diff --git a/.github/workflows/offline-checks.yml b/.github/workflows/offline-checks.yml index fe3cd59..bbacbf1 100644 --- a/.github/workflows/offline-checks.yml +++ b/.github/workflows/offline-checks.yml @@ -23,5 +23,7 @@ jobs: run: python -m pip install -r requirements.txt - name: Test numerical core, parser, and CLI workflows run: python -m unittest discover -s . -p 'test_*.py' -v + - name: Verify research output provenance + run: python scripts/verify_paper_artifacts.py - name: Generate an offline example with its dashboard run: python main_loop.py --mock --difficulty easy --seed 42 --output-dir runs/ci-smoke diff --git a/README.md b/README.md index 6fd357d..69700be 100644 --- a/README.md +++ b/README.md @@ -143,27 +143,38 @@ paths, independently of `--output-dir`. For a reproducible experiment record, retain the JSON logs, exact command, `git rev-parse HEAD`, and `python -m pip freeze` alongside your analysis. -## Results and evidence status - -This repository contains several historical experiment artifacts with different -scopes: - -- The manuscript's preliminary comparison lists **18 attempts / 15 scored runs** - across three OpenAI models, three difficulties, and seeds 42 and 137. -- Checked-in success-rate figures are labeled **180 scored runs across six - models**, with ten runs per model/difficulty combination. This is a different - design from the manuscript's proposed five-seed, two-prompt study. -- Saved Kaggle notebook outputs contain additional exploratory runs under their - own settings. - -The complete per-run JSON archive behind those tables and figures is **not -bundled**, so their aggregates cannot currently be regenerated from a fresh -checkout. The manuscript's six-model, five-seed, two-prompt protocol is a proposed -expansion, not a completed result established here. Historical numerical tables -are preserved; no new provider benchmark was run for this reproducibility update. -The previous README's single-seed, 28-query example is omitted because its source -logs and exact settings are unavailable and it exceeds the current default hard -budget of 24 queries. +## Paper, figures, and archived results + +The [revised manuscript](paper.pdf) analyzes the public +[Kaggle run archive, version 1](https://www.kaggle.com/datasets/jkmiller/elliptic-differential-equation-llm-reasoning/versions/1): +**180 scored runs**, six models, three difficulties, and seeds 1–10 in each cell. +All records use the standard prompt. The archive contains scored observations, +not a complete history of failed or retried attempts. + +The paper's figures are regenerated from the JSON logs, including a correction +to the older strict-threshold chart: GPT-5.4 mini has **2/10** hard successes at +**total absolute coefficient error below 0.01**. This metric is distinct from +function error evaluated on a spatial grid. + +[Reproduction instructions](paper_artifacts/README.md) include the pinned source +URL, archive checksum, figure script, and derived summary. Reproducing these +figures requires no new model calls. Historical provider responses cannot be +recreated exactly from the archive because complete settings and source +revisions were not recorded. + +The research revision adds fixed and randomized numerical controls, paired +statistical comparisons, joint-matrix spectra, and precision/refinement audits. +Fixed sine queries recover all 30 reconstructed tasks at the strict threshold +using the minimum permitted solve budget. This limits claims of a benefit from +adaptive language-model measurement selection on these instances. A separate +300-task fresh-seed validation reports both strong fixed-policy recovery and +failures from zero excitation and numerical sensitivity; the paper keeps those +results distinct from the archive. + +The earlier three-model, two-seed pilot tables are retained separately in +`paper_artifacts/legacy_pilot_tables.tex`. Their source logs have not been recovered, and they are not +pooled with the six-model study. Saved Kaggle notebooks contain further +exploratory outputs under their own settings. ## Repository guide @@ -177,4 +188,5 @@ budget of 24 queries. | `replot.py`, `plot_*.py`, `verify_metacognition.py` | Analysis of saved run logs | | [kaggle-benchmark.ipynb](kaggle-benchmark.ipynb), [kaggle.ipynb](kaggle.ipynb) | Kaggle-specific exploratory notebooks; require their own Kaggle environment | | `test_basis.py`, `test_weak_form.py`, `test_parser.py`, `test_cli.py`, `test_sweep.py` | Offline numerical, protocol, and workflow tests | +| [scripts/reproduce_paper_figures.py](scripts/reproduce_paper_figures.py), [paper_artifacts/](paper_artifacts/) | Reproduce the paper figures and audit the archived study | | [archive/](archive/) | Historical demos and earlier interfaces; outside the supported quick start | diff --git a/algorithm.sty b/algorithm.sty new file mode 100644 index 0000000..843e3d5 --- /dev/null +++ b/algorithm.sty @@ -0,0 +1,79 @@ +% ALGORITHM STYLE -- Released 8 April 1996 +% for LaTeX-2e +% Copyright -- 1994 Peter Williams +% E-mail Peter.Williams@dsto.defence.gov.au +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{algorithm} +\typeout{Document Style `algorithm' - floating environment} + +\RequirePackage{float} +\RequirePackage{ifthen} +\newcommand{\ALG@within}{nothing} +\newboolean{ALG@within} +\setboolean{ALG@within}{false} +\newcommand{\ALG@floatstyle}{ruled} +\newcommand{\ALG@name}{Algorithm} +\newcommand{\listalgorithmname}{List of \ALG@name s} + +% Declare Options +% first appearance +\DeclareOption{plain}{ + \renewcommand{\ALG@floatstyle}{plain} +} +\DeclareOption{ruled}{ + \renewcommand{\ALG@floatstyle}{ruled} +} +\DeclareOption{boxed}{ + \renewcommand{\ALG@floatstyle}{boxed} +} +% then numbering convention +\DeclareOption{part}{ + \renewcommand{\ALG@within}{part} + \setboolean{ALG@within}{true} +} +\DeclareOption{chapter}{ + \renewcommand{\ALG@within}{chapter} + \setboolean{ALG@within}{true} +} +\DeclareOption{section}{ + \renewcommand{\ALG@within}{section} + \setboolean{ALG@within}{true} +} +\DeclareOption{subsection}{ + \renewcommand{\ALG@within}{subsection} + \setboolean{ALG@within}{true} +} +\DeclareOption{subsubsection}{ + \renewcommand{\ALG@within}{subsubsection} + \setboolean{ALG@within}{true} +} +\DeclareOption{nothing}{ + \renewcommand{\ALG@within}{nothing} + \setboolean{ALG@within}{true} +} +\DeclareOption*{\edef\ALG@name{\CurrentOption}} + +% ALGORITHM +% +\ProcessOptions +\floatstyle{\ALG@floatstyle} +\ifthenelse{\boolean{ALG@within}}{ + \ifthenelse{\equal{\ALG@within}{part}} + {\newfloat{algorithm}{htbp}{loa}[part]}{} + \ifthenelse{\equal{\ALG@within}{chapter}} + {\newfloat{algorithm}{htbp}{loa}[chapter]}{} + \ifthenelse{\equal{\ALG@within}{section}} + {\newfloat{algorithm}{htbp}{loa}[section]}{} + \ifthenelse{\equal{\ALG@within}{subsection}} + {\newfloat{algorithm}{htbp}{loa}[subsection]}{} + \ifthenelse{\equal{\ALG@within}{subsubsection}} + {\newfloat{algorithm}{htbp}{loa}[subsubsection]}{} + \ifthenelse{\equal{\ALG@within}{nothing}} + {\newfloat{algorithm}{htbp}{loa}}{} +}{ + \newfloat{algorithm}{htbp}{loa} +} +\floatname{algorithm}{\ALG@name} + +\newcommand{\listofalgorithms}{\listof{algorithm}{\listalgorithmname}} + diff --git a/algorithmic.sty b/algorithmic.sty new file mode 100644 index 0000000..ad61478 --- /dev/null +++ b/algorithmic.sty @@ -0,0 +1,201 @@ +% ALGORITHMIC STYLE -- Released 8 APRIL 1996 +% for LaTeX version 2e +% Copyright -- 1994 Peter Williams +% E-mail PeterWilliams@dsto.defence.gov.au +% +% Modified by Alex Smola (08/2000) +% E-mail Alex.Smola@anu.edu.au +% +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{algorithmic} +\typeout{Document Style `algorithmic' - environment} +% +\RequirePackage{ifthen} +\RequirePackage{calc} +\newboolean{ALC@noend} +\setboolean{ALC@noend}{false} +\newcounter{ALC@line} +\newcounter{ALC@rem} +\newlength{\ALC@tlm} +% +\DeclareOption{noend}{\setboolean{ALC@noend}{true}} +% +\ProcessOptions +% +% ALGORITHMIC +\newcommand{\algorithmicrequire}{\textbf{Require:}} +\newcommand{\algorithmicensure}{\textbf{Ensure:}} +\newcommand{\algorithmiccomment}[1]{\{#1\}} +\newcommand{\algorithmicend}{\textbf{end}} +\newcommand{\algorithmicif}{\textbf{if}} +\newcommand{\algorithmicthen}{\textbf{then}} +\newcommand{\algorithmicelse}{\textbf{else}} +\newcommand{\algorithmicelsif}{\algorithmicelse\ \algorithmicif} +\newcommand{\algorithmicendif}{\algorithmicend\ \algorithmicif} +\newcommand{\algorithmicfor}{\textbf{for}} +\newcommand{\algorithmicforall}{\textbf{for all}} +\newcommand{\algorithmicdo}{\textbf{do}} +\newcommand{\algorithmicendfor}{\algorithmicend\ \algorithmicfor} +\newcommand{\algorithmicwhile}{\textbf{while}} +\newcommand{\algorithmicendwhile}{\algorithmicend\ \algorithmicwhile} +\newcommand{\algorithmicloop}{\textbf{loop}} +\newcommand{\algorithmicendloop}{\algorithmicend\ \algorithmicloop} +\newcommand{\algorithmicrepeat}{\textbf{repeat}} +\newcommand{\algorithmicuntil}{\textbf{until}} + +%changed by alex smola +\newcommand{\algorithmicinput}{\textbf{input}} +\newcommand{\algorithmicoutput}{\textbf{output}} +\newcommand{\algorithmicset}{\textbf{set}} +\newcommand{\algorithmictrue}{\textbf{true}} +\newcommand{\algorithmicfalse}{\textbf{false}} +\newcommand{\algorithmicand}{\textbf{and\ }} +\newcommand{\algorithmicor}{\textbf{or\ }} +\newcommand{\algorithmicfunction}{\textbf{function}} +\newcommand{\algorithmicendfunction}{\algorithmicend\ \algorithmicfunction} +\newcommand{\algorithmicmain}{\textbf{main}} +\newcommand{\algorithmicendmain}{\algorithmicend\ \algorithmicmain} +%end changed by alex smola + +\def\ALC@item[#1]{% +\if@noparitem \@donoparitem + \else \if@inlabel \indent \par \fi + \ifhmode \unskip\unskip \par \fi + \if@newlist \if@nobreak \@nbitem \else + \addpenalty\@beginparpenalty + \addvspace\@topsep \addvspace{-\parskip}\fi + \else \addpenalty\@itempenalty \addvspace\itemsep + \fi + \global\@inlabeltrue +\fi +\everypar{\global\@minipagefalse\global\@newlistfalse + \if@inlabel\global\@inlabelfalse \hskip -\parindent \box\@labels + \penalty\z@ \fi + \everypar{}}\global\@nobreakfalse +\if@noitemarg \@noitemargfalse \if@nmbrlist \refstepcounter{\@listctr}\fi \fi +\sbox\@tempboxa{\makelabel{#1}}% +\global\setbox\@labels + \hbox{\unhbox\@labels \hskip \itemindent + \hskip -\labelwidth \hskip -\ALC@tlm + \ifdim \wd\@tempboxa >\labelwidth + \box\@tempboxa + \else \hbox to\labelwidth {\unhbox\@tempboxa}\fi + \hskip \ALC@tlm}\ignorespaces} +% +\newenvironment{algorithmic}[1][0]{ +\let\@item\ALC@item + \newcommand{\ALC@lno}{% +\ifthenelse{\equal{\arabic{ALC@rem}}{0}} +{{\footnotesize \arabic{ALC@line}:}}{}% +} +\let\@listii\@listi +\let\@listiii\@listi +\let\@listiv\@listi +\let\@listv\@listi +\let\@listvi\@listi +\let\@listvii\@listi + \newenvironment{ALC@g}{ + \begin{list}{\ALC@lno}{ \itemsep\z@ \itemindent\z@ + \listparindent\z@ \rightmargin\z@ + \topsep\z@ \partopsep\z@ \parskip\z@\parsep\z@ + \leftmargin 1em + \addtolength{\ALC@tlm}{\leftmargin} + } + } + {\end{list}} + \newcommand{\ALC@it}{\addtocounter{ALC@line}{1}\addtocounter{ALC@rem}{1}\ifthenelse{\equal{\arabic{ALC@rem}}{#1}}{\setcounter{ALC@rem}{0}}{}\item} + \newcommand{\ALC@com}[1]{\ifthenelse{\equal{##1}{default}}% +{}{\ \algorithmiccomment{##1}}} + \newcommand{\REQUIRE}{\item[\algorithmicrequire]} + \newcommand{\ENSURE}{\item[\algorithmicensure]} + \newcommand{\STATE}{\ALC@it} + \newcommand{\COMMENT}[1]{\algorithmiccomment{##1}} +%changes by alex smola + \newcommand{\INPUT}{\item[\algorithmicinput]} + \newcommand{\OUTPUT}{\item[\algorithmicoutput]} + \newcommand{\SET}{\item[\algorithmicset]} +% \newcommand{\TRUE}{\algorithmictrue} +% \newcommand{\FALSE}{\algorithmicfalse} + \newcommand{\AND}{\algorithmicand} + \newcommand{\OR}{\algorithmicor} + \newenvironment{ALC@func}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@main}{\begin{ALC@g}}{\end{ALC@g}} +%end changes by alex smola + \newenvironment{ALC@if}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@for}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@whl}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@loop}{\begin{ALC@g}}{\end{ALC@g}} + \newenvironment{ALC@rpt}{\begin{ALC@g}}{\end{ALC@g}} + \renewcommand{\\}{\@centercr} + \newcommand{\IF}[2][default]{\ALC@it\algorithmicif\ ##2\ \algorithmicthen% +\ALC@com{##1}\begin{ALC@if}} + \newcommand{\SHORTIF}[2]{\ALC@it\algorithmicif\ ##1\ + \algorithmicthen\ {##2}} + \newcommand{\ELSE}[1][default]{\end{ALC@if}\ALC@it\algorithmicelse% +\ALC@com{##1}\begin{ALC@if}} + \newcommand{\ELSIF}[2][default]% +{\end{ALC@if}\ALC@it\algorithmicelsif\ ##2\ \algorithmicthen% +\ALC@com{##1}\begin{ALC@if}} + \newcommand{\FOR}[2][default]{\ALC@it\algorithmicfor\ ##2\ \algorithmicdo% +\ALC@com{##1}\begin{ALC@for}} + \newcommand{\FORALL}[2][default]{\ALC@it\algorithmicforall\ ##2\ % +\algorithmicdo% +\ALC@com{##1}\begin{ALC@for}} + \newcommand{\SHORTFORALL}[2]{\ALC@it\algorithmicforall\ ##1\ % + \algorithmicdo\ {##2}} + \newcommand{\WHILE}[2][default]{\ALC@it\algorithmicwhile\ ##2\ % +\algorithmicdo% +\ALC@com{##1}\begin{ALC@whl}} + \newcommand{\LOOP}[1][default]{\ALC@it\algorithmicloop% +\ALC@com{##1}\begin{ALC@loop}} +%changed by alex smola + \newcommand{\FUNCTION}[2][default]{\ALC@it\algorithmicfunction\ ##2\ % + \ALC@com{##1}\begin{ALC@func}} + \newcommand{\MAIN}[2][default]{\ALC@it\algorithmicmain\ ##2\ % + \ALC@com{##1}\begin{ALC@main}} +%end changed by alex smola + \newcommand{\REPEAT}[1][default]{\ALC@it\algorithmicrepeat% + \ALC@com{##1}\begin{ALC@rpt}} + \newcommand{\UNTIL}[1]{\end{ALC@rpt}\ALC@it\algorithmicuntil\ ##1} + \ifthenelse{\boolean{ALC@noend}}{ + \newcommand{\ENDIF}{\end{ALC@if}} + \newcommand{\ENDFOR}{\end{ALC@for}} + \newcommand{\ENDWHILE}{\end{ALC@whl}} + \newcommand{\ENDLOOP}{\end{ALC@loop}} + \newcommand{\ENDFUNCTION}{\end{ALC@func}} + \newcommand{\ENDMAIN}{\end{ALC@main}} + }{ + \newcommand{\ENDIF}{\end{ALC@if}\ALC@it\algorithmicendif} + \newcommand{\ENDFOR}{\end{ALC@for}\ALC@it\algorithmicendfor} + \newcommand{\ENDWHILE}{\end{ALC@whl}\ALC@it\algorithmicendwhile} + \newcommand{\ENDLOOP}{\end{ALC@loop}\ALC@it\algorithmicendloop} + \newcommand{\ENDFUNCTION}{\end{ALC@func}\ALC@it\algorithmicendfunction} + \newcommand{\ENDMAIN}{\end{ALC@main}\ALC@it\algorithmicendmain} + } + \renewcommand{\@toodeep}{} + \begin{list}{\ALC@lno}{\setcounter{ALC@line}{0}\setcounter{ALC@rem}{0}% + \itemsep\z@ \itemindent\z@ \listparindent\z@% + \partopsep\z@ \parskip\z@ \parsep\z@% + \labelsep 0.5em \topsep 0.2em% + \ifthenelse{\equal{#1}{0}} + {\labelwidth 0.5em } + {\labelwidth 1.2em } + \leftmargin\labelwidth \addtolength{\leftmargin}{\labelsep} + \ALC@tlm\labelsep + } + } + {\end{list}} + + + + + + + + + + + + + + diff --git a/fancyhdr.sty b/fancyhdr.sty new file mode 100644 index 0000000..b3d811f --- /dev/null +++ b/fancyhdr.sty @@ -0,0 +1,864 @@ +%% +%% This is file `fancyhdr.sty', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% fancyhdr.dtx (with options: `fancyhdr') +%% +%% This is a generated file. +%% +%% This file may be distributed and/or modified under the conditions of +%% the LaTeX Project Public License, either version 1.3 of this license +%% or (at your option) any later version. The latest version of this +%% license is in: +%% +%% http://www.latex-project.org/lppl.txt +%% +%% and version 1.3 or later is part of all distributions of LaTeX version +%% 2005/12/01 or later. +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\NeedsTeXFormat{LaTeX2e}[2018-04-01] +\ProvidesPackage{fancyhdr}% + [2025/02/07 v5.2 + Extensive control of page headers and footers]% +% Copyright (C) 1994-2025 by Pieter van Oostrum +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\ifdefined\NewDocumentCommand\else\RequirePackage{xparse}\fi +\newif\iff@nch@check +\f@nch@checktrue +\DeclareOption{nocheck}{% + \f@nch@checkfalse +} +\let\f@nch@gbl\relax +\newif\iff@nch@compatViii +\DeclareOption{compatV3}{% + \PackageWarningNoLine{fancyhdr}{The `compatV3' option is deprecated.\MessageBreak + It will disappear in one of the following releases.\MessageBreak + Please change your document to work\MessageBreak + without this option} + \let\f@nch@gbl\global + \f@nch@compatViiitrue +} +\newif\iff@nch@twoside +\f@nch@twosidefalse +\DeclareOption{twoside}{% + \if@twoside\else\f@nch@twosidetrue\fi +} +\newcommand\f@nch@def[2]{% + \def\temp@a{#2}\ifx\temp@a\@empty\f@nch@gbl\def#1{}% + \else\f@nch@gbl\def#1{#2\strut}\fi} +\DeclareOption{myheadings}{% + \@ifundefined{chapter}{% + \def\ps@myheadings{\ps@f@nch@fancyproto \let\@mkboth\@gobbletwo + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \let\sectionmark\@gobble + \let\subsectionmark\@gobble + }% + }% + {\def\ps@myheadings{\ps@f@nch@fancyproto \let\@mkboth\@gobbletwo + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \let\chaptermark\@gobble + \let\sectionmark\@gobble + }% + }% +} +\DeclareOption{headings}{% + \@ifundefined{chapter}{% + \if@twoside + \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth} + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \def\sectionmark##1{% + \markboth{\MakeUppercase{% + \ifnum \c@secnumdepth >\z@ \thesection\quad \fi##1}}{}}% + \def\subsectionmark##1{% + \markright{% + \ifnum \c@secnumdepth >\@ne \thesubsection\quad \fi##1}}% + }% + \else + \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth} + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \def\sectionmark##1{% + \markright {\MakeUppercase{% + \ifnum \c@secnumdepth >\z@ \thesection\quad \fi##1}}}% + \let\subsectionmark\@gobble % Not needed but inserted for safety + }% + \fi + }{\if@twoside + \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth} + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \def\chaptermark##1{% + \markboth{\MakeUppercase{% + \ifnum \c@secnumdepth >\m@ne \if@mainmatter + \@chapapp\ \thechapter. \ \fi\fi##1}}{}}% + \def\sectionmark##1{% + \markright {\MakeUppercase{% + \ifnum \c@secnumdepth >\z@ \thesection. \ \fi##1}}}% + }% + \else + \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth} + \fancyhf{} + \fancyhead[LE,RO]{\thepage}% + \fancyhead[RE]{\slshape\leftmark}% + \fancyhead[LO]{\slshape\rightmark}% + \def\chaptermark##1{% + \markright{\MakeUppercase{% + \ifnum \c@secnumdepth >\m@ne \if@mainmatter + \@chapapp\ \thechapter. \ \fi\fi##1}}}% + \let\sectionmark\@gobble % Not needed but inserted for safety + }% + \fi + }% +} +\ProcessOptions* +\newcommand{\f@nch@forc}[3]{\expandafter\f@nchf@rc\expandafter#1\expandafter{#2}{#3}} +\newcommand{\f@nchf@rc}[3]{\def\temp@ty{#2}\ifx\@empty\temp@ty\else + \f@nch@rc#1#2\f@nch@rc{#3}\fi} +\long\def\f@nch@rc#1#2#3\f@nch@rc#4{\def#1{#2}#4\f@nchf@rc#1{#3}{#4}} +\newcommand{\f@nch@for}[3]{\edef\@fortmp{#2}% + \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}} +\newcommand\f@nch@default[3]{% + \edef\temp@a{\lowercase{\edef\noexpand\temp@a{#3}}}\temp@a \def#1{}% + \f@nch@forc\tmpf@ra{#2}% + {\expandafter\f@nch@ifin\tmpf@ra\temp@a{\edef#1{#1\tmpf@ra}}{}}% + \ifx\@empty#1\def#1{#2}\fi} +\newcommand{\f@nch@ifin}[4]{% + \edef\temp@a{#2}\def\temp@b##1#1##2\temp@b{\def\temp@b{##1}}% + \expandafter\temp@b#2#1\temp@b\ifx\temp@a\temp@b #4\else #3\fi} +\newcommand{\fancyhead}[2][]{\f@nch@fancyhf\fancyhead h[#1]{#2}}% +\newcommand{\fancyfoot}[2][]{\f@nch@fancyhf\fancyfoot f[#1]{#2}}% +\newcommand{\fancyhf}[2][]{\f@nch@fancyhf\fancyhf {}[#1]{#2}}% +\newcommand{\fancyheadoffset}[2][]{\f@nch@fancyhfoffs\fancyheadoffset h[#1]{#2}}% +\newcommand{\fancyfootoffset}[2][]{\f@nch@fancyhfoffs\fancyfootoffset f[#1]{#2}}% +\newcommand{\fancyhfoffset}[2][]{\f@nch@fancyhfoffs\fancyhfoffset {}[#1]{#2}}% +\def\f@nch@fancyhf@Echeck#1{% + \if@twoside\else + \iff@nch@twoside\else + \if\f@nch@@eo e% + \PackageWarning{fancyhdr} {\string#1's `E' option without twoside option is useless.\MessageBreak + Please consider using the `twoside' option}% + \fi\fi\fi +} +\long\def\f@nch@fancyhf#1#2[#3]#4{% + \def\temp@c{}% + \f@nch@forc\tmpf@ra{#3}% + {\expandafter\f@nch@ifin\tmpf@ra{eolcrhf,EOLCRHF}% + {}{\edef\temp@c{\temp@c\tmpf@ra}}}% + \ifx\@empty\temp@c\else \PackageError{fancyhdr}{Illegal char `\temp@c' in + \string#1 argument: [#3]}{}% + \fi \f@nch@for\temp@c{#3}% + {\f@nch@default\f@nch@@eo{eo}\temp@c + \f@nch@fancyhf@Echeck{#1}% + \f@nch@default\f@nch@@lcr{lcr}\temp@c + \f@nch@default\f@nch@@hf{hf}{#2\temp@c}% + \f@nch@forc\f@nch@eo\f@nch@@eo + {\f@nch@forc\f@nch@lcr\f@nch@@lcr + {\f@nch@forc\f@nch@hf\f@nch@@hf + {\expandafter\f@nch@def\csname + f@nch@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname {#4}}}}}} +\def\f@nch@fancyhfoffs#1#2[#3]#4{% + \def\temp@c{}% + \f@nch@forc\tmpf@ra{#3}% + {\expandafter\f@nch@ifin\tmpf@ra{eolrhf,EOLRHF}% + {}{\edef\temp@c{\temp@c\tmpf@ra}}}% + \ifx\@empty\temp@c\else \PackageError{fancyhdr}{Illegal char `\temp@c' in + \string#1 argument: [#3]}{}% + \fi \f@nch@for\temp@c{#3}% + {\f@nch@default\f@nch@@eo{eo}\temp@c + \f@nch@fancyhf@Echeck{#1}% + \f@nch@default\f@nch@@lcr{lr}\temp@c + \f@nch@default\f@nch@@hf{hf}{#2\temp@c}% + \f@nch@forc\f@nch@eo\f@nch@@eo + {\f@nch@forc\f@nch@lcr\f@nch@@lcr + {\f@nch@forc\f@nch@hf\f@nch@@hf + {\expandafter\setlength\csname + f@nch@offset@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname {#4}}}}}% + \f@nch@setoffs} +\NewDocumentCommand {\fancyheadwidth}{ s O{} O{} m } + {\f@nch@fancyhfwidth{#1}\fancyheadwidth h[#2][#3]{#4}}% +\NewDocumentCommand {\fancyfootwidth}{ s O{} O{} m } + {\f@nch@fancyhfwidth{#1}\fancyfootwidth f[#2][#3]{#4}}% +\NewDocumentCommand {\fancyhfwidth} { s O{} O{} m } + {\f@nch@fancyhfwidth{#1}\fancyhfwidth {}[#2][#3]{#4}}% +\def\f@nch@fancyhfwidth#1#2#3[#4][#5]#6{% + \setlength\@tempdima{#6}% + \def\temp@c{}% + \f@nch@forc\tmpf@ra{#4}% + {\expandafter\f@nch@ifin\tmpf@ra{eolcrhf,EOLCRHF}% + {}{\edef\temp@c{\temp@c\tmpf@ra}}}% + \ifx\@empty\temp@c\else \PackageError{fancyhdr}{Illegal char `\temp@c' in + \string#2 argument: [#4]}{}% + \fi + \f@nch@for\temp@c{#4}% + {\f@nch@default\f@nch@@eo{eo}\temp@c + \f@nch@fancyhf@Echeck{#2}% + \f@nch@default\f@nch@@lcr{lcr}\temp@c + \f@nch@default\f@nch@@hf{hf}{#3\temp@c}% + \f@nch@forc\f@nch@eo\f@nch@@eo + {\f@nch@forc\f@nch@lcr\f@nch@@lcr + {\f@nch@forc\f@nch@hf\f@nch@@hf + {% + \IfBooleanTF{#1}{% + \expandafter\edef\csname + f@nch@width@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname{\the\@tempdima}% + }% + {% + \expandafter\def\csname + f@nch@width@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname{#6}% + }% + \csname f@nchdrwdt@align@v@\f@nch@hf\endcsname + \edef\f@nch@align@@h{\f@nch@lcr}% + \def\temp@a{#5}% + \ifx\temp@a\@empty \else \f@nchdrwdt@align#5\@nil{#2}\fi + \expandafter\edef\csname + f@nch@align@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname + {\f@nch@align@@v\f@nch@align@@h}}}}}} +\def\f@nch@width@elh{\headwidth} +\def\f@nch@width@ech{\headwidth} +\def\f@nch@width@erh{\headwidth} +\def\f@nch@width@olh{\headwidth} +\def\f@nch@width@och{\headwidth} +\def\f@nch@width@orh{\headwidth} +\def\f@nch@width@elf{\headwidth} +\def\f@nch@width@ecf{\headwidth} +\def\f@nch@width@erf{\headwidth} +\def\f@nch@width@olf{\headwidth} +\def\f@nch@width@ocf{\headwidth} +\def\f@nch@width@orf{\headwidth} +\def\f@nch@align@elh{bl} +\def\f@nch@align@ech{bc} +\def\f@nch@align@erh{br} +\def\f@nch@align@olh{bl} +\def\f@nch@align@och{bc} +\def\f@nch@align@orh{br} +\def\f@nch@align@elf{tl} +\def\f@nch@align@ecf{tc} +\def\f@nch@align@erf{tr} +\def\f@nch@align@olf{tl} +\def\f@nch@align@ocf{tc} +\def\f@nch@align@orf{tr} +\def\f@nchdrwdt@align@v@h{\def\f@nch@align@@v{b}}% +\def\f@nchdrwdt@align@v@f{\def\f@nch@align@@v{t}}% +\long\def\f@nchdrwdt@align#1#2\@nil#3{% + \f@nch@ifin{#1}{TtcbB-}{% + \f@nch@ifin{#1}{-}{}{\def\f@nch@align@@v{#1}}% + \def\@tempa{#2}% + \ifx\@tempa\@empty \else \def\f@nch@align@@h{#2}\fi + }% + {\def\f@nch@align@@h{#1}}% + \expandafter\f@nch@ifin\expandafter{\f@nch@align@@h}{lcrj}{}% + {\PackageError{fancyhdr} + {\string#3: Illegal char `\f@nch@align@@h'\MessageBreak + in alignment argument}{}}% +} +\newcommand{\lhead}[2][\f@nch@olh]% + {\f@nch@def\f@nch@olh{#2}\f@nch@def\f@nch@elh{#1}} +\newcommand{\chead}[2][\f@nch@och]% + {\f@nch@def\f@nch@och{#2}\f@nch@def\f@nch@ech{#1}} +\newcommand{\rhead}[2][\f@nch@orh]% + {\f@nch@def\f@nch@orh{#2}\f@nch@def\f@nch@erh{#1}} +\newcommand{\lfoot}[2][\f@nch@olf]% + {\f@nch@def\f@nch@olf{#2}\f@nch@def\f@nch@elf{#1}} +\newcommand{\cfoot}[2][\f@nch@ocf]% + {\f@nch@def\f@nch@ocf{#2}\f@nch@def\f@nch@ecf{#1}} +\newcommand{\rfoot}[2][\f@nch@orf]% + {\f@nch@def\f@nch@orf{#2}\f@nch@def\f@nch@erf{#1}} +\newlength{\f@nch@headwidth} \let\headwidth\f@nch@headwidth +\newlength{\f@nch@offset@elh} +\newlength{\f@nch@offset@erh} +\newlength{\f@nch@offset@olh} +\newlength{\f@nch@offset@orh} +\newlength{\f@nch@offset@elf} +\newlength{\f@nch@offset@erf} +\newlength{\f@nch@offset@olf} +\newlength{\f@nch@offset@orf} +\newcommand{\headrulewidth}{0.4pt} +\newcommand{\footrulewidth}{0pt} +\@ifundefined{headruleskip}% + {\newcommand{\headruleskip}{0pt}}{} +\@ifundefined{footruleskip}% + {\newcommand{\footruleskip}{.3\normalbaselineskip}}{} +\newcommand{\plainheadrulewidth}{0pt} +\newcommand{\plainfootrulewidth}{0pt} +\newif\if@fancyplain \@fancyplainfalse +\def\fancyplain#1#2{\if@fancyplain#1\else#2\fi} +\headwidth=-123456789sp +\let\f@nch@raggedleft\raggedleft +\let\f@nch@raggedright\raggedright +\let\f@nch@centering\centering +\let\f@nch@everypar\everypar +\ifdefined\ExplSyntaxOn + \ExplSyntaxOn + \providecommand\IfFormatAtLeastTF{\@ifl@t@r\fmtversion} + \IfFormatAtLeastTF{2021-06-01}{ + \def\f@nch@saveclr@parhook #1{ + \expandafter\let\csname f@nch@__hook~#1\expandafter\endcsname + \csname __hook~#1\endcsname + \expandafter\let\csname f@nch@__hook_toplevel~#1\expandafter\endcsname + \csname __hook_toplevel~#1\endcsname + \expandafter\let\csname f@nch@__hook_next~#1\expandafter\endcsname + \csname __hook_next~#1\endcsname + \expandafter\let\csname f@nch@g__hook_#1_code_prop\expandafter\endcsname + \csname g__hook_#1_code_prop\endcsname + \RemoveFromHook{#1}[*] + \ClearHookNext{#1} + } + \def\f@nch@restore@parhook #1{ + \global\expandafter\let\csname __hook~#1\expandafter\endcsname + \csname f@nch@__hook~#1\endcsname + \global\expandafter\let\csname __hook_toplevel~#1\expandafter\endcsname + \csname f@nch@__hook_toplevel~#1\endcsname + \global\expandafter\let\csname __hook_next~#1\expandafter\endcsname + \csname f@nch@__hook_next~#1\endcsname + \global\expandafter\let\csname g__hook_#1_code_prop\expandafter\endcsname + \csname f@nch@g__hook_#1_code_prop\endcsname + } + \def\f@nch@resetpar{ + \f@nch@everypar{} + \f@nch@saveclr@parhook{para/before} + \f@nch@saveclr@parhook{para/begin} + \f@nch@saveclr@parhook{para/end} + \f@nch@saveclr@parhook{para/after} + } + \def\f@nch@restorepar{ + \f@nch@restore@parhook{para/before} + \f@nch@restore@parhook{para/begin} + \f@nch@restore@parhook{para/end} + \f@nch@restore@parhook{para/after} + } + }{ + \def\f@nch@resetpar{ + \f@nch@everypar{} + } + \def\f@nch@restorepar{} + } + \ExplSyntaxOff +\else + \def\f@nch@resetpar{% + \f@nch@everypar{}% + } + \def\f@nch@restorepar{} +\fi +\newcommand\f@nch@noUppercase[2][]{#2} +\def\f@nch@reset{\f@nch@resetpar\restorecr\endlinechar=13 + \catcode`\\=0\catcode`\{=1\catcode`\}=2\catcode`\$=3\catcode`\&=4 + \catcode`\#=6\catcode`\^=7\catcode`\_=8\catcode`\ =10\catcode`\@=11 + \catcode`\:=11\catcode`\~=13\catcode`\%=14 + \catcode0=15 %NULL + \catcode9=10 %TAB + \let\\\@normalcr \let\raggedleft\f@nch@raggedleft + \let\raggedright\f@nch@raggedright \let\centering\f@nch@centering + \def\baselinestretch{1}% + \hsize=\headwidth + \def\nouppercase##1{{% + \let\uppercase\relax\let\MakeUppercase\f@nch@noUppercase + \expandafter\let\csname MakeUppercase \endcsname\relax + \expandafter\def\csname MakeUppercase\space\space\space\endcsname + [####1]####2{####2}% + ##1}}% + \@ifundefined{@normalsize} {\normalsize} % for ucthesis.cls + {\@normalsize}% + } +\newcommand*{\fancycenter}[1][1em]{% + \@ifnextchar[{\f@nch@center{#1}}{\f@nch@center{#1}[3]}% +} +\def\f@nch@center#1[#2]#3#4#5{% + \def\@tempa{#4}\ifx\@tempa\@empty + \hbox to\linewidth{\color@begingroup{#3}\hfil {#5}\color@endgroup}% + \else + \setlength\@tempdima{#1}% + \setlength{\@tempdimb}{#2\@tempdima}% + \@tempdimc \@tempdimb \advance\@tempdimc -\@tempdima + \setlength\@tempskipa{\@tempdimb \@plus 1fil \@minus \@tempdimc}% + \@tempskipb\@tempskipa + \def\@tempa{#3}\ifx\@tempa\@empty + \addtolength\@tempskipa{\z@ \@minus \@tempdima}% + \fi + \def\@tempa{#5}\ifx\@tempa\@empty % empty right + \addtolength\@tempskipb{\z@ \@minus \@tempdima}% + \fi + \settowidth{\@tempdimb}{#3}% + \settowidth{\@tempdimc}{#5}% + \ifdim\@tempdimb>\@tempdimc + \advance\@tempdimb -\@tempdimc + \addtolength\@tempskipb{\@tempdimb \@minus \@tempdimb}% + \else + \advance\@tempdimc -\@tempdimb + \addtolength\@tempskipa{\@tempdimc \@minus \@tempdimc}% + \fi + \hbox to\linewidth{\color@begingroup{#3}\hskip \@tempskipa + {#4}\hskip \@tempskipb {#5}\color@endgroup}% + \fi +} +\newcommand{\f@nch@headinit}{} +\newcommand{\fancyheadinit}[1]{% + \def\f@nch@headinit{#1}% +} +\newcommand{\f@nch@footinit}{} +\newcommand{\fancyfootinit}[1]{% + \def\f@nch@footinit{#1}% +} +\newcommand{\fancyhfinit}[1]{% + \def\f@nch@headinit{#1}% + \def\f@nch@footinit{#1}% +} +\ifdefined\NewMirroredHookPair + \NewMirroredHookPair{fancyhdr/before}{fancyhdr/after} + \NewMirroredHookPair{fancyhdr/head/begin}{fancyhdr/head/end} + \NewMirroredHookPair{fancyhdr/foot/begin}{fancyhdr/foot/end} +\fi +\newlength\f@nch@height +\newlength\f@nch@footalignment +\newif\iff@nch@footalign\f@nch@footalignfalse +\newcommand{\fancyfootalign}[1]{% + \def\temp@a{#1}% + \ifx\temp@a\@empty + \f@nch@footalignfalse + \else + \f@nch@footaligntrue + \setlength\f@nch@footalignment{#1}% + \fi +} +\newcommand\fancyhdrsettoheight[2]{% + \expandafter\ifx\csname f@nch@#2\endcsname\fancyhdrsettoheight + \else\PackageError{fancyhdr}{Unknown parameter #2 in \string\fancyhdrsettoheight}{}\fi + \setbox\@tempboxa\hbox{{\f@nch@checkfalse\csname @#2\endcsname}}% + \setlength{#1}\f@nch@height + \setbox\@tempboxa\box\voidb@x +} +\let\f@nch@oddhead\fancyhdrsettoheight +\let\f@nch@evenhead\fancyhdrsettoheight +\let\f@nch@oddfoot\fancyhdrsettoheight +\let\f@nch@evenfoot\fancyhdrsettoheight +\newcommand\f@nch@vbox[2]{% + \setbox0\vbox{#2}% + \global\f@nch@height=\ht0 + \ifdim\ht0>#1\relax + \iff@nch@check + \dimen0=#1\advance\dimen0-\ht0 + \PackageWarning{fancyhdr}{% + \string#1 is too small (\the#1): \MessageBreak + Make it at least \the\ht0, for example:\MessageBreak + \string\setlength{\string#1}{\the\ht0}% + \iff@nch@compatViii .\MessageBreak + We now make it that large for the rest of the document.\MessageBreak + This may cause the page layout to be inconsistent, however + \fi + \ifx#1\headheight .\MessageBreak + You might also make \topmargin smaller:\MessageBreak + \string\addtolength{\string\topmargin}{\the\dimen0}% + \fi + \@gobble + }% + \iff@nch@compatViii + \dimen0=#1\relax + \global#1=\ht0\relax + \ht0=\dimen0 % + \else + \ht0=#1\relax + \fi + \else + \ht0=#1\relax + \fi + \fi + \box0} +\newcommand\f@nch@head[6]{% + \f@nch@reset + \ifdefined\UseHook\UseHook{fancyhdr/before}\UseHook{fancyhdr/head/begin}\fi + \f@nch@headinit\relax + #1% + \hbox to\headwidth{% + \f@nch@vbox\headheight{% + \f@nch@hfbox{#2}{#3}{#4}{#6}{h}% + \vskip\headruleskip\relax + \headrule + }% + }% + #5% + \ifdefined\UseHook\UseHook{fancyhdr/head/end}\UseHook{fancyhdr/after}\fi + \f@nch@restorepar +} +\newcommand\f@nch@foot[6]{% + \f@nch@reset + \ifdefined\UseHook\UseHook{fancyhdr/before}\UseHook{fancyhdr/foot/begin}\fi + \f@nch@footinit\relax + #1% + \hbox to\headwidth{% + \f@nch@vbox\footskip{% + \setbox0=\vbox{\footrule}\unvbox0 + \vskip\footruleskip + \f@nch@hfbox{#2}{#3}{#4}{#6}{f}% + \iff@nch@footalign \vskip\f@nch@footalignment \fi + }% + }% + #5% + \ifdefined\UseHook\UseHook{fancyhdr/foot/end}\UseHook{fancyhdr/after}\fi + \f@nch@restorepar +} +\newlength\f@nch@widthL +\newlength\f@nch@widthC +\newlength\f@nch@widthR +\newcommand\f@nch@hfbox[5]{% + \setlength\f@nch@widthL{\csname f@nch@width@#4l#5\endcsname}% + \setlength\f@nch@widthC{\csname f@nch@width@#4c#5\endcsname}% + \setlength\f@nch@widthR{\csname f@nch@width@#4r#5\endcsname}% + \let\@tempa\f@nch@hfbox@center + \ifdim \dimexpr \f@nch@widthL+\f@nch@widthC+\f@nch@widthR>\headwidth + \else + \ifdim \dimexpr \f@nch@widthL+0.5\f@nch@widthC>0.5\headwidth + \let \@tempa\f@nch@hfbox@fit + \fi + \ifdim \dimexpr \f@nch@widthR+0.5\f@nch@widthC>0.5\headwidth + \let \@tempa\f@nch@hfbox@fit + \fi + \fi + \@tempa{#1}{#2}{#3}#4#5% +} +\newcommand\f@nch@hfbox@center[5]{% + \hbox to \headwidth{% + \rlap{\f@nch@parbox{#1}\f@nch@widthL{#4}l{#5}}% + \hfill + \f@nch@parbox{#2}\f@nch@widthC{#4}c{#5}% + \hfill + \llap{\f@nch@parbox{#3}\f@nch@widthR{#4}r{#5}}% + }% +} +\newcommand\f@nch@hfbox@fit[5]{% + \hbox to \headwidth{% + \f@nch@parbox{#1}\f@nch@widthL{#4}l{#5}% + \hfill + \f@nch@parbox{#2}\f@nch@widthC{#4}c{#5}% + \hfill + \f@nch@parbox{#3}\f@nch@widthR{#4}r{#5}% + }% +}% +\newcommand\f@nch@parbox[5]{% + \expandafter\expandafter\expandafter\f@nch@parbox@align + \csname f@nch@align@#3#4#5\endcsname + \parbox[\f@nch@align@@v]{#2}% + {% + \f@nch@align@@pre + \f@nch@align@@h\leavevmode\ignorespaces#1% + \f@nch@align@@post + }% +} +\newcommand\f@nch@parbox@align[2]{% + \def\f@nch@align@@pre{}% + \def\f@nch@align@@post{}% + \csname f@nch@parbox@align@v#1\endcsname + \csname f@nch@parbox@align@h#2\endcsname +} +\def\f@nch@parbox@align@vT{\def\f@nch@align@@v{t}\def\f@nch@align@@pre{\vspace{0pt}}} +\def\f@nch@parbox@align@vt{\def\f@nch@align@@v{t}} +\def\f@nch@parbox@align@vc{\def\f@nch@align@@v{c}} +\def\f@nch@parbox@align@vb{\def\f@nch@align@@v{b}} +\def\f@nch@parbox@align@vB{\def\f@nch@align@@v{b}\def\f@nch@align@@post{\vspace{0pt}}} +\def\f@nch@parbox@align@hl{\def\f@nch@align@@h{\raggedright}} +\def\f@nch@parbox@align@hc{\def\f@nch@align@@h{\centering}} +\def\f@nch@parbox@align@hr{\def\f@nch@align@@h{\raggedleft}} +\def\f@nch@parbox@align@hj{\def\f@nch@align@@h{}} +\@ifundefined{@chapapp}{\let\@chapapp\chaptername}{}% +\def\f@nch@initialise{% + \@ifundefined{chapter}% + {\def\sectionmark##1{\markboth{\MakeUppercase{\ifnum \c@secnumdepth>\z@ + \thesection\hskip 1em\relax + \fi ##1}}{}}% + \def\subsectionmark##1{\markright {\ifnum \c@secnumdepth >\@ne + \thesubsection\hskip 1em\relax \fi ##1}}}% + {\def\chaptermark##1{\markboth {\MakeUppercase{\ifnum + \c@secnumdepth>\m@ne \@chapapp\ \thechapter. \ \fi ##1}}{}}% + \def\sectionmark##1{\markright{\MakeUppercase{\ifnum \c@secnumdepth >\z@ + \thesection. \ \fi ##1}}}% + }% + \def\headrule{{\if@fancyplain\let\headrulewidth\plainheadrulewidth\fi + \hrule\@height\headrulewidth\@width\headwidth + \vskip-\headrulewidth}}% + \def\footrule{{\if@fancyplain\let\footrulewidth\plainfootrulewidth\fi + \hrule\@width\headwidth\@height\footrulewidth}}% + \def\headrulewidth{0.4pt}% + \def\footrulewidth{0pt}% + \def\headruleskip{0pt}% + \def\footruleskip{0.3\normalbaselineskip}% + \fancyhf{}% + \if@twoside + \fancyhead[el,or]{\fancyplain{}{\slshape\rightmark}}% + \fancyhead[er,ol]{\fancyplain{}{\slshape\leftmark}}% + \else + \fancyhead[l]{\fancyplain{}{\slshape\rightmark}}% + \fancyhead[r]{\fancyplain{}{\slshape\leftmark}}% + \fi + \fancyfoot[c]{\rmfamily\thepage}% page number +} +\f@nch@initialise +\def\ps@f@nch@fancyproto{% + \ifdim\headwidth<0sp + \global\advance\headwidth123456789sp\global\advance\headwidth\textwidth + \fi + \gdef\ps@f@nch@fancyproto{\@fancyplainfalse\ps@f@nch@fancycore}% + \@fancyplainfalse\ps@f@nch@fancycore +}% +\@namedef{f@nch@ps@f@nch@fancyproto-is-fancyhdr}{} +\def\ps@fancy{\ps@f@nch@fancyproto} +\@namedef{f@nch@ps@fancy-is-fancyhdr}{} +\def\ps@fancyplain{\ps@f@nch@fancyproto \let\ps@plain\ps@plain@fancy} +\def\ps@plain@fancy{\@fancyplaintrue\ps@f@nch@fancycore} +\let\f@nch@ps@empty\ps@empty +\def\ps@f@nch@fancycore{% + \f@nch@ps@empty + \def\@mkboth{\protect\markboth}% + \def\f@nch@oddhead{\f@nch@head\f@nch@Oolh\f@nch@olh\f@nch@och\f@nch@orh\f@nch@Oorh{o}}% + \def\@oddhead{% + \iff@nch@twoside + \ifodd\c@page + \f@nch@oddhead + \else + \@evenhead + \fi + \else + \f@nch@oddhead + \fi + } + \def\f@nch@oddfoot{\f@nch@foot\f@nch@Oolf\f@nch@olf\f@nch@ocf\f@nch@orf\f@nch@Oorf{o}}% + \def\@oddfoot{% + \iff@nch@twoside + \ifodd\c@page + \f@nch@oddfoot + \else + \@evenfoot + \fi + \else + \f@nch@oddfoot + \fi + } + \def\@evenhead{\f@nch@head\f@nch@Oelh\f@nch@elh\f@nch@ech\f@nch@erh\f@nch@Oerh{e}}% + \def\@evenfoot{\f@nch@foot\f@nch@Oelf\f@nch@elf\f@nch@ecf\f@nch@erf\f@nch@Oerf{e}}% +} +\def\f@nch@Oolh{\if@reversemargin\hss\else\relax\fi} +\def\f@nch@Oorh{\if@reversemargin\relax\else\hss\fi} +\let\f@nch@Oelh\f@nch@Oorh +\let\f@nch@Oerh\f@nch@Oolh +\let\f@nch@Oolf\f@nch@Oolh +\let\f@nch@Oorf\f@nch@Oorh +\let\f@nch@Oelf\f@nch@Oelh +\let\f@nch@Oerf\f@nch@Oerh +\def\f@nch@offsolh{\headwidth=\textwidth\advance\headwidth\f@nch@offset@olh + \advance\headwidth\f@nch@offset@orh\hskip-\f@nch@offset@olh} +\def\f@nch@offselh{\headwidth=\textwidth\advance\headwidth\f@nch@offset@elh + \advance\headwidth\f@nch@offset@erh\hskip-\f@nch@offset@elh} +\def\f@nch@offsolf{\headwidth=\textwidth\advance\headwidth\f@nch@offset@olf + \advance\headwidth\f@nch@offset@orf\hskip-\f@nch@offset@olf} +\def\f@nch@offself{\headwidth=\textwidth\advance\headwidth\f@nch@offset@elf + \advance\headwidth\f@nch@offset@erf\hskip-\f@nch@offset@elf} +\def\f@nch@setoffs{% + \f@nch@gbl\let\headwidth\f@nch@headwidth + \f@nch@gbl\def\f@nch@Oolh{\f@nch@offsolh}% + \f@nch@gbl\def\f@nch@Oelh{\f@nch@offselh}% + \f@nch@gbl\def\f@nch@Oorh{\hss}% + \f@nch@gbl\def\f@nch@Oerh{\hss}% + \f@nch@gbl\def\f@nch@Oolf{\f@nch@offsolf}% + \f@nch@gbl\def\f@nch@Oelf{\f@nch@offself}% + \f@nch@gbl\def\f@nch@Oorf{\hss}% + \f@nch@gbl\def\f@nch@Oerf{\hss}% +} +\newif\iff@nch@footnote +\AtBeginDocument{% + \let\latex@makecol\@makecol + \def\@makecol{\ifvoid\footins\f@nch@footnotefalse\else\f@nch@footnotetrue\fi + \let\f@nch@topfloat\@toplist\let\f@nch@botfloat\@botlist\latex@makecol}% +} +\newcommand\iftopfloat[2]{\ifx\f@nch@topfloat\@empty #2\else #1\fi}% +\newcommand\ifbotfloat[2]{\ifx\f@nch@botfloat\@empty #2\else #1\fi}% +\newcommand\iffloatpage[2]{\if@fcolmade #1\else #2\fi}% +\newcommand\iffootnote[2]{\iff@nch@footnote #1\else #2\fi}% +\ifx\@temptokenb\undefined \csname newtoks\endcsname\@temptokenb\fi +\newif\iff@nch@pagestyle@star +\newcommand\fancypagestyle{% + \@ifstar{\f@nch@pagestyle@startrue\f@nch@pagestyle}% + {\f@nch@pagestyle@starfalse\f@nch@pagestyle}% +} +\newcommand\f@nch@pagestyle[1]{% + \@ifnextchar[{\f@nch@@pagestyle{#1}}{\f@nch@@pagestyle{#1}[f@nch@fancyproto]}% +} +\long\def\f@nch@@pagestyle#1[#2]#3{% + \@ifundefined{ps@#2}{% + \PackageError{fancyhdr}{\string\fancypagestyle: Unknown base page style `#2'}{}% + }{% + \@ifundefined{f@nch@ps@#2-is-fancyhdr}{% + \PackageError{fancyhdr}{\string\fancypagestyle: Base page style `#2' is not fancyhdr-based}{}% + }% + {% + \f@nch@pagestyle@setup + \def\temp@b{\@namedef{ps@#1}}% + \expandafter\temp@b\expandafter{\the\@temptokenb + \let\f@nch@gbl\relax\@nameuse{ps@#2}#3\relax}% + \@namedef{f@nch@ps@#1-is-fancyhdr}{}% + }% + }% +} +\newcommand\f@nch@pagestyle@setup{% + \iff@nch@pagestyle@star + \iff@nch@check\@temptokenb={\f@nch@checktrue}\else\@temptokenb={\f@nch@checkfalse}\fi + \@tfor\temp@a:= + \f@nch@olh\f@nch@och\f@nch@orh\f@nch@elh\f@nch@ech\f@nch@erh + \f@nch@olf\f@nch@ocf\f@nch@orf\f@nch@elf\f@nch@ecf\f@nch@erf + \f@nch@width@elh\f@nch@width@ech\f@nch@width@erh\f@nch@width@olh + \f@nch@width@och\f@nch@width@orh\f@nch@width@elf\f@nch@width@ecf + \f@nch@width@erf\f@nch@width@olf\f@nch@width@ocf\f@nch@width@orf + \f@nch@align@elh\f@nch@align@ech\f@nch@align@erh\f@nch@align@olh + \f@nch@align@och\f@nch@align@orh\f@nch@align@elf\f@nch@align@ecf + \f@nch@align@erf\f@nch@align@olf\f@nch@align@ocf\f@nch@align@orf + \f@nch@Oolh\f@nch@Oorh\f@nch@Oelh\f@nch@Oerh + \f@nch@Oolf\f@nch@Oorf\f@nch@Oelf\f@nch@Oerf + \f@nch@headinit\f@nch@footinit + \headrule\headrulewidth\footrule\footrulewidth + \do {% + \toks@=\expandafter\expandafter\expandafter{\temp@a}% + \toks@=\expandafter\expandafter\expandafter{% + \expandafter\expandafter\expandafter\def + \expandafter\expandafter\temp@a\expandafter{\the\toks@}}% + \edef\temp@b{\@temptokenb={\the\@temptokenb\the\toks@}}% + \temp@b + }% + \@tfor\temp@a:= + \f@nch@offset@olh\f@nch@offset@orh\f@nch@offset@elh\f@nch@offset@erh + \f@nch@offset@olf\f@nch@offset@orf\f@nch@offset@elf\f@nch@offset@erf + \do {% + \toks@=\expandafter\expandafter\expandafter{\expandafter\the\temp@a}% + \toks@=\expandafter\expandafter\expandafter{% + \expandafter\expandafter\expandafter\setlength + \expandafter\expandafter\temp@a\expandafter{\the\toks@}}% + \edef\temp@b{\@temptokenb={\the\@temptokenb\the\toks@}}% + \temp@b + }% + \else + \@temptokenb={}% + \fi +} +\newcommand\fancypagestyleassign[2]{% + \@ifundefined{ps@#2}{% + \PackageError{fancyhdr}{\string\fancypagestyleassign: Unknown page style `#2'}{}% + }{% + \expandafter\let + \csname ps@#1\expandafter\endcsname + \csname ps@#2\endcsname + \@ifundefined{f@nch@ps@#2-is-fancyhdr}{% + \expandafter\let\csname f@nch@ps@#1-is-fancyhdr\endcsname\@undefined + }{% + \@namedef{f@nch@ps@#1-is-fancyhdr}{}% + }% + }% +} +\fancypagestyle*{fancydefault}{\f@nch@initialise} +\def\f@nchdrbox@topstrut{\vrule height\ht\strutbox width\z@} +\def\f@nchdrbox@botstrut{\vrule depth\dp\strutbox width\z@} +\def\f@nchdrbox@nostrut{\noalign{\vspace{0pt}}\let\f@nchdrbox@@crstrut\f@nchdrbox@botstrut} +\NewDocumentCommand{\fancyhdrbox}{ O{cl} o m }{% +\begingroup + \let\f@nchdrbox@@pre\f@nchdrbox@topstrut + \let\f@nchdrbox@@postx\f@nchdrbox@botstrut + \let\f@nchdrbox@@posty\relax + \let\f@nchdrbox@@crstrut\strut + \IfNoValueTF{#2}% + {\let\f@nchdrbox@@halignto\@empty}% + {\setlength\@tempdima{#2}% + \def\f@nchdrbox@@halignto{to\@tempdima}}% + \def\@tempa{#1}% + \ifx\@tempa\@empty + \f@nchdrbox@align cl\@nil{#3}% + \else + \f@nchdrbox@align #1\@nil{#3}% + \fi +\endgroup +} +\protected\def\f@nchdrbox@cr{% + {\ifnum0=`}\fi\@ifstar\@f@nchdrbox@xcr\@f@nchdrbox@xcr} + +\def\@f@nchdrbox@xcr{% + \unskip\f@nchdrbox@@crstrut + \@ifnextchar[\@f@nchdrbox@argc{\ifnum0=`{\fi}\cr}% +} + +\def\@f@nchdrbox@argc[#1]{% + \ifnum0=`{\fi}% + \ifdim #1>\z@ + \unskip\@f@nchdrbox@xargc{#1}% + \else + \@f@nchdrbox@yargc{#1}% + \fi} + +\def\@f@nchdrbox@xargc#1{\@tempdima #1\advance\@tempdima \dp \strutbox + \vrule \@height\z@ \@depth\@tempdima \@width\z@ \cr} + +\def\@f@nchdrbox@yargc#1{\cr\noalign{\setlength\@tempdima{#1}\vskip\@tempdima}} +\def\f@nchdrbox@T{\let\f@nchdrbox@@pre\f@nchdrbox@nostrut + \f@nchdrbox@t} +\def\f@nchdrbox@t{\def\f@nchdrbox@@v{t}\def\f@nchdrbox@@h{l}} +\def\f@nchdrbox@c{\def\f@nchdrbox@@v{c}\def\f@nchdrbox@@h{c}} +\def\f@nchdrbox@b{\def\f@nchdrbox@@v{b}\def\f@nchdrbox@@h{l}} +\def\f@nchdrbox@B{\let\f@nchdrbox@@postx\relax + \def\f@nchdrbox@@posty{\vspace{0pt}}% + \f@nchdrbox@b} +\long\def\f@nchdrbox@align#1#2\@nil#3{% + \f@nch@ifin{#1}{TtcbB}{% + \@nameuse{f@nchdrbox@#1}% + \def\@tempa{#2}% + \ifx\@tempa\@empty\else \def\f@nchdrbox@@h{#2}\fi + }% + {\def\f@nchdrbox@@v{c}\def\f@nchdrbox@@h{#1}}% + \expandafter\f@nch@ifin\expandafter{\f@nchdrbox@@h}{lcr}{}% + {\PackageError{fancyhdr}{\string\fancyhdrbox: Illegal char `\f@nchdrbox@@h'\MessageBreak + in alignment argument}{}}% + \let\\\f@nchdrbox@cr + \setbox0=\if \f@nchdrbox@@v t\vtop + \else \vbox + \fi + {% + \ialign \f@nchdrbox@@halignto + \bgroup \relax + {\if \f@nchdrbox@@h l\hskip 1sp\else \hfil \fi + \ignorespaces ##\unskip + \if\f@nchdrbox@@h r\else \hfil \fi + }% + \tabskip\z@skip \cr + \f@nchdrbox@@pre + #3\unskip \f@nchdrbox@@postx + \crcr + \egroup + \f@nchdrbox@@posty + }% + \if\f@nchdrbox@@v c\@tempdima=\ht0\advance\@tempdima\dp0% + \ht0=0.5\@tempdima\dp0=0.5\@tempdima\fi + \leavevmode \box0 +} +\@ifclassloaded{newlfm} +{ + \let\ps@@empty\f@nch@ps@empty + \AtBeginDocument{% + \renewcommand{\@zfancyhead}[5]{\relax\hbox to\headwidth{\f@nch@reset + \@zfancyvbox\headheight{\hbox + {\rlap{\parbox[b]{\headwidth}{\raggedright\f@nch@olh}}\hfill + \parbox[b]{\headwidth}{\centering\f@nch@olh}\hfill + \llap{\parbox[b]{\headwidth}{\raggedleft\f@nch@orh}}}% + \zheadrule}}\relax}% + } +} +{} +\endinput +%% +%% End of file `fancyhdr.sty'. diff --git a/icml2026.bst b/icml2026.bst new file mode 100644 index 0000000..f1a50e8 --- /dev/null +++ b/icml2026.bst @@ -0,0 +1,1443 @@ +%% File: `icml2025.bst' +%% A modification of `plainnl.bst' for use with natbib package +%% +%% Copyright 2010 Hal Daum\'e III +%% Modified by J. Fürnkranz +%% - Changed labels from (X and Y, 2000) to (X & Y, 2000) +%% - Changed References to last name first and abbreviated first names. +%% Modified by Iain Murray 2018 (who suggests adopting a standard .bst in future...) +%% - Made it actually use abbreviated first names +%% +%% Copyright 1993-2007 Patrick W Daly +%% Max-Planck-Institut f\"ur Sonnensystemforschung +%% Max-Planck-Str. 2 +%% D-37191 Katlenburg-Lindau +%% Germany +%% E-mail: daly@mps.mpg.de +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN +%% archives in directory macros/latex/base/lppl.txt; either +%% version 1 of the License, or any later version. +%% + % Version and source file information: + % \ProvidesFile{icml2010.mbs}[2007/11/26 1.93 (PWD)] + % + % BibTeX `plainnat' family + % version 0.99b for BibTeX versions 0.99a or later, + % for LaTeX versions 2.09 and 2e. + % + % For use with the `natbib.sty' package; emulates the corresponding + % member of the `plain' family, but with author-year citations. + % + % With version 6.0 of `natbib.sty', it may also be used for numerical + % citations, while retaining the commands \citeauthor, \citefullauthor, + % and \citeyear to print the corresponding information. + % + % For version 7.0 of `natbib.sty', the KEY field replaces missing + % authors/editors, and the date is left blank in \bibitem. + % + % Includes field EID for the sequence/citation number of electronic journals + % which is used instead of page numbers. + % + % Includes fields ISBN and ISSN. + % + % Includes field URL for Internet addresses. + % + % Includes field DOI for Digital Object Idenfifiers. + % + % Works best with the url.sty package of Donald Arseneau. + % + % Works with identical authors and year are further sorted by + % citation key, to preserve any natural sequence. + % +ENTRY + { address + author + booktitle + chapter + doi + eid + edition + editor + howpublished + institution + isbn + issn + journal + key + month + note + number + organization + pages + publisher + school + series + title + type + url + volume + year + } + {} + { label extra.label sort.label short.list } + +INTEGERS { output.state before.all mid.sentence after.sentence after.block } + +FUNCTION {init.state.consts} +{ #0 'before.all := + #1 'mid.sentence := + #2 'after.sentence := + #3 'after.block := +} + +STRINGS { s t } + +FUNCTION {output.nonnull} +{ 's := + output.state mid.sentence = + { ", " * write$ } + { output.state after.block = + { add.period$ write$ + newline$ + "\newblock " write$ + } + { output.state before.all = + 'write$ + { add.period$ " " * write$ } + if$ + } + if$ + mid.sentence 'output.state := + } + if$ + s +} + +FUNCTION {output} +{ duplicate$ empty$ + 'pop$ + 'output.nonnull + if$ +} + +FUNCTION {output.check} +{ 't := + duplicate$ empty$ + { pop$ "empty " t * " in " * cite$ * warning$ } + 'output.nonnull + if$ +} + +FUNCTION {fin.entry} +{ add.period$ + write$ + newline$ +} + +FUNCTION {new.block} +{ output.state before.all = + 'skip$ + { after.block 'output.state := } + if$ +} + +FUNCTION {new.sentence} +{ output.state after.block = + 'skip$ + { output.state before.all = + 'skip$ + { after.sentence 'output.state := } + if$ + } + if$ +} + +FUNCTION {not} +{ { #0 } + { #1 } + if$ +} + +FUNCTION {and} +{ 'skip$ + { pop$ #0 } + if$ +} + +FUNCTION {or} +{ { pop$ #1 } + 'skip$ + if$ +} + +FUNCTION {new.block.checka} +{ empty$ + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.block.