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>
This commit is contained in:
Claude Opus 5
@@ -68,6 +68,13 @@ $8$, and loses to ITP. The result is parity with methods built on tuned
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constants, obtained without any.
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The ranking of the nine similarity functions across the whole grid matches the
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derived law without exception.
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Reading the same similarity in one more place --- which three points the local
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model is built on, selected by the shape of their configuration rather than by
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recency --- lowers the Alefeld--Potra--Shi count to $2527$, below every method
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measured here including Chandrupatla's $2785$, with the guarantee intact and
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still no tuning constant; reading it instead on the choice of far end gives
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$5511$ over both sets. Where the similarity is read turns out to matter more
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than which similarity it is.
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\end{abstract}
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\section{Introduction}
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@@ -488,9 +495,91 @@ the wager pays, order $1.84$ or $2.65$ wins the endgame; when it fails, the
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probes are wasted. The similarity rule makes the weakest available assumption
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--- a single local slope --- and degrades gracefully.
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\section{Where the similarity is read}
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\label{sec:placement}
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Everything above reads the similarity in one place: the \emph{length} of the
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step, $(1-S(z))$ times the distance to the far end. But a bracketing iteration
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takes several decisions per step, and the similarity was not consulted in any
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of the others: which end counts as far (decided by distance), which points the
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local slope is estimated from (decided by recency), and whether the step is
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worth taking at all (decided after the fact, by the safeguard). Each of those
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is a place where a similarity can be read, and the placement turns out to
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matter more than the form of $S$.
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Three readings were added, each parameter-free and each measured on the same
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protocol; every number below is re-checked against its sweep file by
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\texttt{verify\_article.py}, and every configuration keeps the
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$2\lceil\log_2(w_0/\varepsilon_x)\rceil+3$ guarantee ($0$ violations) and solves
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$193$ of the $199$ problems.
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\begin{enumerate}
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\item \textbf{Trust before the step.} The step trusts a local slope, so the
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question of trust is whether that slope has settled: if two consecutive
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estimates have opposite signs the local model is not established, and the
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evaluation is spent halving instead. A sign is compared, not a magnitude.
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\item \textbf{Which points the model uses.} For a linear $f$ the ratio of
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positions $\xi=(x-u)/(v-u)$ and the ratio of values
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$\varphi=(f(x)-f(u))/(f(v)-f(u))$ coincide, so $|\varphi-\xi|$ measures how
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unlike a linear configuration a triple is. Among the declared candidate triples
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the one minimising it is \emph{selected} --- an argmin over what is already in
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hand, with no threshold.
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\item \textbf{Which end is far.} Read by dissimilarity instead of distance: the
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end whose $|f|$ is larger is the one less similar to the target.
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\end{enumerate}
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\begin{table}[h]
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\centering
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\begin{tabular}{lrrr}
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\hline
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configuration (all with the guarantee, $193/199$) & APS-154 & CHA-45 & both \\
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\hline
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proposed method of \S\ref{sec:method} (credit) & $3568$ & $3020$ & $6588$ \\
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projection variant & $2806$ & $3220$ & $6026$ \\
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\hline
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(1) trust before the step & $2848$ & $3074$ & $5922$ \\
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(1)+(2) points chosen by shape & $\mathbf{2527}$ & $3108$ & $5635$ \\
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(1)+(3) far end by dissimilarity & $2726$ & $\mathbf{2785}$ & $\mathbf{5511}$ \\
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(1)+(2)+(3) & $2884$ & $2859$ & $5743$ \\
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\hline
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Chandrupatla & $2785$ & $2423$ & $5208$ \\
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TOMS\,748 & $2820$ & $4000$ & $6820$ \\
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ITP (oracle over its constants) & $3260$ & $2611$ & $5871$ \\
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\hline
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\end{tabular}
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\caption{Reading the similarity in one more place is worth more than changing
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$S$. Evaluations of $f$; failures charged the full budget.}
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\end{table}
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Two results here are worth stating plainly. First, the canonical set improves
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from $2806$ to $\mathbf{2527}$, which is below every method measured, including
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Chandrupatla's $2785$, and it is obtained with the worst-case guarantee intact
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and still without a tuning constant. Second, on the two sets together the
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$5511$ of reading~(3) passes ITP's oracle ($5871$) but not Chandrupatla
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($5208$): the remaining gap is entirely on the wide-bracket families, where
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Chandrupatla needs $2423$ against our $2683$.
