% Generated companion labels; no other volume's .aux file is needed. \makeatletter \expandafter\gdef\csname r@c:cons-product-param\endcsname{{18.2}{276}{}{corollary.757}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:cons-product-param\endcsname{\default{18.2}\page{276}\abspage{280}\zc@counter{corollary}\thecounter{18.2}\zc@type{corollary}\zc@cntval{2}\zc@pgval{276}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{corollary.757}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:end-margin\endcsname{{19.12}{323}{}{corollary.882}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:end-margin\endcsname{\default{19.12}\page{323}\abspage{327}\zc@counter{corollary}\thecounter{19.12}\zc@type{corollary}\zc@cntval{12}\zc@pgval{323}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{corollary.882}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:end-ramp\endcsname{{19.11}{322}{}{corollary.881}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:end-ramp\endcsname{\default{19.11}\page{322}\abspage{326}\zc@counter{corollary}\thecounter{19.11}\zc@type{corollary}\zc@cntval{11}\zc@pgval{322}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{corollary.881}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:end-residual\endcsname{{19.20}{341}{}{proposition.932}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:end-residual\endcsname{\default{19.20}\page{341}\abspage{345}\zc@counter{proposition}\thecounter{19.20}\zc@type{proposition}\zc@cntval{20}\zc@pgval{341}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{proposition.932}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:extract-onepoint\endcsname{{9.8}{38}{}{corollary.119}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:extract-onepoint\endcsname{\default{9.8}\page{38}\abspage{42}\zc@counter{corollary}\thecounter{9.8}\zc@type{corollary}\zc@cntval{8}\zc@pgval{38}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{corollary.119}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:extract-windows\endcsname{{9.9}{39}{}{corollary.121}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:extract-windows\endcsname{\default{9.9}\page{39}\abspage{43}\zc@counter{corollary}\thecounter{9.9}\zc@type{corollary}\zc@cntval{9}\zc@pgval{39}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{corollary.121}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:met-kernel\endcsname{{13.10}{126}{}{corollary.355}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:met-kernel\endcsname{\default{13.10}\page{126}\abspage{130}\zc@counter{corollary}\thecounter{13.10}\zc@type{corollary}\zc@cntval{10}\zc@pgval{126}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{corollary.355}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:proc-complex-source\endcsname{{8.8}{24}{}{corollary.82}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:proc-complex-source\endcsname{\default{8.8}\page{24}\abspage{28}\zc@counter{corollary}\thecounter{8.8}\zc@type{corollary}\zc@cntval{8}\zc@pgval{24}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{corollary.82}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:res-chains-polar\endcsname{{10.14}{64}{}{corollary.196}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:res-chains-polar\endcsname{\default{10.14}\page{64}\abspage{68}\zc@counter{corollary}\thecounter{10.14}\zc@type{corollary}\zc@cntval{14}\zc@pgval{64}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{corollary.196}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:stab-parity\endcsname{{17.9}{253}{}{corollary.694}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:stab-parity\endcsname{\default{17.9}\page{253}\abspage{257}\zc@counter{corollary}\thecounter{17.9}\zc@type{corollary}\zc@cntval{9}\zc@pgval{253}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{corollary.694}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@c:tools-placements\endcsname{{A.2}{380}{}{corollary.1042}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@c:tools-placements\endcsname{\default{A.2}\page{380}\abspage{384}\zc@counter{corollary}\thecounter{A.2}\zc@type{corollary}\zc@cntval{2}\zc@pgval{380}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{corollary.1042}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@d:coarse-permanent\endcsname{{12.1}{90}{}{definition.258}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@d:coarse-permanent\endcsname{\default{12.1}\page{90}\abspage{94}\zc@counter{definition}\thecounter{12.1}\zc@type{definition}\zc@cntval{1}\zc@pgval{90}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{definition.258}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@d:end-generated\endcsname{{19.1}{303}{}{definition.823}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@d:end-generated\endcsname{\default{19.1}\page{303}\abspage{307}\zc@counter{definition}\thecounter{19.1}\zc@type{definition}\zc@cntval{1}\zc@pgval{303}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{definition.823}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@d:extract-measures\endcsname{{9.1}{29}{}{definition.91}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@d:extract-measures\endcsname{\default{9.1}\page{29}\abspage{33}\zc@counter{definition}\thecounter{9.1}\zc@type{definition}\zc@cntval{1}\zc@pgval{29}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{definition.91}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@d:hist-marks\endcsname{{15.2}{160}{}{definition.458}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@d:hist-marks\endcsname{\default{15.2}\page{160}\abspage{164}\zc@counter{definition}\thecounter{15.2}\zc@type{definition}\zc@cntval{2}\zc@pgval{160}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{definition.458}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@d:iface-local\endcsname{{7.5}{9}{}{definition.30}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@d:iface-local\endcsname{\default{7.5}\page{9}\abspage{13}\zc@counter{definition}\thecounter{7.5}\zc@type{definition}\zc@cntval{5}\zc@pgval{9}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{definition.30}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@d:res-chain\endcsname{{10.6}{53}{}{definition.164}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@d:res-chain\endcsname{\default{10.6}\page{53}\abspage{57}\zc@counter{definition}\thecounter{10.6}\zc@type{definition}\zc@cntval{6}\zc@pgval{53}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{definition.164}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@d:tube-cycle\endcsname{{16.1}{202}{}{definition.572}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@d:tube-cycle\endcsname{\default{16.1}\page{202}\abspage{206}\zc@counter{definition}\thecounter{16.1}\zc@type{definition}\zc@cntval{1}\zc@pgval{202}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{definition.572}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@d:tube-raw\endcsname{{16.7}{213}{}{definition.602}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@d:tube-raw\endcsname{\default{16.7}\page{213}\abspage{217}\zc@counter{definition}\thecounter{16.7}\zc@type{definition}\zc@cntval{7}\zc@pgval{213}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{definition.602}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@d:tube-wick\endcsname{{16.6}{211}{}{definition.597}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@d:tube-wick\endcsname{\default{16.6}\page{211}\abspage{215}\zc@counter{definition}\thecounter{16.6}\zc@type{definition}\zc@cntval{6}\zc@pgval{211}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{definition.597}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-Eout\endcsname{{20.2}{352}{Conventions and the order of choices}{equation.959}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-Eout\endcsname{\default{20.2}\page{352}\abspage{356}\zc@counter{equation}\thecounter{20.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{352}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.959}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-adapted\endcsname{{20.25}{366}{}{equation.1003}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-adapted\endcsname{\default{20.25}\page{366}\abspage{370}\zc@counter{equation}\thecounter{20.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{366}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1003}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-baseline\endcsname{{20.28}{368}{Normalization}{equation.1007}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-baseline\endcsname{\default{20.28}\page{368}\abspage{372}\zc@counter{equation}\thecounter{20.28}\zc@type{equation}\zc@cntval{28}\zc@pgval{368}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1007}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-block\endcsname{{20.6}{353}{The chain in one-sided form}{equation.969}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-block\endcsname{\default{20.6}\page{353}\abspage{357}\zc@counter{equation}\thecounter{20.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{353}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.969}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-cutoff\endcsname{{19.40}{331}{The physical cutoff}{equation.900}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-cutoff\endcsname{\default{19.40}\page{331}\abspage{335}\zc@counter{equation}\thecounter{19.40}\zc@type{equation}\zc@cntval{40}\zc@pgval{331}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.900}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-endpoints\endcsname{{20.5}{353}{The chain in one-sided form}{equation.968}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-endpoints\endcsname{\default{20.5}\page{353}\abspage{357}\zc@counter{equation}\thecounter{20.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{353}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.968}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-entrance\endcsname{{20.23}{364}{Replacement of the physical chain}{equation.998}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-entrance\endcsname{\default{20.23}\page{364}\abspage{368}\zc@counter{equation}\thecounter{20.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{364}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.998}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-every-block\endcsname{{20.34}{372}{The chain with jets}{equation.1022}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-every-block\endcsname{\default{20.34}\page{372}\abspage{376}\zc@counter{equation}\thecounter{20.34}\zc@type{equation}\zc@cntval{34}\zc@pgval{372}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1022}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-generator\endcsname{{18.29}{292}{The tangent generator}{equation.797}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-generator\endcsname{\default{18.29}\page{292}\abspage{296}\zc@counter{equation}\thecounter{18.29}\zc@type{equation}\zc@cntval{29}\zc@pgval{292}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.797}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-homotopy\endcsname{{19.39}{330}{The physical cutoff}{equation.895}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-homotopy\endcsname{\default{19.39}\page{330}\abspage{334}\zc@counter{equation}\thecounter{19.39}\zc@type{equation}\zc@cntval{39}\zc@pgval{330}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.895}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-incoming\endcsname{{20.35}{373}{}{equation.1024}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-incoming\endcsname{\default{20.35}\page{373}\abspage{377}\zc@counter{equation}\thecounter{20.35}\zc@type{equation}\zc@cntval{35}\zc@pgval{373}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1024}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-initial-layer\endcsname{{20.13}{356}{}{equation.981}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-initial-layer\endcsname{\default{20.13}\page{356}\abspage{360}\zc@counter{equation}\thecounter{20.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{356}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.981}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-jet\endcsname{{20.29}{369}{Local sources: proof of part (b)}{equation.1009}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-jet\endcsname{\default{20.29}\page{369}\abspage{373}\zc@counter{equation}\thecounter{20.29}\zc@type{equation}\zc@cntval{29}\zc@pgval{369}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1009}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-jet-block\endcsname{{20.33}{371}{}{equation.1020}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-jet-block\endcsname{\default{20.33}\page{371}\abspage{375}\zc@counter{equation}\thecounter{20.33}\zc@type{equation}\zc@cntval{33}\zc@pgval{371}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1020}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-jet-ratio\endcsname{{20.32}{371}{The chain with jets}{equation.1017}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-jet-ratio\endcsname{\default{20.32}\page{371}\abspage{375}\zc@counter{equation}\thecounter{20.32}\zc@type{equation}\zc@cntval{32}\zc@pgval{371}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1017}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-killing-sum\endcsname{{17.40}{271}{Gain of body weights}{equation.741}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-killing-sum\endcsname{\default{17.40}\page{271}\abspage{275}\zc@counter{equation}\thecounter{17.40}\zc@type{equation}\zc@cntval{40}\zc@pgval{271}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.741}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-layer-length\endcsname{{20.12}{355}{The initial layer}{equation.979}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-layer-length\endcsname{\default{20.12}\page{355}\abspage{359}\zc@counter{equation}\thecounter{20.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{355}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.979}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-layer-step\endcsname{{20.14}{356}{The initial layer}{equation.982}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-layer-step\endcsname{\default{20.14}\page{356}\abspage{360}\zc@counter{equation}\thecounter{20.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{356}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.982}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-local-ratio\endcsname{{20.30}{369}{Local sources: proof of part (b)}{equation.1010}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-local-ratio\endcsname{\default{20.30}\page{369}\abspage{373}\zc@counter{equation}\thecounter{20.30}\zc@type{equation}\zc@cntval{30}\zc@pgval{369}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1010}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-moment-identity\endcsname{{20.36}{375}{All-site moments: proof of part (c)}{equation.1026}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-moment-identity\endcsname{\default{20.36}\page{375}\abspage{379}\zc@counter{equation}\thecounter{20.36}\zc@type{equation}\zc@cntval{36}\zc@pgval{375}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1026}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-normal-pairing\endcsname{{20.11}{355}{The initial layer}{equation.978}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-normal-pairing\endcsname{\default{20.11}\page{355}\abspage{359}\zc@counter{equation}\thecounter{20.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{355}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.978}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-one-sided\endcsname{{20.7}{353}{The chain in one-sided form}{equation.970}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-one-sided\endcsname{\default{20.7}\page{353}\abspage{357}\zc@counter{equation}\thecounter{20.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{353}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.970}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-physical\endcsname{{20.24}{365}{}{equation.1000}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-physical\endcsname{\default{20.24}\page{365}\abspage{369}\zc@counter{equation}\thecounter{20.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{365}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1000}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-prep-gain\endcsname{{20.22}{363}{}{equation.995}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-prep-gain\endcsname{\default{20.22}\page{363}\abspage{367}\zc@counter{equation}\thecounter{20.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{363}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.995}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-projection\endcsname{{20.31}{369}{}{equation.1013}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-projection\endcsname{\default{20.31}\page{369}\abspage{373}\zc@counter{equation}\thecounter{20.31}\zc@type{equation}\zc@cntval{31}\zc@pgval{369}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1013}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-ratio\endcsname{{20.1}{350}{Proof of the interface theorem}{equation.953}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-ratio\endcsname{\default{20.1}\page{350}\abspage{354}\zc@counter{equation}\thecounter{20.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{350}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.953}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-recenter\endcsname{{20.3}{352}{}{equation.965}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-recenter\endcsname{\default{20.3}\page{352}\abspage{356}\zc@counter{equation}\thecounter{20.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{352}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.965}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-recenter-bounds\endcsname{{20.4}{353}{}{equation.966}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-recenter-bounds\endcsname{\default{20.4}\page{353}\abspage{357}\zc@counter{equation}\thecounter{20.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{353}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.966}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-residual\endcsname{{20.15}{358}{The telescoping comparison}{equation.984}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-residual\endcsname{\default{20.15}\page{358}\abspage{362}\zc@counter{equation}\thecounter{20.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{358}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.984}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-returned\endcsname{{20.17}{360}{}{equation.988}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-returned\endcsname{\default{20.17}\page{360}\abspage{364}\zc@counter{equation}\thecounter{20.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{360}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.988}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-returned-density\endcsname{{20.20}{362}{The telescoping comparison}{equation.991}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-returned-density\endcsname{\default{20.20}\page{362}\abspage{366}\zc@counter{equation}\thecounter{20.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{362}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.991}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-sandwich\endcsname{{17.41}{271}{Gain of body weights}{equation.742}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-sandwich\endcsname{\default{17.41}\page{271}\abspage{275}\zc@counter{equation}\thecounter{17.41}\zc@type{equation}\zc@cntval{41}\zc@pgval{271}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.742}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-segment-limit\endcsname{{20.27}{367}{The limit of the adapted chain}{equation.1005}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-segment-limit\endcsname{\default{20.27}\page{367}\abspage{371}\zc@counter{equation}\thecounter{20.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{367}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1005}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-shell-homotopy\endcsname{{20.19}{361}{The telescoping comparison}{equation.990}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-shell-homotopy\endcsname{\default{20.19}\page{361}\abspage{365}\zc@counter{equation}\thecounter{20.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{361}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.990}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-source-ball\endcsname{{20.8}{354}{The chain in one-sided form}{equation.971}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-source-ball\endcsname{\default{20.8}\page{354}\abspage{358}\zc@counter{equation}\thecounter{20.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{354}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.971}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-stokes\endcsname{{20.18}{360}{The telescoping comparison}{equation.989}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-stokes\endcsname{\default{20.18}\page{360}\abspage{364}\zc@counter{equation}\thecounter{20.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{360}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.989}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-telescope\endcsname{{20.16}{359}{}{equation.986}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-telescope\endcsname{\default{20.16}\page{359}\abspage{363}\zc@counter{equation}\thecounter{20.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{359}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.986}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-terminal\endcsname{{20.21}{362}{}{equation.993}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-terminal\endcsname{\default{20.21}\page{362}\abspage{366}\zc@counter{equation}\thecounter{20.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{362}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.993}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-three-pieces\endcsname{{20.26}{366}{The limit of the adapted chain}{equation.1004}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-three-pieces\endcsname{\default{20.26}\page{366}\abspage{370}\zc@counter{equation}\thecounter{20.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{366}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.1004}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-time\endcsname{{18.30}{292}{}{equation.799}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-time\endcsname{\default{18.30}\page{292}\abspage{296}\zc@counter{equation}\thecounter{18.30}\zc@type{equation}\zc@cntval{30}\zc@pgval{292}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.799}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-transpose\endcsname{{20.9}{354}{Three properties of the adapted chain}{equation.973}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-transpose\endcsname{\default{20.9}\page{354}\abspage{358}\zc@counter{equation}\thecounter{20.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{354}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.973}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-unbalanced-gap\endcsname{{20.10}{354}{}{equation.976}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-unbalanced-gap\endcsname{\default{20.10}\page{354}\abspage{358}\zc@counter{equation}\thecounter{20.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{354}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{equation.976}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-weight-crude\endcsname{{17.42}{272}{Gain of body weights}{equation.743}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-weight-crude\endcsname{\default{17.42}\page{272}\abspage{276}\zc@counter{equation}\thecounter{17.42}\zc@type{equation}\zc@cntval{42}\zc@pgval{272}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.743}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-weight-gain\endcsname{{17.39}{270}{}{equation.740}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-weight-gain\endcsname{\default{17.39}\page{270}\abspage{274}\zc@counter{equation}\thecounter{17.39}\zc@type{equation}\zc@cntval{39}\zc@pgval{270}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.740}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:asm-weights\endcsname{{17.38}{270}{Gain of body weights}{equation.738}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:asm-weights\endcsname{\default{17.38}\page{270}\abspage{274}\zc@counter{equation}\thecounter{17.38}\zc@type{equation}\zc@cntval{38}\zc@pgval{270}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.738}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-K-bound\endcsname{{12.17}{105}{The mean shape}{equation.296}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-K-bound\endcsname{\default{12.17}\page{105}\abspage{109}\zc@counter{equation}\thecounter{12.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{105}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.296}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-L1\endcsname{{12.10}{96}{}{equation.278}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-L1\endcsname{\default{12.10}\page{96}\abspage{100}\zc@counter{equation}\thecounter{12.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{96}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.278}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-consistency\endcsname{{12.8}{95}{}{equation.273}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-consistency\endcsname{\default{12.8}\page{95}\abspage{99}\zc@counter{equation}\thecounter{12.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{95}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.273}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-correlation-decay\endcsname{{12.21}{109}{}{equation.304}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-correlation-decay\endcsname{\default{12.21}\page{109}\abspage{113}\zc@counter{equation}\thecounter{12.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{109}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.304}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-iface-limit\endcsname{{12.7}{94}{Permanent moment measures}{equation.269}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-iface-limit\endcsname{\default{12.7}\page{94}\abspage{98}\zc@counter{equation}\thecounter{12.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{94}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.269}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-kl\endcsname{{12.14}{104}{}{equation.293}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-kl\endcsname{\default{12.14}\page{104}\abspage{108}\zc@counter{equation}\thecounter{12.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{104}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.293}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-kl-dlmf\endcsname{{12.15}{104}{The mean shape}{equation.294}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-kl-dlmf\endcsname{\default{12.15}\page{104}\abspage{108}\zc@counter{equation}\thecounter{12.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{104}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.294}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-kl-eigen\endcsname{{12.18}{105}{The mean shape}{equation.297}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-kl-eigen\endcsname{\default{12.18}\page{105}\abspage{109}\zc@counter{equation}\thecounter{12.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{105}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.297}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-kl-inversion\endcsname{{12.16}{105}{The mean shape}{equation.295}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-kl-inversion\endcsname{\default{12.16}\page{105}\abspage{109}\zc@counter{equation}\thecounter{12.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{105}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.295}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-mean-pairing\endcsname{{12.13}{103}{}{equation.291}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-mean-pairing\endcsname{\default{12.13}\page{103}\abspage{107}\zc@counter{equation}\thecounter{12.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{103}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.291}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-mirlin\endcsname{{12.1}{88}{The coarse eigenvector law}{equation.255}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-mirlin\endcsname{\default{12.1}\page{88}\abspage{92}\zc@counter{equation}\thecounter{12.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{88}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.255}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-moments\endcsname{{12.9}{95}{}{equation.275}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-moments\endcsname{\default{12.9}\page{95}\abspage{99}\zc@counter{equation}\thecounter{12.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{95}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.275}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-multiplier\endcsname{{12.6}{93}{Permanent moment measures}{equation.268}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-multiplier\endcsname{\default{12.6}\page{93}\abspage{97}\zc@counter{equation}\thecounter{12.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{93}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.268}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-pairing-recall\endcsname{{12.4}{93}{Permanent moment measures}{equation.265}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-pairing-recall\endcsname{\default{12.4}\page{93}\abspage{97}\zc@counter{equation}\thecounter{12.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{93}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.265}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-permanent-source\endcsname{{12.3}{90}{}{equation.264}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-permanent-source\endcsname{\default{12.3}\page{90}\abspage{94}\zc@counter{equation}\thecounter{12.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{90}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.264}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-permanent-transform\endcsname{{12.2}{90}{}{equation.262}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-permanent-transform\endcsname{\default{12.2}\page{90}\abspage{94}\zc@counter{equation}\thecounter{12.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{90}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.262}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-sdet-derivative\endcsname{{12.5}{93}{Permanent moment measures}{equation.266}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-sdet-derivative\endcsname{\default{12.5}\page{93}\abspage{97}\zc@counter{equation}\thecounter{12.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{93}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.266}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-shape-coupling\endcsname{{12.12}{102}{The infinite wire}{equation.288}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-shape-coupling\endcsname{\default{12.12}\page{102}\abspage{106}\zc@counter{equation}\thecounter{12.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{102}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.288}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-spectral\endcsname{{12.19}{106}{The mean shape}{equation.298}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-spectral\endcsname{\default{12.19}\page{106}\abspage{110}\zc@counter{equation}\thecounter{12.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{106}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.298}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-tensor\endcsname{{12.11}{97}{The coarse law}{equation.279}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-tensor\endcsname{\default{12.11}\page{97}\abspage{101}\zc@counter{equation}\thecounter{12.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{97}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.279}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:coarse-weight\endcsname{{12.20}{107}{The mean shape}{equation.299}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:coarse-weight\endcsname{\default{12.20}\page{107}\abspage{111}\zc@counter{equation}\thecounter{12.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{107}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{equation.299}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-annuli\endcsname{{18.7}{279}{The normal fixed state}{equation.763}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-annuli\endcsname{\default{18.7}\page{279}\abspage{283}\zc@counter{equation}\thecounter{18.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{279}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.763}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-casimir-multiple\endcsname{{18.31}{293}{The tangent generator}{equation.800}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-casimir-multiple\endcsname{\default{18.31}\page{293}\abspage{297}\zc@counter{equation}\thecounter{18.31}\zc@type{equation}\zc@cntval{31}\zc@pgval{293}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.800}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-charge-adjoint\endcsname{{18.12}{282}{The generator}{equation.773}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-charge-adjoint\endcsname{\default{18.12}\page{282}\abspage{286}\zc@counter{equation}\thecounter{18.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{282}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.773}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-collar\endcsname{{18.22}{287}{Identification with the Dirichlet heat}{equation.787}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-collar\endcsname{\default{18.22}\page{287}\abspage{291}\zc@counter{equation}\thecounter{18.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{287}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.787}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-conclusion\endcsname{{18.3}{275}{}{equation.756}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-conclusion\endcsname{\default{18.3}\page{275}\abspage{279}\zc@counter{equation}\thecounter{18.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{275}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.756}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-decomposition\endcsname{{18.8}{279}{The generator}{equation.766}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-decomposition\endcsname{\default{18.8}\page{279}\abspage{283}\zc@counter{equation}\thecounter{18.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{279}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.766}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-duhamel\endcsname{{18.36}{296}{Consistency}{equation.810}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-duhamel\endcsname{\default{18.36}\page{296}\abspage{300}\zc@counter{equation}\thecounter{18.36}\zc@type{equation}\zc@cntval{36}\zc@pgval{296}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.810}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-free-graph\endcsname{{18.26}{290}{}{equation.793}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-free-graph\endcsname{\default{18.26}\page{290}\abspage{294}\zc@counter{equation}\thecounter{18.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{290}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.793}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-gauged\endcsname{{18.23}{289}{The heat over one link}{equation.789}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-gauged\endcsname{\default{18.23}\page{289}\abspage{293}\zc@counter{equation}\thecounter{18.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{289}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.789}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-generator-bound\endcsname{{18.27}{290}{}{equation.794}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-generator-bound\endcsname{\default{18.27}\page{290}\abspage{294}\zc@counter{equation}\thecounter{18.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{290}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.794}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-graph\endcsname{{18.15}{285}{Identification with the Dirichlet heat}{equation.779}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-graph\endcsname{\default{18.15}\page{285}\abspage{289}\zc@counter{equation}\thecounter{18.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{285}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.779}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-graph-velocity\endcsname{{18.19}{286}{Identification with the Dirichlet heat}{equation.784}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-graph-velocity\endcsname{\default{18.19}\page{286}\abspage{290}\zc@counter{equation}\thecounter{18.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{286}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.784}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-homogeneous-chart\endcsname{{18.10}{280}{The generator}{equation.768}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-homogeneous-chart\endcsname{\default{18.10}\page{280}\abspage{284}\zc@counter{equation}\thecounter{18.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{280}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.768}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-horizontal-vacuum\endcsname{{18.34}{295}{}{equation.807}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-horizontal-vacuum\endcsname{\default{18.34}\page{295}\abspage{299}\zc@counter{equation}\thecounter{18.34}\zc@type{equation}\zc@cntval{34}\zc@pgval{295}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.807}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-hyp\endcsname{{18.2}{275}{}{equation.755}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-hyp\endcsname{\default{18.2}\page{275}\abspage{279}\zc@counter{equation}\thecounter{18.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{275}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.755}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-identification\endcsname{{18.18}{286}{}{equation.783}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-identification\endcsname{\default{18.18}\page{286}\abspage{290}\zc@counter{equation}\thecounter{18.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{286}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.783}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-inner-normal\endcsname{{18.32}{294}{}{equation.804}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-inner-normal\endcsname{\default{18.32}\page{294}\abspage{298}\zc@counter{equation}\thecounter{18.32}\zc@type{equation}\zc@cntval{32}\zc@pgval{294}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.804}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-kinetic\endcsname{{18.37}{296}{Consistency}{equation.811}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-kinetic\endcsname{\default{18.37}\page{296}\abspage{300}\zc@counter{equation}\thecounter{18.37}\zc@type{equation}\zc@cntval{37}\zc@pgval{296}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.811}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-moving\endcsname{{18.17}{286}{Identification with the Dirichlet heat}{equation.781}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-moving\endcsname{\default{18.17}\page{286}\abspage{290}\zc@counter{equation}\thecounter{18.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{286}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.781}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-normal-defect\endcsname{{18.33}{294}{}{equation.805}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-normal-defect\endcsname{\default{18.33}\page{294}\abspage{298}\zc@counter{equation}\thecounter{18.33}\zc@type{equation}\zc@cntval{33}\zc@pgval{294}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.805}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-orbit-metric\endcsname{{18.9}{279}{The generator}{equation.767}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-orbit-metric\endcsname{\default{18.9}\page{279}\abspage{283}\zc@counter{equation}\thecounter{18.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{279}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.767}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-pairing\endcsname{{18.6}{278}{}{equation.762}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-pairing\endcsname{\default{18.6}\page{278}\abspage{282}\zc@counter{equation}\thecounter{18.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{278}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.762}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-power\endcsname{{18.1}{275}{}{equation.754}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-power\endcsname{\default{18.1}\page{275}\abspage{279}\zc@counter{equation}\thecounter{18.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{275}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.754}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-pulled\endcsname{{18.16}{285}{Identification with the Dirichlet heat}{equation.780}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-pulled\endcsname{\default{18.16}\page{285}\abspage{289}\zc@counter{equation}\thecounter{18.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{285}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.780}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-raw-metric\endcsname{{18.28}{290}{The heat over one link}{equation.795}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-raw-metric\endcsname{\default{18.28}\page{290}\abspage{294}\zc@counter{equation}\thecounter{18.28}\zc@type{equation}\zc@cntval{28}\zc@pgval{290}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.795}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-relaxation\endcsname{{18.5}{277}{}{equation.761}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-relaxation\endcsname{\default{18.5}\page{277}\abspage{281}\zc@counter{equation}\thecounter{18.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{277}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.761}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-sectorial\endcsname{{18.11}{280}{}{equation.772}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-sectorial\endcsname{\default{18.11}\page{280}\abspage{284}\zc@counter{equation}\thecounter{18.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{280}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.772}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-sep-expansion\endcsname{{18.14}{284}{Gaussian separation}{equation.777}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-sep-expansion\endcsname{\default{18.14}\page{284}\abspage{288}\zc@counter{equation}\thecounter{18.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{284}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.777}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-separation\endcsname{{18.13}{283}{}{equation.776}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-separation\endcsname{\default{18.13}\page{283}\abspage{287}\zc@counter{equation}\thecounter{18.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{283}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.776}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-short-heat\endcsname{{18.25}{290}{}{equation.792}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-short-heat\endcsname{\default{18.25}\page{290}\abspage{294}\zc@counter{equation}\thecounter{18.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{290}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.792}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-state-bounds\endcsname{{18.4}{277}{}{equation.760}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-state-bounds\endcsname{\default{18.4}\page{277}\abspage{281}\zc@counter{equation}\thecounter{18.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{277}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.760}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-steps\endcsname{{18.24}{289}{The heat over one link}{equation.790}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-steps\endcsname{\default{18.24}\page{289}\abspage{293}\zc@counter{equation}\thecounter{18.