From 2e0626d47b0a04a864fde9f82ef4f6f2098e7102 Mon Sep 17 00:00:00 2001 From: Hydrodynamical Date: Tue, 20 Oct 2026 20:40:11 -0400 Subject: [PATCH 1/4] Revise PDE benchmark paper with audited Kaggle figures --- README.md | 45 +- images/paper_actions.pdf | Bin 0 -> 19482 bytes images/paper_calibration.pdf | Bin 0 -> 52539 bytes images/paper_diversity.pdf | Bin 0 -> 29338 bytes images/paper_improvement.pdf | Bin 0 -> 31445 bytes images/paper_success.pdf | Bin 0 -> 19947 bytes paper.pdf | Bin 265182 -> 539137 bytes paper.tex | 1323 +++----- paper_artifacts/README.md | 102 + paper_artifacts/data_manifest.json | 44 + paper_artifacts/legacy_pilot_tables.tex | 54 + paper_artifacts/summary.json | 3811 +++++++++++++++++++++++ references.bib | 116 +- scripts/reproduce_paper_figures.py | 381 +++ test_paper_analysis.py | 61 + 15 files changed, 5068 insertions(+), 869 deletions(-) create mode 100644 images/paper_actions.pdf create mode 100644 images/paper_calibration.pdf create mode 100644 images/paper_diversity.pdf create mode 100644 images/paper_improvement.pdf create mode 100644 images/paper_success.pdf create mode 100644 paper_artifacts/README.md create mode 100644 paper_artifacts/data_manifest.json create mode 100644 paper_artifacts/legacy_pilot_tables.tex create mode 100644 paper_artifacts/summary.json create mode 100644 scripts/reproduce_paper_figures.py create mode 100644 test_paper_analysis.py diff --git a/README.md b/README.md index 6fd357d..fdf973c 100644 --- a/README.md +++ b/README.md @@ -143,27 +143,29 @@ 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 earlier three-model, two-seed pilot tables are retained separately in the +paper's appendix. 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 +179,5 @@ 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/images/paper_actions.pdf b/images/paper_actions.pdf new file mode 100644 index 0000000000000000000000000000000000000000..79fa38c63d7a2e33ced4cfe841e810834016f8ee GIT binary 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-\usepackage{enumitem} +\usepackage{microtype,graphicx,booktabs,tabularx} +\usepackage{xurl} \usepackage{hyperref} -\hypersetup{colorlinks=true,linkcolor=blue!55!black,citecolor=blue!55!black,urlcolor=blue!55!black} -\captionsetup{font=small,labelfont=bf} -\setlist{nosep} -\setlength{\emergencystretch}{2em} +\usepackage[preprint]{icml2026} +\hypersetup{pdfsubject={Research preprint on scientific-agent evaluation}} +\usepackage{amsmath,amssymb,amsthm} +\usepackage{placeins} \newcommand{\dd}{\,\mathrm{d}} \newcommand{\R}{\mathbb{R}} -\newtheorem{remark}{Remark} - -\title{\textbf{Can Language Models Learn a PDE?}\\[5pt] -\large An Interactive Benchmark for Experimental Design\\ -in Weak-Form Coefficient Identification} -\author{Joseph K. Miller\\ -\small Department of Mathematics, Massachusetts Institute of Technology} -\date{\small Experiments: April 2026 \qquad Revised: October 2026} +\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 introduce an interactive benchmark in which language models choose measurements -to identify the coefficients of a one-dimensional elliptic equation. The environment -provides access to a numerical solution, budgeted weak-form queries, and free -diagnostic tools. Numerical linear algebra is delegated to a solver; the model -selects test functions, interprets feedback, and decides when to submit. -We reanalyze a public archive of 180 scored runs from six models, three difficulty -levels, and ten shared problem seeds per configuration. Recovery rates vary -substantially across models and instances, while action counts, within-session -improvement, query-family diversity, and confidence reports reveal differences -beyond the final error. We regenerate the figures from the archived logs and -correct a stale success count in a previously checked-in figure. -The diagnostics describe observable behavior rather than validated cognitive -faculties. We also distinguish numerical conditioning from structural -identifiability: a single solution need not uniquely determine the operator, even -within the benchmark's finite-dimensional coefficient class. The archive and -reproduction script support verification of the reported aggregates; incomplete -attempt histories and provider settings limit broader comparisons. +Interactive scientific benchmarks aim to measure how language models choose +experiments, but recovery scores also depend on the observation model and +numerical implementation. +We audit a weak-form inverse-problem benchmark using 180 archived scored runs +from six models on 30 shared elliptic-equation instances, paired statistical +comparisons, and new numerical controls kept distinct from historical outcomes. +A fixed sine-query design recovers all 30 reconstructed tasks at the strict +error threshold using the protocol's minimum solve budget, so these tasks do +not establish a benefit from adaptive language-model measurement selection. +Free consistency checks supply additional observations, while absolute rounding +and small singular values make recovery sensitive to numerical representation. +An exact nonuniqueness result further shows why full floating-point matrix rank +cannot establish structural identifiability. +The audit provides reproducible diagnostics for separating observable agent +behavior from claims about experimental-design ability. \end{abstract} \section{Introduction} -An experiment can be difficult because the available measurements are poorly -chosen, even when the computations following those measurements are routine. -This distinction motivates a benchmark in which a language model directs an -inverse-problem investigation. It can request measurements, inspect coefficient -estimates, test consistency, and revise its choices. The computational tools -perform integration and least-squares recovery, leaving measurement selection and -the use of feedback as the model's main responsibilities. - -The task builds on the weak formulation of an elliptic boundary value -problem \citep{evans2010pde}. It also connects to active data selection -\citep{mackay1992information}: the informative value of a measurement