% Frozen companion labels: independent compilation needs no companion .aux files.
% The fifth field preserves the existing companion-PDF link target.
\makeatletter
% Part II
\expandafter\gdef\csname r@part:rg\endcsname{{IV}{2}{\contentsname \@mkboth {\MakeUppercase \contentsname }{\MakeUppercase \contentsname }}{part.2}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@part:rg@cref\endcsname{{[part][4][]IV}{[1][2][]2}}
\expandafter\gdef\csname r@ch09:chapter\endcsname{{12}{2}{Fluctuation integration and cluster expansions}{chapter.3}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:chapter@cref\endcsname{{[chapter][12][]12}{[1][2][]2}}
\expandafter\gdef\csname r@ch09:sec:coordinates\endcsname{{12.1}{4}{Coefficient norms and normalized fluctuation coordinates}{section.8}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:sec:coordinates@cref\endcsname{{[section][1][12]12.1}{[1][4][]4}}
\expandafter\gdef\csname r@ch09:def:supported-coefficient-norms\endcsname{{12.1}{4}{}{definition.9}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:def:supported-coefficient-norms@cref\endcsname{{[definition][1][]12.1}{[1][4][]4}}
\expandafter\gdef\csname r@ch09:eq:coefficient-norm\endcsname{{12.1}{4}{}{equation.10}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:coefficient-norm@cref\endcsname{{[equation][1][12]12.1}{[1][4][]4}}
\expandafter\gdef\csname r@ch09:eq:joint-seminorm\endcsname{{12.2}{5}{}{equation.11}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:joint-seminorm@cref\endcsname{{[equation][2][12]12.2}{[1][5][]5}}
\expandafter\gdef\csname r@ch09:lem:local-calculus\endcsname{{12.2}{5}{}{lemma.12}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:lem:local-calculus@cref\endcsname{{[lemma][2][]12.2}{[1][5][]5}}
\expandafter\gdef\csname r@ch09:eq:center-equation\endcsname{{12.3}{7}{Coefficient norms and normalized fluctuation coordinates}{equation.17}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:center-equation@cref\endcsname{{[equation][3][12]12.3}{[1][7][]7}}
\expandafter\gdef\csname r@ch09:lem:centered-white-coarea\endcsname{{12.3}{8}{}{lemma.18}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:lem:centered-white-coarea@cref\endcsname{{[lemma][3][]12.3}{[1][8][]8}}
\expandafter\gdef\csname r@ch09:eq:one-step-conditions\endcsname{{12.4}{9}{}{equation.23}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:one-step-conditions@cref\endcsname{{[equation][4][12]12.4}{[1][9][]9}}
\expandafter\gdef\csname r@ch09:eq:one-step-interaction\endcsname{{12.5}{9}{}{equation.24}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:one-step-interaction@cref\endcsname{{[equation][5][12]12.5}{[1][9][]9}}
\expandafter\gdef\csname r@ch09:eq:one-step-integral\endcsname{{12.6}{9}{}{equation.25}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:one-step-integral@cref\endcsname{{[equation][6][12]12.6}{[1][9][]9}}
\expandafter\gdef\csname r@ch09:eq:resolvent-series\endcsname{{12.7}{10}{Step 2: the inverse square root}{equation.28}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:resolvent-series@cref\endcsname{{[equation][7][12]12.7}{[1][10][]10}}
\expandafter\gdef\csname r@ch09:eq:positive-root\endcsname{{12.8}{11}{Step 4: joint bounds and the positive square root}{equation.31}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:positive-root@cref\endcsname{{[equation][8][12]12.8}{[1][10][]11}}
\expandafter\gdef\csname r@ch09:eq:positive-root-majorant\endcsname{{12.9}{11}{Step 4: joint bounds and the positive square root}{equation.32}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:positive-root-majorant@cref\endcsname{{[equation][9][12]12.9}{[1][11][]11}}
\expandafter\gdef\csname r@ch09:sec:polymers\endcsname{{12.2}{12}{Polymer expansions in the coefficient algebra}{section.36}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:sec:polymers@cref\endcsname{{[section][2][12]12.2}{[1][12][]12}}
\expandafter\gdef\csname r@ch09:lem:tree-sum\endcsname{{12.4}{13}{}{lemma.37}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:lem:tree-sum@cref\endcsname{{[lemma][4][]12.4}{[1][13][]13}}
\expandafter\gdef\csname r@ch09:eq:tree-recursion\endcsname{{12.10}{13}{Polymer expansions in the coefficient algebra}{equation.38}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:tree-recursion@cref\endcsname{{[equation][10][12]12.10}{[1][13][]13}}
\expandafter\gdef\csname r@ch09:lem:penrose\endcsname{{12.5}{13}{}{lemma.39}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:lem:penrose@cref\endcsname{{[lemma][5][]12.5}{[1][13][]13}}
\expandafter\gdef\csname r@ch09:thm:hard-core-log\endcsname{{12.6}{14}{}{theorem.40}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:thm:hard-core-log@cref\endcsname{{[theorem][6][12]12.6}{[1][14][]14}}
\expandafter\gdef\csname r@ch09:eq:activity-bound\endcsname{{12.11}{14}{}{equation.41}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:activity-bound@cref\endcsname{{[equation][11][12]12.11}{[1][14][]14}}
\expandafter\gdef\csname r@ch09:eq:coefficient-to-jet\endcsname{{12.12}{16}{Step 4: derivatives}{equation.50}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:coefficient-to-jet@cref\endcsname{{[equation][12][12]12.12}{[1][16][]16}}
\expandafter\gdef\csname r@ch09:thm:supported-log-holomorphy\endcsname{{12.7}{16}{}{theorem.51}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:thm:supported-log-holomorphy@cref\endcsname{{[theorem][7][12]12.7}{[1][16][]16}}
\expandafter\gdef\csname r@ch09:eq:activity-formula\endcsname{{12.13}{17}{}{equation.52}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:activity-formula@cref\endcsname{{[equation][13][12]12.13}{[1][16][]17}}
\expandafter\gdef\csname r@ch09:eq:supported-log-family-bound\endcsname{{12.14}{17}{}{equation.53}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:supported-log-family-bound@cref\endcsname{{[equation][14][12]12.14}{[1][17][]17}}
\expandafter\gdef\csname r@ch09:eq:supported-log-cauchy\endcsname{{12.15}{17}{}{equation.54}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:supported-log-cauchy@cref\endcsname{{[equation][15][12]12.15}{[1][17][]17}}
\expandafter\gdef\csname r@ch09:cor:supported-mark-two-powers\endcsname{{12.8}{19}{}{corollary.60}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:cor:supported-mark-two-powers@cref\endcsname{{[corollary][8][]12.8}{[1][19][]19}}
\expandafter\gdef\csname r@ch09:eq:supported-mark-two-powers\endcsname{{12.16}{19}{}{equation.61}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:supported-mark-two-powers@cref\endcsname{{[equation][16][12]12.16}{[1][19][]19}}
\expandafter\gdef\csname r@ch09:sec:one-step\endcsname{{12.3}{20}{Connected supports and the effective interaction of one averaging step}{section.62}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:sec:one-step@cref\endcsname{{[section][3][12]12.3}{[1][20][]20}}
\expandafter\gdef\csname r@ch09:lem:supported-activities\endcsname{{12.9}{20}{}{lemma.63}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:lem:supported-activities@cref\endcsname{{[lemma][9][]12.9}{[1][20][]20}}
\expandafter\gdef\csname r@ch09:eq:classical-remainder\endcsname{{12.17}{21}{Step 1: expansion in the fluctuation variables}{equation.69}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:classical-remainder@cref\endcsname{{[equation][17][12]12.17}{[1][21][]21}}
\expandafter\gdef\csname r@ch09:eq:coarea-remainder\endcsname{{12.18}{21}{Step 1: expansion in the fluctuation variables}{equation.70}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:coarea-remainder@cref\endcsname{{[equation][18][12]12.18}{[1][21][]21}}
\expandafter\gdef\csname r@ch09:eq:hull-size\endcsname{{12.19}{22}{Step 3: the supports}{equation.73}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:hull-size@cref\endcsname{{[equation][19][12]12.19}{[1][22][]22}}
\expandafter\gdef\csname r@ch09:eq:hull-payment\endcsname{{12.20}{22}{Step 3: the supports}{equation.74}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:hull-payment@cref\endcsname{{[equation][20][12]12.20}{[1][22][]22}}
\expandafter\gdef\csname r@ch09:eq:one-step-effective-interaction\endcsname{{12.21}{23}{Connected supports and the effective interaction of one averaging step}{equation.76}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:one-step-effective-interaction@cref\endcsname{{[equation][21][12]12.21}{[1][23][]23}}
\expandafter\gdef\csname r@ch09:prop:one-step-interaction\endcsname{{12.10}{23}{}{proposition.77}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:prop:one-step-interaction@cref\endcsname{{[proposition][10][]12.10}{[1][23][]23}}
\expandafter\gdef\csname r@ch09:eq:determinant-series\endcsname{{12.22}{23}{Connected supports and the effective interaction of one averaging step}{equation.80}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:determinant-series@cref\endcsname{{[equation][22][12]12.22}{[1][23][]23}}
\expandafter\gdef\csname r@ch09:prop:one-step-symmetries\endcsname{{12.11}{24}{}{proposition.81}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:prop:one-step-symmetries@cref\endcsname{{[proposition][11][]12.11}{[1][24][]24}}
\expandafter\gdef\csname r@ch09:sec:sequential\endcsname{{12.4}{26}{Several averaging steps}{section.82}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:sec:sequential@cref\endcsname{{[section][4][12]12.4}{[1][26][]26}}
\expandafter\gdef\csname r@ch09:ex:two-steps\endcsname{{12.12}{26}{}{example.83}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:ex:two-steps@cref\endcsname{{[example][12][]12.12}{[1][26][]26}}
\expandafter\gdef\csname r@ch09:thm:sequential-density\endcsname{{12.13}{27}{}{theorem.84}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:thm:sequential-density@cref\endcsname{{[theorem][13][12]12.13}{[1][27][]27}}
\expandafter\gdef\csname r@ch09:eq:sequential-conditions\endcsname{{12.23}{27}{}{equation.85}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:sequential-conditions@cref\endcsname{{[equation][23][12]12.23}{[1][27][]27}}
\expandafter\gdef\csname r@ch09:eq:sequential-interaction\endcsname{{12.24}{28}{}{equation.90}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:sequential-interaction@cref\endcsname{{[equation][24][12]12.24}{[1][28][]28}}
\expandafter\gdef\csname r@ch09:eq:sequential-integral\endcsname{{12.25}{28}{}{equation.91}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:sequential-integral@cref\endcsname{{[equation][25][12]12.25}{[1][28][]28}}
\expandafter\gdef\csname r@ch09:eq:sequential-majorant\endcsname{{12.26}{31}{Step 6: bounds for the fluctuation interaction}{equation.100}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:sequential-majorant@cref\endcsname{{[equation][26][12]12.26}{[1][31][]31}}
\expandafter\gdef\csname r@ch09:sec:insertion\endcsname{{12.5}{32}{Perturbation by an input effective interaction}{section.103}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:sec:insertion@cref\endcsname{{[section][5][12]12.5}{[1][32][]32}}
\expandafter\gdef\csname r@ch09:rem:second-order\endcsname{{12.14}{32}{}{remark.104}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:rem:second-order@cref\endcsname{{[remark][14][]12.14}{[1][32][]32}}
\expandafter\gdef\csname r@ch09:conv:auxiliary-baseline\endcsname{{12.15}{33}{}{definition.105}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:conv:auxiliary-baseline@cref\endcsname{{[definition][15][]12.15}{[1][33][]33}}
\expandafter\gdef\csname r@ch09:eq:auxiliary-classical\endcsname{{12.27}{33}{}{equation.106}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch09:eq:auxiliary-baseline\endcsname{{12.28}{33}{}{equation.107}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch09:lem:marked-insertion\endcsname{{12.16}{33}{}{lemma.108}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:lem:marked-insertion@cref\endcsname{{[lemma][16][]12.16}{[1][33][]33}}
\expandafter\gdef\csname r@ch09:eq:insertion-norm\endcsname{{12.29}{35}{Step 1: supported decompositions}{equation.113}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch09:eq:auxiliary-baseline-norm\endcsname{{12.30}{35}{Step 1: supported decompositions}{equation.114}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:auxiliary-baseline-norm@cref\endcsname{{[equation][30][12]12.30}{[1][35][]35}}
\expandafter\gdef\csname r@ch09:eq:marked-linear-bound\endcsname{{12.31}{36}{Step 3: cancellation of the first order}{equation.117}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:eq:marked-linear-bound@cref\endcsname{{[equation][31][12]12.31}{[1][36][]36}}
\expandafter\gdef\csname r@ch09:rem:order-of-choices\endcsname{{12.17}{36}{}{remark.120}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch09:rem:order-of-choices@cref\endcsname{{[remark][17][]12.17}{[1][36][]36}}
\expandafter\gdef\csname r@ch10:chapter\endcsname{{13}{37}{Conditional measures and analytic estimates}{chapter.121}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:chapter@cref\endcsname{{[chapter][13][]13}{[1][37][]37}}
\expandafter\gdef\csname r@ch10:eq:schematic\endcsname{{13.1}{37}{The exact formula for one step}{equation.123}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch10:sec:coefficients\endcsname{{13.1}{40}{Weighted coefficient bounds and majorant series}{section.129}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:sec:coefficients@cref\endcsname{{[section][1][13]13.1}{[1][40][]40}}
\expandafter\gdef\csname r@ch10:lem:positive-composition\endcsname{{13.1}{40}{}{lemma.130}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:lem:positive-composition@cref\endcsname{{[lemma][1][]13.1}{[1][40][]40}}
\expandafter\gdef\csname r@ch10:eq:row-to-column\endcsname{{13.2}{40}{}{equation.131}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:eq:row-to-column@cref\endcsname{{[equation][2][13]13.2}{[1][40][]40}}
\expandafter\gdef\csname r@ch10:eq:derivative-envelope\endcsname{{13.3}{41}{}{equation.132}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:eq:derivative-envelope@cref\endcsname{{[equation][3][13]13.3}{[1][40][]41}}
\expandafter\gdef\csname r@ch10:sec:gauge\endcsname{{13.2}{42}{Gauge representatives on the reconstruction domain}{section.133}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:sec:gauge@cref\endcsname{{[section][2][13]13.2}{[1][42][]42}}
\expandafter\gdef\csname r@ch10:thm:whole-image-gauge\endcsname{{13.2}{43}{}{theorem.135}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:thm:whole-image-gauge@cref\endcsname{{[theorem][2][13]13.2}{[1][43][]43}}
\expandafter\gdef\csname r@ch10:eq:forest-identification\endcsname{{13.4}{45}{Step 2: the zero-fluctuation reconstruction and the gauge}{equation.142}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:eq:forest-identification@cref\endcsname{{[equation][4][13]13.4}{[1][45][]45}}
\expandafter\gdef\csname r@ch10:sec:cutoff\endcsname{{13.3}{48}{Cutoffs and gauge invariance on overlapping charts}{section.148}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch10:thm:actual-compact-cut@cref\endcsname{{[theorem][4][13]13.4}{[1][49][]49}}
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\expandafter\gdef\csname r@ch10:eq:oscillation\endcsname{{13.6}{51}{Step 1: gluing and normalization}{equation.161}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch10:eq:literal-inheritance\endcsname{{13.7}{52}{Step 3: values of the input effective interaction}{equation.164}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:eq:literal-inheritance@cref\endcsname{{[equation][7][13]13.7}{[1][52][]52}}
\expandafter\gdef\csname r@ch10:sec:iteration\endcsname{{13.4}{54}{Preservation of analytic bounds under iteration}{section.168}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch10:thm:common-analytic-class@cref\endcsname{{[theorem][5][13]13.5}{[1][54][]54}}
\expandafter\gdef\csname r@ch10:eq:recursion-domain\endcsname{{13.8}{55}{}{equation.173}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch10:eq:hull-size\endcsname{{13.10}{56}{Step 1: rates, supports, and the order of choices}{equation.178}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:eq:hull-size@cref\endcsname{{[equation][10][13]13.10}{[1][56][]56}}
\expandafter\gdef\csname r@ch10:eq:root-count\endcsname{{13.11}{58}{Step 2: conversion at the same radius}{equation.180}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch10:eq:root-count@cref\endcsname{{[equation][11][13]13.11}{[1][58][]58}}
\expandafter\gdef\csname r@ch10:eq:inherited-expansion\endcsname{{13.12}{58}{Step 3: the coefficients of the composed input interaction}{equation.182}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch12:lem:box-assignment@cref\endcsname{{[lemma][3][]15.3}{[1][127][]127}}
\expandafter\gdef\csname r@ch12:eq:box-assignment-size\endcsname{{15.2}{127}{}{equation.395}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:box-assignment-size@cref\endcsname{{[equation][2][15]15.2}{[1][127][]127}}
\expandafter\gdef\csname r@ch12:eq:box-assignment-weight\endcsname{{15.3}{127}{}{equation.396}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:box-assignment-weight@cref\endcsname{{[equation][3][15]15.3}{[1][127][]127}}
\expandafter\gdef\csname r@ch12:eq:pre-J-spatial-gates\endcsname{{15.4}{128}{Order of the spatial choices}{equation.398}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch12:lem:product-reserves@cref\endcsname{{[lemma][4][]15.4}{[1][128][]128}}
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\expandafter\gdef\csname r@ch12:eq:actual-product-gates@cref\endcsname{{[equation][6][15]15.6}{[1][129][]129}}
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\expandafter\gdef\csname r@ch12:eq:marked-primitive@cref\endcsname{{[equation][10][15]15.10}{[1][134][]134}}
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\expandafter\gdef\csname r@ch12:eq:principal\endcsname{{15.13}{138}{Step 2: contraction of the pullback}{equation.423}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:principal@cref\endcsname{{[equation][13][15]15.13}{[1][138][]138}}
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\expandafter\gdef\csname r@ch12:eq:aux-disc@cref\endcsname{{[equation][14][15]15.14}{[1][138][]139}}
\expandafter\gdef\csname r@ch12:eq:marked-log\endcsname{{15.15}{139}{Step 3: the Lipschitz bound}{equation.426}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:marked-log@cref\endcsname{{[equation][15][15]15.15}{[1][139][]139}}
\expandafter\gdef\csname r@ch12:eq:two-powers\endcsname{{15.16}{139}{Step 3: the Lipschitz bound}{equation.427}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:two-powers@cref\endcsname{{[equation][16][15]15.16}{[1][139][]139}}
\expandafter\gdef\csname r@ch12:eq:outer-output\endcsname{{15.17}{141}{Step 5: convergence in the norm}{equation.430}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:outer-output@cref\endcsname{{[equation][17][15]15.17}{[1][141][]141}}
\expandafter\gdef\csname r@ch12:eq:suffix\endcsname{{15.18}{142}{Step 6: contraction over finitely many scales}{equation.432}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch12:eq:exponent@cref\endcsname{{[equation][19][15]15.19}{[1][142][]142}}
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\expandafter\gdef\csname r@ch12:sec:kernels@cref\endcsname{{[section][4][15]15.4}{[1][142][]142}}
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\expandafter\gdef\csname r@ch12:thm:white-kernel-limit@cref\endcsname{{[theorem][9][15]15.9}{[1][143][]143}}
\expandafter\gdef\csname r@ch12:lem:backward-law\endcsname{{15.10}{144}{}{lemma.437}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:lem:backward-law@cref\endcsname{{[lemma][10][]15.10}{[1][144][]144}}
\expandafter\gdef\csname r@ch12:eq:cap-iteration\endcsname{{15.20}{147}{Conditional fluctuation kernels and the coarse-level measure}{equation.441}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:cap-iteration@cref\endcsname{{[equation][20][15]15.20}{[1][146][]147}}
\expandafter\gdef\csname r@ch12:eq:calculation-18\endcsname{{15.21}{147}{Conditional fluctuation kernels and the coarse-level measure}{equation.442}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:calculation-18@cref\endcsname{{[equation][21][15]15.21}{[1][147][]147}}
\expandafter\gdef\csname r@ch12:rem:joint-haar-realization\endcsname{{15.11}{147}{}{remark.443}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:rem:joint-haar-realization@cref\endcsname{{[remark][11][]15.11}{[1][147][]147}}
\expandafter\gdef\csname r@ch12:lem:terminal-selector-tv\endcsname{{15.12}{147}{}{lemma.444}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:lem:terminal-selector-tv@cref\endcsname{{[lemma][12][]15.12}{[1][147][]147}}
\expandafter\gdef\csname r@ch12:sec:family\endcsname{{15.5}{148}{A fixed family of reconstruction maps}{section.445}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:sec:family@cref\endcsname{{[section][5][15]15.5}{[1][148][]148}}
\expandafter\gdef\csname r@ch12:eq:order-of-choices\endcsname{{15.22}{148}{A fixed family of reconstruction maps}{equation.446}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:order-of-choices@cref\endcsname{{[equation][22][15]15.22}{[1][148][]148}}
\expandafter\gdef\csname r@ch12:thm:replacement-family\endcsname{{15.13}{148}{}{theorem.447}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:thm:replacement-family@cref\endcsname{{[theorem][13][15]15.13}{[1][148][]148}}
\expandafter\gdef\csname r@ch12:eq:fixed-gauge-map\endcsname{{15.23}{151}{Step 2: the whole sequence in the fixed limiting gauge}{equation.455}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:fixed-gauge-map@cref\endcsname{{[equation][23][15]15.23}{[1][151][]151}}
\expandafter\gdef\csname r@ch12:eq:calculation-20\endcsname{{15.24}{152}{Step 2: the whole sequence in the fixed limiting gauge}{equation.456}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:calculation-20@cref\endcsname{{[equation][24][15]15.24}{[1][152][]152}}
\expandafter\gdef\csname r@ch12:eq:ordinary-mass\endcsname{{15.25}{154}{Step 4: preservation of componentwise integrals}{equation.459}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:ordinary-mass@cref\endcsname{{[equation][25][15]15.25}{[1][154][]154}}
\expandafter\gdef\csname r@ch12:eq:shell-mass\endcsname{{15.26}{154}{Step 4: preservation of componentwise integrals}{equation.460}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:shell-mass@cref\endcsname{{[equation][26][15]15.26}{[1][154][]154}}
\expandafter\gdef\csname r@ch12:eq:column-mass\endcsname{{15.27}{154}{Step 4: preservation of componentwise integrals}{equation.461}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:column-mass@cref\endcsname{{[equation][27][15]15.27}{[1][154][]154}}
\expandafter\gdef\csname r@ch12:sec:prefixes\endcsname{{15.6}{155}{Convergence of finite-scale reconstructions}{section.463}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:sec:prefixes@cref\endcsname{{[section][6][15]15.6}{[1][155][]155}}
\expandafter\gdef\csname r@ch12:thm:representative-prefixes\endcsname{{15.14}{155}{}{theorem.464}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:thm:representative-prefixes@cref\endcsname{{[theorem][14][15]15.14}{[1][155][]155}}
\expandafter\gdef\csname r@ch12:eq:replacement-overlap\endcsname{{15.28}{156}{Convergence of finite-scale reconstructions}{equation.468}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:replacement-overlap@cref\endcsname{{[equation][28][15]15.28}{[1][156][]156}}
\expandafter\gdef\csname r@ch12:eq:calculation-25\endcsname{{15.29}{156}{Convergence of finite-scale reconstructions}{equation.469}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:calculation-25@cref\endcsname{{[equation][29][15]15.29}{[1][156][]156}}
\expandafter\gdef\csname r@ch12:eq:calculation-26\endcsname{{15.30}{157}{Convergence of finite-scale reconstructions}{equation.470}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:calculation-26@cref\endcsname{{[equation][30][15]15.30}{[1][157][]157}}
\expandafter\gdef\csname r@ch12:eq:coordinate-tv\endcsname{{15.31}{157}{Convergence of finite-scale reconstructions}{equation.471}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:coordinate-tv@cref\endcsname{{[equation][31][15]15.31}{[1][157][]157}}
\expandafter\gdef\csname r@ch12:eq:calculation-28\endcsname{{15.32}{157}{Convergence of finite-scale reconstructions}{equation.472}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:calculation-28@cref\endcsname{{[equation][32][15]15.32}{[1][157][]157}}
\expandafter\gdef\csname r@ch12:sec:amplitude\endcsname{{15.7}{157}{Deterministic amplitude bounds for the reconstruction}{section.473}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:sec:amplitude@cref\endcsname{{[section][7][15]15.7}{[1][157][]157}}
\expandafter\gdef\csname r@ch12:lem:deterministic-prolongation-reserve\endcsname{{15.16}{158}{}{lemma.474}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:lem:deterministic-prolongation-reserve@cref\endcsname{{[lemma][16][]15.16}{[1][157][]158}}
\expandafter\gdef\csname r@ch12:eq:majorant-coeff\endcsname{{15.33}{159}{Step 1: the majorant}{equation.476}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:majorant-coeff@cref\endcsname{{[equation][33][15]15.33}{[1][159][]159}}
\expandafter\gdef\csname r@ch12:eq:uniform-free-row\endcsname{{15.34}{161}{Step 3: a uniform bound for the inputs and a first recursion}{equation.479}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch12:eq:forced-ceiling\endcsname{{15.36}{161}{Step 3: a uniform bound for the inputs and a first recursion}{equation.481}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch12:eq:forcing-sum\endcsname{{15.37}{161}{Step 3: a uniform bound for the inputs and a first recursion}{equation.482}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch12:eq:center-products\endcsname{{15.39}{162}{Step 4: the sharper ratio}{equation.485}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch12:eq:compact-coupling-outer-output\endcsname{{15.47}{166}{Step 5: the transformations}{equation.503}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch12:eq:compact-coupling-outer-output@cref\endcsname{{[equation][47][15]15.47}{[1][165][]166}}
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\expandafter\gdef\csname r@ch12:eq:compact-coupling-suffix@cref\endcsname{{[equation][48][15]15.48}{[1][166][]166}}
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\expandafter\gdef\csname r@ch12:eq:compact-coupling-kernel-potential\endcsname{{15.50}{167}{Step 7: kernels and finite-scale reconstructions}{equation.508}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch12:eq:compact-coupling-prefix-telescope\endcsname{{15.52}{167}{Step 7: kernels and finite-scale reconstructions}{equation.510}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@part:reconstruction\endcsname{{V}{170}{Uniformity on compact intervals of couplings}{part.511}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@part:reconstruction@cref\endcsname{{[part][5][]V}{[1][170][]170}}
