Lots of writing, and new figures.

author: Jaron Kent-Dobias <jaron@kent-dobias.com> 2024-08-27 18:35:29 +0200
committer: Jaron Kent-Dobias <jaron@kent-dobias.com> 2024-08-27 18:35:29 +0200
commit: 8cff4a74e2792b6037364aa63bd7d78085724ca5 (patch)
tree: f35b2144ea393fdfd166a22ba4e8ef8ee53fbf92
parent: c0ef21f08ad290bdea74ba39f6002329e4511982 (diff)
download: SciPostPhys_18_158-8cff4a74e2792b6037364aa63bd7d78085724ca5.tar.gz
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SciPostPhys_18_158-8cff4a74e2792b6037364aa63bd7d78085724ca5.zip
19 files changed, 13617 insertions, 306 deletions
diff --git a/SciPost.cls b/SciPost.cls
index ca8b331..2808a40 100644
--- a/SciPost.cls
+++ b/SciPost.cls
@@ -1,5 +1,5 @@
 \NeedsTeXFormat{LaTeX2e}
-\ProvidesClass{SciPost} % SciPost Latex Template (2021-08)
+\ProvidesClass{SciPost} % SciPost Latex Template v1f(2023-10)
 
 
 \LoadClass[11pt,a4paper]{article}
@@ -45,17 +45,13 @@
 \pagestyle{fancy}
 
 \makeatletter
-  \let\ps@plain\ps@fancy
+\let\ps@plain\ps@fancy
 \makeatother
 
 \RequirePackage{xcolor}
 \definecolor{scipostdeepblue}{HTML}{002B49}
-\definecolor{scipostphys}{HTML}{0019A2}
-\definecolor{scipostastro}{HTML}{946E13}
-\definecolor{scipostbio}{HTML}{2B7E13}
-\definecolor{scipostchem}{HTML}{605AAF}
-\definecolor{scipostcompsci}{HTML}{0AADAB}
-\definecolor{scipostmath}{HTML}{A10800}
+\definecolor{scipostblue}{HTML}{0019A2}
+
 
 \RequirePackage{graphicx}
 
@@ -66,101 +62,23 @@
 
 %% Patch lineno when used with amsmath
 \newcommand*\patchAmsMathEnvironmentForLineno[1]{%
-\expandafter\let\csname old#1\expandafter\endcsname\csname #1\endcsname
-\expandafter\let\csname oldend#1\expandafter\endcsname\csname end#1\endcsname
-\renewenvironment{#1}%
-{\linenomath\csname old#1\endcsname}%
-{\csname oldend#1\endcsname\endlinenomath}}%
+    \expandafter\let\csname old#1\expandafter\endcsname\csname #1\endcsname
+    \expandafter\let\csname oldend#1\expandafter\endcsname\csname end#1\endcsname
+    \renewenvironment{#1}%
+    {\linenomath\csname old#1\endcsname}%
+    {\csname oldend#1\endcsname\endlinenomath}}%
 \newcommand*\patchBothAmsMathEnvironmentsForLineno[1]{%
-\patchAmsMathEnvironmentForLineno{#1}%
-\patchAmsMathEnvironmentForLineno{#1*}}%
+    \patchAmsMathEnvironmentForLineno{#1}%
+    \patchAmsMathEnvironmentForLineno{#1*}}%
 \AtBeginDocument{%
-\patchBothAmsMathEnvironmentsForLineno{equation}%
-\patchBothAmsMathEnvironmentsForLineno{align}%
-\patchBothAmsMathEnvironmentsForLineno{flalign}%
-\patchBothAmsMathEnvironmentsForLineno{alignat}%
-\patchBothAmsMathEnvironmentsForLineno{gather}%
-\patchBothAmsMathEnvironmentsForLineno{multline}%
+    \patchBothAmsMathEnvironmentsForLineno{equation}%
+    \patchBothAmsMathEnvironmentsForLineno{align}%
+    \patchBothAmsMathEnvironmentsForLineno{flalign}%
+    \patchBothAmsMathEnvironmentsForLineno{alignat}%
+    \patchBothAmsMathEnvironmentsForLineno{gather}%
+    \patchBothAmsMathEnvironmentsForLineno{multline}%
 }
 %% End patch lineno
 
