paper updated with the placement results and machine-checked: new section 'Where the similarity is read' (2527 on APS below every method measured, 5511 over both sets, the non-composition of the two readings, and the two negatives with their numbers), abstract extended, verify_article.py now also checks every new number against its own sweep file — all green, pdflatex clean

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

article/sim_numerics.tex

@@ -68,6 +68,13 @@ $8$, 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. Where the similarity is read turns out to matter more
+than which similarity it is.
 \end{abstract}
 
 \section{Introduction}
@@ -488,9 +495,91 @@ 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}
+
+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, 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.
+
+\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.
+\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}
+
+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 $2683$.
+
+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.
+
 \section{Negative results}
 
 \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{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 is the right diagnosis of the flat families --- CHA
+improves from $4516$ to $3474$ --- but the canonical set pays $6940$ against
+$2806$. 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