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.sentence.checka} +{ empty$ + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {new.sentence.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {field.or.null} +{ duplicate$ empty$ + { pop$ "" } + 'skip$ + if$ +} + +FUNCTION {emphasize} +{ duplicate$ empty$ + { pop$ "" } + { "\emph{" swap$ * "}" * } + if$ +} + +INTEGERS { nameptr namesleft numnames } + +FUNCTION {format.names} +{ 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't := + nameptr #1 > + { namesleft #1 > + { ", " * t * } + { numnames #2 > + { "," * } + 'skip$ + if$ + t "others" = + { " et~al." * } + { " and " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {format.key} +{ empty$ + { key field.or.null } + { "" } + if$ +} + +FUNCTION {format.authors} +{ author empty$ + { "" } + { author format.names } + if$ +} + +FUNCTION {format.editors} +{ editor empty$ + { "" } + { editor format.names + editor num.names$ #1 > + { " (eds.)" * } + { " (ed.)" * } + if$ + } + if$ +} + +FUNCTION {format.isbn} +{ isbn empty$ + { "" } + { new.block "ISBN " isbn * } + if$ +} + +FUNCTION {format.issn} +{ issn empty$ + { "" } + { new.block "ISSN " issn * } + if$ +} + +FUNCTION {format.url} +{ url empty$ + { "" } + { new.block "URL \url{" url * "}" * } + if$ +} + +FUNCTION {format.doi} +{ doi empty$ + { "" } + { new.block "\doi{" doi * "}" * } + if$ +} + +FUNCTION {format.title} +{ title empty$ + { "" } + { title "t" change.case$ } + if$ +} + +FUNCTION {format.full.names} +{'s := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr + "{vv~}{ll}" format.name$ 't := + nameptr #1 > + { + namesleft #1 > + { ", " * t * } + { + numnames #2 > + { "," * } + 'skip$ + if$ + t "others" = + { " et~al." * } + { " and " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {author.editor.full} +{ author empty$ + { editor empty$ + { "" } + { editor format.full.names } + if$ + } + { author format.full.names } + if$ +} + +FUNCTION {author.full} +{ author empty$ + { "" } + { author format.full.names } + if$ +} + +FUNCTION {editor.full} +{ editor empty$ + { "" } + { editor format.full.names } + if$ +} + +FUNCTION {make.full.names} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.full + { type$ "proceedings" = + 'editor.full + 'author.full + if$ + } + if$ +} + +FUNCTION {output.bibitem} +{ newline$ + "\bibitem[" write$ + label write$ + ")" make.full.names duplicate$ short.list = + { pop$ } + { * } + if$ + "]{" * write$ + cite$ write$ + "}" write$ + newline$ + "" + before.all 'output.state := +} + +FUNCTION {n.dashify} +{ 't := + "" + { t empty$ not } + { t #1 #1 substring$ "-" = + { t #1 #2 substring$ "--" = not + { "--" * + t #2 global.max$ substring$ 't := + } + { { t #1 #1 substring$ "-" = } + { "-" * + t #2 global.max$ substring$ 't := + } + while$ + } + if$ + } + { t #1 #1 substring$ * + t #2 global.max$ substring$ 't := + } + if$ + } + while$ +} + +FUNCTION {format.date} +{ year duplicate$ empty$ + { "empty year in " cite$ * warning$ + pop$ "" } + 'skip$ + if$ + month empty$ + 'skip$ + { month + " " * swap$ * + } + if$ + extra.label * +} + +FUNCTION {format.btitle} +{ title emphasize +} + +FUNCTION {tie.or.space.connect} +{ duplicate$ text.length$ #3 < + { "~" } + { " " } + if$ + swap$ * * +} + +FUNCTION {either.or.check} +{ empty$ + 'pop$ + { "can't use both " swap$ * " fields in " * cite$ * warning$ } + if$ +} + +FUNCTION {format.bvolume} +{ volume empty$ + { "" } + { "volume" volume tie.or.space.connect + series empty$ + 'skip$ + { " of " * series emphasize * } + if$ + "volume and number" number either.or.check + } + if$ +} + +FUNCTION {format.number.series} +{ volume empty$ + { number empty$ + { series field.or.null } + { output.state mid.sentence = + { "number" } + { "Number" } + if$ + number tie.or.space.connect + series empty$ + { "there's a number but no series in " cite$ * warning$ } + { " in " * series * } + if$ + } + if$ + } + { "" } + if$ +} + +FUNCTION {format.edition} +{ edition empty$ + { "" } + { output.state mid.sentence = + { edition "l" change.case$ " edition" * } + { edition "t" change.case$ " edition" * } + if$ + } + if$ +} + +INTEGERS { multiresult } + +FUNCTION {multi.page.check} +{ 't := + #0 'multiresult := + { multiresult not + t empty$ not + and + } + { t #1 #1 substring$ + duplicate$ "-" = + swap$ duplicate$ "," = + swap$ "+" = + or or + { #1 'multiresult := } + { t #2 global.max$ substring$ 't := } + if$ + } + while$ + multiresult +} + +FUNCTION {format.pages} +{ pages empty$ + { "" } + { pages multi.page.check + { "pp.\ " pages n.dashify tie.or.space.connect } + { "pp.\ " pages tie.or.space.connect } + if$ + } + if$ +} + +FUNCTION {format.eid} +{ eid empty$ + { "" } + { "art." eid tie.or.space.connect } + if$ +} + +FUNCTION {format.vol.num.pages} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + pages empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.pages } + { ":\penalty0 " * pages n.dashify * } + if$ + } + if$ +} + +FUNCTION {format.vol.num.eid} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + eid empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.eid } + { ":\penalty0 " * eid * } + if$ + } + if$ +} + +FUNCTION {format.chapter.pages} +{ chapter empty$ + 'format.pages + { type empty$ + { "chapter" } + { type "l" change.case$ } + if$ + chapter tie.or.space.connect + pages empty$ + 'skip$ + { ", " * format.pages * } + if$ + } + if$ +} + +FUNCTION {format.in.ed.booktitle} +{ booktitle empty$ + { "" } + { editor empty$ + { "In " booktitle emphasize * } + { "In " format.editors * ", " * booktitle emphasize * } + if$ + } + if$ +} + +FUNCTION {empty.misc.check} +{ author empty$ title empty$ howpublished empty$ + month empty$ year empty$ note empty$ + and and and and and + key empty$ not and + { "all relevant fields are empty in " cite$ * warning$ } + 'skip$ + if$ +} + +FUNCTION {format.thesis.type} +{ type empty$ + 'skip$ + { pop$ + type "t" change.case$ + } + if$ +} + +FUNCTION {format.tr.number} +{ type empty$ + { "Technical Report" } + 'type + if$ + number empty$ + { "t" change.case$ } + { number tie.or.space.connect } + if$ +} + +FUNCTION {format.article.crossref} +{ key empty$ + { journal empty$ + { "need key or journal for " cite$ * " to crossref " * crossref * + warning$ + "" + } + { "In \emph{" journal * "}" * } + if$ + } + { "In " } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {format.book.crossref} +{ volume empty$ + { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ + "In " + } + { "Volume" volume tie.or.space.connect + " of " * + } + if$ + editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { series empty$ + { "need editor, key, or series for " cite$ * " to crossref " * + crossref * warning$ + "" * + } + { "\emph{" * series * "}" * } + if$ + } + 'skip$ + if$ + } + 'skip$ + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {format.incoll.inproc.crossref} +{ editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { booktitle empty$ + { "need editor, key, or booktitle for " cite$ * " to crossref " * + crossref * warning$ + "" + } + { "In \emph{" booktitle * "}" * } + if$ + } + { "In " } + if$ + } + { "In " } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {article} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { journal emphasize "journal" output.check + eid empty$ + { format.vol.num.pages output } + { format.vol.num.eid output } + if$ + format.date "year" output.check + } + { format.article.crossref output.nonnull + eid empty$ + { format.pages output } + { format.eid output } + if$ + } + if$ + format.issn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {book} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {booklet} +{ output.bibitem + format.authors output + author format.key output + new.block + format.title "title" output.check + howpublished address new.block.checkb + howpublished output + address output + format.date output + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inbook} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + format.chapter.pages "chapter and pages" output.check + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { format.chapter.pages "chapter and pages" output.check + new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {incollection} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.chapter.pages output + new.sentence + publisher "publisher" output.check + address output + format.edition output + format.date "year" output.check + } + { format.incoll.inproc.crossref output.nonnull + format.chapter.pages output + } + if$ + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inproceedings} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.pages output + address empty$ + { organization publisher new.sentence.checkb + organization output + publisher output + format.date "year" output.check + } + { address output.nonnull + format.date "year" output.check + new.sentence + organization output + publisher output + } + if$ + } + { format.incoll.inproc.crossref output.nonnull + format.pages output + } + if$ + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {conference} { inproceedings } + +FUNCTION {manual} +{ output.bibitem + format.authors output + author format.key output + new.block + format.btitle "title" output.check + organization address new.block.checkb + organization output + address output + format.edition output + format.date output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {mastersthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + "Master's thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {misc} +{ output.bibitem + format.authors output + author format.key output + title howpublished new.block.checkb + format.title output + howpublished new.block.checka + howpublished output + format.date output + format.issn output + format.url output + new.block + note output + fin.entry + empty.misc.check +} + +FUNCTION {phdthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.btitle "title" output.check + new.block + "PhD thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {proceedings} +{ output.bibitem + format.editors output + editor format.key output + new.block + format.btitle "title" output.check + format.bvolume output + format.number.series output + address output + format.date "year" output.check + new.sentence + organization output + publisher output + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {techreport} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + format.tr.number output.nonnull + institution "institution" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {unpublished} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + note "note" output.check + format.date output + format.url output + fin.entry +} + +FUNCTION {default.type} { misc } + + +MACRO {jan} {"January"} + +MACRO {feb} {"February"} + +MACRO {mar} {"March"} + +MACRO {apr} {"April"} + +MACRO {may} {"May"} + +MACRO {jun} {"June"} + +MACRO {jul} {"July"} + +MACRO {aug} {"August"} + +MACRO {sep} {"September"} + +MACRO {oct} {"October"} + +MACRO {nov} {"November"} + +MACRO {dec} {"December"} + + + +MACRO {acmcs} {"ACM Computing Surveys"} + +MACRO {acta} {"Acta Informatica"} + +MACRO {cacm} {"Communications of the ACM"} + +MACRO {ibmjrd} {"IBM Journal of Research and Development"} + +MACRO {ibmsj} {"IBM Systems Journal"} + +MACRO {ieeese} {"IEEE Transactions on Software Engineering"} + +MACRO {ieeetc} {"IEEE Transactions on Computers"} + +MACRO {ieeetcad} + {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"} + +MACRO {ipl} {"Information Processing Letters"} + +MACRO {jacm} {"Journal of the ACM"} + +MACRO {jcss} {"Journal of Computer and System Sciences"} + +MACRO {scp} {"Science of Computer Programming"} + +MACRO {sicomp} {"SIAM Journal on Computing"} + +MACRO {tocs} {"ACM Transactions on Computer Systems"} + +MACRO {tods} {"ACM Transactions on Database Systems"} + +MACRO {tog} {"ACM Transactions on Graphics"} + +MACRO {toms} {"ACM Transactions on Mathematical Software"} + +MACRO {toois} {"ACM Transactions on Office Information Systems"} + +MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"} + +MACRO {tcs} {"Theoretical Computer Science"} + + +READ + +FUNCTION {sortify} +{ purify$ + "l" change.case$ +} + +INTEGERS { len } + +FUNCTION {chop.word} +{ 's := + 'len := + s #1 len substring$ = + { s len #1 + global.max$ substring$ } + 's + if$ +} + +FUNCTION {format.lab.names} +{ 's := + s #1 "{vv~}{ll}" format.name$ + s num.names$ duplicate$ + #2 > + { pop$ " et~al." * } + { #2 < + 'skip$ + { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = + { " et~al." * } + { " \& " * s #2 "{vv~}{ll}" format.name$ * } + if$ + } + if$ + } + if$ +} + +FUNCTION {author.key.label} +{ author empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.editor.key.label} +{ author empty$ + { editor empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { editor format.lab.names } + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.key.organization.label} +{ author empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {editor.key.organization.label} +{ editor empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + 'key + if$ + } + { editor format.lab.names } + if$ +} + +FUNCTION {calc.short.authors} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.key.label + { type$ "proceedings" = + 'editor.key.organization.label + { type$ "manual" = + 'author.key.organization.label + 'author.key.label + if$ + } + if$ + } + if$ + 'short.list := +} + +FUNCTION {calc.label} +{ calc.short.authors + short.list + "(" + * + year duplicate$ empty$ + short.list key field.or.null = or + { pop$ "" } + 'skip$ + if$ + * + 'label := +} + +FUNCTION {sort.format.names} +{ 's := + #1 'nameptr := + "" + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { + s nameptr "{vv{ } }{ll{ }}{ f{ }}{ jj{ }}" format.name$ 't := + nameptr #1 > + { + " " * + namesleft #1 = t "others" = and + { "zzzzz" * } + { numnames #2 > nameptr #2 = and + { "zz" * year field.or.null * " " * } + 'skip$ + if$ + t sortify * + } + if$ + } + { t sortify * } + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {sort.format.title} +{ 't := + "A " #2 + "An " #3 + "The " #4 t chop.word + chop.word + chop.word + sortify + #1 global.max$ substring$ +} + +FUNCTION {author.sort} +{ author empty$ + { key empty$ + { "to sort, need author or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.editor.sort} +{ author empty$ + { editor empty$ + { key empty$ + { "to sort, need author, editor, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { editor sort.format.names } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.organization.sort} +{ author empty$ + { organization empty$ + { key empty$ + { "to sort, need author, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {editor.organization.sort} +{ editor empty$ + { organization empty$ + { key empty$ + { "to sort, need editor, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { editor sort.format.names } + if$ +} + + +FUNCTION {presort} +{ calc.label + label sortify + " " + * + type$ "book" = + type$ "inbook" = + or + 'author.editor.sort + { type$ "proceedings" = + 'editor.organization.sort + { type$ "manual" = + 'author.organization.sort + 'author.sort + if$ + } + if$ + } + if$ + " " + * + year field.or.null sortify + * + " " + * + cite$ + * + #1 entry.max$ substring$ + 'sort.label := + sort.label * + #1 entry.max$ substring$ + 'sort.key$ := +} + +ITERATE {presort} + +SORT + +STRINGS { longest.label last.label next.extra } + +INTEGERS { longest.label.width last.extra.num number.label } + +FUNCTION {initialize.longest.label} +{ "" 'longest.label := + #0 int.to.chr$ 'last.label := + "" 'next.extra := + #0 'longest.label.width := + #0 'last.extra.num := + #0 'number.label := +} + +FUNCTION {forward.pass} +{ last.label label = + { last.extra.num #1 + 'last.extra.num := + last.extra.num int.to.chr$ 'extra.label := + } + { "a" chr.to.int$ 'last.extra.num := + "" 'extra.label := + label 'last.label := + } + if$ + number.label #1 + 'number.label := +} + +FUNCTION {reverse.pass} +{ next.extra "b" = + { "a" 'extra.label := } + 'skip$ + if$ + extra.label 'next.extra := + extra.label + duplicate$ empty$ + 'skip$ + { "{\natexlab{" swap$ * "}}" * } + if$ + 'extra.label := + label extra.label * 'label := +} + +EXECUTE {initialize.longest.label} + +ITERATE {forward.pass} + +REVERSE {reverse.pass} + +FUNCTION {bib.sort.order} +{ sort.label 'sort.key$ := +} + +ITERATE {bib.sort.order} + +SORT + +FUNCTION {begin.bib} +{ preamble$ empty$ + 'skip$ + { preamble$ write$ newline$ } + if$ + "\begin{thebibliography}{" number.label int.to.str$ * "}" * + write$ newline$ + "\providecommand{\natexlab}[1]{#1}" + write$ newline$ + "\providecommand{\url}[1]{\texttt{#1}}" + write$ newline$ + "\expandafter\ifx\csname urlstyle\endcsname\relax" + write$ newline$ + " \providecommand{\doi}[1]{doi: #1}\else" + write$ newline$ + " \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi" + write$ newline$ +} + +EXECUTE {begin.bib} + +EXECUTE {init.state.consts} + +ITERATE {call.type$} + +FUNCTION {end.bib} +{ newline$ + "\end{thebibliography}" write$ newline$ +} + +EXECUTE {end.bib} diff --git a/icml2026.sty b/icml2026.sty new file mode 100644 index 0000000..47f1fae --- /dev/null +++ b/icml2026.sty @@ -0,0 +1,767 @@ +% File: icml2026.sty (LaTeX style file for ICML-2026, version of 2025-10-29) + +% This file contains the LaTeX formatting parameters for a two-column +% conference proceedings that is 8.5 inches wide by 11 inches high. +% +% Modified by Hanze Dong, Alberto Bietti, and Felix Berkenkamp, 2025 +% - Revert to times for better compatibility +% - Updated years, volume, location +% - Added preprint version +% - Based on the suggestion from Johan Larsson: +% 1. Added an end-of-document safety check to ensure the affiliations or notice footnote is printed: +% (1) Introduces a flag \newif\ificml@noticeprinted and sets it false by default. +% (2) At end of document, emits a package warning if \printAffiliationsAndNotice{...} was never called. +% 2. \printAffiliationsAndNotice now sets the flag when called: Begins with \global\icml@noticeprintedtrue. +% - Migrated to more recent version of fancyhdr for running title in header +% +% Modified by Johan Larsson, 2025 +% - Use newtx instead of times, aligning serif, sans-serif, typerwriter, +% and math fonts. +% - Use caption package to setup captions instead of manually defining themanually defining them. +% - Formatted icml2026.sty and example_paper.tex +% - Use title case for section title to 2.9 +% - Replace subfigure package with subcaption in example, since it is +% designed to work together with the caption package (which is now required). +% - Remove unused label in example +% +% Modified by Tegan Maharaj and Felix Berkenkamp 2025: changed years, volume, location +% +% Modified by Jonathan Scarlett 2024: changed years, volume, location +% +% Modified by Sivan Sabato 2023: changed years and volume number. +% Modified by Jonathan Scarlett 2023: added page numbers to every page +% +% Modified by Csaba Szepesvari 2022: changed years, PMLR ref. Turned off checking marginparwidth +% as marginparwidth only controls the space available for margin notes and margin notes +% will NEVER be used anyways in submitted versions, so there is no reason one should +% check whether marginparwidth has been tampered with. +% Also removed pdfview=FitH from hypersetup as it did not do its job; the default choice is a bit better +% but of course the double-column format is not supported by this hyperlink preview functionality +% in a completely satisfactory fashion. +% Modified by Gang Niu 2022: Changed color to xcolor +% +% Modified by Iain Murray 2018: changed years, location. Remove affiliation notes when anonymous. +% Move times dependency from .tex to .sty so fewer people delete it. +% +% Modified by Daniel Roy 2017: changed byline to use footnotes for affiliations, and removed emails +% +% Modified by Percy Liang 12/2/2013: changed the year, location from the previous template for ICML 2014 + +% Modified by Fei Sha 9/2/2013: changed the year, location form the previous template for ICML 2013 +% +% Modified by Fei Sha 4/24/2013: (1) remove the extra whitespace after the +% first author's email address (in %the camera-ready version) (2) change the +% Proceeding ... of ICML 2010 to 2014 so PDF's metadata will show up % +% correctly +% +% Modified by Sanjoy Dasgupta, 2013: changed years, location +% +% Modified by Francesco Figari, 2012: changed years, location +% +% Modified by Christoph Sawade and Tobias Scheffer, 2011: added line +% numbers, changed years +% +% Modified by Hal Daume III, 2010: changed years, added hyperlinks +% +% Modified by Kiri Wagstaff, 2009: changed years +% +% Modified by Sam Roweis, 2008: changed years +% +% Modified by Ricardo Silva, 2007: update of the ifpdf verification +% +% Modified by Prasad Tadepalli and Andrew Moore, merely changing years. +% +% Modified by Kristian Kersting, 2005, based on Jennifer Dy's 2004 version +% - running title. If the original title is to long or is breaking a line, +% use \icmltitlerunning{...} in the preamble to supply a shorter form. +% Added fancyhdr package to get a running head. +% - Updated to store the page size because pdflatex does compile the +% page size into the pdf. +% +% Hacked by Terran Lane, 2003: +% - Updated to use LaTeX2e style file conventions (ProvidesPackage, +% etc.) +% - Added an ``appearing in'' block at the base of the first column +% (thus keeping the ``appearing in'' note out of the bottom margin +% where the printer should strip in the page numbers). +% - Added a package option [accepted] that selects between the ``Under +% review'' notice (default, when no option is specified) and the +% ``Appearing in'' notice (for use when the paper has been accepted +% and will appear). +% +% Originally created as: ml2k.sty (LaTeX style file for ICML-2000) +% by P. Langley (12/23/99) + +%%%%%%%%%%%%%%%%%%%% +%% This version of the style file supports both a ``review'' version +%% and a ``final/accepted'' version. The difference is only in the +%% text that appears in the note at the bottom of the first column of +%% the first page. The default behavior is to print a note to the +%% effect that the paper is under review and don't distribute it. The +%% final/accepted version prints an ``Appearing in'' note. To get the +%% latter behavior, in the calling file change the ``usepackage'' line +%% from: +%% \usepackage{icml2025} +%% to +%% \usepackage[accepted]{icml2025} +%%%%%%%%%%%%%%%%%%%% + +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{icml2026}[2025/10/29 v2.0 ICML Conference Style File] + +% Before 2018, \usepackage{times} was in the example TeX, but inevitably +% not everybody did it. +% \RequirePackage[amsthm]{newtx} +% 2025.11.6 revert to times for better compatibility +\RequirePackage{times} + +% Use fancyhdr package +\RequirePackage{fancyhdr} +\RequirePackage{xcolor} % changed from color to xcolor (2021/11/24) +\RequirePackage{algorithm} +\RequirePackage{algorithmic} +\RequirePackage{natbib} +\RequirePackage{eso-pic} % used by \AddToShipoutPicture +\RequirePackage{forloop} +\RequirePackage{url} +\RequirePackage{caption} + +%%%%%%%% Options +\DeclareOption{accepted}{% + \renewcommand{\Notice@String}{\ICML@appearing} + \gdef\isaccepted{1} +} + +% === Preprint option === +\DeclareOption{preprint}{%% + \renewcommand{\Notice@String}{\ICML@preprint}%% + \gdef\ispreprint{1}%% +} + +% Distinct preprint footer text +\newcommand{\ICML@preprint}{% + \textit{Preprint. \today.}% +} + +\DeclareOption{nohyperref}{% + \gdef\nohyperref{1} +} + +% Helper flag: show real authors for accepted or preprint +\newif\ificmlshowauthors +\icmlshowauthorsfalse + +%%%%%%%%%%%%%%%%%%%% +% This string is printed at the bottom of the page for the +% final/accepted version of the ``appearing in'' note. Modify it to +% change that text. +%%%%%%%%%%%%%%%%%%%% +\newcommand{\ICML@appearing}{\textit{Proceedings of the +$\mathit{43}^{rd}$ International Conference on Machine Learning}, +Seoul, South Korea. PMLR 306, 2026. +Copyright 2026 by the author(s).} + +%%%%%%%%%%%%%%%%%%%% +% This string is printed at the bottom of the page for the draft/under +% review version of the ``appearing in'' note. Modify it to change +% that text. +%%%%%%%%%%%%%%%%%%%% +\newcommand{\Notice@String}{Preliminary work. Under review by the +International Conference on Machine Learning (ICML)\@. Do not distribute.} + +% Cause the declared options to actually be parsed and activated +\ProcessOptions\relax + +% After options are processed, decide if authors should be visible +\ifdefined\isaccepted \icmlshowauthorstrue \fi +\ifdefined\ispreprint \icmlshowauthorstrue \fi + +\ifdefined\isaccepted\else\ifdefined\ispreprint\else\ifdefined\hypersetup + \hypersetup{pdfauthor={Anonymous Authors}} +\fi\fi\fi + +\ifdefined\nohyperref\else\ifdefined\hypersetup + \definecolor{mydarkblue}{rgb}{0,0.08,0.45} + \hypersetup{ % + pdftitle={}, + pdfsubject={Proceedings of the International Conference on Machine Learning 2026}, + pdfkeywords={}, + pdfborder=0 0 0, + pdfpagemode=UseNone, + colorlinks=true, + linkcolor=mydarkblue, + citecolor=mydarkblue, + filecolor=mydarkblue, + urlcolor=mydarkblue, + } + \fi +\fi + + + +% Uncomment the following for debugging. It will cause LaTeX to dump +% the version of the ``appearing in'' string that will actually appear +% in the document. +%\typeout{>> Notice string='\Notice@String'} + +% Change citation commands to be more like old ICML styles +\newcommand{\yrcite}[1]{\citeyearpar{#1}} +\renewcommand{\cite}[1]{\citep{#1}} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% to ensure the letter format is used. pdflatex does compile the +% page size into the pdf. This is done using \pdfpagewidth and +% \pdfpageheight. As Latex does not know this directives, we first +% check whether pdflatex or latex is used. +% +% Kristian Kersting 2005 +% +% in order to account for the more recent use of pdfetex as the default +% compiler, I have changed the pdf verification. +% +% Ricardo Silva 2007 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\paperwidth=8.5in +\paperheight=11in + +% old PDFLaTex verification, circa 2005 +% +%\newif\ifpdf\ifx\pdfoutput\undefined +% \pdffalse % we are not running PDFLaTeX +%\else +% \pdfoutput=1 % we are running PDFLaTeX +% \pdftrue +%\fi + +\newif\ifpdf %adapted from ifpdf.sty +\ifx\pdfoutput\undefined +\else + \ifx\pdfoutput\relax + \else + \ifcase\pdfoutput + \else + \pdftrue + \fi + \fi +\fi + +\ifpdf +% \pdfpagewidth=\paperwidth +% \pdfpageheight=\paperheight + \setlength{\pdfpagewidth}{8.5in} + \setlength{\pdfpageheight}{11in} +\fi + +% Physical page layout + +\evensidemargin -0.23in +\oddsidemargin -0.23in +\setlength\textheight{9.0in} +\setlength\textwidth{6.75in} +\setlength\columnsep{0.25in} +\setlength\headheight{10pt} +\setlength\headsep{10pt} +\addtolength{\topmargin}{-20pt} +\addtolength{\topmargin}{-0.29in} + +% Historically many authors tried to include packages like geometry or fullpage, +% which change the page layout. It either makes the proceedings inconsistent, or +% wastes organizers' time chasing authors. So let's nip these problems in the +% bud here. -- Iain Murray 2018. +%\RequirePackage{printlen} +\AtBeginDocument{% +\newif\ifmarginsmessedwith +\marginsmessedwithfalse +\ifdim\oddsidemargin=-16.62178pt \else oddsidemargin has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\headheight=10.0pt \else headheight has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\textheight=650.43pt \else textheight has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\marginparsep=11.0pt \else marginparsep has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\footskip=25.0pt \else footskip has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\hoffset=0.0pt \else hoffset has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\paperwidth=614.295pt \else paperwidth has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\topmargin=-24.95781pt \else topmargin has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\headsep=10.0pt \else headsep has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\textwidth=487.8225pt \else textwidth has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\marginparpush=5.0pt \else marginparpush has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\voffset=0.0pt \else voffset has been altered.\\ \marginsmessedwithtrue\fi +\ifdim\paperheight=794.96999pt \else paperheight has been altered.\\ \marginsmessedwithtrue\fi +\ifmarginsmessedwith + +\textbf{\large \em The page layout violates the ICML style.} + +Please do not change the page layout, or include packages like geometry, +savetrees, or fullpage, which change it for you. + +We're not able to reliably undo arbitrary changes to the style. Please remove +the offending package(s), or layout-changing commands and try again. + +\fi} + + +%% The following is adapted from code in the acmconf.sty conference +%% style file. The constants in it are somewhat magical, and appear +%% to work well with the two-column format on US letter paper that +%% ICML uses, but will break if you change that layout, or if you use +%% a longer block of text for the copyright notice string. Fiddle with +%% them if necessary to get the block to fit/look right. +%% +%% -- Terran Lane, 2003 +%% +%% The following comments are included verbatim from acmconf.sty: +%% +%%% This section (written by KBT) handles the 1" box in the lower left +%%% corner of the left column of the first page by creating a picture, +%%% and inserting the predefined string at the bottom (with a negative +%%% displacement to offset the space allocated for a non-existent +%%% caption). +%%% +\def\ftype@copyrightbox{8} +\def\@copyrightspace{ +\@float{copyrightbox}[b] +\begin{center} +\setlength{\unitlength}{1pc} +\begin{picture}(20,1.5) +\put(0,2.5){\line(1,0){4.818}} +\put(0,0){\parbox[b]{19.75pc}{\small \Notice@String}} +\end{picture} +\end{center} +\end@float} + +\setlength\footskip{25.0pt} +\flushbottom \twocolumn +\sloppy + +% Clear out the addcontentsline command +\def\addcontentsline#1#2#3{} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% commands for formatting paper title, author names, and addresses. + +% box to check the size of the running head +\newbox\titrun + +% general page style +\pagestyle{fancy} +\fancyhf{} +\fancyfoot[C]{\thepage} +% set the width of the head rule to 1 point +\renewcommand{\headrulewidth}{1pt} + +% definition to set the head as running head in the preamble +\def\icmltitlerunning#1{\gdef\@icmltitlerunning{#1}} + +% main definition adapting \icmltitle from 2004 +\long\def\icmltitle#1{% + + %check whether @icmltitlerunning exists + % if not \icmltitle is used as running head + \ifx\undefined\@icmltitlerunning% + \gdef\@icmltitlerunning{#1} + \fi + + %add it to pdf information + \ifdefined\nohyperref\else\ifdefined\hypersetup + \hypersetup{pdftitle={#1}} + \fi\fi + + %get the dimension of the running title + \global\setbox\titrun=\vbox{\small\bf\@icmltitlerunning} + + % error flag + \gdef\@runningtitleerror{0} + + % running title too long + \ifdim\wd\titrun>\textwidth% + \gdef\@runningtitleerror{1}% + % running title breaks a line + \else \ifdim\ht\titrun>6.25pt + \gdef\@runningtitleerror{2}% + \fi + \fi + + % if there is somthing wrong with the running title + \ifnum\@runningtitleerror>0 + \typeout{}% + \typeout{}% + \typeout{*******************************************************}% + \typeout{Title exceeds size limitations for running head.}% + \typeout{Please supply a shorter form for the running head} + \typeout{with \string\icmltitlerunning{...}\space prior to \string\begin{document}}% + \typeout{*******************************************************}% + \typeout{}% + \typeout{}% + % set default running title + \gdef\@icmltitlerunning{Title Suppressed Due to Excessive Size} + \fi + + % no running title on the first page of the paper + \thispagestyle{plain} + + {\center\baselineskip 18pt + \toptitlebar{\Large\bf #1}\bottomtitlebar} +} + +% set running title header +\fancyhead[C]{\small\bf\@icmltitlerunning} + +\gdef\icmlfullauthorlist{} +\newcommand\addstringtofullauthorlist{\g@addto@macro\icmlfullauthorlist} +\newcommand\addtofullauthorlist[1]{% + \ifdefined\icmlanyauthors% + \addstringtofullauthorlist{, #1}% + \else% + \addstringtofullauthorlist{#1}% + \gdef\icmlanyauthors{1}% + \fi% + \ifdefined\hypersetup% + \hypersetup{pdfauthor=\icmlfullauthorlist}% + \fi +} + +\def\toptitlebar{\hrule height1pt \vskip .25in} +\def\bottomtitlebar{\vskip .22in \hrule height1pt \vskip .3in} + +\newenvironment{icmlauthorlist}{% + \setlength\topsep{0pt} + \setlength\parskip{0pt} + \begin{center} + }{% + \end{center} +} + +\newcounter{@affiliationcounter} +\newcommand{\@pa}[1]{% + \ifcsname the@affil#1\endcsname + % do nothing + \else + \ifcsname @icmlsymbol#1\endcsname + % nothing + \else + \stepcounter{@affiliationcounter}% + \newcounter{@affil#1}% + \setcounter{@affil#1}{\value{@affiliationcounter}}% + \fi + \fi% + \ifcsname @icmlsymbol#1\endcsname + \textsuperscript{\csname @icmlsymbol#1\endcsname\,}% + \else + \textsuperscript{\arabic{@affil#1}\,}% + \fi +} + +\newcommand{\icmlauthor}[2]{% + \ificmlshowauthors + \mbox{\bf #1}\,\@for\theaffil:=#2\do{\@pa{\theaffil}} \addtofullauthorlist{#1}% + \else + \ifdefined\@icmlfirsttime\else + \gdef\@icmlfirsttime{1} + \mbox{\bf Anonymous Authors}\@pa{@anon} \addtofullauthorlist{Anonymous Authors} + \fi + \fi +} + +\newcommand{\icmlsetsymbol}[2]{% + \expandafter\gdef\csname @icmlsymbol#1\endcsname{#2} +} + +\newcommand{\icmlaffiliation}[2]{% + \ificmlshowauthors + \ifcsname the@affil#1\endcsname + \expandafter\gdef\csname @affilname\csname the@affil#1\endcsname\endcsname{#2}% + \else + {\bf AUTHORERR: Error in use of \textbackslash{}icmlaffiliation command. Label ``#1'' not mentioned in some \textbackslash{}icmlauthor\{author name\}\{labels here\} command beforehand. } + \typeout{}% + \typeout{}% + \typeout{*******************************************************}% + \typeout{Affiliation label undefined. }% + \typeout{Make sure \string\icmlaffiliation\space follows }% + \typeout{all of \string\icmlauthor\space commands}% + \typeout{*******************************************************}% + \typeout{}% + \typeout{}% + \fi + \else + \expandafter\gdef\csname @affilname1\endcsname{Anonymous Institution, Anonymous City, Anonymous Region, Anonymous Country} + \fi +} + +\newcommand{\icmlcorrespondingauthor}[2]{% + \ificmlshowauthors + \ifdefined\icmlcorrespondingauthor@text + \g@addto@macro\icmlcorrespondingauthor@text{, #1 \textless{}#2\textgreater{}} + \else + \gdef\icmlcorrespondingauthor@text{#1 \textless{}#2\textgreater{}} + \fi + \else + \gdef\icmlcorrespondingauthor@text{Anonymous Author \textless{}anon.email@domain.com\textgreater{}} + \fi +} + +\newcommand{\icmlEqualContribution}{\textsuperscript{*}Equal contribution } + + +% --- ICML 2026: ensure authors do not omit the affiliations/notice footnote --- +\newif\ificml@noticeprinted +\icml@noticeprintedfalse +\AtEndDocument{% + \ificml@noticeprinted\relax\else + \PackageWarningNoLine{icml2026}{% + You did not call \string\printAffiliationsAndNotice{}. If you have no notice,% + call \string\printAffiliationsAndNotice\string{} (empty braces).% + }% + \fi +} + + +\newcounter{@affilnum} +\newcommand{\printAffiliationsAndNotice}[1]{\global\icml@noticeprintedtrue% + \stepcounter{@affiliationcounter}% + {\let\thefootnote\relax\footnotetext{\hspace*{-\footnotesep}\ificmlshowauthors #1\fi% + \forloop{@affilnum}{1}{\value{@affilnum} < \value{@affiliationcounter}}{ + \textsuperscript{\arabic{@affilnum}}\ifcsname @affilname\the@affilnum\endcsname% + \csname @affilname\the@affilnum\endcsname% + \else + {\bf AUTHORERR: Missing \textbackslash{}icmlaffiliation.} + \fi + }.% + \ifdefined\icmlcorrespondingauthor@text + { }Correspondence to: \icmlcorrespondingauthor@text. + \else + {\bf AUTHORERR: Missing \textbackslash{}icmlcorrespondingauthor.} + \fi + + \ \\ + \Notice@String + } + } +} + +\long\def\icmladdress#1{% + {\bf The \textbackslash{}icmladdress command is no longer used. See the example\_paper PDF .tex for usage of \textbackslash{}icmlauther and \textbackslash{}icmlaffiliation.} +} + +%% keywords as first class citizens +\def\icmlkeywords#1{% + \ifdefined\nohyperref\else\ifdefined\hypersetup + \hypersetup{pdfkeywords={#1}} + \fi\fi +} + +% modification to natbib citations +\setcitestyle{authoryear,round,citesep={;},aysep={,},yysep={;}} + +% Redefinition of the abstract environment. +\renewenvironment{abstract} +{% + \centerline{\large\bf Abstract} + \vspace{-0.12in}\begin{quote}} + {\par\end{quote}\vskip 0.12in} + +% numbered section headings with different treatment of numbers + +\def\@startsection#1#2#3#4#5#6{\if@noskipsec \leavevmode \fi + \par \@tempskipa #4\relax + \@afterindenttrue + \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \fi + \if@nobreak \everypar{}\else + \addpenalty{\@secpenalty}\addvspace{\@tempskipa}\fi \@ifstar + {\@ssect{#3}{#4}{#5}{#6}}{\@dblarg{\@sict{#1}{#2}{#3}{#4}{#5}{#6}}}} + +\def\@sict#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth + \def\@svsec{}\else + \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname}\fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \@hangfrom{\hskip #3\relax\@svsec.~}{\interlinepenalty \@M #8\par} + \endgroup + \csname #1mark\endcsname{#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}\else + \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname + {#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}}\fi + \@xsect{#5}} + +\def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth + \def\@svsec{}\else + \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname\hskip 0.4em }\fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par} + \endgroup + \csname #1mark\endcsname{#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}\else + \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname + {#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}}\fi + \@xsect{#5}} + +% section headings with less space above and below them +\def\thesection {\arabic{section}} +\def\thesubsection {\thesection.