 
-\DeclareOption{submission}{
-  \rhead{
-    {\bf \color{scipostdeepblue} ~Submission }
-  }
- }
-
-
-%%%%%%%%%% Physics
-\DeclareOption{Phys}{
-\lhead{
-  \colorbox{scipostphys}{\bf \color{white} ~SciPost Physics }
-}
-}
-
-\DeclareOption{PhysCore}{
-\lhead{
-  \colorbox{scipostphys}{\bf \color{white} ~SciPost Physics Core}
-}
-}
-
-\DeclareOption{PhysLectNotes}{
-  \lhead{
-    \colorbox{scipostphys}{\strut \bf \color{white} ~SciPost Physics Lecture Notes }
-  }
-}
-
-\DeclareOption{PhysProc}{
-\lhead{
-  \colorbox{scipostphys}{\strut \bf \color{white} ~SciPost Physics Proceedings }
-}
-}
-
-\DeclareOption{PhysCodeb}{
-\lhead{
-  \colorbox{scipostphys}{\strut \bf \color{white} ~SciPost Physics Codebases }
-}
-}
-
-%%%%%%%%%%% Astronomy
-\DeclareOption{Astro}{
-\lhead{
-  \colorbox{scipostastro}{\strut \bf \color{white} ~SciPost Astronomy }
-}
-}
-
-
-%%%%%%%%%%% Biology
-\DeclareOption{Bio}{
-\lhead{
-  \colorbox{scipostbio}{\strut \bf \color{white} ~SciPost Biology }
-}
-}
-
-
-%%%%%%%%%%% Chemistry
-\DeclareOption{Chem}{
-\lhead{
-  \colorbox{scipostchem}{\strut \bf \color{white} ~SciPost Chemistry }
-}
-}
-
-
-%%%%%%%%%%% Computer Science
-\DeclareOption{CompSci}{
-\lhead{
-  \colorbox{scipostcompsci}{\strut \bf \color{white} ~SciPost Computer Science }
-}
-}
-
-
-%%%%%%%%%%% Mathematics
-\DeclareOption{Math}{
-\lhead{
-  \colorbox{scipostmath}{\strut \bf \color{white} ~SciPost Mathematics }
-}
-}
-
-
 \ProcessOptions\relax
diff --git a/SciPostPhys_Template.tex b/SciPostPhys.tex
index 0f47a47..88c5a04 100644
--- a/SciPostPhys_Template.tex
+++ b/SciPostPhys.tex
@@ -1,65 +1,13 @@
 % =========================================================================
 % SciPost LaTeX template
-% Version 2021-08
+% Version 2024-07
 %
 % Submissions to SciPost Journals should make use of this template.
 %
 % INSTRUCTIONS: simply look for the `TODO:' tokens and adapt your file.
-%
-% You can also make use of our empty "skeleton" templates for each Journal,
-% e.g. SciPostPhys_skeleton.tex
-% =========================================================================
-
-
-% TODO: uncomment ONE of the class declarations below
-
-% Class declaration format: \documentclass[submission, {DOI label of journal}]{SciPost}
-% where the DOI label of the journal should be one of:
-% Phys          (for SciPost Physics)
-% PhysCore      (for SciPost Physics Core)
-% PhysLectNotes (for SciPost Physics Lecture Notes)
-% PhysProc      (for SciPost Physics Proceedings -> !! Please use the conference-specific template which you will find on our website !!
-% PhysCodeb     (for SciPost Physics Codebases)
-% Astro         (for SciPost Astronomy)
-% Bio           (for SciPost Biology)
-% Chem          (for SciPost Chemistry)
-% CompSci       (for SciPost Computer Science)
-% Math          (for SciPost Mathematics)
-
-
-%% PHYSICS:
-% If you are submitting a paper to SciPost Physics: uncomment next line
-\documentclass[submission, Phys]{SciPost}
-% If you are submitting a paper to SciPost Physics Core: uncomment next line
-%\documentclass[submission, PhysCore]{SciPost}
-% If you are submitting a paper to SciPost Physics Lecture Notes: uncomment next line
-%\documentclass[submission, PhysLectNotes]{SciPost}
-% If you are submitting a paper to SciPost Physics Proceedings: uncomment next line
-%\documentclass[submission, PhysProc]{SciPost}
-% If you are submitting a paper to SciPost Physics Codebases: uncomment next line
-%\documentclass[submission, PhysCodeb]{SciPost}
-
-%% ASTRONOMY:
-% If you are submitting a paper to SciPost Astronomy: uncomment next line
-% \documentclass[submission, Astro]{SciPost}
-
-%% BIOLOGY:
-% If you are submitting a paper to SciPost Biology: uncomment next line
-% \documentclass[submission, Bio]{SciPost}
-
-%% CHEMISTRY:
-% If you are submitting a paper to SciPost Chemistry: uncomment next line
-% \documentclass[submission, Chem]{SciPost}
-
-%% COMPUTER SCIENCE:
-% If you are submitting a paper to SciPost Computer Science: uncomment next line
-% \documentclass[submission, CompSci]{SciPost}
-
-%% MATHEMATICS:
-% If you are submitting a paper to SciPost Mathematics: uncomment next line
-% \documentclass[submission, Math]{SciPost}
-
+% ========================================================================
 
+\documentclass{SciPost}
 
 % Prevent all line breaks in inline equations.
 \binoppenalty=10000
@@ -73,68 +21,112 @@
 }
 
 \usepackage[bitstream-charter]{mathdesign}
-\urlstyle{sf}
+\urlstyle{same}
 
 % Fix \cal and \mathcal characters look (so it's not the same as \mathscr)
 \DeclareSymbolFont{usualmathcal}{OMS}{cmsy}{m}{n}
 \DeclareSymbolFontAlphabet{\mathcal}{usualmathcal}
 
+\fancypagestyle{SPstyle}{
+\fancyhf{}
+\lhead{\colorbox{scipostblue}{\bf \color{white} ~SciPost Physics }}
+\rhead{{\bf \color{scipostdeepblue} ~Submission }}
+\renewcommand{\headrulewidth}{1pt}
+\fancyfoot[C]{\textbf{\thepage}}
+}
+
 \begin{document}
 
-% TODO: write your article's title here.
-% The article title is centered, Large boldface, and should fit in two lines
-\begin{center}{\Large \textbf{
+\pagestyle{SPstyle}
+
+\begin{center}{\Large \textbf{\color{scipostdeepblue}{
+%%%%%%%%%% TODO: Write your article's title here
 Article Title, as descriptive as possible, ideally fitting in two lines (approximately 150 characters) or less\\
-}}\end{center}
+%%%%%%%%%% END TODO: TITLE
+}}}\end{center}
 
-% TODO: write the author list here. Use first name (+ other initials) + surname format.
+\begin{center}\textbf{
+%%%%%%%%%% TODO: AUTHORS
+% Write the author list here. 
+% Use (full) first name (+ middle name initials) + surname format.
 % Separate subsequent authors by a comma, omit comma and use "and" for the last author.
-% Mark the corresponding author with a superscript star.
-\begin{center}
-Dee E. Faa\textsuperscript{1},
-Aah B. Cee\textsuperscript{2} and
-Gee K. See\textsuperscript{3$\star$}
-\end{center}
+% Mark the corresponding author(s) with a superscript symbol in this order
+% \star, \dagger, \ddagger, \circ, \S, \P, \parallel, ...
+Aah B. Cee\textsuperscript{1$\star$},
+Dee E. Faa\textsuperscript{2} and
+Gee K. See\textsuperscript{3$\dagger$}
+%%%%%%%%%% END TODO: AUTHORS
+}\end{center}
 
