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\bibcite{Serafin:2023pkf}{{2}{2024}{{Serafin\ \emph {et~al.}}}{{Serafin, G\'omez-Rocha, More,\ and\ G\l {}azek}}}
\bibcite{Wilson:1994fk}{{3}{1994}{{Wilson\ \emph {et~al.}}}{{Wilson, Walhout, Harindranath, Zhang, Perry,\ and\ Glazek}}}
\bibcite{Shifman:1978bx}{{4}{1979{}}{{Shifman\ \emph {et~al.}}}{{Shifman, Vainshtein,\ and\ Zakharov}}}
\bibcite{Shifman:1978by}{{5}{1979{}}{{Shifman\ \emph {et~al.}}}{{Shifman, Vainshtein,\ and\ Zakharov}}}
\bibcite{Shuryak:2026pqt}{{6}{2026}{{Shuryak\ and\ Zahed}}{{}}}
\bibcite{Casher:1973vh}{{7}{1973}{{Casher\ and\ Susskind}}{{}}}
\bibcite{Kogut:1972di}{{8}{1973}{{Kogut\ and\ Susskind}}{{}}}
\bibcite{Gao:2021dbh}{{9}{2022}{{Gao\ \emph {et~al.}}}{{Gao, Hanlon, Mukherjee, Petreczky, Scior, Syritsyn,\ and\ Zhao}}}
\bibcite{Ji:2024oka}{{10}{2024}{{Ji}}{{}}}
\bibcite{Dirac:1949cp}{{11}{1949}{{Dirac}}{{}}}
\bibcite{ParticleDataGroup:2024cfk}{{12}{2024}{{Navas\ \emph {et~al.}}}{{Navas \emph {et~al.}}}}
\bibcite{Soper:1971wn}{{13}{1971}{{Soper}}{{}}}
\bibcite{Yan:1973qf}{{14}{1973{}}{{Yan}}{{}}}
\bibcite{Yan:1973qg}{{15}{1973{}}{{Yan}}{{}}}
\bibcite{Lepage:1980fj}{{16}{1980}{{Lepage\ and\ Brodsky}}{{}}}
\bibcite{Namyslowski:1985zq}{{17}{1985}{{Namyslowski}}{{}}}
\bibcite{Burkardt:1995ct}{{18}{1996}{{Burkardt}}{{}}}
\bibcite{Brodsky:1997de}{{19}{1998}{{Brodsky\ \emph {et~al.}}}{{Brodsky, Pauli,\ and\ Pinsky}}}
\bibcite{Carbonell:1998rj}{{20}{1998}{{Carbonell\ \emph {et~al.}}}{{Carbonell, Desplanques, Karmanov,\ and\ Mathiot}}}
\bibcite{Brodsky:2014yha}{{21}{2015}{{Brodsky\ \emph {et~al.}}}{{Brodsky, de~Teramond, Dosch,\ and\ Erlich}}}
\bibcite{Ji:2020ect}{{22}{2021}{{Ji\ \emph {et~al.}}}{{Ji, Liu, Liu, Zhang,\ and\ Zhao}}}
\bibcite{Lepage:1982gd}{{23}{1982}{{Lepage\ \emph {et~al.}}}{{Lepage, Brodsky, Huang,\ and\ Mackenzie}}}
\bibcite{Stueckelberg:1938zz}{{24}{1938}{{Stueckelberg}}{{}}}
\bibcite{Glazek:2020xpi}{{25}{2020}{{G\l {}azek}}{{}}}
\bibcite{Kunimasa:1967zza}{{26}{1967}{{Kunimasa\ and\ Goto}}{{}}}
\bibcite{Fritzsch:1972jv}{{27}{1972}{{Fritzsch\ and\ Gell-Mann}}{{}}}
\bibcite{Gell-Mann:1953dcn}{{28}{1953}{{Gell-Mann\ and\ Goldberger}}{{}}}
\bibcite{Weinberg:1995mt}{{29}{1995}{{Weinberg}}{{}}}
\bibcite{Harindranath:1987ex}{{30}{1988}{{Harindranath\ and\ Vary}}{{}}}
\bibcite{Bartnik:1988fu}{{31}{1989}{{Bartnik\ and\ Glazek}}{{}}}
\bibcite{Glazek:1986hs}{{32}{1988}{{Glazek}}{{}}}
\bibcite{Glazek:1997sd}{{33}{1998}{{Glazek}}{{}}}
\bibcite{Glazek:2010zr}{{34}{2010{}}{{Glazek}}{{}}}