\expandafter\gdef\csname r@ch13:chapter\endcsname{{16}{170}{Heat-kernel estimates, interpolation, and concentration}{chapter.512}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:chapter@cref\endcsname{{[chapter][16][]16}{[1][170][]170}}
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\expandafter\gdef\csname r@ch13:sec:concentration@cref\endcsname{{[section][4][16]16.4}{[1][185][]185}}
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\expandafter\gdef\csname r@ch13:eq:two-support-kernels@cref\endcsname{{[equation][15][16]16.15}{[1][185][]185}}
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\expandafter\gdef\csname r@ch13:lem:reconstruction-domain@cref\endcsname{{[lemma][5][]16.5}{[1][186][]187}}
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\expandafter\gdef\csname r@ch13:eq:reserved-product@cref\endcsname{{[equation][16][16]16.16}{[1][187][]187}}
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\expandafter\gdef\csname r@ch13:eq:actual-shell-gates@cref\endcsname{{[equation][17][16]16.17}{[1][187][]187}}
\expandafter\gdef\csname r@ch13:thm:centered-shell\endcsname{{16.6}{189}{}{theorem.563}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:thm:centered-shell@cref\endcsname{{[theorem][6][16]16.6}{[1][189][]189}}
\expandafter\gdef\csname r@ch13:rem:scope\endcsname{{16.7}{190}{}{remark.564}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:rem:scope@cref\endcsname{{[remark][7][]16.7}{[1][190][]190}}
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\expandafter\gdef\csname r@ch13:eq:shell-metric\endcsname{{16.24}{192}{Step 3: a fixed time}{equation.574}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch13:eq:cell-max@cref\endcsname{{[equation][26][16]16.26}{[1][193][]193}}
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\expandafter\gdef\csname r@ch13:eq:dyadic-heat@cref\endcsname{{[equation][27][16]16.27}{[1][193][]193}}
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\expandafter\gdef\csname r@ch13:eq:weighted-sigma@cref\endcsname{{[equation][28][16]16.28}{[1][193][]193}}
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\expandafter\gdef\csname r@ch13:rem:interpolant-derivative\endcsname{{16.10}{198}{}{remark.590}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch13:eq:generator-error@cref\endcsname{{[equation][33][16]16.33}{[1][199][]199}}
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\expandafter\gdef\csname r@ch13:eq:heat-duhamel@cref\endcsname{{[equation][34][16]16.34}{[1][199][]199}}
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\expandafter\gdef\csname r@ch13:eq:differentiated-data@cref\endcsname{{[equation][35][16]16.35}{[1][199][]199}}
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\expandafter\gdef\csname r@ch14:chapter@cref\endcsname{{[chapter][17][]17}{[1][201][]201}}
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\expandafter\gdef\csname r@ch15:thm:regional-gap-and-center\endcsname{{18.12}{254}{}{theorem.728}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:sec:bare\endcsname{{18.7}{262}{The first block integration}{section.752}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:crossing-completion\endcsname{{18.33}{263}{}{equation.759}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:crossing-completion@cref\endcsname{{[equation][33][18]18.33}{[1][263][]263}}
\expandafter\gdef\csname r@ch15:eq:actual-completion\endcsname{{18.34}{263}{}{equation.760}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:actual-completion@cref\endcsname{{[equation][34][18]18.34}{[1][263][]263}}
\expandafter\gdef\csname r@ch15:eq:minimal-varying\endcsname{{18.35}{263}{}{equation.762}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:minimal-varying@cref\endcsname{{[equation][35][18]18.35}{[1][263][]263}}
\expandafter\gdef\csname r@ch15:eq:bare-objective\endcsname{{18.36}{263}{}{equation.763}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-objective@cref\endcsname{{[equation][36][18]18.36}{[1][263][]263}}
\expandafter\gdef\csname r@ch15:eq:varying-restriction\endcsname{{18.37}{264}{The first block integration}{equation.764}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:varying-restriction@cref\endcsname{{[equation][37][18]18.37}{[1][264][]264}}
\expandafter\gdef\csname r@ch15:lem:bare-gap\endcsname{{18.14}{264}{}{lemma.765}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:lem:bare-gap@cref\endcsname{{[lemma][14][]18.14}{[1][264][]264}}
\expandafter\gdef\csname r@ch15:eq:bare-split\endcsname{{18.38}{264}{}{equation.767}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-split@cref\endcsname{{[equation][38][18]18.38}{[1][264][]264}}
\expandafter\gdef\csname r@ch15:eq:bare-identification\endcsname{{18.39}{264}{}{equation.768}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-identification@cref\endcsname{{[equation][39][18]18.39}{[1][264][]264}}
\expandafter\gdef\csname r@ch15:eq:bare-restriction\endcsname{{18.40}{265}{}{equation.770}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-restriction@cref\endcsname{{[equation][40][18]18.40}{[1][265][]265}}
\expandafter\gdef\csname r@ch15:eq:bare-cover\endcsname{{18.41}{265}{}{equation.771}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-cover@cref\endcsname{{[equation][41][18]18.41}{[1][265][]265}}
\expandafter\gdef\csname r@ch15:eq:bare-coercive\endcsname{{18.42}{265}{}{equation.772}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-coercive@cref\endcsname{{[equation][42][18]18.42}{[1][265][]265}}
\expandafter\gdef\csname r@ch15:eq:bare-hessian\endcsname{{18.43}{265}{}{equation.774}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-hessian@cref\endcsname{{[equation][43][18]18.43}{[1][265][]265}}
\expandafter\gdef\csname r@ch15:eq:bare-range\endcsname{{18.44}{265}{}{equation.775}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-range@cref\endcsname{{[equation][44][18]18.44}{[1][265][]265}}
\expandafter\gdef\csname r@ch15:eq:bare-small-data\endcsname{{18.45}{265}{}{equation.777}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-small-data@cref\endcsname{{[equation][45][18]18.45}{[1][265][]265}}
\expandafter\gdef\csname r@ch15:eq:wilson-expansion\endcsname{{18.46}{266}{The first block integration}{equation.778}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:wilson-expansion@cref\endcsname{{[equation][46][18]18.46}{[1][266][]266}}
\expandafter\gdef\csname r@ch15:eq:bare-schur\endcsname{{18.47}{267}{The first block integration}{equation.779}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-schur@cref\endcsname{{[equation][47][18]18.47}{[1][267][]267}}
\expandafter\gdef\csname r@ch15:eq:bare-neumann\endcsname{{18.48}{268}{The first block integration}{equation.780}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:bare-neumann@cref\endcsname{{[equation][48][18]18.48}{[1][268][]268}}
\expandafter\gdef\csname r@ch15:sec:networks\endcsname{{18.8}{269}{Decay of connected boundary perturbations}{section.781}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:sec:networks@cref\endcsname{{[section][8][18]18.8}{[1][269][]269}}
\expandafter\gdef\csname r@ch15:lem:wall-networks\endcsname{{18.15}{270}{}{lemma.782}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:lem:wall-networks@cref\endcsname{{[lemma][15][]18.15}{[1][270][]270}}
\expandafter\gdef\csname r@ch15:eq:input-output-envelope\endcsname{{18.49}{271}{}{equation.786}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:input-output-envelope@cref\endcsname{{[equation][49][18]18.49}{[1][270][]271}}
\expandafter\gdef\csname r@ch15:eq:live-hole-distance\endcsname{{18.50}{271}{Decay of connected boundary perturbations}{equation.787}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:live-hole-distance@cref\endcsname{{[equation][50][18]18.50}{[1][271][]271}}
\expandafter\gdef\csname r@ch15:eq:network-lengths\endcsname{{18.51}{271}{Decay of connected boundary perturbations}{equation.788}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:network-lengths@cref\endcsname{{[equation][51][18]18.51}{[1][271][]271}}
\expandafter\gdef\csname r@ch15:eq:weight-conversion\endcsname{{18.52}{272}{Decay of connected boundary perturbations}{equation.789}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:weight-conversion@cref\endcsname{{[equation][52][18]18.52}{[1][272][]272}}
\expandafter\gdef\csname r@ch15:eq:two-live-lengths\endcsname{{18.53}{272}{Decay of connected boundary perturbations}{equation.790}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:two-live-lengths@cref\endcsname{{[equation][53][18]18.53}{[1][272][]272}}
\expandafter\gdef\csname r@ch15:eq:two-index-weight\endcsname{{18.54}{272}{Decay of connected boundary perturbations}{equation.791}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:two-index-weight@cref\endcsname{{[equation][54][18]18.54}{[1][272][]272}}
\expandafter\gdef\csname r@ch15:eq:marked-majorant\endcsname{{18.55}{273}{Decay of connected boundary perturbations}{equation.792}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:marked-majorant@cref\endcsname{{[equation][55][18]18.55}{[1][273][]273}}
\expandafter\gdef\csname r@ch15:eq:marked-objective\endcsname{{18.56}{273}{Decay of connected boundary perturbations}{equation.793}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:marked-objective@cref\endcsname{{[equation][56][18]18.56}{[1][273][]273}}
% Part III
\expandafter\gdef\csname r@ch16:chapter\endcsname{{19}{2}{Comparison of reconstructions on block regions}{chapter.3}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:chapter@cref\endcsname{{[chapter][19][]19}{[1][2][]2}}
\expandafter\gdef\csname r@ch16:sec:setting\endcsname{{19.1}{3}{Setting and conventions}{section.6}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:sec:setting@cref\endcsname{{[section][1][19]19.1}{[1][3][]3}}
\expandafter\gdef\csname r@ch16:fig:geometry\endcsname{{19.1}{4}{The geometry in the output lattice (schematic). The solid inner rectangle is the fixed complement $H=\Omega ^c$, and its boundary is the interface $W$. The dashed rectangle bounds the enlargement $H_{L_J}$ of $H$ by the reconstruction stencil. The cutoff $\chi $ of \cref {ch16:lem:strict-hybrid-gauge} vanishes inside the inner dotted rectangle and equals one outside the outer dotted rectangle. The black dots are protected vertices, here the endpoints of the paths of two based transports of frozen coordinates. The small dotted rectangle inside $H$ is the complement of the incoming list $D_{\mathrm {in}}$ of \cref {ch16:thm:controlled-perturbation}, which lies within a fixed distance of $H$}{figure.caption.5}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:fig:geometry@cref\endcsname{{[figure][1][19]19.1}{[1][3][]4}}
\expandafter\gdef\csname r@ch16:eq:torus-reserve\endcsname{{19.1}{6}{The torus construction domain}{equation.12}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:torus-reserve@cref\endcsname{{[equation][1][19]19.1}{[1][6][]6}}
\expandafter\gdef\csname r@ch16:eq:hole-sum\endcsname{{19.2}{7}{Coefficient bounds}{equation.14}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:hole-sum@cref\endcsname{{[equation][2][19]19.2}{[1][7][]7}}
\expandafter\gdef\csname r@ch16:eq:implicit-majorant\endcsname{{19.3}{7}{Implicit series}{equation.16}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:implicit-majorant@cref\endcsname{{[equation][3][19]19.3}{[1][7][]7}}
\expandafter\gdef\csname r@ch16:eq:marked-majorant\endcsname{{19.4}{7}{Implicit series}{equation.17}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:marked-majorant@cref\endcsname{{[equation][4][19]19.4}{[1][7][]7}}
\expandafter\gdef\csname r@ch16:lem:matrix-functions\endcsname{{19.1}{8}{}{lemma.19}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:lem:matrix-functions@cref\endcsname{{[lemma][1][]19.1}{[1][8][]8}}
\expandafter\gdef\csname r@ch16:eq:resolvent-series\endcsname{{19.5}{8}{}{equation.22}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:resolvent-series@cref\endcsname{{[equation][5][19]19.5}{[1][8][]8}}
\expandafter\gdef\csname r@ch16:eq:sqrt\endcsname{{19.6}{8}{}{equation.24}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:sqrt@cref\endcsname{{[equation][6][19]19.6}{[1][8][]8}}
\expandafter\gdef\csname r@ch16:eq:sqrt-difference\endcsname{{19.7}{8}{}{equation.25}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:sqrt-difference@cref\endcsname{{[equation][7][19]19.7}{[1][8][]8}}
\expandafter\gdef\csname r@ch16:eq:log-difference\endcsname{{19.8}{8}{}{equation.26}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:log-difference@cref\endcsname{{[equation][8][19]19.8}{[1][8][]8}}
\expandafter\gdef\csname r@ch16:eq:block-inverse\endcsname{{19.9}{8}{}{equation.28}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:block-inverse@cref\endcsname{{[equation][9][19]19.9}{[1][8][]8}}
\expandafter\gdef\csname r@ch16:sec:conditional\endcsname{{19.2}{9}{Conditional minimization and successive substitution}{section.30}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:sec:conditional@cref\endcsname{{[section][2][19]19.2}{[1][9][]9}}
\expandafter\gdef\csname r@ch16:ex:conditional-domain\endcsname{{19.2}{9}{}{example.31}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:ex:conditional-domain@cref\endcsname{{[example][2][]19.2}{[1][9][]9}}
\expandafter\gdef\csname r@ch16:lem:admitted-products\endcsname{{19.3}{10}{}{lemma.32}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:lem:admitted-products@cref\endcsname{{[lemma][3][]19.3}{[1][10][]10}}
\expandafter\gdef\csname r@ch16:eq:conditional-force\endcsname{{19.10}{11}{Step 1: the unperturbed conditional stationary points}{equation.37}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:conditional-force@cref\endcsname{{[equation][10][19]19.10}{[1][11][]11}}
\expandafter\gdef\csname r@ch16:eq:conditional-variation\endcsname{{19.11}{11}{Step 1: the unperturbed conditional stationary points}{equation.38}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:conditional-variation@cref\endcsname{{[equation][11][19]19.11}{[1][11][]11}}
\expandafter\gdef\csname r@ch16:eq:preconditioned-map\endcsname{{19.12}{12}{Step 2: the perturbed conditional stationary points}{equation.40}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:preconditioned-map@cref\endcsname{{[equation][12][19]19.12}{[1][12][]12}}
\expandafter\gdef\csname r@ch16:eq:conditional-force-displacement\endcsname{{19.13}{13}{Step 3: the displacement at fixed coarser coordinates}{equation.42}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:conditional-force-displacement@cref\endcsname{{[equation][13][19]19.13}{[1][13][]13}}
\expandafter\gdef\csname r@ch16:eq:conditional-higher-lipschitz\endcsname{{19.14}{13}{Step 4: the Lipschitz bound in the coarser coordinates}{equation.44}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:conditional-higher-lipschitz@cref\endcsname{{[equation][14][19]19.14}{[1][13][]13}}
\expandafter\gdef\csname r@ch16:eq:sequential-recurrence\endcsname{{19.15}{14}{Step 5: successive substitution}{equation.46}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:sequential-recurrence@cref\endcsname{{[equation][15][19]19.15}{[1][14][]14}}
\expandafter\gdef\csname r@ch16:sec:comparison\endcsname{{19.3}{14}{Comparison of the regional and torus stationary points}{section.49}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:sec:comparison@cref\endcsname{{[section][3][19]19.3}{[1][14][]14}}
\expandafter\gdef\csname r@ch16:thm:nonflat-force-comparison\endcsname{{19.4}{15}{}{theorem.50}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:thm:nonflat-force-comparison@cref\endcsname{{[theorem][4][19]19.4}{[1][15][]15}}
\expandafter\gdef\csname r@ch16:eq:augmented-objectives\endcsname{{19.16}{16}{Step 1: one coordinate list}{equation.58}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:augmented-objectives@cref\endcsname{{[equation][16][19]19.16}{[1][16][]16}}
\expandafter\gdef\csname r@ch16:eq:action-insertion\endcsname{{19.17}{18}{Step 3: the inserted difference}{equation.61}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:action-insertion@cref\endcsname{{[equation][17][19]19.17}{[1][17][]18}}
\expandafter\gdef\csname r@ch16:eq:force-wall\endcsname{{19.18}{18}{Step 4: the gradient formula and the bounds on values}{equation.63}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:force-wall@cref\endcsname{{[equation][18][19]19.18}{[1][18][]18}}
\expandafter\gdef\csname r@ch16:eq:physical-response\endcsname{{19.19}{18}{Step 4: the gradient formula and the bounds on values}{equation.64}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:physical-response@cref\endcsname{{[equation][19][19]19.19}{[1][18][]18}}
\expandafter\gdef\csname r@ch16:eq:unperturbed-center-response\endcsname{{19.20}{19}{Step 5: the coefficient bounds}{equation.66}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:unperturbed-center-response@cref\endcsname{{[equation][20][19]19.20}{[1][19][]19}}
\expandafter\gdef\csname r@ch16:eq:target-change\endcsname{{19.21}{19}{Step 6: the full output}{equation.68}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:target-change@cref\endcsname{{[equation][21][19]19.21}{[1][19][]19}}
\expandafter\gdef\csname r@ch16:eq:value-envelope\endcsname{{19.22}{20}{Step 7: the minimized action}{equation.70}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:value-envelope@cref\endcsname{{[equation][22][19]19.22}{[1][20][]20}}
\expandafter\gdef\csname r@ch16:eq:value-tensors\endcsname{{19.23}{20}{Step 7: the minimized action}{equation.71}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:value-tensors@cref\endcsname{{[equation][23][19]19.23}{[1][20][]20}}
\expandafter\gdef\csname r@ch16:eq:root-fraction\endcsname{{19.24}{20}{Step 7: the minimized action}{equation.72}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:root-fraction@cref\endcsname{{[equation][24][19]19.24}{[1][20][]20}}
\expandafter\gdef\csname r@ch16:eq:full-mixed-hessian\endcsname{{19.25}{21}{Step 8: derivatives in the newly frozen coordinates}{equation.74}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:full-mixed-hessian@cref\endcsname{{[equation][25][19]19.25}{[1][21][]21}}
\expandafter\gdef\csname r@ch16:eq:hessian-interpolation\endcsname{{19.26}{21}{Step 9: the Hessians}{equation.76}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:hessian-interpolation@cref\endcsname{{[equation][26][19]19.26}{[1][21][]21}}
\expandafter\gdef\csname r@ch16:eq:coarea-factor\endcsname{{19.27}{22}{Step 12: logarithms of the Haar disintegration densities}{equation.80}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:coarea-factor@cref\endcsname{{[equation][27][19]19.27}{[1][22][]22}}
\expandafter\gdef\csname r@ch16:sec:gauge\endcsname{{19.4}{23}{A gauge transformation fixing prescribed vertices}{section.81}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:sec:gauge@cref\endcsname{{[section][4][19]19.4}{[1][23][]23}}
\expandafter\gdef\csname r@ch16:lem:strict-hybrid-gauge\endcsname{{19.5}{23}{}{lemma.82}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:lem:strict-hybrid-gauge@cref\endcsname{{[lemma][5][]19.5}{[1][23][]23}}
\expandafter\gdef\csname r@ch16:eq:gauge-log-majorant\endcsname{{19.28}{24}{A gauge transformation fixing prescribed vertices}{equation.87}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:gauge-log-majorant@cref\endcsname{{[equation][28][19]19.28}{[1][24][]24}}
\expandafter\gdef\csname r@ch16:sec:reconstruction\endcsname{{19.5}{26}{The gauge-transformed reconstruction and its input columns}{section.88}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:sec:reconstruction@cref\endcsname{{[section][5][19]19.5}{[1][25][]26}}
\expandafter\gdef\csname r@ch16:thm:hybrid-columns\endcsname{{19.6}{26}{}{theorem.89}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:thm:hybrid-columns@cref\endcsname{{[theorem][6][19]19.6}{[1][26][]26}}
\expandafter\gdef\csname r@ch16:eq:unperturbed-sequential-budget\endcsname{{19.29}{28}{Step 1: the domain of the successive substitutions}{equation.95}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:unperturbed-sequential-budget@cref\endcsname{{[equation][29][19]19.29}{[1][28][]28}}
\expandafter\gdef\csname r@ch16:eq:raw-shell-step\endcsname{{19.30}{28}{Step 2: one scale of the reconstruction}{equation.97}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:raw-shell-step@cref\endcsname{{[equation][30][19]19.30}{[1][28][]28}}
\expandafter\gdef\csname r@ch16:eq:hybrid-split\endcsname{{19.31}{29}{Step 4: the gauge transformation}{equation.100}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:hybrid-split@cref\endcsname{{[equation][31][19]19.31}{[1][29][]29}}
\expandafter\gdef\csname r@ch16:eq:record-integration\endcsname{{19.32}{30}{Step 7: fixed inputs and the newly frozen coordinates}{equation.104}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:record-integration@cref\endcsname{{[equation][32][19]19.32}{[1][30][]30}}
\expandafter\gdef\csname r@ch16:eq:column-local\endcsname{{19.33}{31}{Step 9: the input columns}{equation.107}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:column-local@cref\endcsname{{[equation][33][19]19.33}{[1][31][]31}}
\expandafter\gdef\csname r@ch16:sec:images\endcsname{{19.6}{31}{Construction radii and bounds on the reconstruction majorant}{section.108}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:sec:images@cref\endcsname{{[section][6][19]19.6}{[1][31][]31}}
\expandafter\gdef\csname r@ch16:lem:strict-two-radius-images\endcsname{{19.7}{32}{}{lemma.109}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:lem:strict-two-radius-images@cref\endcsname{{[lemma][7][]19.7}{[1][32][]32}}
\expandafter\gdef\csname r@ch16:eq:zero-exponent-image\endcsname{{19.34}{35}{Step 3: terms of positive degree in the increments}{equation.117}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:zero-exponent-image@cref\endcsname{{[equation][34][19]19.34}{[1][35][]35}}
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\expandafter\gdef\csname r@ch16:eq:perturbed-center-response@cref\endcsname{{[equation][48][19]19.48}{[1][47][]47}}
\expandafter\gdef\csname r@ch16:eq:schur-response\endcsname{{19.49}{47}{Step 9: the differences carry the perturbation}{equation.154}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:schur-response@cref\endcsname{{[equation][49][19]19.49}{[1][47][]47}}
\expandafter\gdef\csname r@ch16:eq:input-difference-norm\endcsname{{19.50}{48}{Step 10: two perturbations}{equation.156}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:input-difference-norm@cref\endcsname{{[equation][50][19]19.50}{[1][48][]48}}
\expandafter\gdef\csname r@ch16:eq:output-lipschitz\endcsname{{19.51}{48}{Step 10: two perturbations}{equation.157}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:output-lipschitz@cref\endcsname{{[equation][51][19]19.51}{[1][48][]48}}
\expandafter\gdef\csname r@ch17:chapter\endcsname{{20}{51}{Fluctuation integration and normalization on block regions}{chapter.161}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:chapter@cref\endcsname{{[chapter][20][]20}{[1][51][]51}}
\expandafter\gdef\csname r@ch17:sec:setting\endcsname{{20.1}{53}{Setting and coefficient calculus}{section.165}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:sec:setting@cref\endcsname{{[section][1][20]20.1}{[1][53][]53}}
\expandafter\gdef\csname r@ch17:eq:auxiliary-product-gate\endcsname{{20.1}{55}{Admissible data}{equation.170}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:auxiliary-product-gate@cref\endcsname{{[equation][1][20]20.1}{[1][55][]55}}
\expandafter\gdef\csname r@ch17:eq:density-sum\endcsname{{20.2}{56}{Density sums and boundary transport}{equation.173}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:density-sum@cref\endcsname{{[equation][2][20]20.2}{[1][56][]56}}
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\expandafter\gdef\csname r@ch17:eq:block-boundary-count\endcsname{{20.5}{57}{Density sums and boundary transport}{equation.176}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:block-boundary-count@cref\endcsname{{[equation][5][20]20.5}{[1][57][]57}}
\expandafter\gdef\csname r@ch17:eq:factorial-series\endcsname{{20.6}{57}{Two series}{equation.178}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:factorial-series@cref\endcsname{{[equation][6][20]20.6}{[1][57][]57}}
\expandafter\gdef\csname r@ch17:lem:coefficient-resolution\endcsname{{20.1}{58}{}{lemma.180}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:lem:coefficient-resolution@cref\endcsname{{[lemma][1][]20.1}{[1][58][]58}}
\expandafter\gdef\csname r@ch17:sec:hessians\endcsname{{20.2}{61}{Conditional minimizers, constrained Hessians, and Haar densities}{section.186}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:sec:hessians@cref\endcsname{{[section][2][20]20.2}{[1][60][]61}}
\expandafter\gdef\csname r@ch17:thm:perturbed-hessian-coarea\endcsname{{20.2}{61}{}{theorem.187}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:thm:perturbed-hessian-coarea@cref\endcsname{{[theorem][2][20]20.2}{[1][61][]61}}
\expandafter\gdef\csname r@ch17:eq:objective\endcsname{{20.7}{61}{}{equation.188}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:objective@cref\endcsname{{[equation][7][20]20.7}{[1][61][]61}}
\expandafter\gdef\csname r@ch17:eq:incoming-wall-gate\endcsname{{20.8}{61}{}{equation.189}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:incoming-wall-gate@cref\endcsname{{[equation][8][20]20.8}{[1][61][]61}}
\expandafter\gdef\csname r@ch17:eq:hessian-bounds\endcsname{{20.9}{62}{}{equation.191}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:hessian-bounds@cref\endcsname{{[equation][9][20]20.9}{[1][62][]62}}
\expandafter\gdef\csname r@ch17:eq:hessian-defect\endcsname{{20.10}{62}{}{equation.193}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:hessian-defect@cref\endcsname{{[equation][10][20]20.10}{[1][62][]62}}
\expandafter\gdef\csname r@ch17:eq:measure-fluctuation\endcsname{{20.11}{63}{}{equation.195}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:measure-fluctuation@cref\endcsname{{[equation][11][20]20.11}{[1][63][]63}}
\expandafter\gdef\csname r@ch17:eq:conditional-force-bound\endcsname{{20.12}{64}{Step 3: conditional minimizers}{equation.200}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:conditional-force-bound@cref\endcsname{{[equation][12][20]20.12}{[1][64][]64}}
\expandafter\gdef\csname r@ch17:eq:sequential-gate\endcsname{{20.13}{65}{Step 4: sequential substitution}{equation.202}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:sequential-gate@cref\endcsname{{[equation][13][20]20.13}{[1][65][]65}}
\expandafter\gdef\csname r@ch17:eq:sequential-displacement\endcsname{{20.14}{65}{Step 4: sequential substitution}{equation.203}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:sequential-displacement@cref\endcsname{{[equation][14][20]20.14}{[1][65][]65}}
