% Frozen companion labels: independent compilation needs no companion .aux files.
% The fifth field preserves the existing companion-PDF link target.
\makeatletter
% Part I
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\expandafter\gdef\csname r@part:intro@cref\endcsname{{[part][1][]I}{[1][2][]2}}
\expandafter\gdef\csname r@s1:chapter\endcsname{{1}{2}{Introduction}{chapter.3}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s1:chapter@cref\endcsname{{[chapter][1][]1}{[1][2][]2}}
\expandafter\gdef\csname r@s1:background\endcsname{{1.1}{2}{Yang--Mills theory and its lattice formulation}{section.4}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s1:background@cref\endcsname{{[section][1][1]1.1}{[1][2][]2}}
\expandafter\gdef\csname r@s1:eq:formal\endcsname{{1.1}{2}{Yang--Mills theory and its lattice formulation}{equation.5}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s1:eq:formal@cref\endcsname{{[equation][1][1]1.1}{[1][2][]2}}
\expandafter\gdef\csname r@s1:eq:coupling\endcsname{{1.2}{2}{Yang--Mills theory and its lattice formulation}{equation.6}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s1:eq:coupling@cref\endcsname{{[equation][2][1]1.2}{[1][2][]2}}
\expandafter\gdef\csname r@s1:previous\endcsname{{1.2}{4}{Previous rigorous results}{section.7}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s1:previous@cref\endcsname{{[section][2][1]1.2}{[1][4][]4}}
\expandafter\gdef\csname r@s1:results\endcsname{{1.3}{6}{Main results}{section.12}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s1:results@cref\endcsname{{[section][3][1]1.3}{[1][6][]6}}
\expandafter\gdef\csname r@s1:relation\endcsname{{1.4}{8}{Relation to previous work}{section.13}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s1:relation@cref\endcsname{{[section][4][1]1.4}{[1][7][]8}}
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\expandafter\gdef\csname r@s1:scope@cref\endcsname{{[section][5][1]1.5}{[1][9][]9}}
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\expandafter\gdef\csname r@s1:organization@cref\endcsname{{[section][6][1]1.6}{[1][9][]9}}
\expandafter\gdef\csname r@s2:chapter\endcsname{{2}{11}{Statement of results}{chapter.16}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:chapter@cref\endcsname{{[chapter][2][]2}{[1][11][]11}}
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\expandafter\gdef\csname r@s2:model@cref\endcsname{{[section][1][2]2.1}{[1][11][]11}}
\expandafter\gdef\csname r@s2:eq:wilson\endcsname{{2.1}{11}{The lattice model}{equation.18}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:eq:wilson@cref\endcsname{{[equation][1][2]2.1}{[1][11][]11}}
\expandafter\gdef\csname r@s2:eq:lattice-energy\endcsname{{2.2}{12}{The Wilson flow}{equation.20}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:eq:lattice-energy@cref\endcsname{{[equation][2][2]2.2}{[1][12][]12}}
\expandafter\gdef\csname r@s2:def:approx\endcsname{{2.1}{12}{}{definition.22}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:def:approx@cref\endcsname{{[definition][1][]2.1}{[1][12][]12}}
\expandafter\gdef\csname r@s2:trajectories\endcsname{{2.2}{12}{The trajectory space}{section.23}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:trajectories@cref\endcsname{{[section][2][2]2.2}{[1][12][]12}}
\expandafter\gdef\csname r@s2:eq:energy\endcsname{{2.3}{13}{The trajectory space}{equation.24}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:eq:energy@cref\endcsname{{[equation][3][2]2.3}{[1][12][]13}}
\expandafter\gdef\csname r@s2:def:trajectories\endcsname{{2.2}{13}{}{definition.25}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:def:trajectories@cref\endcsname{{[definition][2][]2.2}{[1][13][]13}}
\expandafter\gdef\csname r@s2:main\endcsname{{2.3}{13}{The main theorem}{section.26}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:main@cref\endcsname{{[section][3][2]2.3}{[1][13][]13}}
\expandafter\gdef\csname r@s2:thm:main\endcsname{{2.3}{13}{}{theorem.27}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:thm:main@cref\endcsname{{[theorem][3][2]2.3}{[1][13][]13}}
\expandafter\gdef\csname r@s2:eq:loopvector\endcsname{{2.4}{14}{}{equation.29}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:eq:loopvector@cref\endcsname{{[equation][4][2]2.4}{[1][13][]14}}
