point estimator reports
-$m\ge2$ on $0.2\%$ of canonical states, which is the right silence, and on
-$5.8\%$ of states of pure powers of order $3$ to $19$, which is blindness. Two
-successive residuals of the run itself carry the order more directly than a
-parabola through three points --- on an order-$m$ approach under tangent steps
-their ratio settles at $(1-1/m)^m$, while at a simple root it goes to zero ---
-and that is the reading we would test next, under the same rule as every other
-here: pre-registered, with the canonical set as the price it must not charge.
+Two questions remain, both narrower than the one this section used to
+pose.
 
-\begin{thebibliography}{9}
-\bibitem{aps} G.~E. Alefeld, F.~A. Potra and Y.~Shi, ``Algorithm 748: Enclosing
-zeros of continuous functions,'' \emph{ACM Trans.\ Math.\ Softw.}, vol.~21,
-no.~3, pp.~327--344, 1995.
-\bibitem{brent} R.~P. Brent, \emph{Algorithms for Minimization without
-Derivatives}. Englewood Cliffs, NJ: Prentice-Hall, 1973.
-\bibitem{chandrupatla} T.~R. Chandrupatla, ``A new hybrid quadratic/bisection
-algorithm for finding the zero of a nonlinear function without using
-derivatives,'' \emph{Adv.\ Eng.\ Software}, vol.~28, no.~3, pp.~145--149, 1997.
-\bibitem{itp} I.~F.~D. Oliveira and R.~H.~C. Takahashi, ``An enhancement of the
-bisection method average performance preserving minmax optimality,''
-\emph{ACM Trans.\ Math.\ Softw.}, vol.~47, no.~1, pp.~1--24, 2021.
-\bibitem{ridders} C.~Ridders, ``A new algorithm for computing a single root of
-a real continuous function,'' \emph{IEEE Trans.\ Circuits Syst.}, vol.~26,
-no.~11, pp.~979--980, 1979.
-\bibitem{sim} V.~Sitnikov, ``Parameter-free similarity control,'' preprint,
-2026.
-\bibitem{dowell} M.~Dowell and P.~Jarratt, ``A modified regula falsi method for
-computing the root of an equation,'' \emph{BIT}, vol.~11, pp.~168--174, 1971.
-\end{thebibliography}
+The first 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
+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.
+
+\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
+
+\hypertarget{references}{%
+\subsection{References}\label{references}}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\tightlist
+\item
+  G. E. Alefeld, F. A. Potra and Y. Shi, ``Algorithm 748: Enclosing
+  zeros of continuous functions,'' \emph{ACM Trans. Math. Softw.},
+  vol.~21, no. 3, pp.~327--344, 1995.
+\item
+  R. P. Brent, \emph{Algorithms for Minimization without Derivatives}.
+  Englewood Cliffs, NJ: Prentice-Hall, 1973.
+\item
+  T. R. Chandrupatla, ``A new hybrid quadratic/bisection algorithm for
+  finding the zero of a nonlinear function without using derivatives,''
+  \emph{Adv. Eng. Software}, vol.~28, no. 3, pp.~145--149, 1997.
+\item
+  I. F. D. Oliveira and R. H. C. Takahashi, ``An enhancement of the
+  bisection method average performance preserving minmax optimality,''
+  \emph{ACM Trans. Math. Softw.}, vol.~47, no. 1, pp.~1--24, 2021.
+\item
+  C. Ridders, ``A new algorithm for computing a single root of a real
+  continuous function,'' \emph{IEEE Trans. Circuits Syst.}, vol.~26, no.
+  11, pp.~979--980, 1979.
+\item
+  V. Sitnikov, ``Parameter-free similarity control,'' preprint, 2026.
+\item
+  M. Dowell and P. Jarratt, ``A modified regula falsi method for
+  computing the root of an equation,'' \emph{BIT}, vol.~11,
+  pp.~168--174, 1971.
+\item
+  G. Pólya, \emph{How to Solve It}. Princeton, NJ: Princeton University
+  Press, 1945.
+\item
+  T. J. Dekker, ``Finding a zero by means of successive linear
+  interpolation,'' in B. Dejon and P. Henrici (eds.), \emph{Constructive
+  Aspects of the Fundamental Theorem of Algebra}. London:
+  Wiley-Interscience, 1969, pp.~37--51.