\expandafter\gdef\csname r@ch17:eq:prefix-response\endcsname{{20.15}{66}{Step 5: dependence on the coarser coordinates}{equation.205}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:prefix-response@cref\endcsname{{[equation][15][20]20.15}{[1][66][]66}}
\expandafter\gdef\csname r@ch17:eq:action-word-difference\endcsname{{20.16}{66}{Step 6: the unperturbed boundary comparison}{equation.207}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:action-word-difference@cref\endcsname{{[equation][16][20]20.16}{[1][66][]66}}
\expandafter\gdef\csname r@ch17:eq:schur-hessian\endcsname{{20.17}{67}{Step 7: minimization, Schur complements, and restriction}{equation.209}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:schur-hessian@cref\endcsname{{[equation][17][20]20.17}{[1][67][]67}}
\expandafter\gdef\csname r@ch17:eq:restriction-elimination\endcsname{{20.18}{67}{Step 7: minimization, Schur complements, and restriction}{equation.210}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:restriction-elimination@cref\endcsname{{[equation][18][20]20.18}{[1][67][]67}}
\expandafter\gdef\csname r@ch17:eq:square-root\endcsname{{20.19}{68}{Step 10: matrix functions and Haar densities}{equation.214}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:square-root@cref\endcsname{{[equation][19][20]20.19}{[1][68][]68}}
\expandafter\gdef\csname r@ch17:eq:log-difference\endcsname{{20.20}{68}{Step 10: matrix functions and Haar densities}{equation.215}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:log-difference@cref\endcsname{{[equation][20][20]20.20}{[1][68][]68}}
\expandafter\gdef\csname r@ch17:eq:restricted-log\endcsname{{20.21}{69}{Step 10: matrix functions and Haar densities}{equation.216}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:restricted-log@cref\endcsname{{[equation][21][20]20.21}{[1][69][]69}}
\expandafter\gdef\csname r@ch17:eq:free-difference\endcsname{{20.22}{69}{Step 11: the triangular determinant}{equation.218}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:free-difference@cref\endcsname{{[equation][22][20]20.22}{[1][69][]69}}
\expandafter\gdef\csname r@ch17:sec:potential\endcsname{{20.3}{70}{The conditional fluctuation potential}{section.220}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:sec:potential@cref\endcsname{{[section][3][20]20.3}{[1][70][]70}}
\expandafter\gdef\csname r@ch17:lem:complete-white-potential\endcsname{{20.3}{70}{}{lemma.221}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:lem:complete-white-potential@cref\endcsname{{[lemma][3][]20.3}{[1][70][]70}}
\expandafter\gdef\csname r@ch17:eq:potential\endcsname{{20.23}{70}{}{equation.222}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:potential@cref\endcsname{{[equation][23][20]20.23}{[1][70][]70}}
\expandafter\gdef\csname r@ch17:eq:potential-series\endcsname{{20.24}{71}{}{equation.223}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:potential-series@cref\endcsname{{[equation][24][20]20.24}{[1][71][]71}}
\expandafter\gdef\csname r@ch17:eq:enclosure-budgets\endcsname{{20.25}{71}{}{equation.224}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:enclosure-budgets@cref\endcsname{{[equation][25][20]20.25}{[1][71][]71}}
\expandafter\gdef\csname r@ch17:eq:coupling-budgets\endcsname{{20.26}{72}{}{equation.225}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:coupling-budgets@cref\endcsname{{[equation][26][20]20.26}{[1][72][]72}}
\expandafter\gdef\csname r@ch17:eq:curvature\endcsname{{20.27}{72}{}{equation.226}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:curvature@cref\endcsname{{[equation][27][20]20.27}{[1][72][]72}}
\expandafter\gdef\csname r@ch17:eq:cubic-remainder\endcsname{{20.28}{73}{Step 1: the exact formula}{equation.228}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:cubic-remainder@cref\endcsname{{[equation][28][20]20.28}{[1][72][]73}}
\expandafter\gdef\csname r@ch17:eq:residual-jets\endcsname{{20.29}{73}{Step 2: the two series}{equation.230}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:residual-jets@cref\endcsname{{[equation][29][20]20.29}{[1][73][]73}}
\expandafter\gdef\csname r@ch17:rem:profile-jets\endcsname{{20.4}{76}{}{remark.235}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:rem:profile-jets@cref\endcsname{{[remark][4][]20.4}{[1][75][]76}}
\expandafter\gdef\csname r@ch17:lem:ambient-white-comparison\endcsname{{20.5}{76}{}{lemma.236}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:lem:ambient-white-comparison@cref\endcsname{{[lemma][5][]20.5}{[1][76][]76}}
\expandafter\gdef\csname r@ch17:eq:ambient-covariance\endcsname{{20.30}{78}{Step 3: convexity and concentration}{equation.243}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:ambient-covariance@cref\endcsname{{[equation][30][20]20.30}{[1][78][]78}}
\expandafter\gdef\csname r@ch17:eq:ambient-force\endcsname{{20.31}{79}{Step 4: the force}{equation.245}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:ambient-force@cref\endcsname{{[equation][31][20]20.31}{[1][79][]79}}
\expandafter\gdef\csname r@ch17:eq:mean-interpolation\endcsname{{20.32}{79}{Step 5: the means}{equation.247}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:mean-interpolation@cref\endcsname{{[equation][32][20]20.32}{[1][79][]79}}
\expandafter\gdef\csname r@ch17:sec:expansion\endcsname{{20.4}{80}{Cluster expansion with boundary-localized coefficients}{section.250}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:sec:expansion@cref\endcsname{{[section][4][20]20.4}{[1][80][]80}}
\expandafter\gdef\csname r@ch17:def:rooted-norms\endcsname{{20.6}{81}{}{definition.252}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:def:rooted-norms@cref\endcsname{{[definition][6][]20.6}{[1][81][]81}}
\expandafter\gdef\csname r@ch17:lem:decorated-white-primitives\endcsname{{20.7}{81}{}{lemma.253}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:lem:decorated-white-primitives@cref\endcsname{{[lemma][7][]20.7}{[1][81][]81}}
\expandafter\gdef\csname r@ch17:eq:decorated-inherited-gate\endcsname{{20.33}{82}{}{equation.254}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:decorated-inherited-gate@cref\endcsname{{[equation][33][20]20.33}{[1][82][]82}}
\expandafter\gdef\csname r@ch17:eq:cubic-primitive\endcsname{{20.34}{84}{Step 2: a classical monomial}{equation.257}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:cubic-primitive@cref\endcsname{{[equation][34][20]20.34}{[1][84][]84}}
\expandafter\gdef\csname r@ch17:lem:decorated-white-log\endcsname{{20.8}{89}{}{lemma.261}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:lem:decorated-white-log@cref\endcsname{{[lemma][8][]20.8}{[1][89][]89}}
\expandafter\gdef\csname r@ch17:eq:exact-white-two-mark\endcsname{{20.35}{89}{}{equation.265}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:exact-white-two-mark@cref\endcsname{{[equation][35][20]20.35}{[1][89][]89}}
\expandafter\gdef\csname r@ch17:eq:exact-white-tree\endcsname{{20.36}{90}{Step 2: the tree majorant at the first stage}{equation.269}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:exact-white-tree@cref\endcsname{{[equation][36][20]20.36}{[1][90][]90}}
\expandafter\gdef\csname r@ch17:eq:exact-anchor-union\endcsname{{20.37}{91}{Step 4: the networks}{equation.272}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:exact-anchor-union@cref\endcsname{{[equation][37][20]20.37}{[1][91][]91}}
\expandafter\gdef\csname r@ch17:thm:carried-networks\endcsname{{20.9}{93}{}{theorem.277}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:thm:carried-networks@cref\endcsname{{[theorem][9][20]20.9}{[1][93][]93}}
\expandafter\gdef\csname r@ch17:sec:normalization\endcsname{{20.5}{96}{Effective interactions and normalization constants}{section.278}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:sec:normalization@cref\endcsname{{[section][5][20]20.5}{[1][96][]96}}
\expandafter\gdef\csname r@ch17:thm:quantum-ledger\endcsname{{20.10}{96}{}{theorem.279}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:thm:quantum-ledger@cref\endcsname{{[theorem][10][20]20.10}{[1][96][]96}}
\expandafter\gdef\csname r@ch17:eq:auxiliary-factorization\endcsname{{20.38}{97}{Step 1: the exact factorization}{equation.281}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:auxiliary-factorization@cref\endcsname{{[equation][38][20]20.38}{[1][97][]97}}
\expandafter\gdef\csname r@ch17:eq:principal-torus-difference\endcsname{{20.39}{98}{Step 2: the generated term and the torus pullback}{equation.283}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:principal-torus-difference@cref\endcsname{{[equation][39][20]20.39}{[1][98][]98}}
\expandafter\gdef\csname r@ch17:eq:actual-column-ceiling\endcsname{{20.40}{98}{Step 3: the pullback of the boundary function}{equation.285}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:actual-column-ceiling@cref\endcsname{{[equation][40][20]20.40}{[1][98][]98}}
\expandafter\gdef\csname r@ch17:eq:pullback-column\endcsname{{20.41}{99}{Step 3: the pullback of the boundary function}{equation.286}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:pullback-column@cref\endcsname{{[equation][41][20]20.41}{[1][99][]99}}
\expandafter\gdef\csname r@ch17:eq:half-derivative\endcsname{{20.42}{99}{Step 3: the pullback of the boundary function}{equation.287}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:half-derivative@cref\endcsname{{[equation][42][20]20.42}{[1][99][]99}}
\expandafter\gdef\csname r@ch17:def:counts\endcsname{{20.11}{100}{}{definition.289}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:def:counts@cref\endcsname{{[definition][11][]20.11}{[1][100][]100}}
\expandafter\gdef\csname r@ch17:lem:real-cut-integral\endcsname{{20.12}{101}{}{lemma.294}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:lem:real-cut-integral@cref\endcsname{{[lemma][12][]20.12}{[1][100][]101}}
\expandafter\gdef\csname r@ch17:eq:real-vacuum-comparison\endcsname{{20.43}{101}{}{equation.295}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:real-vacuum-comparison@cref\endcsname{{[equation][43][20]20.43}{[1][101][]101}}
\expandafter\gdef\csname r@ch17:thm:real-scalar-ledger\endcsname{{20.13}{102}{}{theorem.296}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:thm:real-scalar-ledger@cref\endcsname{{[theorem][13][20]20.13}{[1][102][]102}}
\expandafter\gdef\csname r@ch17:eq:real-ledger-increment\endcsname{{20.44}{103}{}{equation.297}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:real-ledger-increment@cref\endcsname{{[equation][44][20]20.44}{[1][103][]103}}
\expandafter\gdef\csname r@ch17:eq:freezing-charge\endcsname{{20.45}{103}{}{equation.298}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:freezing-charge@cref\endcsname{{[equation][45][20]20.45}{[1][103][]103}}
\expandafter\gdef\csname r@ch17:sec:iteration\endcsname{{20.6}{105}{Preservation of the analytic bounds under iteration}{section.303}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:sec:iteration@cref\endcsname{{[section][6][20]20.6}{[1][105][]105}}
\expandafter\gdef\csname r@ch17:thm:controlled-kernel\endcsname{{20.14}{105}{}{theorem.304}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:thm:controlled-kernel@cref\endcsname{{[theorem][14][20]20.14}{[1][105][]105}}
\expandafter\gdef\csname r@ch17:tab:parameters\endcsname{{20.1}{108}{The order of the parameter choices in the proof of \cref {ch17:thm:controlled-kernel}. The requirements are stated in Steps 4 to 10}{table.caption.305}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:tab:parameters@cref\endcsname{{[table][1][20]20.1}{[1][107][]108}}
\expandafter\gdef\csname r@ch17:eq:logarithmic-envelope\endcsname{{20.46}{110}{Step 6: output bounds and the width of the frozen layer}{equation.312}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:logarithmic-envelope@cref\endcsname{{[equation][46][20]20.46}{[1][110][]110}}
\expandafter\gdef\csname r@ch17:eq:saved-straddle-budget\endcsname{{20.47}{111}{Step 7: the frozen values and the new boundary-completion differences}{equation.314}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:saved-straddle-budget@cref\endcsname{{[equation][47][20]20.47}{[1][111][]111}}
\expandafter\gdef\csname r@ch17:eq:uniform-logarithmic-gates\endcsname{{20.48}{112}{Step 10: the coupling bound}{equation.318}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:uniform-logarithmic-gates@cref\endcsname{{[equation][48][20]20.48}{[1][112][]112}}
\expandafter\gdef\csname r@ch17:eq:creation-straddle-bound\endcsname{{20.49}{113}{Step 11: the iteration}{equation.320}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:creation-straddle-bound@cref\endcsname{{[equation][49][20]20.49}{[1][113][]113}}
\expandafter\gdef\csname r@ch17:sec:endpoint\endcsname{{20.7}{114}{Summation of truncation errors}{section.322}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:sec:endpoint@cref\endcsname{{[section][7][20]20.7}{[1][114][]114}}
\expandafter\gdef\csname r@ch17:lem:finite-ceiling-endpoint-gates\endcsname{{20.15}{114}{}{lemma.323}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:lem:finite-ceiling-endpoint-gates@cref\endcsname{{[lemma][15][]20.15}{[1][114][]114}}
\expandafter\gdef\csname r@ch17:eq:white-ratio\endcsname{{20.50}{115}{Summation of truncation errors}{equation.327}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:white-ratio@cref\endcsname{{[equation][50][20]20.50}{[1][115][]115}}
\expandafter\gdef\csname r@ch17:eq:exterior-ratio\endcsname{{20.51}{115}{Summation of truncation errors}{equation.328}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:exterior-ratio@cref\endcsname{{[equation][51][20]20.51}{[1][115][]115}}
\expandafter\gdef\csname r@ch18:chapter\endcsname{{21}{117}{Large-field decompositions and conditional densities}{chapter.329}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:chapter@cref\endcsname{{[chapter][21][]21}{[1][117][]117}}
\expandafter\gdef\csname r@ch18:sec:local\endcsname{{21.1}{119}{Local path products and closed cell complexes}{section.331}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:sec:local@cref\endcsname{{[section][1][21]21.1}{[1][119][]119}}
\expandafter\gdef\csname r@ch18:lem:finite-stencils\endcsname{{21.1}{119}{}{lemma.332}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:lem:finite-stencils@cref\endcsname{{[lemma][1][]21.1}{[1][119][]119}}
\expandafter\gdef\csname r@ch18:cor:initial-joint-allowance\endcsname{{21.2}{121}{}{corollary.335}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:cor:initial-joint-allowance@cref\endcsname{{[corollary][2][]21.2}{[1][121][]121}}
\expandafter\gdef\csname r@ch18:thm:fat-cell-admission\endcsname{{21.3}{121}{}{theorem.336}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:thm:fat-cell-admission@cref\endcsname{{[theorem][3][21]21.3}{[1][121][]121}}
\expandafter\gdef\csname r@ch18:eq:axial-cell\endcsname{{21.1}{123}{Step 1: axial frames}{equation.342}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:axial-cell@cref\endcsname{{[equation][1][21]21.1}{[1][123][]123}}
\expandafter\gdef\csname r@ch18:eq:transition\endcsname{{21.2}{123}{Step 2: transitions}{equation.344}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:transition@cref\endcsname{{[equation][2][21]21.2}{[1][123][]123}}
\expandafter\gdef\csname r@ch18:eq:final-sum\endcsname{{21.3}{124}{Step 5: the cells}{equation.348}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:final-sum@cref\endcsname{{[equation][3][21]21.3}{[1][124][]124}}
\expandafter\gdef\csname r@ch18:eq:prescribed-increments\endcsname{{21.4}{124}{Step 6: the relative construction}{equation.350}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:prescribed-increments@cref\endcsname{{[equation][4][21]21.4}{[1][124][]124}}
\expandafter\gdef\csname r@ch18:sec:decompositions\endcsname{{21.2}{126}{Deterministic large-field decompositions}{section.354}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:sec:decompositions@cref\endcsname{{[section][2][21]21.2}{[1][126][]126}}
\expandafter\gdef\csname r@ch18:def:ordered-tests\endcsname{{21.4}{127}{}{definition.358}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:def:ordered-tests@cref\endcsname{{[definition][4][]21.4}{[1][127][]127}}
\expandafter\gdef\csname r@ch18:thm:history-cover\endcsname{{21.5}{128}{}{theorem.363}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:thm:history-cover@cref\endcsname{{[theorem][5][21]21.5}{[1][128][]128}}
\expandafter\gdef\csname r@ch18:lem:frozen-preserving\endcsname{{21.6}{130}{}{lemma.369}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:lem:frozen-preserving@cref\endcsname{{[lemma][6][]21.6}{[1][130][]130}}
\expandafter\gdef\csname r@ch18:eq:native-saving-coordinate\endcsname{{21.5}{131}{Step 2: constants for frozen coordinates}{equation.376}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:native-saving-coordinate@cref\endcsname{{[equation][5][21]21.5}{[1][131][]131}}
\expandafter\gdef\csname r@ch18:eq:actual-saving-gate\endcsname{{21.6}{132}{Step 2: constants for frozen coordinates}{equation.377}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:actual-saving-gate@cref\endcsname{{[equation][6][21]21.6}{[1][131][]132}}
\expandafter\gdef\csname r@ch18:eq:saving-constant\endcsname{{21.7}{133}{Step 4: small representatives}{equation.380}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:saving-constant@cref\endcsname{{[equation][7][21]21.7}{[1][133][]133}}
\expandafter\gdef\csname r@ch18:eq:all-saved-values\endcsname{{21.8}{133}{Step 4: small representatives}{equation.381}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:all-saved-values@cref\endcsname{{[equation][8][21]21.8}{[1][133][]133}}
\expandafter\gdef\csname r@ch18:eq:old-complement-inclusion\endcsname{{21.9}{134}{Step 5: incoming and current boundary distances}{equation.383}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:old-complement-inclusion@cref\endcsname{{[equation][9][21]21.9}{[1][134][]134}}
\expandafter\gdef\csname r@ch18:eq:history-wall-distance\endcsname{{21.10}{134}{Step 5: incoming and current boundary distances}{equation.384}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:history-wall-distance@cref\endcsname{{[equation][10][21]21.10}{[1][134][]134}}
\expandafter\gdef\csname r@ch18:eq:radius-recursion-proof\endcsname{{21.11}{135}{Step 6: locality and counts}{equation.386}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:radius-recursion-proof@cref\endcsname{{[equation][11][21]21.11}{[1][134][]135}}
\expandafter\gdef\csname r@ch18:eq:force-function\endcsname{{21.12}{137}{Cell grids}{equation.390}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:force-function@cref\endcsname{{[equation][12][21]21.12}{[1][137][]137}}
\expandafter\gdef\csname r@ch18:lem:history-input-contract\endcsname{{21.7}{137}{}{lemma.391}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:lem:history-input-contract@cref\endcsname{{[lemma][7][]21.7}{[1][137][]137}}
\expandafter\gdef\csname r@ch18:eq:conditional-force-check\endcsname{{21.13}{137}{}{equation.394}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:conditional-force-check@cref\endcsname{{[equation][13][21]21.13}{[1][137][]137}}
\expandafter\gdef\csname r@ch18:eq:prefix-sequential-check\endcsname{{21.14}{138}{}{equation.396}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:prefix-sequential-check@cref\endcsname{{[equation][14][21]21.14}{[1][138][]138}}
\expandafter\gdef\csname r@ch18:eq:uniform-survivor-bound\endcsname{{21.15}{138}{Step 1: uniform radius on the remaining labels}{equation.399}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:uniform-survivor-bound@cref\endcsname{{[equation][15][21]21.15}{[1][138][]138}}
\expandafter\gdef\csname r@ch18:eq:total-creation-straddle\endcsname{{21.16}{139}{Step 2: new boundary-completion differences}{equation.401}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:total-creation-straddle@cref\endcsname{{[equation][16][21]21.16}{[1][139][]139}}
\expandafter\gdef\csname r@ch18:eq:two-input-image-check\endcsname{{21.17}{140}{Step 4: the image conditions}{equation.404}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:two-input-image-check@cref\endcsname{{[equation][17][21]21.17}{[1][140][]140}}
\expandafter\gdef\csname r@ch18:eq:ambient-physical-gates\endcsname{{21.18}{141}{Step 5: convexity on the enclosing product domain}{equation.406}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:ambient-physical-gates@cref\endcsname{{[equation][18][21]21.18}{[1][141][]141}}
\expandafter\gdef\csname r@ch18:eq:logarithmic-product-gates\endcsname{{21.19}{141}{Step 6: the logarithmic cutoff domain}{equation.408}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:logarithmic-product-gates@cref\endcsname{{[equation][19][21]21.19}{[1][141][]141}}
\expandafter\gdef\csname r@ch18:sec:coverage\endcsname{{21.3}{142}{Coordinate coverage and representatives satisfying the reconstruction identities}{section.409}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:sec:coverage@cref\endcsname{{[section][3][21]21.3}{[1][142][]142}}
\expandafter\gdef\csname r@ch18:lem:relative-output-admission\endcsname{{21.8}{143}{}{lemma.411}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:lem:relative-output-admission@cref\endcsname{{[lemma][8][]21.8}{[1][143][]143}}
\expandafter\gdef\csname r@ch18:thm:all-point-enclosure\endcsname{{21.9}{144}{}{theorem.412}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:thm:all-point-enclosure@cref\endcsname{{[theorem][9][21]21.9}{[1][144][]144}}
\expandafter\gdef\csname r@ch18:eq:small-witness-iterates\endcsname{{21.20}{147}{Step 1: a smallness gauge for an arbitrary fiber point}{equation.418}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:small-witness-iterates@cref\endcsname{{[equation][20][21]21.20}{[1][147][]147}}
\expandafter\gdef\csname r@ch18:eq:root-increment\endcsname{{21.21}{147}{Step 2: the root values}{equation.420}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:root-increment@cref\endcsname{{[equation][21][21]21.21}{[1][147][]147}}
\expandafter\gdef\csname r@ch18:eq:relative-color\endcsname{{21.22}{148}{Step 4: transport to representatives produced from above}{equation.423}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:relative-color@cref\endcsname{{[equation][22][21]21.22}{[1][148][]148}}
\expandafter\gdef\csname r@ch18:eq:unperturbed-covariance\endcsname{{21.23}{149}{Step 4: transport to representatives produced from above}{equation.424}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:unperturbed-covariance@cref\endcsname{{[equation][23][21]21.23}{[1][149][]149}}
\expandafter\gdef\csname r@ch18:eq:center-covariance\endcsname{{21.24}{150}{Step 4: transport to representatives produced from above}{equation.425}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:center-covariance@cref\endcsname{{[equation][24][21]21.24}{[1][149][]150}}
\expandafter\gdef\csname r@ch18:eq:unperturbed-distance\endcsname{{21.25}{150}{Step 4: transport to representatives produced from above}{equation.426}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:unperturbed-distance@cref\endcsname{{[equation][25][21]21.25}{[1][150][]150}}
\expandafter\gdef\csname r@ch18:eq:controlled-enclosure-proof\endcsname{{21.26}{151}{Step 5: the perturbation}{equation.428}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:controlled-enclosure-proof@cref\endcsname{{[equation][26][21]21.26}{[1][151][]151}}
\expandafter\gdef\csname r@ch18:sec:densities\endcsname{{21.4}{155}{Pointwise gauge-invariant densities}{section.433}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:sec:densities@cref\endcsname{{[section][4][21]21.4}{[1][155][]155}}
\expandafter\gdef\csname r@ch18:thm:canonical-density\endcsname{{21.10}{155}{}{theorem.434}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:thm:canonical-density@cref\endcsname{{[theorem][10][21]21.10}{[1][155][]155}}
\expandafter\gdef\csname r@ch18:eq:canonical-local-density\endcsname{{21.27}{157}{Step 3: the outgoing density on a chart}{equation.438}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:canonical-local-density@cref\endcsname{{[equation][27][21]21.27}{[1][157][]157}}
\expandafter\gdef\csname r@ch18:eq:pointwise-raw-covariance\endcsname{{21.28}{158}{Step 4: pointwise covariance}{equation.440}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:pointwise-raw-covariance@cref\endcsname{{[equation][28][21]21.28}{[1][158][]158}}
\expandafter\gdef\csname r@ch18:eq:anchored-endpoint-cancellation\endcsname{{21.29}{158}{Step 5: freezing}{equation.442}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:anchored-endpoint-cancellation@cref\endcsname{{[equation][29][21]21.29}{[1][158][]158}}
\expandafter\gdef\csname r@ch18:sec:freezing\endcsname{{21.5}{160}{Freezing and substitution of frozen coordinates}{section.445}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:sec:freezing@cref\endcsname{{[section][5][21]21.5}{[1][160][]160}}
\expandafter\gdef\csname r@ch18:lem:producer-consumer\endcsname{{21.11}{160}{}{lemma.446}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:lem:producer-consumer@cref\endcsname{{[lemma][11][]21.11}{[1][160][]160}}
\expandafter\gdef\csname r@ch18:eq:full-freeze-identity\endcsname{{21.30}{161}{Freezing and substitution of frozen coordinates}{equation.447}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:full-freeze-identity@cref\endcsname{{[equation][30][21]21.30}{[1][161][]161}}
\expandafter\gdef\csname r@ch18:sec:induction\endcsname{{21.6}{162}{Conditional integration and normalization on large-field decompositions}{section.448}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:sec:induction@cref\endcsname{{[section][6][21]21.6}{[1][162][]162}}
\expandafter\gdef\csname r@ch18:lem:enclosure-comparison\endcsname{{21.12}{162}{}{lemma.449}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:lem:enclosure-comparison@cref\endcsname{{[lemma][12][]21.12}{[1][162][]162}}
\expandafter\gdef\csname r@ch18:eq:implemented-white-product\endcsname{{21.31}{162}{}{equation.450}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:implemented-white-product@cref\endcsname{{[equation][31][21]21.31}{[1][162][]162}}
\expandafter\gdef\csname r@ch18:eq:comparison-hypothesis\endcsname{{21.32}{162}{}{equation.451}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:comparison-hypothesis@cref\endcsname{{[equation][32][21]21.32}{[1][162][]162}}
\expandafter\gdef\csname r@ch18:eq:same-prefactor\endcsname{{21.33}{163}{Conditional integration and normalization on large-field decompositions}{equation.452}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:same-prefactor@cref\endcsname{{[equation][33][21]21.33}{[1][163][]163}}
\expandafter\gdef\csname r@ch18:eq:ambient-finite-normalizer\endcsname{{21.34}{163}{Conditional integration and normalization on large-field decompositions}{equation.453}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:ambient-finite-normalizer@cref\endcsname{{[equation][34][21]21.34}{[1][163][]163}}
\expandafter\gdef\csname r@ch18:thm:slice-induction\endcsname{{21.13}{164}{}{theorem.454}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:thm:slice-induction@cref\endcsname{{[theorem][13][21]21.13}{[1][164][]164}}