\arabic{subsection}} +\def\section{\@startsection{section}{1}{\z@}{-0.12in}{0.02in} + {\large\bf\raggedright}} +\def\subsection{\@startsection{subsection}{2}{\z@}{-0.10in}{0.01in} + {\normalsize\bf\raggedright}} +\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-0.08in}{0.01in} + {\normalsize\sc\raggedright}} +\def\paragraph{\@startsection{paragraph}{4}{\z@}{1.5ex plus + 0.5ex minus .2ex}{-1em}{\normalsize\bf}} +\def\subparagraph{\@startsection{subparagraph}{5}{\z@}{1.5ex plus + 0.5ex minus .2ex}{-1em}{\normalsize\bf}} + +% Footnotes +\footnotesep 6.65pt % +\skip\footins 9pt +\def\footnoterule{\kern-3pt \hrule width 0.8in \kern 2.6pt } +\setcounter{footnote}{0} + +% Lists and paragraphs +\parindent 0pt +\topsep 4pt plus 1pt minus 2pt +\partopsep 1pt plus 0.5pt minus 0.5pt +\itemsep 2pt plus 1pt minus 0.5pt +\parsep 2pt plus 1pt minus 0.5pt +\parskip 6pt + +\leftmargin 2em \leftmargini\leftmargin \leftmarginii 2em +\leftmarginiii 1.5em \leftmarginiv 1.0em \leftmarginv .5em +\leftmarginvi .5em +\labelwidth\leftmargini\advance\labelwidth-\labelsep \labelsep 5pt + +\def\@listi{\leftmargin\leftmargini} +\def\@listii{\leftmargin\leftmarginii + \labelwidth\leftmarginii\advance\labelwidth-\labelsep + \topsep 2pt plus 1pt minus 0.5pt + \parsep 1pt plus 0.5pt minus 0.5pt + \itemsep \parsep} +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii\advance\labelwidth-\labelsep + \topsep 1pt plus 0.5pt minus 0.5pt + \parsep \z@ \partopsep 0.5pt plus 0pt minus 0.5pt + \itemsep \topsep} +\def\@listiv{\leftmargin\leftmarginiv + \labelwidth\leftmarginiv\advance\labelwidth-\labelsep} +\def\@listv{\leftmargin\leftmarginv + \labelwidth\leftmarginv\advance\labelwidth-\labelsep} +\def\@listvi{\leftmargin\leftmarginvi + \labelwidth\leftmarginvi\advance\labelwidth-\labelsep} + +\abovedisplayskip 7pt plus2pt minus5pt% +\belowdisplayskip \abovedisplayskip +\abovedisplayshortskip 0pt plus3pt% +\belowdisplayshortskip 4pt plus3pt minus3pt% + +% Less leading in most fonts (due to the narrow columns) +% The choices were between 1-pt and 1.5-pt leading +\def\@normalsize{\@setsize\normalsize{11pt}\xpt\@xpt} +\def\small{\@setsize\small{10pt}\ixpt\@ixpt} +\def\footnotesize{\@setsize\footnotesize{10pt}\ixpt\@ixpt} +\def\scriptsize{\@setsize\scriptsize{8pt}\viipt\@viipt} +\def\tiny{\@setsize\tiny{7pt}\vipt\@vipt} +\def\large{\@setsize\large{14pt}\xiipt\@xiipt} +\def\Large{\@setsize\Large{16pt}\xivpt\@xivpt} +\def\LARGE{\@setsize\LARGE{20pt}\xviipt\@xviipt} +\def\huge{\@setsize\huge{23pt}\xxpt\@xxpt} +\def\Huge{\@setsize\Huge{28pt}\xxvpt\@xxvpt} + +% Revised formatting for figure captions and table titles. +\captionsetup{ + skip=0.1in, + font=small, + labelfont={it,small}, + labelsep=period +} +\captionsetup[table]{position=above} +\captionsetup[figure]{position=below} + +\def\fnum@figure{Figure \thefigure} +\def\fnum@table{Table \thetable} + +% Strut macros for skipping spaces above and below text in tables. +\def\abovestrut#1{\rule[0in]{0in}{#1}\ignorespaces} +\def\belowstrut#1{\rule[-#1]{0in}{#1}\ignorespaces} + +\def\abovespace{\abovestrut{0.20in}} +\def\aroundspace{\abovestrut{0.20in}\belowstrut{0.10in}} +\def\belowspace{\belowstrut{0.10in}} + +% Various personal itemization commands. +\def\texitem#1{\par\noindent\hangindent 12pt + \hbox to 12pt {\hss #1 ~}\ignorespaces} +\def\icmlitem{\texitem{$\bullet$}} + +% To comment out multiple lines of text. +\long\def\comment#1{} + +%% Line counter (not in final version). Adapted from NIPS style file by Christoph Sawade + +% Vertical Ruler +% This code is, largely, from the CVPR 2010 conference style file +% ----- define vruler +\makeatletter +\newbox\icmlrulerbox +\newcount\icmlrulercount +\newdimen\icmlruleroffset +\newdimen\cv@lineheight +\newdimen\cv@boxheight +\newbox\cv@tmpbox +\newcount\cv@refno +\newcount\cv@tot +% NUMBER with left flushed zeros \fillzeros[] +\newcount\cv@tmpc@ \newcount\cv@tmpc +\def\fillzeros[#1]#2{\cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi + \cv@tmpc=1 % + \loop\ifnum\cv@tmpc@<10 \else \divide\cv@tmpc@ by 10 \advance\cv@tmpc by 1 \fi + \ifnum\cv@tmpc@=10\relax\cv@tmpc@=11\relax\fi \ifnum\cv@tmpc@>10 \repeat + \ifnum#2<0\advance\cv@tmpc1\relax-\fi + \loop\ifnum\cv@tmpc<#1\relax0\advance\cv@tmpc1\relax\fi \ifnum\cv@tmpc<#1 \repeat + \cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi \relax\the\cv@tmpc@}% +% \makevruler[][][][][] +\def\makevruler[#1][#2][#3][#4][#5]{ + \begingroup\offinterlineskip + \textheight=#5\vbadness=10000\vfuzz=120ex\overfullrule=0pt% + \global\setbox\icmlrulerbox=\vbox to \textheight{% + { + \parskip=0pt\hfuzz=150em\cv@boxheight=\textheight + \cv@lineheight=#1\global\icmlrulercount=#2% + \cv@tot\cv@boxheight\divide\cv@tot\cv@lineheight\advance\cv@tot2% + \cv@refno1\vskip-\cv@lineheight\vskip1ex% + \loop\setbox\cv@tmpbox=\hbox to0cm{\hfil {\hfil\fillzeros[#4]\icmlrulercount}}% + \ht\cv@tmpbox\cv@lineheight\dp\cv@tmpbox0pt\box\cv@tmpbox\break + \advance\cv@refno1\global\advance\icmlrulercount#3\relax + \ifnum\cv@refno<\cv@tot\repeat + } + } + \endgroup +}% +\makeatother +% ----- end of vruler + +% \makevruler[][][][][] +\def\icmlruler#1{\makevruler[12pt][#1][1][3][\textheight]\usebox{\icmlrulerbox}} +\AddToShipoutPicture{% + \icmlruleroffset=\textheight + \advance\icmlruleroffset by 5.2pt % top margin + \color[rgb]{.7,.7,.7} + \ificmlshowauthors\else + \AtTextUpperLeft{% + \put(\LenToUnit{-35pt},\LenToUnit{-\icmlruleroffset}){%left ruler + \icmlruler{\icmlrulercount}} + %\put(\LenToUnit{1.04\textwidth},\LenToUnit{-\icmlruleroffset}){%right ruler + % \icmlruler{\icmlrulercount}} + } + \fi +} +\endinput diff --git a/images/paper_actions.pdf b/images/paper_actions.pdf new file mode 100644 index 0000000..79fa38c Binary files /dev/null and b/images/paper_actions.pdf differ diff --git a/images/paper_calibration.pdf b/images/paper_calibration.pdf new file mode 100644 index 0000000..72e466a Binary files /dev/null and b/images/paper_calibration.pdf differ diff --git a/images/paper_confidence_uncertainty.pdf b/images/paper_confidence_uncertainty.pdf new file mode 100644 index 0000000..e89f9df Binary files /dev/null and b/images/paper_confidence_uncertainty.pdf differ diff --git a/images/paper_controls.pdf b/images/paper_controls.pdf new file mode 100644 index 0000000..3e1340f Binary files /dev/null and b/images/paper_controls.pdf differ diff --git a/images/paper_diversity.pdf b/images/paper_diversity.pdf new file mode 100644 index 0000000..da7e54b Binary files /dev/null and b/images/paper_diversity.pdf differ diff --git a/images/paper_improvement.pdf b/images/paper_improvement.pdf new file mode 100644 index 0000000..720e05f Binary files /dev/null and b/images/paper_improvement.pdf differ diff --git a/images/paper_numerical_audit.pdf b/images/paper_numerical_audit.pdf new file mode 100644 index 0000000..0db0dfe Binary files /dev/null and b/images/paper_numerical_audit.pdf differ diff --git a/images/paper_success.pdf b/images/paper_success.pdf new file mode 100644 index 0000000..788be46 Binary files /dev/null and b/images/paper_success.pdf differ diff --git a/paper.pdf b/paper.pdf index 9f44cfc..77a7681 100644 Binary files a/paper.pdf and b/paper.pdf differ diff --git a/paper.tex b/paper.tex index 7d6b1bc..30e45a1 100644 --- a/paper.tex +++ b/paper.tex @@ -1,889 +1,859 @@ -\documentclass[11pt]{article} - -\usepackage[margin=1in]{geometry} -\usepackage{amsmath,amssymb,amsthm} -\usepackage{booktabs} -\usepackage[numbers,sort&compress]{natbib} +\documentclass{article} +\usepackage[T1]{fontenc} +\usepackage{microtype,graphicx,booktabs,tabularx} +\usepackage{xurl} \usepackage{hyperref} -\usepackage{xcolor} -\usepackage{enumitem} -\usepackage{graphicx} - -\hypersetup{colorlinks=true, linkcolor=blue!60!black, citecolor=blue!60!black, urlcolor=blue!60!black} - -\title{\textbf{Can Language Models Learn a PDE?}\\[4pt] -\large A Cognitive Benchmark for Scientific Reasoning\\via Weak-Form Identification of an Unknown Elliptic Equation} -\author{Joseph K. Miller} -\date{} - +\usepackage[preprint]{icml2026} +\hypersetup{pdfsubject={Research preprint on scientific-agent evaluation}} +\usepackage{amsmath,amssymb,amsthm} +\usepackage{placeins} +\usepackage{needspace} \newcommand{\dd}{\,\mathrm{d}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Poly}{\mathcal{P}} +\newtheorem{proposition}{Proposition} +\icmltitlerunning{Can Language Models Learn a PDE? An Audit of Measurement Selection} +\icmlkeywords{scientific agents, experimental design, inverse problems, language model evaluation} \begin{document} -\maketitle +\twocolumn[ +\icmltitle{Can Language Models Learn a PDE?\texorpdfstring{\\}{ } +An Audit of Measurement Selection in Weak-Form Inverse Problems} +\begin{icmlauthorlist} +\icmlauthor{Joseph K. Miller}{mit} +\end{icmlauthorlist} +\icmlaffiliation{mit}{Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA} +\icmlcorrespondingauthor{Joseph K. Miller}{jkm314@mit.edu} +\vskip 0.3in +] +\printAffiliationsAndNotice{} \begin{abstract} -We present a benchmark that measures scientific reasoning in large language -models through the task of identifying an unknown elliptic PDE from its -solution. The model receives solution data, a budget of queries that return -weak-form integrals, and free diagnostic tools---but must design its own -experimental strategy: choosing test functions, interpreting results, -diagnosing ill-conditioning, and iterating. The task produces a -\emph{cognitive profile} spanning eight faculties from the Burnell et -al.\ cognitive taxonomy: mathematical reasoning, metacognitive monitoring, -metacognitive control, executive planning, cognitive flexibility, -in-context learning, attention, and goal maintenance. Each -faculty is measured by a dedicated, mathematically grounded metric with -computable ground truth. A preliminary reported comparison covers three -OpenAI models (GPT-4o, GPT-5.4-mini, GPT-5.4), three difficulty levels, -and two seeds per configuration. In that comparison, GPT-5.4 achieves -total coefficient error below $0.002$ on both medium-difficulty instances, -while GPT-4o fails to produce a scored run on the two hard instances. -The benchmark reveals distinct behavioral failure modes---test function -homogeneity, premature termination, and over-reliance on consistency -checks---that map onto specific cognitive deficits. These observations -are exploratory. The implementation and selected notebook outputs are -publicly available; the underlying run-log archive for the manuscript's -tables is not included in the repository. +Weak measurements of a fixed solution and its source provide a controlled +setting for studying how language models choose informative observations. +We study a one-dimensional elliptic boundary-value problem in which an agent +selects test functions, a tool returns weighted source integrals, and a numerical +solver fits the unknown coefficients by least squares. +Our audit combines 180 archived scored runs from six models on 30 shared tasks +with fixed and randomized query designs, numerical sensitivity tests, and +validation on 300 fresh tasks. +Fixed sine tests recover all 30 archive-derived tasks at the minimum permitted +query count; on fresh tasks, success is 92\%, 91\%, and 98\% across the three +settings, including ten unidentifiable zero-forcing cases. +An exact nonuniqueness result and a consistency--stability analysis distinguish +ambiguity in the inverse problem from sensitivity of its discretization, while +free diagnostic measurements complicate comparisons under a stored-query budget. +These results delimit what coefficient-recovery scores establish about a +language model's measurement-selection ability. \end{abstract} -%---------------------------------------------------------------------- \section{Introduction} -%---------------------------------------------------------------------- - -Benchmarks for scientific reasoning in large language models (LLMs) often -test whether the model can \emph{recall} known facts or \emph{solve} -textbook problems. These tasks measure knowledge retrieval, not the -open-ended reasoning that characterizes real scientific work: forming -hypotheses, designing experiments to test them, interpreting results, and -iterating when the results are surprising. - -Recent work on measuring progress toward AGI has called for benchmarks that -are \emph{targeted to specific cognitive abilities}, \emph{varied in -difficulty}, and \emph{varied in structure}~\citep{burnell2026cognitive}. -The Burnell et al.\ cognitive taxonomy identifies ten faculties underlying -general intelligence, including reasoning, metacognition, executive -functions, and problem solving. Existing benchmarks cover some of these -faculties in isolation, but few provide a single task that engages multiple -faculties simultaneously with mathematically grounded evaluation. - -We propose a benchmark built around a classical problem in applied -mathematics: \emph{identifying an unknown elliptic PDE from its solution}. -The model is given the solution $u(x)$ on a grid, a budget of queries that -probe the PDE through its weak formulation, and a set of computational -tools. It must design a sequence of test functions, interpret the resulting -integrals, and recover the unknown coefficient functions. The task is -\emph{inherently ill-conditioned} in a way that rewards scientific -reasoning: not all test functions are equally informative, and a model that -reasons about \emph{which measurements matter} will significantly -outperform one that collects data mechanically. - -The benchmark produces a cognitive profile across eight faculties, each -measured by a dedicated metric with computable ground truth: - -\begin{enumerate}[nosep] - \item \textbf{Mathematical reasoning}: total coefficient recovery error. - \item \textbf{Metacognitive monitoring}: does the model know which - coefficients it is uncertain about? - \item \textbf{Metacognitive control}: does the model target its queries - toward its most uncertain coefficients? - \item \textbf{Executive planning}: does the model iterate (multiple - solves) or stop prematurely? - \item \textbf{Cognitive flexibility}: can the model switch between test - function families, and does the diversification provide independent - information in the weak-form integral space? - \item \textbf{In-context learning}: does the model improve its coefficient - estimates across successive solves within a single session? - \item \textbf{Attention}: does the model stay focused, or does it waste - turns on unrecognized or malformed actions? - \item \textbf{Goal maintenance}: after solving, does the model submit - promptly, or drift into aimless exploration? -\end{enumerate} - -%---------------------------------------------------------------------- -\section{Problem Setup} -%---------------------------------------------------------------------- - -\subsection{The Unknown PDE} - -The model is asked to identify the coefficient functions $a(x)$, $b(x)$, -$c(x)$, and $f(x)$ in the second-order elliptic boundary value problem -\begin{equation}\label{eq:pde} - -\bigl(a(x)\, u'(x)\bigr)' + b(x)\, u'(x) + c(x)\, u(x) = f(x), - \qquad x \in [0,1], -\end{equation} -with homogeneous Dirichlet conditions $u(0) = u(1) = 0$. Each coefficient -is a linear combination of $N$ shifted Legendre polynomials: -\[ - a(x) = \sum_{j=0}^{N-1} a_j\, P_j(x), - \qquad - b(x) = \sum_{j=0}^{N-1} b_j\, P_j(x), - \qquad \text{etc.} -\] -giving $4N$ unknowns total. The coefficients are drawn randomly with -controlled sparsity and scale, and the PDE is solved numerically to produce -the solution $u(x)$ on a fine grid. - -\subsection{Difficulty Levels} -The number of basis functions scales with difficulty, controlling both the -dimensionality of the inverse problem and the query budget: - -\begin{center} -\begin{tabular}{lcccc} +A scientific agent must choose observations that distinguish plausible +explanations. In an inverse problem, the question is concrete: which measurements +identify the unknown parameters, and how reliably can those parameters be +recovered? A controlled mathematical environment lets us study those choices +while knowing the parameters used to generate the observations. It also makes +the numerical assumptions behind an agent's score available for scrutiny. + +Our setting is a one-dimensional elliptic boundary-value problem, representing +a stationary diffusion--transport--reaction equation. The spatial response is +available through tools, while the diffusion, transport, reaction, and source +fields are hidden. The agent selects functions against which to integrate the +equation. Each choice imposes another weak constraint on the same fixed system; the +source and boundary conditions remain fixed throughout a task. Numerical tools assemble +the resulting linear equations and fit the coefficient fields. The agent chooses what to measure, which diagnostics to request, and when to +stop. + +This setup connects two familiar questions. For PDE and inverse-problem readers, +it asks whether weak measurements of one solution determine the operator and +whether the discrete recovery is stable. For machine-learning readers, it asks +whether a model's sequential choices improve a tool-assisted estimation task +over simple fixed designs. The equation family and parameter count are disclosed, while the supplied +solver handles the coefficient representation. ``Learn'' means identifying +parameters within a session. The spatially +one-dimensional problem is mathematically an ODE boundary-value problem, and +does not test multidimensional PDE structure. + +The benchmark first appeared in a Kaggle write-up \citep{kaggle2026pde}. +We reanalyze its six-model archive and ask: how do the recorded policies differ +on shared tasks; what do simple numerical query designs achieve; and how do +identifiability, numerical error, and the observation budget affect those +comparisons? Our contributions are a precise account of the observation model, +reproducible task-level comparisons, and mathematical and numerical controls. +Fixed sine tests suffice on all 30 archive-derived tasks. Fresh-task validation +and an exact ambiguity construction delimit that finding. The resulting case +study shows why evaluation of scientific agents needs both an inverse-problem +analysis and non-language-model controls. + +\section{Related Work} + +\paragraph{Interactive scientific agents.} +DiscoveryWorld evaluates a broad scientific-discovery loop in simulated worlds +\citep{jansen2024discoveryworld}. BoxingGym studies automated experimental design +and model discovery across generative environments +\citep{gandhi2025boxinggym}. SciGym provides a systems-biology dry lab for +scientific-agent evaluation \citep{duan2025scigym}. Our task shares their +sequential observation--feedback structure. Its narrower contribution is a +finite-dimensional weak-form inverse problem in which the measurement matrix, +coefficient truth, and numerical sensitivities can be audited directly. We do +not claim to introduce interactive scientific evaluation itself. + +\paragraph{Equation discovery and inverse problems.} +LLM-SR uses language models for scientific equation discovery, and LLM-SRBench +evaluates discovery from observations \citep{shojaee2025llmsr,shojaee2025llmsrbench}. +Weak SINDy uses integral formulations for PDE identification +\citep{messenger2021wsindy}. Here the operator family is fixed, and a supplied solver fits coefficients in +a prescribed polynomial basis from the agent's chosen test functions. This is a measurement-selection problem within a specified model, +rather than symbolic structure discovery. Optimal experimental design provides +a relevant reference \citep{mackay1992information,alexanderian2021oed}, but its +criteria require an explicit noise model, admissible designs, and appropriate +rank assumptions. We do not equate expression-family diversity with information +gain. + +\paragraph{Identifiability and confidence.} +Identifiability depends on the observation operator. Results for boundary maps, +reaction--advection--diffusion processes, and parabolic observations +\citep{egger2014simultaneous,browning2024structural,browning2025parabolic} +do not by themselves guarantee uniqueness from one stationary solution and +source. Our analysis distinguishes this structural issue from finite-grid +conditioning. The confidence analysis uses an explicit coefficient-error event, +following the distinction between stated confidence and empirical accuracy +\citep{guo2017calibration}. The cognitive framework of \citet{burnell2026cognitive} contextualizes the +original behavioral labels; our diagnostics describe behavior under a specific +prompted protocol. + +\section{Benchmark and Observation Model} +\label{sec:task} + +\subsection{Forward Problem, Observations, and Recovery} + +The hidden equation is +\begin{equation} + \begin{gathered}-(au')'+bu'+cu=f,\quad x\in(0,1),\\ + u(0)=u(1)=0.\end{gathered} + \label{eq:pde} +\end{equation} +Here \(u\) is the spatial response, primes denote spatial derivatives, and +\(a,b,c,f\) are diffusion, transport, reaction, and forcing, respectively. +The forward problem computes \(u\) from these fields and boundary data. The +inverse problem fixes that response and seeks the fields that produced it. +The supplied generator and solver use the parameterization +\(q(x)=\sum_{j=0}^{N-1}\theta_{q,j}P_j(x)\), for +\(q\in\{a,b,c,f\}\), using ordinary shifted Legendre polynomials on +\([0,1]\). For example, \(P_0(x)=1\) and \(P_1(x)=2x-1\). The fields are therefore +polynomials of degree at most \(N-1\). This basis is orthogonal but not +orthonormal. + +At the continuum level, take smooth coefficients with positive diffusion and an +observed solution \(u\in H_0^1(0,1)\). Here \(H_0^1\) consists of +square-integrable functions with square-integrable first weak derivative and +zero endpoint values. Multiplying Equation~\eqref{eq:pde} by a test function +\(\varphi_k\in H_0^1(0,1)\) and integrating by parts gives +\begin{equation} + \begin{aligned}&\int au'\varphi_k'\dd x+\int bu'\varphi_k\dd x+\int cu\varphi_k\dd x\\ + &\qquad=r_k,\qquad r_k=\int f\varphi_k\dd x. + \end{aligned} + \label{eq:weak} +\end{equation} +All integrals are over \((0,1)\). The endpoint condition removes the boundary +term, so only first derivatives of \(u\) are needed. A query chooses the +weighting function \(\varphi_k\) and observes the source integral \(r_k\), +also called a source moment. For example, \(\varphi_k=\sin(k\pi x)\) +requests a sine moment of the source. + +Each test function supplies one linear equation in the unknown operator +coefficients. Define +\begin{align} + D_{kj}&=\int P_j u'\varphi_k'\dd x,& B_{kj}&=\int P_j u'\varphi_k\dd x,\nonumber\\ + C_{kj}&=\int P_j u\varphi_k\dd x,& F_{kj}&=\int P_j\varphi_k\dd x. + \label{eq:matrices} +\end{align} +After \(K\) queries, let \(A=[D\mid B\mid C]\in\R^{K\times3N}\) and +\(\theta=(\theta_a^\top,\theta_b^\top,\theta_c^\top)^\top\). +The tool separately fits +\begin{equation} + \begin{aligned} + \widehat\theta&\in\arg\min_\theta\|A\theta-r\|_2,\\ + \widehat\theta_f&\in\arg\min_{\theta_f}\|F\theta_f-r\|_2. + \end{aligned} + \label{eq:fits} +\end{equation} +For a nonunique fit, the tool selects the minimum Euclidean norm solution under +its numerical singular-value cutoff. Both fits use the measured vector \(r\), +so operator recovery does not require a preceding source estimate. Source moments provide data in addition to the observed +solution; recovery is not based on \(u\) alone. + +The agent chooses the test functions, rather than matrix entries. The tool +constructs and stores each row internally; the agent receives \(r_k\), then +coefficient estimates and selected diagnostics when it requests a solve. +The full joint matrices and spectra reported below are reconstructed for the +audit. +The tests define identification constraints; the forward solution is computed +by finite differences, rather than expanded in these test functions. The +protocol requires \(K\ge4N\) before a solve, although the operator fit has +\(3N\) unknowns. This rule does not guarantee rank or uniqueness. + +\Needspace{8\baselineskip} +\subsection{Actions, Information, and Budgets} + +Table~\ref{tab:tools} summarizes the interface. The recovered prompt +implementation specifies the equation and parameter count but does not name the +Legendre basis; its coordinate recovery is delegated to the solve tool. The +model can evaluate the +solution and its numerical derivative, choose symbolic expressions, inspect +estimates, and request consistency checks. The parser admits mathematical +expressions including polynomial, trigonometric, exponential, and localized +functions, subject to sampled-value and endpoint checks. Numerical acceptance +does not certify membership in the continuum test space; our predefined controls +use smooth functions. There is no common amplitude normalization. + +\begin{table}[t] +\caption{Session actions. Only accepted stored queries consume the query budget. +Free checks reveal a source moment without adding a solver row.} +\label{tab:tools} +\centering\small +\begin{tabularx}{\columnwidth}{@{}lX@{}} \toprule -\textbf{Difficulty} & $N$ & \textbf{Unknowns} & \textbf{Budget} & \textbf{Turn limit} \\ +Action & Returned information or effect\\ \midrule -Easy & 2 & 8 & 12 & 36 \\ -Medium & 3 & 12 & 18 & 54 \\ -Hard & 4 & 16 & 24 & 72 \\ -Extreme & 8 & 32 & 48 & 144 \\ +\texttt{eval\_solution} & Values of \(u,u'\) at chosen points.\\ +\texttt{QUERY} & \(r_k\); appends rows to \(A,F\).\\ +\texttt{term\_integrals} & The three term integrals before weighting by the coefficient basis.\\ +\texttt{solve} & Least-squares coefficient estimates and changes.\\ +\texttt{check} & Predicted weak left-hand side and actual source moment for a candidate expression.\\ +\texttt{PREDICT} & Submits the latest solve coefficients.\\ \bottomrule -\end{tabular} -\end{center} - -\noindent The budget is set to $1.5\times$ the number of unknowns, -providing enough headroom for the model to solve the system, diagnose -problems, and iterate---but not so much that brute-force data collection -suffices. These are the current runner's defaults: its turn limit is -$3\times$ the budget. Historical results must be compared using their -recorded run settings, which are not recoverable from the tables alone. - -\subsection{Tools Available to the Model} - -The model receives the following tools via a system prompt: +\end{tabularx} +\end{table} -\paragraph{Query tool (\texttt{QUERY}).} -The model submits a test function $\varphi(x)$ as a symbolic expression -and receives the scalar $\int_0^1 f(x)\,\varphi(x)\dd x$. Each query -costs one unit from the budget. Internally, the system also computes and -stores the basis-weighted integrals -\[ - G^{\text{diff}}_{kj} = \int_0^1 P_j(x)\, u'(x)\, \varphi_k'(x)\dd x, - \qquad - G^{\text{adv}}_{kj} = \int_0^1 P_j(x)\, u'(x)\, \varphi_k(x)\dd x, - \qquad - G^{\text{react}}_{kj} = \int_0^1 P_j(x)\, u(x)\, \varphi_k(x)\dd x, -\] -for use by the solve tool. The model never sees these matrices. - -\paragraph{Solve tool (\texttt{COMPUTE: solve}).} -Assembles the $K \times 3N$ matrix $A = [G^{\text{diff}} \mid G^{\text{adv}} -\mid G^{\text{react}}]$ and solves the least-squares problem -$\min_\theta \|A\theta - r\|_2^2$ for the operator coefficients, and -separately recovers $f$ via $\min_f \|G^{\text{src}} f - r\|_2^2$. -Returns the recovered coefficient vectors and, after the first call, the -maximum per-coefficient change from the previous solve (enabling the model -to assess convergence). Free and unlimited, but requires at least $4N$ -stored queries. - -\paragraph{Solution evaluation (\texttt{COMPUTE: eval\_solution}).} -Returns $u(x)$ and $u'(x)$ at arbitrary points. Free. - -\paragraph{Term integral tool (\texttt{COMPUTE: term\_integrals}).} -Returns all three weight integrals simultaneously---diffusion -($\int u'\varphi'\dd x$), advection ($\int u'\varphi\dd x$), and reaction -($\int u\varphi\dd x$)---for a candidate test function, allowing the model -to assess the informativeness of each term \emph{before} spending a query. -Free. - -\paragraph{Consistency check (\texttt{COMPUTE: check}).} -Compares the predicted left-hand side of~\eqref{eq:pde} (from estimated -$\hat a, \hat b, \hat c$) to the actual $\int f\varphi\dd x$ for a new -test function. Free and read-only: the measurement is not added to the -solver's equation system. The system warns the model when zero discrepancy -is expected due to exact determination. This design separates -\emph{information acquisition} (queries, which are budgeted) from -\emph{verification} (checks, which are free), allowing us to measure whether -models engage in self-assessment independently of budget pressure. - -\paragraph{Submission (\texttt{PREDICT}).} -Ends the session and submits the coefficients from the model's most recent -\texttt{COMPUTE: solve} as its answer. The model decides \emph{when} to -stop, but the submitted answer always comes from the solver, eliminating -transcription errors. - -\subsection{The Initial Prompt} - -The standard prompt provides mathematical context about the task: - -\begin{quote} -\small -\texttt{Note: in elliptic inverse problems of this type, the three -coefficient functions a(x), b(x), c(x) are not equally identifiable from -weak-form data. The difficulty of recovering each coefficient depends on -the magnitude of its contribution to the weak-form equations relative to -the others---and this varies with the solution u(x) and the test functions -you choose.} -\end{quote} - -\noindent This tells the model \emph{what} is hard (unequal -identifiability) without telling it \emph{what to do about it}. -A baseline prompt without this context is available for ablation. - -%---------------------------------------------------------------------- -\section{The Ill-Conditioning Problem} -\label{sec:illcond} -%---------------------------------------------------------------------- - -The weak form~\eqref{eq:pde} produces one linear equation per test -function, with three contributions on the left-hand side---diffusion, -advection, and reaction---weighted by the integrals that the -\texttt{term\_integrals} tool returns. - -For a typical test function, these weights have very different magnitudes. -The diffusion weight involves $u'(x)\varphi'(x)$, where both factors are -$O(1)$ for smooth functions. The reaction weight involves -$u(x)\varphi(x)$, and since the solution satisfies homogeneous boundary -conditions and is often small ($|u| \sim 0.01$--$0.3$), this weight is -correspondingly small. As a result: - -\begin{itemize}[nosep] - \item The source $f(x)$ is recovered essentially perfectly (error - $\sim 10^{-11}$) because it appears linearly on the right-hand - side. - \item The diffusion coefficient $a(x)$ dominates every equation and is - recovered with high accuracy. - \item The advection coefficient $b(x)$ has moderate influence. - \item The reaction coefficient $c(x)$ has tiny influence and is the - hardest to recover---errors in $a$ and $b$ are absorbed into the - estimate of $c$. -\end{itemize} - -\noindent This creates a natural \emph{identifiability hierarchy}: -$f \gg a > b > c$. A model that mechanically collects equations -will recover $f$ and $a$ but struggle with $b$ and $c$. A model that -reasons about which test functions amplify the reaction weight will do -better on $c$. This hierarchy is the core discriminating mechanism of -the benchmark. - -Crucially, the benchmark provides a tool for overcoming this -ill-conditioning: \texttt{COMPUTE: term\_integrals}. This tool returns -all three term weights simultaneously---$\int u'\varphi'\dd x$ (diffusion), -$\int u'\varphi\dd x$ (advection), $\int u\varphi\dd x$ -(reaction)---for a candidate test function \emph{before} the model -spends a query. Since the tool is free and unlimited, the optimal -strategy is to screen candidate test functions by their term weights, -discarding those where the reaction weight $\int u\varphi\dd x$ is -negligible relative to the diffusion weight, and committing a query only -when the candidate provides meaningful constraint on the weakly -identifiable coefficients. Favorable test functions exist: concentrating -$\varphi$ near the peak of $u(x)$ amplifies $\int u\varphi\dd x$, while -choosing $\varphi$ with small derivatives suppresses $\int u'\varphi'\dd x$. -The benchmark thus measures not only whether the model \emph{notices} the -ill-conditioning, but whether it connects the diagnosis to the available -tool and uses it for experimental design. - -%---------------------------------------------------------------------- -\section{Cognitive Profile: Metrics} -\label{sec:metrics} -%---------------------------------------------------------------------- - -Following the cognitive taxonomy of Burnell et al.~\citep{burnell2026cognitive}, -we define metrics across eight cognitive faculties that together constitute -a \emph{cognitive profile} for each model. A key advantage of this task is that the ground truth for -each metric is \emph{exactly computable} from the stored $G$-matrices at -every point in the session, without requiring human judgment. - -\subsection{Mathematical Reasoning: Coefficient Error} - -The primary outcome measure. For each coefficient $j \in \{a, b, c, f\}$, -the error is the mean absolute deviation of the recovered Legendre -coefficients from the true values: -\[ - E_j = \frac{1}{N} \sum_{i=0}^{N-1} |\hat\theta_{j,i} - \theta_{j,i}^*|. -\] -The total error is $E = E_a + E_b + E_c + E_f$. We also report pointwise -error $\max_{x \in [0,1]} |j^{\text{pred}}(x) - j^{\text{true}}(x)|$ for -each coefficient function. - -\subsection{Metacognitive Monitoring: $M_k$} - -After each \texttt{COMPUTE: solve}, the model reports its confidence that -each coefficient vector is within 0.1 of the true value: -\texttt{Confidence: a=XX\%, b=XX\%, c=XX\%, f=XX\%}. The true epistemic -state is computable from the Fisher information: after $k$ queries, the -least-squares covariance is $(A_k^\top A_k)^{-1}$, giving per-block -uncertainty -\[ - \sigma_k(j) = \sqrt{\operatorname{mean}\!\bigl( - \operatorname{diag}(A_k^\top A_k)^{-1}\big|_{\text{block }j} - \bigr)}, - \qquad j \in \{a, b, c\}. -\] -Monitoring accuracy is the Spearman rank correlation between the model's -stated uncertainty $(1 - \hat c_k(j))$ and the true uncertainty -$\sigma_k(j)$: -\[ - M_k = \rho_S\!\bigl(\sigma_k,\, 1 - \hat c_k\bigr). -\] -$M_k = 1$ means the model correctly ranks which coefficients are most -uncertain. $M_k \approx 0$ means no self-knowledge. We report the -session mean $\bar M$. - -Note that only the \emph{rankings} of $\sigma_k(j)$ enter the monitoring -metric, not their magnitudes. This is important because $\bigl(A_k^\top A_k\bigr)^{-1}$ -can be highly ill-conditioned---its diagonal entries may fluctuate by orders of -magnitude as rows are added---but the rank ordering of the coefficient blocks is -empirically stable. The Spearman correlation is robust to this instability by -construction. - -\subsection{Metacognitive Control: $C_k$} - -Monitoring is only useful if it drives behavior. After query $k$, we -identify the most uncertain coefficient $j^* = \arg\max_j \sigma_k(j)$ -and compute the information gain from query $k{+}1$ for each coefficient -via a Sherman--Morrison update on $(A_k^\top A_k)^{-1}$: -\[ - \mathrm{IG}_{k+1}(j) = \sigma_k(j) - \sigma_{k+1}(j). -\] -Control efficiency is: -\[ - C_k = \frac{\mathrm{IG}_{k+1}(j^*)}{\max_j\, \mathrm{IG}_{k+1}(j)}. -\] -$C_k = 1$ means the model chose the query that maximally reduces -uncertainty on its hardest coefficient. We report the session mean -$\bar C$. - -\subsection{Executive Planning: Solve Count and Budget Utilization} - -A model that calls \texttt{COMPUTE: solve} multiple times and uses its -full query budget is exhibiting iterative experimental design---the -hallmark of scientific planning. We report: -\begin{itemize}[nosep] - \item \textbf{Solve count}: number of times \texttt{COMPUTE: solve} - was called. - \item \textbf{Budget utilization}: fraction of the query budget used - (queries used / budget). - \item \textbf{Check count}: number of free consistency checks performed. - Since checks do not consume budget, high check counts relative - to queries reflect a behavioral preference for verification over - information acquisition---a signature of inhibitory control - hesitation rather than a direct budget cost. -\end{itemize} - -\subsection{Cognitive Flexibility: Family Entropy and Query Space Conditioning} - -The system classifies each test function into one of $K = 5$ families: -polynomial ($x^m(1{-}x)^n$ products), trigonometric ($\sin$, $\cos$ -combinations), exponential ($e^{\alpha x}$ weighted), localized -(narrow-support functions), and other. Exponential and localized functions -are distinguished by the sign of the exponent: exponential functions -(e.g., $e^x \cdot x(1{-}x)$) have a positive exponent and shift the -integral's center of mass without spatial localization, while localized -functions (e.g., $e^{-10(x-0.3)^2}\sin(\pi x)$) have a negative exponent -and concentrate information in a spatial region. Let $p_i$ be the fraction of -queries in family $i$. The Shannon entropy -\[ - H = -\sum_{i=1}^{K} p_i \log_2 p_i -\] -ranges from $H = 0$ (all queries in one family; total inflexibility) to -$H = \log_2 K \approx 2.32$ (uniform use of all five families). We report -the normalized entropy $H / \log_2 K \in [0,1]$ for comparability. - -Family entropy measures cognitive \emph{strategy}, but not whether -diversification provided independent information. To assess the latter, -we decompose the query matrix $A$ into its three coefficient blocks -$A_{\text{diff}}$, $A_{\text{adv}}$, $A_{\text{react}}$ (each $K \times N$, -where $K$ is the number of queries and $N$ is the number of basis -functions). For each block we compute three SVD-derived quantities: -\begin{itemize}[nosep] - \item \textbf{Log-volume}: $\sum_i \log s_i$ over nonzero singular - values---the log-volume of the image in coefficient space. - \item \textbf{Effective rank}: $|\{i : s_i > \varepsilon\}|$, where - $\varepsilon = 10^{-12}$. - \item \textbf{Condition number}: $s_1 / s_{\min}$, where $s_{\min}$ - is the smallest nonzero singular value. -\end{itemize} -These two metrics are complementary: $H$ measures strategic diversity, -while block conditioning measures the \emph{outcome} of that strategy. -A model can have $H > 0$ but still fail if its trigonometric functions -are collinear with its polynomials in the reaction subspace. - -\subsection{In-Context Learning: Improvement Ratio and Learning Rate} - -A model that calls \texttt{COMPUTE: solve} multiple times has the -opportunity to learn from feedback within the session. Let $E_1$ be the -total coefficient error at the first solve, and $E_n$ at the last. The -\emph{improvement ratio} is -\[ - R = E_1 / E_n. -\] -$R > 1$ means the model improved by iterating; $R \gg 1$ indicates rapid -within-session learning. Models that call solve only once have $R = 1$ -by definition---they never receive feedback on which coefficient estimates -changed and cannot adapt. - -We also report the \emph{learning rate}: the slope of $\log_{10}(E)$ -regressed on query count across all solve points. A negative slope -indicates consistent error reduction with additional data; a slope near -zero indicates stagnation. This metric maps onto Burnell et al.'s -Learning faculty (\S7.4), specifically in-context learning within a -single session. - -\subsection{Attention: Wasted Turn Fraction} - -We define a turn as \emph{wasted} if its \texttt{parsed\_actions} list -contains no recognized action---no QUERY, COMPUTE, or PREDICT. Such -turns arise when the model produces malformed commands (e.g., using -\texttt{\^{}} instead of \texttt{**} for exponentiation) or reasons at -length without acting. We report: -\begin{itemize}[nosep] - \item \textbf{Wasted turn fraction}: wasted turns / total turns. - Values near zero indicate focused, efficient operation. - \item \textbf{Maximum unproductive streak}: the longest consecutive - run of wasted turns, which indicates sustained attention failure - rather than isolated errors. -\end{itemize} -This maps onto Burnell et al.'s Attention faculty (\S7.3), specifically -sustained and selective attention. - -\subsection{Goal Maintenance: Post-Solve Drift} - -After obtaining a coefficient estimate via \texttt{COMPUTE: solve}, the -model should either refine its strategy (add targeted queries, re-solve) -or submit. Failure to do either indicates goal drift. We report: -\begin{itemize}[nosep] - \item \textbf{Tail after last solve}: turns between the model's final - \texttt{COMPUTE: solve} and session end. Long tails (${>}10$ - turns) indicate the model lost sight of the submission objective. - \item \textbf{Maximum gap without progress}: the longest consecutive - stretch of turns (after the first solve) containing neither a - QUERY nor a \texttt{COMPUTE: solve}. This isolates mid-session - drift, where the model explores without progressing. -\end{itemize} -This maps onto Burnell et al.'s Executive Functions faculty (\S7.8), -specifically goal setting and maintenance (\S7.8.1). - -%---------------------------------------------------------------------- -\section{Experimental Design} -%---------------------------------------------------------------------- - -\subsection{Models} - -The preliminary comparison reported in Table~\ref{tab:sweep} covers -GPT-4o, GPT-5.4-mini, and GPT-5.4. An expanded comparison is proposed -across six models spanning three providers and two capability tiers: - -\begin{center} -\small -\begin{tabular}{lll} +A turn is one model response, possibly containing reasoning and tool commands. +The protocol permits at most one stored query per turn; several free +commands can share a response. Submission cannot share a response with a query. +The prompt encourages repeated solves and requests confidence reports. Therefore +solve counts and iteration are behaviors under an instruction to iterate. +A query budget counts stored rows, not all observations: a free check returns +\(\int f\varphi\) for a new expression, and solution evaluations are also free. + +\begin{table}[t] +\caption{Reconstructed task settings. The archive records \(N\), query and turn +budgets. Generator parameters are verified by exact recovery of all 30 archived +coefficient vectors. Sparsity is the generator's zeroing parameter; scale and +basis dimension change together.