depends on -the measurements already collected. A small residual does not necessarily imply -accurate coefficients, and visibly different test functions need not add -independent constraints. These features provide concrete opportunities to study -how models respond to numerical evidence. - -The benchmark was introduced in a Kaggle write-up \citep{kaggle2026pde}, with -behavioral interpretations motivated by the cognitive framework of -\citet{burnell2026cognitive}. Here we organize the task around three contributions: -an instrumented interface for tool-assisted coefficient identification; -a reproducible descriptive analysis of 180 archived scored runs; and an account -of the numerical and interpretive limits of that analysis. The question in the -title concerns identification during a session, not training model weights. -The behavioral measurements are hypotheses about useful aspects of scientific -problem solving, not independent validation of general cognitive abilities. - -\section{Task and numerical interface} +A scientific agent must choose observations that distinguish plausible +explanations. A benchmark can make this process inspectable by placing the agent +in a known inverse problem: the hidden parameters are available to the evaluator, +the measurement actions have mathematical definitions, and the tools expose the +information used by the agent. However, a high recovery score can also reflect +properties of the numerical environment. Before interpreting the score as +experimental-design ability, we must understand which parameters the observations +identify and how numerical error affects their recovery. + +We investigate a benchmark in which a language model identifies the coefficients +of an elliptic differential equation by selecting weak-form test functions. +The solver handles quadrature and least squares; the agent chooses measurements, +inspects estimates, and decides when to stop. The setup is deliberately small +and instrumented. Unlike an open-ended discovery task, it permits direct analysis +of the joint measurement matrix and an exact specification of the correctness +event. ``Learn'' in the title means identification within an interaction, rather +than updating model weights or discovering an unknown equation family. + +The benchmark first appeared in a Kaggle write-up \citep{kaggle2026pde}. +This paper reexamines its public six-model archive rather than treating the +original behavioral labels as established cognitive measurements. +We ask three questions: (i) how do the archived policies differ on shared tasks; +(ii) what recovery is possible with simple numerical query policies; and +(iii) which aspects of the protocol and discretization constrain the interpretation +of those comparisons? + +Our contributions are an auditable formulation of the measurement task; a +reproducible analysis of 180 scored records with comparisons paired by problem +instance; and new controls that expose the distinction between structural +nonuniqueness, numerical conditioning, and measured agent behavior. +An exact polynomial-family result gives a concrete failure of unconditional +identifiability. Reconstructed measurements then quantify numerical and protocol +effects on the actual archived tasks. The fixed sine control succeeds on every reconstructed task, so the archived +results do not establish a benefit from adaptive language-model measurement +selection. The broader contribution is a case study of how omitted numerical +controls can change a scientific-agent benchmark's interpretation. + +\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 and polynomial basis are +known, and the agent chooses test functions while a supplied solver estimates +coefficients. 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{Hidden equation and measurement system} +\subsection{Hidden Equation and Linear Recovery} The hidden boundary value problem is \begin{equation} - -(a(x)u'(x))'+b(x)u'(x)+c(x)u(x)=f(x),\qquad x\in[0,1], - \quad u(0)=u(1)=0. - \label{eq:pde} + \begin{gathered}-(au')'+bu'+cu=f,\quad x\in[0,1],\\ u(0)=u(1)=0.\end{gathered} + \label{eq:pde} \end{equation} -Each coefficient \(q\in\{a,b,c,f\}\) is represented by -\(q(x)=\sum_{j=0}^{N-1}\theta_{q,j}P_j(x)\), where \(P_j\) denotes the -ordinary shifted Legendre polynomial on \([0,1]\). These polynomials are -orthogonal, but are not normalized to have unit \(L^2\) norm. -The environment samples the coefficients and computes a numerical solution. -The model accesses \(u\) and \(u'\) through an evaluation tool. - -For a test function \(\varphi_k\) vanishing at both endpoints, integration by -parts gives +Each field \(q\in\{a,b,c,f\}\) has the form +\(q(x)=\sum_{j=0}^{N-1}\theta_{q,j}P_j(x)\), using ordinary shifted Legendre +polynomials on \([0,1]\). The polynomials are orthogonal but not orthonormal. +The generator samples coefficients and computes \(u\) numerically. +For an admissible test function \(\varphi_k\) vanishing at the endpoints, \begin{equation} - \int_0^1 au'\varphi_k'\dd x+ - \int_0^1 bu'\varphi_k\dd x+ - \int_0^1 cu\varphi_k\dd x - =r_k,\qquad - r_k=\int_0^1 f\varphi_k\dd x. + \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} -Define the basis-weighted matrices +All integrals are over \([0,1]\). Define \begin{align} - D_{kj}&=\int_0^1P_j u'\varphi_k'\dd x,& - B_{kj}&=\int_0^1P_j u'\varphi_k\dd x,\nonumber\\ - C_{kj}&=\int_0^1P_j u\varphi_k\dd x,& - F_{kj}&=\int_0^1P_j\varphi_k\dd x. + 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, the solver separately minimizes -\(\|A\theta-r\|_2\), with \(A=[D\mid B\mid C]\in\R^{K\times3N}\), -and \(\|F\theta_f-r\|_2\). -The model chooses measurements; it does not supply its own least-squares -implementation. The interface requires at least \(4N\) stored queries before -solving. This is a protocol rule, not a guarantee of rank or uniqueness. - -\subsection{Actions and budgets} +The tool separately minimizes \(\|A\theta-r\|_2\) for the operator, where +\(A=[D\mid B\mid C]\in\R^{K\times3N}\), and \(\|F\theta_f-r\|_2\) for +the source. Although the prompt describes \(4N\) unknown coefficients, the +returned source moments already provide the right-hand side of the operator +system. The requirement of \(K\ge4N\) before solving is a protocol rule, +rather than a rank guarantee. + +\subsection{Actions, Information, and Budgets} + +Table~\ref{tab:tools} summarizes the interface. 