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{289}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.790}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-target\endcsname{{18.35}{295}{}{equation.809}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-target\endcsname{\default{18.35}\page{295}\abspage{299}\zc@counter{equation}\thecounter{18.35}\zc@type{equation}\zc@cntval{35}\zc@pgval{295}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.809}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-velocity-bound\endcsname{{18.21}{287}{Identification with the Dirichlet heat}{equation.786}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-velocity-bound\endcsname{\default{18.21}\page{287}\abspage{291}\zc@counter{equation}\thecounter{18.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{287}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.786}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:cons-velocity-form\endcsname{{18.20}{287}{Identification with the Dirichlet heat}{equation.785}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:cons-velocity-form\endcsname{\default{18.20}\page{287}\abspage{291}\zc@counter{equation}\thecounter{18.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{287}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{equation.785}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:contour-V\endcsname{{B.7}{395}{The deformation lemma}{equation.1091}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:contour-V\endcsname{\default{B.7}\page{395}\abspage{399}\zc@counter{equation}\thecounter{B.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{395}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{equation.1091}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:contour-deform-bound\endcsname{{B.8}{395}{}{equation.1095}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:contour-deform-bound\endcsname{\default{B.8}\page{395}\abspage{399}\zc@counter{equation}\thecounter{B.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{395}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{equation.1095}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:contour-edge\endcsname{{B.5}{392}{}{equation.1087}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:contour-edge\endcsname{\default{B.5}\page{392}\abspage{396}\zc@counter{equation}\thecounter{B.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{392}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{equation.1087}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:contour-edge-identity\endcsname{{B.6}{392}{}{equation.1088}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:contour-edge-identity\endcsname{\default{B.6}\page{392}\abspage{396}\zc@counter{equation}\thecounter{B.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{392}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{equation.1088}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:contour-homotopy\endcsname{{B.2}{392}{Real parts along the deformation}{equation.1081}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:contour-homotopy\endcsname{\default{B.2}\page{392}\abspage{396}\zc@counter{equation}\thecounter{B.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{392}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{equation.1081}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:contour-stokes-majorant\endcsname{{B.1}{390}{}{equation.1077}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:contour-stokes-majorant\endcsname{\default{B.1}\page{390}\abspage{394}\zc@counter{equation}\thecounter{B.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{390}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{equation.1077}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:contour-vertex\endcsname{{B.3}{392}{}{equation.1084}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:contour-vertex\endcsname{\default{B.3}\page{392}\abspage{396}\zc@counter{equation}\thecounter{B.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{392}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{equation.1084}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:contour-vertex-bound\endcsname{{B.4}{392}{}{equation.1085}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:contour-vertex-bound\endcsname{\default{B.4}\page{392}\abspage{396}\zc@counter{equation}\thecounter{B.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{392}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{equation.1085}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-BB-deficit\endcsname{{15.35}{175}{The escape function and the onsite weight}{equation.503}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-BB-deficit\endcsname{\default{15.35}\page{175}\abspage{179}\zc@counter{equation}\thecounter{15.35}\zc@type{equation}\zc@cntval{35}\zc@pgval{175}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.503}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-Egeom\endcsname{{19.50}{336}{The energy threshold}{equation.920}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-Egeom\endcsname{\default{19.50}\page{336}\abspage{340}\zc@counter{equation}\thecounter{19.50}\zc@type{equation}\zc@cntval{50}\zc@pgval{336}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.920}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-M-bounds\endcsname{{19.12}{308}{The BB part of the current}{equation.843}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-M-bounds\endcsname{\default{19.12}\page{308}\abspage{312}\zc@counter{equation}\thecounter{19.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{308}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.843}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-actual-collar\endcsname{{19.44}{332}{}{equation.910}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-actual-collar\endcsname{\default{19.44}\page{332}\abspage{336}\zc@counter{equation}\thecounter{19.44}\zc@type{equation}\zc@cntval{44}\zc@pgval{332}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.910}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-barrier\endcsname{{19.62}{346}{}{equation.946}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-barrier\endcsname{\default{19.62}\page{346}\abspage{350}\zc@counter{equation}\thecounter{19.62}\zc@type{equation}\zc@cntval{62}\zc@pgval{346}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.946}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-barrier-duhamel\endcsname{{19.63}{346}{The barrier}{equation.947}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-barrier-duhamel\endcsname{\default{19.63}\page{346}\abspage{350}\zc@counter{equation}\thecounter{19.63}\zc@type{equation}\zc@cntval{63}\zc@pgval{346}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.947}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-boost\endcsname{{15.25}{168}{The boost}{equation.487}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-boost\endcsname{\default{15.25}\page{168}\abspage{172}\zc@counter{equation}\thecounter{15.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{168}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.487}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-boost-conj\endcsname{{15.26}{169}{The boost}{equation.488}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-boost-conj\endcsname{\default{15.26}\page{169}\abspage{173}\zc@counter{equation}\thecounter{15.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{169}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.488}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-canonical-equation\endcsname{{19.37}{326}{The canonical equation}{equation.889}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-canonical-equation\endcsname{\default{19.37}\page{326}\abspage{330}\zc@counter{equation}\thecounter{19.37}\zc@type{equation}\zc@cntval{37}\zc@pgval{326}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.889}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-canonical-frame\endcsname{{19.38}{326}{The canonical equation}{equation.890}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-canonical-frame\endcsname{\default{19.38}\page{326}\abspage{330}\zc@counter{equation}\thecounter{19.38}\zc@type{equation}\zc@cntval{38}\zc@pgval{326}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.890}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-chi0\endcsname{{15.31}{172}{The escape function and the onsite weight}{equation.497}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-chi0\endcsname{\default{15.31}\page{172}\abspage{176}\zc@counter{equation}\thecounter{15.31}\zc@type{equation}\zc@cntval{31}\zc@pgval{172}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.497}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-coercivity\endcsname{{19.26}{315}{}{equation.868}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-coercivity\endcsname{\default{19.26}\page{315}\abspage{319}\zc@counter{equation}\thecounter{19.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{315}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.868}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-commutator\endcsname{{19.52}{339}{The exponent and the body weight of the residual}{equation.925}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-commutator\endcsname{\default{19.52}\page{339}\abspage{343}\zc@counter{equation}\thecounter{19.52}\zc@type{equation}\zc@cntval{52}\zc@pgval{339}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.925}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-conjugated\endcsname{{19.8}{307}{The conjugated generator}{equation.836}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-conjugated\endcsname{\default{19.8}\page{307}\abspage{311}\zc@counter{equation}\thecounter{19.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{307}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.836}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-conjugated-form\endcsname{{19.9}{307}{The conjugated generator}{equation.837}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-conjugated-form\endcsname{\default{19.9}\page{307}\abspage{311}\zc@counter{equation}\thecounter{19.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{307}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.837}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-connection\endcsname{{15.30}{169}{Divided differences}{equation.493}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-connection\endcsname{\default{15.30}\page{169}\abspage{173}\zc@counter{equation}\thecounter{15.30}\zc@type{equation}\zc@cntval{30}\zc@pgval{169}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.493}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-conversion\endcsname{{19.23}{314}{The raw steps}{equation.862}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-conversion\endcsname{\default{19.23}\page{314}\abspage{318}\zc@counter{equation}\thecounter{19.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{314}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.862}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-covariance\endcsname{{19.2}{303}{}{equation.826}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-covariance\endcsname{\default{19.2}\page{303}\abspage{307}\zc@counter{equation}\thecounter{19.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{303}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.826}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-current-identity\endcsname{{19.14}{309}{The BB part of the current}{equation.845}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-current-identity\endcsname{\default{19.14}\page{309}\abspage{313}\zc@counter{equation}\thecounter{19.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{309}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.845}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-curved\endcsname{{19.46}{335}{Curved graphs}{equation.913}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-curved\endcsname{\default{19.46}\page{335}\abspage{339}\zc@counter{equation}\thecounter{19.46}\zc@type{equation}\zc@cntval{46}\zc@pgval{335}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.913}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-damping\endcsname{{15.36}{176}{Parameter balls}{equation.505}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-damping\endcsname{\default{15.36}\page{176}\abspage{180}\zc@counter{equation}\thecounter{15.36}\zc@type{equation}\zc@cntval{36}\zc@pgval{176}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.505}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-davies\endcsname{{19.34}{320}{}{equation.879}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-davies\endcsname{\default{19.34}\page{320}\abspage{324}\zc@counter{equation}\thecounter{19.34}\zc@type{equation}\zc@cntval{34}\zc@pgval{320}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.879}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-defect\endcsname{{15.28}{169}{Divided differences}{equation.491}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-defect\endcsname{\default{15.28}\page{169}\abspage{173}\zc@counter{equation}\thecounter{15.28}\zc@type{equation}\zc@cntval{28}\zc@pgval{169}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.491}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-divided\endcsname{{15.27}{169}{Divided differences}{equation.490}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-divided\endcsname{\default{15.27}\page{169}\abspage{173}\zc@counter{equation}\thecounter{15.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{169}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.490}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-duhamel-eps\endcsname{{19.56}{340}{The exponent and the body weight of the residual}{equation.930}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-duhamel-eps\endcsname{\default{19.56}\page{340}\abspage{344}\zc@counter{equation}\thecounter{19.56}\zc@type{equation}\zc@cntval{56}\zc@pgval{340}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.930}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-early-annuli\endcsname{{19.60}{344}{}{equation.940}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-early-annuli\endcsname{\default{19.60}\page{344}\abspage{348}\zc@counter{equation}\thecounter{19.60}\zc@type{equation}\zc@cntval{60}\zc@pgval{344}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.940}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-energy\endcsname{{19.29}{317}{}{equation.873}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-energy\endcsname{\default{19.29}\page{317}\abspage{321}\zc@counter{equation}\thecounter{19.29}\zc@type{equation}\zc@cntval{29}\zc@pgval{317}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.873}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-energy-form\endcsname{{19.32}{319}{Onsite factors and jet steps}{equation.876}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-energy-form\endcsname{\default{19.32}\page{319}\abspage{323}\zc@counter{equation}\thecounter{19.32}\zc@type{equation}\zc@cntval{32}\zc@pgval{319}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.876}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-escape\endcsname{{15.32}{172}{The escape function and the onsite weight}{equation.498}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-escape\endcsname{\default{15.32}\page{172}\abspage{176}\zc@counter{equation}\thecounter{15.32}\zc@type{equation}\zc@cntval{32}\zc@pgval{172}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.498}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-escape-gradient\endcsname{{15.33}{172}{The escape function and the onsite weight}{equation.499}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-escape-gradient\endcsname{\default{15.33}\page{172}\abspage{176}\zc@counter{equation}\thecounter{15.33}\zc@type{equation}\zc@cntval{33}\zc@pgval{172}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.499}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-exact-tau\endcsname{{19.55}{340}{The exponent and the body weight of the residual}{equation.929}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-exact-tau\endcsname{\default{19.55}\page{340}\abspage{344}\zc@counter{equation}\thecounter{19.55}\zc@type{equation}\zc@cntval{55}\zc@pgval{340}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.929}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-family-norm\endcsname{{19.1}{302}{Parameters}{equation.822}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-family-norm\endcsname{\default{19.1}\page{302}\abspage{306}\zc@counter{equation}\thecounter{19.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{302}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.822}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-ff-current\endcsname{{19.10}{307}{}{equation.840}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-ff-current\endcsname{\default{19.10}\page{307}\abspage{311}\zc@counter{equation}\thecounter{19.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{307}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.840}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-first-step\endcsname{{19.61}{345}{}{equation.943}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-first-step\endcsname{\default{19.61}\page{345}\abspage{349}\zc@counter{equation}\thecounter{19.61}\zc@type{equation}\zc@cntval{61}\zc@pgval{345}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.943}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-full\endcsname{{19.59}{343}{}{equation.937}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-full\endcsname{\default{19.59}\page{343}\abspage{347}\zc@counter{equation}\thecounter{19.59}\zc@type{equation}\zc@cntval{59}\zc@pgval{343}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.937}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-gauge-graph-norm\endcsname{{19.35}{323}{Onsite factors and jet steps}{equation.883}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-gauge-graph-norm\endcsname{\default{19.35}\page{323}\abspage{327}\zc@counter{equation}\thecounter{19.35}\zc@type{equation}\zc@cntval{35}\zc@pgval{323}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.883}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-gauge-growth\endcsname{{19.49}{336}{The energy threshold}{equation.919}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-gauge-growth\endcsname{\default{19.49}\page{336}\abspage{340}\zc@counter{equation}\thecounter{19.49}\zc@type{equation}\zc@cntval{49}\zc@pgval{336}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.919}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-graph-defect\endcsname{{19.45}{333}{The strip of the actual cycle}{equation.911}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-graph-defect\endcsname{\default{19.45}\page{333}\abspage{337}\zc@counter{equation}\thecounter{19.45}\zc@type{equation}\zc@cntval{45}\zc@pgval{333}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.911}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-graph-norm\endcsname{{19.24}{314}{Graph norms}{equation.864}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-graph-norm\endcsname{\default{19.24}\page{314}\abspage{318}\zc@counter{equation}\thecounter{19.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{314}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.864}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-heat-residual\endcsname{{19.53}{339}{}{equation.927}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-heat-residual\endcsname{\default{19.53}\page{339}\abspage{343}\zc@counter{equation}\thecounter{19.53}\zc@type{equation}\zc@cntval{53}\zc@pgval{339}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.927}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-heat-step\endcsname{{19.33}{319}{Onsite factors and jet steps}{equation.877}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-heat-step\endcsname{\default{19.33}\page{319}\abspage{323}\zc@counter{equation}\thecounter{19.33}\zc@type{equation}\zc@cntval{33}\zc@pgval{319}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.877}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-initial-limit\endcsname{{19.57}{340}{The exponent and the body weight of the residual}{equation.931}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-initial-limit\endcsname{\default{19.57}\page{340}\abspage{344}\zc@counter{equation}\thecounter{19.57}\zc@type{equation}\zc@cntval{57}\zc@pgval{340}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.931}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-jet-constant\endcsname{{19.30}{317}{}{equation.874}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-jet-constant\endcsname{\default{19.30}\page{317}\abspage{321}\zc@counter{equation}\thecounter{19.30}\zc@type{equation}\zc@cntval{30}\zc@pgval{317}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.874}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-jet-step\endcsname{{19.25}{314}{Onsite factors and jet steps}{equation.866}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-jet-step\endcsname{\default{19.25}\page{314}\abspage{318}\zc@counter{equation}\thecounter{19.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{314}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.866}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-kick\endcsname{{19.27}{315}{}{equation.870}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-kick\endcsname{\default{19.27}\page{315}\abspage{319}\zc@counter{equation}\thecounter{19.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{315}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.870}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-left\endcsname{{19.20}{313}{The left endpoint}{equation.857}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-left\endcsname{\default{19.20}\page{313}\abspage{317}\zc@counter{equation}\thecounter{19.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{313}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.857}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-logg\endcsname{{15.29}{169}{Divided differences}{equation.492}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-logg\endcsname{\default{15.29}\page{169}\abspage{173}\zc@counter{equation}\thecounter{15.29}\zc@type{equation}\zc@cntval{29}\zc@pgval{169}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.492}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-moving-velocity\endcsname{{19.31}{318}{Onsite factors and jet steps}{equation.875}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-moving-velocity\endcsname{\default{19.31}\page{318}\abspage{322}\zc@counter{equation}\thecounter{19.31}\zc@type{equation}\zc@cntval{31}\zc@pgval{318}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.875}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-normal-block\endcsname{{19.36}{323}{}{equation.885}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-normal-block\endcsname{\default{19.36}\page{323}\abspage{327}\zc@counter{equation}\thecounter{19.36}\zc@type{equation}\zc@cntval{36}\zc@pgval{323}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.885}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-onsite\endcsname{{15.34}{174}{}{equation.502}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-onsite\endcsname{\default{15.34}\page{174}\abspage{178}\zc@counter{equation}\thecounter{15.34}\zc@type{equation}\zc@cntval{34}\zc@pgval{174}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.502}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-ovsyannikov\endcsname{{19.17}{310}{The parameter Hardy space}{equation.849}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-ovsyannikov\endcsname{\default{19.17}\page{310}\abspage{314}\zc@counter{equation}\thecounter{19.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{310}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.849}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-prep-block\endcsname{{19.48}{335}{The prepared norm}{equation.917}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-prep-block\endcsname{\default{19.48}\page{335}\abspage{339}\zc@counter{equation}\thecounter{19.48}\zc@type{equation}\zc@cntval{48}\zc@pgval{335}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.917}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-prep-norm\endcsname{{19.47}{335}{The prepared norm}{equation.916}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-prep-norm\endcsname{\default{19.47}\page{335}\abspage{339}\zc@counter{equation}\thecounter{19.47}\zc@type{equation}\zc@cntval{47}\zc@pgval{335}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.916}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-principal\endcsname{{19.7}{306}{The conjugated generator}{equation.835}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-principal\endcsname{\default{19.7}\page{306}\abspage{310}\zc@counter{equation}\thecounter{19.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{306}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.835}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-raw-steps\endcsname{{19.21}{314}{The raw steps}{equation.860}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-raw-steps\endcsname{\default{19.21}\page{314}\abspage{318}\zc@counter{equation}\thecounter{19.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{314}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.860}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-real-gradient\endcsname{{19.11}{308}{The BB part of the current}{equation.842}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-real-gradient\endcsname{\default{19.11}\page{308}\abspage{312}\zc@counter{equation}\thecounter{19.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{308}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.842}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-remainder-current\endcsname{{19.16}{309}{The BB part of the current}{equation.847}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-remainder-current\endcsname{\default{19.16}\page{309}\abspage{313}\zc@counter{equation}\thecounter{19.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{309}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.847}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-residual-bound\endcsname{{19.58}{341}{}{equation.933}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-residual-bound\endcsname{\default{19.58}\page{341}\abspage{345}\zc@counter{equation}\thecounter{19.58}\zc@type{equation}\zc@cntval{58}\zc@pgval{341}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.933}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-rough-strip\endcsname{{19.43}{331}{}{equation.908}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-rough-strip\endcsname{\default{19.43}\page{331}\abspage{335}\zc@counter{equation}\thecounter{19.43}\zc@type{equation}\zc@cntval{43}\zc@pgval{331}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.908}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-selected\endcsname{{19.4}{303}{}{equation.830}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-selected\endcsname{\default{19.4}\page{303}\abspage{307}\zc@counter{equation}\thecounter{19.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{303}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.830}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-shell\endcsname{{19.51}{337}{}{equation.923}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-shell\endcsname{\default{19.51}\page{337}\abspage{341}\zc@counter{equation}\thecounter{19.51}\zc@type{equation}\zc@cntval{51}\zc@pgval{337}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.923}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-shifted-defect\endcsname{{19.41}{331}{}{equation.904}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-shifted-defect\endcsname{\default{19.41}\page{331}\abspage{335}\zc@counter{equation}\thecounter{19.41}\zc@type{equation}\zc@cntval{41}\zc@pgval{331}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.904}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-similarity\endcsname{{19.22}{314}{The raw steps}{equation.861}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-similarity\endcsname{\default{19.22}\page{314}\abspage{318}\zc@counter{equation}\thecounter{19.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{314}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.861}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-strip\endcsname{{19.42}{331}{}{equation.906}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-strip\endcsname{\default{19.42}\page{331}\abspage{335}\zc@counter{equation}\thecounter{19.42}\zc@type{equation}\zc@cntval{42}\zc@pgval{331}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.906}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-sylvester\endcsname{{19.13}{308}{The BB part of the current}{equation.844}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-sylvester\endcsname{\default{19.13}\page{308}\abspage{312}\zc@counter{equation}\thecounter{19.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{308}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.844}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-tail\endcsname{{19.19}{311}{}{equation.853}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-tail\endcsname{\default{19.19}\page{311}\abspage{315}\zc@counter{equation}\thecounter{19.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{311}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.853}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-taylor\endcsname{{19.28}{316}{Onsite factors and jet steps}{equation.871}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-taylor\endcsname{\default{19.28}\page{316}\abspage{320}\zc@counter{equation}\thecounter{19.28}\zc@type{equation}\zc@cntval{28}\zc@pgval{316}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.871}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-us\endcsname{{19.54}{339}{The exponent and the body weight of the residual}{equation.928}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-us\endcsname{\default{19.54}\page{339}\abspage{343}\zc@counter{equation}\thecounter{19.54}\zc@type{equation}\zc@cntval{54}\zc@pgval{339}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.928}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-ward\endcsname{{19.3}{303}{}{equation.827}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-ward\endcsname{\default{19.3}\page{303}\abspage{307}\zc@counter{equation}\thecounter{19.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{303}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.827}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-ward-current\endcsname{{19.15}{309}{The BB part of the current}{equation.846}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-ward-current\endcsname{\default{19.15}\page{309}\abspage{313}\zc@counter{equation}\thecounter{19.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{309}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.846}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-ward-heat\endcsname{{19.5}{304}{Parameters}{equation.831}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-ward-heat\endcsname{\default{19.5}\page{304}\abspage{308}\zc@counter{equation}\thecounter{19.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{304}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.831}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-ward-ibp\endcsname{{19.6}{305}{Parameters}{equation.832}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-ward-ibp\endcsname{\default{19.6}\page{305}\abspage{309}\zc@counter{equation}\thecounter{19.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{305}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.832}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:end-weighted-heat\endcsname{{19.18}{310}{The weighted heat form}{equation.851}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:end-weighted-heat\endcsname{\default{19.18}\page{310}\abspage{314}\zc@counter{equation}\thecounter{19.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{310}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{equation.851}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-collision\endcsname{{9.10}{34}{Stationary limits, collisions, and Poisson tails}{equation.110}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-collision\endcsname{\default{9.10}\page{34}\abspage{38}\zc@counter{equation}\thecounter{9.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{34}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.110}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-constant\endcsname{{9.13}{36}{The extraction theorem}{equation.114}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-constant\endcsname{\default{9.13}\page{36}\abspage{40}\zc@counter{equation}\thecounter{9.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{36}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.114}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-domination\endcsname{{9.2}{29}{}{equation.93}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-domination\endcsname{\default{9.2}\page{29}\abspage{33}\zc@counter{equation}\thecounter{9.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{29}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.93}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-limits\endcsname{{9.8}{33}{}{equation.107}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-limits\endcsname{\default{9.8}\page{33}\abspage{37}\zc@counter{equation}\thecounter{9.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{33}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.107}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-local-measure\endcsname{{9.1}{29}{}{equation.92}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-local-measure\endcsname{\default{9.1}\page{29}\abspage{33}\zc@counter{equation}\thecounter{9.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{29}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.92}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-mixture-bound\endcsname{{9.7}{31}{}{equation.102}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-mixture-bound\endcsname{\default{9.7}\page{31}\abspage{35}\zc@counter{equation}\thecounter{9.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{31}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.102}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-moments\endcsname{{9.4}{30}{}{equation.96}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-moments\endcsname{\default{9.4}\page{30}\abspage{34}\zc@counter{equation}\thecounter{9.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{30}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.96}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-overlap\endcsname{{9.12}{36}{The extraction theorem}{equation.113}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-overlap\endcsname{\default{9.12}\page{36}\abspage{40}\zc@counter{equation}\thecounter{9.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{36}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.113}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-poisson-compare\endcsname{{9.5}{30}{Weighted moment measures}{equation.97}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-poisson-compare\endcsname{\default{9.5}\page{30}\abspage{34}\zc@counter{equation}\thecounter{9.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{30}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.97}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-rooted-window\endcsname{{9.17}{39}{}{equation.123}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-rooted-window\endcsname{\default{9.17}\page{39}\abspage{43}\zc@counter{equation}\thecounter{9.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{39}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.123}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-smoothing\endcsname{{9.3}{29}{Weighted moment measures}{equation.94}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-smoothing\endcsname{\default{9.3}\page{29}\abspage{33}\zc@counter{equation}\thecounter{9.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{29}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.94}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-source\endcsname{{9.14}{36}{}{equation.116}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-source\endcsname{\default{9.14}\page{36}\abspage{40}\zc@counter{equation}\thecounter{9.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{36}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.116}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-tail\endcsname{{9.9}{33}{}{equation.109}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-tail\endcsname{\default{9.9}\page{33}\abspage{37}\zc@counter{equation}\thecounter{9.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{33}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.109}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-tail-bound\endcsname{{9.11}{35}{Stationary limits, collisions, and Poisson tails}{equation.111}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-tail-bound\endcsname{\default{9.11}\page{35}\abspage{39}\zc@counter{equation}\thecounter{9.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{35}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.111}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-tau\endcsname{{9.6}{31}{Gaussian mixtures}{equation.99}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-tau\endcsname{\default{9.6}\page{31}\abspage{35}\zc@counter{equation}\thecounter{9.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{31}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.99}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-unfolded-at-e\endcsname{{9.15}{38}{Densities and arbitrarily small windows}{equation.120}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-unfolded-at-e\endcsname{\default{9.15}\page{38}\abspage{42}\zc@counter{equation}\thecounter{9.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{38}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.120}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:extract-window\endcsname{{9.16}{39}{}{equation.122}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:extract-window\endcsname{\default{9.16}\page{39}\abspage{43}\zc@counter{equation}\thecounter{9.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{39}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{equation.122}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-B\endcsname{{15.22}{166}{}{equation.481}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-B\endcsname{\default{15.22}\page{166}\abspage{170}\zc@counter{equation}\thecounter{15.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{166}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.481}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-B-det\endcsname{{15.23}{167}{}{equation.482}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-B-det\endcsname{\default{15.23}\page{167}\abspage{171}\zc@counter{equation}\thecounter{15.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{167}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.482}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-B-inverse\endcsname{{15.24}{167}{}{equation.483}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-B-inverse\endcsname{\default{15.24}\page{167}\abspage{171}\zc@counter{equation}\thecounter{15.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{167}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.483}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-EW\endcsname{{15.46}{185}{The a priori bound on partial chains}{equation.527}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-EW\endcsname{\default{15.46}\page{185}\abspage{189}\zc@counter{equation}\thecounter{15.46}\zc@type{equation}\zc@cntval{46}\zc@pgval{185}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.527}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-Gn\endcsname{{15.10}{160}{The anchored identity}{equation.462}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-Gn\endcsname{\default{15.10}\page{160}\abspage{164}\zc@counter{equation}\thecounter{15.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{160}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.462}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-Hp\endcsname{{15.5}{159}{Weighted spaces}{equation.453}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-Hp\endcsname{\default{15.5}\page{159}\abspage{163}\zc@counter{equation}\thecounter{15.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{159}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.453}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-Mn\endcsname{{15.9}{160}{The anchored identity}{equation.461}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-Mn\endcsname{\default{15.9}\page{160}\abspage{164}\zc@counter{equation}\thecounter{15.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{160}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.461}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-Mn-marked\endcsname{{15.49}{186}{Local sources in the anchored identity}{equation.532}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-Mn-marked\endcsname{\default{15.49}\page{186}\abspage{190}\zc@counter{equation}\thecounter{15.49}\zc@type{equation}\zc@cntval{49}\zc@pgval{186}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.532}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-Qm\endcsname{{15.59}{191}{Proof of the theorem}{equation.546}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-Qm\endcsname{\default{15.59}\page{191}\abspage{195}\zc@counter{equation}\thecounter{15.59}\zc@type{equation}\zc@cntval{59}\zc@pgval{191}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.546}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-anchored\endcsname{{15.11}{160}{The anchored identity}{equation.463}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-anchored\endcsname{\default{15.11}\page{160}\abspage{164}\zc@counter{equation}\thecounter{15.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{160}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.463}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-averaging\endcsname{{15.37}{176}{}{equation.508}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-averaging\endcsname{\default{15.37}\page{176}\abspage{180}\zc@counter{equation}\thecounter{15.37}\zc@type{equation}\zc@cntval{37}\zc@pgval{176}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.508}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-blocks\endcsname{{15.2}{158}{The cycle and its coordinates}{equation.449}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-blocks\endcsname{\default{15.2}\page{158}\abspage{162}\zc@counter{equation}\thecounter{15.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{158}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.449}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-boost-props\endcsname{{15.3}{158}{The cycle and its coordinates}{equation.450}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-boost-props\endcsname{\default{15.3}\page{158}\abspage{162}\zc@counter{equation}\thecounter{15.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{158}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.450}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-bound\endcsname{{15.47}{185}{}{equation.529}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-bound\endcsname{\default{15.47}\page{185}\abspage{189}\zc@counter{equation}\thecounter{15.47}\zc@type{equation}\zc@cntval{47}\zc@pgval{185}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.529}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-coef-det\endcsname{{15.39}{180}{}{equation.517}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-coef-det\endcsname{\default{15.39}\page{180}\abspage{184}\zc@counter{equation}\thecounter{15.39}\zc@type{equation}\zc@cntval{39}\zc@pgval{180}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.517}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-coef-inverse\endcsname{{15.40}{180}{}{equation.519}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-coef-inverse\endcsname{\default{15.40}\page{180}\abspage{184}\zc@counter{equation}\thecounter{15.40}\zc@type{equation}\zc@cntval{40}\zc@pgval{180}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.519}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-coef-ratio\endcsname{{15.41}{180}{}{equation.520}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-coef-ratio\endcsname{\default{15.41}\page{180}\abspage{184}\zc@counter{equation}\thecounter{15.41}\zc@type{equation}\zc@cntval{41}\zc@pgval{180}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.520}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-coercive\endcsname{{15.20}{164}{}{equation.478}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-coercive\endcsname{\default{15.20}\page{164}\abspage{168}\zc@counter{equation}\thecounter{15.