} +\label{tab:difficulty} +\centering\small +\begin{tabular}{@{}lrrr@{}} \toprule -\textbf{Provider} & \textbf{Large} & \textbf{Small} \\ + & Easy & Medium & Hard\\ \midrule -Anthropic & Claude Opus 4.6 & Claude Sonnet 4 (2025-05-14) \\ -OpenAI & GPT-5.4 & GPT-5.4-mini \\ -Google & Gemini 3.1 Pro & Gemini 3 Flash \\ +\(N\) & 2 & 3 & 4\\ +Stored-query budget & 12 & 18 & 24\\ +Turn limit & 36 & 54 & 72\\ +Generator sparsity & .3 & .2 & .1\\ +Generator scale & .5 & 1.0 & 1.5\\ +Diffusion parameter \(a_{\min}\) & 1.0 & .5 & .3\\ +Solution grid points & 2001 & 2001 & 4001\\ \bottomrule \end{tabular} -\end{center} - -\noindent Provider backends share tool interfaces, prompt construction, -and parsing logic. The proposed model names reflect the -runner's model registry; this is distinct from the three-model table. The draft -reports \texttt{reasoning\_effort: medium} for GPT-5.4 and GPT-5.4-mini; -GPT-4o has no reasoning-effort override in the registry. The preliminary -comparison uses the standard prompt with mathematical context. - -\subsection{Protocol} - -The preliminary table reports $3 \times 3 \times 2 = 18$ attempted runs: -three models at three difficulty levels (easy, medium, hard), using seeds -42 and 137. Fifteen attempts have reported scores and three have no -score. This matches the batch runner's default model, difficulty, and -seed selection; it is not evidence that the table has been regenerated -with the current code. - -The proposed expanded study would evaluate six models across five seeds, -with and without the mathematical context prompt (baseline ablation), for -a total of $6 \times 3 \times 5 \times 2 = 180$ runs. This expanded sweep -is not reported as completed here. The runner selects one prompt condition -per invocation via \texttt{--baseline}; both conditions require separate -invocations. Budgets and turn limits are auto-scaled from difficulty. Each run -produces a JSON log containing: the full turn-by-turn transcript, every -query and its result, every solve with coefficient errors against ground -truth, every consistency check, every term integral call, the post-hoc -efficiency curve (error at each query count), and the metacognitive metrics -($\sigma$-curves, monitoring reports, control efficiency). - -Kaggle integration is provided in \texttt{kaggle-benchmark.ipynb} and -\texttt{kaggle.ipynb}, using the \texttt{kaggle-benchmarks} SDK and importing -\texttt{dispatch\_turn()} from the main benchmark code. Saved notebook -outputs document additional exploratory runs; they are separate from the -18-attempt table and do not establish completion of the proposed study. -The original draft estimated approximately \$25 for the expanded sweep -(calibrated from Opus 4.6 runs at \$0.85/run); this estimate has not been -revalidated and actual cost depends on the selected models and usage. - -%---------------------------------------------------------------------- +\end{table} + +The generator checks diffusion positivity on 500 spatial points; the parameter +\(a_{\min}\) controls the constant diffusion coefficient, not a certified +pointwise lower bound. Reaction can have either sign, so diffusion positivity +alone does not guarantee a unique continuum forward solution. The evaluation +takes the computed solution as data. + +The reference implementation uses second-order finite differences for the +solution, a fourth-order interior derivative with boundary stencils, and +composite Simpson quadrature on the solution grid. It rounds each stored matrix +entry and right-hand-side value to 12 decimal places. The solve uses unweighted +least squares. The model sees coefficients rounded to eight decimal places; +final scoring uses the internally retained unrounded estimate. Symbolic test +functions are differentiated by a fourth-order centered stencil with step +\(10^{-6}\), independently of the solution-grid spacing. +These choices matter when singular values are small. Table~\ref{tab:difficulty} +also shows that the three difficulty labels are bundled settings; differences +cannot be attributed solely to dimension. + +\subsection{Coefficient Error, Uniqueness, and Stability} +\label{sec:identifiability} + +The score measures error in the prescribed coefficient coordinates: +\begin{equation} + E_q=\max_j|\widehat\theta_{q,j}-\theta_{q,j}|,\qquad + E=\sum_{q\in\{a,b,c,f\}}E_q. + \label{eq:error} +\end{equation} +The primary success event is \(E<0.01\); \(E<0.1\) is a sensitivity check. +This absolute, basis-dependent parameter error is distinct from a PDE residual +or a function error. For each field, +\(\|\widehat q-q\|_{L^\infty(0,1)}\le NE_q\), since \(|P_j|\le1\). +Thus a small coefficient error controls field error here, but the same numerical +thresholds for the two quantities are not equivalent. + +Three questions should be separated. \emph{Structural identifiability} asks +whether different coefficient triples can produce the same \(u,f\), even with +all admissible weak measurements. \emph{Measurement selection} asks whether a +finite set of tests distinguishes triples in the chosen coefficient space. +\emph{Numerical stability} asks how small data and discretization errors affect +the recovered coefficients. Full column rank of an exact moment matrix formed +from the continuum solution would certify uniqueness in the operator coefficient space. +Full rank after numerical approximation and rounding need not do so. +Separate full-rank blocks \(D,B,C\) do not exclude dependencies between blocks. + +To see the role of numerical error, write \(A_h,r_h\) for the assembled system +and \(\theta_*\) for the true operator coefficients. Its consistency defect is +\(e_h=A_h\theta_*-r_h\). The forward solve, derivative approximation, +quadrature, and rounding need not preserve the weak identity exactly, so this +defect can be nonzero even at the truth. If \(A_h\) has full column rank, +exact least squares for that assembled system gives +\begin{equation} + \begin{aligned} + \widehat\theta-\theta_*&=-A_h^\dagger e_h,\\ + \|\widehat\theta-\theta_*\|_2&\le\frac{\|e_h\|_2}{\sigma_{\min}(A_h)}. + \end{aligned} + \label{eq:stability} +\end{equation} +Here \(A_h^\dagger\) is the pseudoinverse. The bound concerns the operator +fit; the score also includes the separately fitted source. A small fitted weak +residual \(\|A_h\widehat\theta-r_h\|_2\) need not imply a small coefficient error: a small singular value can amplify a small +consistency defect. The forward finite-difference residual, fitted weak residual, +and coefficient score therefore answer different questions. + +Let \(\Poly_d\) denote polynomials of degree at most \(d\). The following +example shows an ambiguity that persists even before discretization. + +\begin{proposition}[An admissible nonidentifiable family] +\label{prop:nullity} +Let \(u(x)=x(1-x)\), \(N\ge2\), and +\(T(\delta a,\delta b,\delta c)=-(\delta a u')'+\delta b u'+\delta c u\) +with domain \(\Poly_{N-1}^3\). Then \(T\) has rank \(N+2\) and nullity +\(2N-2\). Every element of its kernel leaves all continuum weak-form +measurements unchanged at fixed \(u,f\). +\end{proposition} + +Even arbitrarily many exact weak measurements cannot distinguish operators +separated by these kernel directions. Appendix~\ref{app:proof} gives the proof +and examples with positive diffusion and nonnegative reaction. This quadratic- +solution family is an admissible counterexample, not a classification of the +30 sampled instances. There is also a positive-probability +zero-excitation event in the generator: each forcing coefficient is zeroed +independently with probability \(s\), giving +\(\Pr(f\equiv0)=s^N\), or 0.09, 0.008, and 0.0001 in the three settings. +When \(f\equiv0\), under homogeneous boundary data the native forward solve returns \(u=0\) when +its matrix is nonsingular. Then every operator design row vanishes, irrespective +of the test function, so no measurement policy can identify \(a,b,c\) from +these observations. The nonzero-source construction in +Proposition~\ref{prop:nullity} is a separate ambiguity mechanism. + +\section{Evaluation Design} +\label{sec:methods} + +\subsection{Archive and Provenance} + +A \emph{task} is one generated coefficient tuple and its numerical solution; +a \emph{run} is one agent's interaction with a task. A \emph{policy} is a rule +for selecting actions, fixed in advance or dependent on feedback. A random +\emph{seed} initializes the reproducible task generator. Thus six models on +30 shared tasks give 180 runs, not 180 different equations. + +The public Kaggle dataset, \emph{Elliptic Differential Equation LLM Reasoning}, +version 1, contains 180 scored logs: six models, three difficulty settings, +and seeds 1--10 in every cell. The same 30 ground-truth coefficient tuples are shared across +models. The archive was released on 16 April 2026; the provider model IDs are +listed in Appendix~\ref{app:repro}. +All logs record the \texttt{standard} prompt, called enhanced in the write-up. +The archive does not contain a baseline-prompt ablation, repeated stochastic +draws per model--task pair, or a complete record of earlier failed attempts and +retries. Rates are consequently conditional on inclusion in the scored archive. + +We verify the archive checksum, cell membership, shared truths, finite scores, +and agreement with the saved sweep summary before analysis. The exact source +revision and temperature are not recorded per run. A public implementation +snapshot postdates some runs. Our new experiments use the current pinned +numerical environment; they do not recreate historical provider calls. + +\subsection{Numerical Controls and Replay} +\label{sec:controls} + +For the controls, we regenerate all 30 tasks from their recorded seeds, verify +exact equality with saved ground-truth coefficient vectors, and evaluate +non-language-model query policies under the same accepted-query limits and +supplied query/solve interface. The policies use no hidden coefficients, solution values, measurement responses, +or diagnostics to select expressions. Fixed sine uses the Dirichlet Laplacian eigenfunctions +\(\varphi_k=\sin(k\pi x)\), \(k=1,\ldots,K_{\max}\), where +\(K_{\max}=6N\) is the +stored-query budget. Boundary Legendre multiplies a polynomial test by the +endpoint-vanishing factor \(x(1-x)\), giving \(x(1-x)P_{k-1}(x)\). +The implementation uses factored expressions for numerical stability. Random mixed draws uniformly +from five admissible families, using ten independently seeded query lists per +setting, reused across the ten PDE seeds. Its fixed parameter ranges and exact +expressions are released in the experiment output. All controls solve at every +query count from \(4N\) through \(6N\); additional solves do not alter their +predefined query lists. These controls test whether simple nonadaptive policies suffice on these +instances; they do not reproduce the models' free-action +or stopping decisions. + +For the numerical audit, we replay the archived query expressions, compare the +returned source moments with the logs, and reconstruct joint matrix spectra. +We report both raw and unit-column-norm spectra, with explicit relative singular +value cutoffs rather than a binary rank label alone. We additionally multiply +each reconstructed row and its source moment by a common amplitude before +12-decimal rounding, leaving the query sets fixed and refitting all stored rows. This audit scores +8-decimal diagnostic estimates, unlike unrounded primary final scores. In exact arithmetic this common scaling leaves least squares unchanged; +its effect here tests the absolute quantization rule. It is a post hoc numerical +sensitivity analysis, rather than an optimized query policy or a model rerun. + +We also refine the forward and quadrature grids together by factors two and +four. Original-resolution primary controls use the native dense solver. +Refinements use an independently assembled banded solver for the same finite- +difference equations; we validate its solution and score sensitivity against +native solves on all 30 tasks. Refinement changes the numerical environment, +rather than adding independent tasks or measuring observation-noise robustness. + +To test specificity to the archived seeds, we additionally evaluate the fixed +sine policy on 100 fresh seeds (1001--1100) in each difficulty, with the original +generator and grid settings. Every planned task is retained, including zero +forcing and numerical failures; there are no retries or filters. This is +validation after design selection, not a preregistered comparison or a new +language-model experiment. We report outcomes on all 100 tasks per setting, +with Wilson intervals and descriptive zero/nonzero-source breakdowns. Reused +seed labels across difficulties are not pooled into independent sampling units. + +\subsection{Task-Level Comparisons and Confidence} + +The comparison unit is the shared PDE seed within each difficulty, not an +individual coefficient, solve, or confidence report. Each model has ten tasks +per setting. We report Wilson 95\% intervals for marginal success proportions +and paired task resampling for success differences (20,000 draws; seed 20261020). Exploratory pairwise tests +condition on discordant tasks, with Holm correction across the 15 model pairs +within each difficulty. There are no repeated model draws from which to estimate +within-task response variance. Empirical paired-bootstrap intervals can +degenerate when all ten observed task differences agree. These intervals describe uncertainty under task +resampling assumptions and do not resolve selection into the scored archive. + +A confidence report states the model's probability that a coefficient block +is accurate. We pair it only with a solve returned on a strictly earlier turn. +Its correctness event is \(E_q<0.1\), separately for +\(q\in\{a,b,c\}\), which differs from final-run success \(E<0.01\). +All reports and coefficient blocks within a run are +kept together when resampling; run-balanced scores avoid giving a model more +weight merely because it reports more often. Action counts, first-to-last solve +improvement, and expression-family entropy are descriptive diagnostics. They do +not independently validate planning, learning, or information gain. + \section{Results} \label{sec:results} -%---------------------------------------------------------------------- - -\subsection{Cross-Model Comparison} - -\paragraph{Evidence status.} -The tables below retain the historical values reported in this draft. -The repository does not include their underlying per-run JSON logs or -sweep summaries, so these values and aggregates cannot be recomputed from -the checkout alone. The checked-in figures likewise do not include a -complete source-run archive. In particular, the checked-in success-rate -figures label 180 scored runs across six models, with ten runs per -model--difficulty combination. That aggregate has a different design -from the proposed five-seed, two-prompt study above. Saved Kaggle notebook -outputs provide additional exploratory evidence under their own recorded settings. -No numerical results have been regenerated for this revision. - -Table~\ref{tab:sweep} shows the preliminary reported results. The identifiability -hierarchy $f \gg a > b > c$ holds across all models and difficulties: -$f$ is recovered to machine precision, $a$ is consistently accurate, and -$c$ is the limiting factor. - -\begin{table}[h] + +\subsection{Archived Recovery Differs, with Limited Resolution} + +Figure~\ref{fig:success} shows the archived success counts. At \(E<0.01\), +Gemini 3.1 Pro succeeds on \(10/10\), \(9/10\), and \(10/10\) easy, medium, +and hard tasks; Claude Opus 4.6 achieves \(10/10\), \(9/10\), and \(8/10\). +GPT-5.4 achieves \(7/10\), \(3/10\), and \(1/10\). The other models also +vary substantially across settings and seeds. Corrections to the earlier write-up figures are documented in +Appendix~\ref{app:repro}. + +With ten tasks, even \(10/10\) has a Wilson 95\% interval of approximately +\([0.72,1.00]\), and \(9/10\) has \([0.60,0.98]\). Thus the heatmap is +not a precise model ranking. All pairwise task-level comparisons and confidence +intervals are released with the analysis; differences in this selected archive +should not be generalized to end-to-end provider reliability. +Only one of 45 exploratory pairwise comparisons has adjusted \(p<0.05\) +under the stated within-difficulty Holm procedure: Gemini 3.1 Pro versus +Claude Sonnet 4 on hard tasks (\(10/10\) versus \(0/10\), adjusted +\(p=0.0293\)). There is no correction across the three difficulty families; +this finding does not establish a global model ordering. + +\begin{figure*}[t] \centering -\caption{Historical reported coefficient error across three models and three -difficulty levels (2 seeds each; 18 attempts, 15 scored). -``---'' indicates the model failed to produce a scored run.}\label{tab:sweep} -\small -\begin{tabular}{llrrrrrrrr} -\toprule -\textbf{Model} & \textbf{Diff} & \textbf{Seed} & $E_a$ & $E_b$ & $E_c$ - & $E_f$ & $E$ & \textbf{Solves} & \textbf{Q used} \\ -\midrule -GPT-4o & easy & 42 & 0.0001 & 0.0004 & 0.0026 & 0.0 & 0.003 & 2 & 11 \\ -GPT-4o & easy & 137 & 0.0000 & 0.0000 & 0.0000 & 0.0 & 0.000 & 3 & 12 \\ -GPT-5.4-mini & easy & 42 & 0.0000 & 0.0002 & 0.0014 & 0.0 & 0.002 & 1 & 10 \\ -GPT-5.4-mini & easy & 137 & 0.0000 & 0.0000 & 0.0001 & 0.0 & 0.000 & 1 & 8 \\ -GPT-5.4 & easy & 42 & 0.0000 & 0.0001 & 0.0004 & 0.0 & 0.001 & 3 & 12 \\ -GPT-5.4 & easy & 137 & 0.0000 & 0.0000 & 0.0000 & 0.0 & 0.000 & 4 & 12 \\ -\midrule -GPT-4o & medium & 42 & --- & --- & --- & --- & --- & 0 & 9 \\ -GPT-4o & medium & 137 & 0.0000 & 0.0002 & 0.0019 & 0.0 & 0.002 & 4 & 18 \\ -GPT-5.4-mini & medium & 42 & 0.1861 & 0.6740 & 1.4416 & 0.0 & 2.302 & 1 & 17 \\ -GPT-5.4-mini & medium & 137 & 0.0229 & 0.2777 & 2.4272 & 0.0 & 2.728 & 3 & 13 \\ -GPT-5.4 & medium & 42 & 0.0001 & 0.0005 & 0.0009 & 0.0 & 0.001 & 3 & 17 \\ -GPT-5.4 & medium & 137 & 0.0000 & 0.0000 & 0.0000 & 0.0 & 0.000 & 4 & 17 \\ -\midrule -GPT-4o & hard & 42 & --- & --- & --- & --- & --- & 0 & 0 \\ -GPT-4o & hard & 137 & --- & --- & --- & --- & --- & 0 & 1 \\ -GPT-5.4-mini & hard & 42 & 0.1275 & 1.4848 & 0.7268 & 0.0 & 2.339 & 1 & 20 \\ -GPT-5.4-mini & hard & 137 & 0.1196 & 1.2099 & 3.5182 & 0.0 & 4.848 & 2 & 21 \\ -GPT-5.4 & hard & 42 & 0.0730 & 2.1305 & 1.4725 & 0.0 & 3.676 & 7 & 23 \\ -GPT-5.4 & hard & 137 & 0.0010 & 0.0424 & 0.2458 & 0.0 & 0.289 & 4 & 24 \\ -\bottomrule -\end{tabular} -\end{table} +\includegraphics[width=0.89\textwidth]{images/paper_success.pdf} +\caption{Archived recovery on 30 shared tasks, with ten scored records per model +and difficulty. The two panels use total maximum-coefficient-error thresholds +\(E<0.01\) and \(E<0.1\). Counts are conditional on the scored archive; unscored +attempts and retries are not represented. The primary threshold was retained +from the legacy success-plot convention; the scorer itself returns continuous errors.} +\label{fig:success} +\end{figure*} + +\subsection{Simple Nonadaptive Queries Suffice on These Tasks} +\label{sec:baseline_results} + +The fixed sine policy reaches \(E<0.01\) on all ten tasks in each difficulty, +already at the minimum \(4N\) stored queries (8, 12, and 16). It retains +30/30 success at the full \(6N\) budget. Boundary Legendre succeeds on 29/30 +at full budget. Random mixed succeeds on 100/100 easy, 92/100 medium, and +85/100 hard policy--task combinations (Table~\ref{tab:controls}); these reuse +ten PDE instances per setting and are not 100 independent tasks. +Across random query-list replicates, hard success ranges from 6/10 to 10/10. +Figure~\ref{fig:controls} shows recovery at every permitted solve count. + +\input{paper_artifacts/control_table.tex} + +\begin{figure*}[t] +\centering +\includegraphics[width=\textwidth]{images/paper_controls.pdf} +\caption{New controls on the 30 reconstructed tasks at the original grid +resolution. Fixed sine and boundary Legendre use one predefined query list per +setting. Random mixed averages ten independently seeded lists, each reused on +the same ten tasks; shading is the minimum--maximum success fraction across +lists, not a confidence interval. No query policy uses feedback or hidden truth.} +\label{fig:controls} +\end{figure*} + +These controls show that the archived tasks can be solved by a simple +nonadaptive design with the supplied numerical tools. They do not show a +statistically established advantage over the best archived model, nor isolate +reasoning from parsing and stopping behavior. Because exact historical internal +matrices were not retained, we compare newly reconstructed controls with +archived outcomes transparently rather than claiming a prospective provider +comparison. The evidence does not support a claim that language-model adaptation +is necessary for recovery on this task set. + +Grid refinement improves numerical errors: for fixed sine at full budget, +median \(E\) changes from \(9.47\times10^{-5}\) to +\(2.18\times10^{-6}\) on easy, \(1.32\times10^{-3}\) to +\(2.82\times10^{-5}\) on medium, and \(5.65\times10^{-4}\) to +\(1.56\times10^{-5}\) on hard when the grid is refined fourfold. +At twofold refinement, random mixed reaches 100/100 hard successes and +boundary Legendre reaches 10/10 medium successes. The original-grid native and +banded solves agree on both success thresholds at every checkpoint across all +360 paired policy--task runs. Thus the banded implementation does not change +these threshold comparisons, although individual numerical errors differ. + +\subsection{Fresh Seeds Expose the Limits of the Archive Result} + +Table~\ref{tab:fresh} validates the fixed sine policy on all 300 fresh tasks. +At the minimum query budget, success is 92/100 easy, 91/100 medium, and +98/100 hard. At full budget, it is 92/100, 85/100, and 93/100, respectively. +All 300 executions complete without errors; no task is retried or excluded. +Seven easy and three medium tasks have zero forcing, exactly zero solutions, +and joint operator rank zero at both budgets. All ten fail, as predicted by the +zero-excitation argument. Excluding them only for a descriptive breakdown, +full-budget recovery is 92/93 easy, 85/97 medium, and 93/100 hard. + +\input{paper_artifacts/sine_generalization_table.tex} + +Thus the archived 30/30 result does not establish perfect generator-wide +recovery. Nonzero-source failures also remain, and increasing the stored-query +budget can reduce the number of coefficient-threshold successes in the same +fixed policy. More rows do not guarantee lower coefficient error in an +inconsistent finite-precision system. This validation evaluates one selected +numerical policy; it does not determine whether adaptive policies improve +recovery on fresh tasks. It does establish why a benchmark needs both simple +controls and a stated treatment of unidentifiable instances. + +\subsection{What Numerical Reconstruction Can Establish} +\label{sec:numerics} + +All 30 ground-truth coefficient vectors regenerate exactly, and the reconstructed reference +implementation reproduces all 2,874 archived query source moments exactly at +the stored precision. Source-moment agreement does not test the historical forward solution or +unretained operator matrix: the source moment depends only on \(f\), the test +function, and quadrature. It does not establish exact historical solve replay: across 687 solves, the median maximum deviation from the logged +8-decimal coefficient vectors is \(2.55\times10^{-5}\), and the largest is +3.50. The archive lacks historical internal matrices and sufficient environment +provenance to localize every discrepancy. New controls and spectral diagnostics +must therefore be identified as reconstructions. + +All 180 reconstructed final stored-query matrices have full column rank at +NumPy's default floating-point tolerance. After unit-column scaling, however, +the median \(\sigma_{\min}/\sigma_{\max}\) is +\(2.26\times10^{-7}\), \(5.53\times10^{-10}\), and +\(1.80\times10^{-10}\) for easy, medium, and hard runs. At a declared +relative cutoff of \(10^{-8}\), 13/60, 42/60, and 37/60 matrices are rank +deficient. These are diagnostics of limited numerical resolution, not continuum +nonuniqueness certificates for the sampled instances. + +\begin{figure*}[t] +\centering +\includegraphics[width=\textwidth]{images/paper_numerical_audit.pdf} +\caption{Numerical audit of reconstructed archived measurements. Left: +unit-column-norm smallest-to-largest singular-value ratios, one point per +model--task query design. The dashed line is a diagnostic cutoff, not a theorem +of nonidentifiability. Right: strict-threshold success after common row/RHS +scaling before absolute 12-decimal rounding, using all stored queries and fixed +archived expression choices. These post hoc solves use reconstructed matrices and 8-decimal diagnostic +estimates, and include later stored queries when a run stopped solving before +its final query; they are not archived final submissions.} +\label{fig:audit} +\end{figure*} + +Common-amplitude scaling changes the reconstruction's \(E<0.01\) counts +(Figure~\ref{fig:audit}). At amplitudes \(10^{-3},1,10^3\), the counts +are respectively 24/50/53 of 60 easy runs, 11/36/48 medium runs, and 19/28/33 +hard runs. No additional test functions are selected: the exact-arithmetic measurements +differ only by a common nonzero factor, while their relative quantization +precision changes. +The changed counts show that absolute rounding and ill conditioning can alter +a success event even when the underlying span of measurements is unchanged. +They do not establish that larger amplitude universally improves recovery. + +\subsection{Information Use and Confidence} + +The archive contains 459 free consistency checks; 457 check expressions are +absent as literal strings from earlier stored-query histories. Literal novelty +does not prove semantic independence, but each check exposes an actual source +moment. Thus stored-query counts alone do not equate the information acquired +by two policies. This behavior is permitted by the protocol. + +Appendix~\ref{app:behavior} reports confidence, action use, improvement between +solves, and expression-family diversity. Confidence reports are matched to the +estimates available when they were made. Repeated reports remain grouped within +their run, and expression diversity is not equated with independent constraints. +These diagnostics describe interaction behavior under the supplied prompts. + +\section{Discussion and Limitations} + +The audit reveals three requirements for interpreting measurement-selection +benchmarks. First, the observation model must distinguish identifiable +parameters from equivalent operators. A finite matrix's numerical rank does not +replace this analysis. Second, normalization, precision, and discretization must +be specified and tested. Equivalent continuum measurement designs can have different rounding errors; +that distinction should be explicit in claims about the ability being measured. +Third, every truth-dependent observation must enter the information accounting, +including diagnostics exposed as free tools. + +Our evidence is deliberately bounded. The model archive contains 30 tasks and one +scored response trajectory per model--task pair, rather than a prospective +all-attempt study. Difficulty settings conflate dimension, scale, sparsity, and +resolution. We cannot reconstruct missing attempt histories or exact provider +settings. The structural proposition concerns an admissible subclass and does +not determine the continuum rank of every sampled operator. Reconstructed +spectra, amplitude tests, and numerical controls inherit the current solver and +must not be presented as historical model reruns. + +A prospective benchmark should preregister the task distribution, retain every +attempt and failure, fix model versions and generation settings, and compare +fixed, randomized, and adaptive designs under a common observation budget. +It should normalize admissible tests, specify a noise or precision model, +characterize joint observability, and include prediction under independent +sources or boundary data where coefficient uniqueness is required. Confidence +scores should use a defined event and a declared reporting schedule. These are +requirements for a stronger experimental claim, rather than quantities that +can be inferred from the present archive. + +\section{Conclusion} -\paragraph{Finding 1: Clear generational hierarchy.} -At easy difficulty, all models succeed. At medium difficulty, GPT-5.4 -achieves error $< 0.002$ on both seeds, while GPT-5.4-mini collapses -to error $> 2$ and GPT-4o fails on one seed. At hard difficulty, only -GPT-5.4 achieves sub-1.0 error (on seed 137). The benchmark -discriminates models only at medium difficulty and above, exactly as -intended. - -\paragraph{Finding 2: The $c(x)$ barrier.} -Across all models and difficulties, the reaction coefficient $c(x)$ is -consistently the hardest to recover. On hard difficulty, $c(x)$ accounts -for 40--73\% of the total error. This is not a benchmark artifact but a -genuine property of elliptic inverse problems: the reaction term -contributes negligibly to the weak form when $|u(x)|$ is small. - -\paragraph{Finding 3: Behavioral divergence.} -The models exhibit qualitatively different strategies. GPT-5.4 calls -\texttt{solve} 3--7 times per run, watches coefficients converge, and -uses 95--100\% of its budget. GPT-5.4-mini typically solves once, relies -heavily on consistency checks rather than collecting new queries, and stops early. GPT-4o fails at -hard difficulty because it cannot reliably format expressions the parser -accepts (using \verb|^| instead of \verb|**|, or appending LaTeX markup). - -\subsection{Behavioral Profiles} - -Table~\ref{tab:behavioral} summarizes the behavioral metrics that -distinguish the models. - -\begin{table}[h] +Weak-form inverse problems make parts of scientific-agent evaluation directly +inspectable. The archived models differ in recovery and interaction behavior, +but an interpretation in terms of experimental-design ability requires controls +for the observation model, numerical conditioning, and information accounting. +A fixed sine design suffices on all 30 archive-derived tasks; fresh seeds +reveal unidentifiable zero-excitation cases and additional coefficient-recovery failures. +Both results make numerical baselines essential to interpreting model scores. The released +analysis provides checks for this benchmark and a concrete +basis for designing a more controlled prospective study. + +\section*{Impact Statement} + +This work studies a small mathematical environment for evaluating scientific +agents. Its immediate use is methodological: making evaluation claims traceable +to observations, numerical assumptions, and complete records. Overinterpreting +such scores could distort comparisons of scientific capabilities; the audit +limits that risk by documenting what the measurements support. We do not infer +readiness for autonomous scientific deployment from these experiments. + +\section*{Reproducibility} + +Code, pinned data provenance, and analysis scripts are available at +\url{https://github.com/Hydrodynamical/PDE_Learning}. +Appendix~\ref{app:repro} gives data and numerical details. The offline analyses +require no new model calls. + +\bibliography{references} +\bibliographystyle{icml2026} + +\clearpage +\appendix +\onecolumn +\section{Structural Nonuniqueness: Proof and Examples} +\label{app:proof} + +\begin{proof}[Proof of Proposition~\ref{prop:nullity}] +Let \(\Poly_d\) denote polynomials of degree at most \(d\). +For \(u=x(1-x)\), all three terms in \(T\) lie in \(\Poly_{N+1}\), +so the rank is at most \(N+2\). The reaction component contains +\(u\Poly_{N-1}\). Taking constant \(\delta a\) gives a nonzero constant +because \(-u''=2\), and taking constant \(\delta b\) gives \(u'=1-2x\). +These span \(\Poly_1\). Polynomial division by the quadratic \(u\) gives +\(\Poly_{N+1}=u\Poly_{N-1}\oplus\Poly_1\), proving surjectivity and rank +\(N+2\). The domain has dimension \(3N\), so rank--nullity gives +\(\dim\ker T=2N-2\). Multiplication by any admissible test function and +integration by parts shows that a kernel element changes no weak measurement. +\end{proof} + +For \(N=2\), \(u=x(1-x)\) and \(f=3\), the operators +\[ + a=\tfrac32+\varepsilon,\qquad b=\varepsilon(4x-2),\qquad c=-8\varepsilon +\] +all satisfy the same equation, since +\(2\varepsilon+\varepsilon(4x-2)(1-2x)-8\varepsilon x(1-x)=0\). +For example \(\varepsilon=-1/8\) preserves positive diffusion and gives +nonnegative reaction. For \(N\ge3\), the base choice +\(a=1,b=0,c=1,f=2+u\) belongs to the class. Sufficiently small perturbations +in every kernel direction preserve positive diffusion and reaction, giving a relatively open \((2N-2)\)-dimensional family in the affine kernel +of admissible coefficient tuples with identical observations. +Neither construction is asserted to occur among the 30 sampled tasks. +The audit script also verifies ranks over exact rational arithmetic for +\(N=2,\ldots,8\); these checks supplement the proof rather than replace it. + +\section{Data, Implementation, and Statistical Details} +\label{app:repro} + +The primary input is the public Kaggle dataset \emph{Elliptic Differential +Equation LLM Reasoning}, version 1. The repository's +\path{paper_artifacts/data_manifest.json} records the dataset URL, version, +SHA-256 digest, expected members, model IDs, seeds, and logged settings. Analysis +scripts validate the pinned archive before reading its JSON members; they do +not execute archive contents. The reference environment is Python 3.12 with +pinned numerical packages in \path{requirements.txt}. Derived JSON files retain +per-task observations, method definitions, and numerical cutoffs. + +\begin{table}[ht] \centering -\caption{Full cognitive profile by model (medians across all scored runs). -$H$ = normalized family entropy; $R$ = improvement ratio; $W$ = wasted -turn fraction; Tail = turns after last solve. Check\% = checks as a -fraction of queries (checks are free and do not consume budget).}\label{tab:behavioral} -\small -\begin{tabular}{lcccccccc} +\caption{Exact provider IDs in the April 2026 archive. Display names describe +that archived experiment, rather than current provider offerings.} +\begin{tabular}{@{}ll@{}} \toprule -\textbf{Model} & \textbf{Solves} & \textbf{Budget} & \textbf{Check\%} - & $H$ & $R$ & $\bar M$ & $W$ & \textbf{Tail} \\ +Display name & Recorded model ID\\ \midrule -GPT-4o & 3 & 96\% & 13\% & 0.30 & 1.0 & --- & 0.05 & 1 \\ -GPT-5.4-mini & 1 & 78\% & 18\% & 0.00 & 1.0 & 0.5 & 0.08 & 2 \\ -GPT-5.4 & 4 & 97\% & 8\% & 0.35 & 4.2 & 0.8 & 0.02 & 1 \\ +Gemini 3.1 Pro & \texttt{gemini-3.1-pro-preview}\\ +Claude Opus 4.6 & \texttt{claude-opus-4-6}\\ +GPT-5.4 & \texttt{gpt-5.4}\\ +Gemini 3 Flash & \texttt{gemini-3-flash-preview}\\ +Claude Sonnet 4 & \texttt{claude-sonnet-4-20250514}\\ +GPT-5.4 mini & \texttt{gpt-5.4-mini}\\ \bottomrule \end{tabular} \end{table} -\noindent GPT-5.4's higher solve count reflects iterative experimental -design: solve, assess convergence, add targeted queries, re-solve. -GPT-5.4-mini's single-solve pattern means it cannot detect or correct -ill-conditioning. The check-to-query ratio reveals that GPT-5.4-mini -performs the most consistency checks relative to queries---a behavioral -signature of verification preference over information acquisition---while -GPT-5.4 checks the least, preferring to expand its query set. - -\paragraph{Finding 4: The $H = 0$ / $C_k = 0.99$ dissociation.} -GPT-5.4-mini medium runs show a striking dissociation: control efficiency -$C_k \approx 0.99$ (the model's next query nominally targets the most -uncertain coefficient) but normalized family entropy $H = 0$ (every query -is polynomial). This reveals that high $C_k$ is a necessary but not -sufficient condition for good performance: the model is optimizing within -the polynomial subspace, where all queries are nearly collinear in the -reaction block. The per-block SVD confirms this---the reaction block -$A_{\text{react}}$ has effective rank 1 despite 17 queries, explaining -why $E_c > 1.4$ persists even when $E_a < 0.2$. - -\paragraph{Finding 5: In-context learning gap.} -GPT-5.4 achieves improvement ratios $R > 4$ on medium difficulty (error -drops from $\sim\!0.5$ at first solve to $\sim\!0.001$ at last), while -GPT-5.4-mini has $R = 1.0$ by construction (single solve). The learning -rate for GPT-5.4 is approximately $-0.10$ decades per query---consistent -reduction across the session. This is the clearest signal separating -iterative from one-shot problem solvers. - -\paragraph{Finding 6: Attention and goal maintenance.} -GPT-5.4 wastes fewer than 2\% of turns on unrecognized actions (median -$W = 0.02$) and submits within 1--2 turns of its final solve (tail -$\leq 2$). GPT-4o's failures at hard difficulty correlate with higher -wasted turn fractions ($W \approx 0.05$), where the expression parser -rejects \texttt{\^{}}-style exponentiation. GPT-5.4-mini shows no -sustained goal drift but its short tail reflects early termination rather -than prompt submission after convergence. - -\subsection{Metacognitive Metrics} - -After each \texttt{COMPUTE: solve}, models report per-coefficient -confidence. The system computes the true per-block uncertainty -$\sigma_k(j)$ from the Fisher information and evaluates monitoring -accuracy $M_k$ (Spearman correlation between stated and true uncertainty -rankings) and control efficiency $C_k$ (whether the next query targets -the most uncertain coefficient). - -\emph{Monitoring accuracy.} GPT-5.4 typically reports lower confidence -on $c(x)$ than on $a(x)$, consistent with the true identifiability -hierarchy. GPT-5.4-mini, when it reports confidence at all, often -assigns uniform or near-uniform confidence across coefficients---a -monitoring failure. - -\emph{Control efficiency.} GPT-5.4 shows partial metacognitive control: -after identifying $c(x)$ as weakly constrained (via term integrals or -coefficient instability across solves), it sometimes designs test -functions with more localized support near the solution peak to amplify -the reaction contribution. However, this behavior is inconsistent across -seeds, and even GPT-5.4 does not reliably close the full -monitoring--control loop. - -%---------------------------------------------------------------------- -\section{Discussion} -%---------------------------------------------------------------------- - -\subsection{What the Benchmark Measures} - -This benchmark occupies a specific point in the space of cognitive -evaluations: - -\begin{enumerate}[nosep] - \item \textbf{Not knowledge recall.} The model is not asked to recite - facts about PDEs. It must \emph{use} the weak form to design - experiments. - \item \textbf{Not tool use alone.} All models have identical tools. - The difference is whether they use them \emph{strategically}. - \item \textbf{Not arithmetic.} The model never solves a linear system - itself; \texttt{COMPUTE: solve} handles that. The model's job - is experimental design and epistemic management. - \item \textbf{Multiple cognitive faculties.} A single session engages - mathematical reasoning, metacognitive monitoring and control, - planning, cognitive flexibility, in-context learning, attention, - and goal maintenance---producing an eight-dimensional cognitive - profile rather than a scalar score. -\end{enumerate} - -\subsection{Connection to the Cognitive Taxonomy} - -Burnell et al.~\citep{burnell2026cognitive} identify metacognition as a -key faculty that existing benchmarks fail to cover adequately. Our -benchmark addresses this gap by providing a task where the correct -metacognitive state is \emph{mathematically derivable} at every point in -the session. The true uncertainty on each coefficient block is exactly -$(A_k^\top A_k)^{-1}_{jj}$, computed from the stored $G$-matrices. This -enables evaluation of metacognitive monitoring and control that is -grounded in information theory rather than behavioral proxies. - -The benchmark covers eight of the ten Burnell et al.\ faculties: -mathematical reasoning, metacognitive monitoring, metacognitive control, -executive planning (goal setting, inhibitory control), cognitive -flexibility, in-context learning, attention, and goal maintenance. -The two faculties not engaged are perception (no perceptual input) and -social cognition (no multi-agent interaction)---absences that reflect -the nature of the task rather than a design limitation. Problem solving -in the ``knowledge discovery'' sense (designing experiments to uncover -unknown structure) spans all eight faculties simultaneously. - -\subsection{Structural Identifiability} - -A natural concern is whether the benchmark is fundamentally fair: can the -coefficient functions be uniquely identified from weak-form data in principle? -The answer is yes. \citet{egger2014simultaneous} prove uniqueness of -simultaneous identification of diffusion and absorption coefficients from the -Dirichlet-to-Neumann map for a quasilinear elliptic problem. -\citet{browning2024structural} study linear reaction--advection--diffusion -systems and show that all three coefficients are structurally identifiable -from solution data under generic conditions, though the identifiability -degrades in specific degenerate cases. -\citet{browning2025parabolic} frame identifiability via the Fredholm -alternative, identifying specific boundary conditions that can cause -non-identifiability. - -In our setting, the weak form with sufficiently many linearly independent -test functions uniquely determines all $4N$ coefficients---the system is -structurally identifiable in the infinite-data limit. The ill-conditioning -documented in Section~\ref{sec:illcond} is a \emph{practical} identifiability -challenge arising from finite budget and the specific test functions chosen, -not a fundamental limitation. The benchmark measures how well models handle -this practical challenge, which is precisely what makes it scientifically -interesting. - -\subsection{Comparison to Existing Benchmarks} - -Existing mathematical reasoning benchmarks test whether models can -\emph{answer} mathematical questions. Our benchmark tests whether models can -\emph{do} mathematics---design experiments, interpret results, diagnose -failures, and adapt strategy within a scientific investigation. - -Recent related work includes HARDMath2~\citep{hardmath2025} (211 problems -covering nonlinear PDEs, boundary layers, and WKB approximation) and -Mamo~\citep{mamo2025} (benchmarking LLMs on constructing ODE and linear -programming models with solver verification). HARDMath2 is static Q\&A with -no interactive probing. Mamo is process-oriented like our benchmark but -focuses on model construction rather than inverse identification. -FrontierMath (Epoch AI) provides extremely hard static problems but no -multi-turn experimental design. - -The multi-turn, tool-using structure of our benchmark is closer to agentic -science benchmarks than to traditional math evaluation, and uniquely provides -mathematically grounded metacognitive metrics as a core output. - -\subsection{Limitations} - -The benchmark evaluates models through text-based tool use, which -conflates mathematical reasoning with parsing ability. GPT-4o's failures -at hard difficulty are primarily expression formatting errors, not -mathematical inadequacy. Future versions could use structured tool -calling to separate these concerns. - -The manuscript's preliminary comparison uses only two seeds per difficulty -level and does not report the proposed cross-provider prompt ablation. Its -underlying run archive is not included, limiting independent verification -of the numerical tables and behavioral aggregates. Hard -difficulty shows high variance (GPT-5.4 achieves 0.29 on one seed and -3.68 on the other), suggesting that some PDE instances are inherently -easier to identify. A larger seed sweep would provide more robust -estimates. - -GPT-5.2 and newer OpenAI models default to \texttt{reasoning\_effort: -none}, which disables chain-of-thought reasoning. The draft reports -\texttt{reasoning\_effort: medium} for GPT-5.4 and GPT-5.4-mini; GPT-4o -has no reasoning-effort override in the runner. -Runs without this override would likely perform significantly worse, -particularly GPT-5.4-mini, and the results should not be compared to -evaluations that use the default API setting. - -Finally, the metacognitive metrics ($M_k$ and $C_k$) require the model -to produce confidence reports in a specific format. Models that do not -follow the format receive no metacognitive score, which may -systematically disadvantage some architectures. - -%---------------------------------------------------------------------- -\section{Conclusion} -%---------------------------------------------------------------------- - -We have presented a benchmark that produces a cognitive profile of -scientific reasoning capabilities in language models, grounded in the -mathematical structure of elliptic inverse problems. The benchmark -cleanly separates models by generation: GPT-5.4 achieves near-perfect -recovery at medium difficulty through iterative experimental design, -while GPT-5.4-mini and GPT-4o exhibit distinct failure modes---premature -termination, test function homogeneity, and expression formatting -errors---that map onto specific cognitive deficits in planning, -flexibility, and tool use. - -The benchmark's key contribution is providing \emph{mathematically -grounded metacognitive evaluation}: the true epistemic state of the model -is exactly computable at every step, enabling precise measurement of -monitoring accuracy and control efficiency without behavioral proxies. -This addresses a gap identified by Burnell et al.\ in the current -landscape of cognitive evaluations for AI systems. - -The code, analysis tools, and selected notebook outputs are available at -\url{https://github.com/Hydrodynamical/PDE_Learning}. The complete -source-run archive for the manuscript's tables and checked-in figures -remains to be published. - -\bibliographystyle{plainnat} - -\begin{thebibliography}{9} - -\bibitem[Burnell et al.(2026)]{burnell2026cognitive} -R.~Burnell, Y.~Yamamori, O.~Firat, K.~Olszewska, S.~Hughes-Fitt, -O.~Kelly, I.~R. Galatzer-Levy, M.~R. Morris, A.~Dafoe, A.~M. Snyder, -N.~D. Goodman, M.~Botvinick, and S.