\bibcite{Glazek:2010zza}{{35}{2010{}}{{Glazek}}{{}}}
\@writefile{toc}{\contentsline {section}{\numberline {}References}{119}{section*.93}\protected@file@percent }
\bibcite{Bjorken:1979hv}{{36}{1979}{{Bjorken}}{{}}}
\bibcite{Wilson:1965zzb}{{37}{1965}{{Wilson}}{{}}}
\bibcite{Wilson:1970tp}{{38}{1970}{{Wilson}}{{}}}
\bibcite{Glazek:1993rc}{{39}{1993}{{Glazek\ and\ Wilson}}{{}}}
\bibcite{Glazek:1994qc}{{40}{1994}{{Glazek\ and\ Wilson}}{{}}}
\bibcite{Glazek:1997gt}{{41}{1998}{{Glazek\ and\ Wilson}}{{}}}
\bibcite{Glazek:2006mg}{{42}{2007}{{Glazek}}{{}}}
\bibcite{Wegner:1994fdg}{{43}{1994}{{Wegner}}{{}}}
\bibcite{Mielke:1998ag}{{44}{1998}{{Mielke}}{{}}}
\bibcite{Wegner_2006}{{45}{2006}{{Wegner}}{{}}}
\bibcite{Kehrein:2006ti}{{46}{2006}{{Kehrein}}{{}}}
\bibcite{PhysRevX.9.021037}{{47}{2019}{{Vogl\ \emph {et~al.}}}{{Vogl, Laurell, Barr,\ and\ Fiete}}}
\bibcite{Deift1983OrdinaryDE}{{48}{1983}{{Deift\ \emph {et~al.}}}{{Deift, Nanda,\ and\ Tomei}}}
\bibcite{doi:10.1137/1030090}{{49}{1988}{{Chu}}{{}}}
\bibcite{BROCKETT199179}{{50}{1991}{{Brockett}}{{}}}
\bibcite{Glazek:2002iw}{{51}{2003}{{Glazek\ and\ Mlynik}}{{}}}
\bibcite{Glazek:2003xr}{{52}{2004}{{Glazek\ and\ Mlynik}}{{}}}
\bibcite{Glazek:2021vnw}{{53}{2021}{{G\l {}azek}}{{}}}
\bibcite{Glazek:2012qj}{{54}{2012}{{Glazek}}{{}}}
\bibcite{Appelquist:1974zd}{{55}{1975}{{Appelquist\ and\ Politzer}}{{}}}
\bibcite{Appelquist:1974yr}{{56}{1975}{{Appelquist\ \emph {et~al.}}}{{Appelquist, De~Rujula, Politzer,\ and\ Glashow}}}
\bibcite{Eichten:1978tg}{{57}{1978}{{Eichten\ \emph {et~al.}}}{{Eichten, Gottfried, Kinoshita, Lane,\ and\ Yan}}}
\bibcite{Eichten:1979ms}{{58}{1980}{{Eichten\ \emph {et~al.}}}{{Eichten, Gottfried, Kinoshita, Lane,\ and\ Yan}}}
\bibcite{Wilson:1974sk}{{59}{1974}{{Wilson}}{{}}}
\bibcite{Bodwin:1994jh}{{60}{1995}{{Bodwin\ \emph {et~al.}}}{{Bodwin, Braaten,\ and\ Lepage}}}
\bibcite{Brambilla:2010cs}{{61}{2011}{{Brambilla\ \emph {et~al.}}}{{Brambilla \emph {et~al.}}}}
\bibcite{Glazek:2014ria}{{62}{2014}{{Glazek}}{{}}}
\bibcite{Glazek:2003ky}{{63}{2004}{{Glazek}}{{}}}
\bibcite{Tamm:1945qv}{{64}{1945}{{Tamm}}{{}}}
\bibcite{Dancoff:1950ud}{{65}{1950}{{Dancoff}}{{}}}
\bibcite{Perry:1990mz}{{66}{1990}{{Perry\ \emph {et~al.}}}{{Perry, Harindranath,\ and\ Wilson}}}
\bibcite{Perry:1991ny}{{67}{1991}{{Perry\ and\ Harindranath}}{{}}}
\bibcite{Melosh:1974cu}{{68}{1974}{{Melosh}}{{}}}
\bibcite{Gell-Mann:1992zxl}{{69}{1992}{{Gell-Mann}}{{}}}
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