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 endpoint and validity checks. It does not impose a common +amplitude normalization. \begin{table}[t] -\centering -\caption{Actions available in a session. Only stored queries consume the query -budget; multiple commands may share a response within a finite turn limit. A free consistency check supplies -a measurement but does not append a row to the solve system.} +\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} -\small -\begin{tabularx}{\linewidth}{@{}lX@{}} +\centering\small +\begin{tabularx}{\columnwidth}{@{}lX@{}} \toprule -Action & Function\\ +Action & Returned information or effect\\ \midrule -\texttt{eval\_solution} & Evaluate \(u\) and its numerical derivative at chosen points.\\ -\texttt{QUERY} & Return \(r_k\) and store the corresponding rows of \(A\) and \(F\).\\ -\texttt{term\_integrals} & Inspect the three unweighted terms \(\int u'\varphi'\), - \(\int u'\varphi\), and \(\int u\varphi\) before querying.\\ -\texttt{solve} & Recover coefficient vectors and report changes from the previous solve.\\ -\texttt{check} & Compare a candidate weak-form measurement with the current estimate.\\ -\texttt{PREDICT} & Submit the coefficient vectors from the most recent solve.\\ +\texttt{eval\_solution} & Values of \(u,u'\) at chosen points.\\ +\texttt{QUERY} & \(r_k\); appends rows to \(A,F\).\\ +\texttt{term\_integrals} & The three unweighted integrals with \(P_0=1\).\\ +\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{tabularx} \end{table} -The protocol permits one stored query per response and prevents submission in the -same response as a query. Tool calls and reasoning share a finite turn budget. -The common instructions encourage repeated solves and ask for coefficient-wise -confidence reports after solving. Consequently, iteration counts reflect behavior -under an explicit instruction to iterate. +The protocol permits at most one stored query per response; 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] -\centering -\caption{Settings recorded in the recovered six-model archive. -Each row contains ten problem seeds per model.} +\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} -\begin{tabular}{@{}lrrrr@{}} +\centering\small +\begin{tabular}{@{}lrrr@{}} \toprule -Difficulty & \(N\) & Total coefficients & Queries & Turns\\ + & Easy & Medium & Hard\\ \midrule -Easy & 2 & 8 & 12 & 36\\ -Medium & 3 & 12 & 18 & 54\\ -Hard & 4 & 16 & 24 & 72\\ +\(N\) & 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{table} -All archived runs record the \texttt{standard} prompt condition. The write-up calls -this the enhanced condition: it adds an initial hint about unequal coefficient -identifiability. Both prompt variants retain the same equation, tools, and -protocol. The recovered archive does not support a baseline-versus-standard -ablation. Its logged basis sizes are \(2,3,4\), correcting the write-up's -\(3,5,8\) description. - -\section{Recovery, conditioning, and identifiability} +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{Error and Identifiability} \label{sec:identifiability} -For each coefficient block, the reported error is the absolute maximum -coefficient deviation, +For each field, the scoring event uses maximum absolute coefficient deviation: \begin{equation} - E_q=\max_{0\leq j

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ksP?%{$oyq)v46<_?Fa<@01$)*AtNIv2Nn>}G13M97nAV-zW@LL diff --git a/paper.tex b/paper.tex index ad72af4..30e45a1 100644 --- a/paper.tex +++ b/paper.tex @@ -7,6 +7,7 @@ \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}} @@ -28,60 +29,62 @@ \printAffiliationsAndNotice{} \begin{abstract} -Interactive scientific benchmarks aim to measure how language models choose -experiments, but recovery scores also depend on the observation model and -numerical implementation. -We audit a weak-form inverse-problem benchmark using 180 archived scored runs -from six models on 30 shared elliptic-equation instances, paired statistical -comparisons, and new numerical controls kept distinct from historical outcomes. -A fixed sine-query design recovers all 30 reconstructed tasks at the strict -error threshold using the protocol's minimum solve budget. On 300 fresh seeds, -minimum-budget success is 92\%, 91\%, and 98\% by difficulty, with ten -unidentifiable zero-forcing cases retained. -Free consistency checks supply additional observations, while absolute rounding -and small singular values make recovery sensitive to numerical representation. -An exact nonuniqueness result further shows why full floating-point matrix rank -cannot establish structural identifiability. +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} A scientific agent must choose observations that distinguish plausible -explanations. A benchmark can make this process inspectable by placing the agent -in a known inverse problem: the hidden parameters are available to the evaluator, -the measurement actions have mathematical definitions, and the tools expose the -information used by the agent. However, a high recovery score can also reflect -properties of the numerical environment. Before interpreting the score as -experimental-design ability, we must understand which parameters the observations -identify and how numerical error affects their recovery. - -We investigate a benchmark in which a language model identifies the coefficients -of an elliptic differential equation by selecting weak-form test functions. -The solver handles quadrature and least squares; the agent chooses measurements, -inspects estimates, and decides when to stop. The setup is deliberately small -and instrumented. Unlike an open-ended discovery task, it permits direct analysis -of the joint measurement matrix and an exact specification of the correctness -event. ``Learn'' in the title means identification within an interaction, rather -than updating