~Legg. -\newblock Measuring progress toward {AGI}: A cognitive framework. -\newblock Google DeepMind, March 2026. - -\bibitem[Flavell(1979)]{flavell1979metacognition} -J.~H. Flavell. -\newblock Metacognition and cognitive monitoring: A new area of -cognitive--developmental inquiry. -\newblock \emph{American Psychologist}, 34(10):906, 1979. - -\bibitem[Nelson(1990)]{nelson1990metamemory} -T.~O. Nelson. -\newblock Metamemory: A theoretical framework and new findings. -\newblock In G.~H. Bower, editor, \emph{The Psychology of Learning and -Motivation}, volume~26, pages 125--173. Academic Press, 1990. - -\bibitem[Morris et al.(2024)]{morris2024levels} -M.~R. Morris, J.~Sohl-Dickstein, N.~Fiedel, T.~Warkentin, A.~Dafoe, -A.~Faust, C.~Farabet, and S.~Legg. -\newblock Position: Levels of {AGI} for operationalizing progress on the -path to {AGI}. -\newblock In \emph{ICML}, 2024. - -\bibitem[Browning et al.(2024)]{browning2024structural} -A.~P. Browning, N.~Tasca, C.~Falc\'o, and R.~E. Baker. -\newblock Structural identifiability analysis of linear - reaction--advection--diffusion processes in mathematical biology. -\newblock \emph{Proc. R. Soc. A}, 480, 2024. - -\bibitem[Browning et al.(2025)]{browning2025parabolic} -A.~P. Browning, N.~Tasca, C.~Falc\'o, and R.~E. Baker. -\newblock Structural identifiability of linear-in-parameter parabolic {PDEs} - through auxiliary elliptic operators. -\newblock \emph{J. Math. Biology}, 2025. - -\bibitem[Egger et al.(2014)]{egger2014simultaneous} -H.~Egger, J.-F. Pietschmann, and M.~Schlottbom. -\newblock Simultaneous identification of diffusion and absorption coefficients - in a quasilinear elliptic problem. -\newblock \emph{Inverse Problems}, 30(3):035009, 2014. - -\bibitem[HARDMath2(2025)]{hardmath2025} -HARDMath2 Benchmark. -\newblock Benchmarking {LLMs} on hard applied mathematics: {PDEs}, boundary - layers, and {WKB} approximation. -\newblock 2025. - -\bibitem[Mamo(2025)]{mamo2025} -Mamo Benchmark. -\newblock Mathematical model construction with {LLMs} and solver verification. -\newblock 2025. - -\end{thebibliography} +OpenAI records specify \texttt{reasoning\_effort=medium}; the other providers +have no recorded override. Temperature is absent. The \emph{PDE Library}, +version 9, is a separately published source snapshot that postdates some runs, +not an exact per-run revision. Exact coefficient regeneration and query-RHS +agreement narrow the reproduction gap, but the solve discrepancies reported in +the main text prohibit a claim of bitwise historical replay. + +The original write-up describes basis sizes 3/5/8 and a held-out-grid function +success criterion; the logs and scoring code establish 2/3/4 and +Equation~\eqref{eq:error}. The regenerated strict-threshold figure also corrects the hard GPT-5.4 mini +count from 0/10 to 2/10, matching the individual logs and saved sweep summary. +These corrections are recorded in the derived summaries. The earlier draft's distinct three-model, two-seed pilot lacks source +logs and used an unverified error norm. Its numeric tables are preserved in +\path{paper_artifacts/legacy_pilot_tables.tex} for provenance, but excluded from +this manuscript's empirical evidence. + +\input{paper_artifacts/statistical_tables.tex} +\FloatBarrier + +\section{Behavioral Figures and Their Interpretation} +\label{app:behavior} + +The figures below regenerate the relevant Kaggle diagnostics from the full +archive. They retain the original operational questions while correcting data +or plotting discrepancies. For action counts, stacked marginal medians need not +sum to the median total action count. For improvement, +\(R=E_{\mathrm{first}}/E_{\mathrm{last}}\); missing ratios, including +single-solve runs, receive the historical display default \(R=1\). Large ratios +can reflect poor initial estimates or the ordinary benefit of additional data. + +Query-family entropy is +\(H=-\sum_{i=1}^5p_i\log_2p_i/\log_2 5\), with \(0\log0=0\). +The five categories are polynomial, trigonometric, exponential, localized, and +other, assigned by a heuristic string classifier. The archived taxonomy is +recorded at the final solve; later queries in six runs are absent from it. +A high entropy does not imply linearly independent or useful measurements. + +The confidence plots use disjoint bins \([l,h)\), with 1 included in the last +bin, and show mean confidence against empirical correctness. Reports are +matched only to strictly earlier solves, since a solve requested in a report's +own response has not yet been returned. The archive contains 733 confidence reports: 707 have a preceding solve and +26 are excluded. Causal matching changes the solve paired to 71 reports relative +to the legacy query-count rule; 827 observations lie on interior bin boundaries +that the old plots could count twice. The retained reports contribute +2,121 dependent observations for \(a,b,c\), excluding \(f\). Individual +coefficient reports are not treated as independent tasks. + +For run \(i\) with \(m_i\) retained coefficient observations, the run-balanced +Brier score is +\[ + \frac{1}{n}\sum_{i=1}^n\frac{1}{m_i}\sum_{r=1}^{m_i}(p_{ir}-y_{ir})^2. +\] +We also report the signed gap obtained by replacing the square with +\(p_{ir}-y_{ir}\); positive values indicate average overconfidence. +This gap is calibration in the large, not expected calibration error or a full +calibration criterion. Intervals resample whole supported runs with 20,000 draws. +There are 179 supported runs: Gemini Flash hard has nine and all other cells ten. +Run balancing does not align reporting stages, query counts, or base accuracies. +Thus these scores cannot compare intrinsic metacognitive ability without a +matched reporting schedule and correctness distribution. + +\begin{figure}[ht] +\centering +\includegraphics[width=\textwidth]{images/paper_confidence_uncertainty.pdf} +\caption{Run-balanced confidence summaries with whole-run percentile-bootstrap +95\% intervals, conditional on having a preceding solve and confidence report. +Brier scores reflect both correctness and stated confidence. Signed gaps measure +average confidence minus correctness; zero alone does not establish calibration. +Reports and coefficient blocks remain clustered within runs.} +\label{fig:confidence_uncertainty} +\end{figure} + +\section{Why Legacy Sensitivity Scores Are Not Uncertainty Estimates} + +The archived code forms a truncated pseudoinverse \(H_k\) of +\(A_k^\top A_k\), discarding singular values below \(10^{-10}\) times the +largest. It summarizes each coefficient block using +\(s_k(q)=\sqrt{\sum_{i\in q}(H_k)_{ii}/N}\). +A truncated inverse can assign zero weight to unresolved directions. Without +an observation-noise model and rank conditions, it is not a calibrated +coefficient covariance. Separate block values also cannot diagnose dependencies +between blocks. + +The monitoring routine ranks values using double \texttt{argsort}, without +averaging ties; it is not standard tie-aware Spearman correlation. The control +routine applies a Sherman--Morrison-style update to a truncated pseudoinverse, +which need not agree with a freshly recomputed truncated inverse, and compares +reductions across blocks for the chosen next row rather than across candidate +queries. We retain these source fields for provenance but do not use them as +evidence of calibrated uncertainty or optimal experimental design. + +\begin{figure}[p] +\centering +\includegraphics[width=.84\textwidth,height=.79\textheight,keepaspectratio]{images/paper_actions.pdf} +\caption{Archived tool use, with one panel per difficulty. Stacked components +are separate marginal medians across ten runs per model. The prompt encourages +iteration and checks supply source moments, so more actions do not establish +better experimental design.} +\label{fig:actions} +\end{figure} +\begin{figure}[p] +\centering +\includegraphics[width=.84\textwidth,height=.79\textheight,keepaspectratio]{images/paper_improvement.pdf} +\caption{First-to-last solve improvement on a logarithmic scale. Points are +runs and black markers are medians. Missing ratios have display value one. +This statistic measures within-session error change, not model-weight learning.} +\label{fig:improvement} +\end{figure} +\begin{figure}[p] +\centering +\includegraphics[width=.84\textwidth,height=.79\textheight,keepaspectratio]{images/paper_diversity.pdf} +\caption{Normalized expression-family entropy. Bars are means, whiskers one +population standard deviation, points individual runs, and black markers medians. +The taxonomy describes expression forms, not measurement independence.} +\label{fig:diversity} +\end{figure} +\begin{figure}[p] +\centering +\includegraphics[width=.84\textwidth,height=.79\textheight,keepaspectratio]{images/paper_calibration.pdf} +\caption{Corrected descriptive confidence curves. The horizontal coordinate +is mean stated confidence in a disjoint width-0.1 bin; the vertical coordinate +is the fraction with \(E_q<0.1\). Marker area grows with bin support. The +707 causally paired reports contribute 2,121 observations, clustered within runs; +these curves alone are not a general calibration ranking.} +\label{fig:calibration} +\end{figure} +\FloatBarrier \end{document} diff --git a/paper_artifacts/README.md b/paper_artifacts/README.md new file mode 100644 index 0000000..556fe03 --- /dev/null +++ b/paper_artifacts/README.md @@ -0,0 +1,198 @@ +# Reproducing the paper figures + +The main analysis uses the public Kaggle dataset +[Elliptic Differential Equation LLM Reasoning, version 1](https://www.kaggle.com/datasets/jkmiller/elliptic-differential-equation-llm-reasoning/versions/1), +released on 16 April 2026. It contains 180 scored run JSON files and a sweep +summary: six models × three difficulties × seeds 1–10. The original +[write-up](https://www.kaggle.com/competitions/kaggle-measuring-agi/writeups/can-language-models-learn-a-pde#3442067) +provides the project context. + +## Run the analysis + +From the repository root, using the Python 3.12 environment in the main README: + +```bash +curl -L --fail \ + 'https://www.kaggle.com/api/v1/datasets/download/jkmiller/elliptic-differential-equation-llm-reasoning?datasetVersionNumber=1' \ + -o /tmp/pde-runs-v1.zip + +python scripts/reproduce_paper_figures.py --archive /tmp/pde-runs-v1.zip +``` + +The script does not make network or model calls. It reads the ZIP without +extracting or executing its contents and verifies the SHA-256 digest against +[data_manifest.json](data_manifest.json). It also checks cell sizes, shared +problem instances, recorded settings, finite scores, and agreement with the +archived sweep summary. A different archive fails validation. + +To verify and regenerate only the numerical summary: + +```bash +python scripts/reproduce_paper_figures.py \ + --archive /tmp/pde-runs-v1.zip --summary-only +``` + +Outputs are [summary.json](summary.json) and five vector figures under `images/`: +`paper_success.pdf`, `paper_actions.pdf`, `paper_improvement.pdf`, +`paper_diversity.pdf`, and `paper_calibration.pdf`. Original PNGs are retained as +historical assets; the manuscript uses the regenerated figures. + +## What is measured + +- **Success:** total absolute coefficient error + `E = sum(max(abs(predicted_q - true_q)))` over `q = a, b, c, f`, with thresholds + `E < 0.01` and `E < 0.1`. These are coefficient errors, not relative or + held-out-grid function errors. +- **Actions:** separate median counts of solves, checks, and term-integral calls. + Their stacked sum need not equal the median total action count. +- **Improvement:** the archived first-to-last solve error ratio. Missing ratios + retain the historical plotting default of one. A large ratio need not imply + accurate final coefficients. +- **Diversity:** archived normalized query-family entropy. Expression categories + are heuristic and do not establish linear independence of measurements. The + archived taxonomy is measured at the final solve; later queries in six runs + do not enter that entropy. +- **Calibration:** complete confidence reports for `a`, `b`, and `c` paired to + the latest strictly preceding solve in the assistant transcript, with correctness + defined by absolute coefficient error below `0.1`. Reports without a preceding + solve are excluded. Bins are disjoint, with confidence one in the final bin. + The summary records bin counts and mean confidence; observations within a run + are dependent. + +The recovered logs correct the older strict-threshold chart: GPT-5.4 mini has +**2/10** hard successes, confirmed by individual logs and the sweep summary. +The archive also establishes basis sizes **2/3/4**, rather than the write-up's +3/5/8. All 180 runs record `prompt_condition: standard`; the write-up calls the +augmented prompt enhanced. + +The confidence reanalysis also fixes two historical plotting issues: matching a +report to a future solve and counting exact confidence-bin endpoints in two bins. +The archive contains 733 reports, of which 707 have a preceding solve and 26 do +not. The 707 paired reports contribute 2,121 coefficient observations. The +reanalysis changes the matched solve for 71 paired reports; 827 paired +coefficient observations lie on interior boundaries that the older code could +count twice. The fixed confidence-order plot and legacy sensitivity-based +monitoring/control statistics are distinct diagnostics and are not substituted +for these calibration curves. + +## Scope and provenance + +The archive includes scored records, not a complete failed-attempt/retry history. +Temperature and exact implementation versions are not recorded per run. The +separate [PDE Library snapshot, version 9](https://www.kaggle.com/datasets/jkmiller/pde-library/versions/9) +postdates some records and is not asserted to be the exact source for every run. +The manifest records its provenance separately. + +The original draft's three-model, two-seed pilot tables are preserved in +[legacy_pilot_tables.tex](legacy_pilot_tables.tex) for provenance. Their +underlying logs have not been recovered. The older draft described their errors +as mean absolute deviations, a convention that cannot be verified and differs +from the main study's maximum-deviation metric; they are not pooled or compared +numerically with the main study. The research revision adds numerical controls without new provider calls. + +## Build the manuscript + +With a TeX installation providing `latexmk`: + +```bash +latexmk -pdf -interaction=nonstopmode -halt-on-error paper.tex +``` + +`paper.tex` uses the single BibTeX source `references.bib`, the official ICML +preprint style, generated research tables, and the regenerated vector figures. +Unverified legacy pilot tables remain outside the manuscript evidence. + +## Research controls and uncertainty analysis + +The October research revision adds three offline experiments. Run these after +setting up the pinned Python 3.12 environment and downloading the pinned ZIP: + +```bash +python scripts/analyze_archive_statistics.py --archive /tmp/pde-runs-v1.zip +python scripts/control_baselines.py --archive /tmp/pde-runs-v1.zip +python scripts/audit_identifiability.py --archive /tmp/pde-runs-v1.zip +python scripts/validate_sine_generalization.py +python scripts/plot_research_audit.py +python scripts/verify_paper_artifacts.py +``` + +These commands make no model or network calls. A full archived-task control experiment takes +roughly one to two minutes; the fresh-seed validation takes about three minutes on the development machine; timings depend on hardware. + +- `statistical_analysis.json` retains Wilson success intervals, all 45 paired + comparisons, discordance counts, exact paired tests, and Holm-adjusted values + within the 15 pairs of each difficulty. Percentile bootstrap intervals use + 20,000 draws, seed `20261020`, and shared tasks within difficulty. Confidence + summaries average within a run before resampling whole runs; reports and + coefficient blocks do not become independent tasks. The estimates condition + on scored archive inclusion and, for confidence, availability of a paired report. +- `control_baselines.json.gz` contains the full new experiment, compressed with + a fixed gzip timestamp; `control_baselines_summary.json` provides a compact + human-readable view. Fixed sine, boundary-weighted Legendre, and ten random + mixed-family query lists are evaluated on the same 30 regenerated truths. + There are 1,080 policy/grid runs across original, 2×, and 4× grid resolutions, + plus 360 native-versus-banded comparisons. The random design lists are reused + across the ten tasks in each setting: 100 outcomes represent ten designs + crossed with ten PDEs. They do not increase the number of independent tasks. +- `identifiability_audit.json` records exact polynomial-rank certificates, + reconstructed joint spectra, 687 solve replays, all 2,874 query-response + comparisons, free-check counts, and common-amplitude quantization tests. + Its CLI reads the validated ZIP directly. An optional `--runs-dir` verifies an + extracted copy byte for byte against the pinned archive before using it. +- Three additional vector figures are `paper_controls.pdf`, + `paper_numerical_audit.pdf`, and `paper_confidence_uncertainty.pdf`. + Generated TeX tables and `research_figure_sources.json` bind their numbers to + the exact derived inputs and plotting script. + +Fixed sine succeeds at `E < 0.01` on all 30 reconstructed tasks already at the +minimum permitted `4N` stored queries. This result limits any claim that adaptive +model measurement selection is necessary on these instances. It is a new +numerical control, not an exact replay of the historical environment. Although +all archived query RHS responses match exactly, reconstructed solve coefficients +can differ materially from their logged values. The paper reports this gap. + +Original-resolution controls use the native dense forward solver and unchanged +query/solve tools. Refinements use an independently assembled equivalent banded +solver, checked against the native solver on every original-resolution task. +Both threshold classifications agree at every checkpoint. Primary control scores +use the internal unrounded final estimate, matching the current submission +flow. Display-rounded scores are also saved, with identical classifications at +both thresholds. The amplitude audit separately scores 8-decimal diagnostic +estimates, using all stored rows; it is not an archived final-submission score. + +Source hashes and executed runtime versions are included in each research +output. `verify_paper_artifacts.py` uses the standard library to reject stale +outputs after an implementation, input, or plotting change. Floating-point +values can still vary across numerical platforms; integrity checks do not claim +bitwise portability of newly rerun solves. + +The public paper uses the unmodified official ICML 2026 template with its +`preprint` option; see [template_provenance.md](template_provenance.md). The main +text follows the eight-page constraint, with references and detailed appendices +following it. It is a public preprint and retains author information and public +repository links. An actual conference submission requires the target year's +instructions and anonymization. + +## Fresh-seed validation after selecting fixed sine + +`scripts/validate_sine_generalization.py` evaluates every seed 1001–1100 in +each of the three difficulty settings, using the native original-resolution +solver and fixed sine queries. This experiment follows design selection on the +archived tasks; it is not a historical held-out study or a formal preregistration. +All 300 planned tasks complete without retries, exclusions, or execution errors. +`sine_generalization.json` retains the complete plan, truths, task outcomes, +zero-source/zero-solution flags, ranks, numerical fingerprints, and provenance. + +Strict-threshold success at the minimum/full budgets is 92/92 of 100 easy, +91/85 medium, and 98/93 hard. Wilson intervals use all 100 tasks per difficulty. +Seven easy and three medium tasks have zero forcing and zero solutions, with +operator rank zero; all fail recovery. These remain in the primary denominators. +The generator assigns zero forcing probability `sparsity**N`. Nonzero-source +failures also remain, so the archived 30/30 result must not be generalized to +perfect generator-wide recovery. Extra stored queries turn six medium and five +hard minimum-budget successes into failures in this fixed policy. + +No new language-model runs are included in this validation. It does not establish +whether adaptation improves results on fresh seeds. Source/runtime hashes and +all raw/displayed scores are retained; the provenance checker verifies this +experiment too. Generated `sine_generalization_table.tex` reports the main rates. diff --git a/paper_artifacts/control_baselines.json.gz b/paper_artifacts/control_baselines.json.gz new file mode 100644 index 0000000..51910ab Binary files /dev/null and b/paper_artifacts/control_baselines.json.gz differ diff --git a/paper_artifacts/control_baselines_summary.json b/paper_artifacts/control_baselines_summary.json new file mode 100644 index 0000000..8cb7bbd --- /dev/null +++ b/paper_artifacts/control_baselines_summary.json @@ -0,0 +1,8758 @@ +{ + "schema_version": 1, + "archive_sha256": "b2f50742cc76acae4a6a8d7b04d434e7735acf107d57f08f499e1dfcf05ce66f", + "dataset_url": "https://www.kaggle.com/datasets/jkmiller/elliptic-differential-equation-llm-reasoning/versions/1", + "experiment_type": "new offline controls on exactly regenerated archived PDE instances", + "historical_replay_limitation": "Exact ground-truth regeneration and first-query RHS replay do not establish identical historical solution grids or numerical implementations for all runs; historical logs lack per-run implementation hashes.", + "protocol": { + "control_seed": 1729, + "random_replicates": 10, + "grid_factors": [ + 1, + 2, + 4 + ], + "unique_instances": 30, + "query_and_identification_implementation": "unchanged ProbeSession.query and ProbeSession.solve", + "minimum_solve_queries": "4N, enforced by the existing interface; fitted LHS has 3N columns and forcing is fitted separately", + "scoring": "Sum over a,b,c,f of maximum absolute coefficient error, using unrounded internal estimates, matching final PREDICT scoring; strict E < threshold. Displayed 8dp scores, matching intermediate historical error curves, are also saved and threshold agreement audited.", + "selection_information": "Only public query budget and independent control RNG. No solution, coefficient, oracle-response, error, rank, or condition information enters selection.", + "random_sampling": "Uniform family choice among sine, boundary-weighted Legendre, exponential, localized Gaussian, boundary-weighted sine. Sine degrees 1..budget and polynomial degrees 0..budget-1 sampled without replacement within each family. Exponential rates uniform[-8,8]; Gaussian centers uniform[.05,.95], log precision uniform[log8,log128].", + "scaling": "Use expressions as generated, with x(1-x) envelope for all families except pure sine; no post-hoc row normalization or tuning.", + "grid_refinement": "Refine forward-solution grid and every query integral together; (n_grid-1)*factor+1 points. This is not a noise robustness test.", + "forward_solver": "Native dense solve at original grid; independently assembled equivalent tridiagonal solve_banded at refined grids, validated against native dense at original grid on all 30 instances.", + "aggregation": "Pooled randomized runs are descriptive. Per-replicate summaries and per-instance success fractions preserve distinct PDE and randomization units; 10 random replicates do not increase the 30 independent PDE instances.", + "design_svd": "Joint [G_diff G_adv G_react] and independently column-normalized version; rank tolerance max(shape)*eps*sigma_max. Full singular spectra are saved." + }, + "environment": { + "python": "3.12.12", + "numpy": "2.5.4", + "scipy": "1.18.1" + }, + "elapsed_seconds": 54.19931433396414, + "source_sha256": { + "scripts/control_baselines.py": "e879bfb0363a22d3429fc62e2dd9fd8adaf333044c1868396581bf9b78ed705e", + "main_loop.py": "b9322fc9aab81265c67883a4c6aba608dff375249ced41d2c315953dac0d01fe", + "pde.py": "580c59d8367ec8b8d1f490ce39cff41dd9cefc934db435d327b5283b466a7c4a", + "basis.py": "5266a1631d7b632645030bbf9e23e02ebe36b3ddc06f5ecbdc1bae81123d8667", + "benchmark.py": "9ad5be152a256c846affee2c32fba246ab99b03a8ba1fa490019120203123cd2", + "expression_parser.py": "fa97b97e71bd8116a2729d209e6a9f5cea8247834d7900255c61b8a888032bc0", + "weak_form.py": "aed59913069e7f3f46e49b7bd98aba9e3c636fcff639259e146e734d320be83e", + "test_functions.py": "4e972186bb125c598024df11626d95f4712188d5601ad709d69b4f6fccd658ad", + "requirements.txt": "39a52b2e7d6304c1c80053325ca16be1542e2993d5299bb11f9807bf875d8bfd" + }, + 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100644 index 0000000..7132637 --- /dev/null +++ b/paper_artifacts/control_table.tex @@ -0,0 +1,24 @@ +% Generated by scripts/plot_research_audit.py; do not edit numbers manually. +\begin{table}[t] +\caption{New offline controls at the original resolution and full stored-query budget. Deterministic policies have ten tasks per difficulty. Random rows summarize ten query-list replicates on the same ten tasks; the denominator 100 is not 100 independent PDEs.} +\label{tab:controls} +\centering\small +\begin{tabular}{@{}llrr@{}} +\toprule +Policy & Setting & Successes & Median $E$\\ +\midrule +Fixed sine & Easy & 10/10 & 9.5e-05\\ +Fixed sine & Medium & 10/10 & 0.0013\\ +Fixed sine & Hard & 10/10 & 0.00057\\ +\addlinespace +Boundary Legendre & Easy & 10/10 & 3.4e-05\\ +Boundary Legendre & Medium & 9/10 & 0.0014\\ +Boundary Legendre & Hard & 10/10 & 0.00057\\ +\addlinespace +Random mixed & Easy & 100/100 & 6.4e-05\\ +Random mixed & Medium & 92/100 & 0.0016\\ +Random mixed & Hard & 85/100 & 0.00097\\ +\addlinespace +\bottomrule +\end{tabular} +\end{table} diff --git a/paper_artifacts/data_manifest.json b/paper_artifacts/data_manifest.json new file mode 100644 index 0000000..6c43d62 --- /dev/null +++ b/paper_artifacts/data_manifest.json @@ -0,0 +1,44 @@ +{ + "schema_version": 1, + "dataset_ref": "jkmiller/elliptic-differential-equation-llm-reasoning", + "dataset_title": "Elliptic Differential Equation LLM Reasoning", + "dataset_version": 1, + "dataset_published_utc": "2026-04-16T19:44:06.873Z", + "dataset_url": "https://www.kaggle.com/datasets/jkmiller/elliptic-differential-equation-llm-reasoning/versions/1", + "download_url": "https://www.kaggle.com/api/v1/datasets/download/jkmiller/elliptic-differential-equation-llm-reasoning?datasetVersionNumber=1", + "archive_sha256": "b2f50742cc76acae4a6a8d7b04d434e7735acf107d57f08f499e1dfcf05ce66f", + "archive_bytes": 1461201, + "archive_members": 181, + "expected_runs": 180, + "expected_scored_runs": 180, + "expected_seeds": [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], + "models": [ + "gemini-3.1-pro-preview", + "claude-opus-4-6", + "gpt-5.4", + "gemini-3-flash-preview", + "claude-sonnet-4-20250514", + "gpt-5.4-mini" + ], + "difficulty_settings": { + "easy": {"n_basis": 2, "unknowns": 8, "max_queries": 12, "max_turns": 36}, + "medium": {"n_basis": 3, "unknowns": 12, "max_queries": 18, "max_turns": 54}, + "hard": {"n_basis": 4, "unknowns": 16, "max_queries": 24, "max_turns": 72} + }, + "prompt_condition": "standard", + "publication_url": "https://www.kaggle.com/competitions/kaggle-measuring-agi/writeups/can-language-models-learn-a-pde#3442067", + "implementation_snapshot": { + "dataset_ref": "jkmiller/pde-library", + "dataset_version": 9, + "published_utc": "2026-04-15T20:31:32.35Z", + "url": "https://www.kaggle.com/datasets/jkmiller/pde-library/versions/9", + "archive_sha256": "0aff72715241e3fc8144cbf0fe9a55c8e700f6368a7400437a4d89b05850e0c3", + "limitation": "This published snapshot postdates most runs; individual logs do not pin implementation versions." + }, + "limitations": [ + "The archive records 180 scored runs, not a complete history of failed attempts or retries.", + "The archive stores the condition name standard; the writeup describes the augmented instructions as enhanced.", + "Kaggle metadata reports the dataset license as Unknown.", + "These data do not recover the preliminary 18-attempt table in the earlier manuscript." + ] +} diff --git a/paper_artifacts/identifiability_audit.json b/paper_artifacts/identifiability_audit.json new file mode 100644 index 0000000..9158064 --- /dev/null +++ b/paper_artifacts/identifiability_audit.json @@ -0,0 +1,53209 @@ +{ + "schema_version": 1, + "archive_sha256": "b2f50742cc76acae4a6a8d7b04d434e7735acf107d57f08f499e1dfcf05ce66f", + "dataset_url": "https://www.kaggle.com/datasets/jkmiller/elliptic-differential-equation-llm-reasoning/versions/1", + "method": { + "input_validation": "ZIP bytes and complete scored-run corpus validated by reproduce_paper_figures.load_and_validate against the pinned dataset manifest.", + "solution_reconstruction": "Current repository pde.solve_elliptic from saved ground truth; no fitting to archived estimates.", + "grids": { + "easy": 2001, + "medium": 2001, + "hard": 4001 + }, + "matrix_quantization": "Each matrix entry and RHS rounded to 12 decimal places, exactly as ProbeSession.query.", + "column_normalization": "Each nonzero column divided by its Euclidean norm, solely for reported diagnostic spectra.", + "rank_warning": "Floating-point rank of a discretized matrix cannot prove structural identifiability of the continuum problem.", + "amplitude_warning": "Sensitivity audit scales precomputed unrounded rows/RHS before 12dp rounding, across all stored rows, then scores coefficient estimates rounded to 8dp. 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three difficulty levels, and two seeds (18 attempts, 15 scored). Values were not regenerated from the six-model archive. +``---'' indicates the model failed to produce a scored run.}\label{tab:pilot} +\scriptsize +\setlength{\tabcolsep}{4pt} +\begin{tabular}{llrrrrrrrr} +\toprule +\textbf{Model} & \textbf{Diff} & \textbf{Seed} & $E_a$ & $E_b$ & $E_c$ + & $E_f$ & $E$ & \textbf{Solves} & \textbf{Q used} \\ +\midrule +GPT-4o & easy & 42 & 0.0001 & 0.0004 & 0.0026 & 0.0 & 0.003 & 2 & 11 \\ +GPT-4o & easy & 137 & 0.0000 & 0.0000 & 0.0000 & 0.0 & 0.000 & 3 & 12 \\ +GPT-5.4-mini & easy & 42 & 0.0000 & 0.0002 & 0.0014 & 0.0 & 0.002 & 1 & 10 \\ +GPT-5.4-mini & easy & 137 & 0.0000 & 0.0000 & 0.0001 & 0.0 & 0.000 & 1 & 8 \\ +GPT-5.4 & easy & 42 & 0.0000 & 0.0001 & 0.0004 & 0.0 & 0.001 & 3 & 12 \\ +GPT-5.4 & easy & 137 & 0.0000 & 0.0000 & 0.0000 & 0.0 & 0.000 & 4 & 12 \\ +\midrule +GPT-4o & medium & 42 & --- & --- & --- & --- & --- & 0 & 9 \\ +GPT-4o & medium & 137 & 0.0000 & 0.0002 & 0.0019 & 0.0 & 0.002 & 4 & 18 \\ +GPT-5.4-mini & medium & 42 & 0.1861 & 0.6740 & 1.4416 & 0.0 & 2.302 & 1 & 17 \\ +GPT-5.4-mini & medium & 137 & 0.0229 & 0.2777 & 2.4272 & 0.0 & 2.728 & 3 & 13 \\ +GPT-5.4 & medium & 42 & 0.0001 & 0.0005 & 0.0009 & 0.0 & 0.001 & 3 & 17 \\ +GPT-5.4 & medium & 137 & 0.0000 & 0.0000 & 0.0000 & 0.0 & 0.000 & 4 & 17 \\ +\midrule +GPT-4o & hard & 42 & --- & --- & --- & --- & --- & 0 & 0 \\ +GPT-4o & hard & 137 & --- & --- & --- & --- & --- & 0 & 1 \\ +GPT-5.4-mini & hard & 42 & 0.1275 & 1.4848 & 0.7268 & 0.0 & 2.339 & 1 & 20 \\ +GPT-5.4-mini & hard & 137 & 0.1196 & 1.2099 & 3.5182 & 0.0 & 4.848 & 2 & 21 \\ +GPT-5.4 & hard & 42 & 0.0730 & 2.1305 & 1.4725 & 0.0 & 3.676 & 7 & 23 \\ +GPT-5.4 & hard & 137 & 0.0010 & 0.0424 & 0.2458 & 0.0 & 0.289 & 4 & 24 \\ +\bottomrule +\end{tabular} +\end{table} + +\begin{table}[htbp] +\centering +\caption{Legacy pilot behavioral summary, preserved from the earlier draft. These medians were not regenerated from the six-model archive. +$H$ = normalized family entropy; $R$ = improvement ratio; $W$ = wasted +turn fraction; Tail = turns after last solve. Check\% = checks as a +fraction of queries (checks are free and do not consume budget).}\label{tab:pilotbehavior} +\scriptsize +\setlength{\tabcolsep}{4pt} +\begin{tabular}{lcccccccc} +\toprule +\textbf{Model} & \textbf{Solves} & \textbf{Budget} & \textbf{Check\%} + & $H$ & $R$ & $\bar M$ & $W$ & \textbf{Tail} \\ +\midrule +GPT-4o & 3 & 96\% & 13\% & 0.30 & 1.0 & --- & 0.05 & 1 \\ +GPT-5.4-mini & 1 & 78\% & 18\% & 0.00 & 1.0 & 0.5 & 0.08 & 2 \\ +GPT-5.4 & 4 & 97\% & 8\% & 0.35 & 4.2 & 0.8 & 0.02 & 1 \\ +\bottomrule +\end{tabular} +\end{table} diff --git a/paper_artifacts/research_figure_sources.json b/paper_artifacts/research_figure_sources.json new file mode 100644 index 0000000..fedee27 --- /dev/null +++ b/paper_artifacts/research_figure_sources.json @@ -0,0 +1,7 @@ +{ + "control_baselines.json.gz": "1f766ee10d8aaa7e07fc8f255d6ee4a4cef76e3751dcc63c426a4790624d7db5", + "identifiability_audit.json": "17cf7ba238b0e7d792e7a25dfb753411c1a44121a02987013ed277c83365770b", + "statistical_analysis.json": "b8a5591c8e7e2f67fbfbf050829f3b8262f7983f80d9dc6a43fef5131441a146", + "sine_generalization.json": "842932165242e5695ad42ab9507dffcc0e772bcac4792fbbddbc9a762f4a1e66", + "plot_script_sha256": "ccf778f38ecd4f04a9876efe78b98d0e5625c4e583610db387fdb154bec56970" +} diff --git a/paper_artifacts/sine_generalization.json b/paper_artifacts/sine_generalization.json new file mode 100644 index 0000000..75b78d6 --- /dev/null +++ b/paper_artifacts/sine_generalization.json @@ -0,0 +1,50368 @@ +{ + "schema_version": 1, + "experiment_type": "fresh-seed validation after design selection", + "protocol": { + "strategy": "fixed_sine", + "first_seed": 1001, + "last_seed": 1100, + "n_seeds_per_difficulty": 100, + "n_planned_tasks": 300, + "query_expression": "sin(k*pi*x), k=1..6N", + "selection_timing": "Fixed sine was selected after its success on the 30 archived instances; this is subsequent validation, not a historical held-out evaluation or formal preregistration.", + "inclusion": "Every seed-difficulty task in the fixed plan, including all-zero forcing, all-zero solutions, generation/solver/identification errors. 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"rank_tolerance": 1.2957228085033144e-14, + "condition_number": 634698.3179866207, + "smallest_singular_value": 3.830833610356013e-06, + "largest_singular_value": 2.4314236489795746 + } + } + }, + "error": null, + "forward_max_abs_u": 0.6694461981612986, + "elapsed_seconds": 1.4643881670199335 + } + ] +} diff --git a/paper_artifacts/sine_generalization_table.tex b/paper_artifacts/sine_generalization_table.tex new file mode 100644 index 0000000..c33a12f --- /dev/null +++ b/paper_artifacts/sine_generalization_table.tex @@ -0,0 +1,18 @@ +% Generated by scripts/plot_research_audit.py. +\begin{table}[t] +\caption{Fixed sine on 100 fresh seeds per setting. All planned tasks remain in the denominator. Parentheses give Wilson 95\% intervals for the minimum-budget rate; zero-source counts are descriptive.} +\label{tab:fresh} +\centering\small +\begin{tabular}{@{}lrrr@{}} +\toprule +Setting & $4N$ successes & $6N$ successes & $f=0$\\ +\midrule +Easy & 92/100 & 92/100 & 7\\ + & (0.85, 0.96) & & \\ +Medium & 91/100 & 85/100 & 3\\ + & (0.84, 0.95) & & \\ +Hard & 98/100 & 93/100 & 0\\ + & (0.93, 0.99) & & \\ +\bottomrule +\end{tabular} +\end{table} diff --git a/paper_artifacts/statistical_analysis.json b/paper_artifacts/statistical_analysis.json new file mode 100644 index 0000000..b901432 --- /dev/null +++ b/paper_artifacts/statistical_analysis.json @@ -0,0 +1,4301 @@ +{ + "schema_version": 1, + "environment": { + "python": "3.12.12", + "numpy": "2.5.4" + }, + "source_sha256": { + "scripts/analyze_archive_statistics.py": "a87ce24b1e3773803be5d65bc896044767722a54881406d41166c8ae71e0bee8", + "scripts/reproduce_paper_figures.py": "abdfe12686a7affcda657bb4e00858a1f4fb40e69aad586b9fa86a32d30a93b7", + "requirements.txt": "39a52b2e7d6304c1c80053325ca16be1542e2993d5299bb11f9807bf875d8bfd" + }, + "archive_sha256": "b2f50742cc76acae4a6a8d7b04d434e7735acf107d57f08f499e1dfcf05ce66f", + "dataset_url": "https://www.kaggle.com/datasets/jkmiller/elliptic-differential-equation-llm-reasoning/versions/1", + "analysis_kind": "exploratory, conditional on inclusion in the scored archive", + "settings": { + "bootstrap_resamples": 20000, + "random_seed": 20261020, + "confidence_level": 0.95, + "bootstrap_quantiles": [ + 0.025, + 0.975 + ], + "bootstrap_quantile_method": "linear", + "success_thresholds": [ + 0.01, + 0.1 + ], + "paired_comparison_threshold": 0.01, + "multiplicity": "Holm correction within all 15 model pairs separately for each difficulty; no correction across difficulties" + }, + "uncertainty_units": { + "success_intervals": "Wilson 95% score intervals over ten task seeds per model/difficulty, conditional on scored archive inclusion; illustrative task-sampling uncertainty under a binomial sampling model", + "paired_comparisons": "Task seed within one difficulty. A draw retains the two models' outcomes on the same task; models are never treated as independent samples on that task.", + "confidence": "One run on one task. First average all causally paired a/b/c observations within the run; then weight supported runs equally. Resample whole supported runs with their reports and coefficient blocks intact, conditional on having a paired report.", + "calibration_gap": "Signed calibration-in-the-large gap: mean within-run confidence minus mean within-run correctness, then averaged equally across runs. Positive means overconfidence; zero alone does not imply calibration, and this is not ECE.", + "confidence_event": "The preceding solve has block maximum absolute coefficient error E_q < 0.1; reports pair only to strictly earlier transcript turns; source f is excluded.", + "scope": "Six model policies evaluated on 30 seed/difficulty tasks (ten per difficulty), with one retained scored record per model/task. No repeated model-sampling runs or complete attempt history are available.", + "limitations": [ + "Intervals do not account for missing attempts, retries, archive selection, implementation drift, or model-sampling variability.", + "Ten tasks per difficulty give coarse tests and potentially degenerate empirical bootstrap intervals; these intervals do not establish universal rates or model rankings.", + "Seed labels are reused across difficulties. Analyses and resampling remain within difficulty and do not assume independence across difficulties.", + "Paired percentile intervals are unadjusted exploratory intervals; only exact paired test p-values receive the stated Holm correction.", + "These resampling distributions describe sensitivity to the ten observed tasks under exchangeable task-sampling assumptions; they do not prove the archive is representative of a target population.", + "Report count is not an independent sample size. Runs without causally paired confidence reports are excluded from confidence estimands and their seeds are listed." + ] + }, + "cells": [ + { + "model": "gemini-3.1-pro-preview", + "difficulty": "easy", + "success": { + "0.01": { + "count": 10, + "n": 10, + "rate": 1.0, + "wilson_95_interval": [ + 0.7224672001371109, + 1.0 + ] + }, + "0.1": { + "count": 10, + "n": 10, + "rate": 1.0, + "wilson_95_interval": [ + 0.7224672001371109, + 1.0 + ] + } + }, + "confidence_report_support": { + "reports": 30, + "reports_paired": 30, + "reports_excluded_no_preceding_solve": 0 + }, + "confidence": { + "n_total_runs": 10, + "n_runs_with_paired_reports": 10, + "seeds_with_paired_reports": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "seeds_without_paired_reports": [], + "coefficient_observations": 90, + "runs": [ + { + "seed": 1, + "coefficient_observations": 9, + "brier_score": 0.13447777777777778, + "mean_confidence": 0.8077777777777777, + "success_fraction": 1.0, + 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"first_only_successes": 0, + "second_only_successes": 2, + "both_successes": 0, + "neither_successes": 8, + "success_rate_difference_first_minus_second": -0.2, + "paired_task_bootstrap_95_interval": [ + -0.5, + 0.0 + ], + "exact_p_two_sided": 0.5, + "holm_adjusted_p_within_difficulty": 1.0 + } + ] +} diff --git a/paper_artifacts/statistical_tables.tex b/paper_artifacts/statistical_tables.tex new file mode 100644 index 0000000..f303ae2 --- /dev/null +++ b/paper_artifacts/statistical_tables.tex @@ -0,0 +1,43 @@ +% Generated by scripts/plot_research_audit.py. +\begin{table}[ht] +\caption{Strict-threshold archived successes with Wilson 95\% intervals. Each cell uses ten scored tasks; intervals cannot account for missing attempts or model-sampling variability.} +\centering\small +\begin{tabular}{@{}lrrr@{}} +\toprule +Model & Easy & Medium & Hard\\ +\midrule +Gemini 3.1 Pro & 10/10 [0.72, 1.00] & 9/10 [0.60, 0.98] & 10/10 [0.72, 1.00]\\ +Claude Opus 4.6 & 10/10 [0.72, 1.00] & 9/10 [0.60, 0.98] & 8/10 [0.49, 0.94]\\ +GPT-5.4 & 7/10 [0.40, 0.89] & 3/10 [0.11, 0.60] & 1/10 [0.02, 0.40]\\ +Gemini 3 Flash & 8/10 [0.49, 0.94] & 7/10 [0.40, 0.89] & 7/10 [0.40, 0.89]\\ +Claude Sonnet 4 & 8/10 [0.49, 0.94] & 6/10 [0.31, 0.83] & 0/10 [0.00, 0.28]\\ +GPT-5.4 mini & 7/10 [0.40, 0.89] & 4/10 [0.17, 0.69] & 2/10 [0.06, 0.51]\\ +\bottomrule +\end{tabular} +\end{table} + +\begin{table}[ht] +\caption{Exploratory exact paired success tests, Holm-adjusted within the 15 model pairs of each difficulty. All 45 comparisons are shown. Values are conditional on the retained scored tasks; no adjustment is made across difficulties.