-% TODO: write all affiliations here.
-% Format: institute, city, country
 \begin{center}
+%%%%%%%%%% TODO: AFFILIATIONS
+% Write all affiliations here.
+% Format: institute, city, country
 {\bf 1} Affiliation1
 \\
 {\bf 2} Affiliation2
 \\
-{\bf 3} Affiliation2
-\\
-% TODO: provide email address of corresponding author
-${}^\star$ {\small \sf CorrespondingAuthor@email.address}
-\end{center}
-
-\begin{center}
-\today
+{\bf 3} Affiliation3
+%%%%%%%%%% END TODO: AFFILIATIONS
+%%%%%%%%%% TODO: EMAIL
+% Provide email address of corresponding author(s)
+\\[\baselineskip]
+$\star$ \href{mailto:email1}{\small email1}\,,\quad
+$\dagger$ \href{mailto:email2}{\small email2}
+%%%%%%%%%% END TODO: EMAIL
 \end{center}
 
-% For convenience during refereeing (optional),
-% you can turn on line numbers by uncommenting the next line:
-%\linenumbers
-% You should run LaTeX twice in order for the line numbers to appear.
 
-\section*{Abstract}
-{\bf
-% TODO: write your abstract here.
-The abstract is in boldface, and should fit in 8 lines.
-It should be written in a clear and accessible style, emphasizing the context, the problem(s) studied, the methods used, the results obtained, the conclusions reached, and the outlook. You can add a table contents, recommended if your paper is more than 6 pages long.
+\section*{\color{scipostdeepblue}{Abstract}}
+\textbf{\boldmath{%
+%%%%%%%%%% TODO: ABSTRACT
+% Write your abstract here.
+The abstract is in boldface, and should fit in 8 lines. It should be written in a clear and accessible style, emphasizing the context, the problem(s) studied, the methods used, the results obtained, the conclusions reached, and the outlook. You can add a table contents, recommended if your paper is more than 6 pages long.
+%%%%%%%%%% END TODO: ABSTRACT
+}}
+
+\vspace{\baselineskip}
+
+%%%%%%%%%% BLOCK: Copyright information
+% This block will be filled during the proof stage, and finilized just before publication.
+% It exists here only as a placeholder, and should not be modified by authors.
+\noindent\textcolor{white!90!black}{%
+\fbox{\parbox{0.975\linewidth}{%
+\textcolor{white!40!black}{\begin{tabular}{lr}%
+  \begin{minipage}{0.6\textwidth}%
+    {\small Copyright attribution to authors. \newline
+    This work is a submission to SciPost Physics. \newline
+    License information to appear upon publication. \newline
+    Publication information to appear upon publication.}
+  \end{minipage} & \begin{minipage}{0.4\textwidth}
+    {\small Received Date \newline Accepted Date \newline Published Date}%
+  \end{minipage}
+\end{tabular}}
+}}
 }
+%%%%%%%%%% BLOCK: Copyright information
+
 
+%%%%%%%%%% TODO: LINENO
+% For convenience during refereeing we turn on line numbers:
+\linenumbers
+% You should run LaTeX twice in order for the line numbers to appear.
+%%%%%%%%%% END TODO: LINENO
 
-% TODO: include a table of contents (optional)
+%%%%%%%%%% TODO: TOC 
 % Guideline: if your paper is longer that 6 pages, include a TOC
 % To remove the TOC, simply cut the following block
 \vspace{10pt}
 \noindent\rule{\textwidth}{1pt}
-\tableofcontents\thispagestyle{fancy}
+\tableofcontents
 \noindent\rule{\textwidth}{1pt}
 \vspace{10pt}
+%%%%%%%%%% END TODO: TOC
+
 
+%%%%%%%%% TODO: CONTENTS 
+% Write your article contents here, starting from first \section.
+% An example structure is given below.
 
 \section{Introduction}
 \label{sec:intro}
@@ -182,16 +174,17 @@ Authors are required to provide funding information, including relevant agencies
 
 
 \begin{appendix}
+\numberwithin{equation}{section}
 
 \section{First appendix}
 Add material which is better left outside the main text in a series of Appendices labeled by capital letters.
 
 \section{About references}
-Your references should start with the comma-separated author list (initials + last name), the publication title in italics, the journal reference with volume in bold, start page number, publication year in parenthesis, completed by the DOI link (linking must be implemented before publication). If using BiBTeX, please use the style files provided  on \url{https://scipost.org/submissions/author_guidelines}. If you are using our \LaTeX template, simply add
+Your references should start with the comma-separated author list (initials + last name), the publication title in italics, the journal reference with volume in bold, start page number, publication year in parenthesis, completed by the DOI link (linking must be implemented before publication). If using BiBTeX, please use the style files provided  on \url{https://scipost.org/submissions/author_guidelines}. If you are using our LaTeX template, simply add
 \begin{verbatim}
 \bibliography{your_bibtex_file}
 \end{verbatim}
-at the end of your document. If you are not using our \LaTeX template, please still use our bibstyle as
+at the end of your document. If you are not using our LaTeX template, please still use our bibstyle as
 \begin{verbatim}
 \bibliographystyle{SciPost_bibstyle}
 \end{verbatim}
@@ -199,20 +192,26 @@ in order to simplify the production of your paper.
 \end{appendix}
 
 
-% TODO:
+%%%%%%%%% END TODO: CONTENTS
+
+
+%%%%%%%%%% TODO: BIBLIOGRAPHY
 % Provide your bibliography here. You have two options:
 
-% FIRST OPTION - write your entries here directly, following the example below, including Author(s), Title, Journal Ref. with year in parentheses at the end, followed by the DOI number.
+%%% FIRST OPTION
+% Write your entries here directly, following the example below, including:
+% Author(s), Title, Journal Ref. with year in parentheses at the end, followed by the DOI number.
+
 %\begin{thebibliography}{99}
 %\bibitem{1931_Bethe_ZP_71} H. A. Bethe, {\it Zur Theorie der Metalle. i. Eigenwerte und Eigenfunktionen der linearen Atomkette}, Zeit. f{\"u}r Phys. {\bf 71}, 205 (1931), \doi{10.1007\%2FBF01341708}.
 %\bibitem{arXiv:1108.2700} P. Ginsparg, {\it It was twenty years ago today... }, \url{http://arxiv.org/abs/1108.2700}.
 %\end{thebibliography}
 