\expandafter\gdef\csname r@s2:rem:ladders\endcsname{{2.4}{14}{}{remark.34}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:rem:ladders@cref\endcsname{{[remark][4][]2.4}{[1][14][]14}}
\expandafter\gdef\csname r@s2:further\endcsname{{2.4}{15}{Further properties of the limit}{section.35}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:further@cref\endcsname{{[section][4][2]2.4}{[1][14][]15}}
\expandafter\gdef\csname r@s2:thm:further\endcsname{{2.5}{15}{}{theorem.36}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:thm:further@cref\endcsname{{[theorem][5][2]2.5}{[1][15][]15}}
\expandafter\gdef\csname r@s2:thm:altflow\endcsname{{2.6}{16}{}{theorem.43}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:thm:altflow@cref\endcsname{{[theorem][6][2]2.6}{[1][16][]16}}
\expandafter\gdef\csname r@s2:largefield\endcsname{{2.5}{16}{The bound on large fields}{section.44}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:largefield@cref\endcsname{{[section][5][2]2.5}{[1][16][]16}}
\expandafter\gdef\csname r@s2:thm:large-field\endcsname{{2.7}{16}{}{theorem.45}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:thm:large-field@cref\endcsname{{[theorem][7][2]2.7}{[1][16][]16}}
\expandafter\gdef\csname r@s2:eq:large-field\endcsname{{2.5}{16}{}{equation.46}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:eq:large-field@cref\endcsname{{[equation][5][2]2.5}{[1][16][]16}}
\expandafter\gdef\csname r@s2:short\endcsname{{2.6}{17}{Short distances}{section.47}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:short@cref\endcsname{{[section][6][2]2.6}{[1][17][]17}}
\expandafter\gdef\csname r@s2:thm:short-distance\endcsname{{2.8}{17}{}{theorem.48}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:thm:short-distance@cref\endcsname{{[theorem][8][2]2.8}{[1][17][]17}}
\expandafter\gdef\csname r@s2:eq:circulation-covariance\endcsname{{2.6}{17}{}{equation.49}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:eq:circulation-covariance@cref\endcsname{{[equation][6][2]2.6}{[1][17][]17}}
\expandafter\gdef\csname r@s2:weak\endcsname{{2.7}{18}{Weak coupling}{section.53}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:weak@cref\endcsname{{[section][7][2]2.7}{[1][18][]18}}
\expandafter\gdef\csname r@s2:thm:flat\endcsname{{2.9}{18}{}{theorem.54}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:thm:flat@cref\endcsname{{[theorem][9][2]2.9}{[1][18][]18}}
\expandafter\gdef\csname r@s2:thm:regular\endcsname{{2.10}{20}{}{theorem.59}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:thm:regular@cref\endcsname{{[theorem][10][2]2.10}{[1][19][]20}}
\expandafter\gdef\csname r@s2:thm:central\endcsname{{2.11}{21}{}{theorem.64}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:thm:central@cref\endcsname{{[theorem][11][2]2.11}{[1][21][]21}}
\expandafter\gdef\csname r@s2:eq:core-ratio\endcsname{{2.7}{21}{}{equation.66}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s2:eq:core-ratio@cref\endcsname{{[equation][7][2]2.7}{[1][21][]21}}
\expandafter\gdef\csname r@s3:chapter\endcsname{{3}{23}{Outline of the proof}{chapter.70}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:chapter@cref\endcsname{{[chapter][3][]3}{[1][23][]23}}
\expandafter\gdef\csname r@s3:strategy\endcsname{{3.1}{23}{Strategy}{section.71}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:strategy@cref\endcsname{{[section][1][3]3.1}{[1][23][]23}}
\expandafter\gdef\csname r@s3:eq:transfer\endcsname{{3.1}{23}{Strategy}{equation.74}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:eq:transfer@cref\endcsname{{[equation][1][3]3.1}{[1][23][]23}}
\expandafter\gdef\csname r@s3:rg\endcsname{{3.2}{24}{The renormalization group on the small-field event}{section.75}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:rg@cref\endcsname{{[section][2][3]3.2}{[1][24][]24}}
\expandafter\gdef\csname r@s3:eq:effective-density\endcsname{{3.2}{26}{Fluctuations and the effective interaction}{equation.79}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:eq:effective-density@cref\endcsname{{[equation][2][3]3.2}{[1][26][]26}}
\expandafter\gdef\csname r@s3:fixed\endcsname{{3.3}{27}{Convergence at fixed resolution and reconstruction}{section.82}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:fixed@cref\endcsname{{[section][3][3]3.3}{[1][27][]27}}