} +\centering\small +\begin{tabular}{@{}llrrr@{}} +\toprule +First model & Second model & Easy & Medium & Hard\\ +\midrule +Gemini 3.1 Pro & Claude Opus 4.6 & 1.0000 & 1.0000 & 1.0000\\ +Gemini 3.1 Pro & GPT-5.4 & 1.0000 & 0.9141 & 0.0547\\ +Gemini 3.1 Pro & Gemini 3 Flash & 1.0000 & 1.0000 & 1.0000\\ +Gemini 3.1 Pro & Claude Sonnet 4 & 1.0000 & 1.0000 & 0.0293\\ +Gemini 3.1 Pro & GPT-5.4 mini & 1.0000 & 0.8750 & 0.1016\\ +Claude Opus 4.6 & GPT-5.4 & 1.0000 & 0.4688 & 0.1719\\ +Claude Opus 4.6 & Gemini 3 Flash & 1.0000 & 1.0000 & 1.0000\\ +Claude Opus 4.6 & Claude Sonnet 4 & 1.0000 & 1.0000 & 0.1016\\ +Claude Opus 4.6 & GPT-5.4 mini & 1.0000 & 1.0000 & 0.2812\\ +GPT-5.4 & Gemini 3 Flash & 1.0000 & 1.0000 & 0.5000\\ +GPT-5.4 & Claude Sonnet 4 & 1.0000 & 1.0000 & 1.0000\\ +GPT-5.4 & GPT-5.4 mini & 1.0000 & 1.0000 & 1.0000\\ +Gemini 3 Flash & Claude Sonnet 4 & 1.0000 & 1.0000 & 0.1719\\ +Gemini 3 Flash & GPT-5.4 mini & 1.0000 & 1.0000 & 0.5000\\ +Claude Sonnet 4 & GPT-5.4 mini & 1.0000 & 1.0000 & 1.0000\\ +\bottomrule +\end{tabular} +\end{table} diff --git a/paper_artifacts/summary.json b/paper_artifacts/summary.json new file mode 100644 index 0000000..41f485c --- /dev/null +++ b/paper_artifacts/summary.json @@ -0,0 +1,3811 @@ +{ + "schema_version": 1, + "archive_sha256": "b2f50742cc76acae4a6a8d7b04d434e7735acf107d57f08f499e1dfcf05ce66f", + "dataset_url": "https://www.kaggle.com/datasets/jkmiller/elliptic-differential-equation-llm-reasoning/versions/1", + "validation": { + "runs": 180, + "scored_runs": 180, + "models": 6, + "difficulties": [ + "easy", + "medium", + "hard" + ], + "seeds_per_cell": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "shared_ground_truth_per_seed_difficulty": true, + "sweep_summary_matches_all_run_scores": true, + "attempts_and_retries_before_archive": "not recorded" + }, + "settings": { + "difficulty_settings": { + "easy": { + "n_basis": 2, + "unknowns": 8, + "max_queries": 12, + "max_turns": 36 + }, + "medium": { + "n_basis": 3, + "unknowns": 12, + "max_queries": 18, + "max_turns": 54 + }, + "hard": { + "n_basis": 4, + "unknowns": 16, + "max_queries": 24, + "max_turns": 72 + } + }, + "prompt_condition": "standard", + "reasoning_effort": { + "openai": "medium", + "anthropic": null, + "google": null + }, + "run_timestamp_range": [ + "2026-04-13T20:57:21", + "2026-04-16T01:38:11" + ], + "run_timestamp_timezone": "not recorded" + }, + "definitions": { + "coefficient_error": "E_j = max_l abs(predicted_theta[j,l] - true_theta[j,l]); absolute coefficient error, not relative error", + "total_error": "E = E_a + E_b + E_c + E_f; strict threshold comparison", + "improvement_ratio": "first solve total error / last solve total error; missing values assigned 1 to retain legacy plotting convention", + "family_entropy": "-sum(p_i log2(p_i)) / log2(5), over the final solve taxonomy: 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"mean_confidence": 0.9338461538461539, + "success_fraction": 0.46153846153846156 + } + ] + } + ], + "calibration_totals": { + "reports": 733, + "reports_paired": 707, + "observations_on_shared_legacy_bin_boundaries": 827, + "coefficient_observations": 2121, + "reports_excluded_no_preceding_solve": 26, + "reports_with_different_solve_from_legacy": 71 + }, + "changes_from_legacy_assets": [ + "GPT-5.4 mini hard success at E < 0.01 is 2/10 in both run logs and sweep_results.json; the checked-in PNG displays 0/10.", + "Calibration uses causally available solve outputs and disjoint confidence bins, replacing forward-looking solve matching and overlapping bin boundaries.", + "Improvement figure shows individual runs and medians on a logarithmic axis; it omits arithmetic-mean bars and standard-deviation whiskers." + ] +} diff --git a/paper_artifacts/template_provenance.md b/paper_artifacts/template_provenance.md new file mode 100644 index 0000000..8aea2a2 --- /dev/null +++ b/paper_artifacts/template_provenance.md @@ -0,0 +1,14 @@ +# Manuscript format + +The public preprint uses the unmodified official ICML 2026 style with its +`preprint` option. This is the latest verified author template at the time of +revision; it does not claim that this work was submitted to or accepted by ICML. +The author information and public repository links remain appropriate for a +public preprint. An actual submission would require the target year's rules and +anonymization. + +Source: https://media.icml.cc/Conferences/ICML2026/Styles/icml2026.zip +Downloaded 20 October 2026. Archive SHA-256: `8b29290f5828e176debb57ea9cc00252502973d55ea561a2f18a7f0a326bfc6c`. + +Copied unchanged: `icml2026.sty`, `icml2026.bst`, `algorithm.sty`, +`algorithmic.sty`, and `fancyhdr.sty`. Their original headers are retained. diff --git a/references.bib b/references.bib index 1ae8547..8486ab9 100644 --- a/references.bib +++ b/references.bib @@ -48,7 +48,8 @@ @article{mackay1992information number = {4}, pages = {590--604}, year = {1992}, - doi = {10.1162/neco.1992.4.4.590} + doi = {10.1162/neco.1992.4.4.590}, + url = {https://doi.org/10.1162/neco.1992.4.4.590} } @article{settles2009active, @@ -73,9 +74,13 @@ @inproceedings{guo2017calibration Weinberger, Kilian Q.}, title = {On Calibration of Modern Neural Networks}, booktitle = {Proceedings of the 34th International Conference on - Machine Learning (ICML)}, + Machine Learning}, pages = {1321--1330}, - year = {2017} + volume = {70}, + series = {Proceedings of Machine Learning Research}, + publisher = {PMLR}, + year = {2017}, + url = {https://proceedings.mlr.press/v70/guo17a.html} } @article{kadavath2022language, @@ -94,3 +99,200 @@ @article{xiong2023llms journal = {arXiv preprint arXiv:2306.13063}, year = {2023} } + +@article{burnell2026cognitive, + author = {Burnell, Ryan and Yamamori, Yumeya and Firat, Orhan and + Olszewska, Kate and Hughes-Fitt, Steph and Kelly, Oran and + Galatzer-Levy, Isaac R. and Morris, Meredith Ringel and + Dafoe, Allan and Snyder, Alison M. and Goodman, Noah D. and + Botvinick, Matthew and Legg, Shane}, + title = {Measuring Progress Toward {AGI}: A Cognitive Framework}, + journal = {arXiv preprint arXiv:2605.28405}, + year = {2026}, + doi = {10.48550/arXiv.2605.28405}, + url = {https://arxiv.org/abs/2605.28405} +} + +@book{evans2010pde, + author = {Evans, Lawrence C.}, + title = {Partial Differential Equations}, + edition = {2}, + series = {Graduate Studies in Mathematics}, + volume = {19}, + publisher = {American Mathematical Society}, + address = {Providence, RI}, + year = {2010}, + isbn = {978-0-8218-4974-3}, + doi = {10.1090/gsm/019}, + url = {https://www.ams.org/books/gsm/019/} +} + +@article{egger2014simultaneous, + author = {Egger, Herbert and Pietschmann, Jan-Frederik and + Schlottbom, Matthias}, + title = {Simultaneous Identification of Diffusion and Absorption + Coefficients in a Quasilinear Elliptic Problem}, + journal = {Inverse Problems}, + volume = {30}, + number = {3}, + pages = {035009}, + year = {2014}, + doi = {10.1088/0266-5611/30/3/035009}, + url = {https://doi.org/10.1088/0266-5611/30/3/035009} +} + +@article{browning2024structural, + author = {Browning, Alexander P. and Ta{\c{s}}c{\u{a}}, Maria and + Falc{\'o}, Carles and Baker, Ruth E.}, + title = {Structural Identifiability Analysis of Linear + Reaction--Advection--Diffusion Processes in Mathematical Biology}, + journal = {Proceedings of the Royal Society A: Mathematical, + Physical and Engineering Sciences}, + volume = {480}, + number = {2286}, + pages = {20230911}, + year = {2024}, + doi = {10.1098/rspa.2023.0911}, + url = {https://doi.org/10.1098/rspa.2023.0911} +} + +@article{browning2025parabolic, + author = {Salmaniw, Yurij and Browning, Alexander P.}, + title = {Structural Identifiability of Linear-in-Parameter Parabolic + {PDEs} Through Auxiliary Elliptic Operators}, + journal = {Journal of Mathematical Biology}, + volume = {91}, + pages = {4}, + year = {2025}, + doi = {10.1007/s00285-025-02225-w}, + url = {https://link.springer.com/article/10.1007/s00285-025-02225-w} +} + +@inproceedings{hardmath2025, + author = {Roggeveen, James and Wang, Erik and Ettel, David and others}, + title = {{HARDMath2}: A Benchmark for Applied Mathematics Built by + Students as Part of a Graduate Class}, + booktitle = {Advances in Neural Information Processing Systems}, + volume = {38}, + year = {2025}, + note = {Datasets and Benchmarks Track}, + doi = {10.52202/085713-0859}, + url = {https://papers.neurips.cc/paper_files/paper/2025/hash/2480068e9fb06c9b2d582007c632b2d7-Abstract-Datasets_and_Benchmarks_Track.html} +} + +@inproceedings{mamo2025, + author = {Huang, Xuhan and Shen, Qingning and Hu, Yan and Gao, Anningzhe + and Wang, Benyou}, + title = {{LLMs} for Mathematical Modeling: Towards Bridging the Gap + between Natural and Mathematical Languages}, + booktitle = {Findings of the Association for Computational Linguistics: + {NAACL} 2025}, + pages = {2678--2710}, + publisher = {Association for Computational Linguistics}, + address = {Albuquerque, New Mexico}, + year = {2025}, + doi = {10.18653/v1/2025.findings-naacl.146}, + url = {https://aclanthology.org/2025.findings-naacl.146/} +} + +@misc{kaggle2026pde, + author = {Miller, Joseph Kentaro}, + title = {Can Language Models Learn a {PDE}?}, + howpublished = {Kaggle Measuring Progress Toward AGI competition writeup}, + year = {2026}, + month = apr, + note = {Published 16 April 2026; released under CC0}, + url = {https://www.kaggle.com/competitions/kaggle-measuring-agi/writeups/can-language-models-learn-a-pde#3442067} +} + +% Focused related-work additions verified against primary sources on 2026-10-20. +% This file is separate from references.bib so the manuscript editor can select +% the works that support the final argument rather than add citations by volume. + +@article{gandhi2025boxinggym, + author = {Gandhi, Kanishk and Li, Michael Y. and Goodyear, Lyle and + Bhatia, Agam and Li, Louise and Bhaskar, Aditi and + Zaman, Mohammed and Goodman, Noah D.}, + title = {{BoxingGym}: Benchmarking Progress in Automated Experimental + Design and Model Discovery}, + journal = {arXiv preprint arXiv:2501.01540}, + year = {2025}, + doi = {10.48550/arXiv.2501.01540}, + url = {https://arxiv.org/abs/2501.01540}, + note = {Version 2, 14 October 2025} +} + +@inproceedings{jansen2024discoveryworld, + author = {Jansen, Peter and C{\^o}t{\'e}, Marc-Alexandre and Khot, Tushar + and Bransom, Erin and Mishra, Bhavana Dalvi and + Majumder, Bodhisattwa Prasad and Tafjord, Oyvind and Clark, Peter}, + title = {{DiscoveryWorld}: A Virtual Environment for Developing and + Evaluating Automated Scientific Discovery Agents}, + booktitle = {Advances in Neural Information Processing Systems}, + volume = {37}, + year = {2024}, + note = {Datasets and Benchmarks Track}, + url = {https://openreview.net/forum?id=cDYqckEt6d} +} + +@inproceedings{duan2025scigym, + author = {Duan, Haonan and Lu, Stephen and Harrigan, Caitlin F. and + Desai, Nishkrit and Lu, Jiarui and Koziarski, Micha{\l} and + Cotta, Leonardo and Maddison, Chris J.}, + title = {Measuring Scientific Capabilities of Language Models with + a Systems Biology Dry Lab}, + booktitle = {Advances in Neural Information Processing Systems}, + volume = {38}, + year = {2025}, + note = {Datasets and Benchmarks Track}, + doi = {10.52202/085713-3733}, + url = {https://proceedings.nips.cc/paper_files/paper/2025/hash/a23760ba51036d530a2656c3835f826c-Abstract-Datasets_and_Benchmarks_Track.html} +} + +@inproceedings{shojaee2025llmsr, + author = {Shojaee, Parshin and Meidani, Kazem and Gupta, Shashank and + Barati Farimani, Amir and Reddy, Chandan K.}, + title = {{LLM-SR}: Scientific Equation Discovery via Programming with + Large Language Models}, + booktitle = {The Thirteenth International Conference on Learning Representations}, + year = {2025}, + url = {https://openreview.net/forum?id=m2nmp8P5in} +} + +@inproceedings{shojaee2025llmsrbench, + author = {Shojaee, Parshin and Nguyen, Ngoc-Hieu and Meidani, Kazem and + Barati Farimani, Amir and Doan, Khoa D and Reddy, Chandan K.}, + title = {{LLM-SRBench}: A New Benchmark for Scientific Equation Discovery + with Large Language Models}, + booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, + pages = {55325--55359}, + volume = {267}, + series = {Proceedings of Machine Learning Research}, + publisher = {PMLR}, + year = {2025}, + url = {https://proceedings.mlr.press/v267/shojaee25a.html} +} + +@article{messenger2021wsindy, + author = {Messenger, Daniel A. and Bortz, David M.}, + title = {Weak {SINDy} for Partial Differential Equations}, + journal = {Journal of Computational Physics}, + volume = {443}, + pages = {110525}, + year = {2021}, + doi = {10.1016/j.jcp.2021.110525}, + url = {https://doi.org/10.1016/j.jcp.2021.110525} +} + +@article{alexanderian2021oed, + author = {Alexanderian, Alen}, + title = {Optimal Experimental Design for Infinite-Dimensional Bayesian + Inverse Problems Governed by {PDEs}: A Review}, + journal = {Inverse Problems}, + volume = {37}, + number = {4}, + pages = {043001}, + year = {2021}, + doi = {10.1088/1361-6420/abe10c}, + url = {https://doi.org/10.1088/1361-6420/abe10c} +} diff --git a/research/related_work.bib b/research/related_work.bib new file mode 100644 index 0000000..d95cb69 --- /dev/null +++ b/research/related_work.bib @@ -0,0 +1,123 @@ +% Focused related-work additions verified against primary sources on 2026-10-20. +% This file is separate from references.bib so the manuscript editor can select +% the works that support the final argument rather than add citations by volume. + +@article{gandhi2025boxinggym, + author = {Gandhi, Kanishk and Li, Michael Y. and Goodyear, Lyle and + Bhatia, Agam and Li, Louise and Bhaskar, Aditi and + Zaman, Mohammed and Goodman, Noah D.}, + title = {{BoxingGym}: Benchmarking Progress in Automated Experimental + Design and Model Discovery}, + journal = {arXiv preprint arXiv:2501.01540}, + year = {2025}, + doi = {10.48550/arXiv.2501.01540}, + url = {https://arxiv.org/abs/2501.01540}, + note = {Version 2, 14 October 2025} +} + +@inproceedings{jansen2024discoveryworld, + author = {Jansen, Peter and C{\^o}t{\'e}, Marc-Alexandre and Khot, Tushar + and Bransom, Erin and Mishra, Bhavana Dalvi and + Majumder, Bodhisattwa Prasad and Tafjord, Oyvind and Clark, Peter}, + title = {{DiscoveryWorld}: A Virtual Environment for Developing and + Evaluating Automated Scientific Discovery Agents}, + booktitle = {Advances in Neural Information Processing Systems}, + volume = {37}, + year = {2024}, + note = {Datasets and Benchmarks Track}, + url = {https://openreview.net/forum?id=cDYqckEt6d} +} + +@inproceedings{duan2025scigym, + author = {Duan, Haonan and Lu, Stephen and Harrigan, Caitlin F. and + Desai, Nishkrit and Lu, Jiarui and Koziarski, Micha{\l} and + Cotta, Leonardo and Maddison, Chris J.}, + title = {Measuring Scientific Capabilities of Language Models with + a Systems Biology Dry Lab}, + booktitle = {Advances in Neural Information Processing Systems}, + volume = {38}, + year = {2025}, + note = {Datasets and Benchmarks Track}, + doi = {10.52202/085713-3733}, + url = {https://proceedings.nips.cc/paper_files/paper/2025/hash/a23760ba51036d530a2656c3835f826c-Abstract-Datasets_and_Benchmarks_Track.html} +} + +@inproceedings{shojaee2025llmsr, + author = {Shojaee, Parshin and Meidani, Kazem and Gupta, Shashank and + Barati Farimani, Amir and Reddy, Chandan K.}, + title = {{LLM-SR}: Scientific Equation Discovery via Programming with + Large Language Models}, + booktitle = {The Thirteenth International Conference on Learning Representations}, + year = {2025}, + url = {https://openreview.net/forum?id=m2nmp8P5in} +} + +@inproceedings{shojaee2025llmsrbench, + author = {Shojaee, Parshin and Nguyen, Ngoc-Hieu and Meidani, Kazem and + Barati Farimani, Amir and Doan, Khoa D and Reddy, Chandan K.}, + title = {{LLM-SRBench}: A New Benchmark for Scientific Equation Discovery + with Large Language Models}, + booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, + pages = {55325--55359}, + volume = {267}, + series = {Proceedings of Machine Learning Research}, + publisher = {PMLR}, + year = {2025}, + url = {https://proceedings.mlr.press/v267/shojaee25a.html} +} + +@article{messenger2021wsindy, + author = {Messenger, Daniel A. and Bortz, David M.}, + title = {Weak {SINDy} for Partial Differential Equations}, + journal = {Journal of Computational Physics}, + volume = {443}, + pages = {110525}, + year = {2021}, + doi = {10.1016/j.jcp.2021.110525}, + url = {https://doi.org/10.1016/j.jcp.2021.110525} +} + +@article{alexanderian2021oed, + author = {Alexanderian, Alen}, + title = {Optimal Experimental Design for Infinite-Dimensional Bayesian + Inverse Problems Governed by {PDEs}: A Review}, + journal = {Inverse Problems}, + volume = {37}, + number = {4}, + pages = {043001}, + year = {2021}, + doi = {10.1088/1361-6420/abe10c}, + url = {https://doi.org/10.1088/1361-6420/abe10c} +} + +% Optional recent comparisons: use if the final paper discusses broader +% scientific-agent evaluation, rather than only inverse-problem measurements. + +@inproceedings{chen2026causalgame, + author = {Chen, Zhenhao and Chen, Yongqiang and Liu, Chenxi and Yu, Junchi + and Song, Xiangchen and Li, Zijian and Li, Jialin and Torr, Philip + and Han, Bo and Zhang, Kun}, + title = {{CausalGame}: Benchmarking Causal Thinking of {LLM} Agents in Games}, + booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, + pages = {13726--13765}, + volume = {306}, + series = {Proceedings of Machine Learning Research}, + publisher = {PMLR}, + year = {2026}, + url = {https://proceedings.mlr.press/v306/chen26l.html} +} + +@inproceedings{wang2026firebench, + author = {Wang, Zhen and Bai, Fan and Luo, Zhongyan and Su, Jinyan and + Sun, Kaiser and Yu, Xinle and Liu, Jieyuan and Zhou, Kun and + Cardie, Claire and Dredze, Mark and Hu, Zhiting and Xing, Eric P.}, + title = {{FIRE-Bench}: Evaluating {AI} Agents on the Rediscovery of + Scientific Insights}, + booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, + pages = {124896--124929}, + volume = {306}, + series = {Proceedings of Machine Learning Research}, + publisher = {PMLR}, + year = {2026}, + url = {https://proceedings.mlr.press/v306/wang26d.html} +} diff --git a/scripts/analyze_archive_statistics.py b/scripts/analyze_archive_statistics.py new file mode 100644 index 0000000..bee2f56 --- /dev/null +++ b/scripts/analyze_archive_statistics.py @@ -0,0 +1,256 @@ +#!/usr/bin/env python3 +"""Exploratory task-level uncertainty for the pinned, selected run archive. + +No network or model calls are made. Example, from any working directory:: + + python scripts/analyze_archive_statistics.py --archive /tmp/reasoning-v1.zip + +This is uncertainty conditional on the scored archive. Unobserved attempts and +model sampling variability cannot be recovered by resampling saved outputs. +""" + +from __future__ import annotations + +import argparse +from itertools import combinations +import hashlib +import platform +import json +import math +from pathlib import Path +from statistics import NormalDist + +import numpy as np + +try: # Support both direct execution and imports from the repository root. + from .reproduce_paper_figures import ( + DIFFICULTIES, ROOT, causal_confidence_pairs, load_and_validate, + ) +except ImportError: + from reproduce_paper_figures import ( + DIFFICULTIES, ROOT, causal_confidence_pairs, load_and_validate, + ) + + +BOOTSTRAP_RESAMPLES = 20_000 +RANDOM_SEED = 20261020 +CONFIDENCE_LEVEL = 0.95 + + +def wilson_interval(successes, n, confidence_level=CONFIDENCE_LEVEL): + """Wilson score interval without continuity correction; no pooled models.""" + if n <= 0 or not 0 <= successes <= n: + raise ValueError("Wilson intervals require 0 <= successes <= n and n > 0.") + z = NormalDist().inv_cdf((1 + confidence_level) / 2) + fraction = successes / n + denominator = 1 + z * z / n + center = (fraction + z * z / (2 * n)) / denominator + radius = z * math.sqrt(fraction * (1 - fraction) / n + z * z / (4 * n * n)) / denominator + return [max(0.0, center - radius), min(1.0, center + radius)] + + +def exact_discordant_p(first_only, second_only): + """Two-sided exact paired binomial/McNemar test on discordant tasks.""" + if min(first_only, second_only) < 0 or any( + int(value) != value for value in (first_only, second_only) + ): + raise ValueError("Discordant counts must be nonnegative integers.") + n = int(first_only + second_only) + if n == 0: + return 1.0 + tail = sum(math.comb(n, k) for k in range(int(min(first_only, second_only)) + 1)) / (2 ** n) + return min(1.0, 2 * tail) + + +def holm_adjust(p_values): + """Holm step-down adjusted p-values in input order.""" + if any(not 0 <= value <= 1 for value in p_values): + raise ValueError("P-values must lie in [0, 1].") + adjusted = [None] * len(p_values) + previous = 0.0 + for order, index in enumerate(sorted(range(len(p_values)), key=p_values.__getitem__)): + previous = max(previous, (len(p_values) - order) * p_values[index]) + adjusted[index] = min(1.0, previous) + return adjusted + + +def percentile_interval(values): + return [float(value) for value in np.quantile(values, [0.025, 0.975], method="linear")] + + +def paired_success_statistics(first, second, bootstrap_indices): + """Join successes by seed, retaining the pair in every bootstrap draw.""" + if not first or first.keys() != second.keys(): + raise ValueError("Paired comparisons require identical nonempty task-seed sets.") + seeds = sorted(first) + a = np.asarray([first[seed] for seed in seeds], dtype=float) + b = np.asarray([second[seed] for seed in seeds], dtype=float) + if not np.isin(a, [0, 1]).all() or not np.isin(b, [0, 1]).all(): + raise ValueError("Success indicators must be binary.") + difference = a - b + first_only = int(np.sum((a == 1) & (b == 0))) + second_only = int(np.sum((a == 0) & (b == 1))) + bootstrap_means = difference[bootstrap_indices].mean(axis=1) + return { + "n_paired_tasks": len(seeds), + "seeds": seeds, + "first_successes": int(a.sum()), + "second_successes": int(b.sum()), + "first_only_successes": first_only, + "second_only_successes": second_only, + "both_successes": int(np.sum((a == 1) & (b == 1))), + "neither_successes": int(np.sum((a == 0) & (b == 0))), + "success_rate_difference_first_minus_second": float(difference.mean()), + "paired_task_bootstrap_95_interval": percentile_interval(bootstrap_means), + "exact_p_two_sided": exact_discordant_p(first_only, second_only), + } + + +def run_confidence_metrics(pairs): + """Reduce dependent reports/blocks to one equally weighted run summary.""" + if not pairs: + return None + values = np.asarray(pairs, dtype=float) + confidence, outcome = values[:, 0], values[:, 1] + if not ((0 <= confidence) & (confidence <= 1)).all() or not np.isin(outcome, [0, 1]).all(): + raise ValueError("Confidence must be in [0, 1] and outcomes must be binary.") + return { + "coefficient_observations": len(pairs), + "brier_score": float(np.mean((confidence - outcome) ** 2)), + "mean_confidence": float(confidence.mean()), + "success_fraction": float(outcome.mean()), + "signed_calibration_in_the_large_gap": float((confidence - outcome).mean()), + } + + +def summarize_run_confidence(run_pairs, rng, resamples=BOOTSTRAP_RESAMPLES): + """Resample whole supported runs, never reports or coefficient blocks. + + Each run contributes its within-run mean, so adding reports in one run does + not increase that run's weight. Runs without causally paired reports are + omitted; the estimand is conditional on reporting and archive inclusion. + """ + supported = [(seed, run_confidence_metrics(pairs)) for seed, pairs in sorted(run_pairs.items()) if pairs] + result = { + "n_total_runs": len(run_pairs), + "n_runs_with_paired_reports": len(supported), + "seeds_with_paired_reports": [seed for seed, _ in supported], + "seeds_without_paired_reports": sorted(seed for seed, pairs in run_pairs.items() if not pairs), + "coefficient_observations": sum(row["coefficient_observations"] for _, row in supported), + "runs": [{"seed": seed, **row} for seed, row in supported], + "run_balanced": {}, + } + if not supported: + return result + metrics = ["brier_score", "mean_confidence", "success_fraction", "signed_calibration_in_the_large_gap"] + values = np.asarray([[row[name] for name in metrics] for _, row in supported]) + indices = rng.integers(0, len(supported), size=(resamples, len(supported))) + bootstrap_means = values[indices].mean(axis=1) + for index, name in enumerate(metrics): + result["run_balanced"][name] = { + "estimate": float(values[:, index].mean()), + "whole_run_bootstrap_95_interval": percentile_interval(bootstrap_means[:, index]), + } + return result + + +def build_analysis(runs, manifest, resamples=BOOTSTRAP_RESAMPLES, seed=RANDOM_SEED): + if resamples <= 0: + raise ValueError("Bootstrap resamples must be positive.") + rng = np.random.default_rng(seed) + analysis = { + "schema_version": 1, + "environment": {"python": platform.python_version(), "numpy": np.__version__}, + "source_sha256": {name: hashlib.sha256((ROOT / name).read_bytes()).hexdigest() + for name in ("scripts/analyze_archive_statistics.py", + "scripts/reproduce_paper_figures.py", "requirements.txt")}, + "archive_sha256": manifest["archive_sha256"], + "dataset_url": manifest["dataset_url"], + "analysis_kind": "exploratory, conditional on inclusion in the scored archive", + "settings": { + "bootstrap_resamples": resamples, + "random_seed": seed, + "confidence_level": CONFIDENCE_LEVEL, + "bootstrap_quantiles": [0.025, 0.975], + "bootstrap_quantile_method": "linear", + "success_thresholds": [0.01, 0.1], + "paired_comparison_threshold": 0.01, + "multiplicity": "Holm correction within all 15 model pairs separately for each difficulty; no correction across difficulties", + }, + "uncertainty_units": { + "success_intervals": "Wilson 95% score intervals over ten task seeds per model/difficulty, conditional on scored archive inclusion; illustrative task-sampling uncertainty under a binomial sampling model", + "paired_comparisons": "Task seed within one difficulty. A draw retains the two models' outcomes on the same task; models are never treated as independent samples on that task.", + "confidence": "One run on one task. First average all causally paired a/b/c observations within the run; then weight supported runs equally. Resample whole supported runs with their reports and coefficient blocks intact, conditional on having a paired report.", + "calibration_gap": "Signed calibration-in-the-large gap: mean within-run confidence minus mean within-run correctness, then averaged equally across runs. Positive means overconfidence; zero alone does not imply calibration, and this is not ECE.", + "confidence_event": "The preceding solve has block maximum absolute coefficient error E_q < 0.1; reports pair only to strictly earlier transcript turns; source f is excluded.", + "scope": "Six model policies evaluated on 30 seed/difficulty tasks (ten per difficulty), with one retained scored record per model/task. No repeated model-sampling runs or complete attempt history are available.", + "limitations": [ + "Intervals do not account for missing attempts, retries, archive selection, implementation drift, or model-sampling variability.", + "Ten tasks per difficulty give coarse tests and potentially degenerate empirical bootstrap intervals; these intervals do not establish universal rates or model rankings.", + "Seed labels are reused across difficulties. Analyses and resampling remain within difficulty and do not assume independence across difficulties.", + "Paired percentile intervals are unadjusted exploratory intervals; only exact paired test p-values receive the stated Holm correction.", + "These resampling distributions describe sensitivity to the ten observed tasks under exchangeable task-sampling assumptions; they do not prove the archive is representative of a target population.", + "Report count is not an independent sample size. Runs without causally paired confidence reports are excluded from confidence estimands and their seeds are listed.", + ], + }, + "cells": [], + "pairwise_success_comparisons": [], + } + for difficulty in DIFFICULTIES: + by_model = {} + for model in manifest["models"]: + cell = [run for run in runs if run["config"]["difficulty"] == difficulty and run["config"]["model"] == model] + by_seed = {run["config"]["seed"]: run for run in cell} + if len(by_seed) != len(cell) or sorted(by_seed) != manifest["expected_seeds"]: + raise ValueError("Cell has duplicate or missing task seeds.") + by_model[model] = { + task_seed: int(run["results"]["coefficient_errors"]["total"] < 0.01) + for task_seed, run in by_seed.items() + } + success = {} + for threshold in (0.01, 0.1): + count = sum(run["results"]["coefficient_errors"]["total"] < threshold for run in cell) + success[str(threshold)] = { + "count": count, "n": len(cell), "rate": count / len(cell), + "wilson_95_interval": wilson_interval(count, len(cell)), + } + confidence_pairs, support = {}, {"reports": 0, "reports_paired": 0, "reports_excluded_no_preceding_solve": 0} + for task_seed, run in sorted(by_seed.items()): + pairs, run_support = causal_confidence_pairs(run) + confidence_pairs[task_seed] = pairs + for name in support: + support[name] += run_support[name] + analysis["cells"].append({ + "model": model, "difficulty": difficulty, "success": success, + "confidence_report_support": support, + "confidence": summarize_run_confidence(confidence_pairs, rng, resamples), + }) + indices = rng.integers(0, len(manifest["expected_seeds"]), size=(resamples, len(manifest["expected_seeds"]))) + comparisons = [{ + "difficulty": difficulty, + "first_model": first, + "second_model": second, + **paired_success_statistics(by_model[first], by_model[second], indices), + } for first, second in combinations(manifest["models"], 2)] + for comparison, adjusted in zip(comparisons, holm_adjust([row["exact_p_two_sided"] for row in comparisons])): + comparison["holm_adjusted_p_within_difficulty"] = adjusted + analysis["pairwise_success_comparisons"].extend(comparisons) + return analysis + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--archive", type=Path, required=True) + parser.add_argument("--manifest", type=Path, default=ROOT / "paper_artifacts/data_manifest.json") + parser.add_argument("--output", type=Path, default=ROOT / "paper_artifacts/statistical_analysis.json") + args = parser.parse_args() + manifest = json.loads(args.manifest.read_text()) + runs = load_and_validate(args.archive, manifest) + analysis = build_analysis(runs, manifest) + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(json.dumps(analysis, indent=2, allow_nan=False) + "\n") + print(f"Analyzed {len(runs)} scored runs; wrote {len(analysis['pairwise_success_comparisons'])} paired comparisons to {args.output}") + + +if __name__ == "__main__": + main() diff --git a/scripts/audit_identifiability.py b/scripts/audit_identifiability.py new file mode 100644 index 0000000..4f2907f --- /dev/null +++ b/scripts/audit_identifiability.py @@ -0,0 +1,345 @@ +#!/usr/bin/env python3 +"""Offline audit of structural ambiguity, joint design spectra, and precision. + +This diagnoses the existing protocol; it neither certifies identifiability of a +sampled continuum problem nor supplies a new model experiment. Saved ground +truth is used only to reconstruct each numerical solution and audit its design. +No API calls are made. Run from the repository root, for example:: + + .venv/bin/python scripts/audit_identifiability.py \ + --archive /path/to/pinned-reasoning-v1.zip + +The exact polynomial calculation is independent of a numerical rank tolerance. +The replay uses the repository's current, dense finite-difference solver, the +archived query expressions, and the protocol's absolute 12-decimal rounding. +Historical solver/library differences can affect ill-conditioned estimates; +replay discrepancies are therefore reported rather than silently tolerated. +""" +from __future__ import annotations + +import argparse +from collections import Counter +from fractions import Fraction +import hashlib +from importlib.metadata import version +import json +from pathlib import Path +import platform +import sys +import zipfile + +import numpy as np + +ROOT = Path(__file__).resolve().parents[1] +sys.path.insert(0, str(ROOT)) +from basis import CoefficientFunction, LegendreBasis +from benchmark import DIFFICULTY_CONFIGS +from expression_parser import make_test_function_from_string +from pde import EllipticPDE, solve_elliptic +from scripts.reproduce_paper_figures import load_and_validate +from weak_form import _composite_simpson + +COEFFICIENTS = ("a", "b", "c", "f") +SCALES = (0.001, 1.0, 1000.0, 1_000_000.0) + + +def protocol_round(array, decimals=12): + """Match the protocol's scalar Python round, not np.round's fast path.""" + values = np.asarray(array) + return np.array([round(float(v), decimals) for v in values.flat]).reshape(values.shape) + + +def exact_rank(matrix): + """Gaussian elimination over Q, for small integer proof certificates.""" + rows = [[Fraction(int(v)) for v in row] for row in matrix] + pivot = 0 + for col in range(len(rows[0])): + choice = next((i for i in range(pivot, len(rows)) if rows[i][col]), None) + if choice is None: + continue + rows[pivot], rows[choice] = rows[choice], rows[pivot] + divisor = rows[pivot][col] + rows[pivot] = [v / divisor for v in rows[pivot]] + for i in range(pivot + 1, len(rows)): + multiplier = rows[i][col] + rows[i] = [v - multiplier * p for v, p in zip(rows[i], rows[pivot])] + pivot += 1 + if pivot == len(rows): + break + return pivot + + +def polynomial_operator_matrix(n_basis): + """Integer monomial matrix for -(a u')' + b u' + c u, u=x(1-x). + + Rows encode powers x^0,...,x^(N+1); columns encode a,b,c coefficients. + The same rank applies in any invertible polynomial basis, including shifted + Legendre. Surjectivity also follows directly: c*u spans u*P_(N-1), while + constant a and b supply span{1,1-2*x}, the remainder upon division by u. + """ + if n_basis < 2: + raise ValueError("The dimension statement requires N >= 2.") + matrix = np.zeros((n_basis + 2, 3 * n_basis), dtype=int) + for j in range(n_basis): + matrix[j, j] = 2 * (j + 1) + if j: + matrix[j - 1, j] = -j + matrix[j, n_basis + j] = 1 + matrix[j + 1, n_basis + j] = -2 + matrix[j + 1, 2 * n_basis + j] = 1 + matrix[j + 2, 2 * n_basis + j] = -1 + return matrix + + +def structural_certificate(): + records = [] + for n in range(2, 9): + matrix = polynomial_operator_matrix(n) + rank = exact_rank(matrix) + assert rank == n + 2 + records.append({"n_basis": n, "exact_rank_over_rationals": rank, + "exact_nullity": 3 * n - rank, + "integer_matrix": matrix.tolist()}) + # The two elementary N=2 directions from the manuscript family. + directions = [[1, 0, -2, 4, -8, 0], [-1, 2, 4, 0, 0, 0]] + matrix = polynomial_operator_matrix(2) + assert all(np.array_equal(matrix @ d, np.zeros(4)) for d in directions) + return { + "solution": "u=x(1-x)", + "coefficient_space": "P_(N-1)^3, any invertible polynomial basis", + "theorem": "For N>=2 the strong operator map has rank N+2 and nullity 2N-2.", + "proof": "The image contains u*P_(N-1) through c, and span{1,u'} through constant a,b; polynomial division by u proves these span P_(N+1).", + "interpretation": "Every null direction preserves u and f for all continuum test functions. Sufficiently small perturbations preserve positive diffusion. For N>=3, base a=1,b=0,c=1,f=2+u also allows an open neighborhood preserving positive reaction.", + "certificates": records, + "n2_monomial_null_directions": directions, + } + + +def pde_from_truth(truth, n_basis): + basis = LegendreBasis(n_basis) + pde = EllipticPDE(**{k: CoefficientFunction(basis, truth[f"{k}_coeffs"]) + for k in COEFFICIENTS}) + return basis, pde + + +def design_row(solution, basis, spec): + tf, error = make_test_function_from_string(spec, domain=solution.pde.domain) + if error: + raise ValueError(f"Cannot replay archived expression {spec!r}: {error}") + x, u, ux = solution.x, solution.u, solution.u_x() + phi, dphi = tf(x), tf.derivative(x, 1) + if not np.all(np.isfinite(phi)) or not np.all(np.isfinite(dphi)): + raise ValueError(f"Non-finite archived expression: {spec!r}") + psi = basis.evaluate(x) + integrate = lambda v: _composite_simpson(v, x[1] - x[0]) + a = [integrate(p * ux * dphi) for p in psi] + b = [integrate(p * ux * phi) for p in psi] + c = [integrate(p * u * phi) for p in psi] + f = [integrate(p * phi) for p in psi] + rhs = integrate(solution.pde.f(x) * phi) + return np.array(a + b + c), np.array(f), float(rhs) + + +def spectra(matrix): + """Numerical diagnostics only: never interpreted as continuum rank proof.""" + n_cols = matrix.shape[1] + singular = np.linalg.svd(matrix, compute_uv=False) + if len(singular) < n_cols: + singular = np.pad(singular, (0, n_cols - len(singular))) + scale = singular[0] if singular.size and singular[0] else 1.0 + return { + "singular_values": singular.tolist(), + "min_over_max_singular_value": float(singular[-1] / scale), + "rank_at_numpy_default_tolerance": int(np.sum(singular > max(matrix.shape) * np.finfo(float).eps * scale)), + "rank_at_relative_tolerance": {str(t): int(np.sum(singular > t * scale)) + for t in (1e-8, 1e-10, 1e-12)}, + } + + +def joint_diagnostics(matrix, rhs, theta): + norms = np.linalg.norm(matrix, axis=0) + balanced = matrix / np.where(norms > 0, norms, 1) + residual = matrix @ theta - rhs + return {"rows": len(rhs), "columns": matrix.shape[1], + "raw": spectra(matrix), "unit_column_norm": spectra(balanced), + "truth_residual_linf": float(np.max(np.abs(residual))), + "truth_residual_l2_over_rhs_l2": float(np.linalg.norm(residual) / max(np.linalg.norm(rhs), np.finfo(float).tiny))} + + +def solve_blocks(matrix, source, rhs): + op = np.linalg.lstsq(matrix, rhs, rcond=None)[0] + forcing = np.linalg.lstsq(source, rhs, rcond=None)[0] + return np.r_[op, forcing] + + +def total_error(estimate, truth): + n = len(truth) // 4 + return float(sum(np.max(np.abs(estimate[i*n:(i+1)*n] - truth[i*n:(i+1)*n])) + for i in range(4))) + + +def load_inputs(archive, manifest, runs_dir=None): + """Validate the pinned ZIP, without extracting or executing its members. + + Optional previously extracted inputs must match the pinned member bytes, + including the complete set of run filenames. Computation always consumes + the validated ZIP content, so directory state cannot alter the audit. + """ + validated_runs = load_and_validate(archive, manifest) + with zipfile.ZipFile(archive) as source: + names = sorted(n for n in source.namelist() if n.startswith("run_") and n.endswith(".json")) + raw_members = {n: source.read(n) for n in names} + if runs_dir is not None: + extracted = {p.name: p.read_bytes() for p in runs_dir.glob("run_*.json")} + if extracted != raw_members: + raise ValueError("Extracted run-file names/bytes do not match the pinned archive.") + return list(zip(names, validated_runs)), { + n: hashlib.sha256(contents).hexdigest() for n, contents in raw_members.items() + } + + +def archive_audit(runs): + solution_cache, row_cache = {}, {} + replay, scale_audit, check_counts = [], [], Counter() + for name, run in runs: + config, truth = run["config"], run["ground_truth"] + difficulty, seed, n = config["difficulty"], config["seed"], config["n_basis"] + key = difficulty, seed + truth_vector = np.concatenate([truth[f"{k}_coeffs"] for k in COEFFICIENTS]) + if key not in solution_cache: + basis, pde = pde_from_truth(truth, n) + grid = DIFFICULTY_CONFIGS[difficulty]["n_grid"] + solution_cache[key] = basis, solve_elliptic(pde, n_grid=grid), truth_vector + print(f"Reconstructed {difficulty} seed {seed}, grid {grid}", flush=True) + basis, solution, saved_truth = solution_cache[key] + if not np.array_equal(truth_vector, saved_truth): + raise ValueError("Archived tasks sharing seed/difficulty disagree on truth.") + + def row(spec): + cache_key = key, spec + if cache_key not in row_cache: + row_cache[cache_key] = design_row(solution, basis, spec) + return row_cache[cache_key] + + rows = [row(q["test_function"]) for q in run["queries"]] + raw_matrix = np.array([r[0] for r in rows]) + raw_source = np.array([r[1] for r in rows]) + raw_rhs = np.array([r[2] for r in rows]) + matrix, source, rhs = (protocol_round(v) for v in (raw_matrix, raw_source, raw_rhs)) + logged_rhs = np.array([q["rhs_value"] for q in run["queries"]]) + solve_records = [] + for solve in run["solves"]: + k = solve.get("queries_at_solve") or solve["n_equations"] + estimate = solve_blocks(matrix[:k], source[:k], rhs[:k]) + logged = np.concatenate([solve["coefficients"][j] for j in COEFFICIENTS]) + solve_records.append({ + "solve_number": solve["solve_number"], "queries_at_solve": k, + "max_coefficient_difference_from_logged_8dp_estimate": float(np.max(np.abs(protocol_round(estimate, 8) - logged))), + "replayed_total_error": total_error(protocol_round(estimate, 8), truth_vector), + "logged_total_error": solve["coeff_errors"]["total"], + "joint": joint_diagnostics(matrix[:k], rhs[:k], truth_vector[:3*n]), + }) + replay.append({ + "run_file": name, "model": config["model"], "difficulty": difficulty, + "seed": seed, "n_basis": n, "n_grid": solution.n_grid, + "query_rhs_max_difference_from_archive": float(np.max(np.abs(rhs - logged_rhs))), + "stored_queries": len(rows), "stored_query_budget": config["max_queries"], + "free_checks": len(run["verifications"]), + "solves": solve_records, + "all_stored_queries": joint_diagnostics(matrix, rhs, truth_vector[:3*n]), + }) + for check in run["verifications"]: + check_counts["free_checks"] += 1 + previous = [q["test_function"] for q in run["queries"] if q["turn"] < check["turn"]] + if check["test_function"] not in previous: + check_counts["expression_absent_from_earlier_stored_queries"] += 1 + # Exactly the same recorded functions and quadrature values, multiplied + # before protocol rounding. This isolates quantization/row weighting; + # it intentionally excludes extra expression-parser roundoff at scale. + scaled_records = [] + for amplitude in SCALES: + estimate = solve_blocks(protocol_round(amplitude * raw_matrix), + protocol_round(amplitude * raw_source), + protocol_round(amplitude * raw_rhs)) + scaled_records.append({"common_amplitude": amplitude, + "total_error": total_error(protocol_round(estimate, 8), truth_vector)}) + scale_audit.append({"run_file": name, "difficulty": difficulty, + "model": config["model"], "seed": seed, + "all_stored_queries": scaled_records}) + return replay, scale_audit, dict(check_counts) + + +def summarize(records, scales, checks): + groups = [] + for difficulty in ("easy", "medium", "hard"): + cell = [r for r in records if r["difficulty"] == difficulty] + s = [r for r in scales if r["difficulty"] == difficulty] + columns = cell[0]["all_stored_queries"]["columns"] + groups.append({ + "difficulty": difficulty, "n_runs": len(cell), "operator_columns": columns, + "all_query_joint_full_rank_at_numpy_tolerance": sum(r["all_stored_queries"]["raw"]["rank_at_numpy_default_tolerance"] == columns for r in cell), + "all_query_joint_rank_deficient_at_relative_1e8": sum(r["all_stored_queries"]["unit_column_norm"]["rank_at_relative_tolerance"]["1e-08"] < columns for r in cell), + "median_unit_column_norm_smallest_over_largest_singular_value": float(np.median([r["all_stored_queries"]["unit_column_norm"]["min_over_max_singular_value"] for r in cell])), + "median_truth_weak_residual_l2_over_rhs_l2": float(np.median([r["all_stored_queries"]["truth_residual_l2_over_rhs_l2"] for r in cell])), + "runs_with_free_checks": sum(r["free_checks"] > 0 for r in cell), + "stored_plus_check_response_count_exceeds_stored_budget": sum(r["stored_queries"] + r["free_checks"] > r["stored_query_budget"] for r in cell), + "median_free_checks": float(np.median([r["free_checks"] for r in cell])), + "amplitude_sensitivity": [{ + "common_amplitude": amplitude, + "median_total_error": float(np.median([r["all_stored_queries"][i]["total_error"] for r in s])), + "successes_at_001": sum(r["all_stored_queries"][i]["total_error"] < .01 for r in s), + "n": len(s), + } for i, amplitude in enumerate(SCALES)], + }) + solves = [s for r in records for s in r["solves"]] + return { + "runs": len(records), "unique_reconstructed_solutions": 30, + "stored_query_rhs_responses": sum(r["stored_queries"] for r in records), + "max_query_rhs_difference_from_archive": max(r["query_rhs_max_difference_from_archive"] for r in records), + "solves_replayed": len(solves), + "max_coefficient_difference_from_logged_estimate": max(s["max_coefficient_difference_from_logged_8dp_estimate"] for s in solves), + "median_coefficient_difference_from_logged_estimate": float(np.median([s["max_coefficient_difference_from_logged_8dp_estimate"] for s in solves])), + "free_measurements": checks, "difficulty_groups": groups, + } + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--archive", type=Path, required=True, help="Unmodified dataset version 1 ZIP; checked against the manifest SHA-256") + parser.add_argument("--manifest", type=Path, default=ROOT / "paper_artifacts/data_manifest.json") + parser.add_argument("--runs-dir", type=Path, help="Optional extracted copy to check byte-for-byte against the pinned ZIP; not used as the audit input") + parser.add_argument("--output", type=Path, default=ROOT / "paper_artifacts/identifiability_audit.json") + args = parser.parse_args() + manifest = json.loads(args.manifest.read_text()) + runs, input_hashes = load_inputs(args.archive, manifest, args.runs_dir) + records, scales, checks = archive_audit(runs) + payload = { + "schema_version": 1, + "archive_sha256": manifest["archive_sha256"], + "dataset_url": manifest["dataset_url"], + "method": { + "input_validation": "ZIP bytes and complete scored-run corpus validated by reproduce_paper_figures.load_and_validate against the pinned dataset manifest.", + "solution_reconstruction": "Current repository pde.solve_elliptic from saved ground truth; no fitting to archived estimates.", + "grids": {d: DIFFICULTY_CONFIGS[d]["n_grid"] for d in ("easy", "medium", "hard")}, + "matrix_quantization": "Each matrix entry and RHS rounded to 12 decimal places, exactly as ProbeSession.query.", + "column_normalization": "Each nonzero column divided by its Euclidean norm, solely for reported diagnostic spectra.", + "rank_warning": "Floating-point rank of a discretized matrix cannot prove structural identifiability of the continuum problem.", + "amplitude_warning": "Sensitivity audit scales precomputed unrounded rows/RHS before 12dp rounding, across all stored rows, then scores coefficient estimates rounded to 8dp. These diagnostic refits differ from the primary final score computed from unrounded internal estimates. It is not a fresh model run, an optimized strategy, or parser-roundoff experiment; archived query sets are held fixed.", + "free_check_warning": "Expression novelty is literal-string novelty, not proof of independent information. Free check returns actual RHS without storing a solve row.", + "source_code_sha256": {name: hashlib.sha256((ROOT / name).read_bytes()).hexdigest() + for name in ("basis.py", "benchmark.py", "pde.py", "main_loop.py", "expression_parser.py", "weak_form.py", "requirements.txt", "scripts/audit_identifiability.py", "scripts/reproduce_paper_figures.py")}, + "input_run_files_sha256": input_hashes, + "runtime": {"python": platform.python_version(), "python_implementation": platform.python_implementation(), + "platform": platform.platform(), "packages": {p: version(p) for p in ("numpy", "scipy", "matplotlib")}}, + }, + "structural": structural_certificate(), + "summary": summarize(records, scales, checks), + "replayed_runs": records, "amplitude_sensitivity": scales, + } + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(json.dumps(payload, indent=2, allow_nan=False) + "\n") + print(json.dumps(payload["summary"], indent=2)) + + +if __name__ == "__main__": + main() diff --git a/scripts/control_baselines.py b/scripts/control_baselines.py new file mode 100644 index 0000000..3c1a92c --- /dev/null +++ b/scripts/control_baselines.py @@ -0,0 +1,407 @@ +#!