-% SECOND OPTION:
-% Use your bibtex library
-% \bibliographystyle{SciPost_bibstyle} % Include this style file here only if you are not using our template
+%%% SECOND OPTION
+% Use your bibtex library, formatted by the SciPost style file.
 \bibliography{SciPost_Example_BiBTeX_File.bib}
 
-\nolinenumbers
+%%%%%%%%%% END TODO: BIBLIOGRAPHY
+
 
 \end{document}
diff --git a/SciPost_Example_BiBTeX_File.bib b/SciPost_Example_BiBTeX_File.bib
index e68ea35..b65766a 100644
--- a/SciPost_Example_BiBTeX_File.bib
+++ b/SciPost_Example_BiBTeX_File.bib
@@ -1,34 +1,31 @@
-
-
-@Article{1931_Bethe_ZP_71,
-  author =       {H. A. Bethe},
-  title = 	 {Zur {T}heorie der {M}etalle.  I.  {E}igenwerte und {E}igenfunktionen der linearen {A}tomkette},
-  journal = 	 {Zeit. f\"ur Physik},
-  year = 	 {1931},
-  volume = 	 {71},
-  pages = 	 {205},
-  doi = {10.1007/BF01341708}
+@article{1931_Bethe_ZP_71,
+  author      =   {H. A. Bethe},
+  title       =   {Zur {T}heorie der {M}etalle.  I.  {E}igenwerte und {E}igenfunktionen der linearen {A}tomkette},
+  journal     =   {Zeit. f\"ur Physik},
+  year        =   {1931},
+  volume      =   {71},
+  pages       =   {205},
+  doi         =   {10.1007/BF01341708}
 }
 
-
 @unpublished{arXiv:1108.2700,
-  author    = {P. Ginsparg},
-  title     = {It was twenty years ago today ...},
-  eprint       = {http://arxiv.org/abs/1108.2700},
+  author      =   {P. Ginsparg},
+  title       =   {It was twenty years ago today ...},
+  eprint      =   {http://arxiv.org/abs/1108.2700},
 }
 
-
 @Article{2016_Piroli_SP_1,
-  author = 	 {Lorenzo Piroli and Pasquale Calabrese and Fabian H. L. Essler},
-  title = 	 {{Quantum quenches to the attractive one-dimensional Bose gas: exact results}},
-  journal = 	 {SciPost Phys.},
-  year = 	 {2016},
-  OPTkey = 	 {},
-  volume = 	 {1},
-  number = 	 {1},
-  pages = 	 {001},
-  OPTmonth = 	 {},
-  OPTnote = 	 {},
-  OPTannote = 	 {}
+  author      =   {Lorenzo Piroli and Pasquale Calabrese and Fabian H. L. Essler},
+  title       =   {{Quantum quenches to the attractive one-dimensional Bose gas: exact results}},
+  journal     =   {SciPost Phys.},
+  year        =   {2016},
+  OPTkey      =   {},
+  volume      =   {1},
+  number      =   {1},
+  pages       =   {001},
+  OPTmonth    =   {},
+  OPTnote     =   {},
+  OPTannote   =   {}
 }
 