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{164}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.478}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-congruence\endcsname{{15.18}{164}{}{equation.476}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-congruence\endcsname{\default{15.18}\page{164}\abspage{168}\zc@counter{equation}\thecounter{15.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{164}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.476}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-core\endcsname{{15.48}{185}{The a priori bound on partial chains}{equation.530}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-core\endcsname{\default{15.48}\page{185}\abspage{189}\zc@counter{equation}\thecounter{15.48}\zc@type{equation}\zc@cntval{48}\zc@pgval{185}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.530}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-core-A\endcsname{{15.52}{187}{The bound on the core}{equation.538}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-core-A\endcsname{\default{15.52}\page{187}\abspage{191}\zc@counter{equation}\thecounter{15.52}\zc@type{equation}\zc@cntval{52}\zc@pgval{187}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.538}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-core-Binv\endcsname{{15.53}{187}{The bound on the core}{equation.539}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-core-Binv\endcsname{\default{15.53}\page{187}\abspage{191}\zc@counter{equation}\thecounter{15.53}\zc@type{equation}\zc@cntval{53}\zc@pgval{187}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.539}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-core-NF\endcsname{{15.55}{188}{The bound on the core}{equation.541}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-core-NF\endcsname{\default{15.55}\page{188}\abspage{192}\zc@counter{equation}\thecounter{15.55}\zc@type{equation}\zc@cntval{55}\zc@pgval{188}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.541}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-core-bound\endcsname{{15.51}{187}{}{equation.537}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-core-bound\endcsname{\default{15.51}\page{187}\abspage{191}\zc@counter{equation}\thecounter{15.51}\zc@type{equation}\zc@cntval{51}\zc@pgval{187}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.537}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-core-marks\endcsname{{15.54}{187}{The bound on the core}{equation.540}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-core-marks\endcsname{\default{15.54}\page{187}\abspage{191}\zc@counter{equation}\thecounter{15.54}\zc@type{equation}\zc@cntval{54}\zc@pgval{187}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.540}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-core-norm\endcsname{{15.60}{191}{Proof of the theorem}{equation.547}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-core-norm\endcsname{\default{15.60}\page{191}\abspage{195}\zc@counter{equation}\thecounter{15.60}\zc@type{equation}\zc@cntval{60}\zc@pgval{191}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.547}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-deficit\endcsname{{15.19}{164}{}{equation.477}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-deficit\endcsname{\default{15.19}\page{164}\abspage{168}\zc@counter{equation}\thecounter{15.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{164}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.477}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-deficit-def\endcsname{{15.17}{164}{}{equation.475}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-deficit-def\endcsname{\default{15.17}\page{164}\abspage{168}\zc@counter{equation}\thecounter{15.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{164}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.475}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-deficit-line\endcsname{{15.21}{165}{Determinant inequalities}{equation.479}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-deficit-line\endcsname{\default{15.21}\page{165}\abspage{169}\zc@counter{equation}\thecounter{15.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{165}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.479}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-far\endcsname{{15.61}{192}{Proof of the theorem}{equation.548}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-far\endcsname{\default{15.61}\page{192}\abspage{196}\zc@counter{equation}\thecounter{15.61}\zc@type{equation}\zc@cntval{61}\zc@pgval{192}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.548}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-far-norm\endcsname{{15.62}{192}{Proof of the theorem}{equation.549}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-far-norm\endcsname{\default{15.62}\page{192}\abspage{196}\zc@counter{equation}\thecounter{15.62}\zc@type{equation}\zc@cntval{62}\zc@pgval{192}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.549}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-ff-body\endcsname{{15.56}{188}{The bound on the core}{equation.542}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-ff-body\endcsname{\default{15.56}\page{188}\abspage{192}\zc@counter{equation}\thecounter{15.56}\zc@type{equation}\zc@cntval{56}\zc@pgval{188}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.542}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-ff-source\endcsname{{15.7}{159}{Sources and marks}{equation.457}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-ff-source\endcsname{\default{15.7}\page{159}\abspage{163}\zc@counter{equation}\thecounter{15.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{159}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.457}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-height\endcsname{{15.13}{161}{}{equation.467}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-height\endcsname{\default{15.13}\page{161}\abspage{165}\zc@counter{equation}\thecounter{15.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{161}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.467}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-hua-G\endcsname{{15.16}{163}{Determinant inequalities}{equation.473}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-hua-G\endcsname{\default{15.16}\page{163}\abspage{167}\zc@counter{equation}\thecounter{15.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{163}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.473}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-hua-inequality\endcsname{{15.15}{162}{Determinant inequalities}{equation.472}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-hua-inequality\endcsname{\default{15.15}\page{162}\abspage{166}\zc@counter{equation}\thecounter{15.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{162}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.472}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-jet-HS\endcsname{{15.50}{186}{}{equation.534}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-jet-HS\endcsname{\default{15.50}\page{186}\abspage{190}\zc@counter{equation}\thecounter{15.50}\zc@type{equation}\zc@cntval{50}\zc@pgval{186}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.534}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-marked-BB\endcsname{{15.43}{183}{Coefficient bounds}{equation.523}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-marked-BB\endcsname{\default{15.43}\page{183}\abspage{187}\zc@counter{equation}\thecounter{15.43}\zc@type{equation}\zc@cntval{43}\zc@pgval{183}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.523}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-marked-chain\endcsname{{15.8}{160}{}{equation.459}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-marked-chain\endcsname{\default{15.8}\page{160}\abspage{164}\zc@counter{equation}\thecounter{15.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{160}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.459}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-marked-onsite\endcsname{{15.42}{182}{}{equation.522}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-marked-onsite\endcsname{\default{15.42}\page{182}\abspage{186}\zc@counter{equation}\thecounter{15.42}\zc@type{equation}\zc@cntval{42}\zc@pgval{182}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.522}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-marked-sdet\endcsname{{15.44}{184}{Coefficient bounds}{equation.524}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-marked-sdet\endcsname{\default{15.44}\page{184}\abspage{188}\zc@counter{equation}\thecounter{15.44}\zc@type{equation}\zc@cntval{44}\zc@pgval{184}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.524}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-minami\endcsname{{15.38}{177}{Spectral averaging}{equation.509}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-minami\endcsname{\default{15.38}\page{177}\abspage{181}\zc@counter{equation}\thecounter{15.38}\zc@type{equation}\zc@cntval{38}\zc@pgval{177}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.509}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-normalized\endcsname{{15.6}{159}{Sources and marks}{equation.456}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-normalized\endcsname{\default{15.6}\page{159}\abspage{163}\zc@counter{equation}\thecounter{15.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{159}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.456}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-odd-trace\endcsname{{15.45}{184}{Coefficient bounds}{equation.525}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-odd-trace\endcsname{\default{15.45}\page{184}\abspage{188}\zc@counter{equation}\thecounter{15.45}\zc@type{equation}\zc@cntval{45}\zc@pgval{184}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.525}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-quotient\endcsname{{15.57}{189}{The bound on the core}{equation.543}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-quotient\endcsname{\default{15.57}\page{189}\abspage{193}\zc@counter{equation}\thecounter{15.57}\zc@type{equation}\zc@cntval{57}\zc@pgval{189}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.543}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-resolvent\endcsname{{15.12}{161}{}{equation.466}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-resolvent\endcsname{\default{15.12}\page{161}\abspage{165}\zc@counter{equation}\thecounter{15.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{161}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.466}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-scales\endcsname{{15.1}{158}{Setting}{equation.447}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-scales\endcsname{\default{15.1}\page{158}\abspage{162}\zc@counter{equation}\thecounter{15.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{158}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.447}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-sdet-bound\endcsname{{15.58}{190}{The bound on the core}{equation.544}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-sdet-bound\endcsname{\default{15.58}\page{190}\abspage{194}\zc@counter{equation}\thecounter{15.58}\zc@type{equation}\zc@cntval{58}\zc@pgval{190}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.544}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-shift\endcsname{{15.14}{161}{}{equation.468}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-shift\endcsname{\default{15.14}\page{161}\abspage{165}\zc@counter{equation}\thecounter{15.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{161}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.468}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:hist-transported\endcsname{{15.4}{158}{The cycle and its coordinates}{equation.451}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:hist-transported\endcsname{\default{15.4}\page{158}\abspage{162}\zc@counter{equation}\thecounter{15.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{158}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{equation.451}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-Z\endcsname{{7.13}{9}{The interface theorem}{equation.29}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-Z\endcsname{\default{7.13}\page{9}\abspage{13}\zc@counter{equation}\thecounter{7.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{9}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.29}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-allsite\endcsname{{7.20}{11}{}{equation.41}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-allsite\endcsname{\default{7.20}\page{11}\abspage{15}\zc@counter{equation}\thecounter{7.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{11}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.41}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-band-lower\endcsname{{7.1}{3}{}{equation.6}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-band-lower\endcsname{\default{7.1}\page{3}\abspage{7}\zc@counter{equation}\thecounter{7.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{3}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.6}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-block-variance\endcsname{{7.6}{5}{Factorial moments of eigenvalue counts}{equation.13}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-block-variance\endcsname{\default{7.6}\page{5}\abspage{9}\zc@counter{equation}\thecounter{7.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{5}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.13}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-blocks\endcsname{{7.5}{4}{Factorial moments of eigenvalue counts}{equation.12}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-blocks\endcsname{\default{7.5}\page{4}\abspage{8}\zc@counter{equation}\thecounter{7.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{4}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.12}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-counts\endcsname{{7.2}{4}{}{equation.9}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-counts\endcsname{\default{7.2}\page{4}\abspage{8}\zc@counter{equation}\thecounter{7.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{4}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.9}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-counts-unfolded\endcsname{{7.3}{4}{}{equation.10}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-counts-unfolded\endcsname{\default{7.3}\page{4}\abspage{8}\zc@counter{equation}\thecounter{7.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{4}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.10}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-dls\endcsname{{7.7}{6}{}{equation.17}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-dls\endcsname{\default{7.7}\page{6}\abspage{10}\zc@counter{equation}\thecounter{7.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{6}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.17}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-energies\endcsname{{7.22}{12}{}{equation.45}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-energies\endcsname{\default{7.22}\page{12}\abspage{16}\zc@counter{equation}\thecounter{7.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{12}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.45}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-examples\endcsname{{7.15}{10}{The interface theorem}{equation.32}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-examples\endcsname{\default{7.15}\page{10}\abspage{14}\zc@counter{equation}\thecounter{7.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{10}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.32}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-intensity\endcsname{{7.8}{6}{}{equation.18}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-intensity\endcsname{\default{7.8}\page{6}\abspage{10}\zc@counter{equation}\thecounter{7.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{6}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.18}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-jets\endcsname{{7.19}{11}{}{equation.39}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-jets\endcsname{\default{7.19}\page{11}\abspage{15}\zc@counter{equation}\thecounter{7.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{11}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.39}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-local\endcsname{{7.14}{9}{}{equation.31}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-local\endcsname{\default{7.14}\page{9}\abspage{13}\zc@counter{equation}\thecounter{7.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{9}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.31}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-multiplier\endcsname{{7.17}{10}{The interface theorem}{equation.34}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-multiplier\endcsname{\default{7.17}\page{10}\abspage{14}\zc@counter{equation}\thecounter{7.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{10}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.34}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-onepoint\endcsname{{7.21}{11}{}{equation.43}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-onepoint\endcsname{\default{7.21}\page{11}\abspage{15}\zc@counter{equation}\thecounter{7.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{11}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.43}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-orbit-resolvent\endcsname{{7.16}{10}{The interface theorem}{equation.33}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-orbit-resolvent\endcsname{\default{7.16}\page{10}\abspage{14}\zc@counter{equation}\thecounter{7.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{10}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.33}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-psss\endcsname{{7.4}{4}{Factorial moments of eigenvalue counts}{equation.11}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-psss\endcsname{\default{7.4}\page{4}\abspage{8}\zc@counter{equation}\thecounter{7.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{4}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.11}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-scalar\endcsname{{7.10}{7}{}{equation.22}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-scalar\endcsname{\default{7.10}\page{7}\abspage{11}\zc@counter{equation}\thecounter{7.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{7}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.22}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-shift\endcsname{{7.11}{7}{}{equation.25}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-shift\endcsname{\default{7.11}\page{7}\abspage{11}\zc@counter{equation}\thecounter{7.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{7}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.25}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-shift-truncated\endcsname{{7.12}{7}{}{equation.27}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-shift-truncated\endcsname{\default{7.12}\page{7}\abspage{11}\zc@counter{equation}\thecounter{7.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{7}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.27}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-source-limit\endcsname{{7.18}{11}{}{equation.37}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-source-limit\endcsname{\default{7.18}\page{11}\abspage{15}\zc@counter{equation}\thecounter{7.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{11}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.37}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:iface-window\endcsname{{7.9}{6}{}{equation.19}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:iface-window\endcsname{\default{7.9}\page{6}\abspage{10}\zc@counter{equation}\thecounter{7.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{6}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{equation.19}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-balanced-contact\endcsname{{11.9}{75}{}{equation.225}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-balanced-contact\endcsname{\default{11.9}\page{75}\abspage{79}\zc@counter{equation}\thecounter{11.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{75}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.225}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-balanced-distinct\endcsname{{11.7}{75}{}{equation.223}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-balanced-distinct\endcsname{\default{11.7}\page{75}\abspage{79}\zc@counter{equation}\thecounter{11.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{75}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.223}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-balanced-ldos\endcsname{{11.8}{75}{}{equation.224}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-balanced-ldos\endcsname{\default{11.8}\page{75}\abspage{79}\zc@counter{equation}\thecounter{11.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{75}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.224}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-contact\endcsname{{11.5}{73}{}{equation.217}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-contact\endcsname{\default{11.5}\page{73}\abspage{77}\zc@counter{equation}\thecounter{11.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{73}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.217}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-contact-multiplier\endcsname{{11.1}{71}{}{equation.211}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-contact-multiplier\endcsname{\default{11.1}\page{71}\abspage{75}\zc@counter{equation}\thecounter{11.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{71}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.211}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-contact-residue\endcsname{{11.6}{74}{}{equation.220}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-contact-residue\endcsname{\default{11.6}\page{74}\abspage{78}\zc@counter{equation}\thecounter{11.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{74}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.220}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-density\endcsname{{11.17}{82}{}{equation.241}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-density\endcsname{\default{11.17}\page{82}\abspage{86}\zc@counter{equation}\thecounter{11.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{82}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.241}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-diagonal\endcsname{{11.11}{77}{}{equation.230}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-diagonal\endcsname{\default{11.11}\page{77}\abspage{81}\zc@counter{equation}\thecounter{11.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{77}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.230}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-even-transform\endcsname{{11.14}{80}{The two-energy intensity}{equation.235}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-even-transform\endcsname{\default{11.14}\page{80}\abspage{84}\zc@counter{equation}\thecounter{11.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{80}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.235}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-even-values\endcsname{{11.15}{80}{The two-energy intensity}{equation.236}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-even-values\endcsname{\default{11.15}\page{80}\abspage{84}\zc@counter{equation}\thecounter{11.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{80}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.236}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-exact\endcsname{{11.20}{85}{}{equation.247}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-exact\endcsname{\default{11.20}\page{85}\abspage{89}\zc@counter{equation}\thecounter{11.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{85}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.247}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-jet-limits\endcsname{{11.3}{72}{Pairings at rank one}{equation.214}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-jet-limits\endcsname{\default{11.3}\page{72}\abspage{76}\zc@counter{equation}\thecounter{11.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{72}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.214}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-laplace\endcsname{{11.24}{86}{The exact participation formula}{equation.251}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-laplace\endcsname{\default{11.24}\page{86}\abspage{90}\zc@counter{equation}\thecounter{11.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{86}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.251}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-laplacian\endcsname{{11.13}{79}{The two-energy intensity}{equation.234}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-laplacian\endcsname{\default{11.13}\page{79}\abspage{83}\zc@counter{equation}\thecounter{11.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{79}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.234}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-mass-split\endcsname{{11.23}{86}{The exact participation formula}{equation.250}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-mass-split\endcsname{\default{11.23}\page{86}\abspage{90}\zc@counter{equation}\thecounter{11.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{86}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.250}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-offdiag-top\endcsname{{11.4}{73}{Pairings at rank one}{equation.215}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-offdiag-top\endcsname{\default{11.4}\page{73}\abspage{77}\zc@counter{equation}\thecounter{11.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{73}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.215}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-rank-two\endcsname{{11.2}{72}{Local sources}{equation.212}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-rank-two\endcsname{\default{11.2}\page{72}\abspage{76}\zc@counter{equation}\thecounter{11.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{72}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.212}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-resolvent-bounds\endcsname{{11.19}{84}{}{equation.245}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-resolvent-bounds\endcsname{\default{11.19}\page{84}\abspage{88}\zc@counter{equation}\thecounter{11.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{84}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.245}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-resolvent-ode\endcsname{{11.18}{84}{}{equation.244}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-resolvent-ode\endcsname{\default{11.18}\page{84}\abspage{88}\zc@counter{equation}\thecounter{11.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{84}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.244}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-split\endcsname{{11.22}{86}{The exact participation formula}{equation.249}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-split\endcsname{\default{11.22}\page{86}\abspage{90}\zc@counter{equation}\thecounter{11.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{86}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.249}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-stationary\endcsname{{11.12}{78}{The two-energy intensity}{equation.232}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-stationary\endcsname{\default{11.12}\page{78}\abspage{82}\zc@counter{equation}\thecounter{11.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{78}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.232}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-time-change\endcsname{{11.16}{80}{The two-energy intensity}{equation.238}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-time-change\endcsname{\default{11.16}\page{80}\abspage{84}\zc@counter{equation}\thecounter{11.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{80}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.238}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-trace-split\endcsname{{11.21}{85}{The exact participation formula}{equation.248}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-trace-split\endcsname{\default{11.21}\page{85}\abspage{89}\zc@counter{equation}\thecounter{11.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{85}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.248}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-virial\endcsname{{11.25}{87}{The exact participation formula}{equation.252}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-virial\endcsname{\default{11.25}\page{87}\abspage{91}\zc@counter{equation}\thecounter{11.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{87}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.252}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:int-ward-identity\endcsname{{11.10}{76}{Local limits at positive damping}{equation.227}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:int-ward-identity\endcsname{\default{11.10}\page{76}\abspage{80}\zc@counter{equation}\thecounter{11.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{76}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{equation.227}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-all-remainders\endcsname{{13.20}{121}{The kernel and the function $I_X$}{equation.342}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-all-remainders\endcsname{\default{13.20}\page{121}\abspage{125}\zc@counter{equation}\thecounter{13.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{121}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.342}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-branch-expansion\endcsname{{13.16}{119}{}{equation.336}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-branch-expansion\endcsname{\default{13.16}\page{119}\abspage{123}\zc@counter{equation}\thecounter{13.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{119}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.336}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-branch-first\endcsname{{13.28}{124}{The kernel and the function $I_X$}{equation.351}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-branch-first\endcsname{\default{13.28}\page{124}\abspage{128}\zc@counter{equation}\thecounter{13.28}\zc@type{equation}\zc@cntval{28}\zc@pgval{124}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.351}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-branch-recursion\endcsname{{13.15}{119}{Expansion of the singlet state}{equation.334}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-branch-recursion\endcsname{\default{13.15}\page{119}\abspage{123}\zc@counter{equation}\thecounter{13.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{119}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.334}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-coefficients\endcsname{{13.24}{123}{}{equation.347}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-coefficients\endcsname{\default{13.24}\page{123}\abspage{127}\zc@counter{equation}\thecounter{13.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{123}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.347}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-d-identities\endcsname{{13.4}{113}{Feynman--Kac preliminaries}{equation.312}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-d-identities\endcsname{\default{13.4}\page{113}\abspage{117}\zc@counter{equation}\thecounter{13.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{113}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.312}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-duhamel\endcsname{{13.10}{117}{}{equation.325}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-duhamel\endcsname{\default{13.10}\page{117}\abspage{121}\zc@counter{equation}\thecounter{13.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{117}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.325}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-envelope\endcsname{{13.9}{116}{}{equation.323}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-envelope\endcsname{\default{13.9}\page{116}\abspage{120}\zc@counter{equation}\thecounter{13.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{116}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.323}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-envelope-function\endcsname{{13.8}{116}{Feynman--Kac preliminaries}{equation.321}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-envelope-function\endcsname{\default{13.8}\page{116}\abspage{120}\zc@counter{equation}\thecounter{13.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{116}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.321}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-fiber\endcsname{{13.22}{122}{The kernel and the function $I_X$}{equation.344}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-fiber\endcsname{\default{13.22}\page{122}\abspage{126}\zc@counter{equation}\thecounter{13.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{122}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.344}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-first-coefficient\endcsname{{13.13}{118}{}{equation.331}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-first-coefficient\endcsname{\default{13.13}\page{118}\abspage{122}\zc@counter{equation}\thecounter{13.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{118}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.331}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-fk-domination\endcsname{{13.5}{113}{Feynman--Kac preliminaries}{equation.313}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-fk-domination\endcsname{\default{13.5}\page{113}\abspage{117}\zc@counter{equation}\thecounter{13.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{113}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.313}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-kernel-coefficients\endcsname{{13.17}{120}{The kernel and the function $I_X$}{equation.338}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-kernel-coefficients\endcsname{\default{13.17}\page{120}\abspage{124}\zc@counter{equation}\thecounter{13.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{120}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.338}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-kernel-expansion\endcsname{{13.18}{120}{}{equation.340}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-kernel-expansion\endcsname{\default{13.18}\page{120}\abspage{124}\zc@counter{equation}\thecounter{13.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{120}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.340}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-laplace-moments\endcsname{{13.26}{124}{The kernel and the function $I_X$}{equation.349}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-laplace-moments\endcsname{\default{13.26}\page{124}\abspage{128}\zc@counter{equation}\thecounter{13.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{124}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.349}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-laplace-powers\endcsname{{13.25}{124}{The kernel and the function $I_X$}{equation.348}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-laplace-powers\endcsname{\default{13.25}\page{124}\abspage{128}\zc@counter{equation}\thecounter{13.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{124}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.348}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-log-tail\endcsname{{13.7}{114}{}{equation.320}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-log-tail\endcsname{\default{13.7}\page{114}\abspage{118}\zc@counter{equation}\thecounter{13.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{114}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.320}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-moments\endcsname{{13.6}{114}{}{equation.318}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-moments\endcsname{\default{13.6}\page{114}\abspage{118}\zc@counter{equation}\thecounter{13.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{114}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.318}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-profile\endcsname{{13.1}{111}{}{equation.308}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-profile\endcsname{\default{13.1}\page{111}\abspage{115}\zc@counter{equation}\thecounter{13.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{111}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.308}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-profile-domination\endcsname{{13.2}{112}{The eigenvector correlation}{equation.309}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-profile-domination\endcsname{\default{13.2}\page{112}\abspage{116}\zc@counter{equation}\thecounter{13.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{112}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.309}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-profile-taylor\endcsname{{13.3}{112}{The eigenvector correlation}{equation.310}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-profile-taylor\endcsname{\default{13.3}\page{112}\abspage{116}\zc@counter{equation}\thecounter{13.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{112}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.310}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-radial\endcsname{{13.21}{122}{The kernel and the function $I_X$}{equation.343}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-radial\endcsname{\default{13.21}\page{122}\abspage{126}\zc@counter{equation}\thecounter{13.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{122}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.343}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-scalar-coefficient\endcsname{{13.23}{122}{The kernel and the function $I_X$}{equation.345}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-scalar-coefficient\endcsname{\default{13.23}\page{122}\abspage{126}\zc@counter{equation}\thecounter{13.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{122}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.345}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-scalar-expansion\endcsname{{13.19}{120}{}{equation.341}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-scalar-expansion\endcsname{\default{13.19}\page{120}\abspage{124}\zc@counter{equation}\thecounter{13.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{120}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.341}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-second-polynomial\endcsname{{13.30}{125}{The kernel and the function $I_X$}{equation.353}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-second-polynomial\endcsname{\default{13.30}\page{125}\abspage{129}\zc@counter{equation}\thecounter{13.30}\zc@type{equation}\zc@cntval{30}\zc@pgval{125}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.353}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-second-terms\endcsname{{13.29}{125}{The kernel and the function $I_X$}{equation.352}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-second-terms\endcsname{\default{13.29}\page{125}\abspage{129}\zc@counter{equation}\thecounter{13.29}\zc@type{equation}\zc@cntval{29}\zc@pgval{125}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.352}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-simplex-moments\endcsname{{13.27}{124}{The kernel and the function $I_X$}{equation.350}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-simplex-moments\endcsname{\default{13.27}\page{124}\abspage{128}\zc@counter{equation}\thecounter{13.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{124}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.350}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-singlet-expansion\endcsname{{13.14}{118}{}{equation.333}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-singlet-expansion\endcsname{\default{13.14}\page{118}\abspage{122}\zc@counter{equation}\thecounter{13.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{118}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.333}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-singlet-operator\endcsname{{13.11}{117}{Expansion of the singlet state}{equation.327}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-singlet-operator\endcsname{\default{13.11}\page{117}\abspage{121}\zc@counter{equation}\thecounter{13.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{117}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.327}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:met-singlet-recursion\endcsname{{13.12}{118}{Expansion of the singlet state}{equation.328}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:met-singlet-recursion\endcsname{\default{13.12}\page{118}\abspage{122}\zc@counter{equation}\thecounter{13.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{118}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{equation.328}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-algebra\endcsname{{8.2}{15}{Ratios with poles at \texorpdfstring {$\pm \iu \eps $}{+-i epsilon}}{equation.52}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-algebra\endcsname{\default{8.2}\page{15}\abspage{19}\zc@counter{equation}\thecounter{8.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{15}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.52}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-cayley\endcsname{{8.3}{15}{}{equation.56}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-cayley\endcsname{\default{8.3}\page{15}\abspage{19}\zc@counter{equation}\thecounter{8.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{15}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.56}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-complexbound\endcsname{{8.14}{23}{Continuation to all source parameters}{equation.81}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-complexbound\endcsname{\default{8.14}\page{23}\abspage{27}\zc@counter{equation}\thecounter{8.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{23}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.81}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-constructR\endcsname{{8.12}{21}{A determining class}{equation.73}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-constructR\endcsname{\default{8.12}\page{21}\abspage{25}\zc@counter{equation}\thecounter{8.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{21}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.73}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-counts\endcsname{{8.4}{17}{Tightness and subsequential limits}{equation.59}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-counts\endcsname{\default{8.4}\page{17}\abspage{21}\zc@counter{equation}\thecounter{8.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{17}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.59}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-cover\endcsname{{8.8}{18}{Tightness and subsequential limits}{equation.64}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-cover\endcsname{\default{8.8}\page{18}\abspage{22}\zc@counter{equation}\thecounter{8.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{18}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.64}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-exp\endcsname{{8.5}{17}{Tightness and subsequential limits}{equation.60}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-exp\endcsname{\default{8.5}\page{17}\abspage{21}\zc@counter{equation}\thecounter{8.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{17}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.60}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-fatou\endcsname{{8.6}{17}{Tightness and subsequential limits}{equation.62}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-fatou\endcsname{\default{8.6}\page{17}\abspage{21}\zc@counter{equation}\thecounter{8.