\expandafter\gdef\csname r@ch18:eq:initial-scalar\endcsname{{21.35}{167}{Part (1): initialization}{equation.461}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:initial-scalar@cref\endcsname{{[equation][35][21]21.35}{[1][167][]167}}
\expandafter\gdef\csname r@ch18:eq:reference-determinant\endcsname{{21.36}{169}{Part (1): the analytic recursion}{equation.463}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:reference-determinant@cref\endcsname{{[equation][36][21]21.36}{[1][169][]169}}
\expandafter\gdef\csname r@ch18:eq:analytic-output-definition\endcsname{{21.37}{170}{Part (1): the analytic recursion}{equation.464}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:analytic-output-definition@cref\endcsname{{[equation][37][21]21.37}{[1][169][]170}}
\expandafter\gdef\csname r@ch18:eq:auxiliary-output\endcsname{{21.38}{170}{Part (1): the analytic recursion}{equation.465}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:auxiliary-output@cref\endcsname{{[equation][38][21]21.38}{[1][170][]170}}
\expandafter\gdef\csname r@ch18:eq:one-triangular-determinant\endcsname{{21.39}{172}{Part (3): density bounds}{equation.468}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:one-triangular-determinant@cref\endcsname{{[equation][39][21]21.39}{[1][172][]172}}
\expandafter\gdef\csname r@ch18:eq:positive-induction-chain\endcsname{{21.40}{173}{Part (3): density bounds}{equation.469}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:positive-induction-chain@cref\endcsname{{[equation][40][21]21.40}{[1][173][]173}}
\expandafter\gdef\csname r@ch18:eq:terminal-source-pullback\endcsname{{21.41}{175}{The remaining averaging steps}{equation.472}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch18:eq:terminal-source-pullback@cref\endcsname{{[equation][41][21]21.41}{[1][175][]175}}
\expandafter\gdef\csname r@ch19:chapter\endcsname{{22}{177}{Exponential-moment bounds between renormalization scales}{chapter.474}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:chapter@cref\endcsname{{[chapter][22][]22}{[1][177][]177}}
\expandafter\gdef\csname r@ch19:sec:setting\endcsname{{22.1}{177}{Setting and the main estimate}{section.475}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:sec:setting@cref\endcsname{{[section][1][22]22.1}{[1][177][]177}}
\expandafter\gdef\csname r@ch19:eq:angle-cost\endcsname{{22.1}{178}{Plaquette energies}{equation.478}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:angle-cost@cref\endcsname{{[equation][1][22]22.1}{[1][178][]178}}
\expandafter\gdef\csname r@ch19:eq:cell-source\endcsname{{22.2}{178}{The plaquette-energy sum}{equation.480}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:cell-source@cref\endcsname{{[equation][2][22]22.2}{[1][178][]178}}
\expandafter\gdef\csname r@ch19:tab:measures\endcsname{{22.1}{180}{Measures used in the proof of \cref {ch19:thm:all-phase-window}}{table.caption.485}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:tab:measures@cref\endcsname{{[table][1][22]22.1}{[1][179][]180}}
\expandafter\gdef\csname r@ch19:thm:all-phase-window\endcsname{{22.1}{180}{}{theorem.489}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:thm:all-phase-window@cref\endcsname{{[theorem][1][22]22.1}{[1][180][]180}}
\expandafter\gdef\csname r@ch19:fig:sets\endcsname{{22.1}{183}{Schematic picture of the sets attached to a recorded bad plaquette $f$, drawn at one scale. The energy-estimate predecessor set $\mathcal B_f$ carries the lower bound for the action. The source plaquette $p_1$ is discarded because its full dependence set $D(p_1)$ meets the excluded neighborhood of $f$; the source plaquette $p_2$ is kept because $D(p_2)$ consists only of links of the block region at the stopping level}{figure.caption.492}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:fig:sets@cref\endcsname{{[figure][1][22]22.1}{[1][183][]183}}
\expandafter\gdef\csname r@ch19:sec:products\endcsname{{22.2}{184}{Product domains along a large-field decomposition}{section.493}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:sec:products@cref\endcsname{{[section][2][22]22.2}{[1][183][]184}}
\expandafter\gdef\csname r@ch19:lem:covered-history-admission\endcsname{{22.3}{184}{}{lemma.494}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:lem:covered-history-admission@cref\endcsname{{[lemma][3][]22.3}{[1][184][]184}}
\expandafter\gdef\csname r@ch19:eq:implemented-history-product\endcsname{{22.3}{184}{}{equation.495}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:implemented-history-product@cref\endcsname{{[equation][3][22]22.3}{[1][184][]184}}
\expandafter\gdef\csname r@ch19:eq:logarithmic-product-gates\endcsname{{22.4}{185}{Product domains along a large-field decomposition}{equation.496}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:logarithmic-product-gates@cref\endcsname{{[equation][4][22]22.4}{[1][185][]185}}
\expandafter\gdef\csname r@ch19:eq:ambient-finite-normalizer\endcsname{{22.5}{185}{Product domains along a large-field decomposition}{equation.497}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:ambient-finite-normalizer@cref\endcsname{{[equation][5][22]22.5}{[1][185][]185}}
\expandafter\gdef\csname r@ch19:eq:history-freeze-gates\endcsname{{22.6}{186}{Product domains along a large-field decomposition}{equation.498}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:history-freeze-gates@cref\endcsname{{[equation][6][22]22.6}{[1][186][]186}}
\expandafter\gdef\csname r@ch19:eq:history-conditional-force\endcsname{{22.7}{186}{Product domains along a large-field decomposition}{equation.499}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:history-conditional-force@cref\endcsname{{[equation][7][22]22.7}{[1][186][]186}}
\expandafter\gdef\csname r@ch19:sec:energy\endcsname{{22.3}{187}{Energy transport, feasible configurations, and the reference normalization}{section.500}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:sec:energy@cref\endcsname{{[section][3][22]22.3}{[1][187][]187}}
\expandafter\gdef\csname r@ch19:thm:classical-source-payment\endcsname{{22.4}{187}{}{theorem.501}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:thm:classical-source-payment@cref\endcsname{{[theorem][4][22]22.4}{[1][187][]187}}
\expandafter\gdef\csname r@ch19:eq:cone-transport\endcsname{{22.8}{188}{}{Item.502}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:cone-transport@cref\endcsname{{[equation][8][22]22.8}{[1][188][]188}}
\expandafter\gdef\csname r@ch19:eq:two-lower-bounds\endcsname{{22.9}{189}{}{Item.503}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:two-lower-bounds@cref\endcsname{{[equation][9][22]22.9}{[1][189][]189}}
\expandafter\gdef\csname r@ch19:eq:reference-identity\endcsname{{22.10}{189}{}{Item.504}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:reference-identity@cref\endcsname{{[equation][10][22]22.10}{[1][189][]189}}
\expandafter\gdef\csname r@ch19:rem:surface-term\endcsname{{22.5}{190}{}{remark.505}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:rem:surface-term@cref\endcsname{{[remark][5][]22.5}{[1][190][]190}}
\expandafter\gdef\csname r@ch19:eq:cone-bootstrap\endcsname{{22.11}{191}{Energy transport, feasible configurations, and the reference normalization}{equation.506}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:cone-bootstrap@cref\endcsname{{[equation][11][22]22.11}{[1][191][]191}}
\expandafter\gdef\csname r@ch19:eq:full-value-step\endcsname{{22.12}{192}{Energy transport, feasible configurations, and the reference normalization}{equation.507}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:full-value-step@cref\endcsname{{[equation][12][22]22.12}{[1][192][]192}}
\expandafter\gdef\csname r@ch19:eq:base-floor\endcsname{{22.13}{193}{Energy transport, feasible configurations, and the reference normalization}{equation.508}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:base-floor@cref\endcsname{{[equation][13][22]22.13}{[1][193][]193}}
\expandafter\gdef\csname r@ch19:sec:completion\endcsname{{22.4}{195}{The minimum of the Wilson action over omitted links}{section.509}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:sec:completion@cref\endcsname{{[section][4][22]22.4}{[1][195][]195}}
\expandafter\gdef\csname r@ch19:lem:wilson-minimum\endcsname{{22.6}{196}{}{lemma.510}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:lem:wilson-minimum@cref\endcsname{{[lemma][6][]22.6}{[1][196][]196}}
\expandafter\gdef\csname r@ch19:lem:compact-wilson-completion\endcsname{{22.7}{196}{}{lemma.511}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:lem:compact-wilson-completion@cref\endcsname{{[lemma][7][]22.7}{[1][196][]196}}
\expandafter\gdef\csname r@ch19:sec:source\endcsname{{22.5}{198}{Domination of the plaquette-energy sum by the Wilson minimum}{section.512}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:sec:source@cref\endcsname{{[section][5][22]22.5}{[1][198][]198}}
\expandafter\gdef\csname r@ch19:lem:compact-wilson-source-cap\endcsname{{22.8}{198}{}{lemma.513}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:lem:compact-wilson-source-cap@cref\endcsname{{[lemma][8][]22.8}{[1][198][]198}}
\expandafter\gdef\csname r@ch19:eq:arbitrary-completion-source-cap\endcsname{{22.14}{198}{}{equation.514}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:arbitrary-completion-source-cap@cref\endcsname{{[equation][14][22]22.14}{[1][198][]198}}
\expandafter\gdef\csname r@ch19:eq:compact-wilson-source-cap\endcsname{{22.15}{198}{}{equation.515}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:compact-wilson-source-cap@cref\endcsname{{[equation][15][22]22.15}{[1][198][]198}}
\expandafter\gdef\csname r@ch19:eq:source-cap-pushforward\endcsname{{22.16}{199}{}{equation.516}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:source-cap-pushforward@cref\endcsname{{[equation][16][22]22.16}{[1][198][]199}}
\expandafter\gdef\csname r@ch19:lem:all-face-source-cap\endcsname{{22.9}{201}{}{corollary.523}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:lem:all-face-source-cap@cref\endcsname{{[corollary][9][]22.9}{[1][201][]201}}
\expandafter\gdef\csname r@ch19:sec:envelope\endcsname{{22.6}{202}{Gauge-invariant domination of the densities}{section.524}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:sec:envelope@cref\endcsname{{[section][6][22]22.6}{[1][202][]202}}
\expandafter\gdef\csname r@ch19:thm:invariant-wilson-envelope\endcsname{{22.10}{202}{}{theorem.525}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:thm:invariant-wilson-envelope@cref\endcsname{{[theorem][10][22]22.10}{[1][202][]202}}
\expandafter\gdef\csname r@ch19:eq:completed-source-envelope\endcsname{{22.17}{203}{}{equation.526}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:completed-source-envelope@cref\endcsname{{[equation][17][22]22.17}{[1][203][]203}}
\expandafter\gdef\csname r@ch19:sec:summation\endcsname{{22.7}{206}{Summation over large-field decompositions}{section.534}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:sec:summation@cref\endcsname{{[section][7][22]22.7}{[1][206][]206}}
\expandafter\gdef\csname r@ch19:eq:window-endpoint-gates\endcsname{{22.18}{207}{Step 1: ceilings and the minimum side}{equation.536}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:window-endpoint-gates@cref\endcsname{{[equation][18][22]22.18}{[1][207][]207}}
\expandafter\gdef\csname r@ch19:eq:aligned-source-pullback\endcsname{{22.19}{207}{Step 2: the stopping level}{equation.538}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:aligned-source-pullback@cref\endcsname{{[equation][19][22]22.19}{[1][207][]207}}
\expandafter\gdef\csname r@ch19:eq:label-log\endcsname{{22.20}{208}{Step 4: assignment of the corrections}{equation.541}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:label-log@cref\endcsname{{[equation][20][22]22.20}{[1][208][]208}}
\expandafter\gdef\csname r@ch19:tab:activation-charges\endcsname{{22.2}{209}{All corrections assigned to one recorded bad plaquette $f$ created at side $n_f$. Here $L_f=1+\log x_f$, and the later RG steps include empty steps}{table.caption.542}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:tab:activation-charges@cref\endcsname{{[table][2][22]22.2}{[1][208][]209}}
\expandafter\gdef\csname r@ch19:eq:charge-budget\endcsname{{22.21}{210}{Step 4: assignment of the corrections}{equation.543}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:charge-budget@cref\endcsname{{[equation][21][22]22.21}{[1][210][]210}}
\expandafter\gdef\csname r@ch19:eq:history-moment\endcsname{{22.22}{210}{Step 5: absorption and the bound for one decomposition}{equation.545}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:history-moment@cref\endcsname{{[equation][22][22]22.22}{[1][210][]210}}
\expandafter\gdef\csname r@ch19:sec:chessboard\endcsname{{22.8}{211}{Chessboard estimates with regularity indicators}{section.548}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:sec:chessboard@cref\endcsname{{[section][8][22]22.8}{[1][211][]211}}
\expandafter\gdef\csname r@ch19:lem:marked-chessboard\endcsname{{22.11}{212}{}{lemma.549}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:lem:marked-chessboard@cref\endcsname{{[lemma][11][]22.11}{[1][212][]212}}
\expandafter\gdef\csname r@ch19:sec:prefix\endcsname{{22.9}{214}{Domination of successive small-field restrictions}{section.550}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:sec:prefix@cref\endcsname{{[section][9][22]22.9}{[1][213][]214}}
\expandafter\gdef\csname r@ch19:lem:original-prefix-onsets\endcsname{{22.12}{214}{}{lemma.551}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:lem:original-prefix-onsets@cref\endcsname{{[lemma][12][]22.12}{[1][214][]214}}
\expandafter\gdef\csname r@ch19:eq:prefix-positivity-gates\endcsname{{22.23}{216}{Step 2: the common minimum side}{equation.554}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:prefix-positivity-gates@cref\endcsname{{[equation][23][22]22.23}{[1][216][]216}}
\expandafter\gdef\csname r@ch19:eq:prefix-common-ratio\endcsname{{22.24}{217}{Step 4: one RG step}{equation.557}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:eq:prefix-common-ratio@cref\endcsname{{[equation][24][22]22.24}{[1][217][]217}}
\expandafter\gdef\csname r@ch19:lem:finite-ceiling-windows-and-prefixes\endcsname{{22.13}{219}{}{lemma.562}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch19:lem:finite-ceiling-windows-and-prefixes@cref\endcsname{{[lemma][13][]22.13}{[1][219][]219}}
\expandafter\gdef\csname r@ch20:chapter\endcsname{{23}{221}{Uniform bounds for the small-field event}{chapter.563}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:chapter@cref\endcsname{{[chapter][23][]23}{[1][221][]221}}
\expandafter\gdef\csname r@ch20:eq:intro-bound\endcsname{{23.1}{221}{Uniform bounds for the small-field event}{equation.564}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:intro-bound@cref\endcsname{{[equation][1][23]23.1}{[1][221][]221}}
\expandafter\gdef\csname r@ch20:sec:elementary\endcsname{{23.1}{222}{Elementary inequalities}{section.565}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:sec:elementary@cref\endcsname{{[section][1][23]23.1}{[1][222][]222}}
\expandafter\gdef\csname r@ch20:lem:scalar-gates\endcsname{{23.1}{222}{}{lemma.566}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:lem:scalar-gates@cref\endcsname{{[lemma][1][]23.1}{[1][222][]222}}
\expandafter\gdef\csname r@ch20:eq:tail-ratios\endcsname{{23.2}{223}{Elementary inequalities}{equation.567}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:tail-ratios@cref\endcsname{{[equation][2][23]23.2}{[1][223][]223}}
\expandafter\gdef\csname r@ch20:sec:parameters\endcsname{{23.2}{223}{Choice of the parameters}{section.568}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:sec:parameters@cref\endcsname{{[section][2][23]23.2}{[1][223][]223}}
\expandafter\gdef\csname r@ch20:thm:common-parameters\endcsname{{23.2}{224}{}{theorem.569}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:thm:common-parameters@cref\endcsname{{[theorem][2][23]23.2}{[1][224][]224}}
\expandafter\gdef\csname r@ch20:eq:entropy-per-cell\endcsname{{23.3}{230}{}{equation.580}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:entropy-per-cell@cref\endcsname{{[equation][3][23]23.3}{[1][230][]230}}
\expandafter\gdef\csname r@ch20:eq:center-majorant\endcsname{{23.4}{231}{}{equation.581}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:center-majorant@cref\endcsname{{[equation][4][23]23.4}{[1][231][]231}}
\expandafter\gdef\csname r@ch20:rem:flow-exponents\endcsname{{23.3}{231}{}{remark.583}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:rem:flow-exponents@cref\endcsname{{[remark][3][]23.3}{[1][231][]231}}
\expandafter\gdef\csname r@ch20:tab:parameter-order\endcsname{{23.1}{232}{The order of the choices in \cref {ch20:thm:common-parameters}. Each row uses only the quantities of the earlier rows and the constants of the cited results, evaluated on their larger construction domains. The amplitude $A_{\mathrm w}$ may be fixed at any stage before item (8)}{table.caption.582}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:tab:parameter-order@cref\endcsname{{[table][1][23]23.1}{[1][231][]232}}
\expandafter\gdef\csname r@ch20:lem:unrestricted-window-onsets\endcsname{{23.4}{232}{}{lemma.584}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:lem:unrestricted-window-onsets@cref\endcsname{{[lemma][4][]23.4}{[1][232][]232}}
\expandafter\gdef\csname r@ch20:thm:finite-ceiling-parameters\endcsname{{23.5}{233}{}{theorem.585}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:thm:finite-ceiling-parameters@cref\endcsname{{[theorem][5][23]23.5}{[1][233][]233}}
\expandafter\gdef\csname r@ch20:sec:action\endcsname{{23.3}{234}{Bounds under the Wilson measure}{section.586}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:sec:action@cref\endcsname{{[section][3][23]23.3}{[1][234][]234}}
\expandafter\gdef\csname r@ch20:thm:shell-induction\endcsname{{23.6}{235}{}{theorem.587}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:thm:shell-induction@cref\endcsname{{[theorem][6][23]23.6}{[1][235][]235}}
\expandafter\gdef\csname r@ch20:eq:first-failure\endcsname{{23.5}{235}{}{equation.588}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:first-failure@cref\endcsname{{[equation][5][23]23.5}{[1][235][]235}}
\expandafter\gdef\csname r@ch20:eq:bare-start\endcsname{{23.6}{235}{Bounds under the Wilson measure}{equation.589}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:bare-start@cref\endcsname{{[equation][6][23]23.6}{[1][235][]235}}
\expandafter\gdef\csname r@ch20:eq:current-chessboard\endcsname{{23.7}{236}{Bounds under the Wilson measure}{equation.590}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:current-chessboard@cref\endcsname{{[equation][7][23]23.7}{[1][236][]236}}
\expandafter\gdef\csname r@ch20:thm:unmarked-coarse-action-moments\endcsname{{23.7}{236}{}{theorem.591}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:thm:unmarked-coarse-action-moments@cref\endcsname{{[theorem][7][23]23.7}{[1][236][]236}}
\expandafter\gdef\csname r@ch20:eq:unmarked-predecessor-moment\endcsname{{23.8}{237}{}{equation.592}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:unmarked-predecessor-moment@cref\endcsname{{[equation][8][23]23.8}{[1][236][]237}}
\expandafter\gdef\csname r@ch20:eq:unmarked-all-face-moment\endcsname{{23.9}{237}{}{equation.593}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:unmarked-all-face-moment@cref\endcsname{{[equation][9][23]23.9}{[1][236][]237}}
\expandafter\gdef\csname r@ch20:cor:joint-unconditional-bad-cells\endcsname{{23.8}{238}{}{corollary.594}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:cor:joint-unconditional-bad-cells@cref\endcsname{{[corollary][8][]23.8}{[1][238][]238}}
\expandafter\gdef\csname r@ch20:eq:joint-unconditional-bad-cells\endcsname{{23.10}{238}{}{equation.595}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:joint-unconditional-bad-cells@cref\endcsname{{[equation][10][23]23.10}{[1][238][]238}}
\expandafter\gdef\csname r@ch20:eq:unconditional-factorial-moments\endcsname{{23.11}{238}{}{equation.596}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:unconditional-factorial-moments@cref\endcsname{{[equation][11][23]23.11}{[1][238][]238}}
\expandafter\gdef\csname r@ch20:eq:unconditional-binomial-envelope\endcsname{{23.12}{238}{}{equation.597}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:unconditional-binomial-envelope@cref\endcsname{{[equation][12][23]23.12}{[1][238][]238}}
\expandafter\gdef\csname r@ch20:sec:small-field\endcsname{{23.4}{240}{Probability of the small-field event}{section.598}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:sec:small-field@cref\endcsname{{[section][4][23]23.4}{[1][239][]240}}
\expandafter\gdef\csname r@ch20:thm:literal-payment\endcsname{{23.9}{240}{}{theorem.599}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:thm:literal-payment@cref\endcsname{{[theorem][9][23]23.9}{[1][240][]240}}
\expandafter\gdef\csname r@ch20:thm:finite-ceiling-payment\endcsname{{23.10}{241}{}{theorem.600}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:thm:finite-ceiling-payment@cref\endcsname{{[theorem][10][23]23.10}{[1][241][]241}}
\expandafter\gdef\csname r@ch20:eq:finite-ceiling-absorbed\endcsname{{23.13}{241}{}{equation.601}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:finite-ceiling-absorbed@cref\endcsname{{[equation][13][23]23.13}{[1][241][]241}}
\expandafter\gdef\csname r@ch20:eq:finite-ceiling-unabsorbed\endcsname{{23.14}{241}{}{equation.602}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:finite-ceiling-unabsorbed@cref\endcsname{{[equation][14][23]23.14}{[1][241][]241}}
\expandafter\gdef\csname r@ch20:eq:curvature-payment\endcsname{{23.15}{241}{Probability of the small-field event}{equation.603}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:curvature-payment@cref\endcsname{{[equation][15][23]23.15}{[1][241][]241}}
\expandafter\gdef\csname r@ch20:eq:unabsorbed-entropy\endcsname{{23.16}{242}{Probability of the small-field event}{equation.604}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:unabsorbed-entropy@cref\endcsname{{[equation][16][23]23.16}{[1][241][]242}}
\expandafter\gdef\csname r@ch20:eq:unabsorbed-curvature\endcsname{{23.17}{242}{Probability of the small-field event}{equation.605}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:unabsorbed-curvature@cref\endcsname{{[equation][17][23]23.17}{[1][242][]242}}
\expandafter\gdef\csname r@ch20:eq:first-rejection-inclusion\endcsname{{23.18}{242}{Probability of the small-field event}{equation.606}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:first-rejection-inclusion@cref\endcsname{{[equation][18][23]23.18}{[1][242][]242}}
\expandafter\gdef\csname r@ch20:eq:one-macro-rejection\endcsname{{23.19}{242}{Probability of the small-field event}{equation.607}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:one-macro-rejection@cref\endcsname{{[equation][19][23]23.19}{[1][242][]242}}
\expandafter\gdef\csname r@ch20:eq:sum-white-rejections\endcsname{{23.20}{242}{Probability of the small-field event}{equation.608}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:sum-white-rejections@cref\endcsname{{[equation][20][23]23.20}{[1][242][]242}}
\expandafter\gdef\csname r@ch20:cor:payment-all-powers\endcsname{{23.11}{243}{}{corollary.609}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:cor:payment-all-powers@cref\endcsname{{[corollary][11][]23.11}{[1][243][]243}}
\expandafter\gdef\csname r@ch20:sec:log-squared\endcsname{{23.5}{244}{Two-sided bounds of logarithm-squared order}{section.610}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:sec:log-squared@cref\endcsname{{[section][5][23]23.5}{[1][244][]244}}
\expandafter\gdef\csname r@ch20:thm:literal-rejection-log-squared\endcsname{{23.12}{245}{}{theorem.611}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:thm:literal-rejection-log-squared@cref\endcsname{{[theorem][12][23]23.12}{[1][244][]245}}
\expandafter\gdef\csname r@ch20:eq:literal-rejection-log-squared\endcsname{{23.21}{245}{}{equation.612}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:literal-rejection-log-squared@cref\endcsname{{[equation][21][23]23.21}{[1][245][]245}}
\expandafter\gdef\csname r@ch20:eq:rejection-local-box\endcsname{{23.22}{246}{Two-sided bounds of logarithm-squared order}{equation.613}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:rejection-local-box@cref\endcsname{{[equation][22][23]23.22}{[1][245][]246}}
\expandafter\gdef\csname r@ch20:eq:rejection-exact-haar-mass\endcsname{{23.23}{246}{Two-sided bounds of logarithm-squared order}{equation.614}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:rejection-exact-haar-mass@cref\endcsname{{[equation][23][23]23.23}{[1][246][]246}}
\expandafter\gdef\csname r@ch20:eq:rejection-box-mass-lower\endcsname{{23.24}{247}{Two-sided bounds of logarithm-squared order}{equation.615}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:rejection-box-mass-lower@cref\endcsname{{[equation][24][23]23.24}{[1][247][]247}}
\expandafter\gdef\csname r@ch20:cor:finite-ceiling-rejection-order\endcsname{{23.13}{248}{}{corollary.616}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:cor:finite-ceiling-rejection-order@cref\endcsname{{[corollary][13][]23.13}{[1][247][]248}}
\expandafter\gdef\csname r@ch20:sec:parameter-proof\endcsname{{23.6}{248}{Proof of the choice of parameters}{section.617}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:sec:parameter-proof@cref\endcsname{{[section][6][23]23.6}{[1][248][]248}}
\expandafter\gdef\csname r@ch20:eq:torus-spatial-gate\endcsname{{23.25}{249}{Step 1: the choice of $J$}{equation.619}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:torus-spatial-gate@cref\endcsname{{[equation][25][23]23.25}{[1][249][]249}}
\expandafter\gdef\csname r@ch20:eq:torus-enlarged-center\endcsname{{23.26}{250}{Step 1: the choice of $J$}{equation.620}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:torus-enlarged-center@cref\endcsname{{[equation][26][23]23.26}{[1][250][]250}}
\expandafter\gdef\csname r@ch20:eq:outer-torus-domains\endcsname{{23.27}{251}{Step 2: larger domains, solution neighborhoods, and rates}{equation.622}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:outer-torus-domains@cref\endcsname{{[equation][27][23]23.27}{[1][251][]251}}
\expandafter\gdef\csname r@ch20:eq:interpolation-choice\endcsname{{23.28}{252}{Step 2: larger domains, solution neighborhoods, and rates}{equation.623}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:interpolation-choice@cref\endcsname{{[equation][28][23]23.28}{[1][252][]252}}
\expandafter\gdef\csname r@ch20:eq:conditional-force-gate\endcsname{{23.29}{253}{Step 3: perturbation bounds, the fluctuation radius, and the final perturbation bound}{equation.625}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:conditional-force-gate@cref\endcsname{{[equation][29][23]23.29}{[1][253][]253}}