+
diff --git a/figs/action_1.pdf b/figs/action_1.pdf
index 2a87561..9b7e848 100644
--- a/figs/action_1.pdf
+++ b/figs/action_1.pdf
diff --git a/figs/action_3.pdf b/figs/action_3.pdf
index 54825b1..693d7c0 100644
--- a/figs/action_3.pdf
+++ b/figs/action_3.pdf
diff --git a/figs/bar.pdf b/figs/bar.pdf
index bee8e41..e7e3af6 100644
--- a/figs/bar.pdf
+++ b/figs/bar.pdf
diff --git a/figs/coexist.pdf b/figs/coexist.pdf
index 3b8f5e0..bf7e744 100644
--- a/figs/coexist.pdf
+++ b/figs/coexist.pdf
diff --git a/figs/connected.pdf b/figs/connected.pdf
index c847fa9..85abb5f 100644
--- a/figs/connected.pdf
+++ b/figs/connected.pdf
diff --git a/figs/gone.pdf b/figs/gone.pdf
index 2602e05..ec3174e 100644
--- a/figs/gone.pdf
+++ b/figs/gone.pdf
diff --git a/figs/phases_1.pdf b/figs/phases_1.pdf
index 4a8c9c9..c39222d 100644
--- a/figs/phases_1.pdf
+++ b/figs/phases_1.pdf
diff --git a/figs/phases_2.pdf b/figs/phases_2.pdf
index a73666e..9680f37 100644
--- a/figs/phases_2.pdf
+++ b/figs/phases_2.pdf
diff --git a/figs/phases_3.pdf b/figs/phases_3.pdf
index e7a55fb..7c0fda2 100644
--- a/figs/phases_3.pdf
+++ b/figs/phases_3.pdf
diff --git a/figs/regime_1.pdf b/figs/regime_1.pdf
new file mode 100644
index 0000000..04a8da5
--- /dev/null
+++ b/figs/regime_1.pdf
diff --git a/figs/regime_2.pdf b/figs/regime_2.pdf
new file mode 100644
index 0000000..f3473cc
--- /dev/null
+++ b/figs/regime_2.pdf
diff --git a/figs/regime_3.pdf b/figs/regime_3.pdf
new file mode 100644
index 0000000..255b960
--- /dev/null
+++ b/figs/regime_3.pdf
diff --git a/figs/shattered.pdf b/figs/shattered.pdf
index 6e82041..839ee07 100644
--- a/figs/shattered.pdf
+++ b/figs/shattered.pdf
diff --git a/figures.nb b/figures.nb
new file mode 100644
index 0000000..843299e
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+++ b/figures.nb
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+      "Label" -> {"XYLabel"}, "Ball" -> {"InterpolatedBall"}|>, 
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+       "PlotRange" -> {{-0.025, 1.025}, {-0.249, 0.049}}, 
+       "Frame" -> {{True, True}, {True, True}}, "AxesOrigin" -> {0, 0}, 
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+       "LabelStyle" -> {FontFamily -> "BitstreamCharter", FontSize -> 12, 
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+          RGBColor[0.528488, 0.470624, 0.701351], 
+          AbsoluteThickness[2]], 
+         Directive[
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+          RGBColor[0.922526, 0.385626, 0.209179], 
+          AbsoluteThickness[2]], 
+         Directive[
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+          RGBColor[0.560181, 0.691569, 0.194885], 
+          AbsoluteThickness[2]], 
+         Directive[
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+          RGBColor[0.880722, 0.611041, 0.142051], 
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+          RGBColor[0.368417, 0.506779, 0.709798], 
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+           Identity[
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+     "Meta" -> <|
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+       Plot, "GroupHighlight" -> False|>|>, "DynamicHighlight"]],
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+  Axes->{True, True},
+  AxesLabel->{None, None},
+  AxesOrigin->{0, 0},
+  DisplayFunction->Identity,
+  Epilog->{
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+                    GraphicsBox[{{
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+                    GrayLevel[0]]], 
+                    PointSize[0.5], 
+                    AbsoluteThickness[1.6], 
+                    RGBColor[0.368417, 0.506779, 0.709798]], {}}}, 
+                    AspectRatio -> Full, ImageSize -> {20, 12.5}, 
+                    PlotRangePadding -> None, ImagePadding -> Automatic, 
+                    BaselinePosition -> (Scaled[0.116] -> Baseline)], #}, {
+                    GraphicsBox[{{
+                    Directive[
+                    EdgeForm[
+                    Directive[
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+                    GrayLevel[0]]], 
+                    PointSize[0.5], 
+                    AbsoluteThickness[1.6], 
+                    RGBColor[0.880722, 0.611041, 0.142051]], {
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+                    GrayLevel[0]]], 
+                    PointSize[0.5], 
+                    AbsoluteThickness[1.6], 
+                    RGBColor[0.880722, 0.611041, 0.142051]], {}}}, 
+                    AspectRatio -> Full, ImageSize -> {20, 12.5}, 
+                    PlotRangePadding -> None, ImagePadding -> Automatic, 
+                    BaselinePosition -> (Scaled[0.116] -> Baseline)], #2}, {
+                    GraphicsBox[{{
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+                    GrayLevel[0]]], 
+                    PointSize[0.5], 
+                    AbsoluteThickness[1.6], 
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+                    GrayLevel[0]]], 
+                    PointSize[0.5], 
+                    AbsoluteThickness[1.6], 
+                    RGBColor[0.560181, 0.691569, 0.194885]], {}}}, 
+                    AspectRatio -> Full, ImageSize -> {20, 12.5}, 
+                    PlotRangePadding -> None, ImagePadding -> Automatic, 
+                    BaselinePosition -> (Scaled[0.116] -> Baseline)], #3}, {
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+                    GrayLevel[0]]], 
+                    PointSize[0.5], 
+                    AbsoluteThickness[1.6], 
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+                    AbsoluteThickness[1.6], 
+                    RGBColor[0.922526, 0.385626, 0.209179]], {}}}, 
+                    AspectRatio -> Full, ImageSize -> {20, 12.5}, 
+                    PlotRangePadding -> None, ImagePadding -> Automatic, 
+                    BaselinePosition -> (Scaled[0.116] -> Baseline)], #4}, {
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+                    Directive[
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+                    Directive[
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+                    GrayLevel[0]]], 
+                    PointSize[0.5], 
+                    AbsoluteThickness[1.6], 
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+                    GrayLevel[0]]], 
+                    PointSize[0.5], 
+                    AbsoluteThickness[1.6], 
+                    RGBColor[0.528488, 0.470624, 0.701351]], {}}}, 
+                    AspectRatio -> Full, ImageSize -> {20, 12.5}, 
+                    PlotRangePadding -> None, ImagePadding -> Automatic, 
+                    BaselinePosition -> (Scaled[0.116] -> Baseline)], #5}}, 
+                    GridBoxAlignment -> {
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+                    AutoDelete -> False, 
+                    GridBoxDividers -> {
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+                    GridBoxItemSize -> {
+                    "Columns" -> {{All}}, "Rows" -> {{All}}}, 
+                    GridBoxSpacings -> {
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+                    GridBoxAlignment -> {
+                    "Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> 
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+                    GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}],
+                    "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Center}}}, 
+                AutoDelete -> False, 
+                GridBoxItemSize -> {
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+                GridBoxSpacings -> {
+                 "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], 
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+                 TemplateBox[<|
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+                 
+                 TemplateBox[<|
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+       GrayLevel[0], ScriptLevel -> 1, StripOnInput -> False], 
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+     Scaled[{0.85, 0.91}]]},
+  Frame->{{True, True}, {True, True}},
+  FrameLabel->{{
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+     FormBox[
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+  FrameStyle->GrayLevel[0],
+  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
+  GridLines->{None, None},
+  GridLinesStyle->Directive[
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+  ImagePadding->All,
+  ImageSize->317,
+  LabelStyle->{FontFamily -> "BitstreamCharter", FontSize -> 12, 
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+  Method->{
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+    "DefaultGraphicsInteraction" -> {
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+      "Effects" -> {
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+  PlotRange->{{-0.025, 1.025}, {-0.249, 0.049}},
+  PlotRangeClipping->True,
+  PlotRangePadding->{{0, 0}, {0, 0}},
+  Prolog->{
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+]
+*)
+
diff --git a/topology.bib b/topology.bib
index 8c17a2c..88f1a7f 100644
--- a/topology.bib
+++ b/topology.bib
@@ -110,6 +110,35 @@
  issn = {1742-5468}
 }
 