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{17}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.62}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-jensen\endcsname{{8.9}{19}{}{equation.67}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-jensen\endcsname{\default{8.9}\page{19}\abspage{23}\zc@counter{equation}\thecounter{8.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{19}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.67}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-jet-values\endcsname{{8.16}{25}{The rank-one derivatives}{equation.86}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-jet-values\endcsname{\default{8.16}\page{25}\abspage{29}\zc@counter{equation}\thecounter{8.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{25}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.86}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-pairs\endcsname{{8.7}{18}{Tightness and subsequential limits}{equation.63}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-pairs\endcsname{\default{8.7}\page{18}\abspage{22}\zc@counter{equation}\thecounter{8.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{18}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.63}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-parameters\endcsname{{8.13}{22}{Continuation to all source parameters}{equation.78}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-parameters\endcsname{\default{8.13}\page{22}\abspage{26}\zc@counter{equation}\thecounter{8.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{22}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.78}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-ratio\endcsname{{8.1}{15}{Ratios with poles at \texorpdfstring {$\pm \iu \eps $}{+-i epsilon}}{equation.51}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-ratio\endcsname{\default{8.1}\page{15}\abspage{19}\zc@counter{equation}\thecounter{8.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{15}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.51}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-source-law\endcsname{{8.15}{25}{Proof of parts (a) and (b)}{equation.84}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-source-law\endcsname{\default{8.15}\page{25}\abspage{29}\zc@counter{equation}\thecounter{8.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{25}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.84}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-tail\endcsname{{8.10}{19}{Tightness and subsequential limits}{equation.69}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-tail\endcsname{\default{8.10}\page{19}\abspage{23}\zc@counter{equation}\thecounter{8.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{19}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.69}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:proc-weighted-approx\endcsname{{8.11}{20}{}{equation.72}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:proc-weighted-approx\endcsname{\default{8.11}\page{20}\abspage{24}\zc@counter{equation}\thecounter{8.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{20}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{equation.72}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:rep-kick\endcsname{{B.9}{398}{}{equation.1101}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:rep-kick\endcsname{\default{B.9}\page{398}\abspage{402}\zc@counter{equation}\thecounter{B.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{398}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{equation.1101}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-T\endcsname{{10.5}{44}{The limiting semigroups}{equation.134}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-T\endcsname{\default{10.5}\page{44}\abspage{48}\zc@counter{equation}\thecounter{10.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{44}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.134}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-action-entries\endcsname{{10.27}{56}{The Casimir operator in the polar chart}{equation.176}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-action-entries\endcsname{\default{10.27}\page{56}\abspage{60}\zc@counter{equation}\thecounter{10.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{56}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.176}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-carre\endcsname{{10.26}{56}{}{equation.175}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-carre\endcsname{\default{10.26}\page{56}\abspage{60}\zc@counter{equation}\thecounter{10.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{56}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.175}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-chain-bound\endcsname{{10.45}{66}{The permanent residue}{equation.201}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-chain-bound\endcsname{\default{10.45}\page{66}\abspage{70}\zc@counter{equation}\thecounter{10.45}\zc@type{equation}\zc@cntval{45}\zc@pgval{66}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.201}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-chain-pieces\endcsname{{10.36}{63}{The permanent residue}{equation.190}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-chain-pieces\endcsname{\default{10.36}\page{63}\abspage{67}\zc@counter{equation}\thecounter{10.36}\zc@type{equation}\zc@cntval{36}\zc@pgval{63}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.190}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-charge\endcsname{{10.21}{54}{Charges}{equation.166}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-charge\endcsname{\default{10.21}\page{54}\abspage{58}\zc@counter{equation}\thecounter{10.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{54}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.166}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-coefficients\endcsname{{10.4}{43}{Scaled coordinates}{equation.132}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-coefficients\endcsname{\default{10.4}\page{43}\abspage{47}\zc@counter{equation}\thecounter{10.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{43}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.132}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-component-equation\endcsname{{10.41}{64}{}{equation.197}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-component-equation\endcsname{\default{10.41}\page{64}\abspage{68}\zc@counter{equation}\thecounter{10.41}\zc@type{equation}\zc@cntval{41}\zc@pgval{64}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.197}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-components\endcsname{{10.31}{60}{Freezing at the noncompact boundary}{equation.182}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-components\endcsname{\default{10.31}\page{60}\abspage{64}\zc@counter{equation}\thecounter{10.31}\zc@type{equation}\zc@cntval{31}\zc@pgval{60}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.182}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-coupling\endcsname{{10.15}{50}{The scalar residue}{equation.156}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-coupling\endcsname{\default{10.15}\page{50}\abspage{54}\zc@counter{equation}\thecounter{10.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{50}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.156}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-d-polar\endcsname{{10.17}{52}{The polar chart}{equation.160}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-d-polar\endcsname{\default{10.17}\page{52}\abspage{56}\zc@counter{equation}\thecounter{10.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{52}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.160}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-entries-product\endcsname{{10.25}{56}{}{equation.174}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-entries-product\endcsname{\default{10.25}\page{56}\abspage{60}\zc@counter{equation}\thecounter{10.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{56}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.174}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-envelope\endcsname{{10.10}{48}{}{equation.148}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-envelope\endcsname{\default{10.10}\page{48}\abspage{52}\zc@counter{equation}\thecounter{10.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{48}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.148}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-envelope-original\endcsname{{10.11}{48}{}{equation.149}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-envelope-original\endcsname{\default{10.11}\page{48}\abspage{52}\zc@counter{equation}\thecounter{10.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{48}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.149}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-exit\endcsname{{10.39}{64}{}{equation.194}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-exit\endcsname{\default{10.39}\page{64}\abspage{68}\zc@counter{equation}\thecounter{10.39}\zc@type{equation}\zc@cntval{39}\zc@pgval{64}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.194}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-flat-limit\endcsname{{10.6}{44}{}{equation.139}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-flat-limit\endcsname{\default{10.6}\page{44}\abspage{48}\zc@counter{equation}\thecounter{10.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{44}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.139}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-frozen-chain\endcsname{{10.37}{63}{The permanent residue}{equation.191}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-frozen-chain\endcsname{\default{10.37}\page{63}\abspage{67}\zc@counter{equation}\thecounter{10.37}\zc@type{equation}\zc@cntval{37}\zc@pgval{63}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.191}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-frozen-limit\endcsname{{10.46}{69}{The permanent residue}{equation.202}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-frozen-limit\endcsname{\default{10.46}\page{69}\abspage{73}\zc@counter{equation}\thecounter{10.46}\zc@type{equation}\zc@cntval{46}\zc@pgval{69}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.202}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-frozen-source\endcsname{{10.34}{61}{}{equation.186}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-frozen-source\endcsname{\default{10.34}\page{61}\abspage{65}\zc@counter{equation}\thecounter{10.34}\zc@type{equation}\zc@cntval{34}\zc@pgval{61}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.186}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-generator-limit\endcsname{{10.33}{61}{}{equation.185}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-generator-limit\endcsname{\default{10.33}\page{61}\abspage{65}\zc@counter{equation}\thecounter{10.33}\zc@type{equation}\zc@cntval{33}\zc@pgval{61}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.185}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-gronwall\endcsname{{10.42}{65}{The permanent residue}{equation.198}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-gronwall\endcsname{\default{10.42}\page{65}\abspage{69}\zc@counter{equation}\thecounter{10.42}\zc@type{equation}\zc@cntval{42}\zc@pgval{65}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.198}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-h-transform\endcsname{{10.7}{44}{}{equation.141}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-h-transform\endcsname{\default{10.7}\page{44}\abspage{48}\zc@counter{equation}\thecounter{10.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{44}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.141}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-hierarchy\endcsname{{10.8}{46}{}{equation.144}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-hierarchy\endcsname{\default{10.8}\page{46}\abspage{50}\zc@counter{equation}\thecounter{10.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{46}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.144}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-left-even\endcsname{{10.23}{55}{}{equation.171}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-left-even\endcsname{\default{10.23}\page{55}\abspage{59}\zc@counter{equation}\thecounter{10.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{55}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.171}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-left-odd\endcsname{{10.24}{55}{}{equation.172}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-left-odd\endcsname{\default{10.24}\page{55}\abspage{59}\zc@counter{equation}\thecounter{10.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{55}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.172}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-middle\endcsname{{10.14}{49}{}{equation.155}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-middle\endcsname{\default{10.14}\page{49}\abspage{53}\zc@counter{equation}\thecounter{10.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{49}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.155}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-pair\endcsname{{10.9}{46}{}{equation.145}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-pair\endcsname{\default{10.9}\page{46}\abspage{50}\zc@counter{equation}\thecounter{10.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{46}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.145}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-permanent\endcsname{{10.35}{63}{}{equation.189}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-permanent\endcsname{\default{10.35}\page{63}\abspage{67}\zc@counter{equation}\thecounter{10.35}\zc@type{equation}\zc@cntval{35}\zc@pgval{63}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.189}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-polar\endcsname{{10.22}{54}{}{equation.169}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-polar\endcsname{\default{10.22}\page{54}\abspage{58}\zc@counter{equation}\thecounter{10.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{54}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.169}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-polar-form\endcsname{{10.28}{58}{The Casimir operator in the polar chart}{equation.177}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-polar-form\endcsname{\default{10.28}\page{58}\abspage{62}\zc@counter{equation}\thecounter{10.28}\zc@type{equation}\zc@cntval{28}\zc@pgval{58}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.177}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-polar-form2\endcsname{{10.29}{59}{The Casimir operator in the polar chart}{equation.178}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-polar-form2\endcsname{\default{10.29}\page{59}\abspage{63}\zc@counter{equation}\thecounter{10.29}\zc@type{equation}\zc@cntval{29}\zc@pgval{59}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.178}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-polar-integral\endcsname{{10.18}{53}{The polar chart}{equation.161}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-polar-integral\endcsname{\default{10.18}\page{53}\abspage{57}\zc@counter{equation}\thecounter{10.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{53}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.161}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-polar-test\endcsname{{10.30}{60}{The Casimir operator in the polar chart}{equation.179}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-polar-test\endcsname{\default{10.30}\page{60}\abspage{64}\zc@counter{equation}\thecounter{10.30}\zc@type{equation}\zc@cntval{30}\zc@pgval{60}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.179}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-polynomial-bound\endcsname{{10.44}{66}{The permanent residue}{equation.200}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-polynomial-bound\endcsname{\default{10.44}\page{66}\abspage{70}\zc@counter{equation}\thecounter{10.44}\zc@type{equation}\zc@cntval{44}\zc@pgval{66}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.200}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-radial-moments\endcsname{{10.38}{63}{}{equation.193}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-radial-moments\endcsname{\default{10.38}\page{63}\abspage{67}\zc@counter{equation}\thecounter{10.38}\zc@type{equation}\zc@cntval{38}\zc@pgval{63}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.193}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-rescaling\endcsname{{10.3}{43}{Scaled coordinates}{equation.131}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-rescaling\endcsname{\default{10.3}\page{43}\abspage{47}\zc@counter{equation}\thecounter{10.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{43}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.131}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-scalar\endcsname{{10.12}{49}{}{equation.152}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-scalar\endcsname{\default{10.12}\page{49}\abspage{53}\zc@counter{equation}\thecounter{10.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{49}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.152}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-scalar-generator\endcsname{{10.32}{61}{Freezing at the noncompact boundary}{equation.183}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-scalar-generator\endcsname{\default{10.32}\page{61}\abspage{65}\zc@counter{equation}\thecounter{10.32}\zc@type{equation}\zc@cntval{32}\zc@pgval{61}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.183}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-scaled-kernel\endcsname{{10.16}{51}{The scalar residue}{equation.157}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-scaled-kernel\endcsname{\default{10.16}\page{51}\abspage{55}\zc@counter{equation}\thecounter{10.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{51}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.157}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-scaled-sde\endcsname{{10.1}{43}{Scaled coordinates}{equation.129}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-scaled-sde\endcsname{\default{10.1}\page{43}\abspage{47}\zc@counter{equation}\thecounter{10.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{43}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.129}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-scaled-states\endcsname{{10.2}{43}{Scaled coordinates}{equation.130}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-scaled-states\endcsname{\default{10.2}\page{43}\abspage{47}\zc@counter{equation}\thecounter{10.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{43}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.130}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-side\endcsname{{10.13}{49}{}{equation.154}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-side\endcsname{\default{10.13}\page{49}\abspage{53}\zc@counter{equation}\thecounter{10.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{49}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.154}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-source\endcsname{{10.19}{53}{The polar chart}{equation.162}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-source\endcsname{\default{10.19}\page{53}\abspage{57}\zc@counter{equation}\thecounter{10.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{53}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.162}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-source-polar\endcsname{{10.20}{53}{The polar chart}{equation.163}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-source-polar\endcsname{\default{10.20}\page{53}\abspage{57}\zc@counter{equation}\thecounter{10.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{53}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.163}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-stopped-fk\endcsname{{10.43}{65}{The permanent residue}{equation.199}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-stopped-fk\endcsname{\default{10.43}\page{65}\abspage{69}\zc@counter{equation}\thecounter{10.43}\zc@type{equation}\zc@cntval{43}\zc@pgval{65}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.199}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:res-tight\endcsname{{10.40}{64}{}{equation.195}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:res-tight\endcsname{\default{10.40}\page{64}\abspage{68}\zc@counter{equation}\thecounter{10.40}\zc@type{equation}\zc@cntval{40}\zc@pgval{64}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{equation.195}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-V-ratio\endcsname{{14.40}{155}{The crossover}{equation.443}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-V-ratio\endcsname{\default{14.40}\page{155}\abspage{159}\zc@counter{equation}\thecounter{14.40}\zc@type{equation}\zc@cntval{40}\zc@pgval{155}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.443}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-angular\endcsname{{14.31}{147}{The stationary pairing}{equation.421}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-angular\endcsname{\default{14.31}\page{147}\abspage{151}\zc@counter{equation}\thecounter{14.31}\zc@type{equation}\zc@cntval{31}\zc@pgval{147}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.421}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-approx\endcsname{{14.27}{145}{}{equation.415}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-approx\endcsname{\default{14.27}\page{145}\abspage{149}\zc@counter{equation}\thecounter{14.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{145}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.415}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-balanced\endcsname{{14.3}{131}{The pair measure}{equation.365}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-balanced\endcsname{\default{14.3}\page{131}\abspage{135}\zc@counter{equation}\thecounter{14.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{131}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.365}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-bessel\endcsname{{14.29}{146}{}{equation.419}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-bessel\endcsname{\default{14.29}\page{146}\abspage{150}\zc@counter{equation}\thecounter{14.29}\zc@type{equation}\zc@cntval{29}\zc@pgval{146}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.419}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-bessel-identities\endcsname{{14.17}{138}{}{equation.392}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-bessel-identities\endcsname{\default{14.17}\page{138}\abspage{142}\zc@counter{equation}\thecounter{14.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{138}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.392}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-bv-fourier\endcsname{{14.37}{153}{Large-scale covariance}{equation.438}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-bv-fourier\endcsname{\default{14.37}\page{153}\abspage{157}\zc@counter{equation}\thecounter{14.37}\zc@type{equation}\zc@cntval{37}\zc@pgval{153}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.438}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-circle\endcsname{{14.30}{146}{The stationary pairing}{equation.420}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-circle\endcsname{\default{14.30}\page{146}\abspage{150}\zc@counter{equation}\thecounter{14.30}\zc@type{equation}\zc@cntval{30}\zc@pgval{146}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.420}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-cov-fourier\endcsname{{14.7}{133}{}{equation.374}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-cov-fourier\endcsname{\default{14.7}\page{133}\abspage{137}\zc@counter{equation}\thecounter{14.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{133}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.374}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-filter-tail\endcsname{{14.12}{136}{Laplace transforms of positive measures}{equation.384}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-filter-tail\endcsname{\default{14.12}\page{136}\abspage{140}\zc@counter{equation}\thecounter{14.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{136}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.384}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-flat-constant\endcsname{{14.20}{140}{}{equation.400}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-flat-constant\endcsname{\default{14.20}\page{140}\abspage{144}\zc@counter{equation}\thecounter{14.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{140}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.400}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-fractional\endcsname{{14.38}{154}{Large-scale covariance}{equation.439}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-fractional\endcsname{\default{14.38}\page{154}\abspage{158}\zc@counter{equation}\thecounter{14.38}\zc@type{equation}\zc@cntval{38}\zc@pgval{154}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.439}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-gap\endcsname{{14.19}{140}{}{equation.398}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-gap\endcsname{\default{14.19}\page{140}\abspage{144}\zc@counter{equation}\thecounter{14.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{140}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.398}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-growth\endcsname{{14.11}{136}{Laplace transforms of positive measures}{equation.383}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-growth\endcsname{\default{14.11}\page{136}\abspage{140}\zc@counter{equation}\thecounter{14.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{136}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.383}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-gue\endcsname{{14.4}{132}{}{equation.368}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-gue\endcsname{\default{14.4}\page{132}\abspage{136}\zc@counter{equation}\thecounter{14.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{132}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.368}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-kernel-bound\endcsname{{14.25}{143}{}{equation.411}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-kernel-bound\endcsname{\default{14.25}\page{143}\abspage{147}\zc@counter{equation}\thecounter{14.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{143}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.411}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-laplace\endcsname{{14.6}{133}{}{equation.372}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-laplace\endcsname{\default{14.6}\page{133}\abspage{137}\zc@counter{equation}\thecounter{14.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{133}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.372}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-laplace-smoothed\endcsname{{14.9}{134}{The Bartlett spectrum}{equation.376}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-laplace-smoothed\endcsname{\default{14.9}\page{134}\abspage{138}\zc@counter{equation}\thecounter{14.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{134}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.376}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-large-damping\endcsname{{14.35}{151}{}{equation.433}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-large-damping\endcsname{\default{14.35}\page{151}\abspage{155}\zc@counter{equation}\thecounter{14.35}\zc@type{equation}\zc@cntval{35}\zc@pgval{151}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.433}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-largecov-exact\endcsname{{14.36}{153}{Large-scale covariance}{equation.437}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-largecov-exact\endcsname{\default{14.36}\page{153}\abspage{157}\zc@counter{equation}\thecounter{14.36}\zc@type{equation}\zc@cntval{36}\zc@pgval{153}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.437}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-m-endpoints\endcsname{{14.32}{147}{}{equation.424}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-m-endpoints\endcsname{\default{14.32}\page{147}\abspage{151}\zc@counter{equation}\thecounter{14.32}\zc@type{equation}\zc@cntval{32}\zc@pgval{147}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.424}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-multiplier\endcsname{{14.21}{140}{}{equation.402}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-multiplier\endcsname{\default{14.21}\page{140}\abspage{144}\zc@counter{equation}\thecounter{14.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{140}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.402}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-pairing\endcsname{{14.26}{145}{The stationary approximation}{equation.413}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-pairing\endcsname{\default{14.26}\page{145}\abspage{149}\zc@counter{equation}\thecounter{14.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{145}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.413}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-pairing-explicit\endcsname{{14.33}{148}{}{equation.426}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-pairing-explicit\endcsname{\default{14.33}\page{148}\abspage{152}\zc@counter{equation}\thecounter{14.33}\zc@type{equation}\zc@cntval{33}\zc@pgval{148}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.426}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-parseval\endcsname{{14.5}{133}{}{equation.370}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-parseval\endcsname{\default{14.5}\page{133}\abspage{137}\zc@counter{equation}\thecounter{14.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{133}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.370}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-physics\endcsname{{14.34}{150}{}{equation.429}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-physics\endcsname{\default{14.34}\page{150}\abspage{154}\zc@counter{equation}\thecounter{14.34}\zc@type{equation}\zc@cntval{34}\zc@pgval{150}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.429}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-pq\endcsname{{14.15}{137}{The stationary approximation}{equation.388}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-pq\endcsname{\default{14.15}\page{137}\abspage{141}\zc@counter{equation}\thecounter{14.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{137}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.388}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-scaled-scalar\endcsname{{14.14}{137}{The stationary approximation}{equation.387}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-scaled-scalar\endcsname{\default{14.14}\page{137}\abspage{141}\zc@counter{equation}\thecounter{14.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{137}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.387}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-scaling\endcsname{{14.13}{137}{The stationary approximation}{equation.386}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-scaling\endcsname{\default{14.13}\page{137}\abspage{141}\zc@counter{equation}\thecounter{14.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{137}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.386}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-second-jet\endcsname{{14.2}{131}{The pair measure}{equation.364}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-second-jet\endcsname{\default{14.2}\page{131}\abspage{135}\zc@counter{equation}\thecounter{14.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{131}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.364}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-simplex\endcsname{{14.28}{145}{The stationary approximation}{equation.416}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-simplex\endcsname{\default{14.28}\page{145}\abspage{149}\zc@counter{equation}\thecounter{14.28}\zc@type{equation}\zc@cntval{28}\zc@pgval{145}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.416}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-state\endcsname{{14.16}{137}{The stationary approximation}{equation.389}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-state\endcsname{\default{14.16}\page{137}\abspage{141}\zc@counter{equation}\thecounter{14.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{137}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.389}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-state-identity\endcsname{{14.22}{142}{}{equation.404}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-state-identity\endcsname{\default{14.22}\page{142}\abspage{146}\zc@counter{equation}\thecounter{14.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{142}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.404}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-state-tail\endcsname{{14.23}{142}{}{equation.405}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-state-tail\endcsname{\default{14.23}\page{142}\abspage{146}\zc@counter{equation}\thecounter{14.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{142}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.405}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-transient\endcsname{{14.24}{143}{}{equation.409}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-transient\endcsname{\default{14.24}\page{143}\abspage{147}\zc@counter{equation}\thecounter{14.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{143}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.409}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-translation-bound\endcsname{{14.1}{129}{}{equation.362}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-translation-bound\endcsname{\default{14.1}\page{129}\abspage{133}\zc@counter{equation}\thecounter{14.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{129}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.362}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-variance-bartlett\endcsname{{14.8}{134}{The Bartlett spectrum}{equation.375}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-variance-bartlett\endcsname{\default{14.8}\page{134}\abspage{138}\zc@counter{equation}\thecounter{14.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{134}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.375}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-variance-fourier\endcsname{{14.10}{135}{The Bartlett spectrum}{equation.377}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-variance-fourier\endcsname{\default{14.10}\page{135}\abspage{139}\zc@counter{equation}\thecounter{14.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{135}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.377}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-w\endcsname{{14.18}{138}{}{equation.394}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-w\endcsname{\default{14.18}\page{138}\abspage{142}\zc@counter{equation}\thecounter{14.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{138}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.394}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:scales-wire\endcsname{{14.39}{154}{}{equation.442}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:scales-wire\endcsname{\default{14.39}\page{154}\abspage{158}\zc@counter{equation}\thecounter{14.39}\zc@type{equation}\zc@cntval{39}\zc@pgval{154}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{equation.442}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:sigma-chains-elliptic\endcsname{{C.7}{405}{Semigroup products at all times}{equation.1115}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:sigma-chains-elliptic\endcsname{\default{C.7}\page{405}\abspage{409}\zc@counter{equation}\thecounter{C.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{405}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{equation.1115}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:sigma-chains-interior\endcsname{{C.6}{403}{Weights}{equation.1112}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:sigma-chains-interior\endcsname{\default{C.6}\page{403}\abspage{407}\zc@counter{equation}\thecounter{C.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{403}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{equation.1112}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:sigma-coercive\endcsname{{C.4}{401}{The form}{equation.1108}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:sigma-coercive\endcsname{\default{C.4}\page{401}\abspage{405}\zc@counter{equation}\thecounter{C.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{401}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{equation.1108}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:sigma-domain\endcsname{{C.5}{401}{The form}{equation.1109}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:sigma-domain\endcsname{\default{C.5}\page{401}\abspage{405}\zc@counter{equation}\thecounter{C.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{401}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{equation.1109}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:sigma-generator-local\endcsname{{C.1}{400}{Local structure}{equation.1104}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:sigma-generator-local\endcsname{\default{C.1}\page{400}\abspage{404}\zc@counter{equation}\thecounter{C.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{400}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{equation.1104}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:sigma-n-small\endcsname{{C.2}{400}{Local structure}{equation.1105}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:sigma-n-small\endcsname{\default{C.2}\page{400}\abspage{404}\zc@counter{equation}\thecounter{C.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{400}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{equation.1105}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:sigma-potential-bounds\endcsname{{C.3}{400}{Local structure}{equation.1106}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:sigma-potential-bounds\endcsname{\default{C.3}\page{400}\abspage{404}\zc@counter{equation}\thecounter{C.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{400}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{equation.1106}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-Pi-x\endcsname{{17.10}{248}{The decomposition of the step}{equation.681}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-Pi-x\endcsname{\default{17.10}\page{248}\abspage{252}\zc@counter{equation}\thecounter{17.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{248}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.681}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-R0\endcsname{{17.24}{259}{The uncharged part}{equation.709}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-R0\endcsname{\default{17.24}\page{259}\abspage{263}\zc@counter{equation}\thecounter{17.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{259}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.709}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-R0-row\endcsname{{17.25}{259}{The uncharged part}{equation.710}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-R0-row\endcsname{\default{17.25}\page{259}\abspage{263}\zc@counter{equation}\thecounter{17.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{259}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.710}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-anti-marg\endcsname{{17.19}{254}{}{equation.697}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-anti-marg\endcsname{\default{17.19}\page{254}\abspage{258}\zc@counter{equation}\thecounter{17.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{254}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.697}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-anti-odd\endcsname{{17.18}{254}{}{equation.696}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-anti-odd\endcsname{\default{17.18}\page{254}\abspage{258}\zc@counter{equation}\thecounter{17.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{254}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.696}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-blocks\endcsname{{17.12}{249}{}{equation.684}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-blocks\endcsname{\default{17.12}\page{249}\abspage{253}\zc@counter{equation}\thecounter{17.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{249}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.684}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-body-odd-marg\endcsname{{17.22}{256}{}{equation.703}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-body-odd-marg\endcsname{\default{17.22}\page{256}\abspage{260}\zc@counter{equation}\thecounter{17.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{256}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.703}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-body-parity\endcsname{{17.20}{256}{}{equation.701}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-body-parity\endcsname{\default{17.20}\page{256}\abspage{260}\zc@counter{equation}\thecounter{17.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{256}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.701}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-body-parity-bounds\endcsname{{17.21}{256}{}{equation.702}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-body-parity-bounds\endcsname{\default{17.21}\page{256}\abspage{260}\zc@counter{equation}\thecounter{17.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{256}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.702}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-degree-one\endcsname{{17.11}{248}{The decomposition of the step}{equation.682}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-degree-one\endcsname{\default{17.11}\page{248}\abspage{252}\zc@counter{equation}\thecounter{17.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{248}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.682}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-dirichlet\endcsname{{17.16}{252}{The geodesic difference energy}{equation.690}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-dirichlet\endcsname{\default{17.16}\page{252}\abspage{256}\zc@counter{equation}\thecounter{17.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{252}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.690}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-dirichlet-one\endcsname{{17.17}{252}{The geodesic difference energy}{equation.691}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-dirichlet-one\endcsname{\default{17.17}\page{252}\abspage{256}\zc@counter{equation}\thecounter{17.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{252}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.691}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-energy\endcsname{{17.13}{250}{}{equation.687}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-energy\endcsname{\default{17.13}\page{250}\abspage{254}\zc@counter{equation}\thecounter{17.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{250}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.687}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-energy-gradient\endcsname{{17.14}{250}{The geodesic difference energy}{equation.688}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-energy-gradient\endcsname{\default{17.14}\page{250}\abspage{254}\zc@counter{equation}\thecounter{17.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{250}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.688}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-energy-pointwise\endcsname{{17.15}{250}{The geodesic difference energy}{equation.689}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-energy-pointwise\endcsname{\default{17.15}\page{250}\abspage{254}\zc@counter{equation}\thecounter{17.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{250}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.689}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-exponent\endcsname{{17.23}{258}{The uncharged part}{equation.704}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-exponent\endcsname{\default{17.23}\page{258}\abspage{262}\zc@counter{equation}\thecounter{17.