\expandafter\gdef\csname r@ch20:eq:saving-constants\endcsname{{23.30}{254}{Step 3: perturbation bounds, the fluctuation radius, and the final perturbation bound}{equation.626}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:saving-constants@cref\endcsname{{[equation][30][23]23.30}{[1][254][]254}}
\expandafter\gdef\csname r@ch20:eq:zero-white-sum\endcsname{{23.31}{255}{Step 4: boundary radii, removed layers, and frozen coordinates}{equation.628}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:zero-white-sum@cref\endcsname{{[equation][31][23]23.31}{[1][255][]255}}
\expandafter\gdef\csname r@ch20:eq:complete-freeze-budget\endcsname{{23.32}{256}{Step 4: boundary radii, removed layers, and frozen coordinates}{equation.629}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:complete-freeze-budget@cref\endcsname{{[equation][32][23]23.32}{[1][256][]256}}
\expandafter\gdef\csname r@ch20:eq:center-margin\endcsname{{23.33}{256}{Step 5: the majorant of the reconstruction maps, the working radius, and the support radius}{equation.631}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:center-margin@cref\endcsname{{[equation][33][23]23.33}{[1][256][]256}}
\expandafter\gdef\csname r@ch20:eq:relative-collar-gates\endcsname{{23.34}{257}{Step 5: the majorant of the reconstruction maps, the working radius, and the support radius}{equation.632}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:relative-collar-gates@cref\endcsname{{[equation][34][23]23.34}{[1][257][]257}}
\expandafter\gdef\csname r@ch20:eq:db-whole-image-gate\endcsname{{23.35}{257}{Step 5: the majorant of the reconstruction maps, the working radius, and the support radius}{equation.633}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:db-whole-image-gate@cref\endcsname{{[equation][35][23]23.35}{[1][257][]257}}
\expandafter\gdef\csname r@ch20:eq:physical-enclosure\endcsname{{23.36}{257}{Step 5: the majorant of the reconstruction maps, the working radius, and the support radius}{equation.634}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:physical-enclosure@cref\endcsname{{[equation][36][23]23.36}{[1][257][]257}}
\expandafter\gdef\csname r@ch20:eq:fat-cell-specialization\endcsname{{23.37}{258}{Step 6: widths, the curvature threshold, and the large-field decompositions}{equation.636}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:fat-cell-specialization@cref\endcsname{{[equation][37][23]23.37}{[1][258][]258}}
\expandafter\gdef\csname r@ch20:eq:coverage-counts\endcsname{{23.38}{260}{Step 6: widths, the curvature threshold, and the large-field decompositions}{equation.637}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:coverage-counts@cref\endcsname{{[equation][38][23]23.38}{[1][260][]260}}
\expandafter\gdef\csname r@ch20:eq:log-family-gates\endcsname{{23.39}{261}{Step 7: the coupling bound}{equation.639}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:log-family-gates@cref\endcsname{{[equation][39][23]23.39}{[1][261][]261}}
\expandafter\gdef\csname r@ch20:eq:cap-ceiling\endcsname{{23.40}{261}{Step 7: the coupling bound}{equation.640}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:cap-ceiling@cref\endcsname{{[equation][40][23]23.40}{[1][261][]261}}
\expandafter\gdef\csname r@ch20:eq:actual-torus-ball\endcsname{{23.41}{262}{Step 8: the effective interaction on the torus}{equation.642}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:actual-torus-ball@cref\endcsname{{[equation][41][23]23.41}{[1][262][]262}}
\expandafter\gdef\csname r@ch20:eq:actual-stopped-input\endcsname{{23.42}{262}{Step 8: the effective interaction on the torus}{equation.643}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:actual-stopped-input@cref\endcsname{{[equation][42][23]23.42}{[1][262][]262}}
\expandafter\gdef\csname r@ch20:eq:actual-saved-values\endcsname{{23.43}{263}{Step 9: the hypotheses of the regional estimates}{equation.645}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:actual-saved-values@cref\endcsname{{[equation][43][23]23.43}{[1][263][]263}}
\expandafter\gdef\csname r@ch20:eq:input-wall-compatibility\endcsname{{23.44}{264}{Step 9: the hypotheses of the regional estimates}{equation.646}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:input-wall-compatibility@cref\endcsname{{[equation][44][23]23.44}{[1][264][]264}}
\expandafter\gdef\csname r@ch20:eq:covered-point-enclosure\endcsname{{23.45}{264}{Step 9: the hypotheses of the regional estimates}{equation.647}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:covered-point-enclosure@cref\endcsname{{[equation][45][23]23.45}{[1][264][]264}}
\expandafter\gdef\csname r@ch20:eq:complete-curvature\endcsname{{23.46}{264}{Step 9: the hypotheses of the regional estimates}{equation.648}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:complete-curvature@cref\endcsname{{[equation][46][23]23.46}{[1][264][]264}}
\expandafter\gdef\csname r@ch20:eq:finite-white-normalizer\endcsname{{23.47}{265}{Step 9: the hypotheses of the regional estimates}{equation.649}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:finite-white-normalizer@cref\endcsname{{[equation][47][23]23.47}{[1][265][]265}}
\expandafter\gdef\csname r@ch20:eq:full-macro-source-phase\endcsname{{23.48}{266}{Step 10: stopping levels and the insertion interval}{equation.651}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:full-macro-source-phase@cref\endcsname{{[equation][48][23]23.48}{[1][265][]266}}
\expandafter\gdef\csname r@ch20:eq:compact-source-cap\endcsname{{23.49}{266}{Step 10: stopping levels and the insertion interval}{equation.652}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:compact-source-cap@cref\endcsname{{[equation][49][23]23.49}{[1][266][]266}}
\expandafter\gdef\csname r@ch20:eq:compact-source-budget\endcsname{{23.50}{267}{Step 10: stopping levels and the insertion interval}{equation.653}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:compact-source-budget@cref\endcsname{{[equation][50][23]23.50}{[1][266][]267}}
\expandafter\gdef\csname r@ch20:eq:endpoint-coupling-gate\endcsname{{23.51}{267}{Step 11: the minimum side}{equation.655}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:endpoint-coupling-gate@cref\endcsname{{[equation][51][23]23.51}{[1][267][]267}}
\expandafter\gdef\csname r@ch20:eq:whole-flat-side\endcsname{{23.52}{268}{Step 11: the minimum side}{equation.656}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:whole-flat-side@cref\endcsname{{[equation][52][23]23.52}{[1][268][]268}}
\expandafter\gdef\csname r@ch20:eq:side-uniform-forced-cap\endcsname{{23.53}{269}{Step 12: the reconstruction maps and the identity of coarse fields}{equation.658}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:side-uniform-forced-cap@cref\endcsname{{[equation][53][23]23.53}{[1][269][]269}}
\expandafter\gdef\csname r@ch20:sec:finite-ceiling\endcsname{{23.7}{270}{Parameters below an arbitrary coupling ceiling}{section.659}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:sec:finite-ceiling@cref\endcsname{{[section][7][23]23.7}{[1][270][]270}}
\expandafter\gdef\csname r@ch20:sec:large-J\endcsname{{23.8}{272}{Every sufficiently large number of averaging steps}{section.660}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:sec:large-J@cref\endcsname{{[section][8][23]23.8}{[1][272][]272}}
\expandafter\gdef\csname r@ch20:lem:all-large-macro-depths\endcsname{{23.14}{272}{}{lemma.661}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:lem:all-large-macro-depths@cref\endcsname{{[lemma][14][]23.14}{[1][272][]272}}
\expandafter\gdef\csname r@ch20:eq:pre-J-lower-bounds\endcsname{{23.54}{272}{Every sufficiently large number of averaging steps}{equation.662}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:eq:pre-J-lower-bounds@cref\endcsname{{[equation][54][23]23.54}{[1][272][]272}}
\expandafter\gdef\csname r@ch20:cor:finite-ceiling-macro-depths\endcsname{{23.15}{274}{}{corollary.663}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch20:cor:finite-ceiling-macro-depths@cref\endcsname{{[corollary][15][]23.15}{[1][274][]274}}
% Part IV
\expandafter\gdef\csname r@part:flow\endcsname{{VII}{2}{\contentsname \@mkboth {\MakeUppercase \contentsname }{\MakeUppercase \contentsname }}{part.2}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@part:flow@cref\endcsname{{[part][7][]VII}{[1][2][]2}}
\expandafter\gdef\csname r@ch21:chapter\endcsname{{24}{2}{Local Wilson flow and continuum solutions}{chapter.3}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:chapter@cref\endcsname{{[chapter][24][]24}{[1][2][]2}}
\expandafter\gdef\csname r@ch21:eq:intro-equation\endcsname{{24.1}{2}{Local Wilson flow and continuum solutions}{equation.4}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:intro-equation@cref\endcsname{{[equation][1][24]24.1}{[1][2][]2}}
\expandafter\gdef\csname r@ch21:eq:intro-decomposition\endcsname{{24.2}{2}{Local Wilson flow and continuum solutions}{equation.5}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:intro-decomposition@cref\endcsname{{[equation][2][24]24.2}{[1][2][]2}}
\expandafter\gdef\csname r@ch21:sec:equation\endcsname{{24.1}{3}{The local equation for the Wilson flow}{section.6}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:sec:equation@cref\endcsname{{[section][1][24]24.1}{[1][3][]3}}
\expandafter\gdef\csname r@ch21:eq:c1a-norm\endcsname{{24.3}{3}{Notation}{equation.8}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:c1a-norm@cref\endcsname{{[equation][3][24]24.3}{[1][3][]3}}
\expandafter\gdef\csname r@ch21:lem:stencil-analytic\endcsname{{24.1}{4}{}{lemma.10}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:lem:stencil-analytic@cref\endcsname{{[lemma][1][]24.1}{[1][4][]4}}
\expandafter\gdef\csname r@ch21:lem:wilson-local-algebra\endcsname{{24.2}{6}{}{lemma.17}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:lem:wilson-local-algebra@cref\endcsname{{[lemma][2][]24.2}{[1][6][]6}}
\expandafter\gdef\csname r@ch21:eq:Na-form\endcsname{{24.4}{7}{}{equation.22}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:Na-form@cref\endcsname{{[equation][4][24]24.4}{[1][7][]7}}
\expandafter\gdef\csname r@ch21:eq:curl-adjoint\endcsname{{24.5}{8}{Step 1: Hodge identities}{equation.26}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:curl-adjoint@cref\endcsname{{[equation][5][24]24.5}{[1][8][]8}}
\expandafter\gdef\csname r@ch21:eq:wilson-coordinate-force\endcsname{{24.6}{9}{Step 3: the group gradient in logarithmic coordinates}{equation.29}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:wilson-coordinate-force@cref\endcsname{{[equation][6][24]24.6}{[1][9][]9}}
\expandafter\gdef\csname r@ch21:eq:physical-cubic-series\endcsname{{24.7}{11}{Step 6: the remainder}{equation.33}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:physical-cubic-series@cref\endcsname{{[equation][7][24]24.7}{[1][11][]11}}
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\expandafter\gdef\csname r@ch22:eq:endpoint-near-smooth-compact@cref\endcsname{{[equation][37][25]25.37}{[1][75][]75}}
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\expandafter\gdef\csname r@ch22:eq:matrix-in-initial-frame@cref\endcsname{{[equation][40][25]25.40}{[1][77][]77}}
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\expandafter\gdef\csname r@ch22:eq:twice-differentiated-residual@cref\endcsname{{[equation][41][25]25.41}{[1][78][]78}}
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\expandafter\gdef\csname r@ch22:eq:C2-smooth-family-convergence@cref\endcsname{{[equation][42][25]25.42}{[1][79][]79}}
\expandafter\gdef\csname r@ch22:eq:L1-generator-convergence\endcsname{{25.43}{79}{Parameter derivatives in a fixed initial gauge}{equation.193}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch22:eq:L1-generator-convergence@cref\endcsname{{[equation][43][25]25.43}{[1][79][]79}}
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\expandafter\gdef\csname r@ch22:eq:root-ODE-consistency@cref\endcsname{{[equation][44][25]25.44}{[1][80][]80}}
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\expandafter\gdef\csname r@ch22:sec:modified@cref\endcsname{{[section][8][25]25.8}{[1][81][]81}}
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\expandafter\gdef\csname r@ch22:lem:alternative-smooth-reference@cref\endcsname{{[lemma][14][]25.14}{[1][81][]81}}
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\expandafter\gdef\csname r@ch22:eq:alternative-reference-bound@cref\endcsname{{[equation][45][25]25.45}{[1][82][]82}}
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\expandafter\gdef\csname r@ch22:lem:alternative-uniform-handoff@cref\endcsname{{[lemma][15][]25.15}{[1][83][]83}}
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\expandafter\gdef\csname r@part:stochastic@cref\endcsname{{[part][8][]VIII}{[1][86][]86}}
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\expandafter\gdef\csname r@ch23:chapter@cref\endcsname{{[chapter][26][]26}{[1][86][]86}}
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\expandafter\gdef\csname r@ch23:eq:implemented-gates@cref\endcsname{{[equation][1][26]26.1}{[1][91][]91}}
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\expandafter\gdef\csname r@ch23:rem:base-point@cref\endcsname{{[remark][3][]26.3}{[1][96][]97}}
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\expandafter\gdef\csname r@ch23:sec:deterministic@cref\endcsname{{[section][2][26]26.2}{[1][97][]97}}
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\expandafter\gdef\csname r@ch23:lem:physical-heat\endcsname{{26.4}{98}{}{lemma.242}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:physical-heat@cref\endcsname{{[lemma][4][]26.4}{[1][98][]98}}
\expandafter\gdef\csname r@ch23:lem:pair-schur\endcsname{{26.5}{99}{}{lemma.244}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch23:lem:marked-calculus\endcsname{{26.6}{100}{}{lemma.247}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch23:eq:background-differences@cref\endcsname{{[equation][12][26]26.12}{[1][101][]101}}
\expandafter\gdef\csname r@ch23:eq:profile-bounds\endcsname{{26.13}{102}{Step 5: the convolution integrals}{equation.254}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:profile-bounds@cref\endcsname{{[equation][13][26]26.13}{[1][102][]102}}
\expandafter\gdef\csname r@ch23:lem:coefficient-cover\endcsname{{26.7}{103}{}{lemma.256}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:coefficient-cover@cref\endcsname{{[lemma][7][]26.7}{[1][102][]103}}
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\expandafter\gdef\csname r@ch23:eq:normalized-cover@cref\endcsname{{[equation][14][26]26.14}{[1][103][]103}}
\expandafter\gdef\csname r@ch23:eq:cover-source-derivatives\endcsname{{26.15}{103}{Covering numbers of coefficient rows}{equation.258}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:cover-source-derivatives@cref\endcsname{{[equation][15][26]26.15}{[1][103][]103}}
\expandafter\gdef\csname r@ch23:eq:normalized-modulus\endcsname{{26.16}{104}{Covering numbers of coefficient rows}{equation.259}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:normalized-modulus@cref\endcsname{{[equation][16][26]26.16}{[1][104][]104}}
\expandafter\gdef\csname r@ch23:sec:maxima\endcsname{{26.3}{105}{Concentration and maximal inequalities}{section.260}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:sec:maxima@cref\endcsname{{[section][3][26]26.3}{[1][105][]105}}
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\expandafter\gdef\csname r@ch23:lem:coefficient-chaining@cref\endcsname{{[lemma][8][]26.8}{[1][105][]105}}
\expandafter\gdef\csname r@ch23:lem:input-maxima\endcsname{{26.9}{106}{}{lemma.262}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:input-maxima@cref\endcsname{{[lemma][9][]26.9}{[1][106][]106}}
\expandafter\gdef\csname r@ch23:sec:linear\endcsname{{26.4}{109}{The linear fluctuation and the mixed Duhamel terms}{section.273}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:sec:linear@cref\endcsname{{[section][4][26]26.4}{[1][109][]109}}
\expandafter\gdef\csname r@ch23:thm:fresh-and-mixed-rows\endcsname{{26.10}{110}{}{theorem.274}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch23:thm:centered-residual@cref\endcsname{{[theorem][11][26]26.11}{[1][114][]114}}
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\expandafter\gdef\csname r@ch23:thm:fresh-square@cref\endcsname{{[theorem][12][26]26.12}{[1][118][]118}}
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\expandafter\gdef\csname r@ch23:rem:centering-scope@cref\endcsname{{[remark][13][]26.13}{[1][124][]124}}
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\expandafter\gdef\csname r@ch24:tab:constants\endcsname{{27.1}{127}{Dependence of constants in \cref {ch24:chapter}. No constant depends on $K$, on the scale indices $i$ and $j$, on the number of preceding averaging steps, on $a/\delta $, or on the values of the coarser fields}{table.caption.321}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch24:lem:geometric-convolution@cref\endcsname{{[lemma][2][]27.2}{[1][133][]133}}
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\expandafter\gdef\csname r@ch24:eq:actual-kernel-concentration@cref\endcsname{{[equation][7][27]27.7}{[1][134][]134}}
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\expandafter\gdef\csname r@ch24:eq:uniform-white-covariance@cref\endcsname{{[equation][8][27]27.8}{[1][134][]134}}
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\expandafter\gdef\csname r@ch24:eq:uniform-linear-coefficient@cref\endcsname{{[equation][9][27]27.9}{[1][134][]134}}
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\expandafter\gdef\csname r@ch24:eq:uniform-reconstructed-covariance@cref\endcsname{{[equation][10][27]27.10}{[1][134][]134}}
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\expandafter\gdef\csname r@ch24:lem:quadratic-mean@cref\endcsname{{[lemma][4][]27.4}{[1][136][]136}}
\expandafter\gdef\csname r@ch24:eq:unweighted-quantum\endcsname{{27.12}{136}{}{equation.358}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:unweighted-quantum@cref\endcsname{{[equation][12][27]27.12}{[1][136][]136}}
\expandafter\gdef\csname r@ch24:eq:unweighted-classical\endcsname{{27.13}{136}{}{equation.359}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:unweighted-classical@cref\endcsname{{[equation][13][27]27.13}{[1][136][]136}}
\expandafter\gdef\csname r@ch24:eq:nuclear\endcsname{{27.14}{136}{Conditional means of quadratic functions}{equation.360}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:nuclear@cref\endcsname{{[equation][14][27]27.14}{[1][136][]136}}
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\expandafter\gdef\csname r@ch24:eq:B-mean-bounded\endcsname{{27.16}{137}{Conditional means of quadratic functions}{equation.362}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:B-mean-bounded@cref\endcsname{{[equation][16][27]27.16}{[1][137][]137}}
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\expandafter\gdef\csname r@ch24:thm:ancestor-gram-rows@cref\endcsname{{[theorem][5][27]27.5}{[1][138][]138}}
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\expandafter\gdef\csname r@ch24:eq:source-coefficients@cref\endcsname{{[equation][17][27]27.17}{[1][140][]140}}
\expandafter\gdef\csname r@ch24:eq:value-source-white\endcsname{{27.18}{140}{Step 1: sources and late rows}{equation.371}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:value-source-white@cref\endcsname{{[equation][18][27]27.18}{[1][140][]140}}
\expandafter\gdef\csname r@ch24:eq:slot-gram\endcsname{{27.19}{140}{Step 1: sources and late rows}{equation.372}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:slot-gram@cref\endcsname{{[equation][19][27]27.19}{[1][140][]140}}
\expandafter\gdef\csname r@ch24:eq:late-source-row\endcsname{{27.20}{140}{Step 1: sources and late rows}{equation.373}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:late-source-row@cref\endcsname{{[equation][20][27]27.20}{[1][140][]140}}
\expandafter\gdef\csname r@ch24:eq:J-formula\endcsname{{27.21}{141}{Step 2: the outer heat semigroup at small times}{equation.375}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:J-formula@cref\endcsname{{[equation][21][27]27.21}{[1][141][]141}}
\expandafter\gdef\csname r@ch24:eq:retained-mesh-log\endcsname{{27.22}{141}{Step 4: quadratic coefficients and weights}{equation.378}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:retained-mesh-log@cref\endcsname{{[equation][22][27]27.22}{[1][141][]141}}
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\expandafter\gdef\csname r@ch24:eq:crude-source@cref\endcsname{{[equation][23][27]27.23}{[1][142][]142}}
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\expandafter\gdef\csname r@ch24:eq:weighted-mean-mesh@cref\endcsname{{[equation][24][27]27.24}{[1][142][]142}}
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\expandafter\gdef\csname r@ch24:eq:mean-majorant\endcsname{{27.26}{143}{Step 7: bounds on the coarser fields and summability}{equation.385}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch24:sec:ordered\endcsname{{27.3}{143}{Maximal estimates and ultraviolet tails of the ordered terms}{section.386}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:sec:ordered@cref\endcsname{{[section][3][27]27.3}{[1][143][]143}}
\expandafter\gdef\csname r@ch24:thm:ordered-rough-return\endcsname{{27.6}{144}{}{theorem.387}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:thm:ordered-rough-return@cref\endcsname{{[theorem][6][27]27.6}{[1][143][]144}}
\expandafter\gdef\csname r@ch24:eq:sample-cover-bounds\endcsname{{27.27}{145}{Step 2: a deterministic net}{equation.393}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:sample-cover-bounds@cref\endcsname{{[equation][27][27]27.27}{[1][145][]145}}
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\expandafter\gdef\csname r@ch24:eq:microscopic-cover-ratio@cref\endcsname{{[equation][28][27]27.28}{[1][145][]145}}
\expandafter\gdef\csname r@ch24:eq:net-cardinality\endcsname{{27.29}{146}{Step 2: a deterministic net}{equation.395}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:net-cardinality@cref\endcsname{{[equation][29][27]27.29}{[1][146][]146}}
\expandafter\gdef\csname r@ch24:eq:ordered-telescope\endcsname{{27.30}{147}{Step 4: the three orders and the telescope}{equation.398}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:ordered-telescope@cref\endcsname{{[equation][30][27]27.30}{[1][147][]147}}
\expandafter\gdef\csname r@ch24:sec:predictable\endcsname{{27.4}{148}{The predictable term on the finest lattice}{section.400}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:sec:predictable@cref\endcsname{{[section][4][27]27.4}{[1][148][]148}}
\expandafter\gdef\csname r@ch24:thm:full-smoother-return\endcsname{{27.7}{148}{}{theorem.401}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:thm:full-smoother-return@cref\endcsname{{[theorem][7][27]27.7}{[1][148][]148}}
\expandafter\gdef\csname r@ch24:eq:static-coefficient\endcsname{{27.31}{150}{Step 1: the propagated quadratic coefficient}{equation.408}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:static-coefficient@cref\endcsname{{[equation][31][27]27.31}{[1][150][]150}}
\expandafter\gdef\csname r@ch24:eq:static-micro\endcsname{{27.32}{151}{Step 3: the centered term}{equation.411}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:static-micro@cref\endcsname{{[equation][32][27]27.32}{[1][150][]151}}
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\expandafter\gdef\csname r@ch24:eq:uniform-static-mean\endcsname{{27.34}{152}{Step 4: the drifts and the coarsest term}{equation.414}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch24:eq:bounded-smoother\endcsname{{27.35}{152}{Step 5: the first difference, including the smallest times}{equation.416}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:bounded-smoother@cref\endcsname{{[equation][35][27]27.35}{[1][152][]152}}
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\expandafter\gdef\csname r@ch24:eq:mean-score-covariance\endcsname{{27.38}{154}{Step 7: the cubic term and the conditional mean}{equation.421}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:mean-score-covariance@cref\endcsname{{[equation][38][27]27.38}{[1][154][]154}}
\expandafter\gdef\csname r@ch24:eq:mean-explicit-target-derivative\endcsname{{27.39}{154}{Step 7: the cubic term and the conditional mean}{equation.422}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch24:eq:physical-mean\endcsname{{27.41}{155}{Step 7: the cubic term and the conditional mean}{equation.424}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:physical-mean@cref\endcsname{{[equation][41][27]27.41}{[1][155][]155}}
\expandafter\gdef\csname r@ch24:rem:product-application\endcsname{{27.8}{155}{}{remark.426}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:rem:product-application@cref\endcsname{{[remark][8][]27.8}{[1][155][]155}}
\expandafter\gdef\csname r@ch25:chapter\endcsname{{28}{156}{Linear and quadratic flow data and convergence of Wilson loops}{chapter.427}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:chapter@cref\endcsname{{[chapter][28][]28}{[1][156][]156}}
\expandafter\gdef\csname r@ch25:sec:setting\endcsname{{28.1}{158}{Setting and main results}{section.431}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:sec:setting@cref\endcsname{{[section][1][28]28.1}{[1][157][]158}}
\expandafter\gdef\csname r@ch25:thm:selected-loop-limit\endcsname{{28.1}{160}{}{theorem.438}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:thm:selected-loop-limit@cref\endcsname{{[theorem][1][28]28.1}{[1][160][]160}}
\expandafter\gdef\csname r@ch25:thm:original-loop-stitching\endcsname{{28.2}{160}{}{theorem.439}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:thm:original-loop-stitching@cref\endcsname{{[theorem][2][28]28.2}{[1][160][]160}}
\expandafter\gdef\csname r@ch25:sec:deterministic\endcsname{{28.2}{161}{Deterministic estimates}{section.440}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:sec:deterministic@cref\endcsname{{[section][2][28]28.2}{[1][161][]161}}
\expandafter\gdef\csname r@ch25:lem:elementary-estimates\endcsname{{28.3}{161}{}{lemma.442}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:lem:elementary-estimates@cref\endcsname{{[lemma][3][]28.3}{[1][161][]161}}
\expandafter\gdef\csname r@ch25:eq:cardinal-elementary\endcsname{{28.1}{161}{}{equation.443}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:cardinal-elementary@cref\endcsname{{[equation][1][28]28.1}{[1][161][]161}}
\expandafter\gdef\csname r@ch25:eq:cardinal-sampling\endcsname{{28.2}{162}{}{equation.444}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:cardinal-sampling@cref\endcsname{{[equation][2][28]28.2}{[1][161][]162}}
\expandafter\gdef\csname r@ch25:eq:coarse-heat-short\endcsname{{28.3}{162}{}{equation.445}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:coarse-heat-short@cref\endcsname{{[equation][3][28]28.3}{[1][161][]162}}