+@article{Folena_2020_Rethinking,
+ author = {Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico},
+ title = {Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed $p$-Spin Model},
+ journal = {Physical Review X},
+ publisher = {American Physical Society},
+ year = {2020},
+ month = {8},
+ volume = {10},
+ pages = {031045},
+ url = {https://link.aps.org/doi/10.1103/PhysRevX.10.031045},
+ doi = {10.1103/PhysRevX.10.031045},
+ issue = {3},
+ numpages = {26}
+}
+
+@article{Folena_2021_Gradient,
+ author = {Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico},
+ title = {Gradient descent dynamics in the mixed $p$-spin spherical model: finite-size simulations and comparison with mean-field integration},
+ journal = {Journal of Statistical Mechanics: Theory and Experiment},
+ publisher = {IOP Publishing},
+ year = {2021},
+ month = {3},
+ number = {3},
+ volume = {2021},
+ pages = {033302},
+ url = {https://doi.org/10.1088%2F1742-5468%2Fabe29f},
+ doi = {10.1088/1742-5468/abe29f}
+}
+
 @article{Folena_2023_On,
  author = {Folena, Giampaolo and Zamponi, Francesco},
  title = {On weak ergodicity breaking in mean-field spin glasses},
diff --git a/topology.tex b/topology.tex
index 85d1518..65c717e 100644
--- a/topology.tex
+++ b/topology.tex
@@ -1,4 +1,4 @@
-\documentclass[submission, Phys]{SciPost}
+\documentclass{SciPost}
 
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
@@ -9,16 +9,23 @@
 
 \urlstyle{sf}
 
+\fancypagestyle{SPstyle}{
+\fancyhf{}
+\lhead{\colorbox{scipostblue}{\bf \color{white} ~SciPost Physics }}
+\rhead{{\bf \color{scipostdeepblue} ~Submission }}
+\renewcommand{\headrulewidth}{1pt}
+\fancyfoot[C]{\textbf{\thepage}}
+}
+
 % Fix \cal and \mathcal characters look (so it's not the same as \mathscr)
 \DeclareSymbolFont{usualmathcal}{OMS}{cmsy}{m}{n}
 \DeclareSymbolFontAlphabet{\mathcal}{usualmathcal}
 
 \hypersetup{
-  colorlinks=true,
-  urlcolor={blue!50!black},
-  citecolor={blue!50!black},
-  filecolor={blue!50!black},
-  linkcolor={blue!50!black}
+    colorlinks,
+    linkcolor={red!50!black},
+    citecolor={blue!50!black},
+    urlcolor={blue!80!black}
 }
 
 \author{\footnote{\url{jaron.kent-dobias@roma1.infn.it}}}
@@ -26,47 +33,32 @@
 
 \begin{document}
 
-\begin{center}{\Large \textbf{
+\pagestyle{SPstyle}
+
+\begin{center}{\Large \textbf{\color{scipostdeepblue}{
   On the topology of solutions to random continuous constraint satisfaction problems\\
-}}\end{center}
+}}}\end{center}
 
-% TODO: write the author list here. Use first name (+ other initials) + surname format.
-% Separate subsequent authors by a comma, omit comma and use "and" for the last author.
-% Mark the corresponding author with a superscript star.
 \begin{center}
-\bf Jaron Kent-Dobias$^*$
+  \textbf{Jaron Kent-Dobias$^\star$}
 \end{center}
 
-% TODO: write all affiliations here.
-% Format: institute, city, country
 \begin{center}
 Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, Italy
-\\
-% TODO: provide email address of corresponding author
-${}^\star$ {\small \sf jaron.kent-dobias@roma1.infn.it}
-\end{center}
-
-\begin{center}
-\today
+\\[\baselineskip]
+$\star$ \href{mailto:jaron.kent-dobias@roma1.infn.it}{\small jaron.kent-dobias@roma1.infn.it}
 \end{center}
 
-% For convenience during refereeing (optional),
-% you can turn on line numbers by uncommenting the next line:
-%\linenumbers
-% You should run LaTeX twice in order for the line numbers to appear.
-
-\section*{Abstract}
-{\bf
-% TODO: write your abstract here.
+\section*{\color{scipostdeepblue}{Abstract}}
+\textbf{\boldmath{%
 We consider the set of solutions to $M$ random polynomial equations with
 independent Gaussian coefficients and a target value $V_0$ on the $(N-1)$-sphere. When solutions
 exist, they form a manifold. We compute the average Euler characteristic of
 this manifold in the limit of large $N$, and find different behavior
 depending on the ratio $\alpha=M/N$. When $\alpha<\alpha_\text{onset}$, the
 average characteristic is 2 and there is a single connected component, while
-for $\alpha_\text{onset}<\alpha<\alpha_\text{shatter}$ a large connected
-component coexists with many disconnected components. When
-$\alpha>\alpha_\text{shatter}$ the large connected component vanishes, and the
+for $\alpha_\text{onset}<\alpha<\alpha_\text{shatter}$ many large connected components coexist. When
+$\alpha>\alpha_\text{shatter}$ the large connected components vanish, replaced by small fragments, and the
 entire manifold vanishes for $\alpha>\alpha_\text{\textsc{sat}}$. In the
 limit $\alpha\to0$ there is a correspondence between this problem and the
 topology of constant-energy level sets in the spherical spin glasses. We
@@ -74,16 +66,45 @@ conjecture that the energy $E_\text{shatter}$ associated with the vanishing of
 the large connected component corresponds to the asymptotic limit of gradient
 descent from a random initial condition.
 }
+}
+
+\vspace{\baselineskip}
+
+%%%%%%%%%% BLOCK: Copyright information
+% This block will be filled during the proof stage, and finilized just before publication.
+% It exists here only as a placeholder, and should not be modified by authors.
+\noindent\textcolor{white!90!black}{%
+\fbox{\parbox{0.975\linewidth}{%
+\textcolor{white!40!black}{\begin{tabular}{lr}%
+  \begin{minipage}{0.6\textwidth}%
+    {\small Copyright attribution to authors. \newline
+    This work is a submission to SciPost Physics. \newline
+    License information to appear upon publication. \newline
+    Publication information to appear upon publication.}
+  \end{minipage} & \begin{minipage}{0.4\textwidth}
+    {\small Received Date \newline Accepted Date \newline Published Date}%
+  \end{minipage}
+\end{tabular}}
+}}
+}
+%%%%%%%%%% BLOCK: Copyright information
 