model weights or discovering an unknown equation family. +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}. -This paper reexamines its public six-model archive rather than treating the -original behavioral labels as established cognitive measurements. -We ask three questions: (i) how do the archived policies differ on shared tasks; -(ii) what recovery is possible with simple numerical query policies; and -(iii) which aspects of the protocol and discretization constrain the interpretation -of those comparisons? - -Our contributions are an auditable formulation of the measurement task; a -reproducible analysis of 180 scored records with comparisons paired by problem -instance; and new controls that expose the distinction between structural -nonuniqueness, numerical conditioning, and measured agent behavior. -An exact polynomial-family result gives a concrete failure of unconditional -identifiability. Reconstructed measurements then quantify numerical and protocol -effects on the actual archived tasks. The fixed sine control succeeds on all 30 archive-derived tasks, so the archived -results do not establish a benefit from adaptive language-model measurement -selection. The broader contribution is a case study of how omitted numerical -controls can change a scientific-agent benchmark's interpretation. +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} @@ -100,9 +103,8 @@ \section{Related Work} 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 and polynomial basis are -known, and the agent chooses test functions while a supplied solver estimates -coefficients. This is a measurement-selection problem within a specified model, +\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 @@ -124,43 +126,87 @@ \section{Related Work} \section{Benchmark and Observation Model} \label{sec:task} -\subsection{Hidden Equation and Linear Recovery} +\subsection{Forward Problem, Observations, and Recovery} -The hidden boundary value problem is +The hidden equation is \begin{equation} - \begin{gathered}-(au')'+bu'+cu=f,\quad x\in[0,1],\\ u(0)=u(1)=0.\end{gathered} + \begin{gathered}-(au')'+bu'+cu=f,\quad x\in(0,1),\\ + u(0)=u(1)=0.\end{gathered} \label{eq:pde} \end{equation} -Each field \(q\in\{a,b,c,f\}\) has the form -\(q(x)=\sum_{j=0}^{N-1}\theta_{q,j}P_j(x)\), using ordinary shifted Legendre -polynomials on \([0,1]\). The polynomials are orthogonal but not orthonormal. -The generator samples coefficients and computes \(u\) numerically. -For an admissible test function \(\varphi_k\) vanishing at the endpoints, +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} + \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 \([0,1]\). Define +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} -The tool separately minimizes \(\|A\theta-r\|_2\) for the operator, where -\(A=[D\mid B\mid C]\in\R^{K\times3N}\), and \(\|F\theta_f-r\|_2\) for -the source. Although the prompt describes \(4N\) unknown coefficients, the -returned source moments already provide the right-hand side of the operator -system. The requirement of \(K\ge4N\) before solving is a protocol rule, -rather than a rank guarantee. - +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 model can evaluate the +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 endpoint and validity checks. It does not impose a common -amplitude normalization. +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. @@ -173,7 +219,7 @@ \subsection{Actions, Information, and Budgets} \midrule \texttt{eval\_solution} & Values of \(u,u'\) at chosen points.\\ \texttt{QUERY} & \(r_k\); appends rows to \(A,F\).\\ -\texttt{term\_integrals} & The three unweighted integrals with \(P_0=1\).\\ +\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.\\ @@ -181,7 +227,8 @@ \subsection{Actions, Information, and Budgets} \end{tabularx} \end{table} -The protocol permits at most one stored query per response; several free +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. @@ -210,6 +257,12 @@ \subsection{Actions, Information, and Budgets} \end{tabular} \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 @@ -222,25 +275,53 @@ \subsection{Actions, Information, and Budgets} also shows that the three difficulty labels are bundled settings; differences cannot be attributed solely to dimension. -\subsection{Error and Identifiability} +\subsection{Coefficient Error, Uniqueness, and Stability} \label{sec:identifiability} -For each field, the scoring event uses maximum absolute coefficient deviation: +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} -Our primary threshold is \(E<0.01\); \(E<0.1\) is a sensitivity analysis. -These are not function-error thresholds on held-out points. The separately -reported function error is bounded by \(NE\), since \(|P_j(x)|\le1\), but -equal numerical thresholds are not equivalent. +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. -Full column rank of the joint matrix \(A\), rather than each coefficient block, -is necessary and sufficient for unique unconstrained least-squares operator -coefficients. Small singular values can amplify discretization and rounding -errors. Floating-point rank is a property of the reconstructed finite matrix; -it is not a uniqueness theorem for the underlying observation operator. +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} @@ -251,11 +332,11 @@ \subsection{Error and Identifiability} measurements unchanged at fixed \(u,f\). \end{proposition} -A proof and explicit elliptic examples appear in Appendix~\ref{app:proof}. -This is a counterexample within the permitted class, rather than a claim that -the 30 sampled instances belong to this quadratic-solution family. It rules out -unconditional identifiability. Separate audits are needed to characterize -recoverability on the sampled tasks. There is also a positive-probability +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. @@ -270,9 +351,15 @@ \section{Evaluation Design} \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 coefficient truths are shared across +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. @@ -290,13 +377,15 @@ \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 