\documentclass[10pt,a4paper]{article} \usepackage[a4paper,textwidth=14cm,textheight=22.5cm]{geometry} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage{amsmath,amssymb,mathtools} \usepackage{booktabs,tabularx,array} \usepackage{fvextra} \usepackage{float} \usepackage{xcolor} \usepackage{newunicodechar} \usepackage[hidelinks]{hyperref} \floatstyle{plaintop} \newfloat{listing}{tbp}{lol} \floatname{listing}{Listing} \fvset{fontsize=\small,breaklines=true,frame=lines} \newunicodechar{¬}{\ensuremath{\neg}} \newunicodechar{¹}{\textsuperscript{1}} \newunicodechar{α}{\ensuremath{\alpha}} \newunicodechar{β}{\ensuremath{\beta}} \newunicodechar{θ}{\ensuremath{\theta}} \newunicodechar{κ}{\ensuremath{\kappa}} \newunicodechar{σ}{\ensuremath{\sigma}} \newunicodechar{τ}{\ensuremath{\tau}} \newunicodechar{φ}{\ensuremath{\varphi}} \newunicodechar{χ}{\ensuremath{\chi}} \newunicodechar{⁻}{\textsuperscript{-}} \newunicodechar{₁}{\textsubscript{1}} \newunicodechar{₂}{\textsubscript{2}} \newunicodechar{ₐ}{\textsubscript{a}} \newunicodechar{ₗ}{\textsubscript{l}} \newunicodechar{ℂ}{\ensuremath{\mathbb{C}}} \newunicodechar{ℕ}{\ensuremath{\mathbb{N}}} \newunicodechar{ℚ}{\ensuremath{\mathbb{Q}}} \newunicodechar{ℝ}{\ensuremath{\mathbb{R}}} \newunicodechar{ℤ}{\ensuremath{\mathbb{Z}}} \newunicodechar{→}{\ensuremath{\to}} \newunicodechar{∀}{\ensuremath{\forall}} \newunicodechar{∃}{\ensuremath{\exists}} \newunicodechar{∏}{\ensuremath{\prod}} \newunicodechar{∑}{\ensuremath{\sum}} \newunicodechar{∧}{\ensuremath{\wedge}} \newunicodechar{≃}{\ensuremath{\simeq}} \newunicodechar{≠}{\ensuremath{\neq}} \newunicodechar{≤}{\ensuremath{\le}} \newunicodechar{⊤}{\ensuremath{\top}} \newunicodechar{𝓞}{\ensuremath{\mathcal{O}}} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \newcommand{\C}{\mathbb C} \newcommand{\R}{\mathbb R} \newcommand{\N}{\mathbb Z_{\ge0}} \newcommand{\cR}{\mathcal R} \newcommand{\cP}{\mathcal P} \newcommand{\cK}{\mathcal K} \DeclareMathOperator{\conv}{conv} \newenvironment{paperquote}[1]{\begin{quote}\small\renewcommand{\theequation}{#1}}{\end{quote}} \sloppy \setlength{\emergencystretch}{3em} \makeatletter \DeclareRobustCommand{\_}{\ifmmode\nfss@text{\textunderscore}\else\textunderscore\allowbreak\fi} \DeclareRobustCommand{\hash}[1]{\hash@#1\@nil} \def\hash@#1{\ifx#1\@nil\else#1\allowbreak\expandafter\hash@\fi} \makeatother \title{Formalizing the interpolation lemma of the openai/math Landau--Siegel paper and substituting it into a Comparator-accepted proof} \author{Joseph M. Reilly \\ Northeastern University} \date{} \begin{document} \maketitle \input{abstract} \input{sec1-intro} \input{sec2-background} \input{sec3-faithfulness} \input{sec4-gap} \input{sec5-lemma3} \input{sec6-reroute} \input{sec7-corpus} \input{sec8-threats} \input{sec9-artifacts} \input{sec10-ai} \bibliographystyle{plainurl} \bibliography{refs} \appendix \input{appendix-78-listings} \input{appendix-reasoning} \end{document}