 
-% TODO: include a table of contents (optional)
+%%%%%%%%%% TODO: LINENO
+% For convenience during refereeing we turn on line numbers:
+%\linenumbers
+% You should run LaTeX twice in order for the line numbers to appear.
+%%%%%%%%%% END TODO: LINENO
+
+%%%%%%%%%% TODO: TOC 
 % Guideline: if your paper is longer that 6 pages, include a TOC
 % To remove the TOC, simply cut the following block
 \vspace{10pt}
 \noindent\rule{\textwidth}{1pt}
-\tableofcontents\thispagestyle{fancy}
+\tableofcontents
 \noindent\rule{\textwidth}{1pt}
 \vspace{10pt}
+%%%%%%%%%% END TODO: TOC
 
 \section{Introduction}
 
@@ -232,8 +253,6 @@ is the vector of partial derivatives with respect to all $N+M+1$ variables.
 This integral is now in a form where standard techniques from mean-field theory
 can be applied to calculate it.
 
-\subsubsection{Calculation of the average Euler characteristic}
-
 To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$ functions and determinant to integral form, with
 \begin{align}
   \delta\big(\partial L(\mathbf x,\pmb\omega)\big)
@@ -377,55 +396,6 @@ These transition values of the target $V_0$ correspond with transition values in
 \end{figure}
 
 
-
-
-\subsubsection{How to interpret the average Euler characteristic}
-
-It is not straightforward to directly use the average Euler characteristic to
-infer something about the number of connected components in the set of
-solutions. To understand why, a simple example is helpful. Consider the set of
-solutions on the sphere $\|\mathbf x\|^2=N$ that satisfy the single quadratic
-constraint
-\begin{equation}
-  0=\sum_{i=1}^N\sigma_ix_i^2
-\end{equation}
-where each $\sigma_i$ is taken to be $\pm1$ with equal probability. If we take $\mathbf x$ to be ordered such that all terms with $\sigma_i=+1$ come first, this gives
-\begin{equation}
-  0=\sum_{i=1}^{N_+}x_i^2-\sum_{i=N_++1}^Nx_i^2
-\end{equation}
-where $N_+$ is the number of terms with $\sigma_i=+1$. The topology of the resulting manifold can be found by adding and subtracting this constraint from the spherical one, which gives
-\begin{align}
-  \frac12=\sum_{i=1}^{N_+}x_i^2
-  \qquad
-  \frac12=\sum_{i=N_++1}^{N}x_i^2
-\end{align}
-These are two independent equations for spheres of radius $1/\sqrt2$, one of
-dimension $N_+$ and the other of dimension $N-N_+$. Therefore, the topology of
-the configuration space is that of $S^{N_+-1}\times S^{N-N_+-1}$. The Euler
-characteristic of a product space is the product of the Euler characteristics,
-and so we have $\chi(\Omega)=\chi(S^{N_+-1})\chi(S^{N-N_+-1})$.
-
-What is the average value of the Euler characteristic over values of
-$\sigma_i$? First, recall that the Euler characteristic of a sphere $S^d$ is 2
-when $d$ is even and 0 when $d$ is odd. When $N$ is odd, any value
-of $N_+$ will result in one of the two spheres in the product to be
-odd-dimensional, and therefore $\chi(\Omega)=0$, as is always true for
-odd-dimensional manifolds. When $N$ is even, there are two possibilities: when $N_+$ is even then both spheres are odd-dimensional, while when $N_+$ is odd then both spheres are even-dimensional.
-The number of terms $N_+$ with $\sigma_i=+1$ is distributed with the binomial distribution
-\begin{equation}
-  P(N_+)=\frac1{2^N}\binom{N}{N_+}
-\end{equation}
-Therefore, the average Euler characteristic for even $N$ is
-\begin{equation}
-  \overline{\chi(\Omega)}
-  =\sum_{N_+=0}^NP(N_+)\chi(S^{N_+-1})\chi(S^{N-N_+-1})
-  =\frac4{2^N}\sum_{n=0}^{N/2}\binom{N}{2n}
-  =2
-\end{equation}
-Thus we find the average Euler characteristic in this simple example is 2
-despite the fact that the possible manifolds resulting from the constraints
-have characteristics of either 0 or 4.
-
 The many positive values the Euler characteristic can take is one problem, but
 a larger one is the many negative values it can take. A manifold with many
 handles will have a very negative Euler characteristic, and our calculation
@@ -439,6 +409,8 @@ that in the complex region, the large-deviation function in $m$ that $\mathcal
 S_\Omega(m)$ represents breaks down due to the vanishingly small probability of
 finding any stationary points.
 
+
+
 \subsection{Complexity of a linear test function}
 
 One way to definitely narrow possible interpretations of the average Euler
@@ -493,6 +465,109 @@ Appendix~\ref{sec:complexity.details}. The result is that, to largest order in
 $N$, the logarithm of the average number of stationary points is the same as
 the logarithm of the average Euler characteristic.
 