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{258}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.704}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-fki\endcsname{{17.26}{260}{Holonomy}{equation.714}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-fki\endcsname{\default{17.26}\page{260}\abspage{264}\zc@counter{equation}\thecounter{17.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{260}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.714}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-four-term\endcsname{{17.4}{242}{}{equation.666}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-four-term\endcsname{\default{17.4}\page{242}\abspage{246}\zc@counter{equation}\thecounter{17.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{242}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.666}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-frame\endcsname{{17.9}{245}{}{equation.676}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-frame\endcsname{\default{17.9}\page{245}\abspage{249}\zc@counter{equation}\thecounter{17.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{245}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.676}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-higher-move\endcsname{{17.30}{264}{Higher words}{equation.724}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-higher-move\endcsname{\default{17.30}\page{264}\abspage{268}\zc@counter{equation}\thecounter{17.30}\zc@type{equation}\zc@cntval{30}\zc@pgval{264}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.724}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-higher-term\endcsname{{17.29}{264}{Higher words}{equation.723}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-higher-term\endcsname{\default{17.29}\page{264}\abspage{268}\zc@counter{equation}\thecounter{17.29}\zc@type{equation}\zc@cntval{29}\zc@pgval{264}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.723}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-hol\endcsname{{17.28}{261}{}{equation.718}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-hol\endcsname{\default{17.28}\page{261}\abspage{265}\zc@counter{equation}\thecounter{17.28}\zc@type{equation}\zc@cntval{28}\zc@pgval{261}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.718}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-hol-formula\endcsname{{17.27}{261}{Holonomy}{equation.715}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-hol-formula\endcsname{\default{17.27}\page{261}\abspage{265}\zc@counter{equation}\thecounter{17.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{261}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.715}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-kernel\endcsname{{17.5}{243}{The kernel in midpoint coordinates}{equation.668}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-kernel\endcsname{\default{17.5}\page{243}\abspage{247}\zc@counter{equation}\thecounter{17.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{243}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.668}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-kick\endcsname{{17.2}{241}{The reference contraction}{equation.662}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-kick\endcsname{\default{17.2}\page{241}\abspage{245}\zc@counter{equation}\thecounter{17.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{241}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.662}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-midframe\endcsname{{17.6}{244}{Midpoint frames}{equation.670}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-midframe\endcsname{\default{17.6}\page{244}\abspage{248}\zc@counter{equation}\thecounter{17.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{244}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.670}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-midpoint\endcsname{{17.8}{244}{}{equation.673}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-midpoint\endcsname{\default{17.8}\page{244}\abspage{248}\zc@counter{equation}\thecounter{17.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{244}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.673}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-product-defects\endcsname{{17.3}{242}{The reference contraction}{equation.663}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-product-defects\endcsname{\default{17.3}\page{242}\abspage{246}\zc@counter{equation}\thecounter{17.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{242}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.663}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-right\endcsname{{17.36}{268}{The source}{equation.734}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-right\endcsname{\default{17.36}\page{268}\abspage{272}\zc@counter{equation}\thecounter{17.36}\zc@type{equation}\zc@cntval{36}\zc@pgval{268}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.734}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-scalar\endcsname{{17.1}{241}{The reference contraction}{equation.661}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-scalar\endcsname{\default{17.1}\page{241}\abspage{245}\zc@counter{equation}\thecounter{17.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{241}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.661}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-src-decomposition\endcsname{{17.35}{267}{The source}{equation.733}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-src-decomposition\endcsname{\default{17.35}\page{267}\abspage{271}\zc@counter{equation}\thecounter{17.35}\zc@type{equation}\zc@cntval{35}\zc@pgval{267}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.733}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-src-factor\endcsname{{17.34}{267}{The source}{equation.732}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-src-factor\endcsname{\default{17.34}\page{267}\abspage{271}\zc@counter{equation}\thecounter{17.34}\zc@type{equation}\zc@cntval{34}\zc@pgval{267}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.732}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-src-multiplier\endcsname{{17.32}{266}{The source}{equation.730}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-src-multiplier\endcsname{\default{17.32}\page{266}\abspage{270}\zc@counter{equation}\thecounter{17.32}\zc@type{equation}\zc@cntval{32}\zc@pgval{266}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.730}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-src-parts\endcsname{{17.31}{266}{The source}{equation.728}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-src-parts\endcsname{\default{17.31}\page{266}\abspage{270}\zc@counter{equation}\thecounter{17.31}\zc@type{equation}\zc@cntval{31}\zc@pgval{266}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.728}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-src-split\endcsname{{17.33}{266}{The source}{equation.731}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-src-split\endcsname{\default{17.33}\page{266}\abspage{270}\zc@counter{equation}\thecounter{17.33}\zc@type{equation}\zc@cntval{33}\zc@pgval{266}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.731}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-swap\endcsname{{17.7}{244}{Midpoint frames}{equation.671}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-swap\endcsname{\default{17.7}\page{244}\abspage{248}\zc@counter{equation}\thecounter{17.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{244}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.671}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:stab-weight-split\endcsname{{17.37}{268}{Proof of the main theorem}{equation.736}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:stab-weight-split\endcsname{\default{17.37}\page{268}\abspage{272}\zc@counter{equation}\thecounter{17.37}\zc@type{equation}\zc@cntval{37}\zc@pgval{268}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{equation.736}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-Pi\endcsname{{A.12}{382}{An operator-valued Dirichlet-form bound}{equation.1046}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-Pi\endcsname{\default{A.12}\page{382}\abspage{386}\zc@counter{equation}\thecounter{A.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{382}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1046}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-defect-B\endcsname{{A.4}{379}{}{equation.1036}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-defect-B\endcsname{\default{A.4}\page{379}\abspage{383}\zc@counter{equation}\thecounter{A.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{379}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1036}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-defect-conclusion\endcsname{{A.3}{379}{}{equation.1034}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-defect-conclusion\endcsname{\default{A.3}\page{379}\abspage{383}\zc@counter{equation}\thecounter{A.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{379}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1034}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-defect-cross\endcsname{{A.7}{379}{Perturbations of a contraction}{equation.1039}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-defect-cross\endcsname{\default{A.7}\page{379}\abspage{383}\zc@counter{equation}\thecounter{A.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{379}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1039}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-defect-expansion\endcsname{{A.6}{379}{Perturbations of a contraction}{equation.1038}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-defect-expansion\endcsname{\default{A.6}\page{379}\abspage{383}\zc@counter{equation}\thecounter{A.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{379}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1038}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-defect-square\endcsname{{A.8}{379}{Perturbations of a contraction}{equation.1040}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-defect-square\endcsname{\default{A.8}\page{379}\abspage{383}\zc@counter{equation}\thecounter{A.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{379}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1040}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-defect-step\endcsname{{A.9}{380}{Perturbations of a contraction}{equation.1041}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-defect-step\endcsname{\default{A.9}\page{380}\abspage{384}\zc@counter{equation}\thecounter{A.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{380}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1041}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-defect-sums\endcsname{{A.5}{379}{}{equation.1037}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-defect-sums\endcsname{\default{A.5}\page{379}\abspage{383}\zc@counter{equation}\thecounter{A.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{379}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1037}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-defects\endcsname{{A.1}{378}{Perturbations of a contraction}{equation.1030}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-defects\endcsname{\default{A.1}\page{378}\abspage{382}\zc@counter{equation}\thecounter{A.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{378}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1030}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-dirichlet-form\endcsname{{A.19}{383}{An operator-valued Dirichlet-form bound}{equation.1056}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-dirichlet-form\endcsname{\default{A.19}\page{383}\abspage{387}\zc@counter{equation}\thecounter{A.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{383}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1056}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-dirichlet-identity\endcsname{{A.20}{383}{An operator-valued Dirichlet-form bound}{equation.1057}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-dirichlet-identity\endcsname{\default{A.20}\page{383}\abspage{387}\zc@counter{equation}\thecounter{A.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{383}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1057}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-ellipsoid\endcsname{{A.27}{388}{}{equation.1073}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-ellipsoid\endcsname{\default{A.27}\page{388}\abspage{392}\zc@counter{equation}\thecounter{A.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{388}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1073}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-form-bound\endcsname{{A.2}{378}{}{equation.1033}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-form-bound\endcsname{\default{A.2}\page{378}\abspage{382}\zc@counter{equation}\thecounter{A.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{378}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1033}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-loc\endcsname{{A.21}{384}{Poisson transforms of translation-bounded measures}{equation.1059}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-loc\endcsname{\default{A.21}\page{384}\abspage{388}\zc@counter{equation}\thecounter{A.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{384}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1059}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-placements\endcsname{{A.10}{380}{}{equation.1043}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-placements\endcsname{\default{A.10}\page{380}\abspage{384}\zc@counter{equation}\thecounter{A.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{380}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1043}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-placements-B\endcsname{{A.11}{381}{}{equation.1044}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-placements-B\endcsname{\default{A.11}\page{381}\abspage{385}\zc@counter{equation}\thecounter{A.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{381}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1044}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-poisson-decay\endcsname{{A.22}{385}{Poisson transforms of translation-bounded measures}{equation.1065}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-poisson-decay\endcsname{\default{A.22}\page{385}\abspage{389}\zc@counter{equation}\thecounter{A.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{385}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1065}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-poisson-fubini\endcsname{{A.25}{385}{Poisson transforms of translation-bounded measures}{equation.1068}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-poisson-fubini\endcsname{\default{A.25}\page{385}\abspage{389}\zc@counter{equation}\thecounter{A.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{385}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1068}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-poisson-tail\endcsname{{A.23}{385}{Poisson transforms of translation-bounded measures}{equation.1066}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-poisson-tail\endcsname{\default{A.23}\page{385}\abspage{389}\zc@counter{equation}\thecounter{A.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{385}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1066}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-poisson-test\endcsname{{A.24}{385}{Poisson transforms of translation-bounded measures}{equation.1067}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-poisson-test\endcsname{\default{A.24}\page{385}\abspage{389}\zc@counter{equation}\thecounter{A.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{385}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1067}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-row-T\endcsname{{A.14}{382}{}{equation.1049}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-row-T\endcsname{\default{A.14}\page{382}\abspage{386}\zc@counter{equation}\thecounter{A.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{382}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1049}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-row-bound\endcsname{{A.16}{382}{}{equation.1052}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-row-bound\endcsname{\default{A.16}\page{382}\abspage{386}\zc@counter{equation}\thecounter{A.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{382}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1052}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-row-bound-square\endcsname{{A.17}{382}{}{equation.1053}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-row-bound-square\endcsname{\default{A.17}\page{382}\abspage{386}\zc@counter{equation}\thecounter{A.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{382}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1053}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-row-hyp\endcsname{{A.13}{382}{}{equation.1048}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-row-hyp\endcsname{\default{A.13}\page{382}\abspage{386}\zc@counter{equation}\thecounter{A.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{382}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1048}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-row-identity\endcsname{{A.15}{382}{}{equation.1051}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-row-identity\endcsname{\default{A.15}\page{382}\abspage{386}\zc@counter{equation}\thecounter{A.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{382}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1051}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-row-schur\endcsname{{A.18}{383}{An operator-valued Dirichlet-form bound}{equation.1055}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-row-schur\endcsname{\default{A.18}\page{383}\abspage{387}\zc@counter{equation}\thecounter{A.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{383}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1055}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tools-submean\endcsname{{A.26}{387}{}{equation.1071}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tools-submean\endcsname{\default{A.26}\page{387}\abspage{391}\zc@counter{equation}\thecounter{A.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{387}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{equation.1071}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-ad-trace\endcsname{{16.4}{197}{Super vector spaces}{equation.558}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-ad-trace\endcsname{\default{16.4}\page{197}\abspage{201}\zc@counter{equation}\thecounter{16.4}\zc@type{equation}\zc@cntval{4}\zc@pgval{197}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.558}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-c\endcsname{{16.13}{202}{Root spaces and the tubular map}{equation.571}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-c\endcsname{\default{16.13}\page{202}\abspage{206}\zc@counter{equation}\thecounter{16.13}\zc@type{equation}\zc@cntval{13}\zc@pgval{202}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.571}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-carre\endcsname{{16.17}{204}{The harmonic half-density}{equation.578}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-carre\endcsname{\default{16.17}\page{204}\abspage{208}\zc@counter{equation}\thecounter{16.17}\zc@type{equation}\zc@cntval{17}\zc@pgval{204}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.578}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-carre-split\endcsname{{16.27}{208}{The decomposition of the ambient Laplacian}{equation.591}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-carre-split\endcsname{\default{16.27}\page{208}\abspage{212}\zc@counter{equation}\thecounter{16.27}\zc@type{equation}\zc@cntval{27}\zc@pgval{208}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.591}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-casimir\endcsname{{16.30}{209}{}{equation.595}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-casimir\endcsname{\default{16.30}\page{209}\abspage{213}\zc@counter{equation}\thecounter{16.30}\zc@type{equation}\zc@cntval{30}\zc@pgval{209}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.595}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-closed-contraction\endcsname{{16.52}{237}{}{equation.653}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-closed-contraction\endcsname{\default{16.52}\page{237}\abspage{241}\zc@counter{equation}\thecounter{16.52}\zc@type{equation}\zc@cntval{52}\zc@pgval{237}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.653}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-compact-commutator\endcsname{{16.46}{230}{Mehler kernels with amplitudes}{equation.637}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-compact-commutator\endcsname{\default{16.46}\page{230}\abspage{234}\zc@counter{equation}\thecounter{16.46}\zc@type{equation}\zc@cntval{46}\zc@pgval{230}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.637}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-conjugacy\endcsname{{16.21}{205}{}{equation.583}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-conjugacy\endcsname{\default{16.21}\page{205}\abspage{209}\zc@counter{equation}\thecounter{16.21}\zc@type{equation}\zc@cntval{21}\zc@pgval{205}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.583}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-cross\endcsname{{16.19}{205}{}{equation.581}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-cross\endcsname{\default{16.19}\page{205}\abspage{209}\zc@counter{equation}\thecounter{16.19}\zc@type{equation}\zc@cntval{19}\zc@pgval{205}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.581}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-decomposition\endcsname{{16.29}{209}{The decomposition of the ambient Laplacian}{equation.593}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-decomposition\endcsname{\default{16.29}\page{209}\abspage{213}\zc@counter{equation}\thecounter{16.29}\zc@type{equation}\zc@cntval{29}\zc@pgval{209}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.593}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-derivation-split\endcsname{{16.25}{207}{The decomposition of the ambient Laplacian}{equation.589}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-derivation-split\endcsname{\default{16.25}\page{207}\abspage{211}\zc@counter{equation}\thecounter{16.25}\zc@type{equation}\zc@cntval{25}\zc@pgval{207}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.589}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-differential\endcsname{{16.14}{203}{}{equation.574}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-differential\endcsname{\default{16.14}\page{203}\abspage{207}\zc@counter{equation}\thecounter{16.14}\zc@type{equation}\zc@cntval{14}\zc@pgval{203}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.574}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-endpoint-duality\endcsname{{16.39}{215}{The adapted transfer operator and the normal gauge}{equation.608}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-endpoint-duality\endcsname{\default{16.39}\page{215}\abspage{219}\zc@counter{equation}\thecounter{16.39}\zc@type{equation}\zc@cntval{39}\zc@pgval{215}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.608}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-exp-density\endcsname{{16.10}{200}{The homogeneous superspace and its invariant volume}{equation.566}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-exp-density\endcsname{\default{16.10}\page{200}\abspage{204}\zc@counter{equation}\thecounter{16.10}\zc@type{equation}\zc@cntval{10}\zc@pgval{200}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.566}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-exp-rule\endcsname{{16.18}{204}{The harmonic half-density}{equation.579}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-exp-rule\endcsname{\default{16.18}\page{204}\abspage{208}\zc@counter{equation}\thecounter{16.18}\zc@type{equation}\zc@cntval{18}\zc@pgval{204}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.579}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-four-term\endcsname{{16.48}{233}{Four-term bounds}{equation.643}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-four-term\endcsname{\default{16.48}\page{233}\abspage{237}\zc@counter{equation}\thecounter{16.48}\zc@type{equation}\zc@cntval{48}\zc@pgval{233}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.643}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-gamma\endcsname{{16.35}{214}{The adapted transfer operator and the normal gauge}{equation.604}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-gamma\endcsname{\default{16.35}\page{214}\abspage{218}\zc@counter{equation}\thecounter{16.35}\zc@type{equation}\zc@cntval{35}\zc@pgval{214}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.604}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-gauge-change\endcsname{{16.8}{199}{The homogeneous superspace and its invariant volume}{equation.564}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-gauge-change\endcsname{\default{16.8}\page{199}\abspage{203}\zc@counter{equation}\thecounter{16.8}\zc@type{equation}\zc@cntval{8}\zc@pgval{199}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.564}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-gradient-split\endcsname{{16.26}{208}{The decomposition of the ambient Laplacian}{equation.590}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-gradient-split\endcsname{\default{16.26}\page{208}\abspage{212}\zc@counter{equation}\thecounter{16.26}\zc@type{equation}\zc@cntval{26}\zc@pgval{208}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.590}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-graph-contraction\endcsname{{16.53}{237}{}{equation.654}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-graph-contraction\endcsname{\default{16.53}\page{237}\abspage{241}\zc@counter{equation}\thecounter{16.53}\zc@type{equation}\zc@cntval{53}\zc@pgval{237}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.654}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-graph-defect\endcsname{{16.50}{234}{}{equation.648}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-graph-defect\endcsname{\default{16.50}\page{234}\abspage{238}\zc@counter{equation}\thecounter{16.50}\zc@type{equation}\zc@cntval{50}\zc@pgval{234}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.648}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-half-density\endcsname{{16.11}{201}{Root spaces and the tubular map}{equation.569}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-half-density\endcsname{\default{16.11}\page{201}\abspage{205}\zc@counter{equation}\thecounter{16.11}\zc@type{equation}\zc@cntval{11}\zc@pgval{201}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.569}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-half-weight\endcsname{{16.34}{213}{The adapted transfer operator and the normal gauge}{equation.603}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-half-weight\endcsname{\default{16.34}\page{213}\abspage{217}\zc@counter{equation}\thecounter{16.34}\zc@type{equation}\zc@cntval{34}\zc@pgval{213}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.603}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-harmonic\endcsname{{16.20}{205}{}{equation.582}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-harmonic\endcsname{\default{16.20}\page{205}\abspage{209}\zc@counter{equation}\thecounter{16.20}\zc@type{equation}\zc@cntval{20}\zc@pgval{205}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.582}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-heat-kernel\endcsname{{16.1}{195}{Two scales}{equation.552}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-heat-kernel\endcsname{\default{16.1}\page{195}\abspage{199}\zc@counter{equation}\thecounter{16.1}\zc@type{equation}\zc@cntval{1}\zc@pgval{195}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.552}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-horizontal\endcsname{{16.24}{207}{The decomposition of the ambient Laplacian}{equation.588}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-horizontal\endcsname{\default{16.24}\page{207}\abspage{211}\zc@counter{equation}\thecounter{16.24}\zc@type{equation}\zc@cntval{24}\zc@pgval{207}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.588}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-horizontal-weak\endcsname{{16.28}{208}{The decomposition of the ambient Laplacian}{equation.592}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-horizontal-weak\endcsname{\default{16.28}\page{208}\abspage{212}\zc@counter{equation}\thecounter{16.28}\zc@type{equation}\zc@cntval{28}\zc@pgval{208}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.592}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-interpolation\endcsname{{16.43}{224}{Setting}{equation.625}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-interpolation\endcsname{\default{16.43}\page{224}\abspage{228}\zc@counter{equation}\thecounter{16.43}\zc@type{equation}\zc@cntval{43}\zc@pgval{224}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.625}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-laplace-symbol\endcsname{{16.16}{204}{The harmonic half-density}{equation.577}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-laplace-symbol\endcsname{\default{16.16}\page{204}\abspage{208}\zc@counter{equation}\thecounter{16.16}\zc@type{equation}\zc@cntval{16}\zc@pgval{204}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.577}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-map\endcsname{{16.12}{201}{Root spaces and the tubular map}{equation.570}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-map\endcsname{\default{16.12}\page{201}\abspage{205}\zc@counter{equation}\thecounter{16.12}\zc@type{equation}\zc@cntval{12}\zc@pgval{201}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.570}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-maurer-cartan\endcsname{{16.7}{199}{The homogeneous superspace and its invariant volume}{equation.563}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-maurer-cartan\endcsname{\default{16.7}\page{199}\abspage{203}\zc@counter{equation}\thecounter{16.7}\zc@type{equation}\zc@cntval{7}\zc@pgval{199}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.563}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-mehler\endcsname{{16.32}{212}{The normal oscillator}{equation.599}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-mehler\endcsname{\default{16.32}\page{212}\abspage{216}\zc@counter{equation}\thecounter{16.32}\zc@type{equation}\zc@cntval{32}\zc@pgval{212}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.599}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-mu\endcsname{{16.40}{216}{The exact quadratic normal transfer}{equation.613}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-mu\endcsname{\default{16.40}\page{216}\abspage{220}\zc@counter{equation}\thecounter{16.40}\zc@type{equation}\zc@cntval{40}\zc@pgval{216}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.613}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-normal-coordinates\endcsname{{16.41}{217}{The exact quadratic normal transfer}{equation.614}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-normal-coordinates\endcsname{\default{16.41}\page{217}\abspage{221}\zc@counter{equation}\thecounter{16.41}\zc@type{equation}\zc@cntval{41}\zc@pgval{217}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.614}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-normal-defect\endcsname{{16.55}{238}{The normal contraction and the Ward identity}{equation.656}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-normal-defect\endcsname{\default{16.55}\page{238}\abspage{242}\zc@counter{equation}\thecounter{16.55}\zc@type{equation}\zc@cntval{55}\zc@pgval{238}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.656}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-nu\endcsname{{16.9}{199}{The homogeneous superspace and its invariant volume}{equation.565}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-nu\endcsname{\default{16.9}\page{199}\abspage{203}\zc@counter{equation}\thecounter{16.9}\zc@type{equation}\zc@cntval{9}\zc@pgval{199}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.565}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-odd-commutator\endcsname{{16.47}{231}{Mehler kernels with amplitudes}{equation.638}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-odd-commutator\endcsname{\default{16.47}\page{231}\abspage{235}\zc@counter{equation}\thecounter{16.47}\zc@type{equation}\zc@cntval{47}\zc@pgval{231}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.638}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-odd-wick\endcsname{{16.42}{218}{The exact quadratic normal transfer}{equation.616}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-odd-wick\endcsname{\default{16.42}\page{218}\abspage{222}\zc@counter{equation}\thecounter{16.42}\zc@type{equation}\zc@cntval{42}\zc@pgval{218}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.616}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-pairing\endcsname{{16.31}{211}{The normal oscillator}{equation.598}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-pairing\endcsname{\default{16.31}\page{211}\abspage{215}\zc@counter{equation}\thecounter{16.31}\zc@type{equation}\zc@cntval{31}\zc@pgval{211}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.598}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-pairing-ad\endcsname{{16.36}{214}{The adapted transfer operator and the normal gauge}{equation.605}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-pairing-ad\endcsname{\default{16.36}\page{214}\abspage{218}\zc@counter{equation}\thecounter{16.36}\zc@type{equation}\zc@cntval{36}\zc@pgval{214}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.605}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-pairing-factor\endcsname{{16.37}{214}{The adapted transfer operator and the normal gauge}{equation.606}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-pairing-factor\endcsname{\default{16.37}\page{214}\abspage{218}\zc@counter{equation}\thecounter{16.37}\zc@type{equation}\zc@cntval{37}\zc@pgval{214}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.606}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-partial-traces\endcsname{{16.22}{205}{The harmonic half-density}{equation.584}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-partial-traces\endcsname{\default{16.22}\page{205}\abspage{209}\zc@counter{equation}\thecounter{16.22}\zc@type{equation}\zc@cntval{22}\zc@pgval{205}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.584}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-phase\endcsname{{16.49}{234}{The frozen normal transfer with cutoffs}{equation.646}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-phase\endcsname{\default{16.49}\page{234}\abspage{238}\zc@counter{equation}\thecounter{16.49}\zc@type{equation}\zc@cntval{49}\zc@pgval{234}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.646}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-raw-kernel\endcsname{{16.33}{213}{The adapted transfer operator and the normal gauge}{equation.601}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-raw-kernel\endcsname{\default{16.33}\page{213}\abspage{217}\zc@counter{equation}\thecounter{16.33}\zc@type{equation}\zc@cntval{33}\zc@pgval{213}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.601}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-resolvent-integral\endcsname{{16.23}{205}{The harmonic half-density}{equation.585}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-resolvent-integral\endcsname{\default{16.23}\page{205}\abspage{209}\zc@counter{equation}\thecounter{16.23}\zc@type{equation}\zc@cntval{23}\zc@pgval{205}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.585}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-sdim-hom\endcsname{{16.2}{197}{Super vector spaces}{equation.556}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-sdim-hom\endcsname{\default{16.2}\page{197}\abspage{201}\zc@counter{equation}\thecounter{16.2}\zc@type{equation}\zc@cntval{2}\zc@pgval{197}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.556}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-symmetric-pair\endcsname{{16.5}{198}{The stabilizer of a root point}{equation.560}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-symmetric-pair\endcsname{\default{16.5}\page{198}\abspage{202}\zc@counter{equation}\thecounter{16.5}\zc@type{equation}\zc@cntval{5}\zc@pgval{198}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.560}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-table\endcsname{{16.38}{214}{The adapted transfer operator and the normal gauge}{equation.607}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-table\endcsname{\default{16.38}\page{214}\abspage{218}\zc@counter{equation}\thecounter{16.38}\zc@type{equation}\zc@cntval{38}\zc@pgval{214}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.607}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-tensor-trace\endcsname{{16.3}{197}{Super vector spaces}{equation.557}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-tensor-trace\endcsname{\default{16.3}\page{197}\abspage{201}\zc@counter{equation}\thecounter{16.3}\zc@type{equation}\zc@cntval{3}\zc@pgval{197}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.557}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-unbalanced-defect\endcsname{{16.54}{237}{}{equation.655}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-unbalanced-defect\endcsname{\default{16.54}\page{237}\abspage{241}\zc@counter{equation}\thecounter{16.54}\zc@type{equation}\zc@cntval{54}\zc@pgval{237}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.655}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-unimodular\endcsname{{16.6}{198}{The stabilizer of a root point}{equation.561}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-unimodular\endcsname{\default{16.6}\page{198}\abspage{202}\zc@counter{equation}\thecounter{16.6}\zc@type{equation}\zc@cntval{6}\zc@pgval{198}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.561}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-vacuum-gaussian\endcsname{{16.45}{229}{Mehler kernels with amplitudes}{equation.636}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-vacuum-gaussian\endcsname{\default{16.45}\page{229}\abspage{233}\zc@counter{equation}\thecounter{16.45}\zc@type{equation}\zc@cntval{45}\zc@pgval{229}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.636}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-vacuum-residual\endcsname{{16.51}{234}{}{equation.649}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-vacuum-residual\endcsname{\default{16.51}\page{234}\abspage{238}\zc@counter{equation}\thecounter{16.51}\zc@type{equation}\zc@cntval{51}\zc@pgval{234}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.649}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-volume\endcsname{{16.15}{203}{}{equation.575}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-volume\endcsname{\default{16.15}\page{203}\abspage{207}\zc@counter{equation}\thecounter{16.15}\zc@type{equation}\zc@cntval{15}\zc@pgval{203}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.575}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@e:tube-word-vacuum\endcsname{{16.44}{225}{}{equation.628}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@e:tube-word-vacuum\endcsname{\default{16.44}\page{225}\abspage{229}\zc@counter{equation}\thecounter{16.44}\zc@type{equation}\zc@cntval{44}\zc@pgval{225}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{equation.628}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@f:asm-chain\endcsname{{20.1}{351}{The decomposition of the chain in \eqref {e:asm-one-sided}. The right block is attached to the tube by \zcref {t:end-entrance}, the remaining $N - m_W$ physical steps are compared with adapted steps by \zcref {l:asm-telescope}, and the left endpoint is the constant function $1$. In the balanced adapted chain, the shaded initial layers of $n_h$ steps (\zcref {l:asm-initial-layer}) move the endpoints to the normal fixed state. The outer portions of length about $\theta N$ suppress the unbalanced signatures and prepare body weights, and the balanced tangent variables converge to the continuum semigroup. The two ticks show sites that carry the local sources of \zcref {t:interface}(b), which \zcref {s:asm-b} calls marks}{figure.caption.954}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@f:asm-chain\endcsname{\default{20.1}\page{351}\abspage{355}\zc@counter{figure}\thecounter{20.1}\zc@type{figure}\zc@cntval{1}\zc@pgval{351}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{figure.caption.954}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@f:tube-scales\endcsname{{16.1}{195}{The saddle manifold (solid), the region within normal distance of order $h$ of it where the normal variables of the chain are concentrated (shaded), and the normal cutoff at a fixed radius $r$ (dashed). Each link moves the supermatrix by $h^2$. The normal coordinate relaxes after about $h^{-2}$ links, during which the tangent coordinate moves by about $h$; over the $h^{-4}$ links of the chain the tangent coordinate moves by a distance of order one}{figure.caption.553}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@f:tube-scales\endcsname{\default{16.1}\page{195}\abspage{199}\zc@counter{figure}\thecounter{16.1}\zc@type{figure}\zc@cntval{1}\zc@pgval{195}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{figure.caption.553}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-Egeom\endcsname{{{{(ii)}}}{351}{Conventions and the order of choices}{Item.957}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-Egeom\endcsname{\default{{{(ii)}}}\page{351}\abspage{355}\zc@counter{enumi}\thecounter{{{(ii)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{351}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.957}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-Eout\endcsname{{{{(iii)}}}{351}{Conventions and the order of choices}{Item.958}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-Eout\endcsname{\default{{{(iii)}}}\page{351}\abspage{355}\zc@counter{enumi}\thecounter{{{(iii)}}}\zc@type{item}\zc@cntval{3}\zc@pgval{351}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.958}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-Estar\endcsname{{{{(iv)}}}{352}{Conventions and the order of choices}{Item.960}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-Estar\endcsname{\default{{{(iv)}}}\page{352}\abspage{356}\zc@counter{enumi}\thecounter{{{(iv)}}}\zc@type{item}\zc@cntval{4}\zc@pgval{352}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.960}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-W0\endcsname{{{{(vii)}}}{352}{Conventions and the order of choices}{Item.963}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-W0\endcsname{\default{{{(vii)}}}\page{352}\abspage{356}\zc@counter{enumi}\thecounter{{{(vii)}}}\zc@type{item}\zc@cntval{7}\zc@pgval{352}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.963}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-homotopy-bounds\endcsname{{{{(c)}}}{330}{}{Item.899}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-homotopy-bounds\endcsname{\default{{{(c)}}}\page{330}\abspage{334}\zc@counter{enumi}\thecounter{{{(c)}}}\zc@type{item}\zc@cntval{3}\zc@pgval{330}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.899}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-homotopy-ends\endcsname{{{{(b)}}}{330}{}{Item.898}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-homotopy-ends\endcsname{\default{{{(b)}}}\page{330}\abspage{334}\zc@counter{enumi}\thecounter{{{(b)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{330}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.898}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-homotopy-frame\endcsname{{{{(a)}}}{330}{}{Item.897}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-homotopy-frame\endcsname{\default{{{(a)}}}\page{330}\abspage{334}\zc@counter{enumi}\thecounter{{{(a)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{330}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.897}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-jet-block-a\endcsname{{{{(a)}}}{371}{}{Item.1019}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-jet-block-a\endcsname{\default{{{(a)}}}\page{371}\abspage{375}\zc@counter{enumi}\thecounter{{{(a)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{371}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.1019}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-jet-block-b\endcsname{{{{(b)}}}{371}{}{Item.1021}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-jet-block-b\endcsname{\default{{{(b)}}}\page{371}\abspage{375}\zc@counter{enumi}\thecounter{{{(b)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{371}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.1021}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-radius\endcsname{{{{(vi)}}}{352}{Conventions