\expandafter\gdef\csname r@ch25:eq:geometric-sums\endcsname{{28.4}{162}{}{equation.446}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:geometric-sums@cref\endcsname{{[equation][4][28]28.4}{[1][161][]162}}
\expandafter\gdef\csname r@ch25:eq:cutoff-linear\endcsname{{28.5}{162}{Cardinal interpolation}{equation.447}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:cutoff-linear@cref\endcsname{{[equation][5][28]28.5}{[1][162][]162}}
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\expandafter\gdef\csname r@ch25:lem:fixed-smooth-prefix@cref\endcsname{{[lemma][4][]28.4}{[1][162][]162}}
\expandafter\gdef\csname r@ch25:sec:ordered\endcsname{{28.3}{164}{The four-term decomposition and the ordered bilinear terms}{section.450}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:sec:ordered@cref\endcsname{{[section][3][28]28.3}{[1][164][]164}}
\expandafter\gdef\csname r@ch25:eq:quadratic-increment\endcsname{{28.6}{164}{The four-term decomposition and the ordered bilinear terms}{equation.451}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch26:eq:tail-transfer@cref\endcsname{{[equation][14][29]29.14}{[1][208][]208}}
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\expandafter\gdef\csname r@ch26:eq:finite-scale-tail@cref\endcsname{{[equation][15][29]29.15}{[1][208][]208}}
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\expandafter\gdef\csname r@ch26:sec:measurable-maps@cref\endcsname{{[section][4][29]29.4}{[1][212][]212}}
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\expandafter\gdef\csname r@ch26:thm:energy-constrained-matching@cref\endcsname{{[theorem][19][29]29.19}{[1][212][]212}}
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\expandafter\gdef\csname r@ch26:eq:energy-constrained-feasibility@cref\endcsname{{[equation][17][29]29.17}{[1][212][]213}}
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\expandafter\gdef\csname r@ch26:eq:matching-coordinate-error@cref\endcsname{{[equation][18][29]29.18}{[1][214][]214}}
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\expandafter\gdef\csname r@ch26:eq:matching-energy-domination@cref\endcsname{{[equation][19][29]29.19}{[1][214][]214}}
\expandafter\gdef\csname r@ch26:sec:unconditioning\endcsname{{29.5}{216}{Removal of the conditioning}{section.611}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:unconditioning@cref\endcsname{{[section][5][29]29.5}{[1][215][]216}}
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\expandafter\gdef\csname r@ch26:thm:orbit-limit@cref\endcsname{{[theorem][20][29]29.20}{[1][215][]216}}
\expandafter\gdef\csname r@ch26:sec:coupling-bound\endcsname{{29.6}{217}{Couplings below a fixed bound}{section.617}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:coupling-bound@cref\endcsname{{[section][6][29]29.6}{[1][217][]217}}
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\expandafter\gdef\csname r@ch26:thm:finite-ceiling-matching@cref\endcsname{{[theorem][21][29]29.21}{[1][217][]217}}
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\expandafter\gdef\csname r@ch26:thm:matched-paths@cref\endcsname{{[theorem][23][29]29.23}{[1][219][]219}}
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\expandafter\gdef\csname r@ch26:sec:refinement-sequences@cref\endcsname{{[section][7][29]29.7}{[1][220][]220}}
\expandafter\gdef\csname r@ch26:thm:finite-ceiling-tier-one\endcsname{{29.24}{220}{}{theorem.634}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:finite-ceiling-tier-one@cref\endcsname{{[theorem][24][29]29.24}{[1][220][]220}}
\expandafter\gdef\csname r@ch26:thm:tier-one\endcsname{{29.25}{221}{}{theorem.639}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:tier-one@cref\endcsname{{[theorem][25][29]29.25}{[1][221][]221}}
\expandafter\gdef\csname r@ch26:cor:common-ladder-family\endcsname{{29.26}{222}{}{corollary.640}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:cor:common-ladder-family@cref\endcsname{{[corollary][26][]29.26}{[1][222][]222}}
\expandafter\gdef\csname r@ch26:sec:full-sequence\endcsname{{29.8}{223}{The sequence \texorpdfstring {$nM^k$}{nM\^{}k}}{section.643}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:full-sequence@cref\endcsname{{[section][8][29]29.8}{[1][223][]223}}
\expandafter\gdef\csname r@ch26:lem:consecutive-depth-intersection\endcsname{{29.27}{223}{}{lemma.644}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:lem:consecutive-depth-intersection@cref\endcsname{{[lemma][27][]29.27}{[1][223][]223}}
\expandafter\gdef\csname r@ch26:eq:common-original-cutoffs\endcsname{{29.20}{223}{}{equation.645}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:common-original-cutoffs@cref\endcsname{{[equation][20][29]29.20}{[1][223][]223}}
\expandafter\gdef\csname r@ch26:thm:full-M-ladder\endcsname{{29.28}{223}{}{theorem.646}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:full-M-ladder@cref\endcsname{{[theorem][28][29]29.28}{[1][223][]223}}
\expandafter\gdef\csname r@ch26:rem:scope-full-sequence\endcsname{{29.29}{224}{}{remark.652}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:rem:scope-full-sequence@cref\endcsname{{[remark][29][]29.29}{[1][224][]224}}
\expandafter\gdef\csname r@ch26:chapter-properties\endcsname{{30}{227}{Further properties of the continuum measure}{chapter.659}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:chapter-properties@cref\endcsname{{[chapter][30][]30}{[1][227][]227}}
\expandafter\gdef\csname r@ch26:sec:properties-setting\endcsname{{30.1}{228}{Setting and notation}{section.667}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:properties-setting@cref\endcsname{{[section][1][30]30.1}{[1][228][]228}}
\expandafter\gdef\csname r@ch26:eq:properties-failure-bound\endcsname{{30.1}{229}{Conditioned measures}{equation.669}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:properties-failure-bound@cref\endcsname{{[equation][1][30]30.1}{[1][229][]229}}
\expandafter\gdef\csname r@ch26:eq:conditioning-inequality\endcsname{{30.2}{229}{Conditioned measures}{equation.670}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:conditioning-inequality@cref\endcsname{{[equation][2][30]30.2}{[1][229][]229}}
\expandafter\gdef\csname r@ch26:eq:properties-radius-tightness\endcsname{{30.3}{230}{Flow data and reference solutions}{equation.674}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:properties-radius-tightness@cref\endcsname{{[equation][3][30]30.3}{[1][230][]230}}
\expandafter\gdef\csname r@ch26:sec:dilation\endcsname{{30.2}{230}{Exact dilation}{section.675}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:dilation@cref\endcsname{{[section][2][30]30.2}{[1][230][]230}}
\expandafter\gdef\csname r@ch26:thm:exact-L-dilation\endcsname{{30.1}{230}{}{theorem.676}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:exact-L-dilation@cref\endcsname{{[theorem][1][30]30.1}{[1][230][]230}}
\expandafter\gdef\csname r@ch26:eq:dilated-orbit-law\endcsname{{30.4}{231}{}{equation.677}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:dilated-orbit-law@cref\endcsname{{[equation][4][30]30.4}{[1][230][]231}}
\expandafter\gdef\csname r@ch26:eq:dilated-energy\endcsname{{30.5}{231}{}{equation.678}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:dilated-energy@cref\endcsname{{[equation][5][30]30.5}{[1][231][]231}}
\expandafter\gdef\csname r@ch26:eq:dilated-holonomy\endcsname{{30.6}{232}{Step 2: the continuum dilation}{equation.681}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:dilated-holonomy@cref\endcsname{{[equation][6][30]30.6}{[1][232][]232}}
\expandafter\gdef\csname r@ch26:rem:dilation-scope\endcsname{{30.2}{233}{}{remark.684}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:rem:dilation-scope@cref\endcsname{{[remark][2][]30.2}{[1][232][]233}}
\expandafter\gdef\csname r@ch26:sec:paths\endcsname{{30.3}{233}{Energy and loop processes at positive times}{section.685}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:paths@cref\endcsname{{[section][3][30]30.3}{[1][233][]233}}
\expandafter\gdef\csname r@ch26:lem:positive-window-observable-modulus\endcsname{{30.3}{233}{}{lemma.686}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:lem:positive-window-observable-modulus@cref\endcsname{{[lemma][3][]30.3}{[1][233][]233}}
\expandafter\gdef\csname r@ch26:eq:positive-window-modulus\endcsname{{30.7}{233}{}{equation.687}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:positive-window-modulus@cref\endcsname{{[equation][7][30]30.7}{[1][233][]233}}
\expandafter\gdef\csname r@ch26:eq:positive-window-time-derivative\endcsname{{30.8}{235}{Step 3: the time derivative}{equation.691}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:positive-window-time-derivative@cref\endcsname{{[equation][8][30]30.8}{[1][235][]235}}
\expandafter\gdef\csname r@ch26:thm:actual-observable-paths\endcsname{{30.4}{236}{}{theorem.693}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:actual-observable-paths@cref\endcsname{{[theorem][4][30]30.4}{[1][235][]236}}
\expandafter\gdef\csname r@ch26:eq:selected-orbit-bounded-payment\endcsname{{30.9}{237}{Step 2: comparison of the limits for Borel tests}{equation.696}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:selected-orbit-bounded-payment@cref\endcsname{{[equation][9][30]30.9}{[1][236][]237}}
\expandafter\gdef\csname r@ch26:sec:sobolev\endcsname{{30.4}{238}{Convergence in the Sobolev topology}{section.699}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:sobolev@cref\endcsname{{[section][4][30]30.4}{[1][238][]238}}
\expandafter\gdef\csname r@ch26:thm:sobolev-image-limit\endcsname{{30.5}{238}{}{theorem.700}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:sobolev-image-limit@cref\endcsname{{[theorem][5][30]30.5}{[1][238][]238}}
\expandafter\gdef\csname r@ch26:sec:symmetry\endcsname{{30.5}{240}{Spatial isometries and center symmetry}{section.701}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:symmetry@cref\endcsname{{[section][5][30]30.5}{[1][239][]240}}
\expandafter\gdef\csname r@ch26:thm:spatial-center-symmetry\endcsname{{30.6}{240}{}{theorem.702}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:spatial-center-symmetry@cref\endcsname{{[theorem][6][30]30.6}{[1][240][]240}}
\expandafter\gdef\csname r@ch26:eq:center-word-sign\endcsname{{30.10}{241}{Step 1: lattice symmetries}{equation.704}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:center-word-sign@cref\endcsname{{[equation][10][30]30.10}{[1][241][]241}}
\expandafter\gdef\csname r@ch26:eq:continuum-center-holonomy\endcsname{{30.11}{242}{Step 2: the continuum actions}{equation.706}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:continuum-center-holonomy@cref\endcsname{{[equation][11][30]30.11}{[1][242][]242}}
\expandafter\gdef\csname r@ch26:sec:properties-full-sequence\endcsname{{30.6}{243}{Collected conclusions and the full sequence}{section.709}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:properties-full-sequence@cref\endcsname{{[section][6][30]30.6}{[1][243][]243}}
\expandafter\gdef\csname r@ch26:cor:finite-ceiling-observable-upgrades\endcsname{{30.7}{243}{}{corollary.710}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:cor:finite-ceiling-observable-upgrades@cref\endcsname{{[corollary][7][]30.7}{[1][243][]243}}
\expandafter\gdef\csname r@ch26:sec:compact-coupling-addendum\endcsname{{30.7}{245}{Continuity and uniform convergence in the coupling}{section.711}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:compact-coupling-addendum@cref\endcsname{{[section][7][30]30.7}{[1][245][]245}}
\expandafter\gdef\csname r@ch26:lem:bounded-input-energy-comparison\endcsname{{30.8}{245}{}{lemma.712}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:lem:bounded-input-energy-comparison@cref\endcsname{{[lemma][8][]30.8}{[1][245][]245}}
\expandafter\gdef\csname r@ch26:lem:compact-coupling-smooth-prefix-approximation\endcsname{{30.9}{246}{}{lemma.713}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:lem:compact-coupling-smooth-prefix-approximation@cref\endcsname{{[lemma][9][]30.9}{[1][246][]246}}
\expandafter\gdef\csname r@ch26:eq:compact-coupling-smooth-prefix-error\endcsname{{30.12}{246}{}{equation.714}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:compact-coupling-smooth-prefix-error@cref\endcsname{{[equation][12][30]30.12}{[1][246][]246}}
\expandafter\gdef\csname r@ch26:thm:positive-coupling-uniform-UV\endcsname{{30.10}{247}{}{theorem.715}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:positive-coupling-uniform-UV@cref\endcsname{{[theorem][10][30]30.10}{[1][247][]247}}
\expandafter\gdef\csname r@ch26:rem:compact-coupling-scope\endcsname{{30.11}{248}{}{remark.720}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:rem:compact-coupling-scope@cref\endcsname{{[remark][11][]30.11}{[1][248][]248}}
\expandafter\gdef\csname r@ch26:eq:finite-ceiling-uniform-feasibility\endcsname{{30.13}{250}{Step 3: uniform feasibility at a fixed level}{equation.724}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:finite-ceiling-uniform-feasibility@cref\endcsname{{[equation][13][30]30.13}{[1][250][]250}}
\expandafter\gdef\csname r@ch26:eq:finite-ceiling-matching-energy\endcsname{{30.14}{251}{Step 4: the diagonal choice}{equation.726}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:finite-ceiling-matching-energy@cref\endcsname{{[equation][14][30]30.14}{[1][251][]251}}
\expandafter\gdef\csname r@ch26:cor:continuous-torus-side\endcsname{{30.12}{253}{}{corollary.729}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:cor:continuous-torus-side@cref\endcsname{{[corollary][12][]30.12}{[1][253][]253}}
\expandafter\gdef\csname r@ch26:sec:bounded-quartic-flow-addendum\endcsname{{30.8}{254}{Modified smoothing flows}{section.730}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:bounded-quartic-flow-addendum@cref\endcsname{{[section][8][30]30.8}{[1][254][]254}}
\expandafter\gdef\csname r@ch26:thm:bounded-quartic-flow-universality\endcsname{{30.13}{254}{}{theorem.731}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:bounded-quartic-flow-universality@cref\endcsname{{[theorem][13][30]30.13}{[1][254][]254}}
\expandafter\gdef\csname r@ch26:eq:alternative-original-comparison\endcsname{{30.15}{254}{}{equation.732}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:alternative-original-comparison@cref\endcsname{{[equation][15][30]30.15}{[1][254][]254}}
\expandafter\gdef\csname r@ch26:eq:alternative-orbit-fidelity\endcsname{{30.16}{255}{}{equation.733}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:alternative-orbit-fidelity@cref\endcsname{{[equation][16][30]30.16}{[1][255][]255}}
\expandafter\gdef\csname r@ch26:sec:alternative-observable-paths-addendum\endcsname{{30.9}{257}{Energy and loop processes for the modified flows}{section.737}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:alternative-observable-paths-addendum@cref\endcsname{{[section][9][30]30.9}{[1][256][]257}}
\expandafter\gdef\csname r@ch26:lem:alternative-energy-handoff\endcsname{{30.14}{257}{}{lemma.738}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:lem:alternative-energy-handoff@cref\endcsname{{[lemma][14][]30.14}{[1][257][]257}}
\expandafter\gdef\csname r@ch26:thm:bounded-quartic-observable-paths\endcsname{{30.15}{258}{}{theorem.739}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:bounded-quartic-observable-paths@cref\endcsname{{[theorem][15][30]30.15}{[1][258][]258}}
\expandafter\gdef\csname r@ch26:eq:alternative-energy-path-comparison\endcsname{{30.17}{258}{}{equation.740}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:alternative-energy-path-comparison@cref\endcsname{{[equation][17][30]30.17}{[1][258][]258}}
\expandafter\gdef\csname r@ch26:rem:modified-flow-scope\endcsname{{30.16}{260}{}{remark.744}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:rem:modified-flow-scope@cref\endcsname{{[remark][16][]30.16}{[1][260][]260}}
% Part V
\expandafter\gdef\csname r@part:shortdistance\endcsname{{X}{2}{\contentsname \@mkboth {\MakeUppercase \contentsname }{\MakeUppercase \contentsname }}{part.2}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@part:shortdistance@cref\endcsname{{[part][10][]X}{[1][2][]2}}
\expandafter\gdef\csname r@ch28:chapter\endcsname{{31}{2}{Reconstruction estimates and divergence-free currents}{chapter.3}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:chapter@cref\endcsname{{[chapter][31][]31}{[1][2][]2}}
\expandafter\gdef\csname r@ch28:sec:setting\endcsname{{31.1}{4}{Setting and conventions}{section.4}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:sec:setting@cref\endcsname{{[section][1][31]31.1}{[1][4][]4}}
\expandafter\gdef\csname r@ch28:sec:free\endcsname{{31.2}{6}{Divergence-free currents and the Maxwell covariance}{section.7}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:sec:free@cref\endcsname{{[section][2][31]31.2}{[1][6][]6}}
\expandafter\gdef\csname r@ch28:def:closed-current-target\endcsname{{31.1}{6}{}{definition.8}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:def:closed-current-target@cref\endcsname{{[definition][1][]31.1}{[1][6][]6}}
\expandafter\gdef\csname r@ch28:lem:free-current-covariance\endcsname{{31.2}{7}{}{lemma.9}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:lem:free-current-covariance@cref\endcsname{{[lemma][2][]31.2}{[1][7][]7}}
\expandafter\gdef\csname r@ch28:eq:cycle-Fourier\endcsname{{31.1}{9}{Step 4: the lattice covariance}{equation.19}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:cycle-Fourier@cref\endcsname{{[equation][1][31]31.1}{[1][9][]9}}
\expandafter\gdef\csname r@ch28:eq:current-attachment-error\endcsname{{31.2}{10}{Step 5: the limit $a\downarrow 0$}{equation.21}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:current-attachment-error@cref\endcsname{{[equation][2][31]31.2}{[1][10][]10}}
\expandafter\gdef\csname r@ch28:eq:lattice-current-bound\endcsname{{31.3}{10}{Step 5: the limit $a\downarrow 0$}{equation.22}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:lattice-current-bound@cref\endcsname{{[equation][3][31]31.3}{[1][10][]10}}
\expandafter\gdef\csname r@ch28:sec:linear\endcsname{{31.3}{11}{Linearized reconstruction and nonlinear error terms}{section.24}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:sec:linear@cref\endcsname{{[section][3][31]31.3}{[1][11][]11}}
\expandafter\gdef\csname r@ch28:thm:one-shot-and-source-sum\endcsname{{31.3}{11}{}{theorem.25}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:thm:one-shot-and-source-sum@cref\endcsname{{[theorem][3][31]31.3}{[1][11][]11}}
\expandafter\gdef\csname r@ch28:rem:circulation-only\endcsname{{31.4}{13}{}{remark.31}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:rem:circulation-only@cref\endcsname{{[remark][4][]31.4}{[1][13][]13}}
\expandafter\gdef\csname r@ch28:eq:iterated-positive-series\endcsname{{31.4}{16}{Step 5: iterated majorants and the remainder bound}{equation.37}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:iterated-positive-series@cref\endcsname{{[equation][4][31]31.4}{[1][16][]16}}
\expandafter\gdef\csname r@ch28:eq:two-leg-source\endcsname{{31.5}{17}{Step 5: iterated majorants and the remainder bound}{equation.38}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:two-leg-source@cref\endcsname{{[equation][5][31]31.5}{[1][16][]17}}
\expandafter\gdef\csname r@ch28:eq:finite-source-telescope\endcsname{{31.6}{17}{Step 6: the field $\widehat {\mathcal E}_j^a$}{equation.40}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:finite-source-telescope@cref\endcsname{{[equation][6][31]31.6}{[1][17][]17}}
\expandafter\gdef\csname r@ch28:eq:extension-majorant\endcsname{{31.7}{17}{Step 6: the field $\widehat {\mathcal E}_j^a$}{equation.41}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:extension-majorant@cref\endcsname{{[equation][7][31]31.7}{[1][17][]17}}
\expandafter\gdef\csname r@ch28:sec:means\endcsname{{31.4}{18}{Conditional current coefficients and means}{section.42}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:sec:means@cref\endcsname{{[section][4][31]31.4}{[1][18][]18}}
\expandafter\gdef\csname r@ch28:thm:predictable-rows-and-means\endcsname{{31.5}{18}{}{theorem.43}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:thm:predictable-rows-and-means@cref\endcsname{{[theorem][5][31]31.5}{[1][18][]18}}
\expandafter\gdef\csname r@ch28:eq:largest-integrated-coupling\endcsname{{31.8}{20}{Step 1: the kernels and the threshold}{equation.49}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:largest-integrated-coupling@cref\endcsname{{[equation][8][31]31.8}{[1][20][]20}}
\expandafter\gdef\csname r@ch28:eq:cutoff-envelope\endcsname{{31.9}{21}{Step 1: the kernels and the threshold}{equation.50}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:cutoff-envelope@cref\endcsname{{[equation][9][31]31.9}{[1][20][]21}}
\expandafter\gdef\csname r@ch28:eq:joint-fine-output\endcsname{{31.10}{21}{Step 2: coefficients at a general coarse field}{equation.52}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:joint-fine-output@cref\endcsname{{[equation][10][31]31.10}{[1][21][]21}}
\expandafter\gdef\csname r@ch28:eq:normalized-white-matrix\endcsname{{31.11}{21}{Step 2: coefficients at a general coarse field}{equation.53}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:normalized-white-matrix@cref\endcsname{{[equation][11][31]31.11}{[1][21][]21}}
\expandafter\gdef\csname r@ch28:eq:score-degree-decomposition\endcsname{{31.12}{22}{Step 4: coupling powers in the potential}{equation.56}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:score-degree-decomposition@cref\endcsname{{[equation][12][31]31.12}{[1][22][]22}}
\expandafter\gdef\csname r@ch28:eq:rooted-score-Ltwo\endcsname{{31.13}{23}{Step 5: the mixed derivative of the potential}{equation.58}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:rooted-score-Ltwo@cref\endcsname{{[equation][13][31]31.13}{[1][23][]23}}
\expandafter\gdef\csname r@ch28:eq:available-score-bound\endcsname{{31.14}{23}{Step 5: the mixed derivative of the potential}{equation.59}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:available-score-bound@cref\endcsname{{[equation][14][31]31.14}{[1][23][]23}}
\expandafter\gdef\csname r@ch28:eq:output-mean-derivative\endcsname{{31.15}{23}{Step 6: general equivariant outputs}{equation.61}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:output-mean-derivative@cref\endcsname{{[equation][15][31]31.15}{[1][23][]23}}
\expandafter\gdef\csname r@ch28:sec:covariance\endcsname{{31.5}{25}{Covariance identity for the linearized reconstruction}{section.63}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:sec:covariance@cref\endcsname{{[section][5][31]31.5}{[1][25][]25}}
\expandafter\gdef\csname r@ch28:thm:maxwell-calibration\endcsname{{31.6}{26}{}{theorem.64}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:thm:maxwell-calibration@cref\endcsname{{[theorem][6][31]31.6}{[1][26][]26}}
\expandafter\gdef\csname r@ch28:rem:linearized-only\endcsname{{31.7}{27}{}{remark.65}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:rem:linearized-only@cref\endcsname{{[remark][7][]31.7}{[1][27][]27}}
\expandafter\gdef\csname r@ch28:eq:tangent-quadratic-telescope\endcsname{{31.16}{28}{Step 2: the energy identity}{equation.68}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:tangent-quadratic-telescope@cref\endcsname{{[equation][16][31]31.16}{[1][27][]28}}
\expandafter\gdef\csname r@ch28:eq:tangent-target\endcsname{{31.17}{28}{Step 2: the energy identity}{equation.69}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:tangent-target@cref\endcsname{{[equation][17][31]31.17}{[1][28][]28}}
\expandafter\gdef\csname r@ch28:eq:retained-current-row\endcsname{{31.18}{29}{Step 5: the coarse contribution}{equation.73}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:retained-current-row@cref\endcsname{{[equation][18][31]31.18}{[1][29][]29}}
\expandafter\gdef\csname r@ch28:eq:fixed-side-covariance-bound\endcsname{{31.19}{29}{Step 5: the coarse contribution}{equation.74}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:fixed-side-covariance-bound@cref\endcsname{{[equation][19][31]31.19}{[1][29][]29}}
\expandafter\gdef\csname r@ch28:sec:scales\endcsname{{31.6}{29}{The coarse-field contribution and summation over scales}{section.76}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:sec:scales@cref\endcsname{{[section][6][31]31.6}{[1][29][]29}}
\expandafter\gdef\csname r@ch28:lem:top-current-and-caps\endcsname{{31.8}{30}{}{lemma.77}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:lem:top-current-and-caps@cref\endcsname{{[lemma][8][]31.8}{[1][30][]30}}
\expandafter\gdef\csname r@ch28:eq:actual-forced-recurrence\endcsname{{31.20}{31}{Step 1: a recurrence}{equation.79}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:actual-forced-recurrence@cref\endcsname{{[equation][20][31]31.20}{[1][30][]31}}
\expandafter\gdef\csname r@ch28:eq:uniform-forced-cap\endcsname{{31.21}{31}{Step 3: the rate $\alpha _*^j$}{equation.82}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:uniform-forced-cap@cref\endcsname{{[equation][21][31]31.21}{[1][31][]31}}
\expandafter\gdef\csname r@ch28:eq:affine-white-cut\endcsname{{31.22}{32}{Step 4: the sum of the bounds}{equation.84}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:affine-white-cut@cref\endcsname{{[equation][22][31]31.22}{[1][31][]32}}
\expandafter\gdef\csname r@ch28:eq:fixed-coarse-top\endcsname{{31.23}{32}{Step 6: the coarse-field contribution}{equation.87}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:fixed-coarse-top@cref\endcsname{{[equation][23][31]31.23}{[1][32][]32}}
\expandafter\gdef\csname r@ch28:eq:centered-triangle-sum\endcsname{{31.24}{33}{Step 7: conditional means and centered remainders}{equation.89}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:centered-triangle-sum@cref\endcsname{{[equation][24][31]31.24}{[1][32][]33}}
\expandafter\gdef\csname r@ch28:rem:forced-cap-alternative\endcsname{{31.9}{33}{}{remark.90}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:rem:forced-cap-alternative@cref\endcsname{{[remark][9][]31.9}{[1][33][]33}}
\expandafter\gdef\csname r@ch28:eq:cap-input-enclosure\endcsname{{31.25}{33}{}{equation.91}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:eq:cap-input-enclosure@cref\endcsname{{[equation][25][31]31.25}{[1][33][]33}}
\expandafter\gdef\csname r@ch28:sec:fixed-coupling\endcsname{{31.7}{34}{Validity at each fixed positive coupling}{section.92}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:sec:fixed-coupling@cref\endcsname{{[section][7][31]31.7}{[1][33][]34}}
\expandafter\gdef\csname r@ch28:lem:finite-ceiling-current-suppliers\endcsname{{31.10}{34}{}{lemma.93}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch28:lem:finite-ceiling-current-suppliers@cref\endcsname{{[lemma][10][]31.10}{[1][33][]34}}
\expandafter\gdef\csname r@ch29:chapter\endcsname{{32}{36}{Gaussian short-distance limits and asymptotic independence}{chapter.99}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:chapter@cref\endcsname{{[chapter][32][]32}{[1][36][]36}}