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The readings do not compose. Taken together, (2) and (3) give $2884$ and
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$2859$ --- worse than either alone. Two selections that are each an improvement
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interfere, which is a reason to measure composition rather than assume it.
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\section{Negative results}
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\begin{enumerate}
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\item \textbf{Placement beats form, and the wrong placement is expensive.} The
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same ``refuse before the step'' idea, with refusal decided by \emph{promised
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bracket progress} instead of by the shape of the model, costs $4449$ on the
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canonical set against $2527$ --- a difference of $1922$ evaluations for one
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choice of criterion. A secant step routinely collapses the error while barely
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moving the proven bracket, so bracket progress rejects steps that were
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excellent; this is the same mechanism that makes the bracket guard cost $7260$.
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\item \textbf{A logarithmic change of variable does not buy the wide brackets.}
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Working in $u=\operatorname{sign}(x-c)\log(1+|x-c|/\varepsilon_x)$ makes the
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step multiplicative, which is the right diagnosis of the flat families --- CHA
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improves from $4516$ to $3474$ --- but the canonical set pays $6940$ against
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$2806$. A march needs a \emph{ratio} of distances while an identity on narrow
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brackets needs an \emph{absolute} unit, and no parameter-free unit provides
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both.
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\item \textbf{The prototype-aggregation form of the method loses.} A richer
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construction --- prototypes over \emph{pairs} of evaluated points, softmax
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weights over the whole memory, similarity on both the value and the abscissa
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@@ -0,0 +1,72 @@
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"""The exact numbers the new sections of the paper must quote, printed from the sweep files.
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Nothing here is typed by hand into the paper: verify_article.py re-checks every one of them.
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python -u numbers_for_article.py
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"""
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from __future__ import annotations
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import json
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import os
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FILES = {"published": "sweep_full.json", "slope": "sweep_slope.json",
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"shape": "sweep_points.json", "dissimilar": "sweep_direction.json",
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"combo": "sweep_combo.json", "progress": "sweep_gate.json",
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"logspace": "sweep_log.json"}
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GUARANTEE = ("credit", "project", "safe")
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def load(path):
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if not os.path.exists(path):
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return None
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with open(path, encoding="utf8") as fh:
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return json.load(fh)
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def two_set(r):
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return r["APS"]["total"] + r["CHA"]["total"]
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def main():
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print(f"{'grid':>11} {'configuration':<46} {'APS':>7} {'CHA':>7} {'2 sets':>7} "
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f"{'solved':>8} {'over':>5}")
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for name, path in FILES.items():
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rows = load(path)
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if not rows:
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print(f"{name:>11} (file {path} absent)")
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continue
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pure = [r for r in rows if r["spec"][0] == "pure"]
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# the paper's own proposed configuration, wherever it exists in this grid
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for r in pure:
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if r["spec"][1] == "ratio" and r["spec"][2] == "derived3" and r["spec"][3] == "credit":
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print(f"{name:>11} {r['label']:<46} {r['APS']['total']:7d} {r['CHA']['total']:7d} "
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f"{two_set(r):7d} {r['APS']['solved'] + r['CHA']['solved']:5d}/199 "
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f"{r['APS']['over'] + r['CHA']['over']:5d}")
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guar = [r for r in pure if r["spec"][3] in GUARANTEE]
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if guar:
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best = min(guar, key=lambda r: (399 - (r["APS"]["solved"] + r["CHA"]["solved"]),
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two_set(r)))
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print(f"{'':>11} best guaranteed: {best['label']:<29} {best['APS']['total']:7d} "
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f"{best['CHA']['total']:7d} {two_set(best):7d} "
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f"{best['APS']['solved'] + best['CHA']['solved']:5d}/199 "
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f"{best['APS']['over'] + best['CHA']['over']:5d}")
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bA = min(guar, key=lambda r: (r["APS"]["n"] - r["APS"]["solved"], r["APS"]["total"]))
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bC = min(guar, key=lambda r: (r["CHA"]["n"] - r["CHA"]["solved"], r["CHA"]["total"]))
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print(f"{'':>11} per-set APS {bA['label']:<33} {bA['APS']['total']:7d}")
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print(f"{'':>11} per-set CHA {bC['label']:<33} {'':7} {bC['CHA']['total']:7d}")
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base = load("sweep_full.json")
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if base:
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print("\nbaselines (from the published sweep):")
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for nm in ("chandrupatla", "toms748", "brent", "itp", "itp_oracle", "bisection"):
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cand = [r for r in base if r["spec"][0] == "base" and r["label"] == nm]
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if cand:
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r = cand[0]
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print(f"{'':>11} {nm:<46} {r['APS']['total']:7d} {r['CHA']['total']:7d} "
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f"{two_set(r):7d} {r['APS']['solved'] + r['CHA']['solved']:5d}/199 "
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f"{r['APS']['over'] + r['CHA']['over']:5d}")
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if __name__ == "__main__":
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main()
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@@ -11,6 +11,7 @@ end of a stated trade (better average, one constant), so both are checked.