and the order of choices}{Item.962}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-radius\endcsname{\default{{{(vi)}}}\page{352}\abspage{356}\zc@counter{enumi}\thecounter{{{(vi)}}}\zc@type{item}\zc@cntval{6}\zc@pgval{352}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.962}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-rank\endcsname{{{{(v)}}}{352}{Conventions and the order of choices}{Item.961}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-rank\endcsname{\default{{{(v)}}}\page{352}\abspage{356}\zc@counter{enumi}\thecounter{{{(v)}}}\zc@type{item}\zc@cntval{5}\zc@pgval{352}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.961}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:asm-universal\endcsname{{{{(i)}}}{351}{Conventions and the order of choices}{Item.956}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:asm-universal\endcsname{\default{{{(i)}}}\page{351}\abspage{355}\zc@counter{enumi}\thecounter{{{(i)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{351}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.956}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:cons-contraction\endcsname{{{{(N1)}}}{274}{Inputs from Sections~\ref {s:tube} and~\ref {s:stab}}{Item.748}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:cons-contraction\endcsname{\default{{{(N1)}}}\page{274}\abspage{278}\zc@counter{enumi}\thecounter{{{(N1)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{274}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.748}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:cons-graph\endcsname{{{{(N2)}}}{274}{Inputs from Sections~\ref {s:tube} and~\ref {s:stab}}{Item.749}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:cons-graph\endcsname{\default{{{(N2)}}}\page{274}\abspage{278}\zc@counter{enumi}\thecounter{{{(N2)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{274}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.749}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:cons-localization\endcsname{{{{(N3)}}}{274}{Inputs from Sections~\ref {s:tube} and~\ref {s:stab}}{Item.750}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:cons-localization\endcsname{\default{{{(N3)}}}\page{274}\abspage{278}\zc@counter{enumi}\thecounter{{{(N3)}}}\zc@type{item}\zc@cntval{3}\zc@pgval{274}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.750}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:cons-sect-chart\endcsname{{{{(a)}}}{280}{}{Item.770}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:cons-sect-chart\endcsname{\default{{{(a)}}}\page{280}\abspage{284}\zc@counter{enumi}\thecounter{{{(a)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{280}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.770}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:cons-sect-form\endcsname{{{{(b)}}}{280}{}{Item.771}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:cons-sect-form\endcsname{\default{{{(b)}}}\page{280}\abspage{284}\zc@counter{enumi}\thecounter{{{(b)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{280}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.771}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:cons-stability\endcsname{{{{(N4)}}}{275}{Inputs from Sections~\ref {s:tube} and~\ref {s:stab}}{Item.751}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:cons-stability\endcsname{\default{{{(N4)}}}\page{275}\abspage{279}\zc@counter{enumi}\thecounter{{{(N4)}}}\zc@type{item}\zc@cntval{4}\zc@pgval{275}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.751}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:contour-deform-heat\endcsname{{{{(ii)}}}{395}{}{Item.1094}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:contour-deform-heat\endcsname{\default{{{(ii)}}}\page{395}\abspage{399}\zc@counter{enumi}\thecounter{{{(ii)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{395}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.1094}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:contour-deform-massive\endcsname{{{{(i)}}}{395}{}{Item.1093}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:contour-deform-massive\endcsname{\default{{{(i)}}}\page{395}\abspage{399}\zc@counter{enumi}\thecounter{{{(i)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{395}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.1093}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:contour-real-edge\endcsname{{{{(b)}}}{392}{}{Item.1086}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:contour-real-edge\endcsname{\default{{{(b)}}}\page{392}\abspage{396}\zc@counter{enumi}\thecounter{{{(b)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{392}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.1086}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:contour-real-pinned\endcsname{{{{(c)}}}{392}{}{Item.1089}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:contour-real-pinned\endcsname{\default{{{(c)}}}\page{392}\abspage{396}\zc@counter{enumi}\thecounter{{{(c)}}}\zc@type{item}\zc@cntval{3}\zc@pgval{392}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.1089}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:contour-real-vertex\endcsname{{{{(a)}}}{392}{}{Item.1083}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:contour-real-vertex\endcsname{\default{{{(a)}}}\page{392}\abspage{396}\zc@counter{enumi}\thecounter{{{(a)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{392}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.1083}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:end-height-bb\endcsname{{{{(a)}}}{326}{}{Item.892}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:end-height-bb\endcsname{\default{{{(a)}}}\page{326}\abspage{330}\zc@counter{enumi}\thecounter{{{(a)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{326}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.892}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:end-height-ff\endcsname{{{{(b)}}}{327}{}{Item.893}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:end-height-ff\endcsname{\default{{{(b)}}}\page{327}\abspage{331}\zc@counter{enumi}\thecounter{{{(b)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{327}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.893}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:end-shifted-collar\endcsname{{{{(d)}}}{332}{}{Item.909}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:end-shifted-collar\endcsname{\default{{{(d)}}}\page{332}\abspage{336}\zc@counter{enumi}\thecounter{{{(d)}}}\zc@type{item}\zc@cntval{4}\zc@pgval{332}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.909}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:end-shifted-cont\endcsname{{{{(c)}}}{331}{}{Item.907}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:end-shifted-cont\endcsname{\default{{{(c)}}}\page{331}\abspage{335}\zc@counter{enumi}\thecounter{{{(c)}}}\zc@type{item}\zc@cntval{3}\zc@pgval{331}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.907}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:end-shifted-defect\endcsname{{{{(a)}}}{331}{}{Item.903}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:end-shifted-defect\endcsname{\default{{{(a)}}}\page{331}\abspage{335}\zc@counter{enumi}\thecounter{{{(a)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{331}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.903}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:end-shifted-onsite\endcsname{{{{(b)}}}{331}{}{Item.905}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:end-shifted-onsite\endcsname{\default{{{(b)}}}\page{331}\abspage{335}\zc@counter{enumi}\thecounter{{{(b)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{331}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.905}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:end-ward-a\endcsname{{{{(a)}}}{303}{}{Item.825}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:end-ward-a\endcsname{\default{{{(a)}}}\page{303}\abspage{307}\zc@counter{enumi}\thecounter{{{(a)}}}\zc@type{item}\zc@cntval{1}\zc@pgval{303}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.825}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:end-ward-b\endcsname{{{{(b)}}}{303}{}{Item.828}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:end-ward-b\endcsname{\default{{{(b)}}}\page{303}\abspage{307}\zc@counter{enumi}\thecounter{{{(b)}}}\zc@type{item}\zc@cntval{2}\zc@pgval{303}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.828}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@i:end-ward-c\endcsname{{{{(c)}}}{303}{}{Item.829}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@i:end-ward-c\endcsname{\default{{{(c)}}}\page{303}\abspage{307}\zc@counter{enumi}\thecounter{{{(c)}}}\zc@type{item}\zc@cntval{3}\zc@pgval{303}\zc@enclval{}\zc@pgfmt{2}\anchor{Item.829}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-homotopy\endcsname{{19.16}{330}{}{lemma.896}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-homotopy\endcsname{\default{19.16}\page{330}\abspage{334}\zc@counter{lemma}\thecounter{19.16}\zc@type{lemma}\zc@cntval{16}\zc@pgval{330}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.896}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-incoming\endcsname{{20.12}{373}{}{lemma.1023}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-incoming\endcsname{\default{20.12}\page{373}\abspage{377}\zc@counter{lemma}\thecounter{20.12}\zc@type{lemma}\zc@cntval{12}\zc@pgval{373}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{lemma.1023}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-initial-layer\endcsname{{20.3}{356}{}{lemma.980}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-initial-layer\endcsname{\default{20.3}\page{356}\abspage{360}\zc@counter{lemma}\thecounter{20.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{356}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{lemma.980}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-jet-block\endcsname{{20.11}{371}{}{lemma.1018}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-jet-block\endcsname{\default{20.11}\page{371}\abspage{375}\zc@counter{lemma}\thecounter{20.11}\zc@type{lemma}\zc@cntval{11}\zc@pgval{371}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{lemma.1018}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-prep-gain\endcsname{{20.7}{363}{}{lemma.994}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-prep-gain\endcsname{\default{20.7}\page{363}\abspage{367}\zc@counter{lemma}\thecounter{20.7}\zc@type{lemma}\zc@cntval{7}\zc@pgval{363}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{lemma.994}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-recenter\endcsname{{20.1}{352}{}{lemma.964}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-recenter\endcsname{\default{20.1}\page{352}\abspage{356}\zc@counter{lemma}\thecounter{20.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{352}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{lemma.964}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-return\endcsname{{20.5}{360}{}{lemma.987}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-return\endcsname{\default{20.5}\page{360}\abspage{364}\zc@counter{lemma}\thecounter{20.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{360}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{lemma.987}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-telescope\endcsname{{20.4}{358}{}{lemma.985}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-telescope\endcsname{\default{20.4}\page{358}\abspage{362}\zc@counter{lemma}\thecounter{20.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{358}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{lemma.985}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-terminal\endcsname{{20.6}{362}{}{lemma.992}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-terminal\endcsname{\default{20.6}\page{362}\abspage{366}\zc@counter{lemma}\thecounter{20.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{362}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{lemma.992}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-time\endcsname{{18.8}{292}{}{lemma.798}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-time\endcsname{\default{18.8}\page{292}\abspage{296}\zc@counter{lemma}\thecounter{18.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{292}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{lemma.798}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-unbalanced\endcsname{{20.2}{354}{}{lemma.975}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-unbalanced\endcsname{\default{20.2}\page{354}\abspage{358}\zc@counter{lemma}\thecounter{20.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{354}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{lemma.975}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:asm-weight-gain\endcsname{{17.20}{270}{}{lemma.739}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:asm-weight-gain\endcsname{\default{17.20}\page{270}\abspage{274}\zc@counter{lemma}\thecounter{17.20}\zc@type{lemma}\zc@cntval{20}\zc@pgval{270}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.739}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:coarse-L1\endcsname{{12.6}{96}{}{lemma.277}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:coarse-L1\endcsname{\default{12.6}\page{96}\abspage{100}\zc@counter{lemma}\thecounter{12.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{96}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{lemma.277}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:coarse-cameron-martin\endcsname{{12.8}{99}{}{lemma.282}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:coarse-cameron-martin\endcsname{\default{12.8}\page{99}\abspage{103}\zc@counter{lemma}\thecounter{12.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{99}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{lemma.282}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:coarse-densities\endcsname{{12.4}{95}{}{lemma.272}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:coarse-densities\endcsname{\default{12.4}\page{95}\abspage{99}\zc@counter{lemma}\thecounter{12.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{95}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{lemma.272}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:coarse-kl\endcsname{{12.11}{104}{}{lemma.292}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:coarse-kl\endcsname{\default{12.11}\page{104}\abspage{108}\zc@counter{lemma}\thecounter{12.11}\zc@type{lemma}\zc@cntval{11}\zc@pgval{104}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{lemma.292}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:coarse-mean-pairing\endcsname{{12.10}{103}{}{lemma.290}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:coarse-mean-pairing\endcsname{\default{12.10}\page{103}\abspage{107}\zc@counter{lemma}\thecounter{12.10}\zc@type{lemma}\zc@cntval{10}\zc@pgval{103}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{lemma.290}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:coarse-permanent\endcsname{{12.2}{90}{}{lemma.259}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:coarse-permanent\endcsname{\default{12.2}\page{90}\abspage{94}\zc@counter{lemma}\thecounter{12.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{90}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{lemma.259}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:coarse-second-moment\endcsname{{12.12}{108}{}{lemma.300}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:coarse-second-moment\endcsname{\default{12.12}\page{108}\abspage{112}\zc@counter{lemma}\thecounter{12.12}\zc@type{lemma}\zc@cntval{12}\zc@pgval{108}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{lemma.300}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:coarse-side\endcsname{{12.9}{101}{}{lemma.284}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:coarse-side\endcsname{\default{12.9}\page{101}\abspage{105}\zc@counter{lemma}\thecounter{12.9}\zc@type{lemma}\zc@cntval{9}\zc@pgval{101}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{lemma.284}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:cons-fixed-state\endcsname{{18.3}{277}{}{lemma.759}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:cons-fixed-state\endcsname{\default{18.3}\page{277}\abspage{281}\zc@counter{lemma}\thecounter{18.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{277}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{lemma.759}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:cons-horizontal-vacuum\endcsname{{18.11}{295}{}{lemma.806}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:cons-horizontal-vacuum\endcsname{\default{18.11}\page{295}\abspage{299}\zc@counter{lemma}\thecounter{18.11}\zc@type{lemma}\zc@cntval{11}\zc@pgval{295}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{lemma.806}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:cons-identification\endcsname{{18.6}{286}{}{lemma.782}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:cons-identification\endcsname{\default{18.6}\page{286}\abspage{290}\zc@counter{lemma}\thecounter{18.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{286}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{lemma.782}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:cons-inner-normal\endcsname{{18.10}{294}{}{lemma.803}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:cons-inner-normal\endcsname{\default{18.10}\page{294}\abspage{298}\zc@counter{lemma}\thecounter{18.10}\zc@type{lemma}\zc@cntval{10}\zc@pgval{294}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{lemma.803}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:cons-product\endcsname{{18.1}{275}{}{lemma.753}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:cons-product\endcsname{\default{18.1}\page{275}\abspage{279}\zc@counter{lemma}\thecounter{18.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{275}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{lemma.753}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:cons-sectorial\endcsname{{18.4}{280}{}{lemma.769}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:cons-sectorial\endcsname{\default{18.4}\page{280}\abspage{284}\zc@counter{lemma}\thecounter{18.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{280}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{lemma.769}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:cons-separation\endcsname{{18.5}{283}{}{lemma.775}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:cons-separation\endcsname{\default{18.5}\page{283}\abspage{287}\zc@counter{lemma}\thecounter{18.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{283}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{lemma.775}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:cons-short-heat\endcsname{{18.7}{290}{}{lemma.791}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:cons-short-heat\endcsname{\default{18.7}\page{290}\abspage{294}\zc@counter{lemma}\thecounter{18.7}\zc@type{lemma}\zc@cntval{7}\zc@pgval{290}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{lemma.791}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:contour-deform\endcsname{{B.4}{395}{}{lemma.1092}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:contour-deform\endcsname{\default{B.4}\page{395}\abspage{399}\zc@counter{lemma}\thecounter{B.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{395}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{lemma.1092}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:contour-flat\endcsname{{B.2}{391}{}{lemma.1079}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:contour-flat\endcsname{\default{B.2}\page{391}\abspage{395}\zc@counter{lemma}\thecounter{B.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{391}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{lemma.1079}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:contour-real\endcsname{{B.3}{392}{}{lemma.1082}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:contour-real\endcsname{\default{B.3}\page{392}\abspage{396}\zc@counter{lemma}\thecounter{B.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{392}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{lemma.1082}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:contour-stokes\endcsname{{B.1}{389}{}{lemma.1076}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:contour-stokes\endcsname{\default{B.1}\page{389}\abspage{393}\zc@counter{lemma}\thecounter{B.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{389}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{lemma.1076}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-barrier\endcsname{{19.24}{346}{}{lemma.945}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-barrier\endcsname{\default{19.24}\page{346}\abspage{350}\zc@counter{lemma}\thecounter{19.24}\zc@type{lemma}\zc@cntval{24}\zc@pgval{346}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.945}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-coercivity\endcsname{{19.6}{315}{}{lemma.867}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-coercivity\endcsname{\default{19.6}\page{315}\abspage{319}\zc@counter{lemma}\thecounter{19.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{315}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.867}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-continuation\endcsname{{19.26}{348}{}{lemma.951}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-continuation\endcsname{\default{19.26}\page{348}\abspage{352}\zc@counter{lemma}\thecounter{19.26}\zc@type{lemma}\zc@cntval{26}\zc@pgval{348}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.951}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-davies\endcsname{{19.9}{320}{}{lemma.878}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-davies\endcsname{\default{19.9}\page{320}\abspage{324}\zc@counter{lemma}\thecounter{19.9}\zc@type{lemma}\zc@cntval{9}\zc@pgval{320}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.878}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-early\endcsname{{19.22}{344}{}{lemma.939}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-early\endcsname{\default{19.22}\page{344}\abspage{348}\zc@counter{lemma}\thecounter{19.22}\zc@type{lemma}\zc@cntval{22}\zc@pgval{344}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.939}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-energy\endcsname{{19.8}{317}{}{lemma.872}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-energy\endcsname{\default{19.8}\page{317}\abspage{321}\zc@counter{lemma}\thecounter{19.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{317}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.872}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-ff-current\endcsname{{19.3}{307}{}{lemma.839}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-ff-current\endcsname{\default{19.3}\page{307}\abspage{311}\zc@counter{lemma}\thecounter{19.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{307}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.839}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-ff-strip\endcsname{{15.8}{173}{}{lemma.500}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-ff-strip\endcsname{\default{15.8}\page{173}\abspage{177}\zc@counter{lemma}\thecounter{15.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{173}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.500}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-first-step\endcsname{{19.23}{345}{}{lemma.942}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-first-step\endcsname{\default{19.23}\page{345}\abspage{349}\zc@counter{lemma}\thecounter{19.23}\zc@type{lemma}\zc@cntval{23}\zc@pgval{345}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.942}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-heat-residual\endcsname{{19.19}{339}{}{lemma.926}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-heat-residual\endcsname{\default{19.19}\page{339}\abspage{343}\zc@counter{lemma}\thecounter{19.19}\zc@type{lemma}\zc@cntval{19}\zc@pgval{339}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.926}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-height\endcsname{{19.15}{326}{}{lemma.891}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-height\endcsname{\default{19.15}\page{326}\abspage{330}\zc@counter{lemma}\thecounter{19.15}\zc@type{lemma}\zc@cntval{15}\zc@pgval{326}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.891}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-kick\endcsname{{19.7}{315}{}{lemma.869}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-kick\endcsname{\default{19.7}\page{315}\abspage{319}\zc@counter{lemma}\thecounter{19.7}\zc@type{lemma}\zc@cntval{7}\zc@pgval{315}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.869}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-kinetic\endcsname{{15.7}{170}{}{lemma.495}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-kinetic\endcsname{\default{15.7}\page{170}\abspage{174}\zc@counter{lemma}\thecounter{15.7}\zc@type{lemma}\zc@cntval{7}\zc@pgval{170}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.495}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-onsite\endcsname{{15.9}{174}{}{lemma.501}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-onsite\endcsname{\default{15.9}\page{174}\abspage{178}\zc@counter{lemma}\thecounter{15.9}\zc@type{lemma}\zc@cntval{9}\zc@pgval{174}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.501}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-radii\endcsname{{19.25}{348}{}{lemma.949}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-radii\endcsname{\default{19.25}\page{348}\abspage{352}\zc@counter{lemma}\thecounter{19.25}\zc@type{lemma}\zc@cntval{25}\zc@pgval{348}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.949}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-raw-norms\endcsname{{19.5}{312}{}{lemma.855}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-raw-norms\endcsname{\default{19.5}\page{312}\abspage{316}\zc@counter{lemma}\thecounter{19.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{312}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.855}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-real-family\endcsname{{19.14}{324}{}{lemma.886}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-real-family\endcsname{\default{19.14}\page{324}\abspage{328}\zc@counter{lemma}\thecounter{19.14}\zc@type{lemma}\zc@cntval{14}\zc@pgval{324}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.886}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-shell\endcsname{{19.18}{337}{}{lemma.922}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-shell\endcsname{\default{19.18}\page{337}\abspage{341}\zc@counter{lemma}\thecounter{19.18}\zc@type{lemma}\zc@cntval{18}\zc@pgval{337}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.922}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:end-ward\endcsname{{19.2}{303}{}{lemma.824}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:end-ward\endcsname{\default{19.2}\page{303}\abspage{307}\zc@counter{lemma}\thecounter{19.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{303}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{lemma.824}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:extract-gaussian\endcsname{{9.3}{31}{}{lemma.100}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:extract-gaussian\endcsname{\default{9.3}\page{31}\abspage{35}\zc@counter{lemma}\thecounter{9.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{31}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{lemma.100}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:extract-limits\endcsname{{9.5}{33}{}{lemma.104}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:extract-limits\endcsname{\default{9.5}\page{33}\abspage{37}\zc@counter{lemma}\thecounter{9.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{33}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{lemma.104}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:extract-mixture\endcsname{{9.4}{31}{}{lemma.101}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:extract-mixture\endcsname{\default{9.4}\page{31}\abspage{35}\zc@counter{lemma}\thecounter{9.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{31}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{lemma.101}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:extract-moments\endcsname{{9.2}{30}{}{lemma.95}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:extract-moments\endcsname{\default{9.2}\page{30}\abspage{34}\zc@counter{lemma}\thecounter{9.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{30}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{lemma.95}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-averaging\endcsname{{15.10}{176}{}{lemma.507}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-averaging\endcsname{\default{15.10}\page{176}\abspage{180}\zc@counter{lemma}\thecounter{15.10}\zc@type{lemma}\zc@cntval{10}\zc@pgval{176}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.507}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-bb\endcsname{{15.6}{166}{}{lemma.480}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-bb\endcsname{\default{15.6}\page{166}\abspage{170}\zc@counter{lemma}\thecounter{15.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{166}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.480}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-coefficients\endcsname{{15.12}{180}{}{lemma.515}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-coefficients\endcsname{\default{15.12}\page{180}\abspage{184}\zc@counter{lemma}\thecounter{15.12}\zc@type{lemma}\zc@cntval{12}\zc@pgval{180}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.515}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-coercivity\endcsname{{15.5}{164}{}{lemma.474}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-coercivity\endcsname{\default{15.5}\page{164}\abspage{168}\zc@counter{lemma}\thecounter{15.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{164}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.474}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-core\endcsname{{15.16}{187}{}{lemma.536}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-core\endcsname{\default{15.16}\page{187}\abspage{191}\zc@counter{lemma}\thecounter{15.16}\zc@type{lemma}\zc@cntval{16}\zc@pgval{187}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.536}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-hua\endcsname{{15.4}{162}{}{lemma.469}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-hua\endcsname{\default{15.4}\page{162}\abspage{166}\zc@counter{lemma}\thecounter{15.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{162}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.469}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-jets\endcsname{{15.15}{186}{}{lemma.533}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-jets\endcsname{\default{15.15}\page{186}\abspage{190}\zc@counter{lemma}\thecounter{15.15}\zc@type{lemma}\zc@cntval{15}\zc@pgval{186}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.533}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-logdet\endcsname{{15.3}{161}{}{lemma.465}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-logdet\endcsname{\default{15.3}\page{161}\abspage{165}\zc@counter{lemma}\thecounter{15.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{161}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.465}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-marked-onsite\endcsname{{15.13}{182}{}{lemma.521}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-marked-onsite\endcsname{\default{15.13}\page{182}\abspage{186}\zc@counter{lemma}\thecounter{15.13}\zc@type{lemma}\zc@cntval{13}\zc@pgval{182}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.521}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-pairing\endcsname{{15.1}{159}{}{lemma.454}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-pairing\endcsname{\default{15.1}\page{159}\abspage{163}\zc@counter{lemma}\thecounter{15.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{159}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.454}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:hist-profile\endcsname{{15.11}{178}{}{lemma.510}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:hist-profile\endcsname{\default{15.11}\page{178}\abspage{182}\zc@counter{lemma}\thecounter{15.11}\zc@type{lemma}\zc@cntval{11}\zc@pgval{178}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{lemma.510}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:iface-counts\endcsname{{7.2}{4}{}{lemma.8}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:iface-counts\endcsname{\default{7.2}\page{4}\abspage{8}\zc@counter{lemma}\thecounter{7.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{4}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{lemma.8}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:iface-density\endcsname{{7.3}{6}{}{lemma.16}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:iface-density\endcsname{\default{7.3}\page{6}\abspage{10}\zc@counter{lemma}\thecounter{7.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{6}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{lemma.16}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:iface-gaussian\endcsname{{7.4}{7}{}{lemma.21}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:iface-gaussian\endcsname{\default{7.4}\page{7}\abspage{11}\zc@counter{lemma}\thecounter{7.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{7}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{lemma.21}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:iface-variance\endcsname{{7.1}{3}{}{lemma.5}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:iface-variance\endcsname{\default{7.1}\page{3}\abspage{7}\zc@counter{lemma}\thecounter{7.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{3}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{lemma.5}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:int-contact\endcsname{{11.2}{73}{}{lemma.216}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:int-contact\endcsname{\default{11.2}\page{73}\abspage{77}\zc@counter{lemma}\thecounter{11.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{73}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{lemma.216}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:int-contact-residue\endcsname{{11.4}{74}{}{lemma.219}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:int-contact-residue\endcsname{\default{11.4}\page{74}\abspage{78}\zc@counter{lemma}\thecounter{11.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{74}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{lemma.219}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:int-continuity\endcsname{{11.9}{80}{}{lemma.237}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:int-continuity\endcsname{\default{11.9}\page{80}\abspage{84}\zc@counter{lemma}\thecounter{11.9}\zc@type{lemma}\zc@cntval{9}\zc@pgval{80}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{lemma.237}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:int-density\endcsname{{11.10}{82}{}{lemma.240}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:int-density\endcsname{\default{11.10}\page{82}\abspage{86}\zc@counter{lemma}\thecounter{11.10}\zc@type{lemma}\zc@cntval{10}\zc@pgval{82}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{lemma.240}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:int-equicontinuity\endcsname{{11.6}{76}{}{lemma.226}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:int-equicontinuity\endcsname{\default{11.6}\page{76}\abspage{80}\zc@counter{lemma}\thecounter{11.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{76}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{lemma.226}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:int-even\endcsname{{11.8}{78}{}{lemma.233}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:int-even\endcsname{\default{11.8}\page{78}\abspage{82}\zc@counter{lemma}\thecounter{11.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{78}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{lemma.233}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:int-resolvent\endcsname{{11.11}{84}{}{lemma.243}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:int-resolvent\endcsname{\default{11.11}\page{84}\abspage{88}\zc@counter{lemma}\thecounter{11.11}\zc@type{lemma}\zc@cntval{11}\zc@pgval{84}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{lemma.243}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:int-sources\endcsname{{11.1}{71}{}{lemma.206}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:int-sources\endcsname{\default{11.1}\page{71}\abspage{75}\zc@counter{lemma}\thecounter{11.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{71}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{lemma.206}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:met-branch\endcsname{{13.6}{119}{}{lemma.335}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:met-branch\endcsname{\default{13.6}\page{119}\abspage{123}\zc@counter{lemma}\thecounter{13.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{119}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{lemma.335}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:met-coefficients\endcsname{{13.8}{123}{}{lemma.346}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:met-coefficients\endcsname{\default{13.8}\page{123}\abspage{127}\zc@counter{lemma}\thecounter{13.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{123}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{lemma.346}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:met-duhamel\endcsname{{13.4}{117}{}{lemma.324}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:met-duhamel\endcsname{\default{13.4}\page{117}\abspage{121}\zc@counter{lemma}\thecounter{13.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{117}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{lemma.324}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:met-envelope\endcsname{{13.3}{116}{}{lemma.322}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:met-envelope\endcsname{\default{13.3}\page{116}\abspage{120}\zc@counter{lemma}\thecounter{13.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{116}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{lemma.322}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:met-ito\endcsname{{13.2}{114}{}{lemma.314}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:met-ito\endcsname{\default{13.2}\page{114}\abspage{118}\zc@counter{lemma}\thecounter{13.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{114}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{lemma.314}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:met-profile\endcsname{{13.1}{111}{}{lemma.307}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:met-profile\endcsname{\default{13.1}\page{111}\abspage{115}\zc@counter{lemma}\thecounter{13.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{111}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{lemma.307}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:met-singlet\endcsname{{13.5}{118}{}{lemma.329}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:met-singlet\endcsname{\default{13.5}\page{118}\abspage{122}\zc@counter{lemma}\thecounter{13.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{118}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{lemma.329}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:proc-algebra\endcsname{{8.1}{15}{}{lemma.53}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:proc-algebra\endcsname{\default{8.1}\page{15}\abspage{19}\zc@counter{lemma}\thecounter{8.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{15}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{lemma.53}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:proc-continuation\endcsname{{8.7}{23}{}{lemma.80}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:proc-continuation\endcsname{\default{8.7}\page{23}\abspage{27}\zc@counter{lemma}\thecounter{8.7}\zc@type{lemma}\zc@cntval{7}\zc@pgval{23}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{lemma.80}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:proc-entire\endcsname{{8.6}{22}{}{lemma.79}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:proc-entire\endcsname{\default{8.6}\page{22}\abspage{26}\zc@counter{lemma}\thecounter{8.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{22}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{lemma.79}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:proc-moments\endcsname{{8.3}{19}{}{lemma.65}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:proc-moments\endcsname{\default{8.3}\page{19}\abspage{23}\zc@counter{lemma}\thecounter{8.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{19}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{lemma.65}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:proc-rational\endcsname{{8.4}{20}{}{lemma.71}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:proc-rational\endcsname{\default{8.4}\page{20}\abspage{24}\zc@counter{lemma}\thecounter{8.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{20}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{lemma.71}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:rep-kick\endcsname{{B.5}{398}{}{lemma.1100}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:rep-kick\endcsname{\default{B.5}\page{398}\abspage{402}\zc@counter{lemma}\thecounter{B.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{398}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{lemma.1100}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-coupling\endcsname{{10.5}{49}{}{lemma.153}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-coupling\endcsname{\default{10.5}\page{49}\abspage{53}\zc@counter{lemma}\thecounter{10.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{49}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.153}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-entries\endcsname{{10.9}{56}{}{lemma.173}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-entries\endcsname{\default{10.9}\page{56}\abspage{60}\zc@counter{lemma}\thecounter{10.9}\zc@type{lemma}\zc@cntval{9}\zc@pgval{56}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.173}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-envelope\endcsname{{10.3}{48}{}{lemma.147}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-envelope\endcsname{\default{10.3}\page{48}\abspage{52}\zc@counter{lemma}\thecounter{10.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{48}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.147}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-freezing\endcsname{{10.11}{61}{}{lemma.184}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-freezing\endcsname{\default{10.11}\page{61}\abspage{65}\zc@counter{lemma}\thecounter{10.11}\zc@type{lemma}\zc@cntval{11}\zc@pgval{61}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.184}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-hierarchy\endcsname{{10.2}{46}{}{lemma.143}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-hierarchy\endcsname{\default{10.2}\page{46}\abspage{50}\zc@counter{lemma}\thecounter{10.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{46}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.143}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-limit-semigroups\endcsname{{10.1}{44}{}{lemma.135}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-limit-semigroups\endcsname{\default{10.1}\page{44}\abspage{48}\zc@counter{lemma}\thecounter{10.