\expandafter\gdef\csname r@ch29:sec:setting\endcsname{{32.1}{37}{Setting and inputs}{section.100}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:sec:setting@cref\endcsname{{[section][1][32]32.1}{[1][37][]37}}
\expandafter\gdef\csname r@ch29:def:current-chain\endcsname{{32.1}{37}{}{definition.101}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:def:current-chain@cref\endcsname{{[definition][1][]32.1}{[1][37][]37}}
\expandafter\gdef\csname r@ch29:rem:fixed-current-output\endcsname{{32.2}{38}{}{remark.102}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:rem:fixed-current-output@cref\endcsname{{[remark][2][]32.2}{[1][38][]38}}
\expandafter\gdef\csname r@ch29:eq:fixed-extension-ceiling\endcsname{{32.1}{39}{}{equation.103}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:fixed-extension-ceiling@cref\endcsname{{[equation][1][32]32.1}{[1][38][]39}}
\expandafter\gdef\csname r@ch29:prop:inputs\endcsname{{32.3}{39}{}{proposition.104}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:prop:inputs@cref\endcsname{{[proposition][3][]32.3}{[1][39][]39}}
\expandafter\gdef\csname r@ch29:sec:stein\endcsname{{32.2}{40}{Integration by parts and conditional characteristic functions}{section.110}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:sec:stein@cref\endcsname{{[section][2][32]32.2}{[1][40][]40}}
\expandafter\gdef\csname r@ch29:lem:fiber-and-martingale-stein\endcsname{{32.4}{40}{}{lemma.111}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:lem:fiber-and-martingale-stein@cref\endcsname{{[lemma][4][]32.4}{[1][40][]40}}
\expandafter\gdef\csname r@ch29:eq:interior-ibp\endcsname{{32.2}{41}{Integration by parts and conditional characteristic functions}{equation.112}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:interior-ibp@cref\endcsname{{[equation][2][32]32.2}{[1][41][]41}}
\expandafter\gdef\csname r@ch29:eq:stein-split\endcsname{{32.3}{41}{Integration by parts and conditional characteristic functions}{equation.113}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:stein-split@cref\endcsname{{[equation][3][32]32.3}{[1][41][]41}}
\expandafter\gdef\csname r@ch29:eq:cutoff-score-mean\endcsname{{32.4}{41}{Integration by parts and conditional characteristic functions}{equation.114}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:cutoff-score-mean@cref\endcsname{{[equation][4][32]32.4}{[1][41][]41}}
\expandafter\gdef\csname r@ch29:lem:characteristic-telescope\endcsname{{32.5}{41}{}{lemma.115}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:lem:characteristic-telescope@cref\endcsname{{[lemma][5][]32.5}{[1][41][]41}}
\expandafter\gdef\csname r@ch29:eq:one-step-characteristic\endcsname{{32.5}{42}{Integration by parts and conditional characteristic functions}{equation.116}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:one-step-characteristic@cref\endcsname{{[equation][5][32]32.5}{[1][42][]42}}
\expandafter\gdef\csname r@ch29:sec:cutoff\endcsname{{32.3}{42}{Telescoping circulations and dimension-dependent cutoffs}{section.117}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:sec:cutoff@cref\endcsname{{[section][3][32]32.3}{[1][42][]42}}
\expandafter\gdef\csname r@ch29:lem:current-telescope-and-cutoff\endcsname{{32.6}{42}{}{lemma.118}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:lem:current-telescope-and-cutoff@cref\endcsname{{[lemma][6][]32.6}{[1][42][]42}}
\expandafter\gdef\csname r@ch29:eq:couplings-and-radii\endcsname{{32.6}{44}{Telescoping circulations and dimension-dependent cutoffs}{equation.119}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:couplings-and-radii@cref\endcsname{{[equation][6][32]32.6}{[1][44][]44}}
\expandafter\gdef\csname r@ch29:eq:mean-slack\endcsname{{32.7}{44}{Telescoping circulations and dimension-dependent cutoffs}{equation.120}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:mean-slack@cref\endcsname{{[equation][7][32]32.7}{[1][44][]44}}
\expandafter\gdef\csname r@ch29:eq:cutoff-probability\endcsname{{32.8}{45}{The probability that the cutoff is active}{equation.123}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:cutoff-probability@cref\endcsname{{[equation][8][32]32.8}{[1][45][]45}}
\expandafter\gdef\csname r@ch29:eq:score-oscillation\endcsname{{32.9}{45}{The error terms with the dimension retained}{equation.125}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:score-oscillation@cref\endcsname{{[equation][9][32]32.9}{[1][45][]45}}
\expandafter\gdef\csname r@ch29:eq:dimension-score\endcsname{{32.10}{45}{The error terms with the dimension retained}{equation.126}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:dimension-score@cref\endcsname{{[equation][10][32]32.10}{[1][45][]45}}
\expandafter\gdef\csname r@ch29:eq:tail-envelope\endcsname{{32.11}{46}{The deterministic envelope}{equation.128}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:tail-envelope@cref\endcsname{{[equation][11][32]32.11}{[1][45][]46}}
\expandafter\gdef\csname r@ch29:sec:gaussian\endcsname{{32.4}{46}{Gaussian approximation of the circulations}{section.129}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:sec:gaussian@cref\endcsname{{[section][4][32]32.4}{[1][46][]46}}
\expandafter\gdef\csname r@ch29:lem:smoothing\endcsname{{32.7}{46}{}{lemma.130}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:lem:smoothing@cref\endcsname{{[lemma][7][]32.7}{[1][46][]46}}
\expandafter\gdef\csname r@ch29:eq:smoothing-bound\endcsname{{32.12}{46}{}{equation.131}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:smoothing-bound@cref\endcsname{{[equation][12][32]32.12}{[1][46][]46}}
\expandafter\gdef\csname r@ch29:eq:quadratic-truncation\endcsname{{32.13}{46}{}{equation.132}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:quadratic-truncation@cref\endcsname{{[equation][13][32]32.13}{[1][46][]46}}
\expandafter\gdef\csname r@ch29:thm:every-fixed-side-gaussian\endcsname{{32.8}{47}{}{theorem.133}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:thm:every-fixed-side-gaussian@cref\endcsname{{[theorem][8][32]32.8}{[1][47][]47}}
\expandafter\gdef\csname r@ch29:eq:row-square-sum\endcsname{{32.14}{48}{Step 1: sums over the scales}{equation.136}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:row-square-sum@cref\endcsname{{[equation][14][32]32.14}{[1][48][]48}}
\expandafter\gdef\csname r@ch29:eq:log-geometric\endcsname{{32.15}{48}{Step 1: sums over the scales}{equation.137}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:log-geometric@cref\endcsname{{[equation][15][32]32.15}{[1][48][]48}}
\expandafter\gdef\csname r@ch29:eq:small-variation-sum\endcsname{{32.16}{49}{Step 1: sums over the scales}{equation.138}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:small-variation-sum@cref\endcsname{{[equation][16][32]32.16}{[1][48][]49}}
\expandafter\gdef\csname r@ch29:eq:main-stein-sum\endcsname{{32.17}{49}{Step 1: sums over the scales}{equation.139}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:main-stein-sum@cref\endcsname{{[equation][17][32]32.17}{[1][49][]49}}
\expandafter\gdef\csname r@ch29:eq:cap-in-current-units\endcsname{{32.18}{49}{Step 2: the variation of the coefficients}{equation.141}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:cap-in-current-units@cref\endcsname{{[equation][18][32]32.18}{[1][49][]49}}
\expandafter\gdef\csname r@ch29:eq:background-sum\endcsname{{32.19}{49}{Step 2: the variation of the coefficients}{equation.142}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:background-sum@cref\endcsname{{[equation][19][32]32.19}{[1][49][]49}}
\expandafter\gdef\csname r@ch29:eq:averaged-remainder\endcsname{{32.20}{50}{Step 3: the remainder}{equation.144}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:averaged-remainder@cref\endcsname{{[equation][20][32]32.20}{[1][49][]50}}
\expandafter\gdef\csname r@ch29:eq:nonmartingale-bound\endcsname{{32.21}{50}{Step 3: the remainder}{equation.145}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:nonmartingale-bound@cref\endcsname{{[equation][21][32]32.21}{[1][50][]50}}
\expandafter\gdef\csname r@ch29:eq:flat-row-error\endcsname{{32.22}{50}{Step 4: coefficients at the zero field}{equation.147}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:flat-row-error@cref\endcsname{{[equation][22][32]32.22}{[1][50][]50}}
\expandafter\gdef\csname r@ch29:eq:deterministic-current-error\endcsname{{32.23}{51}{Step 5: characteristic functions}{equation.149}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:deterministic-current-error@cref\endcsname{{[equation][23][32]32.23}{[1][51][]51}}
\expandafter\gdef\csname r@ch29:eq:current-characteristic-limit\endcsname{{32.24}{51}{Step 5: characteristic functions}{equation.150}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:current-characteristic-limit@cref\endcsname{{[equation][24][32]32.24}{[1][51][]51}}
\expandafter\gdef\csname r@ch29:rem:fixed-coupling-scope\endcsname{{32.9}{52}{}{remark.152}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:rem:fixed-coupling-scope@cref\endcsname{{[remark][9][]32.9}{[1][52][]52}}
\expandafter\gdef\csname r@ch29:lem:fixed-prefix-current-mixing\endcsname{{32.10}{52}{}{lemma.153}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:lem:fixed-prefix-current-mixing@cref\endcsname{{[lemma][10][]32.10}{[1][52][]52}}
\expandafter\gdef\csname r@ch29:eq:conditional-prefix-gaussian\endcsname{{32.25}{52}{}{equation.154}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:conditional-prefix-gaussian@cref\endcsname{{[equation][25][32]32.25}{[1][52][]52}}
\expandafter\gdef\csname r@ch29:eq:prefix-bounded-multiplier\endcsname{{32.26}{52}{}{equation.155}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:prefix-bounded-multiplier@cref\endcsname{{[equation][26][32]32.26}{[1][52][]52}}
\expandafter\gdef\csname r@ch29:sec:flow\endcsname{{32.5}{54}{Wilson flow and removal of the conditioning}{section.161}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:sec:flow@cref\endcsname{{[section][5][32]32.5}{[1][54][]54}}
\expandafter\gdef\csname r@ch29:lem:actual-flow-trace-reduction\endcsname{{32.11}{55}{}{lemma.162}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:lem:actual-flow-trace-reduction@cref\endcsname{{[lemma][11][]32.11}{[1][55][]55}}
\expandafter\gdef\csname r@ch29:eq:three-flow-powers\endcsname{{32.27}{57}{Step 3: the circulation of the flowed field}{equation.169}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:three-flow-powers@cref\endcsname{{[equation][27][32]32.27}{[1][57][]57}}
\expandafter\gdef\csname r@ch29:lem:finite-ceiling-prefix-gaussian\endcsname{{32.12}{58}{}{lemma.172}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:lem:finite-ceiling-prefix-gaussian@cref\endcsname{{[lemma][12][]32.12}{[1][58][]58}}
\expandafter\gdef\csname r@ch29:sec:continuum\endcsname{{32.6}{59}{The continuum limit}{section.173}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:sec:continuum@cref\endcsname{{[section][6][32]32.6}{[1][58][]59}}
\expandafter\gdef\csname r@ch29:thm:short-distance-gaussian\endcsname{{32.13}{59}{}{theorem.174}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:thm:short-distance-gaussian@cref\endcsname{{[theorem][13][32]32.13}{[1][59][]59}}
\expandafter\gdef\csname r@ch29:sec:variance\endcsname{{32.7}{60}{A uniform short-distance variance lower bound}{section.175}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:sec:variance@cref\endcsname{{[section][7][32]32.7}{[1][60][]60}}
\expandafter\gdef\csname r@ch29:cor:short-distance-variance-liminf\endcsname{{32.14}{60}{}{corollary.176}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:cor:short-distance-variance-liminf@cref\endcsname{{[corollary][14][]32.14}{[1][60][]60}}
\expandafter\gdef\csname r@ch29:eq:variance-liminf\endcsname{{32.28}{60}{}{equation.177}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:variance-liminf@cref\endcsname{{[equation][28][32]32.28}{[1][60][]60}}
\expandafter\gdef\csname r@ch29:sec:macroscopic\endcsname{{32.8}{61}{Asymptotic independence from macroscopic Wilson loops}{section.178}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:sec:macroscopic@cref\endcsname{{[section][8][32]32.8}{[1][60][]61}}
\expandafter\gdef\csname r@ch29:lem:same-history-macroscopic-prefix\endcsname{{32.15}{61}{}{lemma.179}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:lem:same-history-macroscopic-prefix@cref\endcsname{{[lemma][15][]32.15}{[1][61][]61}}
\expandafter\gdef\csname r@ch29:eq:macroscopic-prefix-approximation\endcsname{{32.29}{61}{}{equation.180}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:macroscopic-prefix-approximation@cref\endcsname{{[equation][29][32]32.29}{[1][61][]61}}
\expandafter\gdef\csname r@ch29:eq:two-grade-prefix-tail\endcsname{{32.30}{61}{Asymptotic independence from macroscopic Wilson loops}{equation.181}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:eq:two-grade-prefix-tail@cref\endcsname{{[equation][30][32]32.30}{[1][61][]61}}
\expandafter\gdef\csname r@ch29:eq:uniform-prefix-loop-modulus\endcsname{{32.31}{62}{Asymptotic independence from macroscopic Wilson loops}{equation.182}{3d-ym-torus-volume-v.pdf}}
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\expandafter\gdef\csname r@ch29:eq:prefix-error-with-radius@cref\endcsname{{[equation][32][32]32.32}{[1][62][]62}}
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\expandafter\gdef\csname r@ch29:thm:short-distance-orbit-borel-mixing\endcsname{{32.17}{63}{}{theorem.187}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch29:thm:short-distance-orbit-borel-mixing@cref\endcsname{{[theorem][17][32]32.17}{[1][63][]63}}
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\expandafter\gdef\csname r@part:weak@cref\endcsname{{[part][11][]XI}{[1][67][]67}}
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\expandafter\gdef\csname r@ch27:chapter@cref\endcsname{{[chapter][33][]33}{[1][67][]67}}
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\expandafter\gdef\csname r@ch27:lem:smooth-charts@cref\endcsname{{[lemma][1][]33.1}{[1][70][]70}}
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\expandafter\gdef\csname r@ch27:eq:weighted-density\endcsname{{33.4}{77}{}{equation.221}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:weighted-density@cref\endcsname{{[equation][4][33]33.4}{[1][77][]77}}
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\expandafter\gdef\csname r@ch27:lem:pointwise-laplace@cref\endcsname{{[lemma][5][]33.5}{[1][77][]77}}
\expandafter\gdef\csname r@ch27:eq:exponential-difference\endcsname{{33.6}{78}{Laplace asymptotics in normal coordinates}{equation.228}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:exponential-difference@cref\endcsname{{[equation][6][33]33.6}{[1][78][]78}}
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\expandafter\gdef\csname r@ch27:eq:pointwise-laplace@cref\endcsname{{[equation][7][33]33.7}{[1][79][]79}}
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\expandafter\gdef\csname r@ch27:sec:central@cref\endcsname{{[section][5][33]33.5}{[1][81][]81}}
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\expandafter\gdef\csname r@ch27:thm:central-stiff-chart@cref\endcsname{{[theorem][8][33]33.8}{[1][86][]86}}
\expandafter\gdef\csname r@ch27:eq:stiff-stationarity\endcsname{{33.12}{89}{Step 4: stationarity and the gap}{equation.245}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:stiff-stationarity@cref\endcsname{{[equation][12][33]33.12}{[1][89][]89}}
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\expandafter\gdef\csname r@ch27:sec:normalizer@cref\endcsname{{[section][6][33]33.6}{[1][90][]90}}
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\expandafter\gdef\csname r@ch27:lem:regular-flat-tube@cref\endcsname{{[lemma][9][]33.9}{[1][91][]91}}
\expandafter\gdef\csname r@ch27:thm:sharp-normalizer\endcsname{{33.10}{93}{}{theorem.251}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:thm:sharp-normalizer@cref\endcsname{{[theorem][10][33]33.10}{[1][93][]93}}
\expandafter\gdef\csname r@ch27:eq:bare-constant-quartic\endcsname{{33.13}{95}{Step 1: constant configurations}{equation.256}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:bare-constant-quartic@cref\endcsname{{[equation][13][33]33.13}{[1][94][]95}}
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\expandafter\gdef\csname r@ch27:eq:central-bare-bound@cref\endcsname{{[equation][14][33]33.14}{[1][95][]95}}
\expandafter\gdef\csname r@ch27:eq:dyadic-cone\endcsname{{33.15}{95}{Step 3: the nine-dimensional estimate}{equation.260}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:dyadic-cone@cref\endcsname{{[equation][15][33]33.15}{[1][95][]95}}
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\expandafter\gdef\csname r@ch27:sec:covariance@cref\endcsname{{[section][7][33]33.7}{[1][97][]97}}
\expandafter\gdef\csname r@ch27:thm:flowed-log-covariance\endcsname{{33.11}{97}{}{theorem.264}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:thm:flowed-log-covariance@cref\endcsname{{[theorem][11][33]33.11}{[1][97][]97}}
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\expandafter\gdef\csname r@ch27:eq:charged-quadratic@cref\endcsname{{[equation][16][33]33.16}{[1][99][]99}}
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\expandafter\gdef\csname r@ch27:eq:charged-covariance@cref\endcsname{{[equation][17][33]33.17}{[1][100][]100}}
\expandafter\gdef\csname r@ch27:eq:lattice-maxwell-exponent\endcsname{{33.18}{102}{Step 8: the lattice covariance}{equation.279}{3d-ym-torus-volume-v.pdf}}
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\expandafter\gdef\csname r@ch27:eq:lattice-sine-bounds@cref\endcsname{{[equation][19][33]33.19}{[1][103][]103}}
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\expandafter\gdef\csname r@ch27:rem:scope@cref\endcsname{{[remark][13][]33.13}{[1][111][]112}}
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\expandafter\gdef\csname r@ch27:lem:abelian-reference@cref\endcsname{{[lemma][14][]33.14}{[1][112][]112}}
\expandafter\gdef\csname r@ch30:chapter\endcsname{{34}{114}{Uniform fluctuation bounds and comparison of holonomies}{chapter.306}{3d-ym-torus-volume-v.pdf}}
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\expandafter\gdef\csname r@ch30:lem:component-metric-suppliers\endcsname{{34.3}{126}{}{lemma.346}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:lem:component-metric-suppliers@cref\endcsname{{[lemma][3][]34.3}{[1][126][]126}}
\expandafter\gdef\csname r@ch30:eq:local-scales\endcsname{{34.17}{127}{}{equation.353}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:local-scales@cref\endcsname{{[equation][17][34]34.17}{[1][127][]127}}
\expandafter\gdef\csname r@ch30:eq:metric-cover\endcsname{{34.18}{127}{}{equation.354}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:metric-cover@cref\endcsname{{[equation][18][34]34.18}{[1][127][]127}}
\expandafter\gdef\csname r@ch30:eq:mixed-rows\endcsname{{34.19}{128}{Heat estimates on a common mesh and fluctuation tails}{equation.355}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:mixed-rows@cref\endcsname{{[equation][19][34]34.19}{[1][128][]128}}
\expandafter\gdef\csname r@ch30:eq:support-cap\endcsname{{34.20}{129}{Heat estimates on a common mesh and fluctuation tails}{equation.356}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:support-cap@cref\endcsname{{[equation][20][34]34.20}{[1][129][]129}}
\expandafter\gdef\csname r@ch30:eq:ancestor-normalization\endcsname{{34.21}{129}{Heat estimates on a common mesh and fluctuation tails}{equation.357}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:ancestor-normalization@cref\endcsname{{[equation][21][34]34.21}{[1][129][]129}}
\expandafter\gdef\csname r@ch30:lem:zero-white-and-chaining\endcsname{{34.4}{130}{}{lemma.358}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:lem:zero-white-and-chaining@cref\endcsname{{[lemma][4][]34.4}{[1][130][]130}}
\expandafter\gdef\csname r@ch30:eq:classical-adjacent\endcsname{{34.22}{131}{Heat estimates on a common mesh and fluctuation tails}{equation.362}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:classical-adjacent@cref\endcsname{{[equation][22][34]34.22}{[1][130][]131}}
\expandafter\gdef\csname r@ch30:eq:det-ancestor\endcsname{{34.23}{131}{Heat estimates on a common mesh and fluctuation tails}{equation.363}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:det-ancestor@cref\endcsname{{[equation][23][34]34.23}{[1][131][]131}}
\expandafter\gdef\csname r@ch30:eq:det-rough\endcsname{{34.24}{131}{Heat estimates on a common mesh and fluctuation tails}{equation.364}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:det-rough@cref\endcsname{{[equation][24][34]34.24}{[1][131][]131}}
\expandafter\gdef\csname r@ch30:lem:anchored-chaining\endcsname{{34.5}{132}{}{lemma.365}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:lem:anchored-chaining@cref\endcsname{{[lemma][5][]34.5}{[1][132][]132}}
\expandafter\gdef\csname r@ch30:lem:cutoff-logarithms\endcsname{{34.6}{133}{}{lemma.368}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:lem:cutoff-logarithms@cref\endcsname{{[lemma][6][]34.6}{[1][133][]133}}
\expandafter\gdef\csname r@ch30:sec:maxima\endcsname{{34.3}{134}{Conditional bounds uniform in the coupling}{section.369}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:sec:maxima@cref\endcsname{{[section][3][34]34.3}{[1][133][]134}}
\expandafter\gdef\csname r@ch30:thm:uniform-component-maxima\endcsname{{34.7}{134}{}{theorem.370}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:thm:uniform-component-maxima@cref\endcsname{{[theorem][7][34]34.7}{[1][134][]134}}
\expandafter\gdef\csname r@ch30:eq:uniform-U\endcsname{{34.25}{137}{Step 5: summation over levels}{equation.376}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:uniform-U@cref\endcsname{{[equation][25][34]34.25}{[1][137][]137}}
\expandafter\gdef\csname r@ch30:eq:uniform-first\endcsname{{34.26}{137}{Step 5: summation over levels}{equation.377}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:uniform-first@cref\endcsname{{[equation][26][34]34.26}{[1][137][]137}}
\expandafter\gdef\csname r@ch30:sec:frame\endcsname{{34.4}{138}{The initial gauge frame along the flow}{section.379}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:sec:frame@cref\endcsname{{[section][4][34]34.4}{[1][138][]138}}
\expandafter\gdef\csname r@ch30:thm:root-matrix-modulus\endcsname{{34.8}{139}{}{remark.380}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:thm:root-matrix-modulus@cref\endcsname{{[remark][8][]34.8}{[1][139][]139}}
\expandafter\gdef\csname r@ch30:sec:paired\endcsname{{34.5}{139}{Comparison of linear and quadratic flow data}{section.381}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:sec:paired@cref\endcsname{{[section][5][34]34.5}{[1][139][]139}}
\expandafter\gdef\csname r@ch30:thm:paired-enhancement-return\endcsname{{34.9}{140}{}{theorem.382}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:thm:paired-enhancement-return@cref\endcsname{{[theorem][9][34]34.9}{[1][139][]140}}
\expandafter\gdef\csname r@ch30:eq:paired-field-moments\endcsname{{34.27}{142}{Step 1: moments of the fields}{equation.384}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:paired-field-moments@cref\endcsname{{[equation][27][34]34.27}{[1][141][]142}}
\expandafter\gdef\csname r@ch30:eq:marked-cubic\endcsname{{34.28}{142}{Step 2: exact telescopes and the classical cubic difference}{equation.386}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:marked-cubic@cref\endcsname{{[equation][28][34]34.28}{[1][142][]142}}
\expandafter\gdef\csname r@ch30:eq:paired-quantum\endcsname{{34.29}{142}{Step 3: decomposition of the pulled-back forms}{equation.388}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:paired-quantum@cref\endcsname{{[equation][29][34]34.29}{[1][142][]142}}
\expandafter\gdef\csname r@ch30:eq:paired-ancestor\endcsname{{34.30}{143}{Step 3: decomposition of the pulled-back forms}{equation.389}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:paired-ancestor@cref\endcsname{{[equation][30][34]34.30}{[1][143][]143}}
\expandafter\gdef\csname r@ch30:eq:paired-sizes\endcsname{{34.31}{143}{Step 4: sizes of the terms and summation}{equation.391}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:paired-sizes@cref\endcsname{{[equation][31][34]34.31}{[1][143][]143}}
\expandafter\gdef\csname r@ch30:eq:paired-U\endcsname{{34.32}{144}{Step 4: sizes of the terms and summation}{equation.392}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:paired-U@cref\endcsname{{[equation][32][34]34.32}{[1][143][]144}}
\expandafter\gdef\csname r@ch30:eq:paired-X\endcsname{{34.33}{144}{Step 4: sizes of the terms and summation}{equation.393}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:paired-X@cref\endcsname{{[equation][33][34]34.33}{[1][144][]144}}
\expandafter\gdef\csname r@ch30:eq:paired-three\endcsname{{34.34}{144}{Step 4: sizes of the terms and summation}{equation.394}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:paired-three@cref\endcsname{{[equation][34][34]34.34}{[1][144][]144}}
\expandafter\gdef\csname r@ch30:eq:dangerous-smoother\endcsname{{34.35}{144}{Step 5: the cross products with the predictable term}{equation.396}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:dangerous-smoother@cref\endcsname{{[equation][35][34]34.35}{[1][144][]144}}
\expandafter\gdef\csname r@ch30:sec:transport\endcsname{{34.6}{146}{Holonomy comparison at the weak-coupling scale}{section.398}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:sec:transport@cref\endcsname{{[section][6][34]34.6}{[1][146][]146}}
\expandafter\gdef\csname r@ch30:thm:scaled-matrix-tail\endcsname{{34.10}{146}{}{theorem.399}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:thm:scaled-matrix-tail@cref\endcsname{{[theorem][10][34]34.10}{[1][146][]146}}
\expandafter\gdef\csname r@ch30:eq:enh-natural\endcsname{{34.36}{147}{Holonomy comparison at the weak-coupling scale}{equation.400}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:enh-natural@cref\endcsname{{[equation][36][34]34.36}{[1][147][]147}}
\expandafter\gdef\csname r@ch30:eq:radius-payment\endcsname{{34.37}{147}{Holonomy comparison at the weak-coupling scale}{equation.401}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:radius-payment@cref\endcsname{{[equation][37][34]34.37}{[1][147][]147}}
\expandafter\gdef\csname r@ch30:thm:terminal-frozen-matrix-tail\endcsname{{34.11}{148}{}{theorem.402}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:thm:terminal-frozen-matrix-tail@cref\endcsname{{[theorem][11][34]34.11}{[1][147][]148}}
\expandafter\gdef\csname r@ch30:eq:tower\endcsname{{34.38}{149}{Step 2: transfer of conditional estimates}{equation.405}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:tower@cref\endcsname{{[equation][38][34]34.38}{[1][149][]149}}
\expandafter\gdef\csname r@ch30:eq:terminal-field-moments\endcsname{{34.39}{149}{Step 3: moments of the fields}{equation.407}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:terminal-field-moments@cref\endcsname{{[equation][39][34]34.39}{[1][149][]149}}