/usr/bin/env python3 +"""Run predefined, offline query controls on the 30 archived PDE instances. + + python scripts/control_baselines.py --archive /tmp/pde-runs-v1.zip + +The archive SHA-256 is checked before reading it. No network or model calls are +made. Query lists depend only on the public budget and the control RNG seed; +they never depend on PDE coefficients, solutions, errors, or archived queries. +All strategies use the unchanged ProbeSession.query/solve implementation. + +These are new runs on reconstructed instances, not historical model runs. The +generator is checked against all archived coefficient vectors. The original +finite-difference solution is checked against a banded implementation of the +same equations; the latter makes the grid-refinement experiment affordable. +""" + +from __future__ import annotations + +import argparse +from collections import defaultdict +import gzip +import hashlib +import json +import math +from pathlib import Path +import platform +from statistics import mean, median +import sys +import time + +import numpy as np +from scipy.linalg import solve_banded + + +ROOT = Path(__file__).resolve().parents[1] +if str(ROOT) not in sys.path: + sys.path.insert(0, str(ROOT)) + +from basis import LegendreBasis +from benchmark import DIFFICULTY_CONFIGS +from main_loop import ProbeSession +from pde import EllipticSolution, make_random_elliptic_pde, solve_elliptic +from scripts.reproduce_paper_figures import load_and_validate + + +COEFFICIENTS = ("a", "b", "c", "f") +DIFFICULTIES = ("easy", "medium", "hard") +STRATEGIES = ("fixed_sine", "boundary_legendre", "random_mixed") +DEFAULT_CONTROL_SEED = 1729 + + +def polynomial_expression(degree: int) -> str: + """x(1-x) P_degree(2x-1), factored to avoid monomial cancellation. + + The roots and leading coefficient are properties of the public Legendre + family, independent of the coefficient basis of an individual hidden PDE. + """ + if degree < 0: + raise ValueError("Polynomial degree must be nonnegative.") + roots = np.real_if_close(np.polynomial.legendre.Legendre.basis(degree).roots()) + if np.iscomplexobj(roots): + raise ValueError("Legendre roots unexpectedly have nonzero imaginary parts.") + leading = math.comb(2 * degree, degree) / 2**degree + factors = "".join(f"*(2*x-1-({root:.17g}))" for root in roots) + return f"{leading:.17g}*x*(1-x){factors}" + + +def query_list(strategy: str, budget: int, replicate: int = 0, + control_seed: int = DEFAULT_CONTROL_SEED) -> list[str]: + """Generate a complete query policy before any session is constructed. + + Random policies use separate SeedSequence streams by budget and replicate. + Families are sampled uniformly: sine, boundary-weighted Legendre, + exponential, Gaussian, and boundary-weighted sine. Discrete functions are + sampled without replacement within each family; continuous parameters make + exponential/Gaussian repeats probability zero. All streams are reused across + the ten instances of a difficulty, so seeds are paired when aggregating. + """ + if budget < 1 or replicate < 0: + raise ValueError("Budget must be positive and replicate nonnegative.") + if strategy == "fixed_sine": + return [f"sin({k}*pi*x)" for k in range(1, budget + 1)] + if strategy == "boundary_legendre": + return [polynomial_expression(k) for k in range(budget)] + if strategy != "random_mixed": + raise ValueError(f"Unknown strategy: {strategy}") + rng = np.random.default_rng(np.random.SeedSequence([control_seed, budget, replicate])) + discrete = {name: list(rng.permutation(budget)) for name in ("sine", "polynomial", "weighted_sine")} + result = [] + for _ in range(budget): + family = ("sine", "polynomial", "exponential", "gaussian", "weighted_sine")[int(rng.integers(5))] + if family == "sine": + result.append(f"sin({discrete[family].pop() + 1}*pi*x)") + elif family == "polynomial": + result.append(polynomial_expression(discrete[family].pop())) + elif family == "exponential": + rate = float(rng.uniform(-8, 8)) + result.append(f"x*(1-x)*exp(({rate:.17g})*(x-0.5))") + elif family == "gaussian": + center = float(rng.uniform(0.05, 0.95)) + precision = float(np.exp(rng.uniform(np.log(8), np.log(128)))) + result.append(f"x*(1-x)*exp(-({precision:.17g})*(x-({center:.17g}))**2)") + else: + result.append(f"x*(1-x)*sin({discrete[family].pop() + 1}*pi*x)") + return result + + +def reconstruct_instance(run: dict): + """Regenerate from public configuration; hidden truth only audits equality.""" + config = run["config"] + settings = DIFFICULTY_CONFIGS[config["difficulty"]] + basis = LegendreBasis(settings["n_basis"], domain=(0.0, 1.0)) + pde = make_random_elliptic_pde( + basis, rng=np.random.default_rng(config["seed"]), + a_min=settings["a_min"], sparsity=settings["sparsity"], scale=settings["scale"], + ) + exact = all(np.array_equal(getattr(pde, name).coeffs, + run["ground_truth"][name + "_coeffs"]) + for name in COEFFICIENTS) + if not exact: + raise ValueError("Current generator does not exactly regenerate archived coefficients; cannot silently substitute instances.") + return basis, pde + + +def solve_banded_equivalent(pde, n_grid: int) -> EllipticSolution: + """Solve the exact second-order difference equations used in pde.py. + + This changes only the linear algebra storage/solver, not stencil, data, + differentiation, quadrature, rounding, or the identification solver. + """ + if n_grid < 3: + raise ValueError("At least three grid points are required.") + x = np.linspace(*pde.domain, n_grid) + h = x[1] - x[0] + a_half = pde.a((x[:-1] + x[1:]) / 2) + b, c, f = pde.b(x)[1:-1], pde.c(x)[1:-1], pde.f(x)[1:-1] + lower = -a_half[:-1] / h**2 - b / (2 * h) + diagonal = (a_half[:-1] + a_half[1:]) / h**2 + c + upper = -a_half[1:] / h**2 + b / (2 * h) + bands = np.zeros((3, n_grid - 2)) + bands[0, 1:] = upper[:-1] + bands[1] = diagonal + bands[2, :-1] = lower[1:] + rhs = f.copy() + rhs[0] -= lower[0] * pde.bc_left + rhs[-1] -= upper[-1] * pde.bc_right + u = np.empty(n_grid) + u[0], u[-1] = pde.bc_left, pde.bc_right + u[1:-1] = solve_banded((1, 1), bands, rhs, check_finite=True) + residual = (lower * u[:-2] + diagonal * u[1:-1] + upper * u[2:]) - f + return EllipticSolution(pde=pde, x=x, u=u, residual=residual) + + +def design_diagnostics(history: list[dict]) -> dict: + """Joint LHS sensitivity, unlike separate block condition numbers.""" + matrices = [np.asarray([row[key] for row in history]) + for key in ("_G_diff", "_G_adv", "_G_react")] + matrix = np.hstack(matrices) + + def summarize(array): + singular = np.linalg.svd(array, compute_uv=False) + tolerance = max(array.shape) * np.finfo(float).eps * singular[0] + rank = int(np.sum(singular > tolerance)) + return { + "singular_values": singular.tolist(), "rank": rank, + "n_columns": int(array.shape[1]), "rank_tolerance": float(tolerance), + "condition_number": float(singular[0] / singular[-1]) if singular[-1] > 0 else None, + "smallest_singular_value": float(singular[-1]), + } + + norms = np.linalg.norm(matrix, axis=0) + return {"raw": summarize(matrix), + "column_normalized": summarize(matrix / np.where(norms > 0, norms, 1)), + "column_l2_norms": norms.tolist()} + + +def evaluate_policy(session: ProbeSession, specs: list[str], truth: dict) -> list[dict]: + """Use existing query/solve; hidden truth enters only post-solve scoring.""" + checkpoints = [] + first_solve = 4 * session.basis.n_basis + for spec in specs: + response = session.query(spec) + if response["status"] != "ok": + raise ValueError(f"Control query failed: {spec}: {response}") + if session.queries_used < first_solve: + continue + result = session.solve() + if result["status"] != "ok": + raise ValueError(f"Control solve failed: {result}") + errors = {name: float(np.max(np.abs(np.asarray(session.last_solve_coeffs[name]) + - truth[name + "_coeffs"]))) + for name in COEFFICIENTS} + displayed_errors = {name: float(np.max(np.abs(np.asarray(result[name + "_coeffs"]) + - truth[name + "_coeffs"]))) + for name in COEFFICIENTS} + checkpoints.append({ + "queries": session.queries_used, + "coefficient_errors": {**errors, "total": sum(errors.values())}, + "displayed_8dp_coefficient_errors": {**displayed_errors, "total": sum(displayed_errors.values())}, + "coefficient_estimates": {name: list(session.last_solve_coeffs[name]) for name in COEFFICIENTS}, + "joint_design": design_diagnostics(session.history), + }) + return checkpoints + + +def error_description(values): + values = [float(v) for v in values] + return {"n": len(values), "mean": mean(values), "median": median(values), + "min": min(values), "max": max(values), + "success_e_lt_0.01": sum(v < 0.01 for v in values), + "success_e_lt_0.1": sum(v < 0.1 for v in values)} + + +def aggregate(records: list[dict]) -> list[dict]: + """Keep instance and randomization variation distinct in the output. + + Deterministic policies have one run per PDE. Random policies report pooled + runs descriptively, plus each replicate's ten-instance summary and each + instance's randomized success fraction. No pooled run is called a new PDE. + """ + groups = defaultdict(list) + for record in records: + groups[(record["difficulty"], record["strategy"], record["grid_factor"])].append(record) + output = [] + for (difficulty, strategy, factor), group in sorted(groups.items()): + checkpoints = [] + for index in range(len(group[0]["checkpoints"])): + rows = [record["checkpoints"][index] for record in group] + checkpoints.append({"queries": rows[0]["queries"], + "total_error": error_description(row["coefficient_errors"]["total"] for row in rows), + "joint_rank_deficient_runs": sum(row["joint_design"]["raw"]["rank"] + < row["joint_design"]["raw"]["n_columns"] for row in rows)}) + per_replicate = [] + for replicate in sorted({record["replicate"] for record in group}): + subset = [record for record in group if record["replicate"] == replicate] + per_replicate.append({"replicate": replicate, + "minimum_queries": error_description(r["checkpoints"][0]["coefficient_errors"]["total"] for r in subset), + "full_budget": error_description(r["checkpoints"][-1]["coefficient_errors"]["total"] for r in subset)}) + per_instance = [] + for seed in sorted({record["seed"] for record in group}): + subset = [record for record in group if record["seed"] == seed] + per_instance.append({"seed": seed, + "full_budget": error_description(r["checkpoints"][-1]["coefficient_errors"]["total"] for r in subset)}) + output.append({"difficulty": difficulty, "strategy": strategy, "grid_factor": factor, + "n_instances": len(per_instance), "n_replicates": len(per_replicate), + "checkpoints": checkpoints, "per_replicate": per_replicate, + "per_instance": per_instance}) + return output + + +def sensitivity_summary(sensitivities): + output = [] + for difficulty in DIFFICULTIES: + for strategy in STRATEGIES: + subset = [row for row in sensitivities if row["difficulty"] == difficulty and row["strategy"] == strategy] + if not subset: + continue + all_checks = [check for row in subset for check in row["checkpoints"]] + final_checks = [row["checkpoints"][-1] for row in subset] + def describe(checks): + return {"n_comparisons": len(checks), + "max_absolute_total_error_change": max(row["absolute_total_error_change"] for row in checks), + "threshold_0.01_changes": sum(row["threshold_0.01_changed"] for row in checks), + "threshold_0.1_changes": sum(row["threshold_0.1_changed"] for row in checks)} + output.append({"difficulty": difficulty, "strategy": strategy, + "all_checkpoints": describe(all_checks), "full_budget": describe(final_checks)}) + return output + + +def write_json(path: Path, value): + """Write readable JSON or deterministic gzip according to the file suffix.""" + data = (json.dumps(value, indent=2, allow_nan=False) + "\n").encode() + path.parent.mkdir(parents=True, exist_ok=True) + path.write_bytes(gzip.compress(data, mtime=0) if path.suffix == ".gz" else data) + + +def compact_summary(result): + excluded = {"runs", "query_policies", "native_vs_banded_score_sensitivity"} + return {key: value for key, value in result.items() if key not in excluded} + + +def run_controls(archive: Path, manifest: dict, replicates: int, grid_factors: list[int], + control_seed: int = DEFAULT_CONTROL_SEED) -> dict: + if replicates < 1 or not grid_factors or any(factor < 1 for factor in grid_factors): + raise ValueError("Positive replicate count and grid factors are required.") + if len(set(grid_factors)) != len(grid_factors): + raise ValueError("Grid factors must be distinct.") + runs = load_and_validate(archive, manifest) + by_instance = {} + for run in runs: + by_instance.setdefault((run["config"]["difficulty"], run["config"]["seed"]), run) + # Policies are fixed before constructing any session or generating its PDE. + policies = {(strategy, budget, replicate): query_list(strategy, budget, replicate, control_seed) + for budget in sorted({run["config"]["max_queries"] for run in runs}) + for strategy in STRATEGIES + for replicate in range(replicates if strategy == "random_mixed" else 1)} + records, audits, sensitivities = [], [], [] + start_time = time.perf_counter() + for (difficulty, seed), run in sorted(by_instance.items()): + basis, pde = reconstruct_instance(run) + default_grid = DIFFICULTY_CONFIGS[difficulty]["n_grid"] + native = solve_elliptic(pde, n_grid=default_grid) + banded = solve_banded_equivalent(pde, default_grid) + discrepancy = float(np.max(np.abs(native.u - banded.u))) + if not np.allclose(native.u, banded.u, rtol=2e-8, atol=1e-10): + raise ValueError("Banded solution failed agreement with the native dense solver.") + replay = ProbeSession(pde, native, basis, difficulty, seed, max_queries=run["config"]["max_queries"]) + first_query = run["queries"][0] + replay_response = replay.query(first_query["test_function"]) + rhs_discrepancy = abs(replay_response["integral_f_phi"] - first_query["rhs_value"]) + if rhs_discrepancy != 0: + raise ValueError("First archived query RHS did not replay exactly.") + audits.append({"difficulty": difficulty, "seed": seed, + "coefficient_regeneration_exact": True, + "native_grid_points": default_grid, + "native_banded_max_solution_difference": discrepancy, + "first_archived_rhs_exact": True}) + budget = run["config"]["max_queries"] + for factor in grid_factors: + # Default-resolution controls use the native implementation itself. + solution = native if factor == 1 else solve_banded_equivalent(pde, (default_grid - 1) * factor + 1) + for strategy in STRATEGIES: + for replicate in range(replicates if strategy == "random_mixed" else 1): + session = ProbeSession(pde, solution, basis, difficulty, seed, max_queries=budget) + checkpoints = evaluate_policy(session, policies[(strategy, budget, replicate)], run["ground_truth"]) + records.append({"difficulty": difficulty, "seed": seed, "n_basis": basis.n_basis, + "strategy": strategy, "replicate": replicate, + "grid_factor": factor, "n_grid": len(solution.x), "budget": budget, + "checkpoints": checkpoints}) + if factor == 1: + alternative = ProbeSession(pde, banded, basis, difficulty, seed, max_queries=budget) + banded_checks = evaluate_policy(alternative, policies[(strategy, budget, replicate)], run["ground_truth"]) + sensitivities.append({ + "difficulty": difficulty, "seed": seed, "strategy": strategy, + "replicate": replicate, + "checkpoints": [{"queries": left["queries"], + "native_total_error": left["coefficient_errors"]["total"], + "banded_total_error": right["coefficient_errors"]["total"], + "absolute_total_error_change": abs(left["coefficient_errors"]["total"] - right["coefficient_errors"]["total"]), + "threshold_0.01_changed": (left["coefficient_errors"]["total"] < .01) != (right["coefficient_errors"]["total"] < .01), + "threshold_0.1_changed": (left["coefficient_errors"]["total"] < .1) != (right["coefficient_errors"]["total"] < .1)} + for left, right in zip(checkpoints, banded_checks)], + }) + print(f"Completed {difficulty} seed {seed}: {len(records)} policy/grid runs", file=sys.stderr, flush=True) + import scipy + source_paths = ("scripts/control_baselines.py", "main_loop.py", "pde.py", "basis.py", "benchmark.py", + "expression_parser.py", "weak_form.py", "test_functions.py", "requirements.txt") + all_checkpoints = [check for record in records for check in record["checkpoints"]] + rounding_summary = { + "n_checkpoints": len(all_checkpoints), + "max_absolute_total_error_change": max(abs(row["coefficient_errors"]["total"] + - row["displayed_8dp_coefficient_errors"]["total"]) for row in all_checkpoints), + "threshold_agreement": {str(threshold): all((row["coefficient_errors"]["total"] < threshold) + == (row["displayed_8dp_coefficient_errors"]["total"] < threshold) + for row in all_checkpoints) for threshold in (.01, .1)}, + } + return { + "schema_version": 1, "archive_sha256": manifest["archive_sha256"], + "dataset_url": manifest["dataset_url"], + "experiment_type": "new offline controls on exactly regenerated archived PDE instances", + "historical_replay_limitation": "Exact ground-truth regeneration and first-query RHS replay do not establish identical historical solution grids or numerical implementations for all runs; historical logs lack per-run implementation hashes.", + "protocol": { + "control_seed": control_seed, "random_replicates": replicates, "grid_factors": grid_factors, + "unique_instances": len(by_instance), "query_and_identification_implementation": "unchanged ProbeSession.query and ProbeSession.solve", + "minimum_solve_queries": "4N, enforced by the existing interface; fitted LHS has 3N columns and forcing is fitted separately", + "scoring": "Sum over a,b,c,f of maximum absolute coefficient error, using unrounded internal estimates, matching final PREDICT scoring; strict E < threshold. Displayed 8dp scores, matching intermediate historical error curves, are also saved and threshold agreement audited.", + "selection_information": "Only public query budget and independent control RNG. No solution, coefficient, oracle-response, error, rank, or condition information enters selection.", + "random_sampling": "Uniform family choice among sine, boundary-weighted Legendre, exponential, localized Gaussian, boundary-weighted sine. Sine degrees 1..budget and polynomial degrees 0..budget-1 sampled without replacement within each family. Exponential rates uniform[-8,8]; Gaussian centers uniform[.05,.95], log precision uniform[log8,log128].", + "scaling": "Use expressions as generated, with x(1-x) envelope for all families except pure sine; no post-hoc row normalization or tuning.", + "grid_refinement": "Refine forward-solution grid and every query integral together; (n_grid-1)*factor+1 points. This is not a noise robustness test.", + "forward_solver": "Native dense solve at original grid; independently assembled equivalent tridiagonal solve_banded at refined grids, validated against native dense at original grid on all 30 instances.", + "aggregation": "Pooled randomized runs are descriptive. Per-replicate summaries and per-instance success fractions preserve distinct PDE and randomization units; 10 random replicates do not increase the 30 independent PDE instances.", + "design_svd": "Joint [G_diff G_adv G_react] and independently column-normalized version; rank tolerance max(shape)*eps*sigma_max. Full singular spectra are saved.", + }, + "environment": {"python": platform.python_version(), "numpy": np.__version__, "scipy": scipy.__version__}, + "elapsed_seconds": time.perf_counter() - start_time, + "source_sha256": {path: hashlib.sha256((ROOT / path).read_bytes()).hexdigest() for path in source_paths}, + "regeneration_audit": audits, + "display_rounding_score_sensitivity": rounding_summary, + "native_vs_banded_score_sensitivity_summary": sensitivity_summary(sensitivities), + "native_vs_banded_score_sensitivity": sensitivities, + "query_policies": [{"strategy": strategy, "budget": budget, "replicate": replicate, "queries": specs} + for (strategy, budget, replicate), specs in sorted(policies.items())], + "summary": aggregate(records), "runs": records, + } + + +def main(argv=None): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--archive", type=Path, required=True) + parser.add_argument("--manifest", type=Path, default=ROOT / "paper_artifacts/data_manifest.json") + parser.add_argument("--output", type=Path, default=ROOT / "paper_artifacts/control_baselines.json.gz", + help="Full output, compressed deterministically when the filename ends .gz") + parser.add_argument("--summary-output", type=Path, default=ROOT / "paper_artifacts/control_baselines_summary.json") + parser.add_argument("--replicates", type=int, default=10) + parser.add_argument("--grid-factors", type=int, nargs="+", default=[1, 2, 4]) + parser.add_argument("--control-seed", type=int, default=DEFAULT_CONTROL_SEED) + args = parser.parse_args(argv) + manifest = json.loads(args.manifest.read_text()) + result = run_controls(args.archive, manifest, args.replicates, args.grid_factors, args.control_seed) + write_json(args.output, result) + write_json(args.summary_output, compact_summary(result)) + print(f"Saved {len(result['runs'])} new control runs to {args.output}") + + +if __name__ == "__main__": + main() diff --git a/scripts/plot_research_audit.py b/scripts/plot_research_audit.py new file mode 100644 index 0000000..dfb72bf --- /dev/null +++ b/scripts/plot_research_audit.py @@ -0,0 +1,221 @@ +#!/usr/bin/env python3 +"""Plot new offline controls and audit results, keeping data sources separate. + +Run the three analysis scripts first; this script only reads their derived data. +It makes no network or provider calls. Figure-source checksums are saved beside +its TeX tables so figures can be traced to an exact analysis output. +""" +from __future__ import annotations + +import argparse +import gzip +import hashlib +import json +from pathlib import Path + +import matplotlib +matplotlib.use('Agg') +import matplotlib.pyplot as plt +import numpy as np + +ROOT = Path(__file__).resolve().parents[1] +DIFFICULTIES = ('easy', 'medium', 'hard') +COLORS = ('#0072B2', '#D55E00', '#009E73') +LABELS = { + 'gemini-3.1-pro-preview': 'Gemini 3.1 Pro', + 'claude-opus-4-6': 'Claude Opus 4.6', + 'gpt-5.4': 'GPT-5.4', + 'gemini-3-flash-preview': 'Gemini 3 Flash', + 'claude-sonnet-4-20250514': 'Claude Sonnet 4', + 'gpt-5.4-mini': 'GPT-5.4 mini', +} +STRATEGY_LABELS = {'fixed_sine': 'Fixed sine', 'boundary_legendre': 'Boundary Legendre', + 'random_mixed': 'Random mixed'} + + +def read_json(path): + opener = gzip.open if path.suffix == '.gz' else open + with opener(path, 'rt') as source: + return json.load(source) + + +def configure(): + plt.rcParams.update({'font.size': 9, 'axes.labelsize': 9, 'axes.titlesize': 10, + 'legend.fontsize': 8, 'xtick.labelsize': 8, 'ytick.labelsize': 8, + 'pdf.fonttype': 42, 'ps.fonttype': 42, + 'axes.spines.top': False, 'axes.spines.right': False, + 'savefig.bbox': 'tight', 'savefig.pad_inches': .03}) + + +def plot_numerics(audit, output): + fig, axes = plt.subplots(1, 2, figsize=(6.75, 2.55), layout='constrained') + rng = np.random.default_rng(108) + for index, difficulty in enumerate(DIFFICULTIES): + values = [row['all_stored_queries']['unit_column_norm']['min_over_max_singular_value'] + for row in audit['replayed_runs'] if row['difficulty'] == difficulty] + axes[0].scatter(index + rng.uniform(-.17, .17, len(values)), values, + s=12, alpha=.7, color=COLORS[index], edgecolor='none') + axes[0].plot([index - .2, index + .2], [np.median(values)] * 2, color='black', lw=1.6) + axes[0].axhline(1e-8, ls='--', color='.4', lw=.8) + axes[0].set(yscale='log', ylabel=r'$\sigma_{\min}/\sigma_{\max}$ (unit columns)', + xticks=range(3), xticklabels=[d.title() for d in DIFFICULTIES]) + axes[0].grid(axis='y', alpha=.2) + for index, row in enumerate(audit['summary']['difficulty_groups']): + entries = row['amplitude_sensitivity'] + axes[1].plot([v['common_amplitude'] for v in entries], + [v['successes_at_001'] / v['n'] for v in entries], + marker=('o','s','^')[index], color=COLORS[index], label=row['difficulty'].title()) + axes[1].set(xscale='log', xlabel='Common amplitude before 12dp rounding', + ylabel=r'Refit success fraction ($E<0.01$)', ylim=(0, 1.04), + xticks=[1e-3, 1, 1e3, 1e6]) + axes[1].legend(loc='lower right', frameon=False) + axes[1].grid(alpha=.2) + fig.savefig(output / 'paper_numerical_audit.pdf') + plt.close(fig) + + +def plot_controls(controls, output): + fig, axes = plt.subplots(1, 3, figsize=(6.75, 2.45), sharey=True, layout='constrained') + for ax, difficulty in zip(axes, DIFFICULTIES): + for index, strategy in enumerate(STRATEGY_LABELS): + cell = next(row for row in controls['summary'] if row['difficulty'] == difficulty + and row['strategy'] == strategy and row['grid_factor'] == 1) + points = cell['checkpoints'] + xs = [p['queries'] for p in points] + ys = [p['total_error']['success_e_lt_0.01'] / p['total_error']['n'] for p in points] + ax.plot(xs, ys, color=COLORS[index], ls=('-', '--', ':')[index], + lw=1.6, marker=('o','s','^')[index], ms=3, label=STRATEGY_LABELS[strategy]) + if strategy == 'random_mixed': + runs = [r for r in controls['runs'] if r['difficulty'] == difficulty + and r['strategy'] == strategy and r['grid_factor'] == 1] + by_rep = np.asarray([[np.mean([r['checkpoints'][j]['coefficient_errors']['total'] < .01 + for r in runs if r['replicate'] == rep]) + for j in range(len(points))] + for rep in range(controls['protocol']['random_replicates'])]) + ax.fill_between(xs, by_rep.min(axis=0), by_rep.max(axis=0), + color=COLORS[index], alpha=.12, linewidth=0) + ax.set(title=difficulty.title(), xlabel='Stored queries', ylim=(-.04, 1.06)) + ax.set_xticks([xs[0], (xs[0]+xs[-1])//2, xs[-1]]) + ax.grid(alpha=.2) + axes[0].set_ylabel(r'Success fraction ($E<0.01$)') + handles, labels = axes[0].get_legend_handles_labels() + fig.legend(handles, labels, loc='outside lower center', ncol=3, frameon=False) + fig.savefig(output / 'paper_controls.pdf') + plt.close(fig) + + +def plot_confidence(statistics, output): + fig, axes = plt.subplots(2, 3, figsize=(6.75, 4.25), layout='constrained') + for col, difficulty in enumerate(DIFFICULTIES): + cells = [row for row in statistics['cells'] if row['difficulty'] == difficulty] + for row_index, metric in enumerate(('brier_score','signed_calibration_in_the_large_gap')): + ax = axes[row_index, col] + for index, cell in enumerate(cells): + data = cell['confidence']['run_balanced'][metric] + value = data['estimate'] + lo, hi = data['whole_run_bootstrap_95_interval'] + ax.errorbar(value, index, xerr=[[max(0, value-lo)], [max(0, hi-value)]], + fmt='o', ms=3, capsize=2, color=COLORS[col], lw=1) + ax.set(yticks=range(len(cells)), ylim=(len(cells)-.5, -.5)) + ax.set_yticklabels([LABELS[c['model']] for c in cells] if col == 0 else []) + if row_index == 0: + ax.set_title(difficulty.title()) + ax.set_xlabel('Run-balanced Brier score') + ax.set_xlim(0, .7) + else: + ax.axvline(0, color='.5', ls='--', lw=.7) + ax.set_xlabel('Confidence − correctness') + ax.set_xlim(-.6, .8) + ax.grid(axis='x', alpha=.2) + fig.savefig(output / 'paper_confidence_uncertainty.pdf') + plt.close(fig) + + +def make_tables(controls, statistics, destination): + lines = [r'% Generated by scripts/plot_research_audit.py; do not edit numbers manually.', + r'\begin{table}[t]', r'\caption{New offline controls at the original resolution and full stored-query budget. Deterministic policies have ten tasks per difficulty. Random rows summarize ten query-list replicates on the same ten tasks; the denominator 100 is not 100 independent PDEs.}', + r'\label{tab:controls}', r'\centering\small', + r'\begin{tabular}{@{}llrr@{}}', r'\toprule', + r'Policy & Setting & Successes & Median $E$\\', r'\midrule'] + for strategy in STRATEGY_LABELS: + for difficulty in DIFFICULTIES: + row = next(r for r in controls['summary'] if r['difficulty'] == difficulty + and r['strategy'] == strategy and r['grid_factor'] == 1) + scores = row['checkpoints'][-1]['total_error'] + lines.append(f"{STRATEGY_LABELS[strategy]} & {difficulty.title()} & " + f"{scores['success_e_lt_0.01']}/{scores['n']} & {scores['median']:.2g}\\\\") + lines.append(r'\addlinespace') + lines += [r'\bottomrule', r'\end{tabular}', r'\end{table}', ''] + destination.write_text('\n'.join(lines)) + lines = [r'% Generated by scripts/plot_research_audit.py.', + r'\begin{table}[ht]', r'\caption{Strict-threshold archived successes with Wilson 95\% intervals. Each cell uses ten scored tasks; intervals cannot account for missing attempts or model-sampling variability.}', + r'\centering\small', r'\begin{tabular}{@{}lrrr@{}}', r'\toprule', + r'Model & Easy & Medium & Hard\\', r'\midrule'] + for model, label in LABELS.items(): + values = [] + for difficulty in DIFFICULTIES: + cell = next(c for c in statistics['cells'] if c['model'] == model and c['difficulty'] == difficulty)['success']['0.01'] + lo, hi = cell['wilson_95_interval'] + values.append(f"{cell['count']}/10 [{lo:.2f}, {hi:.2f}]") + lines.append(label+' & '+' & '.join(values)+r'\\') + lines += [r'\bottomrule', r'\end{tabular}', r'\end{table}', '', + r'\begin{table}[ht]', r'\caption{Exploratory exact paired success tests, Holm-adjusted within the 15 model pairs of each difficulty. All 45 comparisons are shown. Values are conditional on the retained scored tasks; no adjustment is made across difficulties.}', + r'\centering\small', r'\begin{tabular}{@{}llrrr@{}}', r'\toprule', + r'First model & Second model & Easy & Medium & Hard\\', r'\midrule'] + pairs = [(r['first_model'],r['second_model']) for r in statistics['pairwise_success_comparisons'] if r['difficulty']=='easy'] + for first, second in pairs: + values=[] + for difficulty in DIFFICULTIES: + row=next(r for r in statistics['pairwise_success_comparisons'] if r['difficulty']==difficulty and r['first_model']==first and r['second_model']==second) + values.append(f"{row['holm_adjusted_p_within_difficulty']:.4f}") + lines.append(LABELS[first]+' & '+LABELS[second]+' & '+' & '.join(values)+r'\\') + lines += [r'\bottomrule',r'\end{tabular}',r'\end{table}',''] + destination.with_name('statistical_tables.tex').write_text('\n'.join(lines)) + + +def generalization_table(data, destination): + lines = [r'% Generated by scripts/plot_research_audit.py.', + r'\begin{table}[t]', + r'\caption{Fixed sine on 100 fresh seeds per setting. All planned tasks remain in the denominator. Parentheses give Wilson 95\% intervals for the minimum-budget rate; zero-source counts are descriptive.}', + r'\label{tab:fresh}',r'\centering\small', + r'\begin{tabular}{@{}lrrr@{}}',r'\toprule', + r'Setting & $4N$ successes & $6N$ successes & $f=0$\\',r'\midrule'] + for row in data['summary']: + minimum, full = row['minimum_queries'], row['full_budget'] + lo, hi = minimum['wilson_95_interval'] + lines.append(f"{row['difficulty'].title()} & {minimum['successes_e_lt_0.01']}/100 & " + f"{full['successes_e_lt_0.01']}/100 & {row['zero_source_tasks']}\\\\") + lines.append(f" & ({lo:.2f}, {hi:.2f}) & & \\\\") + lines += [r'\bottomrule',r'\end{tabular}',r'\end{table}',''] + destination.write_text('\n'.join(lines)) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + default_controls = ROOT / 'paper_artifacts/control_baselines.json.gz' + if not default_controls.exists(): + default_controls = ROOT / 'paper_artifacts/control_baselines.json' + parser.add_argument('--controls', type=Path, default=default_controls) + parser.add_argument('--audit', type=Path, default=ROOT/'paper_artifacts/identifiability_audit.json') + parser.add_argument('--statistics', type=Path, default=ROOT/'paper_artifacts/statistical_analysis.json') + parser.add_argument('--generalization', type=Path, default=ROOT/'paper_artifacts/sine_generalization.json') + parser.add_argument('--output-dir', type=Path, default=ROOT/'images') + args=parser.parse_args() + controls, audit, statistics = [read_json(p) for p in (args.controls,args.audit,args.statistics)] + if len({v['archive_sha256'] for v in (controls,audit,statistics)}) != 1: + raise ValueError('Derived inputs do not refer to the same pinned archive.') + generalization = read_json(args.generalization) + configure() + args.output_dir.mkdir(parents=True,exist_ok=True) + plot_numerics(audit,args.output_dir) + plot_controls(controls,args.output_dir) + plot_confidence(statistics,args.output_dir) + make_tables(controls,statistics,ROOT/'paper_artifacts/control_table.tex') + generalization_table(generalization,ROOT/'paper_artifacts/sine_generalization_table.tex') + provenance={p.name:hashlib.sha256(p.read_bytes()).hexdigest() for p in (args.controls,args.audit,args.statistics,args.generalization)} + provenance['plot_script_sha256']=hashlib.sha256(Path(__file__).read_bytes()).hexdigest() + (ROOT/'paper_artifacts/research_figure_sources.json').write_text(json.dumps(provenance,indent=2)+'\n') + print('Saved three audit figures and numerical tables.') + +if __name__=='__main__': + main() diff --git a/scripts/reproduce_paper_figures.py b/scripts/reproduce_paper_figures.py new file mode 100644 index 0000000..47bddbc --- /dev/null +++ b/scripts/reproduce_paper_figures.py @@ -0,0 +1,381 @@ +#!/usr/bin/env python3 +"""Audit the archived Kaggle study and regenerate its publication figures offline. + +Download separately (the script never makes network or model API calls):: + + curl -L --fail 'https://www.kaggle.com/api/v1/datasets/download/jkmiller/elliptic-differential-equation-llm-reasoning?datasetVersionNumber=1' -o /tmp/pde-runs-v1.zip + python scripts/reproduce_paper_figures.py --archive /tmp/pde-runs-v1.zip + +Only the standard library, NumPy, and Matplotlib are required. Run from any +directory; default output paths resolve relative to this repository. The input +archive is checked against paper_artifacts/data_manifest.json before parsing. +No archive member is extracted or executed. +""" + +from __future__ import annotations + +import argparse +from collections import Counter +import hashlib +import json +import math +from pathlib import Path +import re +from statistics import mean, median, pstdev +import zipfile + + +ROOT = Path(__file__).resolve().parents[1] +DIFFICULTIES = ("easy", "medium", "hard") +COEFFICIENTS = ("a", "b", "c", "f") +MODEL_LABELS = { + "gemini-3.1-pro-preview": "Gemini 3.1\nPro", + "claude-opus-4-6": "Claude\nOpus 4.6", + "gpt-5.4": "GPT-5.4", + "gemini-3-flash-preview": "Gemini 3\nFlash", + "claude-sonnet-4-20250514": "Claude\nSonnet 4", + "gpt-5.4-mini": "GPT-5.4\nmini", +} +MODEL_COLORS = ("#3274A1", "#C47E16", "#238B72", "#81A9D6", "#D4A329", "#76BEA4") +CONFIDENCE = re.compile( + r"Confidence:\s*a=(\d+)%,\s*b=(\d+)%,\s*c=(\d+)%,\s*f=(\d+)%", re.I +) + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def describe(values): + """Population SD describes the archived runs, not estimation uncertainty.""" + values = [float(v) for v in values] + return { + "n": len(values), + "mean": mean(values), + "median": median(values), + "population_sd": pstdev(values), + "min": min(values), + "max": max(values), + } + + +def load_and_validate(archive, manifest): + digest = hashlib.sha256(archive.read_bytes()).hexdigest() + require(digest == manifest["archive_sha256"], "Archive SHA-256 does not match the pinned dataset version.") + with zipfile.ZipFile(archive) as source: + require(len(source.infolist()) == manifest["archive_members"], "Unexpected archive member count.") + names = sorted(n for n in source.namelist() if n.startswith("run_") and n.endswith(".json")) + runs = [json.loads(source.read(n)) for n in names] + sweep = json.loads(source.read("sweep_results.json")) + require(len(runs) == manifest["expected_runs"], "Unexpected run count.") + require(len(sweep) == len(runs), "Sweep summary count differs from run-log count.") + require({r["config"]["model"] for r in runs} == set(manifest["models"]), "Unexpected model set.") + require({r["config"]["difficulty"] for r in runs} == set(DIFFICULTIES), "Unexpected difficulty set.") + keys = [(r["config"]["model"], r["config"]["difficulty"], r["config"]["seed"]) for r in runs] + require(len(keys) == len(set(keys)), "Duplicate model/difficulty/seed record.") + summaries = {(r["model"], r["difficulty"], r["seed"]): r for r in sweep} + for run in runs: + cfg = run["config"] + settings = manifest["difficulty_settings"][cfg["difficulty"]] + require(all(cfg[k] == v for k, v in settings.items()), "Run settings differ from the manifest.") + require(cfg["prompt_condition"] == manifest["prompt_condition"], "Unexpected prompt condition.") + expected_effort = "medium" if cfg["provider"] == "openai" else None + require(cfg["reasoning_effort"] == expected_effort, "Unexpected reasoning-effort setting.") + errors = run["results"]["coefficient_errors"] + require(all(isinstance(errors[k], (int, float)) and math.isfinite(errors[k]) and errors[k] >= 0 + for k in (*COEFFICIENTS, "total")), "A run lacks a finite nonnegative score.") + require(math.isclose(sum(errors[k] for k in COEFFICIENTS), errors["total"], rel_tol=1e-12, abs_tol=1e-12), + "Component errors do not sum to the reported total.") + key = (cfg["model"], cfg["difficulty"], cfg["seed"]) + require(key in summaries and summaries[key]["total_coeff_error"] == errors["total"], + "Sweep summary score disagrees with its run log.") + for difficulty in DIFFICULTIES: + for model in manifest["models"]: + cell = [r for r in runs if r["config"]["difficulty"] == difficulty and r["config"]["model"] == model] + require(sorted(r["config"]["seed"] for r in cell) == manifest["expected_seeds"], + "A model/difficulty cell does not contain the ten expected seeds.") + for seed in manifest["expected_seeds"]: + truths = {json.dumps(r["ground_truth"], sort_keys=True) for r in runs + if r["config"]["difficulty"] == difficulty and r["config"]["seed"] == seed} + require(len(truths) == 1, "Models do not share ground truth for a seed/difficulty pair.") + return runs + + +def causal_confidence_pairs(run): + """Pair reports to previously returned solve outputs using transcript turns. + + Archived reports omit turn numbers, but their exact confidence strings occur + in the assistant transcript in the same order. A solve in the report's own + response is not yet observable to the model and is therefore excluded. + """ + reports = run.get("metacognitive", {}).get("confidence_reports", []) + occurrences = [(turn["turn"], match) for turn in run["turns"] + if turn.get("role") == "assistant" + for match in CONFIDENCE.finditer(turn.get("content", ""))] + require(len(occurrences) == len(reports), "Confidence reports do not match transcript occurrence counts.") + pairs, stats = [], Counter(reports=len(reports)) + for (turn_number, occurrence), report in zip(occurrences, reports): + require([int(v) / 100 for v in occurrence.groups()] == [report[k] for k in COEFFICIENTS], + "Archived confidence values disagree with the transcript.") + eligible = [s for s in run.get("solves", []) + if s["turn"] < turn_number and (s.get("queries_at_solve") or s["n_equations"]) <= report["query_count"]] + if not eligible: + stats["reports_excluded_no_preceding_solve"] += 1 + continue + solve = max(eligible, key=lambda s: (s["turn"], s.get("solve_number", 0))) + stats["reports_paired"] += 1 + # Audit the legacy forward-looking matching and overlapping boundaries. + legacy = next((s for s in run["solves"] + if (s.get("queries_at_solve") or s["n_equations"]) >= report["query_count"]), run["solves"][-1]) + if legacy.get("solve_number") != solve.get("solve_number"): + stats["reports_with_different_solve_from_legacy"] += 1 + for coefficient in ("a", "b", "c"): + confidence = float(report[coefficient]) + require(0 <= confidence <= 1, "Confidence is outside [0, 1].") + error = float(solve["coeff_errors"][coefficient]) + pairs.append((confidence, int(error < 0.1))) + if 0 < confidence < 1 and math.isclose(confidence * 10, round(confidence * 10), abs_tol=1e-9): + stats["observations_on_shared_legacy_bin_boundaries"] += 1 + stats["coefficient_observations"] = len(pairs) + return pairs, stats + + +def calibration_summary(pairs): + """Disjoint bins [lo, hi), with 1 included in the final bin.""" + bins = [[] for _ in range(10)] + for confidence, success in pairs: + index = min(int(confidence * 10), 9) + bins[index].append((confidence, success)) + return [{ + "lower": i / 10, + "upper": (i + 1) / 10, + "upper_inclusive": i == 9, + "n": len(values), + "successes": sum(v[1] for v in values), + "mean_confidence": mean(v[0] for v in values) if values else None, + "success_fraction": mean(v[1] for v in values) if values else None, + } for i, values in enumerate(bins)] + + +def build_summary(runs, manifest): + summary = { + "schema_version": 1, + "archive_sha256": manifest["archive_sha256"], + "dataset_url": manifest["dataset_url"], + "validation": { + "runs": len(runs), "scored_runs": len(runs), "models": len(manifest["models"]), + "difficulties": list(DIFFICULTIES), "seeds_per_cell": manifest["expected_seeds"], + "shared_ground_truth_per_seed_difficulty": True, + "sweep_summary_matches_all_run_scores": True, + "attempts_and_retries_before_archive": "not recorded", + }, + "settings": { + "difficulty_settings": manifest["difficulty_settings"], + "prompt_condition": manifest["prompt_condition"], + "reasoning_effort": {"openai": "medium", "anthropic": None, "google": None}, + "run_timestamp_range": [min(r["config"]["timestamp"] for r in runs), max(r["config"]["timestamp"] for r in runs)], + "run_timestamp_timezone": "not recorded", + }, + "definitions": { + "coefficient_error": "E_j = max_l abs(predicted_theta[j,l] - true_theta[j,l]); absolute coefficient error, not relative error", + "total_error": "E = E_a + E_b + E_c + E_f; strict threshold comparison", + "improvement_ratio": "first solve total error / last solve total error; missing values assigned 1 to retain legacy plotting convention", + "family_entropy": "-sum(p_i log2(p_i)) / log2(5), over the final solve taxonomy: polynomial/trigonometric/exponential/localized/other; later queries are omitted", + "action_stack": "separate