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{44}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.135}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-permanent\endcsname{{10.12}{63}{}{lemma.188}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-permanent\endcsname{\default{10.12}\page{63}\abspage{67}\zc@counter{lemma}\thecounter{10.12}\zc@type{lemma}\zc@cntval{12}\zc@pgval{63}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.188}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-polar\endcsname{{10.7}{54}{}{lemma.168}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-polar\endcsname{\default{10.7}\page{54}\abspage{58}\zc@counter{lemma}\thecounter{10.7}\zc@type{lemma}\zc@cntval{7}\zc@pgval{54}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.168}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-radial\endcsname{{10.13}{63}{}{lemma.192}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-radial\endcsname{\default{10.13}\page{63}\abspage{67}\zc@counter{lemma}\thecounter{10.13}\zc@type{lemma}\zc@cntval{13}\zc@pgval{63}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.192}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:res-stabilizer\endcsname{{10.8}{55}{}{lemma.170}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:res-stabilizer\endcsname{\default{10.8}\page{55}\abspage{59}\zc@counter{lemma}\thecounter{10.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{55}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{lemma.170}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-approx\endcsname{{14.10}{145}{}{lemma.414}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-approx\endcsname{\default{14.10}\page{145}\abspage{149}\zc@counter{lemma}\thecounter{14.10}\zc@type{lemma}\zc@cntval{10}\zc@pgval{145}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.414}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-bessel\endcsname{{14.11}{146}{}{lemma.418}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-bessel\endcsname{\default{14.11}\page{146}\abspage{150}\zc@counter{lemma}\thecounter{14.11}\zc@type{lemma}\zc@cntval{11}\zc@pgval{146}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.418}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-bounds\endcsname{{14.9}{143}{}{lemma.406}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-bounds\endcsname{\default{14.9}\page{143}\abspage{147}\zc@counter{lemma}\thecounter{14.9}\zc@type{lemma}\zc@cntval{9}\zc@pgval{143}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.406}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-filter\endcsname{{14.5}{136}{}{lemma.379}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-filter\endcsname{\default{14.5}\page{136}\abspage{140}\zc@counter{lemma}\thecounter{14.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{136}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.379}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-fourier\endcsname{{14.4}{133}{}{lemma.369}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-fourier\endcsname{\default{14.4}\page{133}\abspage{137}\zc@counter{lemma}\thecounter{14.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{133}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.369}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-gue\endcsname{{14.3}{132}{}{lemma.367}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-gue\endcsname{\default{14.3}\page{132}\abspage{136}\zc@counter{lemma}\thecounter{14.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{132}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.367}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-l2\endcsname{{14.7}{140}{}{lemma.395}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-l2\endcsname{\default{14.7}\page{140}\abspage{144}\zc@counter{lemma}\thecounter{14.7}\zc@type{lemma}\zc@cntval{7}\zc@pgval{140}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.395}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-limit\endcsname{{14.8}{142}{}{lemma.403}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-limit\endcsname{\default{14.8}\page{142}\abspage{146}\zc@counter{lemma}\thecounter{14.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{142}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.403}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-pair\endcsname{{14.1}{129}{}{lemma.361}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-pair\endcsname{\default{14.1}\page{129}\abspage{133}\zc@counter{lemma}\thecounter{14.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{129}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.361}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-pair-transform\endcsname{{14.2}{130}{}{lemma.363}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-pair-transform\endcsname{\default{14.2}\page{130}\abspage{134}\zc@counter{lemma}\thecounter{14.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{130}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.363}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-pairing\endcsname{{14.12}{147}{}{lemma.422}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-pairing\endcsname{\default{14.12}\page{147}\abspage{151}\zc@counter{lemma}\thecounter{14.12}\zc@type{lemma}\zc@cntval{12}\zc@pgval{147}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.422}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:scales-state\endcsname{{14.6}{138}{}{lemma.390}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:scales-state\endcsname{\default{14.6}\page{138}\abspage{142}\zc@counter{lemma}\thecounter{14.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{138}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{lemma.390}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-blocks\endcsname{{17.6}{249}{}{lemma.683}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-blocks\endcsname{\default{17.6}\page{249}\abspage{253}\zc@counter{lemma}\thecounter{17.6}\zc@type{lemma}\zc@cntval{6}\zc@pgval{249}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.683}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-body-parity\endcsname{{17.12}{256}{}{lemma.700}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-body-parity\endcsname{\default{17.12}\page{256}\abspage{260}\zc@counter{lemma}\thecounter{17.12}\zc@type{lemma}\zc@cntval{12}\zc@pgval{256}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.700}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-chord\endcsname{{17.2}{244}{}{lemma.672}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-chord\endcsname{\default{17.2}\page{244}\abspage{248}\zc@counter{lemma}\thecounter{17.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{244}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.672}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-degree-one\endcsname{{17.16}{262}{}{lemma.720}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-degree-one\endcsname{\default{17.16}\page{262}\abspage{266}\zc@counter{lemma}\thecounter{17.16}\zc@type{lemma}\zc@cntval{16}\zc@pgval{262}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.720}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-degree-one-parity\endcsname{{17.10}{254}{}{lemma.695}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-degree-one-parity\endcsname{\default{17.10}\page{254}\abspage{258}\zc@counter{lemma}\thecounter{17.10}\zc@type{lemma}\zc@cntval{10}\zc@pgval{254}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.695}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-energy\endcsname{{17.7}{250}{}{lemma.686}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-energy\endcsname{\default{17.7}\page{250}\abspage{254}\zc@counter{lemma}\thecounter{17.7}\zc@type{lemma}\zc@cntval{7}\zc@pgval{250}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.686}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-frame\endcsname{{17.3}{245}{}{lemma.675}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-frame\endcsname{\default{17.3}\page{245}\abspage{249}\zc@counter{lemma}\thecounter{17.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{245}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.675}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-higher\endcsname{{17.17}{264}{}{lemma.722}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-higher\endcsname{\default{17.17}\page{264}\abspage{268}\zc@counter{lemma}\thecounter{17.17}\zc@type{lemma}\zc@cntval{17}\zc@pgval{264}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.722}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-hol\endcsname{{17.15}{261}{}{lemma.717}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-hol\endcsname{\default{17.15}\page{261}\abspage{265}\zc@counter{lemma}\thecounter{17.15}\zc@type{lemma}\zc@cntval{15}\zc@pgval{261}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.717}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-holonomy\endcsname{{17.14}{261}{}{lemma.716}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-holonomy\endcsname{\default{17.14}\page{261}\abspage{265}\zc@counter{lemma}\thecounter{17.14}\zc@type{lemma}\zc@cntval{14}\zc@pgval{261}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.716}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-moment\endcsname{{17.4}{246}{}{lemma.677}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-moment\endcsname{\default{17.4}\page{246}\abspage{250}\zc@counter{lemma}\thecounter{17.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{246}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.677}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-nor\endcsname{{17.18}{265}{}{lemma.726}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-nor\endcsname{\default{17.18}\page{265}\abspage{269}\zc@counter{lemma}\thecounter{17.18}\zc@type{lemma}\zc@cntval{18}\zc@pgval{265}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.726}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-parity\endcsname{{17.8}{252}{}{lemma.693}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-parity\endcsname{\default{17.8}\page{252}\abspage{256}\zc@counter{lemma}\thecounter{17.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{252}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.693}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-schur\endcsname{{17.11}{255}{}{lemma.699}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-schur\endcsname{\default{17.11}\page{255}\abspage{259}\zc@counter{lemma}\thecounter{17.11}\zc@type{lemma}\zc@cntval{11}\zc@pgval{255}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.699}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-src\endcsname{{17.19}{266}{}{lemma.729}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-src\endcsname{\default{17.19}\page{266}\abspage{270}\zc@counter{lemma}\thecounter{17.19}\zc@type{lemma}\zc@cntval{19}\zc@pgval{266}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.729}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-uncharged\endcsname{{17.13}{260}{}{lemma.711}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-uncharged\endcsname{\default{17.13}\page{260}\abspage{264}\zc@counter{lemma}\thecounter{17.13}\zc@type{lemma}\zc@cntval{13}\zc@pgval{260}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.711}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:stab-words\endcsname{{17.5}{247}{}{lemma.679}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:stab-words\endcsname{\default{17.5}\page{247}\abspage{251}\zc@counter{lemma}\thecounter{17.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{247}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{lemma.679}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tools-defect\endcsname{{A.1}{378}{}{lemma.1031}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tools-defect\endcsname{\default{A.1}\page{378}\abspage{382}\zc@counter{lemma}\thecounter{A.1}\zc@type{lemma}\zc@cntval{1}\zc@pgval{378}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{lemma.1031}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tools-dirichlet\endcsname{{A.3}{382}{}{lemma.1047}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tools-dirichlet\endcsname{\default{A.3}\page{382}\abspage{386}\zc@counter{lemma}\thecounter{A.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{382}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{lemma.1047}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tools-poisson\endcsname{{A.4}{384}{}{lemma.1060}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tools-poisson\endcsname{\default{A.4}\page{384}\abspage{388}\zc@counter{lemma}\thecounter{A.4}\zc@type{lemma}\zc@cntval{4}\zc@pgval{384}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{lemma.1060}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tools-submean\endcsname{{A.5}{387}{}{lemma.1070}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tools-submean\endcsname{\default{A.5}\page{387}\abspage{391}\zc@counter{lemma}\thecounter{A.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{387}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{lemma.1070}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-amplitude\endcsname{{16.18}{228}{}{lemma.631}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-amplitude\endcsname{\default{16.18}\page{228}\abspage{232}\zc@counter{lemma}\thecounter{16.18}\zc@type{lemma}\zc@cntval{18}\zc@pgval{228}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.631}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-casimir\endcsname{{16.5}{209}{}{lemma.594}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-casimir\endcsname{\default{16.5}\page{209}\abspage{213}\zc@counter{lemma}\thecounter{16.5}\zc@type{lemma}\zc@cntval{5}\zc@pgval{209}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.594}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-class\endcsname{{16.8}{215}{}{lemma.609}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-class\endcsname{\default{16.8}\page{215}\abspage{219}\zc@counter{lemma}\thecounter{16.8}\zc@type{lemma}\zc@cntval{8}\zc@pgval{215}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.609}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-faces\endcsname{{16.17}{226}{}{lemma.629}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-faces\endcsname{\default{16.17}\page{226}\abspage{230}\zc@counter{lemma}\thecounter{16.17}\zc@type{lemma}\zc@cntval{17}\zc@pgval{226}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.629}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-harmonic\endcsname{{16.3}{205}{}{lemma.580}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-harmonic\endcsname{\default{16.3}\page{205}\abspage{209}\zc@counter{lemma}\thecounter{16.3}\zc@type{lemma}\zc@cntval{3}\zc@pgval{205}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.580}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-localization\endcsname{{16.13}{221}{}{lemma.620}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-localization\endcsname{\default{16.13}\page{221}\abspage{225}\zc@counter{lemma}\thecounter{16.13}\zc@type{lemma}\zc@cntval{13}\zc@pgval{221}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.620}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-odd-gaussian\endcsname{{16.19}{231}{}{lemma.640}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-odd-gaussian\endcsname{\default{16.19}\page{231}\abspage{235}\zc@counter{lemma}\thecounter{16.19}\zc@type{lemma}\zc@cntval{19}\zc@pgval{231}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.640}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-polynomial\endcsname{{16.15}{223}{}{lemma.624}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-polynomial\endcsname{\default{16.15}\page{223}\abspage{227}\zc@counter{lemma}\thecounter{16.15}\zc@type{lemma}\zc@cntval{15}\zc@pgval{223}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.624}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-powers\endcsname{{16.21}{234}{}{lemma.647}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-powers\endcsname{\default{16.21}\page{234}\abspage{238}\zc@counter{lemma}\thecounter{16.21}\zc@type{lemma}\zc@cntval{21}\zc@pgval{234}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.647}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-quadratic\endcsname{{16.10}{217}{}{lemma.615}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-quadratic\endcsname{\default{16.10}\page{217}\abspage{221}\zc@counter{lemma}\thecounter{16.10}\zc@type{lemma}\zc@cntval{10}\zc@pgval{217}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.615}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-slow\endcsname{{16.14}{223}{}{lemma.623}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-slow\endcsname{\default{16.14}\page{223}\abspage{227}\zc@counter{lemma}\thecounter{16.14}\zc@type{lemma}\zc@cntval{14}\zc@pgval{223}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.623}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-supercharge\endcsname{{16.12}{219}{}{lemma.619}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-supercharge\endcsname{\default{16.12}\page{219}\abspage{223}\zc@counter{lemma}\thecounter{16.12}\zc@type{lemma}\zc@cntval{12}\zc@pgval{219}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.619}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-volume\endcsname{{16.2}{203}{}{lemma.573}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-volume\endcsname{\default{16.2}\page{203}\abspage{207}\zc@counter{lemma}\thecounter{16.2}\zc@type{lemma}\zc@cntval{2}\zc@pgval{203}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.573}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@l:tube-word\endcsname{{16.16}{225}{}{lemma.627}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@l:tube-word\endcsname{\default{16.16}\page{225}\abspage{229}\zc@counter{lemma}\thecounter{16.16}\zc@type{lemma}\zc@cntval{16}\zc@pgval{225}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{lemma.627}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:asm-adapted\endcsname{{20.9}{366}{}{proposition.1002}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:asm-adapted\endcsname{\default{20.9}\page{366}\abspage{370}\zc@counter{proposition}\thecounter{20.9}\zc@type{proposition}\zc@cntval{9}\zc@pgval{366}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{proposition.1002}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:asm-jets\endcsname{{20.10}{369}{}{proposition.1011}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:asm-jets\endcsname{\default{20.10}\page{369}\abspage{373}\zc@counter{proposition}\thecounter{20.10}\zc@type{proposition}\zc@cntval{10}\zc@pgval{369}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{proposition.1011}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:asm-physical\endcsname{{20.8}{365}{}{proposition.999}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:asm-physical\endcsname{\default{20.8}\page{365}\abspage{369}\zc@counter{proposition}\thecounter{20.8}\zc@type{proposition}\zc@cntval{8}\zc@pgval{365}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{proposition.999}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:coarse-endpoints\endcsname{{12.14}{109}{}{proposition.303}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:coarse-endpoints\endcsname{\default{12.14}\page{109}\abspage{113}\zc@counter{proposition}\thecounter{12.14}\zc@type{proposition}\zc@cntval{14}\zc@pgval{109}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{proposition.303}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:coarse-source-limit\endcsname{{12.3}{93}{}{proposition.267}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:coarse-source-limit\endcsname{\default{12.3}\page{93}\abspage{97}\zc@counter{proposition}\thecounter{12.3}\zc@type{proposition}\zc@cntval{3}\zc@pgval{93}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{proposition.267}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:cons-consistency\endcsname{{18.12}{295}{}{proposition.808}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:cons-consistency\endcsname{\default{18.12}\page{295}\abspage{299}\zc@counter{proposition}\thecounter{18.12}\zc@type{proposition}\zc@cntval{12}\zc@pgval{295}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{proposition.808}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:end-coupled\endcsname{{19.13}{323}{}{proposition.884}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:end-coupled\endcsname{\default{19.13}\page{323}\abspage{327}\zc@counter{proposition}\thecounter{19.13}\zc@type{proposition}\zc@cntval{13}\zc@pgval{323}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{proposition.884}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:end-escape\endcsname{{19.4}{311}{}{proposition.852}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:end-escape\endcsname{\default{19.4}\page{311}\abspage{315}\zc@counter{proposition}\thecounter{19.4}\zc@type{proposition}\zc@cntval{4}\zc@pgval{311}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{proposition.852}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:end-shifted\endcsname{{19.17}{331}{}{proposition.902}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:end-shifted\endcsname{\default{19.17}\page{331}\abspage{335}\zc@counter{proposition}\thecounter{19.17}\zc@type{proposition}\zc@cntval{17}\zc@pgval{331}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{proposition.902}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:int-balanced\endcsname{{11.5}{75}{}{proposition.222}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:int-balanced\endcsname{\default{11.5}\page{75}\abspage{79}\zc@counter{proposition}\thecounter{11.5}\zc@type{proposition}\zc@cntval{5}\zc@pgval{75}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{proposition.222}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:int-diagonal\endcsname{{11.7}{77}{}{proposition.229}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:int-diagonal\endcsname{\default{11.7}\page{77}\abspage{81}\zc@counter{proposition}\thecounter{11.7}\zc@type{proposition}\zc@cntval{7}\zc@pgval{77}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{proposition.229}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:int-exact\endcsname{{11.12}{85}{}{proposition.246}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:int-exact\endcsname{\default{11.12}\page{85}\abspage{89}\zc@counter{proposition}\thecounter{11.12}\zc@type{proposition}\zc@cntval{12}\zc@pgval{85}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{proposition.246}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:met-kernel\endcsname{{13.7}{120}{}{proposition.339}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:met-kernel\endcsname{\default{13.7}\page{120}\abspage{124}\zc@counter{proposition}\thecounter{13.7}\zc@type{proposition}\zc@cntval{7}\zc@pgval{120}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{proposition.339}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:proc-determination\endcsname{{8.5}{21}{}{proposition.74}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:proc-determination\endcsname{\default{8.5}\page{21}\abspage{25}\zc@counter{proposition}\thecounter{8.5}\zc@type{proposition}\zc@cntval{5}\zc@pgval{21}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{proposition.74}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:proc-subsequences\endcsname{{8.2}{17}{}{proposition.61}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:proc-subsequences\endcsname{\default{8.2}\page{17}\abspage{21}\zc@counter{proposition}\thecounter{8.2}\zc@type{proposition}\zc@cntval{2}\zc@pgval{17}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{proposition.61}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:scales-scalar\endcsname{{14.14}{151}{}{proposition.430}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:scales-scalar\endcsname{\default{14.14}\page{151}\abspage{155}\zc@counter{proposition}\thecounter{14.14}\zc@type{proposition}\zc@cntval{14}\zc@pgval{151}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{proposition.430}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:scales-wire\endcsname{{14.15}{154}{}{proposition.441}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:scales-wire\endcsname{\default{14.15}\page{154}\abspage{158}\zc@counter{proposition}\thecounter{14.15}\zc@type{proposition}\zc@cntval{15}\zc@pgval{154}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{proposition.441}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@p:tube-gap\endcsname{{16.22}{237}{}{proposition.652}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@p:tube-gap\endcsname{\default{16.22}\page{237}\abspage{241}\zc@counter{proposition}\thecounter{16.22}\zc@type{proposition}\zc@cntval{22}\zc@pgval{237}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{proposition.652}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:coarse-mean-physics\endcsname{{12.13}{109}{}{remark.301}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:coarse-mean-physics\endcsname{\default{12.13}\page{109}\abspage{113}\zc@counter{remark}\thecounter{12.13}\zc@type{remark}\zc@cntval{13}\zc@pgval{109}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{remark.301}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:coarse-rv\endcsname{{12.7}{98}{}{remark.280}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:coarse-rv\endcsname{\default{12.7}\page{98}\abspage{102}\zc@counter{remark}\thecounter{12.7}\zc@type{remark}\zc@cntval{7}\zc@pgval{98}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{remark.280}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:cons-link-variance\endcsname{{18.9}{293}{}{remark.801}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:cons-link-variance\endcsname{\default{18.9}\page{293}\abspage{297}\zc@counter{remark}\thecounter{18.9}\zc@type{remark}\zc@cntval{9}\zc@pgval{293}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{remark.801}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:cons-transpose\endcsname{{18.13}{297}{}{remark.812}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:cons-transpose\endcsname{\default{18.13}\page{297}\abspage{301}\zc@counter{remark}\thecounter{18.13}\zc@type{remark}\zc@cntval{13}\zc@pgval{297}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{remark.812}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:end-universal\endcsname{{19.10}{321}{}{remark.880}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:end-universal\endcsname{\default{19.10}\page{321}\abspage{325}\zc@counter{remark}\thecounter{19.10}\zc@type{remark}\zc@cntval{10}\zc@pgval{321}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{remark.880}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:extract-uniform\endcsname{{9.7}{38}{}{remark.117}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:extract-uniform\endcsname{\default{9.7}\page{38}\abspage{42}\zc@counter{remark}\thecounter{9.7}\zc@type{remark}\zc@cntval{7}\zc@pgval{38}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{remark.117}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:iface-energies\endcsname{{7.7}{11}{}{remark.44}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:iface-energies\endcsname{\default{7.7}\page{11}\abspage{15}\zc@counter{remark}\thecounter{7.7}\zc@type{remark}\zc@cntval{7}\zc@pgval{11}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{remark.44}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:iface-onsite\endcsname{{7.9}{12}{}{remark.47}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:iface-onsite\endcsname{\default{7.9}\page{12}\abspage{16}\zc@counter{remark}\thecounter{7.9}\zc@type{remark}\zc@cntval{9}\zc@pgval{12}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{remark.47}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:iface-uses\endcsname{{7.8}{12}{}{remark.46}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:iface-uses\endcsname{\default{7.8}\page{12}\abspage{16}\zc@counter{remark}\thecounter{7.8}\zc@type{remark}\zc@cntval{8}\zc@pgval{12}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{remark.46}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:int-gue\endcsname{{11.3}{74}{}{remark.218}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:int-gue\endcsname{\default{11.3}\page{74}\abspage{78}\zc@counter{remark}\thecounter{11.3}\zc@type{remark}\zc@cntval{3}\zc@pgval{74}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{remark.218}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:int-metallic\endcsname{{11.13}{87}{}{remark.253}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:int-metallic\endcsname{\default{11.13}\page{87}\abspage{91}\zc@counter{remark}\thecounter{11.13}\zc@type{remark}\zc@cntval{13}\zc@pgval{87}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{remark.253}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:met-km\endcsname{{13.12}{128}{}{remark.358}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:met-km\endcsname{\default{13.12}\page{128}\abspage{132}\zc@counter{remark}\thecounter{13.12}\zc@type{remark}\zc@cntval{12}\zc@pgval{128}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{remark.358}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:met-script\endcsname{{13.9}{126}{}{remark.354}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:met-script\endcsname{\default{13.9}\page{126}\abspage{130}\zc@counter{remark}\thecounter{13.9}\zc@type{remark}\zc@cntval{9}\zc@pgval{126}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{remark.354}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:met-traces\endcsname{{13.11}{127}{}{remark.357}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:met-traces\endcsname{\default{13.11}\page{127}\abspage{131}\zc@counter{remark}\thecounter{13.11}\zc@type{remark}\zc@cntval{11}\zc@pgval{127}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{remark.357}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:proc-pair\endcsname{{8.9}{26}{}{remark.87}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:proc-pair\endcsname{\default{8.9}\page{26}\abspage{30}\zc@counter{remark}\thecounter{8.9}\zc@type{remark}\zc@cntval{9}\zc@pgval{26}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{remark.87}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:res-polar-check\endcsname{{10.10}{60}{}{remark.180}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:res-polar-check\endcsname{\default{10.10}\page{60}\abspage{64}\zc@counter{remark}\thecounter{10.10}\zc@type{remark}\zc@cntval{10}\zc@pgval{60}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{remark.180}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:scales-physics\endcsname{{14.13}{149}{}{remark.428}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:scales-physics\endcsname{\default{14.13}\page{149}\abspage{153}\zc@counter{remark}\thecounter{14.13}\zc@type{remark}\zc@cntval{13}\zc@pgval{149}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{remark.428}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:stab-fock\endcsname{{16.11}{219}{}{remark.617}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:stab-fock\endcsname{\default{16.11}\page{219}\abspage{223}\zc@counter{remark}\thecounter{16.11}\zc@type{remark}\zc@cntval{11}\zc@pgval{219}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{remark.617}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:tools-ellipsoid\endcsname{{A.6}{388}{}{remark.1072}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:tools-ellipsoid\endcsname{\default{A.6}\page{388}\abspage{392}\zc@counter{remark}\thecounter{A.6}\zc@type{remark}\zc@cntval{6}\zc@pgval{388}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{remark.1072}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:tube-four-term-reduction\endcsname{{16.20}{233}{}{remark.644}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:tube-four-term-reduction\endcsname{\default{16.20}\page{233}\abspage{237}\zc@counter{remark}\thecounter{16.20}\zc@type{remark}\zc@cntval{20}\zc@pgval{233}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{remark.644}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:tube-harmonic-checks\endcsname{{16.4}{206}{}{remark.586}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:tube-harmonic-checks\endcsname{\default{16.4}\page{206}\abspage{210}\zc@counter{remark}\thecounter{16.4}\zc@type{remark}\zc@cntval{4}\zc@pgval{206}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{remark.586}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:tube-transport\endcsname{{16.9}{216}{}{remark.610}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:tube-transport\endcsname{\default{16.9}\page{216}\abspage{220}\zc@counter{remark}\thecounter{16.9}\zc@type{remark}\zc@cntval{9}\zc@pgval{216}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{remark.610}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@r:tube-two-descriptions\endcsname{{16.11}{219}{}{remark.617}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@r:tube-two-descriptions\endcsname{\default{16.11}\page{219}\abspage{223}\zc@counter{remark}\thecounter{16.11}\zc@type{remark}\zc@cntval{11}\zc@pgval{219}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{remark.617}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-a\endcsname{{20.5}{364}{The source limit: proof of part (a)}{section.996}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-a\endcsname{\default{20.5}\page{364}\abspage{368}\zc@counter{section}\thecounter{20.5}\zc@type{section}\zc@cntval{5}\zc@pgval{364}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{section.996}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-adapted\endcsname{{20.3}{354}{Three properties of the adapted chain}{section.972}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-adapted\endcsname{\default{20.3}\page{354}\abspage{358}\zc@counter{section}\thecounter{20.3}\zc@type{section}\zc@cntval{3}\zc@pgval{354}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{section.972}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-b\endcsname{{20.6}{369}{Local sources: proof of part (b)}{section.1008}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-b\endcsname{\default{20.6}\page{369}\abspage{373}\zc@counter{section}\thecounter{20.6}\zc@type{section}\zc@cntval{6}\zc@pgval{369}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{section.1008}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-c\endcsname{{20.7}{374}{All-site moments: proof of part (c)}{section.1025}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-c\endcsname{\default{20.7}\page{374}\abspage{378}\zc@counter{section}\thecounter{20.7}\zc@type{section}\zc@cntval{7}\zc@pgval{374}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{section.1025}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-calibration\endcsname{{20.5}{368}{Normalization}{subsection*.1006}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-calibration\endcsname{\default{20.5}\page{368}\abspage{372}\zc@counter{section}\thecounter{20.5}\zc@type{section}\zc@cntval{5}\zc@pgval{368}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{subsection*.1006}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-choices\endcsname{{20.1}{351}{Conventions and the order of choices}{section.955}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-choices\endcsname{\default{20.1}\page{351}\abspage{355}\zc@counter{section}\thecounter{20.1}\zc@type{section}\zc@cntval{1}\zc@pgval{351}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{section.955}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-d\endcsname{{20.8}{376}{One-point normalization: proof of part (d)}{section.1027}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-d\endcsname{\default{20.8}\page{376}\abspage{380}\zc@counter{section}\thecounter{20.8}\zc@type{section}\zc@cntval{8}\zc@pgval{376}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{section.1027}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-initial-layer\endcsname{{20.3}{355}{The initial layer}{subsection*.977}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-initial-layer\endcsname{\default{20.3}\page{355}\abspage{359}\zc@counter{section}\thecounter{20.3}\zc@type{section}\zc@cntval{3}\zc@pgval{355}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{subsection*.977}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-one-sided\endcsname{{20.2}{353}{The chain in one-sided form}{section.967}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-one-sided\endcsname{\default{20.2}\page{353}\abspage{357}\zc@counter{section}\thecounter{20.2}\zc@type{section}\zc@cntval{2}\zc@pgval{353}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{section.967}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-post-mark\endcsname{{20.6}{371}{The chain with jets}{subsection*.1016}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-post-mark\endcsname{\default{20.6}\page{371}\abspage{375}\zc@counter{section}\thecounter{20.6}\zc@type{section}\zc@cntval{6}\zc@pgval{371}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{subsection*.1016}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:asm-telescope\endcsname{{20.4}{357}{The telescoping comparison}{section.983}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:asm-telescope\endcsname{\default{20.4}\page{357}\abspage{361}\zc@counter{section}\thecounter{20.4}\zc@type{section}\zc@cntval{4}\zc@pgval{357}\zc@enclval{{20}{0}}\zc@pgfmt{2}\anchor{section.983}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:assembly\endcsname{{20}{350}{Proof of the interface theorem}{chapter.952}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:assembly\endcsname{\default{20}\page{350}\abspage{354}\zc@counter{chapter}\thecounter{20}\zc@type{chapter}\zc@cntval{20}\zc@pgval{350}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.952}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:coarse\endcsname{{12}{88}{The coarse eigenvector law}{chapter.254}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:coarse\endcsname{\default{12}\page{88}\abspage{92}\zc@counter{chapter}\thecounter{12}\zc@type{chapter}\zc@cntval{12}\zc@pgval{88}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.254}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:coarse-endpoints\endcsname{{12.7}{109}{Endpoints of the coarse law}{section.302}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:coarse-endpoints\endcsname{\default{12.7}\page{109}\abspage{113}\zc@counter{section}\thecounter{12.7}\zc@type{section}\zc@cntval{7}\zc@pgval{109}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{section.302}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:coarse-law\endcsname{{12.3}{96}{The coarse law}{section.276}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:coarse-law\endcsname{\default{12.3}\page{96}\abspage{100}\zc@counter{section}\thecounter{12.3}\zc@type{section}\zc@cntval{3}\zc@pgval{96}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{section.276}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:coarse-mean\endcsname{{12.6}{103}{The mean shape}{section.289}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:coarse-mean\endcsname{\default{12.6}\page{103}\abspage{107}\zc@counter{section}\thecounter{12.6}\zc@type{section}\zc@cntval{6}\zc@pgval{103}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{section.289}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:coarse-moments\endcsname{{12.2}{94}{All fixed moments}{section.271}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:coarse-moments\endcsname{\default{12.2}\page{94}\abspage{98}\zc@counter{section}\thecounter{12.2}\zc@type{section}\zc@cntval{2}\zc@pgval{94}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{section.271}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:coarse-permanent\endcsname{{12.1}{89}{Permanent moment measures}{section.257}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:coarse-permanent\endcsname{\default{12.1}\page{89}\abspage{93}\zc@counter{section}\thecounter{12.1}\zc@type{section}\zc@cntval{1}\zc@pgval{89}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{section.257}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:coarse-rooted\endcsname{{12.4}{99}{Rooted laws}{section.281}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:coarse-rooted\endcsname{\default{12.4}\page{99}\abspage{103}\zc@counter{section}\thecounter{12.4}\zc@type{section}\zc@cntval{4}\zc@pgval{99}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{section.281}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:coarse-wire\endcsname{{12.5}{101}{The infinite wire}{section.283}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:coarse-wire\endcsname{\default{12.5}\page{101}\abspage{105}\zc@counter{section}\thecounter{12.5}\zc@type{section}\zc@cntval{5}\zc@pgval{101}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{section.283}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:cons\endcsname{{18}{273}{Consistency on the invariant normal line}{chapter.744}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:cons\endcsname{\default{18}\page{273}\abspage{277}\zc@counter{chapter}\thecounter{18}\zc@type{chapter}\zc@cntval{18}\zc@pgval{273}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.744}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:cons-consistency\endcsname{{18.5}{293}{Consistency}{section.802}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:cons-consistency\endcsname{\default{18.5}\page{293}\abspage{297}\zc@counter{section}\thecounter{18.5}\zc@type{section}\zc@cntval{5}\zc@pgval{293}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{section.802}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:cons-heat\endcsname{{18.3}{279}{The heat semigroup on the tube}{section.764}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:cons-heat\endcsname{\default{18.3}\page{279}\abspage{283}\zc@counter{section}\thecounter{18.3}\zc@type{section}\zc@cntval{3}\zc@pgval{279}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{section.764}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:cons-layer\endcsname{{18.6}{298}{The initial layer}{section.814}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:cons-layer\endcsname{\default{18.6}\page{298}\abspage{302}\zc@counter{section}\thecounter{18.6}\zc@type{section}\zc@cntval{6}\zc@pgval{298}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{section.814}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:cons-product\endcsname{{18.1}{275}{The Trotter--Kurtz approximation theorem}{section.752}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:cons-product\endcsname{\default{18.1}\page{275}\abspage{279}\zc@counter{section}\thecounter{18.1}\zc@type{section}\zc@cntval{1}\zc@pgval{275}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{section.752}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:cons-state\endcsname{{18.2}{277}{The normal fixed state}{section.758}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:cons-state\endcsname{\default{18.2}\page{277}\abspage{281}\zc@counter{section}\thecounter{18.2}\zc@type{section}\zc@cntval{2}\zc@pgval{277}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{section.758}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:cons-tangent\endcsname{{18.4}{292}{The tangent generator}{section.796}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:cons-tangent\endcsname{\default{18.4}\page{292}\abspage{296}\zc@counter{section}\thecounter{18.4}\zc@type{section}\zc@cntval{4}\zc@pgval{292}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{section.796}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:contour\endcsname{{B}{389}{The product contour}{appendix.1074}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:contour\endcsname{\default{B}\page{389}\abspage{393}\zc@counter{chapter}\thecounter{B}\zc@type{appendix}\zc@cntval{2}\zc@pgval{389}\zc@enclval{{1}}\zc@pgfmt{2}\anchor{appendix.1074}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:contour-anchored\endcsname{{B.7}{398}{The deformation in the anchored chain}{section.1098}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:contour-anchored\endcsname{\default{B.7}\page{398}\abspage{402}\zc@counter{section}\thecounter{B.7}\zc@type{appendix}\zc@cntval{7}\zc@pgval{398}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{section.1098}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:contour-deform\endcsname{{B.4}{395}{The deformation lemma}{section.1090}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:contour-deform\endcsname{\default{B.4}\page{395}\abspage{399}\zc@counter{section}\thecounter{B.4}\zc@type{appendix}\zc@cntval{4}\zc@pgval{395}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{section.1090}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:contour-flat\endcsname{{B.2}{390}{Gaussian integrals on the flat cycle}{section.1078}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:contour-flat\endcsname{\default{B.2}\page{390}\abspage{394}\zc@counter{section}\thecounter{B.2}\zc@type{appendix}\zc@cntval{2}\zc@pgval{390}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{section.1078}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:contour-heat\endcsname{{B.6}{397}{The heat kernel on the