\expandafter\gdef\csname r@ch30:eq:terminal-quantum\endcsname{{34.40}{150}{Step 4: the pulled-back forms}{equation.409}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:terminal-quantum@cref\endcsname{{[equation][40][34]34.40}{[1][150][]150}}
\expandafter\gdef\csname r@ch30:eq:terminal-ancestor\endcsname{{34.41}{150}{Step 4: the pulled-back forms}{equation.410}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:terminal-ancestor@cref\endcsname{{[equation][41][34]34.41}{[1][150][]150}}
\expandafter\gdef\csname r@ch30:eq:terminal-sum\endcsname{{34.42}{150}{Step 5: sizes and summation}{equation.412}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:terminal-sum@cref\endcsname{{[equation][42][34]34.42}{[1][150][]150}}
\expandafter\gdef\csname r@ch30:eq:terminal-enh\endcsname{{34.43}{152}{Step 7: the difference of flow data}{equation.415}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:terminal-enh@cref\endcsname{{[equation][43][34]34.43}{[1][152][]152}}
\expandafter\gdef\csname r@ch30:sec:gaussian\endcsname{{34.7}{153}{Linear Gaussian response near a fixed flat connection}{section.417}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:sec:gaussian@cref\endcsname{{[section][7][34]34.7}{[1][153][]153}}
\expandafter\gdef\csname r@ch30:lem:flat-gaussian-response-tail\endcsname{{34.12}{153}{}{lemma.418}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:lem:flat-gaussian-response-tail@cref\endcsname{{[lemma][12][]34.12}{[1][153][]153}}
\expandafter\gdef\csname r@ch30:eq:flat-linear-base\endcsname{{34.44}{154}{Step 1: the base point and the linear recursion}{equation.420}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:flat-linear-base@cref\endcsname{{[equation][44][34]34.44}{[1][154][]154}}
\expandafter\gdef\csname r@ch30:eq:flat-base-bounds\endcsname{{34.45}{154}{Step 1: the base point and the linear recursion}{equation.421}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:flat-base-bounds@cref\endcsname{{[equation][45][34]34.45}{[1][154][]154}}
\expandafter\gdef\csname r@ch30:eq:flat-tangent-recursion\endcsname{{34.46}{155}{Step 1: the base point and the linear recursion}{equation.422}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:flat-tangent-recursion@cref\endcsname{{[equation][46][34]34.46}{[1][154][]155}}
\expandafter\gdef\csname r@ch30:eq:flat-linear-field\endcsname{{34.47}{155}{Step 1: the base point and the linear recursion}{equation.423}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:flat-linear-field@cref\endcsname{{[equation][47][34]34.47}{[1][155][]155}}
\expandafter\gdef\csname r@ch30:eq:linear-enhanced-pair\endcsname{{34.48}{155}{Step 2: first-order identities on the common lattice}{equation.425}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-enhanced-pair@cref\endcsname{{[equation][48][34]34.48}{[1][155][]155}}
\expandafter\gdef\csname r@ch30:eq:linear-adjacent\endcsname{{34.49}{156}{Step 2: first-order identities on the common lattice}{equation.426}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-adjacent@cref\endcsname{{[equation][49][34]34.49}{[1][156][]156}}
\expandafter\gdef\csname r@ch30:eq:linear-fresh-rough\endcsname{{34.50}{156}{Step 3: linear rows and the background on two lattices}{equation.428}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-fresh-rough@cref\endcsname{{[equation][50][34]34.50}{[1][156][]156}}
\expandafter\gdef\csname r@ch30:eq:linear-background-four\endcsname{{34.51}{157}{Step 3: linear rows and the background on two lattices}{equation.429}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-background-four@cref\endcsname{{[equation][51][34]34.51}{[1][157][]157}}
\expandafter\gdef\csname r@ch30:eq:linear-old-fresh\endcsname{{34.52}{157}{Step 3: linear rows and the background on two lattices}{equation.430}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-old-fresh@cref\endcsname{{[equation][52][34]34.52}{[1][157][]157}}
\expandafter\gdef\csname r@ch30:eq:linear-fresh-defect\endcsname{{34.53}{157}{Step 3: linear rows and the background on two lattices}{equation.431}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-fresh-defect@cref\endcsname{{[equation][53][34]34.53}{[1][157][]157}}
\expandafter\gdef\csname r@ch30:eq:linear-classical-row\endcsname{{34.54}{157}{Step 4: the pulled-back forms at first order}{equation.433}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-classical-row@cref\endcsname{{[equation][54][34]34.54}{[1][157][]157}}
\expandafter\gdef\csname r@ch30:eq:linear-ancestor-identity\endcsname{{34.55}{158}{Step 4: the pulled-back forms at first order}{equation.434}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-ancestor-identity@cref\endcsname{{[equation][55][34]34.55}{[1][157][]158}}
\expandafter\gdef\csname r@ch30:eq:linear-rough-innovation\endcsname{{34.56}{159}{Step 4: the pulled-back forms at first order}{equation.435}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-rough-innovation@cref\endcsname{{[equation][56][34]34.56}{[1][159][]159}}
\expandafter\gdef\csname r@ch30:eq:linear-static-innovation\endcsname{{34.57}{159}{Step 4: the pulled-back forms at first order}{equation.436}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-static-innovation@cref\endcsname{{[equation][57][34]34.57}{[1][159][]159}}
\expandafter\gdef\csname r@ch30:eq:linear-cubic-sum\endcsname{{34.58}{159}{Step 4: the pulled-back forms at first order}{equation.437}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-cubic-sum@cref\endcsname{{[equation][58][34]34.58}{[1][159][]159}}
\expandafter\gdef\csname r@ch30:eq:linear-smoother-tail\endcsname{{34.59}{159}{Step 5: summation of both components}{equation.439}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-smoother-tail@cref\endcsname{{[equation][59][34]34.59}{[1][159][]159}}
\expandafter\gdef\csname r@ch30:eq:linear-first-tail\endcsname{{34.60}{160}{Step 5: summation of both components}{equation.440}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-first-tail@cref\endcsname{{[equation][60][34]34.60}{[1][159][]160}}
\expandafter\gdef\csname r@ch30:eq:linear-classical-smoother\endcsname{{34.61}{160}{Step 5: summation of both components}{equation.441}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-classical-smoother@cref\endcsname{{[equation][61][34]34.61}{[1][160][]160}}
\expandafter\gdef\csname r@ch30:eq:linear-enhanced-tail\endcsname{{34.62}{160}{Step 5: summation of both components}{equation.442}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-enhanced-tail@cref\endcsname{{[equation][62][34]34.62}{[1][160][]160}}
\expandafter\gdef\csname r@ch30:eq:linear-flow-response\endcsname{{34.63}{161}{Step 6: the derivative of the flow}{equation.444}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:linear-flow-response@cref\endcsname{{[equation][63][34]34.63}{[1][161][]161}}
\expandafter\gdef\csname r@ch30:sec:flat\endcsname{{34.8}{162}{Flat support of weak-coupling subsequential limits}{section.446}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:sec:flat@cref\endcsname{{[section][8][34]34.8}{[1][162][]162}}
\expandafter\gdef\csname r@ch30:thm:flat-supported-cluster-laws\endcsname{{34.13}{163}{}{theorem.447}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:thm:flat-supported-cluster-laws@cref\endcsname{{[theorem][13][34]34.13}{[1][162][]163}}
\expandafter\gdef\csname r@ch30:eq:small-coupling-energy-tail\endcsname{{34.64}{163}{}{equation.449}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:small-coupling-energy-tail@cref\endcsname{{[equation][64][34]34.64}{[1][163][]163}}
\expandafter\gdef\csname r@ch30:lem:terminal-energy-and-skeleton\endcsname{{34.14}{164}{}{lemma.453}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:lem:terminal-energy-and-skeleton@cref\endcsname{{[lemma][14][]34.14}{[1][163][]164}}
\expandafter\gdef\csname r@ch30:eq:zero-white-energy-identity\endcsname{{34.65}{164}{}{equation.454}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:eq:zero-white-energy-identity@cref\endcsname{{[equation][65][34]34.65}{[1][164][]164}}
\expandafter\gdef\csname r@ch30:lem:early-energy-domination\endcsname{{34.15}{165}{}{lemma.456}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:lem:early-energy-domination@cref\endcsname{{[lemma][15][]34.15}{[1][165][]165}}
\expandafter\gdef\csname r@ch30:tab:products\endcsname{{34.2}{171}{The tools used for the sixteen ordered products and the eight cross terms in \eqref {ch30:eq:bilinear-expansion}. Both orders of every pair are estimated separately. In the fourth column, $s$, $z$, $r$, and $u$ stand for the uncentered summands $\tilde s_j$, $\tilde z_j$, $\tilde r_j$, and $\tilde u_j$ of the deterministic reconstruction in the proof of \cref {ch30:lem:flat-gaussian-response-tail}, whose values at zero fluctuation are $(0,0,R_j,U_j)$ and whose first variations are $(\xi _j,0,\rho _j,\upsilon _j)$; they are not the centered summands of \eqref {ch30:eq:increments}}{table.caption.326}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:tab:products@cref\endcsname{{[table][2][34]34.2}{[1][121][]171}}
\expandafter\gdef\csname r@ch31:chapter\endcsname{{35}{172}{Normal fluctuations near regular commuting holonomies}{chapter.458}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch31:chapter@cref\endcsname{{[chapter][35][]35}{[1][172][]172}}
\expandafter\gdef\csname r@ch31:sec:results\endcsname{{35.1}{174}{Setting and main results}{section.459}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch31:sec:results@cref\endcsname{{[section][1][35]35.1}{[1][173][]174}}
\expandafter\gdef\csname r@ch31:def:regular-observables\endcsname{{35.1}{175}{}{definition.463}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch31:def:regular-observables@cref\endcsname{{[definition][1][]35.1}{[1][175][]175}}
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\expandafter\gdef\csname r@ch32:eq:quartic-upper@cref\endcsname{{[equation][10][36]36.10}{[1][227][]227}}
\expandafter\gdef\csname r@ch32:eq:closed-cochain\endcsname{{36.11}{228}{Step 4: the first-order term of the minimizer}{equation.613}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:closed-cochain@cref\endcsname{{[equation][11][36]36.11}{[1][228][]228}}
\expandafter\gdef\csname r@ch32:eq:harmonic-part\endcsname{{36.12}{228}{Step 4: the first-order term of the minimizer}{equation.614}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:harmonic-part@cref\endcsname{{[equation][12][36]36.12}{[1][228][]228}}
\expandafter\gdef\csname r@ch32:sec:holonomy\endcsname{{36.3}{229}{Holonomies near the trivial flat configuration}{section.617}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:sec:holonomy@cref\endcsname{{[section][3][36]36.3}{[1][229][]229}}
\expandafter\gdef\csname r@ch32:thm:central-matrix-transfer\endcsname{{36.3}{229}{}{theorem.618}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:thm:central-matrix-transfer@cref\endcsname{{[theorem][3][36]36.3}{[1][229][]229}}
\expandafter\gdef\csname r@ch32:eq:flat-value\endcsname{{36.13}{230}{Step 1: deterministic regularity}{equation.620}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:flat-value@cref\endcsname{{[equation][13][36]36.13}{[1][230][]230}}
\expandafter\gdef\csname r@ch32:sec:measure\endcsname{{36.4}{232}{The quartic reference measure}{section.624}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:sec:measure@cref\endcsname{{[section][4][36]36.4}{[1][232][]232}}
\expandafter\gdef\csname r@ch32:lem:quartic-intensity\endcsname{{36.4}{232}{}{lemma.625}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:lem:quartic-intensity@cref\endcsname{{[lemma][4][]36.4}{[1][232][]232}}
\expandafter\gdef\csname r@ch32:eq:normal-quartic\endcsname{{36.14}{233}{Infinite mass}{equation.627}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:normal-quartic@cref\endcsname{{[equation][14][36]36.14}{[1][232][]233}}
\expandafter\gdef\csname r@ch32:rem:small-ball\endcsname{{36.5}{234}{}{remark.629}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:rem:small-ball@cref\endcsname{{[remark][5][]36.5}{[1][233][]234}}
\expandafter\gdef\csname r@ch32:eq:radial-density\endcsname{{36.15}{234}{}{equation.630}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:radial-density@cref\endcsname{{[equation][15][36]36.15}{[1][234][]234}}
\expandafter\gdef\csname r@ch32:sec:integration\endcsname{{36.5}{234}{Integration over the transverse modes}{section.631}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:sec:integration@cref\endcsname{{[section][5][36]36.5}{[1][234][]234}}
\expandafter\gdef\csname r@ch32:thm:common-scalar-core\endcsname{{36.6}{234}{}{theorem.632}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:thm:common-scalar-core@cref\endcsname{{[theorem][6][36]36.6}{[1][234][]234}}
\expandafter\gdef\csname r@ch32:eq:complete-amplitude\endcsname{{36.16}{235}{Step 1: bounds on the density}{equation.634}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:complete-amplitude@cref\endcsname{{[equation][16][36]36.16}{[1][235][]235}}
\expandafter\gdef\csname r@ch32:eq:stiff-bounds\endcsname{{36.17}{235}{Step 1: bounds on the density}{equation.635}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:stiff-bounds@cref\endcsname{{[equation][17][36]36.17}{[1][235][]235}}
\expandafter\gdef\csname r@ch32:eq:normalizer\endcsname{{36.18}{235}{Step 1: bounds on the density}{equation.636}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:normalizer@cref\endcsname{{[equation][18][36]36.18}{[1][235][]235}}
\expandafter\gdef\csname r@ch32:eq:off-flat\endcsname{{36.19}{236}{Step 2: reduction to a neighborhood of the trivial flat configuration}{equation.638}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:off-flat@cref\endcsname{{[equation][19][36]36.19}{[1][236][]236}}
\expandafter\gdef\csname r@ch32:eq:near-identity-output\endcsname{{36.20}{236}{Step 3: the output coordinates}{equation.640}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:near-identity-output@cref\endcsname{{[equation][20][36]36.20}{[1][236][]236}}
\expandafter\gdef\csname r@ch32:eq:inverse-output\endcsname{{36.21}{237}{Step 3: the output coordinates}{equation.641}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:inverse-output@cref\endcsname{{[equation][21][36]36.21}{[1][236][]237}}
\expandafter\gdef\csname r@ch32:eq:stiff-tail\endcsname{{36.22}{237}{Step 4: the transverse tail and the change of variables}{equation.643}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:stiff-tail@cref\endcsname{{[equation][22][36]36.22}{[1][237][]237}}
\expandafter\gdef\csname r@ch32:eq:core-change\endcsname{{36.23}{237}{Step 4: the transverse tail and the change of variables}{equation.644}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:core-change@cref\endcsname{{[equation][23][36]36.23}{[1][237][]237}}
\expandafter\gdef\csname r@ch32:eq:scaled-inverse\endcsname{{36.24}{237}{Step 4: the transverse tail and the change of variables}{equation.645}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:scaled-inverse@cref\endcsname{{[equation][24][36]36.24}{[1][237][]237}}
\expandafter\gdef\csname r@ch32:eq:constant-phase-error\endcsname{{36.25}{238}{Step 5: comparison of the density}{equation.647}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:constant-phase-error@cref\endcsname{{[equation][25][36]36.25}{[1][237][]238}}
\expandafter\gdef\csname r@ch32:eq:stiff-phase-error\endcsname{{36.26}{238}{Step 5: comparison of the density}{equation.648}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:stiff-phase-error@cref\endcsname{{[equation][26][36]36.26}{[1][238][]238}}
\expandafter\gdef\csname r@ch32:eq:exponential-difference\endcsname{{36.27}{238}{Step 5: comparison of the density}{equation.649}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:exponential-difference@cref\endcsname{{[equation][27][36]36.27}{[1][238][]238}}
\expandafter\gdef\csname r@ch32:eq:compact-phase-majorant\endcsname{{36.28}{238}{Step 5: comparison of the density}{equation.650}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:compact-phase-majorant@cref\endcsname{{[equation][28][36]36.28}{[1][238][]238}}
\expandafter\gdef\csname r@ch32:eq:unnormalized-core\endcsname{{36.29}{238}{Step 6: integration and the common factor}{equation.652}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:unnormalized-core@cref\endcsname{{[equation][29][36]36.29}{[1][238][]238}}
\expandafter\gdef\csname r@ch32:sec:continuum\endcsname{{36.6}{239}{Passage to the continuum measure}{section.653}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:sec:continuum@cref\endcsname{{[section][6][36]36.6}{[1][239][]239}}
\expandafter\gdef\csname r@ch32:thm:core-ratios-and-ward\endcsname{{36.7}{239}{}{theorem.654}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:thm:core-ratios-and-ward@cref\endcsname{{[theorem][7][36]36.7}{[1][239][]239}}
\expandafter\gdef\csname r@ch32:eq:rare-matrix-event\endcsname{{36.30}{240}{Step 1: comparison under the conditioned Wilson measure}{equation.656}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:rare-matrix-event@cref\endcsname{{[equation][30][36]36.30}{[1][240][]240}}
\expandafter\gdef\csname r@ch32:eq:continuous-transfer\endcsname{{36.31}{240}{Step 1: comparison under the conditioned Wilson measure}{equation.657}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:continuous-transfer@cref\endcsname{{[equation][31][36]36.31}{[1][240][]240}}
\expandafter\gdef\csname r@ch32:eq:original-core\endcsname{{36.32}{241}{Step 2: removing the conditioning}{equation.659}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:original-core@cref\endcsname{{[equation][32][36]36.32}{[1][241][]241}}
\expandafter\gdef\csname r@ch32:eq:original-positive-mass\endcsname{{36.33}{241}{Step 3: bounds and ratios at finite $K$}{equation.661}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:original-positive-mass@cref\endcsname{{[equation][33][36]36.33}{[1][241][]241}}
\expandafter\gdef\csname r@ch32:sec:rates\endcsname{{36.7}{242}{Rates, central triples, and several times}{section.664}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:sec:rates@cref\endcsname{{[section][7][36]36.7}{[1][242][]242}}
\expandafter\gdef\csname r@ch32:conv:core-strengthening-notation\endcsname{{36.8}{242}{}{definition.665}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:conv:core-strengthening-notation@cref\endcsname{{[definition][8][]36.8}{[1][242][]242}}
\expandafter\gdef\csname r@ch32:lem:quantitative-rare-transfer\endcsname{{36.9}{243}{}{lemma.667}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:lem:quantitative-rare-transfer@cref\endcsname{{[lemma][9][]36.9}{[1][243][]243}}
\expandafter\gdef\csname r@ch32:eq:quantitative-rare-transfer\endcsname{{36.34}{243}{}{equation.668}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:quantitative-rare-transfer@cref\endcsname{{[equation][34][36]36.34}{[1][243][]243}}
\expandafter\gdef\csname r@ch32:thm:core-ratio-rates\endcsname{{36.10}{243}{}{theorem.669}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:thm:core-ratio-rates@cref\endcsname{{[theorem][10][36]36.10}{[1][243][]243}}
\expandafter\gdef\csname r@ch32:eq:finite-core-rate\endcsname{{36.35}{244}{Rates for compactly supported Lipschitz tests}{equation.670}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:finite-core-rate@cref\endcsname{{[equation][35][36]36.35}{[1][244][]244}}
\expandafter\gdef\csname r@ch32:lem:finite-center-invariance\endcsname{{36.11}{245}{}{lemma.672}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:lem:finite-center-invariance@cref\endcsname{{[lemma][11][]36.11}{[1][245][]245}}
\expandafter\gdef\csname r@ch32:thm:all-central-triples\endcsname{{36.12}{246}{}{theorem.673}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:thm:all-central-triples@cref\endcsname{{[theorem][12][36]36.12}{[1][246][]246}}
\expandafter\gdef\csname r@ch32:def:finite-time-core-insertion\endcsname{{36.13}{247}{}{definition.675}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:def:finite-time-core-insertion@cref\endcsname{{[definition][13][]36.13}{[1][247][]247}}
\expandafter\gdef\csname r@ch32:lem:localized-stiff-core-moment\endcsname{{36.14}{248}{}{lemma.676}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:lem:localized-stiff-core-moment@cref\endcsname{{[lemma][14][]36.14}{[1][248][]248}}
\expandafter\gdef\csname r@ch32:thm:finite-time-core\endcsname{{36.15}{249}{}{theorem.677}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:thm:finite-time-core@cref\endcsname{{[theorem][15][36]36.15}{[1][249][]249}}
\expandafter\gdef\csname r@ch32:cor:endpoint-joint-core\endcsname{{36.15}{249}{}{theorem.677}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:cor:endpoint-joint-core@cref\endcsname{{[theorem][15][36]36.15}{[1][249][]249}}
\expandafter\gdef\csname r@ch32:eq:joint-central-flat-jet\endcsname{{36.36}{249}{Step 1: the zero-fluctuation holonomies near the diagonal}{equation.679}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:joint-central-flat-jet@cref\endcsname{{[equation][36][36]36.36}{[1][249][]249}}
\expandafter\gdef\csname r@ch32:eq:zero-white-diagonal-error\endcsname{{36.37}{250}{Step 1: the zero-fluctuation holonomies near the diagonal}{equation.680}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:zero-white-diagonal-error@cref\endcsname{{[equation][37][36]36.37}{[1][250][]250}}
\expandafter\gdef\csname r@ch32:eq:zero-white-joint-scalar\endcsname{{36.38}{250}{Step 1: the zero-fluctuation holonomies near the diagonal}{equation.681}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:zero-white-joint-scalar@cref\endcsname{{[equation][38][36]36.38}{[1][250][]250}}
\expandafter\gdef\csname r@ch32:eq:finite-joint-transfer\endcsname{{36.39}{251}{Step 2: comparison at all listed times}{equation.683}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:finite-joint-transfer@cref\endcsname{{[equation][39][36]36.39}{[1][251][]251}}
\expandafter\gdef\csname r@ch32:eq:original-joint-core-rate\endcsname{{36.40}{251}{Step 3: Lipschitz tests at finite cutoff}{equation.685}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:original-joint-core-rate@cref\endcsname{{[equation][40][36]36.40}{[1][251][]251}}
\expandafter\gdef\csname r@ch32:sec:paths\endcsname{{36.8}{252}{Holonomy paths on a time interval}{section.688}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:sec:paths@cref\endcsname{{[section][8][36]36.8}{[1][252][]252}}
\expandafter\gdef\csname r@ch32:lem:uniform-time-central-jet\endcsname{{36.16}{252}{}{lemma.689}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:lem:uniform-time-central-jet@cref\endcsname{{[lemma][16][]36.16}{[1][252][]252}}
\expandafter\gdef\csname r@ch32:def:central-quotient-path\endcsname{{36.17}{254}{}{definition.690}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:def:central-quotient-path@cref\endcsname{{[definition][17][]36.17}{[1][254][]254}}
\expandafter\gdef\csname r@ch32:thm:central-quotient-path\endcsname{{36.18}{254}{}{theorem.691}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:thm:central-quotient-path@cref\endcsname{{[theorem][18][36]36.18}{[1][254][]254}}
\expandafter\gdef\csname r@ch32:eq:path-transfer\endcsname{{36.41}{256}{Step 2: comparison with the zero-fluctuation path}{equation.694}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:path-transfer@cref\endcsname{{[equation][41][36]36.41}{[1][256][]256}}
\expandafter\gdef\csname r@ch32:eq:path-constant\endcsname{{36.42}{256}{Step 3: the weighted zero-fluctuation path is nearly constant}{equation.696}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:path-constant@cref\endcsname{{[equation][42][36]36.42}{[1][256][]256}}
\expandafter\gdef\csname r@ch32:eq:path-finite\endcsname{{36.43}{256}{Step 3: the weighted zero-fluctuation path is nearly constant}{equation.697}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:path-finite@cref\endcsname{{[equation][43][36]36.43}{[1][256][]256}}
\expandafter\gdef\csname r@ch32:sec:comparison\endcsname{{36.9}{257}{Dilation statistics of comparison measures}{section.700}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:sec:comparison@cref\endcsname{{[section][9][36]36.9}{[1][257][]257}}
\expandafter\gdef\csname r@ch32:thm:corrected-reference-ward\endcsname{{36.19}{258}{}{theorem.701}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:thm:corrected-reference-ward@cref\endcsname{{[theorem][19][36]36.19}{[1][258][]258}}
\expandafter\gdef\csname r@ch32:eq:gaussian-supremum\endcsname{{36.44}{260}{Gaussian holonomies with Gaussian constant modes}{equation.705}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:gaussian-supremum@cref\endcsname{{[equation][44][36]36.44}{[1][260][]260}}
\expandafter\gdef\csname r@ch32:eq:gaussian-core-tail\endcsname{{36.45}{260}{Gaussian holonomies with Gaussian constant modes}{equation.706}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:gaussian-core-tail@cref\endcsname{{[equation][45][36]36.45}{[1][260][]260}}
\expandafter\gdef\csname r@ch32:eq:bgm-coordinate-window\endcsname{{36.46}{261}{The mixture over the cone of commuting triples}{equation.708}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:bgm-coordinate-window@cref\endcsname{{[equation][46][36]36.46}{[1][261][]261}}
\expandafter\gdef\csname r@ch32:eq:cone-volume\endcsname{{36.47}{261}{The mixture over the cone of commuting triples}{equation.709}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:cone-volume@cref\endcsname{{[equation][47][36]36.47}{[1][261][]261}}
\expandafter\gdef\csname r@ch32:eq:orthogonal-output\endcsname{{36.48}{261}{The mixture over the cone of commuting triples}{equation.710}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:orthogonal-output@cref\endcsname{{[equation][48][36]36.48}{[1][261][]261}}
\expandafter\gdef\csname r@ch32:eq:bgm-quartic\endcsname{{36.49}{261}{The mixture over the cone of commuting triples}{equation.711}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:bgm-quartic@cref\endcsname{{[equation][49][36]36.49}{[1][261][]261}}
\expandafter\gdef\csname r@ch32:eq:rescaled-bg-density\endcsname{{36.50}{262}{The mixture over the cone of commuting triples}{equation.712}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:rescaled-bg-density@cref\endcsname{{[equation][50][36]36.50}{[1][261][]262}}
\expandafter\gdef\csname r@ch32:eq:bundle-ward\endcsname{{36.51}{262}{The mixture over the cone of commuting triples}{equation.713}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:bundle-ward@cref\endcsname{{[equation][51][36]36.51}{[1][262][]262}}
\expandafter\gdef\csname r@ch32:rem:bundle-small-ball\endcsname{{36.20}{263}{}{remark.714}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:rem:bundle-small-ball@cref\endcsname{{[remark][20][]36.20}{[1][263][]263}}
\expandafter\gdef\csname r@ch32:sec:proof-central\endcsname{{36.10}{263}{The main result on central holonomies}{section.715}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:sec:proof-central@cref\endcsname{{[section][10][36]36.10}{[1][263][]263}}
\makeatother