coefficient truths, and evaluate +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 chooses -\(\varphi_k=\sin(k\pi x)\), \(k=1,\ldots,B\), where \(B=6N\) is the -stored-query budget. Boundary Legendre chooses \(x(1-x)P_{k-1}(x)\), using -factored expressions for numerical stability. Random mixed draws uniformly +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 @@ -315,8 +404,6 @@ \subsection{Numerical Controls and Replay} 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. -\subsection{Statistical Unit and Behavioral Diagnostics} - 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- @@ -333,6 +420,8 @@ \subsection{Statistical Unit and Behavioral Diagnostics} 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 @@ -343,9 +432,11 @@ \subsection{Statistical Unit and Behavioral Diagnostics} 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. -For confidence, a report is paired only with a solve returned on a strictly -earlier transcript turn. The event is \(E_q<0.1\), separately for -\(q\in\{a,b,c\}\). All reports and coefficient blocks within a run are +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 @@ -360,9 +451,8 @@ \subsection{Archived Recovery Differs, with Limited Resolution} 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. The regenerated figure corrects -the write-up asset's hard GPT-5.4 mini count from \(0/10\) to \(2/10\), -which agrees with both individual logs and the archived sweep summary. +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 @@ -456,7 +546,7 @@ \subsection{Fresh Seeds Expose the Limits of the Archive Result} \subsection{What Numerical Reconstruction Can Establish} \label{sec:numerics} -All 30 coefficient truths regenerate exactly, and the reconstructed reference +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 @@ -507,15 +597,11 @@ \subsection{Information Use and Confidence} moment. Thus stored-query counts alone do not equate the information acquired by two policies. This behavior is permitted by the protocol. -The confidence analysis finds 733 reports, of which 707 follow a solve and -26 are excluded. The retained reports yield 2,121 coefficient observations. -Causal matching changes the solve assigned to 71 reports relative to the legacy -query-count rule; 827 observations lie on interior confidence-bin boundaries -that the older plots could count twice. Corrected calibration curves and the -Kaggle action, improvement, and diversity figures appear in -Appendix~\ref{app:behavior}. Their interpretation remains descriptive: repeated -reports are dependent, prompt instructions affect action use, and expression -families are not independent measurement directions. +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} @@ -523,16 +609,15 @@ \section{Discussion and Limitations} 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. Changing a mathematical representation without adding -information should not silently change the claimed ability being measured. +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. This spatially one-dimensional equation is mathematically an ODE -boundary-value problem; the task does not test multidimensional PDE structure. We cannot reconstruct missing attempt histories or exact provider +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 @@ -555,7 +640,7 @@ \section{Conclusion} 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 numerical failures. +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. @@ -571,11 +656,10 @@ \section*{Impact Statement} \section*{Reproducibility} -Code, analysis scripts, pinned archive provenance, and derived summaries are -available at \url{https://github.com/Hydrodynamical/PDE_Learning}. -The new controls and analyses in this revision use no provider calls. Appendix~\ref{app:repro} -gives data and numerical details. The PDF is a public preprint using the ICML -2026 format; it is not an anonymized conference submission. +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} @@ -653,8 +737,9 @@ \section{Data, Implementation, and Statistical Details} 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}. These corrections are recorded in the derived -summaries. The earlier draft's distinct three-model, two-seed pilot lacks source +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. @@ -683,7 +768,10 @@ \section{Behavioral Figures and Their Interpretation} 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 corrected 707 reports contribute +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.

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zDL+n9F0NKC?Nly*pa_uGMx4rhrJn?@&Zhuw2_-zl~s5=yh zfBlfPJ)lth8xg(i#tG%lbURQEMH1+`-`Y|rgFes~`av=Dm%zIjcHRI9yf@XVff9H> z-Ks$_SOOo^SUUuUYP-kUVNfE0-E|uI-FuaNIst)iH`vh%7!H+C1tVZ2jDiI)8pgm_ zsD^Pc9wxvTh$<;EpJ*iTSB|8TQx^QU!Jk5Q9=h#Sk)w< z`=_j$E8)@uR?U;}%bTs5FX2OlI;*Wql`HzsAtp0ZtE(EH@MN{EO51kjhNdKZQ$J$( z)<4Rh#mZfsp76gVdWD~$gnm|HbtGfrpHUKDQIdD@jA#m^LK-ju(au2lqZvT%qg}yv zc^h43I+_i{J4(ExxsV6>K=h&o5|g$^pO$;a`yU+-Yb2NWItLvLHa9mgG72RnMNdWw DO3>LX diff --git a/paper.tex b/paper.tex index 9480a29..ad72af4 100644 --- a/paper.tex +++ b/paper.tex @@ -35,14 +35,13 @@ from six models on 30 shared elliptic-equation instances, paired statistical comparisons, and new numerical controls kept distinct from historical outcomes. A fixed sine-query design recovers all 30 reconstructed tasks at the strict -error threshold using the protocol's minimum solve budget, so these tasks do -not establish a benefit from adaptive language-model measurement selection. +error threshold using the protocol's minimum solve budget. On 300 fresh seeds, +minimum-budget success is 92\%, 91\%, and 98\% by difficulty, with ten +unidentifiable zero-forcing cases retained. Free consistency checks supply additional observations, while absolute rounding and small singular values make recovery sensitive to numerical representation. An exact nonuniqueness result further shows why full floating-point matrix rank cannot establish structural identifiability. -The audit provides reproducible diagnostics for separating observable agent -behavior from claims about experimental-design ability. \end{abstract} \section{Introduction} @@ -79,7 +78,7 @@ \section{Introduction} nonuniqueness, numerical conditioning, and measured agent behavior. An exact polynomial-family result gives a concrete failure of unconditional identifiability. Reconstructed measurements then quantify numerical and protocol -effects on the actual archived tasks. The fixed sine control succeeds on every reconstructed task, so the archived +effects on the actual archived tasks. The fixed sine control succeeds on all 30 archive-derived tasks, so the archived results do not establish a benefit from adaptive language-model measurement selection. The broader contribution is a case study of how omitted numerical controls can change a scientific-agent benchmark's interpretation. @@ -256,7 +255,15 @@ \subsection{Error and Identifiability} This is a counterexample within the permitted class, rather than a claim that the 30 sampled instances belong to this quadratic-solution family. It rules out unconditional identifiability. Separate audits are needed to characterize -recoverability on the sampled tasks. +recoverability on the sampled tasks. 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} @@ -317,6 +324,15 @@ \subsection{Statistical Unit and Behavioral Diagnostics} 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. + 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 @@ -415,6 +431,28 @@ \subsection{Simple Nonadaptive Queries Suffice on These Tasks} 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} @@ -490,7 +528,7 @@ \section{Discussion and Limitations} Third, every truth-dependent observation must enter the information accounting, including diagnostics exposed as free tools. -Our evidence is deliberately bounded. The archive contains 30 tasks and one +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. This spatially one-dimensional equation is mathematically an ODE @@ -516,8 +554,9 @@ \section{Conclusion} 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 every reconstructed task, making numerical -baselines essential to the interpretation of model scores. The released +A fixed sine design suffices on all 30 archive-derived tasks; fresh seeds +reveal unidentifiable zero-excitation cases and additional numerical 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. diff --git a/paper_artifacts/README.md b/paper_artifacts/README.md index e6d321b..556fe03 100644 --- a/paper_artifacts/README.md +++ b/paper_artifacts/README.md @@ -111,12 +111,13 @@ setting up the pinned Python 3.12 environment and downloading the pinned ZIP: 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 control experiment takes -roughly one to two minutes on the development machine; timings depend on hardware. +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 @@ -171,3 +172,27 @@ 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/research_figure_sources.json b/paper_artifacts/research_figure_sources.json index a645929..fedee27 100644 --- a/paper_artifacts/research_figure_sources.json +++ b/paper_artifacts/research_figure_sources.json @@ -2,5 +2,6 @@ "control_baselines.json.gz": "1f766ee10d8aaa7e07fc8f255d6ee4a4cef76e3751dcc63c426a4790624d7db5", "identifiability_audit.json": "17cf7ba238b0e7d792e7a25dfb753411c1a44121a02987013ed277c83365770b", "statistical_analysis.json": "b8a5591c8e7e2f67fbfbf050829f3b8262f7983f80d9dc6a43fef5131441a146", - "plot_script_sha256": "527f150fb9025ffcba11dd673f610957dbec790786e5ec0424879d1a15b70ee8" + "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. 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. 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"smallest_singular_value": 2.001230878059774e-05, + "largest_singular_value": 2.549427667249901 + } + } + }, + "full_budget": { + "queries": 24, + "coefficient_errors": { + "a": 1.208068160370007e-05, + "b": 0.00012407296528338385, + "c": 0.00021292436175157636, + "f": 2.893241202173158e-12, + "total": 0.00034907801153190147 + }, + "displayed_8dp_coefficient_errors": { + "a": 1.208e-05, + "b": 0.00012406950351562251, + "c": 0.00021292126498528852, + "f": 2.3979387364647664e-09, + "total": 0.0003490731664396475 + }, + "coefficient_estimates": { + "a": [ + 0.48736137091294995, + 1.208068160370007e-05, + -0.006751653079739144, + -0.055551526946044275 + ], + "b": [ + -1.4495329165382322, + 0.3450474280182615, + -1.2924821129090434, + 0.7061820908986588 + ], + "c": [ + 0.04929233568623931, + -0.2799411769032337, + -0.0001720244803680611, + -0.902567799278436 + ], + "f": [ + 1.157422997601115, + -1.1619594175726888e-12, + 1.2526187982969288, + 1.128177171394091 + ] + }, + "joint_design": { + "raw": { + "rank": 12, + "n_columns": 12, + "rank_tolerance": 3.483079272865397e-14, + "condition_number": 862798.5642541355, + "smallest_singular_value": 7.5753460721057465e-06, + "largest_singular_value": 6.535997714741042 + }, + "column_normalized": { + "rank": 12, + "n_columns": 12, + "rank_tolerance": 1.3587581311910953e-14, + "condition_number": 89462.46345701921, + "smallest_singular_value": 2.8500326546911525e-05, + "largest_singular_value": 2.549709422216187 + } + } + }, + "error": null, + "forward_max_abs_u": 0.23073773355375102, + "elapsed_seconds": 1.2824940830469131 + }, + { + "difficulty": "hard", + "seed": 1100, + "status": "ok", + "stage": "complete", + "config": { + "n_basis": 4, + "sparsity": 0.1, + "scale": 1.5, + "a_min": 0.3, + "n_grid": 4001, + "n_solution_points": 21, + "domain": [ + 0, + 1 + ], + "bc_left": 0, + "bc_right": 0, + "minimum_queries": 16, + "budget": 24 + }, + "ground_truth": { + "a_coeffs": [ + 0.3465179872945448, + 0.1347748445789095, + 0.08928798843656942, + 0.0 + ], + "b_coeffs": [ + -0.7849523818488366, + 0.0, + -1.6455051817975044, + 0.03588156004406973 + ], + "c_coeffs": [ + -0.22078917351468483, + 1.2552417757157306, + 0.9742713150513727, + -1.1456603140125543 + ], + "f_coeffs": [ + 0.9693371311512613, + 1.5714031652554956, + -1.6515705417358733, + -0.0 + ] + }, + "zero_source": false, + "zero_solution": false, + "minimum_queries": { + "queries": 16, + "coefficient_errors": { + "a": 1.766725715063311e-05, + "b": 0.00016997387571016098, + "c": 0.00046415169643920784, + "f": 