cycle}{section.1097}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:contour-heat\endcsname{\default{B.6}\page{397}\abspage{401}\zc@counter{section}\thecounter{B.6}\zc@type{appendix}\zc@cntval{6}\zc@pgval{397}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{section.1097}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:contour-kick\endcsname{{B.8}{398}{The source factor on the saddle manifold}{section.1099}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:contour-kick\endcsname{\default{B.8}\page{398}\abspage{402}\zc@counter{section}\thecounter{B.8}\zc@type{appendix}\zc@cntval{8}\zc@pgval{398}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{section.1099}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:contour-proof\endcsname{{B.5}{396}{Proof of \texorpdfstring {\zcref {p:rep-contour}}{the product-contour identity}}{section.1096}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:contour-proof\endcsname{\default{B.5}\page{396}\abspage{400}\zc@counter{section}\thecounter{B.5}\zc@type{appendix}\zc@cntval{5}\zc@pgval{396}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{section.1096}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:contour-real\endcsname{{B.3}{392}{Real parts along the deformation}{section.1080}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:contour-real\endcsname{\default{B.3}\page{392}\abspage{396}\zc@counter{section}\thecounter{B.3}\zc@type{appendix}\zc@cntval{3}\zc@pgval{392}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{section.1080}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:contour-stokes\endcsname{{B.1}{389}{A Stokes lemma for real cycles}{section.1075}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:contour-stokes\endcsname{\default{B.1}\page{389}\abspage{393}\zc@counter{section}\thecounter{B.1}\zc@type{appendix}\zc@cntval{1}\zc@pgval{389}\zc@enclval{{2}{1}}\zc@pgfmt{2}\anchor{section.1075}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end\endcsname{{19}{299}{The endpoints of the chain}{chapter.815}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end\endcsname{\default{19}\page{299}\abspage{303}\zc@counter{chapter}\thecounter{19}\zc@type{chapter}\zc@cntval{19}\zc@pgval{299}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.815}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end-attach\endcsname{{19.7}{342}{The early barrier and the attachment of the endpoint}{section.934}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end-attach\endcsname{\default{19.7}\page{342}\abspage{346}\zc@counter{section}\thecounter{19.7}\zc@type{section}\zc@cntval{7}\zc@pgval{342}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{section.934}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end-canonical\endcsname{{19.5}{326}{Canonical coordinates near the saddle manifold}{section.887}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end-canonical\endcsname{\default{19.5}\page{326}\abspage{330}\zc@counter{section}\thecounter{19.5}\zc@type{section}\zc@cntval{5}\zc@pgval{326}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{section.887}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end-cycle\endcsname{{15.3}{168}{The cycle near the saddle manifold}{subsection*.485}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end-cycle\endcsname{\default{15.3}\page{168}\abspage{172}\zc@counter{section}\thecounter{15.3}\zc@type{section}\zc@cntval{3}\zc@pgval{168}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{subsection*.485}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end-energy\endcsname{{19.4}{313}{A weighted energy estimate for short blocks}{section.858}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end-energy\endcsname{\default{19.4}\page{313}\abspage{317}\zc@counter{section}\thecounter{19.4}\zc@type{section}\zc@cntval{4}\zc@pgval{313}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{section.858}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end-escape\endcsname{{19.2}{306}{The escape ramp}{section.833}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end-escape\endcsname{\default{19.2}\page{306}\abspage{310}\zc@counter{section}\thecounter{19.2}\zc@type{section}\zc@cntval{2}\zc@pgval{306}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{section.833}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end-families\endcsname{{19.1}{302}{Families generated from the endpoint and their equivariance}{section.819}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end-families\endcsname{\default{19.1}\page{302}\abspage{306}\zc@counter{section}\thecounter{19.1}\zc@type{section}\zc@cntval{1}\zc@pgval{302}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{section.819}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end-onsite\endcsname{{15.3}{172}{The escape function and the onsite weight}{subsection*.496}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end-onsite\endcsname{\default{15.3}\page{172}\abspage{176}\zc@counter{section}\thecounter{15.3}\zc@type{section}\zc@cntval{3}\zc@pgval{172}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{subsection*.496}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end-raw\endcsname{{19.3}{312}{The endpoint factors on the tube}{section.854}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end-raw\endcsname{\default{19.3}\page{312}\abspage{316}\zc@counter{section}\thecounter{19.3}\zc@type{section}\zc@cntval{3}\zc@pgval{312}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{section.854}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:end-residual\endcsname{{19.6}{335}{The energy threshold and the physical residual}{section.914}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:end-residual\endcsname{\default{19.6}\page{335}\abspage{339}\zc@counter{section}\thecounter{19.6}\zc@type{section}\zc@cntval{6}\zc@pgval{335}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{section.914}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:extract\endcsname{{9}{28}{Weighted moment measures and extraction of eigenvector moments}{chapter.88}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:extract\endcsname{\default{9}\page{28}\abspage{32}\zc@counter{chapter}\thecounter{9}\zc@type{chapter}\zc@cntval{9}\zc@pgval{28}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.88}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:extract-limits\endcsname{{9.3}{32}{Stationary limits, collisions, and Poisson tails}{section.103}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:extract-limits\endcsname{\default{9.3}\page{32}\abspage{36}\zc@counter{section}\thecounter{9.3}\zc@type{section}\zc@cntval{3}\zc@pgval{32}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{section.103}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:extract-measures\endcsname{{9.1}{29}{Weighted moment measures}{section.90}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:extract-measures\endcsname{\default{9.1}\page{29}\abspage{33}\zc@counter{section}\thecounter{9.1}\zc@type{section}\zc@cntval{1}\zc@pgval{29}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{section.90}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:extract-mixtures\endcsname{{9.2}{31}{Gaussian mixtures}{section.98}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:extract-mixtures\endcsname{\default{9.2}\page{31}\abspage{35}\zc@counter{section}\thecounter{9.2}\zc@type{section}\zc@cntval{2}\zc@pgval{31}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{section.98}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:extract-theorem\endcsname{{9.4}{36}{The extraction theorem}{section.112}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:extract-theorem\endcsname{\default{9.4}\page{36}\abspage{40}\zc@counter{section}\thecounter{9.4}\zc@type{section}\zc@cntval{4}\zc@pgval{36}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{section.112}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:extract-windows\endcsname{{9.5}{38}{Densities and arbitrarily small windows}{section.118}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:extract-windows\endcsname{\default{9.5}\page{38}\abspage{42}\zc@counter{section}\thecounter{9.5}\zc@type{section}\zc@cntval{5}\zc@pgval{38}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{section.118}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:hist\endcsname{{15}{157}{A priori bounds on transfer chains}{chapter.445}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:hist\endcsname{\default{15}\page{157}\abspage{161}\zc@counter{chapter}\thecounter{15}\zc@type{chapter}\zc@cntval{15}\zc@pgval{157}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.445}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:hist-averaging\endcsname{{15.4}{176}{Spectral averaging}{section.506}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:hist-averaging\endcsname{\default{15.4}\page{176}\abspage{180}\zc@counter{section}\thecounter{15.4}\zc@type{section}\zc@cntval{4}\zc@pgval{176}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{section.506}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:hist-bound\endcsname{{15.6}{185}{The a priori bound on partial chains}{section.526}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:hist-bound\endcsname{\default{15.6}\page{185}\abspage{189}\zc@counter{section}\thecounter{15.6}\zc@type{section}\zc@cntval{6}\zc@pgval{185}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{section.526}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:hist-coefficients\endcsname{{15.5}{180}{Coefficient bounds}{section.514}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:hist-coefficients\endcsname{\default{15.5}\page{180}\abspage{184}\zc@counter{section}\thecounter{15.5}\zc@type{section}\zc@cntval{5}\zc@pgval{180}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{section.514}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:hist-det\endcsname{{15.2}{161}{Determinant inequalities}{section.464}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:hist-det\endcsname{\default{15.2}\page{161}\abspage{165}\zc@counter{section}\thecounter{15.2}\zc@type{section}\zc@cntval{2}\zc@pgval{161}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{section.464}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:hist-setting\endcsname{{15.1}{158}{Setting}{section.446}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:hist-setting\endcsname{\default{15.1}\page{158}\abspage{162}\zc@counter{section}\thecounter{15.1}\zc@type{section}\zc@cntval{1}\zc@pgval{158}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{section.446}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:hist-step\endcsname{{15.3}{168}{One step on the cycle}{section.484}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:hist-step\endcsname{\default{15.3}\page{168}\abspage{172}\zc@counter{section}\thecounter{15.3}\zc@type{section}\zc@cntval{3}\zc@pgval{168}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{section.484}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:iface-counts\endcsname{{7.2}{4}{Factorial moments of eigenvalue counts}{section.7}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:iface-counts\endcsname{\default{7.2}\page{4}\abspage{8}\zc@counter{section}\thecounter{7.2}\zc@type{section}\zc@cntval{2}\zc@pgval{4}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{section.7}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:iface-density\endcsname{{7.3}{5}{The expected density}{section.14}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:iface-density\endcsname{\default{7.3}\page{5}\abspage{9}\zc@counter{section}\thecounter{7.3}\zc@type{section}\zc@cntval{3}\zc@pgval{5}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{section.14}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:iface-gaussian\endcsname{{7.4}{7}{The scalar Gaussian component}{section.20}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:iface-gaussian\endcsname{\default{7.4}\page{7}\abspage{11}\zc@counter{section}\thecounter{7.4}\zc@type{section}\zc@cntval{4}\zc@pgval{7}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{section.20}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:iface-theorem\endcsname{{7.5}{9}{The interface theorem}{section.28}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:iface-theorem\endcsname{\default{7.5}\page{9}\abspage{13}\zc@counter{section}\thecounter{7.5}\zc@type{section}\zc@cntval{5}\zc@pgval{9}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{section.28}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:iface-variance\endcsname{{7.1}{2}{The variance profile}{section.4}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:iface-variance\endcsname{\default{7.1}\page{2}\abspage{6}\zc@counter{section}\thecounter{7.1}\zc@type{section}\zc@cntval{1}\zc@pgval{2}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{section.4}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:int-balanced\endcsname{{11.3}{74}{Local limits at positive damping}{section.221}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:int-balanced\endcsname{\default{11.3}\page{74}\abspage{78}\zc@counter{section}\thecounter{11.3}\zc@type{section}\zc@cntval{3}\zc@pgval{74}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{section.221}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:int-exact\endcsname{{11.7}{83}{The exact participation formula}{section.242}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:int-exact\endcsname{\default{11.7}\page{83}\abspage{87}\zc@counter{section}\thecounter{11.7}\zc@type{section}\zc@cntval{7}\zc@pgval{83}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{section.242}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:int-ipr\endcsname{{11.6}{82}{Boundary control and the inverse participation ratio}{section.239}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:int-ipr\endcsname{\default{11.6}\page{82}\abspage{86}\zc@counter{section}\thecounter{11.6}\zc@type{section}\zc@cntval{6}\zc@pgval{82}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{section.239}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:int-pairings\endcsname{{11.2}{72}{Pairings at rank one}{section.213}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:int-pairings\endcsname{\default{11.2}\page{72}\abspage{76}\zc@counter{section}\thecounter{11.2}\zc@type{section}\zc@cntval{2}\zc@pgval{72}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{section.213}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:int-profile\endcsname{{11.4}{77}{The profile}{section.228}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:int-profile\endcsname{\default{11.4}\page{77}\abspage{81}\zc@counter{section}\thecounter{11.4}\zc@type{section}\zc@cntval{4}\zc@pgval{77}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{section.228}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:int-sources\endcsname{{11.1}{71}{Local sources}{section.205}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:int-sources\endcsname{\default{11.1}\page{71}\abspage{75}\zc@counter{section}\thecounter{11.1}\zc@type{section}\zc@cntval{1}\zc@pgval{71}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{section.205}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:int-twoenergy\endcsname{{11.5}{78}{The two-energy intensity}{section.231}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:int-twoenergy\endcsname{\default{11.5}\page{78}\abspage{82}\zc@counter{section}\thecounter{11.5}\zc@type{section}\zc@cntval{5}\zc@pgval{78}\zc@enclval{{11}{0}}\zc@pgfmt{2}\anchor{section.231}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:intensity\endcsname{{11}{70}{Local intensities and the inverse participation ratio}{chapter.203}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:intensity\endcsname{\default{11}\page{70}\abspage{74}\zc@counter{chapter}\thecounter{11}\zc@type{chapter}\zc@cntval{11}\zc@pgval{70}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.203}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:interface\endcsname{{7}{2}{Inputs and the interface theorem}{chapter.3}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:interface\endcsname{\default{7}\page{2}\abspage{6}\zc@counter{chapter}\thecounter{7}\zc@type{chapter}\zc@cntval{7}\zc@pgval{2}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.3}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:met-fk\endcsname{{13.2}{113}{Feynman--Kac preliminaries}{section.311}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:met-fk\endcsname{\default{13.2}\page{113}\abspage{117}\zc@counter{section}\thecounter{13.2}\zc@type{section}\zc@cntval{2}\zc@pgval{113}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{section.311}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:met-modes\endcsname{{13.5}{127}{The coefficients and the diffusion modes}{section.356}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:met-modes\endcsname{\default{13.5}\page{127}\abspage{131}\zc@counter{section}\thecounter{13.5}\zc@type{section}\zc@cntval{5}\zc@pgval{127}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{section.356}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:met-profile\endcsname{{13.1}{111}{The eigenvector correlation}{section.306}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:met-profile\endcsname{\default{13.1}\page{111}\abspage{115}\zc@counter{section}\thecounter{13.1}\zc@type{section}\zc@cntval{1}\zc@pgval{111}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{section.306}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:met-scalar\endcsname{{13.4}{120}{The kernel and the function $I_X$}{section.337}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:met-scalar\endcsname{\default{13.4}\page{120}\abspage{124}\zc@counter{section}\thecounter{13.4}\zc@type{section}\zc@cntval{4}\zc@pgval{120}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{section.337}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:met-singlet\endcsname{{13.3}{117}{Expansion of the singlet state}{section.326}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:met-singlet\endcsname{\default{13.3}\page{117}\abspage{121}\zc@counter{section}\thecounter{13.3}\zc@type{section}\zc@cntval{3}\zc@pgval{117}\zc@enclval{{13}{0}}\zc@pgfmt{2}\anchor{section.326}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:metallic\endcsname{{13}{111}{The metallic expansion}{chapter.305}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:metallic\endcsname{\default{13}\page{111}\abspage{115}\zc@counter{chapter}\thecounter{13}\zc@type{chapter}\zc@cntval{13}\zc@pgval{111}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.305}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:proc-continuation\endcsname{{8.4}{22}{Continuation to all source parameters}{section.77}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:proc-continuation\endcsname{\default{8.4}\page{22}\abspage{26}\zc@counter{section}\thecounter{8.4}\zc@type{section}\zc@cntval{4}\zc@pgval{22}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{section.77}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:proc-determination\endcsname{{8.3}{20}{A determining class}{section.70}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:proc-determination\endcsname{\default{8.3}\page{20}\abspage{24}\zc@counter{section}\thecounter{8.3}\zc@type{section}\zc@cntval{3}\zc@pgval{20}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{section.70}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:proc-jets\endcsname{{8.6}{25}{The rank-one derivatives}{section.85}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:proc-jets\endcsname{\default{8.6}\page{25}\abspage{29}\zc@counter{section}\thecounter{8.6}\zc@type{section}\zc@cntval{6}\zc@pgval{25}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{section.85}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:proc-proof\endcsname{{8.5}{24}{Proof of parts (a) and (b)}{section.83}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:proc-proof\endcsname{\default{8.5}\page{24}\abspage{28}\zc@counter{section}\thecounter{8.5}\zc@type{section}\zc@cntval{5}\zc@pgval{24}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{section.83}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:proc-ratios\endcsname{{8.1}{15}{Ratios with poles at \texorpdfstring {$\pm \iu \eps $}{+-i epsilon}}{section.50}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:proc-ratios\endcsname{\default{8.1}\page{15}\abspage{19}\zc@counter{section}\thecounter{8.1}\zc@type{section}\zc@cntval{1}\zc@pgval{15}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{section.50}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:proc-tight\endcsname{{8.2}{17}{Tightness and subsequential limits}{section.58}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:proc-tight\endcsname{\default{8.2}\page{17}\abspage{21}\zc@counter{section}\thecounter{8.2}\zc@type{section}\zc@cntval{2}\zc@pgval{17}\zc@enclval{{8}{0}}\zc@pgfmt{2}\anchor{section.58}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:process\endcsname{{8}{14}{The limiting point process}{chapter.49}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:process\endcsname{\default{8}\page{14}\abspage{18}\zc@counter{chapter}\thecounter{8}\zc@type{chapter}\zc@cntval{8}\zc@pgval{14}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.49}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:regularity\endcsname{{C}{400}{Proof of the regular realization}{appendix.1102}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:regularity\endcsname{\default{C}\page{400}\abspage{404}\zc@counter{chapter}\thecounter{C}\zc@type{appendix}\zc@cntval{3}\zc@pgval{400}\zc@enclval{{1}}\zc@pgfmt{2}\anchor{appendix.1102}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:res-chart\endcsname{{10.6}{52}{The rank-one superspace in polar coordinates}{section.158}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:res-chart\endcsname{\default{10.6}\page{52}\abspage{56}\zc@counter{section}\thecounter{10.6}\zc@type{section}\zc@cntval{6}\zc@pgval{52}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{section.158}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:res-diffusions\endcsname{{10.1}{42}{The radial and the singlet diffusions}{section.126}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:res-diffusions\endcsname{\default{10.1}\page{42}\abspage{46}\zc@counter{section}\thecounter{10.1}\zc@type{section}\zc@cntval{1}\zc@pgval{42}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{section.126}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:res-envelope\endcsname{{10.4}{48}{A tail envelope}{section.146}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:res-envelope\endcsname{\default{10.4}\page{48}\abspage{52}\zc@counter{section}\thecounter{10.4}\zc@type{section}\zc@cntval{4}\zc@pgval{48}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{section.146}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:res-hierarchy\endcsname{{10.3}{46}{The moment densities as Brownian functionals}{section.142}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:res-hierarchy\endcsname{\default{10.3}\page{46}\abspage{50}\zc@counter{section}\thecounter{10.3}\zc@type{section}\zc@cntval{3}\zc@pgval{46}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{section.142}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:res-permanent\endcsname{{10.7}{62}{The permanent residue}{section.187}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:res-permanent\endcsname{\default{10.7}\page{62}\abspage{66}\zc@counter{section}\thecounter{10.7}\zc@type{section}\zc@cntval{7}\zc@pgval{62}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{section.187}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:res-scalar\endcsname{{10.5}{49}{The scalar residue}{section.150}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:res-scalar\endcsname{\default{10.5}\page{49}\abspage{53}\zc@counter{section}\thecounter{10.5}\zc@type{section}\zc@cntval{5}\zc@pgval{49}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{section.150}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:res-scaling\endcsname{{10.2}{42}{The boundary at infinity}{section.127}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:res-scaling\endcsname{\default{10.2}\page{42}\abspage{46}\zc@counter{section}\thecounter{10.2}\zc@type{section}\zc@cntval{2}\zc@pgval{42}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{section.127}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:residue\endcsname{{10}{41}{The noncompact boundary and Brownian functionals}{chapter.124}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:residue\endcsname{\default{10}\page{41}\abspage{45}\zc@counter{chapter}\thecounter{10}\zc@type{chapter}\zc@cntval{10}\zc@pgval{41}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.124}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:scales\endcsname{{14}{129}{Spectral fluctuations across scales}{chapter.359}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:scales\endcsname{\default{14}\page{129}\abspage{133}\zc@counter{chapter}\thecounter{14}\zc@type{chapter}\zc@cntval{14}\zc@pgval{129}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.359}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:scales-crossover\endcsname{{14.8}{154}{The crossover}{section.440}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:scales-crossover\endcsname{\default{14.8}\page{154}\abspage{158}\zc@counter{section}\thecounter{14.8}\zc@type{section}\zc@cntval{8}\zc@pgval{154}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{section.440}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:scales-endpoints\endcsname{{14.6}{152}{The endpoints}{section.435}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:scales-endpoints\endcsname{\default{14.6}\page{152}\abspage{156}\zc@counter{section}\thecounter{14.6}\zc@type{section}\zc@cntval{6}\zc@pgval{152}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{section.435}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:scales-filter\endcsname{{14.3}{135}{Laplace transforms of positive measures}{section.378}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:scales-filter\endcsname{\default{14.3}\page{135}\abspage{139}\zc@counter{section}\thecounter{14.3}\zc@type{section}\zc@cntval{3}\zc@pgval{135}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{section.378}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:scales-fourier\endcsname{{14.2}{132}{The Bartlett spectrum}{section.366}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:scales-fourier\endcsname{\default{14.2}\page{132}\abspage{136}\zc@counter{section}\thecounter{14.2}\zc@type{section}\zc@cntval{2}\zc@pgval{132}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{section.366}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:scales-largecov\endcsname{{14.7}{153}{Large-scale covariance}{section.436}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:scales-largecov\endcsname{\default{14.7}\page{153}\abspage{157}\zc@counter{section}\thecounter{14.7}\zc@type{section}\zc@cntval{7}\zc@pgval{153}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{section.436}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:scales-pair\endcsname{{14.1}{129}{The pair measure}{section.360}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:scales-pair\endcsname{\default{14.1}\page{129}\abspage{133}\zc@counter{section}\thecounter{14.1}\zc@type{section}\zc@cntval{1}\zc@pgval{129}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{section.360}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:scales-pairing\endcsname{{14.5}{146}{The stationary pairing}{section.417}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:scales-pairing\endcsname{\default{14.5}\page{146}\abspage{150}\zc@counter{section}\thecounter{14.5}\zc@type{section}\zc@cntval{5}\zc@pgval{146}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{section.417}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:scales-stationary\endcsname{{14.4}{137}{The stationary approximation}{section.385}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:scales-stationary\endcsname{\default{14.4}\page{137}\abspage{141}\zc@counter{section}\thecounter{14.4}\zc@type{section}\zc@cntval{4}\zc@pgval{137}\zc@enclval{{14}{0}}\zc@pgfmt{2}\anchor{section.385}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:sigma-chains\endcsname{{C.6}{405}{Semigroup products at all times}{section.1114}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:sigma-chains\endcsname{\default{C.6}\page{405}\abspage{409}\zc@counter{section}\thecounter{C.6}\zc@type{appendix}\zc@cntval{6}\zc@pgval{405}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{section.1114}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:sigma-core\endcsname{{C.3}{402}{The core}{section.1110}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:sigma-core\endcsname{\default{C.3}\page{402}\abspage{406}\zc@counter{section}\thecounter{C.3}\zc@type{appendix}\zc@cntval{3}\zc@pgval{402}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{section.1110}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:sigma-form\endcsname{{C.2}{401}{The form}{section.1107}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:sigma-form\endcsname{\default{C.2}\page{401}\abspage{405}\zc@counter{section}\thecounter{C.2}\zc@type{appendix}\zc@cntval{2}\zc@pgval{401}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{section.1107}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:sigma-local\endcsname{{C.1}{400}{Local structure}{section.1103}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:sigma-local\endcsname{\default{C.1}\page{400}\abspage{404}\zc@counter{section}\thecounter{C.1}\zc@type{appendix}\zc@cntval{1}\zc@pgval{400}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{section.1103}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:sigma-regular-proof\endcsname{{C}{400}{Proof of the regular realization}{appendix.1102}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:sigma-regular-proof\endcsname{\default{C}\page{400}\abspage{404}\zc@counter{chapter}\thecounter{C}\zc@type{appendix}\zc@cntval{3}\zc@pgval{400}\zc@enclval{{1}}\zc@pgfmt{2}\anchor{appendix.1102}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:sigma-symmetry-proof\endcsname{{C.5}{404}{Symmetry, the split pairing, and small damping}{section.1113}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:sigma-symmetry-proof\endcsname{\default{C.5}\page{404}\abspage{408}\zc@counter{section}\thecounter{C.5}\zc@type{appendix}\zc@cntval{5}\zc@pgval{404}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{section.1113}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:sigma-weights-proof\endcsname{{C.4}{402}{Weights}{section.1111}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:sigma-weights-proof\endcsname{\default{C.4}\page{402}\abspage{406}\zc@counter{section}\thecounter{C.4}\zc@type{appendix}\zc@cntval{4}\zc@pgval{402}\zc@enclval{{3}{1}}\zc@pgfmt{2}\anchor{section.1111}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab\endcsname{{17}{239}{Stability of the transfer kernel}{chapter.657}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab\endcsname{\default{17}\page{239}\abspage{243}\zc@counter{chapter}\thecounter{17}\zc@type{chapter}\zc@cntval{17}\zc@pgval{239}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.657}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab-energy\endcsname{{17.4}{250}{The geodesic difference energy}{section.685}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab-energy\endcsname{\default{17.4}\page{250}\abspage{254}\zc@counter{section}\thecounter{17.4}\zc@type{section}\zc@cntval{4}\zc@pgval{250}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{section.685}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab-frame\endcsname{{17.3}{245}{The frame factorization and scalar amplitudes}{section.674}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab-frame\endcsname{\default{17.3}\page{245}\abspage{249}\zc@counter{section}\thecounter{17.3}\zc@type{section}\zc@cntval{3}\zc@pgval{245}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{section.674}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab-kernel\endcsname{{17.2}{243}{The kernel in midpoint coordinates}{section.667}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab-kernel\endcsname{\default{17.2}\page{243}\abspage{247}\zc@counter{section}\thecounter{17.2}\zc@type{section}\zc@cntval{2}\zc@pgval{243}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{section.667}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab-main\endcsname{{17.1}{239}{The reference contraction and the main theorem}{section.658}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab-main\endcsname{\default{17.1}\page{239}\abspage{243}\zc@counter{section}\thecounter{17.1}\zc@type{section}\zc@cntval{1}\zc@pgval{239}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{section.658}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab-parity\endcsname{{17.5}{252}{Marginals and midpoint parity}{section.692}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab-parity\endcsname{\default{17.5}\page{252}\abspage{256}\zc@counter{section}\thecounter{17.5}\zc@type{section}\zc@cntval{5}\zc@pgval{252}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{section.692}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab-proof\endcsname{{17.8}{268}{Proof of the main theorem}{section.735}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab-proof\endcsname{\default{17.8}\page{268}\abspage{272}\zc@counter{section}\thecounter{17.8}\zc@type{section}\zc@cntval{8}\zc@pgval{268}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{section.735}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab-uncharged\endcsname{{17.6}{255}{The uncharged part}{section.698}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab-uncharged\endcsname{\default{17.6}\page{255}\abspage{259}\zc@counter{section}\thecounter{17.6}\zc@type{section}\zc@cntval{6}\zc@pgval{255}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{section.698}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab-weight-gain\endcsname{{17.9}{270}{Gain of body weights}{section.737}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab-weight-gain\endcsname{\default{17.9}\page{270}\abspage{274}\zc@counter{section}\thecounter{17.9}\zc@type{section}\zc@cntval{9}\zc@pgval{270}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{section.737}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:stab-words\endcsname{{17.7}{260}{Holonomy, charged words, the frozen normal transfer and the source}{section.712}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:stab-words\endcsname{\default{17.7}\page{260}\abspage{264}\zc@counter{section}\thecounter{17.7}\zc@type{section}\zc@cntval{7}\zc@pgval{260}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{section.712}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tools\endcsname{{A}{378}{General tools}{appendix.1028}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tools\endcsname{\default{A}\page{378}\abspage{382}\zc@counter{chapter}\thecounter{A}\zc@type{appendix}\zc@cntval{1}\zc@pgval{378}\zc@enclval{{1}}\zc@pgfmt{2}\anchor{appendix.1028}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tools-defect\endcsname{{A.1}{378}{Perturbations of a contraction}{section.1029}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tools-defect\endcsname{\default{A.1}\page{378}\abspage{382}\zc@counter{section}\thecounter{A.1}\zc@type{appendix}\zc@cntval{1}\zc@pgval{378}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{section.1029}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tools-dirichlet\endcsname{{A.2}{382}{An operator-valued Dirichlet-form bound}{section.1045}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tools-dirichlet\endcsname{\default{A.2}\page{382}\abspage{386}\zc@counter{section}\thecounter{A.2}\zc@type{appendix}\zc@cntval{2}\zc@pgval{382}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{section.1045}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tools-poisson\endcsname{{A.3}{384}{Poisson transforms of translation-bounded measures}{section.1058}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tools-poisson\endcsname{\default{A.3}\page{384}\abspage{388}\zc@counter{section}\thecounter{A.3}\zc@type{appendix}\zc@cntval{3}\zc@pgval{384}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{section.1058}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tools-submean\endcsname{{A.4}{387}{A submean inequality for holomorphic families}{section.1069}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tools-submean\endcsname{\default{A.4}\page{387}\abspage{391}\zc@counter{section}\thecounter{A.4}\zc@type{appendix}\zc@cntval{4}\zc@pgval{387}\zc@enclval{{1}{1}}\zc@pgfmt{2}\anchor{section.1069}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube\endcsname{{16}{194}{The normal tube and the normal oscillator}{chapter.550}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube\endcsname{\default{16}\page{194}\abspage{198}\zc@counter{chapter}\thecounter{16}\zc@type{chapter}\zc@cntval{16}\zc@pgval{194}\zc@enclval{{0}}\zc@pgfmt{2}\anchor{chapter.550}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-amplitude\endcsname{{16.8}{227}{Mehler kernels with amplitudes}{section.630}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-amplitude\endcsname{\default{16.8}\page{227}\abspage{231}\zc@counter{section}\thecounter{16.8}\zc@type{section}\zc@cntval{8}\zc@pgval{227}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.630}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-cutoff\endcsname{{16.7}{221}{Cutoffs and Mehler defects}{section.621}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-cutoff\endcsname{\default{16.7}\page{221}\abspage{225}\zc@counter{section}\thecounter{16.7}\zc@type{section}\zc@cntval{7}\zc@pgval{221}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.621}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-gap\endcsname{{16.10}{236}{The normal contraction and the Ward identity}{section.651}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-gap\endcsname{\default{16.10}\page{236}\abspage{240}\zc@counter{section}\thecounter{16.10}\zc@type{section}\zc@cntval{10}\zc@pgval{236}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.651}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-gauge\endcsname{{16.5}{212}{The adapted transfer operator and the normal gauge}{section.600}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-gauge\endcsname{\default{16.5}\page{212}\abspage{216}\zc@counter{section}\thecounter{16.5}\zc@type{section}\zc@cntval{5}\zc@pgval{212}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.600}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-geometry\endcsname{{16.3}{201}{Tubular coordinates around the saddle manifold}{section.567}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-geometry\endcsname{\default{16.3}\page{201}\abspage{205}\zc@counter{section}\thecounter{16.3}\zc@type{section}\zc@cntval{3}\zc@pgval{201}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.567}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-operator\endcsname{{16.3}{207}{The decomposition of the ambient Laplacian}{subsection*.587}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-operator\endcsname{\default{16.3}\page{207}\abspage{211}\zc@counter{section}\thecounter{16.3}\zc@type{section}\zc@cntval{3}\zc@pgval{207}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{subsection*.587}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-oscillator\endcsname{{16.4}{211}{The normal oscillator}{section.596}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-oscillator\endcsname{\default{16.4}\page{211}\abspage{215}\zc@counter{section}\thecounter{16.4}\zc@type{section}\zc@cntval{4}\zc@pgval{211}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.596}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-powers\endcsname{{16.9}{232}{Graph powers of the frozen normal transfer}{section.641}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-powers\endcsname{\default{16.9}\page{232}\abspage{236}\zc@counter{section}\thecounter{16.9}\zc@type{section}\zc@cntval{9}\zc@pgval{232}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.641}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-prelim\endcsname{{16.2}{196}{Super vector spaces and homogeneous superspaces}{section.554}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-prelim\endcsname{\default{16.2}\page{196}\abspage{200}\zc@counter{section}\thecounter{16.2}\zc@type{section}\zc@cntval{2}\zc@pgval{196}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.554}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-quadratic\endcsname{{16.6}{216}{The quadratic normal transfer and the supercharge}{section.611}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-quadratic\endcsname{\default{16.6}\page{216}\abspage{220}\zc@counter{section}\thecounter{16.6}\zc@type{section}\zc@cntval{6}\zc@pgval{216}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.611}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@s:tube-scales\endcsname{{16.1}{194}{Two scales}{section.551}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@s:tube-scales\endcsname{\default{16.1}\page{194}\abspage{198}\zc@counter{section}\thecounter{16.1}\zc@type{section}\zc@cntval{1}\zc@pgval{194}\zc@enclval{{16}{0}}\zc@pgfmt{2}\anchor{section.551}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@t:coarse-moments\endcsname{{12.5}{95}{}{theorem.274}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@t:coarse-moments\endcsname{\default{12.5}\page{95}\abspage{99}\zc@counter{theorem}\thecounter{12.5}\zc@type{theorem}\zc@cntval{5}\zc@pgval{95}\zc@enclval{{12}{0}}\zc@pgfmt{2}\anchor{theorem.274}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@t:cons-limit\endcsname{{18.14}{297}{}{theorem.813}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@t:cons-limit\endcsname{\default{18.14}\page{297}\abspage{301}\zc@counter{theorem}\thecounter{18.14}\zc@type{theorem}\zc@cntval{14}\zc@pgval{297}\zc@enclval{{18}{0}}\zc@pgfmt{2}\anchor{theorem.813}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@t:end-entrance\endcsname{{19.21}{343}{}{theorem.936}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@t:end-entrance\endcsname{\default{19.21}\page{343}\abspage{347}\zc@counter{theorem}\thecounter{19.21}\zc@type{theorem}\zc@cntval{21}\zc@pgval{343}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{theorem.936}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@t:extract-main\endcsname{{9.6}{36}{}{theorem.115}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@t:extract-main\endcsname{\default{9.6}\page{36}\abspage{40}\zc@counter{theorem}\thecounter{9.6}\zc@type{theorem}\zc@cntval{6}\zc@pgval{36}\zc@enclval{{9}{0}}\zc@pgfmt{2}\anchor{theorem.115}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@t:hist-bound\endcsname{{15.14}{185}{}{theorem.528}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@t:hist-bound\endcsname{\default{15.14}\page{185}\abspage{189}\zc@counter{theorem}\thecounter{15.14}\zc@type{theorem}\zc@cntval{14}\zc@pgval{185}\zc@enclval{{15}{0}}\zc@pgfmt{2}\anchor{theorem.528}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@t:interface\endcsname{{7.6}{10}{}{theorem.35}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@t:interface\endcsname{\default{7.6}\page{10}\abspage{14}\zc@counter{theorem}\thecounter{7.6}\zc@type{theorem}\zc@cntval{6}\zc@pgval{10}\zc@enclval{{7}{0}}\zc@pgfmt{2}\anchor{theorem.35}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@t:res-scalar\endcsname{{10.4}{49}{}{theorem.151}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@t:res-scalar\endcsname{\default{10.4}\page{49}\abspage{53}\zc@counter{theorem}\thecounter{10.4}\zc@type{theorem}\zc@cntval{4}\zc@pgval{49}\zc@enclval{{10}{0}}\zc@pgfmt{2}\anchor{theorem.151}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@t:stab-main\endcsname{{17.1}{242}{}{theorem.665}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@t:stab-main\endcsname{\default{17.1}\page{242}\abspage{246}\zc@counter{theorem}\thecounter{17.1}\zc@type{theorem}\zc@cntval{1}\zc@pgval{242}\zc@enclval{{17}{0}}\zc@pgfmt{2}\anchor{theorem.665}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname r@tab:end-constants\endcsname{{19.1}{300}{The constants of \zcref {s:end}, in the order in which they are chosen. Here $s_0 = \inf _{|E| \le 1/2} s_E$}{table.19.1}{lopatto-critical-bandwidth-volume-ii.pdf}} \expandafter\gdef\csname Z@R@tab:end-constants\endcsname{\default{19.1}\page{300}\abspage{304}\zc@counter{table}\thecounter{19.1}\zc@type{table}\zc@cntval{1}\zc@pgval{300}\zc@enclval{{19}{0}}\zc@pgfmt{2}\anchor{table.19.1}\externaldocument{lopatto-critical-bandwidth-volume-ii}\url{lopatto-critical-bandwidth-volume-ii.pdf}} \makeatother