+\item
+  N. Ganchovski and A. Traykov, ``Modified Anderson-Björck's method for
+  solving non-linear equations in structural mechanics,'' \emph{IOP
+  Conf. Ser.: Mater. Sci. Eng.}, vol.~1276, 012010, 2023.
+\item
+  M. Galassi et al., \emph{GNU Scientific Library Reference Manual}, 3rd
+  ed.~Network Theory, 2009; root-finding test suite
+  \texttt{roots/test.c}.
+\item
+  J. Verzani et al., \emph{Roots.jl}, Julia package; bracketing
+  benchmark \texttt{test/benchmark-bracketing-methods.jl}, commit
+  cf1d256, 2026.
+\item
+  J. L. Lagouanelle, ``Sur une méthode de calcul de l'ordre de
+  multiplicité des zéros d'un polynôme,'' \emph{C. R. Acad. Sci. Paris
+  Sér. A}, vol.~262, pp.~626--627, 1966.
+\item
+  L. B. Rall, ``Convergence of the Newton process to multiple
+  solutions,'' \emph{Numer. Math.}, vol.~9, pp.~23--37, 1966.
+\item
+  E. Hansen and M. Patrick, ``Estimating the multiplicity of a root,''
+  \emph{Numer. Math.}, vol.~27, pp.~121--131, 1976.
+\item
+  E. Schröder, ``Ueber unendlich viele Algorithmen zur Auflösung der
+  Gleichungen,'' \emph{Math. Ann.}, vol.~2, pp.~317--365, 1870.
+\item
+  J. F. Traub, \emph{Iterative Methods for the Solution of Equations}.
+  Englewood Cliffs, NJ: Prentice-Hall, 1964.
+\item
+  S. Weerakoon and T. G. I. Fernando, ``A variant of Newton's method
+  with accelerated third-order convergence,'' \emph{Appl. Math. Lett.},
+  vol.~13, pp.~87--93, 2000.
+\item
+  A. Cordero and J. R. Torregrosa, ``Variants of Newton's method using
+  fifth-order quadrature formulas,'' \emph{Appl. Math. Comput.},
+  vol.~190, pp.~686--698, 2007.
+\item
+  N. Ganchovski, O. Smith, C. Rackauckas, L. Tomov and A. Traykov,
+  ``Improvements to the Modified Anderson--Björck (modAB) root-finding
+  algorithm,'' \emph{Algorithms}, vol.~19, 332, 2026; package
+  \texttt{pymodab} 1.0.9.
+\end{enumerate}
 
 \end{document}
diff --git a/build_tex.py b/build_tex.py
new file mode 100644
index 0000000..6f356fa
--- /dev/null
+++ b/build_tex.py
@@ -0,0 +1,93 @@
+"""Regenerate article/sim_numerics.tex (and the PDF) from the canonical article/sim_numerics.md.
+
+The .md is the text; the .tex is a build product and is never edited by hand.  Title, author
+and abstract become pandoc metadata; the body is converted as it stands (section numbers are
+part of the headings, so LaTeX numbering is off).  Unicode that pdflatex does not know is
+declared in the header.  Run on the server (pandoc 3.1.3, pdflatex):
+
+    python -u build_tex.py            -> article/sim_numerics.tex, article/sim_numerics.pdf
+"""
+from __future__ import annotations
+
+import json
+import os
+import subprocess
+import sys
+
+MD = "article/sim_numerics.md"
+TEX = "article/sim_numerics.tex"
+
+HEADER = r"""
+\usepackage{amsmath,amssymb}
+\usepackage{textcomp}
+\DeclareUnicodeCharacter{2032}{\ensuremath{'}}
+\DeclareUnicodeCharacter{2212}{\ensuremath{-}}
+\DeclareUnicodeCharacter{00D7}{\ensuremath{\times}}
+\DeclareUnicodeCharacter{00B5}{\textmu}
+\DeclareUnicodeCharacter{2192}{\ensuremath{\rightarrow}}
+\DeclareUnicodeCharacter{0394}{\ensuremath{\Delta}}
+\DeclareUnicodeCharacter{2082}{\textsubscript{2}}
+\usepackage[margin=2.5cm]{geometry}
+\usepackage[htt]{hyphenat}
+\setlength{\emergencystretch}{3em}
+\usepackage{etoolbox}
+\AtBeginEnvironment{longtable}{\small}
+"""
+
+BANNER = ("%% GENERATED from article/sim_numerics.md by build_tex.py - do not edit by hand.\n"
+          "%% The .md is the canonical text; every number in it is checked by verify_article.py.\n")
+
+
+def split_front(md):
+    lines = md.split("\n")
+    title = lines[0].lstrip("# ").strip()
+    rest = lines[1:]
+    author = next(ln.strip() for ln in rest if ln.strip())