1.7863488466218769e-12, + "total": 0.0006517928310863508 + }, + "displayed_8dp_coefficient_errors": { + "a": 1.7667294544831957e-05, + "b": 0.00016997179750433133, + "c": 0.00046415571573055203, + "f": 4.744504344955658e-09, + "total": 0.0006517995522840603 + }, + "coefficient_estimates": { + "a": [ + 0.34650032003739417, + 0.13476678518520768, + 0.08928313951893045, + 2.017658050060828e-06 + ], + "b": [ + -0.7850148641506075, + -8.827036623449045e-06, + -1.6453352079217942, + 0.03587421216308872 + ], + "c": [ + -0.22075605017873623, + 1.2547776240192914, + 0.9742591438139896, + -1.1454897380221285 + ], + "f": [ + 0.9693371311512401, + 1.571403165253932, + -1.6515705417376596, + 7.011001031103375e-13 + ] + }, + "joint_design": { + "raw": { + "rank": 12, + "n_columns": 12, + "rank_tolerance": 3.6962292553041564e-14, + "condition_number": 13127318.730609043, + "smallest_singular_value": 7.925426851471851e-07, + "largest_singular_value": 10.403960435539828 + }, + "column_normalized": { + "rank": 12, + "n_columns": 12, + "rank_tolerance": 8.574404687775556e-15, + "condition_number": 986121.277315129, + "smallest_singular_value": 2.447447809229354e-06, + "largest_singular_value": 2.4134803597993644 + } + } + }, + "full_budget": { + "queries": 24, + "coefficient_errors": { + "a": 3.799584542801826e-05, + "b": 0.00038111104714189104, + "c": 0.0010749895272637033, + "f": 1.3522516439934407e-12, + "total": 0.0014940964211858643 + }, + "displayed_8dp_coefficient_errors": { + "a": 3.7997294544789284e-05, + "b": 0.0003811117975043299, + "c": 0.0010749857157306142, + "f": 4.744504344955658e-09, + "total": 0.0014940995522840783 + }, + "coefficient_estimates": { + "a": [ + 0.3464799914491168, + 0.13475716352940967, + 0.08927778162594152, + 4.788290835327152e-06 + ], + "b": [ + -0.7850984649728308, + -1.6159350960199717e-05, + -1.6451240707503625, + 0.03586400457267026 + ], + "c": [ + -0.2207304363628116, + 1.2541667861884669, + 0.9742519197097851, + -1.145285571045833 + ], + "f": [ + 0.9693371311513995, + 1.5714031652541434, + -1.6515705417368682, + 1.1854169903348926e-12 + ] + }, + "joint_design": { + "raw": { + "rank": 12, + "n_columns": 12, + "rank_tolerance": 5.845343358803939e-14, + "condition_number": 8713661.694590427, + "smallest_singular_value": 1.2588032780804154e-06, + "largest_singular_value": 10.968785905234176 + }, + "column_normalized": { + "rank": 12, + "n_columns": 12, + "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/scripts/plot_research_audit.py b/scripts/plot_research_audit.py index 03be1c0..dfb72bf 100644 --- a/scripts/plot_research_audit.py +++ b/scripts/plot_research_audit.py @@ -173,6 +173,23 @@ def make_tables(controls, statistics, destination): 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' @@ -181,18 +198,21 @@ def main(): 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') - provenance={p.name:hashlib.sha256(p.read_bytes()).hexdigest() for p in (args.controls,args.audit,args.statistics)} + 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.') 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 index e186f35..69bf1c3 100644 --- a/scripts/verify_paper_artifacts.py +++ b/scripts/verify_paper_artifacts.py @@ -30,6 +30,12 @@ def main(): 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'] 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() From 9eb8b1d4393fbdb02af48d842de97cf5646d4952 Mon Sep 17 00:00:00 2001 From: Hydrodynamical Date: Tue, 20 Oct 2026 21:41:54 -0400 Subject: [PATCH 4/4] Clarify PDE formulation and measurement protocol for both audiences --- paper.pdf | Bin 473889 -> 488291 bytes paper.tex | 338 ++++++++++++++++++++++++++++++++++-------------------- 2 files changed, 213 insertions(+), 125 deletions(-) diff --git a/paper.pdf b/paper.pdf index cdbb9fb0c3840dc2b3c24d76df0c80c5c876f2f0..77a7681d7b8af32c504078133c0024e7b23c9825 100644 GIT binary patch delta 168494 zcmZtMLv$qyurA;o+qP}HW81cqj?Eo(Y}+nE zh$jbQ(!&*63&_9ZER+=Na~6h4hCT|3C-1)J(yCc;$ z!q?uUf!!o)Mj5?{ah+xOLwGCW`Og?Dir%Hu^2eZ)ts>YK*!1x>4eY_ao<(x_e(O#? zSHLZZQSA&X%aPkRn9P|h4o@6z&K5?<$ix7{H){nS#v{OGC9OvqDvt54%R zz;fyMe0|~qZv1@h58lM{7@KzTjft7*EY{z?9((w_a-}_qO@Bv()%|k{6<#d2sfs1~ 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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/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. 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. 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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. 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"neither_successes": 0, + "success_rate_difference_first_minus_second": 1.0, + "paired_task_bootstrap_95_interval": [ + 1.0, + 1.0 + ], + "exact_p_two_sided": 0.001953125, + "holm_adjusted_p_within_difficulty": 0.029296875 + }, + { + "difficulty": "hard", + "first_model": "gemini-3.1-pro-preview", + "second_model": "gpt-5.4-mini", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 10, + "second_successes": 2, + "first_only_successes": 8, + "second_only_successes": 0, + "both_successes": 2, + "neither_successes": 0, + "success_rate_difference_first_minus_second": 0.8, + "paired_task_bootstrap_95_interval": [ + 0.5, + 1.0 + ], + "exact_p_two_sided": 0.0078125, + "holm_adjusted_p_within_difficulty": 0.1015625 + }, + { + "difficulty": "hard", + "first_model": "claude-opus-4-6", + "second_model": "gpt-5.4", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 8, + "second_successes": 1, + "first_only_successes": 7, + "second_only_successes": 0, + "both_successes": 1, + "neither_successes": 2, + "success_rate_difference_first_minus_second": 0.7, + "paired_task_bootstrap_95_interval": [ + 0.4, + 1.0 + ], + "exact_p_two_sided": 0.015625, + "holm_adjusted_p_within_difficulty": 0.171875 + }, + { + "difficulty": "hard", + "first_model": "claude-opus-4-6", + "second_model": "gemini-3-flash-preview", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 8, + "second_successes": 7, + "first_only_successes": 3, + "second_only_successes": 2, + "both_successes": 5, + "neither_successes": 0, + "success_rate_difference_first_minus_second": 0.1, + "paired_task_bootstrap_95_interval": [ + -0.3, + 0.5 + ], + "exact_p_two_sided": 