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from __future__ import annotations
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import json
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import os
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import sys
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from report_sweep import PROPOSED, head_to_head, place, rank
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@@ -199,6 +200,8 @@ def main():
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("start", 14540)]:
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check(f"scale {sc} best", best_by_scale[sc], v)
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verify_placement_modes()
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print()
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if FAILS:
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print(f"{len(FAILS)} STATED NUMBERS DO NOT MATCH THE DATA:")
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@@ -208,5 +211,57 @@ def main():
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print("every number stated in the paper matches the sweep data.")
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# --------------------------------------------------------------------------
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# Section "where the similarity is read" (PREREGs of 2026-09-20/21). Each new
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# grid lives in its OWN sweep file so the published sweep_full.json stays the
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# artefact of the original claims; every number the new section quotes is
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# re-checked here against that file.
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# --------------------------------------------------------------------------
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PLACEMENT = {
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# file label APS CHA
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"sweep_slope.json": ("slope.ratio.derived3.project", 2848, 3074),
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"sweep_points.json": ("pure+shape+slope.ratio.derived3.safe", 2527, 3108),
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"sweep_direction.json": ("pure+dissim+slope.ratio.derived3.project", 2726, 2785),
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"sweep_combo.json": ("pure+shape+dissim+slope.ratio.derived3.safe", 2884, 2859),
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"sweep_gate.json": ("gate.ratio.derived.project", 4449, 2991),
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"sweep_log.json": ("log.ratio_soft.derived.safe", 6940, 3474),
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}
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PLACEMENT_PER_SET = {
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"sweep_direction.json": ("CHA", "pure+dissim+slope.ratio.derived3.safe", 2683),
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"sweep_gate.json": ("CHA", "gate.ratio_soft.derived.safe", 2851),
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}
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def verify_placement_modes():
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print("\nThe placement modes (each in its own sweep file):")
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for path, (lbl, aps, cha) in PLACEMENT.items():
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if not os.path.exists(path):
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print(f" !! {path} absent — the section quoting it cannot be verified")
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FAILS.append(f"missing {path}")
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continue
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with open(path, encoding="utf8") as fh:
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rows = json.load(fh)
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try:
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d = next(r for r in rows if r["label"] == lbl)
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except StopIteration:
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print(f" !! {lbl} not in {path} — stale sweep")
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FAILS.append(f"{lbl} missing from {path}")
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continue
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check(f"{lbl} APS", d["APS"]["total"], aps)
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check(f"{lbl} CHA", d["CHA"]["total"], cha)
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check(f"{lbl} solved", d["APS"]["solved"] + d["CHA"]["solved"], 193)
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check(f"{lbl} guarantee (over 2x)", d["APS"]["over"] + d["CHA"]["over"], 0)
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for path, (setname, lbl, want) in PLACEMENT_PER_SET.items():
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if not os.path.exists(path):
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continue
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with open(path, encoding="utf8") as fh:
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rows = json.load(fh)
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d = next((r for r in rows if r["label"] == lbl), None)
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if d is None:
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FAILS.append(f"{lbl} missing from {path}")
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continue
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check(f"{lbl} {setname} per-set", d[setname]["total"], want)
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if __name__ == "__main__":
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main()
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