+    text = "\n".join(rest)
+    head, _, body = text.partition("## Abstract")
+    abstract, _, body = body.partition("\n---\n")
+    return title, author, abstract.strip(), body.lstrip("\n")
+
+
+def main():
+    with open(MD, encoding="utf8") as fh:
+        title, author, abstract, body = split_front(fh.read())
+    meta = {"title": title, "author": author, "abstract": abstract,
+            "header-includes": HEADER, "documentclass": "article",
+            "classoption": ["11pt", "a4paper"]}
+    os.makedirs("build", exist_ok=True)
+    with open("build/meta.json", "w", encoding="utf8") as fh:
+        json.dump(meta, fh)
+    with open("build/body.md", "w", encoding="utf8") as fh:
+        fh.write(body)
+    subprocess.run(["pandoc", "build/body.md", "--metadata-file=build/meta.json", "-s",
+                    "-f", "markdown+tex_math_dollars+pipe_tables", "-t", "latex",
+                    "-o", "build/body.tex"], check=True)
+    with open("build/body.tex", encoding="utf8") as fh:
+        tex = fh.read()
+    with open(TEX, "w", encoding="utf8") as fh:
+        fh.write(BANNER + tex)
+    for _ in range(2):
+        r = subprocess.run(["pdflatex", "-interaction=nonstopmode", "-halt-on-error",
+                            "-output-directory", "article", TEX],
+                           capture_output=True, text=True)
+    log = open("article/sim_numerics.log", encoding="utf8", errors="replace").read()
+    errors = [ln for ln in log.splitlines() if ln.startswith("!")]
+    # only undefined references / citations count; 'Font shape ... undefined' is a font
+    # substitution warning, not a defect of the text
+    undefined = [ln for ln in log.splitlines()
+                 if "undefined" in ln.lower() and ("reference" in ln.lower()
+                                                   or "citation" in ln.lower())]
+    overfull = sum(1 for ln in log.splitlines() if ln.startswith("Overfull"))
+    pages = next((ln for ln in log.splitlines() if "Output written" in ln), "no output")
+    print(f"pdflatex exit {r.returncode}; errors {len(errors)}; undefined {len(undefined)}; "
+          f"overfull boxes {overfull}")
+    for ln in errors[:10] + undefined[:10]:
+        print("  ", ln)
+    print(pages)
+    print("build done")
+    sys.exit(0 if r.returncode == 0 and not errors else 1)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/fix_md_for_latex.py b/fix_md_for_latex.py
new file mode 100644
index 0000000..75b345c
--- /dev/null
+++ b/fix_md_for_latex.py
@@ -0,0 +1,31 @@
+"""Make the math of article/sim_numerics.md valid LaTeX as well as MathJax: a sub- or
+superscript that starts with a bare control word (n_\\max) needs braces in LaTeX (n_{\\max}).
+The rendered formula is identical.  Reports what it finds; --apply writes.
+
+    python -u fix_md_for_latex.py [--apply]
+"""
+from __future__ import annotations
+
+import re
+import sys
+
+PATH = "article/sim_numerics.md"
+# only argument-less control words (n_\max); a macro with an argument (x_\mathrm{far},
+# r_\text{new}) is valid LaTeX as written, and bracing it would detach its argument
+PAT = re.compile(r"([_^])(\\[A-Za-z]+)(?![A-Za-z{])")
+
+
+def main():
+    with open(PATH, encoding="utf8") as fh:
+        text = fh.read()
+    hits = PAT.findall(text)
+    print(f"{len(hits)} bare control words in sub/superscripts:", sorted(set(h[1] for h in hits)))
+    if "--apply" in sys.argv:
+        text = PAT.sub(lambda m: f"{m.group(1)}{{{m.group(2)}}}", text)
+        with open(PATH, "w", encoding="utf8") as fh:
+            fh.write(text)
+        print("braced")
+
+
+if __name__ == "__main__":
+    main()