1.0, + "holm_adjusted_p_within_difficulty": 1.0 + }, + { + "difficulty": "hard", + "first_model": "claude-opus-4-6", + "second_model": "claude-sonnet-4-20250514", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 8, + "second_successes": 0, + "first_only_successes": 8, + "second_only_successes": 0, + "both_successes": 0, + "neither_successes": 2, + "success_rate_difference_first_minus_second": 0.8, + "paired_task_bootstrap_95_interval": [ + 0.5, + 1.0 + ], + "exact_p_two_sided": 0.0078125, + "holm_adjusted_p_within_difficulty": 0.1015625 + }, + { + "difficulty": "hard", + "first_model": "claude-opus-4-6", + "second_model": "gpt-5.4-mini", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 8, + "second_successes": 2, + "first_only_successes": 6, + "second_only_successes": 0, + "both_successes": 2, + "neither_successes": 2, + "success_rate_difference_first_minus_second": 0.6, + "paired_task_bootstrap_95_interval": [ + 0.3, + 0.9 + ], + "exact_p_two_sided": 0.03125, + "holm_adjusted_p_within_difficulty": 0.28125 + }, + { + "difficulty": "hard", + "first_model": "gpt-5.4", + "second_model": "gemini-3-flash-preview", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 1, + "second_successes": 7, + "first_only_successes": 1, + "second_only_successes": 7, + "both_successes": 0, + "neither_successes": 2, + "success_rate_difference_first_minus_second": -0.6, + "paired_task_bootstrap_95_interval": [ + -1.0, + -0.2 + ], + "exact_p_two_sided": 0.0703125, + "holm_adjusted_p_within_difficulty": 0.5 + }, + { + "difficulty": "hard", + "first_model": "gpt-5.4", + "second_model": "claude-sonnet-4-20250514", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 1, + "second_successes": 0, + "first_only_successes": 1, + "second_only_successes": 0, + "both_successes": 0, + "neither_successes": 9, + "success_rate_difference_first_minus_second": 0.1, + "paired_task_bootstrap_95_interval": [ + 0.0, + 0.3 + ], + "exact_p_two_sided": 1.0, + "holm_adjusted_p_within_difficulty": 1.0 + }, + { + "difficulty": "hard", + "first_model": "gpt-5.4", + "second_model": "gpt-5.4-mini", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 1, + "second_successes": 2, + "first_only_successes": 1, + "second_only_successes": 2, + "both_successes": 0, + "neither_successes": 7, + "success_rate_difference_first_minus_second": -0.1, + "paired_task_bootstrap_95_interval": [ + -0.4, + 0.2 + ], + "exact_p_two_sided": 1.0, + "holm_adjusted_p_within_difficulty": 1.0 + }, + { + "difficulty": "hard", + "first_model": "gemini-3-flash-preview", + "second_model": "claude-sonnet-4-20250514", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 7, + "second_successes": 0, + "first_only_successes": 7, + "second_only_successes": 0, + "both_successes": 0, + "neither_successes": 3, + "success_rate_difference_first_minus_second": 0.7, + "paired_task_bootstrap_95_interval": [ + 0.4, + 1.0 + ], + "exact_p_two_sided": 0.015625, + "holm_adjusted_p_within_difficulty": 0.171875 + }, + { + "difficulty": "hard", + "first_model": "gemini-3-flash-preview", + "second_model": "gpt-5.4-mini", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 7, + "second_successes": 2, + "first_only_successes": 5, + "second_only_successes": 0, + "both_successes": 2, + "neither_successes": 3, + "success_rate_difference_first_minus_second": 0.5, + "paired_task_bootstrap_95_interval": [ + 0.2, + 0.8 + ], + "exact_p_two_sided": 0.0625, + "holm_adjusted_p_within_difficulty": 0.5 + }, + { + "difficulty": "hard", + "first_model": "claude-sonnet-4-20250514", + "second_model": "gpt-5.4-mini", + "n_paired_tasks": 10, + "seeds": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10 + ], + "first_successes": 0, + "second_successes": 2, + "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/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 1ba5e42..8486ab9 100644 --- a/references.bib +++ b/references.bib @@ -204,3 +204,95 @@ @misc{kaggle2026pde 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..03be1c0 --- /dev/null +++ b/scripts/plot_research_audit.py @@ -0,0 +1,201 @@ +#!/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 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('--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.') + 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') + provenance={p.name:hashlib.sha256(p.read_bytes()).hexdigest() for p in (args.controls,args.audit,args.statistics)} + 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/verify_paper_artifacts.py b/scripts/verify_paper_artifacts.py new file mode 100644 index 0000000..e186f35 --- /dev/null +++ b/scripts/verify_paper_artifacts.py @@ -0,0 +1,52 @@ +#!/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()) + 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() From ac126c74fc0e0daa47f74ae3263a49b2aef8baa0 Mon Sep 17 00:00:00 2001 From: Hydrodynamical Date: Tue, 20 Oct 2026 21:14:08 -0400 Subject: [PATCH 3/4] Validate fixed-query recovery on 300 fresh task seeds --- README.md | 5 +- images/paper_confidence_uncertainty.pdf | Bin 24113 -> 24113 bytes images/paper_controls.pdf | Bin 20096 -> 20096 bytes images/paper_numerical_audit.pdf | Bin 22479 -> 22479 bytes paper.pdf | Bin 469330 -> 473889 bytes paper.tex | 57 +- paper_artifacts/README.md | 29 +- paper_artifacts/research_figure_sources.json | 3 +- paper_artifacts/sine_generalization.json | 50368 ++++++++++++++++ paper_artifacts/sine_generalization_table.tex | 18 + scripts/plot_research_audit.py | 22 +- scripts/validate_sine_generalization.py | 207 + scripts/verify_paper_artifacts.py | 6 + test_sine_generalization.py | 45 + 14 files changed, 50746 insertions(+), 14 deletions(-) create mode 100644 paper_artifacts/sine_generalization.json create mode 100644 paper_artifacts/sine_generalization_table.tex create mode 100644 scripts/validate_sine_generalization.py create mode 100644 test_sine_generalization.py diff --git a/README.md b/README.md index b1cf92a..69700be 100644 --- a/README.md +++ b/README.md @@ -166,7 +166,10 @@ 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. +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`. 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