+\subsection{How to interpret these calculations}
+
+It is not straightforward to directly use the average Euler characteristic to
+infer something about the number of connected components in the set of
+solutions. To understand why, a simple example is helpful. Consider the set of
+solutions on the sphere $\|\mathbf x\|^2=N$ that satisfy the single quadratic
+constraint
+\begin{equation}
+  0=\sum_{i=1}^N\sigma_ix_i^2
+\end{equation}
+where each $\sigma_i$ is taken to be $\pm1$ with equal probability. If we take $\mathbf x$ to be ordered such that all terms with $\sigma_i=+1$ come first, this gives
+\begin{equation}
+  0=\sum_{i=1}^{N_+}x_i^2-\sum_{i=N_++1}^Nx_i^2
+\end{equation}
+where $N_+$ is the number of terms with $\sigma_i=+1$. The topology of the resulting manifold can be found by adding and subtracting this constraint from the spherical one, which gives
+\begin{align}
+  \frac12=\sum_{i=1}^{N_+}x_i^2
+  \qquad
+  \frac12=\sum_{i=N_++1}^{N}x_i^2
+\end{align}
+These are two independent equations for spheres of radius $1/\sqrt2$, one of
+dimension $N_+$ and the other of dimension $N-N_+$. Therefore, the topology of
+the configuration space is that of $S^{N_+-1}\times S^{N-N_+-1}$. The Euler
+characteristic of a product space is the product of the Euler characteristics,
+and so we have $\chi(\Omega)=\chi(S^{N_+-1})\chi(S^{N-N_+-1})$.
+
+What is the average value of the Euler characteristic over values of
+$\sigma_i$? First, recall that the Euler characteristic of a sphere $S^d$ is 2
+when $d$ is even and 0 when $d$ is odd. When $N$ is odd, any value
+of $N_+$ will result in one of the two spheres in the product to be
+odd-dimensional, and therefore $\chi(\Omega)=0$, as is always true for
+odd-dimensional manifolds. When $N$ is even, there are two possibilities: when $N_+$ is even then both spheres are odd-dimensional, while when $N_+$ is odd then both spheres are even-dimensional.
+The number of terms $N_+$ with $\sigma_i=+1$ is distributed with the binomial distribution
+\begin{equation}
+  P(N_+)=\frac1{2^N}\binom{N}{N_+}
+\end{equation}
+Therefore, the average Euler characteristic for even $N$ is
+\begin{equation}
+  \overline{\chi(\Omega)}
+  =\sum_{N_+=0}^NP(N_+)\chi(S^{N_+-1})\chi(S^{N-N_+-1})
+  =\frac4{2^N}\sum_{n=0}^{N/2}\binom{N}{2n}
+  =2
+\end{equation}
+Thus we find the average Euler characteristic in this simple example is 2
+despite the fact that the possible manifolds resulting from the constraints
+have characteristics of either 0 or 4.
+
+\begin{figure}
+  \includegraphics{figs/regime_1.pdf}
+  \hspace{-3em}
+  \includegraphics{figs/regime_2.pdf}
+  \hspace{-3em}
+  \includegraphics{figs/regime_3.pdf}
+
+  \caption{
+    \textbf{Behavior of the action in our three nontrivial regimes.} The effective action $\mathcal S_\Omega$ as a function of overlap $m$ with
+    the height axis for our model with $f(q)=\frac12q^3$ and $\alpha=\frac12$
+    at three different target values $V_0$. \textbf{Left: the connected
+    regime.} The action is maximized with $\mathcal S_\Omega(m^*)=0$, and no
+    stationary points are found with overlap less than $m_\text{min}$ with a
+    random point on the sphere. \textbf{Center: the onset regime.} The action
+    is maximized with $\mathcal S_\Omega(m_\text{min})>0$, and is positive up
+    to $m_\text{max}$. No stationary points are found with overlap less than $m_\text{min}$. \textbf{Right: the shattered regime.} The action is
+    maximized with $\mathcal S_\Omega(0)>0$, and is positive up to
+    $m_\text{max}$.
+  }
+\end{figure}
+
+\paragraph{The connected regime: \boldmath{$V_0^2<V_\text{on}$}.}
+
+In our calculation above, $\overline{\chi(\Omega)}=2$ could mean a fine-tuned
+average like this, or it could indicate the presence of manifold homeomorphic
+to $S^{N-M-1}$ for even $N-M-1$. In either case it strongly indicates a single
+connected component, whose few minima and maxima are usually found at the
+latitude $m^*$. Randomly chosen points on the sphere have a typical nearest
+overlap $m^*$ with the solution manifold, but can never have a smaller overlap than
+$m_\text{min}$, indicating that the manifold is extensive.
+
+\paragraph{The onset regime: \boldmath{$V_\text{on}^2<V_0^2<V_\text{sh}$}.}
+
+In this regime $\log\overline{\chi(\Omega)}=O(N)$ but a minimum overlap
+$m_\text{min}>0$ still exists. The minimum overlap indicates that the solution
+manifold is exclusively made up of extensive components, because the existence
+of small components would lead to stationary points near the equator with
+respect to a randomly chosen axis $\mathbf x_0$. The solution manifold is
+homeomorphic to a topological space with large Euler characteristic like the
+product of union of many spheres. In the former case we would have one
+topologically nontrivial connected component, while in the latter we would have
+many simple disconnected components; the reality could be a combination of the
+two. It the framework of this calculation, it is not possible to distinguish between
+these scenarios. In any case, the minima and maxima of the height on the
+solution manifold are typically found at latitude $m_\text{min}$ but are found
+in exponential number up to the latitude $m_\text{max}$.
+
+\paragraph{The shattered regime: \boldmath{$V_\text{sh}^2<V_0^2<V_\text{\textsc{sat}}^2$}.}
+
+Here $\log\overline{\chi(\Omega)}=O(N)$ and the primary contribution to the
+number of stationary points and to the Euler characteristic comes from the
+equator. This indicates the presence of a large number of small disconnected
+components to the solution manifold. While most minima and maxima of the height
+are located at the equator $m=0$, they are found in exponential number up to
+the latitude $m_\text{max}$.
+
 \section{Results}
 
 \subsection{Topology of solutions to many equations and the satisfiability transition}
@@ -545,6 +620,8 @@ topology of the solution manifold.
 
 \subsection{Topology of level sets of the spherical spin glasses and the dynamic threshold}
 
+\cite{Folena_2020_Rethinking, Folena_2021_Gradient}
+
 As indicated earlier, for $M=1$ the solution manifold corresponds to the energy
 level set of a spherical spin glass with energy density $E=\sqrt NV_0$. All the
 results from the previous sections follow, and can be translated to the spin
author	Jaron Kent-Dobias <jaron@kent-dobias.com>	2024-08-27 18:35:29 +0200
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