per-action medians, summed; not the median total action count", + "population_sd": "ddof=0 across the ten archived runs; descriptive variation, not a confidence interval", + "calibration": "pool a/b/c reports; pair with latest solve on an earlier transcript turn with queries_at_solve <= report query count; omit reports without prior solve; event is E_j < 0.1", + "calibration_bins": "disjoint [lo,hi), with 1 in the final bin; plot mean stated confidence against empirical success fraction; marker area in points squared is 5 + 1.1 times bin count", + }, + "cells": [], + "calibration_totals": {}, + "changes_from_legacy_assets": [ + "GPT-5.4 mini hard success at E < 0.01 is 2/10 in both run logs and sweep_results.json; the checked-in PNG displays 0/10.", + "Calibration uses causally available solve outputs and disjoint confidence bins, replacing forward-looking solve matching and overlapping bin boundaries.", + "Improvement figure shows individual runs and medians on a logarithmic axis; it omits arithmetic-mean bars and standard-deviation whiskers." + ], + } + total_calibration = Counter() + for difficulty in DIFFICULTIES: + for model in manifest["models"]: + cell_runs = [r for r in runs if r["config"]["difficulty"] == difficulty and r["config"]["model"] == model] + errors = [r["results"]["coefficient_errors"]["total"] for r in cell_runs] + metrics = [r["behavioral_metrics"] for r in cell_runs] + improvement = [m.get("improvement_ratio") if m.get("improvement_ratio") is not None else 1.0 for m in metrics] + pairs, counts = [], Counter() + for run in cell_runs: + run_pairs, run_counts = causal_confidence_pairs(run) + pairs.extend(run_pairs) + counts.update(run_counts) + total_calibration.update(counts) + summary["cells"].append({ + "model": model, "difficulty": difficulty, "n": len(cell_runs), + "success": {str(t): {"count": sum(e < t for e in errors), "n": len(errors), "rate": sum(e < t for e in errors) / len(errors)} for t in (0.01, 0.1)}, + "total_error": describe(errors), + "coefficient_errors": {k: describe([r["results"]["coefficient_errors"][k] for r in cell_runs]) for k in COEFFICIENTS}, + "solves": describe([m.get("solve_count", 0) or 0 for m in metrics]), + "checks": describe([m.get("check_count", 0) or 0 for m in metrics]), + "term_integrals": describe([m.get("term_integral_count", 0) or 0 for m in metrics]), + "budget_utilization": describe([m["budget_utilization"] for m in metrics]), + "improvement_ratio": describe(improvement), + "improvement_ratio_imputed_runs": sum(m.get("improvement_ratio") is None for m in metrics), + "family_entropy": describe([m.get("family_entropy_normalized", 0) or 0 for m in metrics]), + "calibration_support": dict(counts), + "calibration_bins": calibration_summary(pairs), + }) + summary["calibration_totals"] = dict(total_calibration) + return summary + + +def plot_figures(runs, summary, manifest, output): + import numpy as np + import matplotlib + matplotlib.use("Agg") + import matplotlib.pyplot as plt + from matplotlib.colors import Normalize + + plt.rcParams.update({ + "font.family": "DejaVu Sans", "font.size": 8.5, + "axes.titlesize": 9, "axes.labelsize": 8.5, + "xtick.labelsize": 8, "ytick.labelsize": 8, + "legend.fontsize": 7.5, "axes.spines.top": False, + "axes.spines.right": False, "pdf.fonttype": 42, + }) + output.mkdir(parents=True, exist_ok=True) + models = manifest["models"] + cells = {(c["model"], c["difficulty"]): c for c in summary["cells"]} + labels = [MODEL_LABELS[m] for m in models] + pdf_metadata = {"Creator": "scripts/reproduce_paper_figures.py", "CreationDate": None, "ModDate": None} + + def save(fig, name): + fig.savefig(output / name, metadata=pdf_metadata) + plt.close(fig) + + def three_panels(ylabel, *, top=0.96): + fig, axes = plt.subplots(3, 1, figsize=(6.4, 7.3), sharex=True) + fig.subplots_adjust(left=0.12, right=0.98, bottom=0.09, top=top, hspace=0.34) + for index, (ax, difficulty) in enumerate(zip(axes, DIFFICULTIES)): + cfg = manifest["difficulty_settings"][difficulty] + ax.set_title(f"({chr(97 + index)}) {difficulty.capitalize()} ($N={cfg['n_basis']}$; query budget {cfg['max_queries']})", loc="left", pad=4) + ax.set_ylabel(ylabel) + ax.set_xticks(range(len(models)), labels) + ax.grid(axis="y", color="0.91", linewidth=0.6, zorder=0) + return fig, axes + + # Two threshold views share a color scale and annotate exact numerators. + fig, axes = plt.subplots(2, 1, figsize=(6.4, 5.1)) + fig.subplots_adjust(left=0.26, right=0.90, top=0.94, bottom=0.09, hspace=0.33) + for ax, threshold in zip(axes, (0.01, 0.1)): + counts = np.array([[cells[(m, d)]["success"][str(threshold)]["count"] for d in DIFFICULTIES] for m in models]) + im = ax.imshow(counts / 10, cmap="Blues", norm=Normalize(0, 1), aspect="auto") + ax.set_xticks(range(3), [d.capitalize() for d in DIFFICULTIES]) + ax.set_yticks(range(6), [MODEL_LABELS[m].replace("\n", " ") for m in models]) + ax.set_title(f"{'(a) Main threshold' if threshold == .01 else '(b) Sensitivity threshold'}: total error $E < {threshold}$", loc="left", pad=5) + for row in range(6): + for col in range(3): + ax.text(col, row, f"{counts[row, col]}/10", ha="center", va="center", fontsize=8, + color="white" if counts[row, col] >= 7 else "#183047") + ax.tick_params(length=0) + color_ax = fig.add_axes([0.93, 0.22, 0.018, 0.56]) + fig.colorbar(im, cax=color_ax, ticks=[0, .5, 1]) + color_ax.set_title("Rate", fontsize=8, pad=5) + save(fig, "paper_success.pdf") + + fig, axes = three_panels("Action count", top=0.92) + action_names = (("solves", "Solve"), ("checks", "Check"), ("term_integrals", "Term integral")) + action_colors = ("#329367", "#D5A229", "#35A9BC") + for ax, difficulty in zip(axes, DIFFICULTIES): + bottom = np.zeros(len(models)) + maximum_stack = max(sum(cells[(m, difficulty)][key]["median"] for key, _ in action_names) for m in models) + for (key, label), color in zip(action_names, action_colors): + values = np.array([cells[(m, difficulty)][key]["median"] for m in models]) + ax.bar(range(len(models)), values, bottom=bottom, color=color, edgecolor="white", linewidth=.5, label=label, zorder=3) + for x, value, start in zip(range(len(models)), values, bottom): + if value >= maximum_stack * .10: + ax.text(x, start + value / 2, f"{value:g}", ha="center", va="center", fontsize=7.5) + bottom += values + ax.set_ylim(0, max(bottom) * 1.13) + fig.legend(*axes[0].get_legend_handles_labels(), loc="upper center", ncol=3, frameon=False) + save(fig, "paper_actions.pdf") + + fig, axes = three_panels("First / last solve error") + rng = np.random.default_rng(2026) + for ax, difficulty in zip(axes, DIFFICULTIES): + all_values = [] + for x, (model, color) in enumerate(zip(models, MODEL_COLORS)): + values = [r["behavioral_metrics"].get("improvement_ratio") for r in runs + if r["config"]["model"] == model and r["config"]["difficulty"] == difficulty] + values = np.array([v if v is not None else 1 for v in values]) + require(np.all(values > 0), "Log-scale improvement plot requires positive ratios.") + all_values.extend(values) + ax.scatter(x + rng.uniform(-.17, .17, len(values)), values, s=18, color=color, alpha=.7, linewidths=.3, edgecolors="white", zorder=3) + med = float(np.median(values)) + ax.plot([x - .28, x + .28], [med, med], color="#202020", linewidth=1.8, zorder=4) + ax.axhline(1, color="0.5", linestyle="--", linewidth=.7) + ax.set_yscale("log") + ax.set_ylim(min(.5, min(all_values) / 1.7), max(all_values) * 2) + save(fig, "paper_improvement.pdf") + + fig, axes = three_panels("Normalized family entropy") + rng = np.random.default_rng(2026) + for ax, difficulty in zip(axes, DIFFICULTIES): + for x, (model, color) in enumerate(zip(models, MODEL_COLORS)): + values = [r["behavioral_metrics"].get("family_entropy_normalized", 0) or 0 for r in runs + if r["config"]["model"] == model and r["config"]["difficulty"] == difficulty] + stats = cells[(model, difficulty)]["family_entropy"] + ax.bar(x, stats["mean"], color=color, alpha=.7, zorder=2) + ax.errorbar(x, stats["mean"], yerr=stats["population_sd"], color="#444444", capsize=2, linewidth=.8, zorder=3) + ax.scatter(x + rng.uniform(-.17, .17, len(values)), values, s=13, color=color, alpha=.9, linewidths=.3, edgecolors="white", zorder=4) + ax.plot([x - .28, x + .28], [stats["median"], stats["median"]], color="#202020", linewidth=1.7, zorder=5) + ax.set_ylim(0, 1) + save(fig, "paper_diversity.pdf") + + fig, axes = plt.subplots(3, 1, figsize=(6.4, 7.3), sharex=True, sharey=True) + fig.subplots_adjust(left=.12, right=.97, top=.88, bottom=.08, hspace=.29) + for index, (ax, difficulty) in enumerate(zip(axes, DIFFICULTIES)): + support = sum(cells[(m, difficulty)]["calibration_support"].get("coefficient_observations", 0) for m in models) + ax.set_title(f"({chr(97 + index)}) {difficulty.capitalize()} ({support} coefficient reports)", loc="left", pad=4) + ax.plot([0, 1], [0, 1], "--", color="0.6", linewidth=.8, zorder=1) + for model, color in zip(models, MODEL_COLORS): + bins = [b for b in cells[(model, difficulty)]["calibration_bins"] if b["n"]] + x = [b["mean_confidence"] for b in bins] + y = [b["success_fraction"] for b in bins] + ax.plot(x, y, color=color, linewidth=.9, alpha=.9) + ax.scatter(x, y, s=[5 + b["n"] * 1.1 for b in bins], color=color, alpha=.75, edgecolor="white", linewidth=.4, zorder=3, + label=MODEL_LABELS[model].replace("\n", " ")) + ax.set_xlim(-.02, 1.02) + ax.set_ylim(-.06, 1.06) + ax.set_ylabel("Fraction with $E_j < 0.1$") + ax.grid(color="0.92", linewidth=.6) + axes[-1].set_xlabel("Mean stated confidence within bin") + handles, legend_labels = axes[0].get_legend_handles_labels() + # Uniform legend markers retain model identity; areas in panels show support. + from matplotlib.lines import Line2D + handles = [Line2D([], [], color=c, marker="o", markersize=4, linewidth=.9) for c in MODEL_COLORS] + fig.legend(handles, legend_labels, loc="upper center", ncol=3, frameon=False) + fig.text(.5, .905, "Marker area increases with bin support; exact counts are in summary.json", ha="center", fontsize=7) + save(fig, "paper_calibration.pdf") + + +def main(): + parser = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter) + parser.add_argument("--archive", type=Path, required=True, help="Downloaded, unmodified Kaggle dataset version 1 ZIP") + parser.add_argument("--manifest", type=Path, default=ROOT / "paper_artifacts/data_manifest.json") + parser.add_argument("--output-dir", type=Path, default=ROOT / "images") + parser.add_argument("--summary", type=Path, default=ROOT / "paper_artifacts/summary.json") + parser.add_argument("--summary-only", action="store_true", help="Audit and write JSON without importing NumPy/Matplotlib") + args = parser.parse_args() + manifest = json.loads(args.manifest.read_text()) + runs = load_and_validate(args.archive, manifest) + summary = build_summary(runs, manifest) + args.summary.parent.mkdir(parents=True, exist_ok=True) + args.summary.write_text(json.dumps(summary, indent=2, allow_nan=False) + "\n") + print(f"Verified {len(runs)} scored runs against the pinned archive SHA-256; wrote {args.summary}") + print("Calibration support:", json.dumps(summary["calibration_totals"], sort_keys=True)) + if not args.summary_only: + plot_figures(runs, summary, manifest, args.output_dir) + print(f"Wrote five publication figures to {args.output_dir}") + + +if __name__ == "__main__": + main() diff --git a/scripts/validate_sine_generalization.py b/scripts/validate_sine_generalization.py new file mode 100644 index 0000000..ddeee89 --- /dev/null +++ b/scripts/validate_sine_generalization.py @@ -0,0 +1,207 @@ +#!/usr/bin/env python3 +"""Fresh-seed validation after selecting fixed sine on the archived tasks. + + python scripts/validate_sine_generalization.py + +The fixed plan contains seeds 1001--1100 in each difficulty, including zero +forcing and any failed tasks. No retries, exclusions, policy changes, archive +selection, network access, or model calls are made. Original-resolution native +forward solves and the existing query/identification interface are retained. +""" + +from __future__ import annotations + +import argparse +from collections import Counter +import hashlib +import json +from pathlib import Path +import platform +from statistics import NormalDist, median +import sys +import time + +import numpy as np +import scipy + + +ROOT = Path(__file__).resolve().parents[1] +if str(ROOT) not in sys.path: + sys.path.insert(0, str(ROOT)) + +from basis import LegendreBasis +from benchmark import DIFFICULTY_CONFIGS +from main_loop import ProbeSession +from pde import make_random_elliptic_pde, solve_elliptic +from scripts.control_baselines import evaluate_policy, query_list + + +DIFFICULTIES = ("easy", "medium", "hard") +FIRST_SEED, LAST_SEED = 1001, 1100 + + +def validation_plan(): + """Return the complete fixed plan, in deterministic execution order.""" + return [{"difficulty": difficulty, "seed": seed} + for difficulty in DIFFICULTIES + for seed in range(FIRST_SEED, LAST_SEED + 1)] + + +def wilson_interval(successes: int, n: int): + if not 0 <= successes <= n or n < 1: + raise ValueError("Wilson interval requires 0 <= successes <= positive n.") + z = NormalDist().inv_cdf(.975) + p = successes / n + divisor = 1 + z**2 / n + center = (p + z**2 / (2 * n)) / divisor + radius = z * np.sqrt(p * (1 - p) / n + z**2 / (4 * n**2)) / divisor + return [float(max(0, center - radius)), float(min(1, center + radius))] + + +def concise_checkpoint(checkpoint): + diagnostics = {} + for kind in ("raw", "column_normalized"): + full = checkpoint["joint_design"][kind] + diagnostics[kind] = {key: value for key, value in full.items() if key != "singular_values"} + diagnostics[kind]["largest_singular_value"] = full["singular_values"][0] + return {"queries": checkpoint["queries"], + "coefficient_errors": checkpoint["coefficient_errors"], + "displayed_8dp_coefficient_errors": checkpoint["displayed_8dp_coefficient_errors"], + "coefficient_estimates": checkpoint["coefficient_estimates"], + "joint_design": diagnostics} + + +def evaluate_task(task): + """Retain an outcome and failure stage for every generated instance.""" + difficulty, seed = task["difficulty"], task["seed"] + settings = DIFFICULTY_CONFIGS[difficulty] + n_basis, n_grid = settings["n_basis"], settings["n_grid"] + result = {**task, "status": "pending", "stage": "generation", + "config": {**settings, "domain": [0, 1], "bc_left": 0, "bc_right": 0, + "minimum_queries": 4 * n_basis, "budget": 6 * n_basis}, + "ground_truth": None, "zero_source": None, "zero_solution": None, + "minimum_queries": None, "full_budget": None, "error": None} + started = time.perf_counter() + try: + basis = LegendreBasis(n_basis, domain=(0.0, 1.0)) + pde = make_random_elliptic_pde( + basis, rng=np.random.default_rng(seed), a_min=settings["a_min"], + sparsity=settings["sparsity"], scale=settings["scale"], + ) + result["ground_truth"] = {name + "_coeffs": getattr(pde, name).coeffs.tolist() + for name in ("a", "b", "c", "f")} + result["zero_source"] = bool(np.all(pde.f.coeffs == 0)) + result["stage"] = "native_forward_solve" + solution = solve_elliptic(pde, n_grid=n_grid) + if not np.all(np.isfinite(solution.u)): + raise ValueError("Native forward solution contains nonfinite values.") + result["zero_solution"] = bool(np.all(solution.u == 0)) + result["forward_max_abs_u"] = float(np.max(np.abs(solution.u))) + result["stage"] = "query_and_identification" + session = ProbeSession(pde, solution, basis, difficulty, seed, max_queries=6 * n_basis) + checkpoints = evaluate_policy(session, query_list("fixed_sine", 6 * n_basis), result["ground_truth"]) + result["minimum_queries"] = concise_checkpoint(checkpoints[0]) + result["full_budget"] = concise_checkpoint(checkpoints[-1]) + if not all(np.isfinite(result[key]["coefficient_errors"]["total"]) + for key in ("minimum_queries", "full_budget")): + raise ValueError("Coefficient recovery contains a nonfinite score.") + result["status"], result["stage"] = "ok", "complete" + except Exception as error: + result["status"] = "error" + result["error"] = {"type": type(error).__name__, "message": str(error)} + result["elapsed_seconds"] = time.perf_counter() - started + return result + + +def execute_plan(plan, runner=evaluate_task, progress=False): + """Never drop or retry a planned task, even on an unexpected runner failure.""" + outcomes = [] + for index, task in enumerate(plan, 1): + try: + outcome = runner(task) + except Exception as error: + outcome = {**task, "status": "error", "stage": "unexpected_runner_failure", + "ground_truth": None, "zero_source": None, "zero_solution": None, + "minimum_queries": None, "full_budget": None, + "error": {"type": type(error).__name__, "message": str(error)}} + outcomes.append(outcome) + if progress and index % 10 == 0: + print(f"Completed {index}/{len(plan)} fresh tasks: {task['difficulty']} seed {task['seed']}", + file=sys.stderr, flush=True) + return outcomes + + +def summarize(outcomes): + summary = [] + for difficulty in DIFFICULTIES: + rows = [row for row in outcomes if row["difficulty"] == difficulty] + if not rows: + continue + cell = {"difficulty": difficulty, "n_planned_tasks": len(rows), + "status_counts": dict(Counter(row["status"] for row in rows)), + "zero_source_tasks": sum(row.get("zero_source") is True for row in rows), + "zero_solution_tasks": sum(row.get("zero_solution") is True for row in rows), + "zero_source_seeds": [row["seed"] for row in rows if row.get("zero_source") is True], + "failed_tasks": [{"seed": row["seed"], "stage": row["stage"], "error": row["error"]} + for row in rows if row["status"] != "ok"]} + for checkpoint in ("minimum_queries", "full_budget"): + good = [row for row in rows if row["status"] == "ok"] + successes = sum(row[checkpoint]["coefficient_errors"]["total"] < .01 for row in good) + cell[checkpoint] = { + "successes_e_lt_0.01": successes, "denominator_all_planned_tasks": len(rows), + "success_fraction": successes / len(rows), "wilson_95_interval": wilson_interval(successes, len(rows)), + "successes_e_lt_0.1": sum(row[checkpoint]["coefficient_errors"]["total"] < .1 for row in good), + "median_error_completed_tasks": median(row[checkpoint]["coefficient_errors"]["total"] for row in good) if good else None, + "failure_seeds_including_errors": [row["seed"] for row in rows if row["status"] != "ok" + or row[checkpoint]["coefficient_errors"]["total"] >= .01], + "zero_source_subset_descriptive": { + "n": cell["zero_source_tasks"], + "successes": sum(row.get("zero_source") is True and row[checkpoint]["coefficient_errors"]["total"] < .01 for row in good)}, + "nonzero_source_subset_descriptive": { + "n": sum(row.get("zero_source") is False for row in rows), + "successes": sum(row.get("zero_source") is False and row[checkpoint]["coefficient_errors"]["total"] < .01 for row in good)}, + } + summary.append(cell) + return summary + + +def main(argv=None): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--output", type=Path, default=ROOT / "paper_artifacts/sine_generalization.json") + args = parser.parse_args(argv) + started = time.perf_counter() + plan = validation_plan() + outcomes = execute_plan(plan, progress=True) + environment = {"python": platform.python_version(), "numpy": np.__version__, "scipy": scipy.__version__, + "machine": platform.machine(), "platform": platform.platform()} + sources = ("scripts/validate_sine_generalization.py", "scripts/control_baselines.py", "basis.py", "benchmark.py", + "pde.py", "main_loop.py", "expression_parser.py", "weak_form.py", "test_functions.py", "requirements.txt") + hashes = {source: hashlib.sha256((ROOT / source).read_bytes()).hexdigest() for source in sources} + digest = lambda value: hashlib.sha256(json.dumps(value, sort_keys=True, separators=(",", ":")).encode()).hexdigest() + result = { + "schema_version": 1, + "experiment_type": "fresh-seed validation after design selection", + "protocol": { + "strategy": "fixed_sine", "first_seed": FIRST_SEED, "last_seed": LAST_SEED, + "n_seeds_per_difficulty": 100, "n_planned_tasks": len(plan), "query_expression": "sin(k*pi*x), k=1..6N", + "selection_timing": "Fixed sine was selected after its success on the 30 archived instances; this is subsequent validation, not a historical held-out evaluation or formal preregistration.", + "inclusion": "Every seed-difficulty task in the fixed plan, including all-zero forcing, all-zero solutions, generation/solver/identification errors. No retries or exclusions.", + "environment": "Original-resolution native solve_elliptic and unchanged ProbeSession.query/solve; no banded substitution or grid refinement.", + "scoring": "Sum of per-field maximum absolute coefficient errors from unrounded internal estimates; E<0.01 primary. Displayed 8dp errors also retained.", + "units": "100 tasks per difficulty. The same seed labels are reused across difficulties, with related generator RNG streams; no pooled 300-task independent-binomial interval is claimed.", + "intervals": "Wilson 95% marginal intervals per difficulty over all 100 planned tasks; execution failures count as non-successes. Assumes seed-generated task sampling and conditions on this generator and fixed design.", + "zero_source_subsets": "Descriptive counts only; zero-source tasks remain in all primary denominators.", + "provider_calls": 0, + }, + "environment": environment, "runtime_sha256": digest(environment), + "source_sha256": hashes, "source_bundle_sha256": digest(hashes), + "elapsed_seconds": time.perf_counter() - started, + "plan": plan, "summary": summarize(outcomes), "tasks": outcomes, + } + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(json.dumps(result, indent=2, allow_nan=False) + "\n") + print(json.dumps(result["summary"], indent=2)) + + +if __name__ == "__main__": + main() diff --git a/scripts/verify_paper_artifacts.py b/scripts/verify_paper_artifacts.py new file mode 100644 index 0000000..69bf1c3 --- /dev/null +++ b/scripts/verify_paper_artifacts.py @@ -0,0 +1,58 @@ +#!/usr/bin/env python3 +"""Fail if checked-in research outputs no longer match their inputs or code. + +This integrity check uses only the standard library and requires no archive, +network, numerical runtime, or provider credentials. It checks provenance rather +than asserting bitwise portability of floating-point experiments. +""" +from pathlib import Path +import gzip +import hashlib +import json + +ROOT = Path(__file__).resolve().parents[1] +ARTIFACTS = ROOT / 'paper_artifacts' + + +def digest(path): + return hashlib.sha256(path.read_bytes()).hexdigest() + + +def check(condition, message): + if not condition: + raise ValueError(message) + + +def main(): + manifest = json.loads((ARTIFACTS/'data_manifest.json').read_text()) + with gzip.open(ARTIFACTS/'control_baselines.json.gz', 'rt') as source: + controls = json.load(source) + compact = json.loads((ARTIFACTS/'control_baselines_summary.json').read_text()) + audit = json.loads((ARTIFACTS/'identifiability_audit.json').read_text()) + stats = json.loads((ARTIFACTS/'statistical_analysis.json').read_text()) + fresh = json.loads((ARTIFACTS/'sine_generalization.json').read_text()) + for path, expected in fresh['source_sha256'].items(): + check(digest(ROOT/path) == expected, f'Fresh-seed output stale after changing {path}') + check(len(fresh['tasks']) == len(fresh['plan']) == 300, 'Missing fresh-seed outcomes') + check({(r['difficulty'],r['seed']) for r in fresh['tasks']} == + {(r['difficulty'],r['seed']) for r in fresh['plan']}, 'Fresh-seed task membership differs from plan') + for name, data in [('controls',controls), ('audit',audit), ('statistics',stats)]: + check(data['archive_sha256'] == manifest['archive_sha256'], f'{name}: archive mismatch') + hashes = data['method']['source_code_sha256'] if name == 'audit' else data['source_sha256'] + check(bool(hashes), f'{name}: no source provenance') + for path, expected in hashes.items(): + check(digest(ROOT/path) == expected, f'{name}: stale output after changing {path}') + for key in ('protocol','source_sha256','summary','regeneration_audit'): + check(compact[key] == controls[key], f'Compact control output differs: {key}') + check(len(controls['runs']) == 1080, 'Unexpected control policy/grid run count') + check(len(audit['replayed_runs']) == 180, 'Unexpected archive replay count') + check(len(stats['pairwise_success_comparisons']) == 45, 'Missing paired comparisons') + figure_sources = json.loads((ARTIFACTS/'research_figure_sources.json').read_text()) + for name, expected in figure_sources.items(): + path = ROOT/'scripts/plot_research_audit.py' if name == 'plot_script_sha256' else ARTIFACTS/name + check(digest(path) == expected, f'Research figures/tables are stale after changing {path.name}') + print('Verified source, input, summary, and research-figure provenance.') + + +if __name__ == '__main__': + main() diff --git a/test_archive_statistics.py b/test_archive_statistics.py new file mode 100644 index 0000000..12c2a24 --- /dev/null +++ b/test_archive_statistics.py @@ -0,0 +1,85 @@ +"""Task-pairing and dependence checks for exploratory archive statistics.""" + +import unittest + +import numpy as np + +from scripts.analyze_archive_statistics import ( + exact_discordant_p, holm_adjust, paired_success_statistics, + run_confidence_metrics, summarize_run_confidence, wilson_interval, +) + + +class TestPairedArchiveStatistics(unittest.TestCase): + def test_pairs_are_joined_by_seed_not_input_order(self): + result = paired_success_statistics( + {2: 0, 1: 1}, {1: 1, 2: 1}, np.array([[0, 0], [1, 1]]) + ) + self.assertEqual(result["seeds"], [1, 2]) + self.assertEqual(result["first_only_successes"], 0) + self.assertEqual(result["second_only_successes"], 1) + self.assertEqual(result["success_rate_difference_first_minus_second"], -0.5) + np.testing.assert_allclose(result["paired_task_bootstrap_95_interval"], [-0.975, -0.025]) + + def test_unmatched_seeds_are_rejected(self): + with self.assertRaisesRegex(ValueError, "identical"): + paired_success_statistics({1: 1, 2: 0}, {1: 1, 3: 0}, np.array([[0, 1]])) + + def test_exact_test_handles_zero_and_extreme_discordance(self): + self.assertEqual(exact_discordant_p(0, 0), 1.0) + self.assertEqual(exact_discordant_p(3, 0), 0.25) + self.assertEqual(exact_discordant_p(0, 10), 2 / 1024) + self.assertEqual(exact_discordant_p(3, 3), 1.0) + + def test_holm_is_monotone_in_sorted_p_and_preserves_input_order(self): + np.testing.assert_allclose(holm_adjust([0.04, 0.01, 0.03]), [0.06, 0.03, 0.06]) + + def test_wilson_retains_uncertainty_when_all_ten_tasks_succeed(self): + lower, upper = wilson_interval(10, 10) + self.assertAlmostEqual(lower, 0.7224672001371107) + self.assertAlmostEqual(upper, 1.0) + lower_zero, upper_zero = wilson_interval(0, 10) + self.assertAlmostEqual(lower_zero, 0.0) + self.assertAlmostEqual(upper_zero, 1 - lower) + + +class TestRunBalancedConfidence(unittest.TestCase): + def test_replicating_reports_does_not_reweight_a_run(self): + short = {1: [(0.9, 1)], 2: [(0.9, 0)]} + duplicated = {1: [(0.9, 1)] * 100, 2: [(0.9, 0)]} + a = summarize_run_confidence(short, np.random.default_rng(17), 200) + b = summarize_run_confidence(duplicated, np.random.default_rng(17), 200) + self.assertEqual(a["n_runs_with_paired_reports"], 2) + self.assertEqual(b["coefficient_observations"], 101) + for name in a["run_balanced"]: + self.assertAlmostEqual(a["run_balanced"][name]["estimate"], b["run_balanced"][name]["estimate"]) + np.testing.assert_allclose( + a["run_balanced"][name]["whole_run_bootstrap_95_interval"], + b["run_balanced"][name]["whole_run_bootstrap_95_interval"], + ) + self.assertAlmostEqual(a["run_balanced"]["brier_score"]["estimate"], 0.41) + self.assertAlmostEqual(a["run_balanced"]["signed_calibration_in_the_large_gap"]["estimate"], 0.4) + + def test_bootstrap_draws_retain_all_blocks_within_a_run(self): + # Both runs average to the same metrics despite opposite block outcomes. + result = summarize_run_confidence( + {1: [(0.8, 1), (0.8, 0)], 2: [(0.8, 0), (0.8, 1)]}, + np.random.default_rng(42), 200, + ) + np.testing.assert_allclose( + result["run_balanced"]["brier_score"]["whole_run_bootstrap_95_interval"], + [0.34, 0.34], + ) + + def test_missing_reports_are_explicit_and_not_imputed(self): + result = summarize_run_confidence({1: [], 2: [(1.0, 1)]}, np.random.default_rng(0), 10) + self.assertEqual(result["seeds_without_paired_reports"], [1]) + self.assertEqual(result["n_runs_with_paired_reports"], 1) + self.assertEqual(result["run_balanced"]["brier_score"]["estimate"], 0.0) + empty = summarize_run_confidence({1: []}, np.random.default_rng(0), 10) + self.assertEqual(empty["run_balanced"], {}) + self.assertIsNone(run_confidence_metrics([])) + + +if __name__ == "__main__": + unittest.main() diff --git a/test_control_baselines.py b/test_control_baselines.py new file mode 100644 index 0000000..917e459 --- /dev/null +++ b/test_control_baselines.py @@ -0,0 +1,125 @@ +"""Scientific integrity checks for offline, hidden-truth-independent controls.""" + +import unittest +import gzip +import json +from pathlib import Path +import tempfile + +import numpy as np + +from basis import CoefficientFunction, LegendreBasis +from expression_parser import make_test_function_from_string +from main_loop import ProbeSession +from pde import EllipticPDE, solve_elliptic +from scripts.control_baselines import ( + aggregate, design_diagnostics, evaluate_policy, polynomial_expression, + query_list, reconstruct_instance, solve_banded_equivalent, write_json, +) + + +class TestControlPolicies(unittest.TestCase): + def test_queries_are_valid_unique_and_reproducible(self): + for strategy in ("fixed_sine", "boundary_legendre", "random_mixed"): + specs = query_list(strategy, 24, replicate=3) + self.assertEqual(specs, query_list(strategy, 24, replicate=3)) + self.assertEqual(len(specs), 24) + self.assertEqual(len(set(specs)), 24) + self.assertFalse(any("j" in spec for spec in specs)) + for spec in specs: + function, error = make_test_function_from_string(spec) + self.assertIsNone(error, (spec, error)) + self.assertIsNotNone(function) + self.assertNotEqual(query_list("random_mixed", 24, 2), query_list("random_mixed", 24, 3)) + + def test_factored_legendre_matches_stable_recurrence(self): + x = np.linspace(0, 1, 257) + for degree in (0, 1, 5, 15, 23): + function, error = make_test_function_from_string(polynomial_expression(degree)) + self.assertIsNone(error) + expected = x * (1 - x) * np.polynomial.legendre.Legendre.basis(degree)(2 * x - 1) + np.testing.assert_allclose(function(x), expected, atol=5e-14, rtol=1e-10) + + def test_policy_does_not_accept_instance_or_solution_information(self): + import inspect + self.assertEqual(tuple(inspect.signature(query_list).parameters), + ("strategy", "budget", "replicate", "control_seed")) + + +class TestNumericalControls(unittest.TestCase): + def make_pde(self): + basis = LegendreBasis(2) + functions = [CoefficientFunction(basis, coeff) for coeff in + ([1.3, .1], [.2, -.1], [.5, .2], [1, -.4])] + return basis, EllipticPDE(*functions, bc_left=.3, bc_right=-.2) + + def test_banded_system_matches_native_including_nonzero_boundaries(self): + _, pde = self.make_pde() + for n_grid in (3, 17, 101): + dense = solve_elliptic(pde, n_grid) + banded = solve_banded_equivalent(pde, n_grid) + np.testing.assert_allclose(banded.u, dense.u, rtol=1e-12, atol=1e-13) + self.assertLess(np.max(np.abs(banded.residual)), 1e-10) + + def test_reconstruction_rejects_changed_archived_truth(self): + session = ProbeSession.from_difficulty("easy", seed=1) + run = {"config": {"difficulty": "easy", "seed": 1}, + "ground_truth": {name + "_coeffs": getattr(session.pde, name).coeffs.tolist() + for name in ("a", "b", "c", "f")}} + reconstruct_instance(run) + run["ground_truth"]["a_coeffs"][0] += .001 + with self.assertRaisesRegex(ValueError, "exactly regenerate"): + reconstruct_instance(run) + + def test_evaluation_respects_native_minimum_and_full_budget(self): + basis, pde = self.make_pde() + solution = solve_elliptic(pde, 101) + session = ProbeSession(pde, solution, basis, "easy", 1, max_queries=12) + truth = {name + "_coeffs": getattr(pde, name).coeffs.tolist() for name in ("a", "b", "c", "f")} + checkpoints = evaluate_policy(session, query_list("fixed_sine", 12), truth) + self.assertEqual([row["queries"] for row in checkpoints], list(range(8, 13))) + self.assertEqual(session.queries_used, 12) + errors = checkpoints[-1]["coefficient_errors"] + self.assertAlmostEqual(errors["total"], sum(errors[name] for name in ("a", "b", "c", "f"))) + displayed = checkpoints[-1]["displayed_8dp_coefficient_errors"] + self.assertLessEqual(abs(errors["total"] - displayed["total"]), 2.01e-8) + + def test_joint_rank_reveals_cross_block_dependence(self): + history = [{"_G_diff": [1, 0], "_G_adv": [1, 0], "_G_react": [0, 1]}, + {"_G_diff": [0, 1], "_G_adv": [0, 1], "_G_react": [1, 0]}] + diagnostic = design_diagnostics(history) + self.assertEqual(diagnostic["raw"]["rank"], 2) + self.assertEqual(diagnostic["raw"]["n_columns"], 6) + self.assertEqual(diagnostic["column_normalized"]["rank"], 2) + + def test_random_replicates_are_not_counted_as_additional_pdes(self): + records = [] + for seed in (1, 2): + for replicate in (0, 1): + checkpoint = {"queries": 8, "coefficient_errors": {"total": .001 if replicate == 0 else .1}, + "joint_design": {"raw": {"rank": 6, "n_columns": 6}}} + records.append({"difficulty": "easy", "strategy": "random_mixed", "grid_factor": 1, + "seed": seed, "replicate": replicate, "checkpoints": [checkpoint]}) + summary, = aggregate(records) + self.assertEqual(summary["n_instances"], 2) + self.assertEqual(summary["n_replicates"], 2) + self.assertEqual(summary["checkpoints"][0]["total_error"]["n"], 4) + self.assertEqual(summary["checkpoints"][0]["total_error"]["success_e_lt_0.01"], 2) + for instance in summary["per_instance"]: + self.assertEqual(instance["full_budget"]["n"], 2) + self.assertEqual(instance["full_budget"]["success_e_lt_0.01"], 1) + + def test_compressed_output_is_readable_and_deterministic(self): + value = {"score": .001, "nested": {"n": 30}} + with tempfile.TemporaryDirectory() as directory: + plain, zipped = Path(directory) / "out.json", Path(directory) / "out.json.gz" + write_json(plain, value) + write_json(zipped, value) + first = zipped.read_bytes() + write_json(zipped, value) + self.assertEqual(zipped.read_bytes(), first) + self.assertEqual(json.loads(gzip.decompress(first)), json.loads(plain.read_text())) + + +if __name__ == "__main__": + unittest.main() diff --git a/test_identifiability.py b/test_identifiability.py new file mode 100644 index 0000000..422eddf --- /dev/null +++ b/test_identifiability.py @@ -0,0 +1,106 @@ +"""Proof and numeric sanity checks for the paper's identifiability audit.""" +import unittest +from unittest.mock import patch +from pathlib import Path +import tempfile +import zipfile + +import numpy as np +from numpy.polynomial import Polynomial + +from basis import CoefficientFunction, LegendreBasis +from pde import EllipticPDE, solve_elliptic +from scripts.audit_identifiability import ( + design_row, exact_rank, joint_diagnostics, load_inputs, polynomial_operator_matrix, + spectra, structural_certificate, +) + + +class StructuralAmbiguityTests(unittest.TestCase): + def test_exact_nullity_in_all_archived_classes_and_larger_classes(self): + for n in range(2, 9): + with self.subTest(n=n): + matrix = polynomial_operator_matrix(n) + self.assertEqual(exact_rank(matrix), n + 2) + self.assertEqual(3 * n - exact_rank(matrix), 2 * n - 2) + + def test_integer_matrix_matches_independent_polynomial_differentiation(self): + """Check the certificate against calculus, not a second rank routine.""" + u = Polynomial([0, 1, -1]) + for n in (2, 3, 4, 8): + matrix = polynomial_operator_matrix(n) + for block in range(3): + for j in range(n): + p = Polynomial([0] * j + [1]) + image = (-(p * u.deriv()).deriv() if block == 0 else + p * u.deriv() if block == 1 else p * u) + expected = np.pad(image.coef, (0, n + 2 - len(image.coef))) + np.testing.assert_array_equal(matrix[:, block * n + j], expected) + + def test_ellipticity_and_same_source_for_concrete_pair(self): + # epsilon=-1/8 keeps a>0 and c>=0; both operators give f=2. + x = np.linspace(0, 1, 1001) + epsilon = -.125 + a = 1 + epsilon + b, c = epsilon * (4*x - 2), -8 * epsilon + self.assertGreater(a, 0) + self.assertGreaterEqual(c, 0) + np.testing.assert_allclose(2*a + b*(1 - 2*x) + c*x*(1-x), 2, + rtol=0, atol=1e-15) + certificate = structural_certificate() + self.assertEqual([r["exact_nullity"] for r in certificate["certificates"][:3]], [2, 4, 6]) + + +class NumericalAuditTests(unittest.TestCase): + def test_unpinned_archive_is_rejected_before_computation(self): + with tempfile.TemporaryDirectory() as directory: + archive = Path(directory) / "invalid.zip" + archive.write_bytes(b"unverified input") + with self.assertRaisesRegex(ValueError, "Archive SHA-256"): + load_inputs(archive, {"archive_sha256": "0" * 64}) + + def test_optional_extracted_copy_must_match_pinned_member_bytes(self): + with tempfile.TemporaryDirectory() as directory: + root = Path(directory) + archive = root / "sample.zip" + with zipfile.ZipFile(archive, "w") as source: + source.writestr("run_example.json", b'{"example":true}\n') + extracted = root / "runs" + extracted.mkdir() + (extracted / "run_example.json").write_bytes(b'{"example":true}\n') + with patch("scripts.audit_identifiability.load_and_validate", return_value=[{"example": True}]): + runs, hashes = load_inputs(archive, {}, extracted) + self.assertEqual(runs, [("run_example.json", {"example": True})]) + self.assertEqual(len(hashes["run_example.json"]), 64) + (extracted / "run_example.json").write_bytes(b'{"example":false}\n') + with self.assertRaisesRegex(ValueError, "names/bytes"): + load_inputs(archive, {}, extracted) + + def test_joint_rank_exposes_dependency_missed_by_separate_blocks(self): + full_rank_block = np.array([[1., 0.], [0., 1.], [1., 1.]]) + joint = np.hstack([full_rank_block] * 3) + self.assertEqual(np.linalg.matrix_rank(full_rank_block), 2) + summary = joint_diagnostics(joint, np.ones(3), np.zeros(6)) + self.assertEqual(summary["raw"]["rank_at_numpy_default_tolerance"], 2) + self.assertEqual(len(summary["raw"]["singular_values"]), 6) + + def test_column_normalization_does_not_claim_exact_structural_rank(self): + result = spectra(np.diag([1., 1e-11])) + self.assertEqual(result["rank_at_numpy_default_tolerance"], 2) + self.assertEqual(result["rank_at_relative_tolerance"]["1e-08"], 1) + + def test_reconstructed_row_matches_analytic_forcing_integral(self): + basis = LegendreBasis(2) + coefficient = lambda values: CoefficientFunction(basis, values) + pde = EllipticPDE(a=coefficient([1, 0]), b=coefficient([0, 0]), + c=coefficient([0, 0]), f=coefficient([2, 0])) + solution = solve_elliptic(pde, n_grid=101) + matrix, source, rhs = design_row(solution, basis, "x*(1-x)") + self.assertAlmostEqual(rhs, 1/3, places=13) + self.assertAlmostEqual(source[0], 1/6, places=13) + # The expression parser differentiates numerically at step 1e-5. + self.assertAlmostEqual(matrix[0], rhs, delta=1e-10) + + +if __name__ == "__main__": + unittest.main() diff --git a/test_paper_analysis.py b/test_paper_analysis.py new file mode 100644 index 0000000..1eda4eb --- /dev/null +++ b/test_paper_analysis.py @@ -0,0 +1,61 @@ +"""Offline checks for the paper's archive validation and calibration analysis.""" + +from pathlib import Path +import tempfile +import unittest + +from scripts.reproduce_paper_figures import ( + calibration_summary, causal_confidence_pairs, load_and_validate, +) + + +class TestPaperCalibration(unittest.TestCase): + def test_confidence_boundaries_are_counted_once(self): + bins = calibration_summary([(0.0, 1), (0.1, 0), (0.5, 1), (1.0, 0)]) + self.assertEqual(sum(row['n'] for row in bins), 4) + self.assertEqual([row['n'] for row in bins], [1, 1, 0, 0, 0, 1, 0, 0, 0, 1]) + self.assertEqual(bins[-1]['mean_confidence'], 1.0) + self.assertTrue(bins[-1]['upper_inclusive']) + + def test_reports_use_only_solve_outputs_available_on_an_earlier_turn(self): + report = {'query_count': 8, 'a': 0.8, 'b': 0.7, 'c': 0.6, 'f': 0.9} + text = 'Confidence: a=80%, b=70%, c=60%, f=90%' + run = { + 'metacognitive': {'confidence_reports': [report.copy() for _ in range(4)]}, + 'turns': [{'turn': turn, 'role': 'assistant', 'content': text} for turn in (1, 2, 3, 5)], + 'solves': [ + {'turn': 2, 'solve_number': 1, 'queries_at_solve': 8, + 'coeff_errors': {'a': 0.2, 'b': 0.05, 'c': 0.01}}, + {'turn': 4, 'solve_number': 2, 'queries_at_solve': 8, + 'coeff_errors': {'a': 0.05, 'b': 0.05, 'c': 0.01}}, + ], + } + pairs, support = causal_confidence_pairs(run) + self.assertEqual(support['reports_excluded_no_preceding_solve'], 2) + self.assertEqual(support['reports_paired'], 2) + self.assertEqual(pairs, [(0.8, 0), (0.7, 1), (0.6, 1), + (0.8, 1), (0.7, 1), (0.6, 1)]) + + def test_archive_report_values_must_match_transcript(self): + run = { + 'metacognitive': {'confidence_reports': [ + {'query_count': 8, 'a': 0.8, 'b': 0.7, 'c': 0.6, 'f': 0.9}]}, + 'turns': [{'turn': 1, 'role': 'assistant', + 'content': 'Confidence: a=99%, b=70%, c=60%, f=90%'}], + 'solves': [], + } + with self.assertRaisesRegex(ValueError, 'disagree with the transcript'): + causal_confidence_pairs(run) + + +class TestPinnedArchive(unittest.TestCase): + def test_wrong_archive_fails_before_any_zip_member_is_read(self): + with tempfile.TemporaryDirectory() as directory: + archive = Path(directory) / 'untrusted.zip' + archive.write_bytes(b'not the pinned archive') + with self.assertRaisesRegex(ValueError, 'SHA-256'): + load_and_validate(archive, {'archive_sha256': '0' * 64}) + + +if __name__ == '__main__': + unittest.main() diff --git a/test_sine_generalization.py b/test_sine_generalization.py new file mode 100644 index 0000000..b497105 --- /dev/null +++ b/test_sine_generalization.py @@ -0,0 +1,45 @@ +"""Inclusion and fixed-plan checks for subsequent fresh-seed validation.""" + +import unittest + +from scripts.validate_sine_generalization import execute_plan, summarize, validation_plan + + +class TestFreshSineValidation(unittest.TestCase): + def test_plan_is_fixed_complete_and_has_no_archived_seed(self): + plan = validation_plan() + self.assertEqual(plan, validation_plan()) + self.assertEqual(len(plan), 300) + self.assertEqual(len({(row["difficulty"], row["seed"]) for row in plan}), 300) + for difficulty in ("easy", "medium", "hard"): + self.assertEqual([row["seed"] for row in plan if row["difficulty"] == difficulty], + list(range(1001, 1101))) + + def test_zero_source_and_execution_failure_both_remain_in_primary_denominator(self): + plan = [{"difficulty": "easy", "seed": seed} for seed in (1001, 1002, 1003)] + called = [] + def runner(task): + called.append(task["seed"]) + if task["seed"] == 1002: + raise RuntimeError("Synthetic solver failure") + error = 2.0 if task["seed"] == 1001 else .0001 + checkpoint = {"coefficient_errors": {"total": error}} + return {**task, "status": "ok", "stage": "complete", "error": None, + "zero_source": task["seed"] == 1001, "zero_solution": task["seed"] == 1001, + "minimum_queries": checkpoint, "full_budget": checkpoint} + outcomes = execute_plan(plan, runner=runner) + self.assertEqual(called, [1001, 1002, 1003]) + self.assertEqual(len(outcomes), 3) + cell, = summarize(outcomes) + self.assertEqual(cell["n_planned_tasks"], 3) + self.assertEqual(cell["status_counts"], {"ok": 2, "error": 1}) + self.assertEqual(cell["zero_source_tasks"], 1) + for stage in ("minimum_queries", "full_budget"): + self.assertEqual(cell[stage]["successes_e_lt_0.01"], 1) + self.assertEqual(cell[stage]["denominator_all_planned_tasks"], 3) + self.assertEqual(cell[stage]["failure_seeds_including_errors"], [1001, 1002]) + self.assertEqual(cell[stage]["zero_source_subset_descriptive"], {"n": 1, "successes": 0}) + + +if __name__ == "__main__": + unittest.main()