\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{booktabs,longtable,array} \usepackage{enumitem} \usepackage{xcolor} \usepackage{needspace} \usepackage{xurl} \usepackage{tikz} \usetikzlibrary{arrows.meta,calc,positioning} \usepackage{hyperref} \hypersetup{colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue,pdfborder={0 0 0}} \setcounter{tocdepth}{2} % The unified archival edition uses a book class, while the two public % companion papers use an article class. Preserve chapter numbering in the % archive and use ordinary section numbering in the public papers. \makeatletter \@ifundefined{c@chapter} {\newtheorem{theorem}{Theorem}[section]} {\newtheorem{theorem}{Theorem}[chapter]} \makeatother \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{certificate}[theorem]{Certificate} \newtheorem{proofcandidate}[theorem]{Candidate statement} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{openproblem}[theorem]{Open problem} \newtheorem{conjecture}[theorem]{Conjecture} % Exact integer/symbolic calculations retain their source alias but receive a % conventional, evidence-matched public name. \newtheorem{computation}[theorem]{Exact computation} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} % These source aliases are retained for label stability. Publicly they are % ordinary remarks stating limitations or typing boundaries. \newtheorem{scopefirewall}[theorem]{Remark} \newtheorem{warning}[theorem]{Remark} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\C}{\mathbb C} \newcommand{\F}{\mathbb F} \newcommand{\FA}{{\mathbb F}_{A}} \newcommand{\K}{\mathbb K} \newcommand{\N}{\mathbb N} \newcommand{\Lam}{\Lambda} \newcommand{\Sk}{\mathcal S} \newcommand{\cA}{\mathcal A} \newcommand{\cB}{\mathcal B} \newcommand{\cC}{\mathcal C} \newcommand{\cD}{\mathcal D} \newcommand{\cE}{\mathcal E} \newcommand{\cG}{\mathcal G} \newcommand{\cH}{\mathcal H} \newcommand{\cI}{\mathcal I} \newcommand{\cJ}{\mathcal J} \newcommand{\cL}{\mathcal L} \newcommand{\cM}{\mathcal M} \newcommand{\cO}{\mathcal O} \newcommand{\cP}{\mathcal P} \newcommand{\cQ}{\mathcal Q} \newcommand{\cR}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cT}{\mathcal T} \newcommand{\cX}{\mathcal X} \newcommand{\wh}{\widehat} \newcommand{\MCG}{\operatorname{MCG}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\Sp}{\operatorname{Sp}} \newcommand{\Tor}{\operatorname{Tor}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\im}{\operatorname{im}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\rad}{\operatorname{rad}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\End}{\operatorname{End}} \newcommand{\Ann}{\operatorname{Ann}} \newcommand{\Span}{\operatorname{Span}} \newcommand{\id}{\operatorname{id}} \newcommand{\sat}{\operatorname{sat}}