\expandafter\gdef\csname r@s3:largefield\endcsname{{3.4}{28}{The bound on large fields}{section.85}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:largefield@cref\endcsname{{[section][4][3]3.4}{[1][28][]28}}
\expandafter\gdef\csname r@s3:flow\endcsname{{3.5}{30}{The gradient flow}{section.89}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:flow@cref\endcsname{{[section][5][3]3.5}{[1][30][]30}}
\expandafter\gdef\csname r@s3:stochastic\endcsname{{3.6}{31}{Stochastic estimates and convergence of flowed loops}{section.90}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:stochastic@cref\endcsname{{[section][6][3]3.6}{[1][31][]31}}
\expandafter\gdef\csname r@s3:construction\endcsname{{3.7}{32}{Construction of the limiting measure}{section.91}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:construction@cref\endcsname{{[section][7][3]3.7}{[1][31][]32}}
\expandafter\gdef\csname r@s3:short\endcsname{{3.8}{32}{Short distances}{section.92}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:short@cref\endcsname{{[section][8][3]3.8}{[1][32][]32}}
\expandafter\gdef\csname r@s3:weak\endcsname{{3.9}{33}{Weak coupling}{section.93}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:weak@cref\endcsname{{[section][9][3]3.9}{[1][33][]33}}
\expandafter\gdef\csname r@s3:constants\endcsname{{3.10}{34}{Order of choices and dependence of constants}{section.94}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@s3:constants@cref\endcsname{{[section][10][3]3.10}{[1][34][]34}}
\expandafter\gdef\csname r@ch01:chapter\endcsname{{4}{35}{Notation and conventions}{chapter.95}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:chapter@cref\endcsname{{[chapter][4][]4}{[1][35][]35}}
\expandafter\gdef\csname r@ch01:sec:model\endcsname{{4.1}{35}{The periodic lattice and the Wilson measure}{section.96}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:sec:model@cref\endcsname{{[section][1][4]4.1}{[1][35][]35}}
\expandafter\gdef\csname r@ch01:def:ensemble\endcsname{{4.1}{35}{}{definition.97}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:def:ensemble@cref\endcsname{{[definition][1][]4.1}{[1][35][]35}}
\expandafter\gdef\csname r@ch01:rem:normalization\endcsname{{4.2}{36}{}{remark.98}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:rem:normalization@cref\endcsname{{[remark][2][]4.2}{[1][36][]36}}
\expandafter\gdef\csname r@ch01:eq:lattice-gauge\endcsname{{4.1}{36}{Gauge transformations}{equation.100}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:eq:lattice-gauge@cref\endcsname{{[equation][1][4]4.1}{[1][36][]36}}
\expandafter\gdef\csname r@ch01:eq:haar-coordinates\endcsname{{4.2}{37}{The Lie algebra}{equation.102}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:eq:haar-coordinates@cref\endcsname{{[equation][2][4]4.2}{[1][37][]37}}
\expandafter\gdef\csname r@ch01:sec:hierarchy\endcsname{{4.2}{37}{Lattice refinements and the block averaging map}{section.103}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:sec:hierarchy@cref\endcsname{{[section][2][4]4.2}{[1][37][]37}}
\expandafter\gdef\csname r@ch01:def:block-map\endcsname{{4.3}{38}{}{definition.104}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:def:block-map@cref\endcsname{{[definition][3][]4.3}{[1][38][]38}}
\expandafter\gdef\csname r@ch01:eq:ordinary-map\endcsname{{4.3}{38}{}{equation.105}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:eq:ordinary-map@cref\endcsname{{[equation][3][4]4.3}{[1][38][]38}}
\expandafter\gdef\csname r@ch01:eq:haar-disintegration\endcsname{{4.4}{40}{Constraint coordinates}{equation.107}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:eq:haar-disintegration@cref\endcsname{{[equation][4][4]4.4}{[1][39][]40}}
\expandafter\gdef\csname r@ch01:sec:measures\endcsname{{4.3}{40}{Effective actions, the small-field event, and conditioned measures}{section.108}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:sec:measures@cref\endcsname{{[section][3][4]4.3}{[1][40][]40}}
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\expandafter\gdef\csname r@ch01:eq:literal-event@cref\endcsname{{[equation][6][4]4.6}{[1][42][]42}}
\expandafter\gdef\csname r@ch01:lem:positive-mass\endcsname{{4.5}{42}{}{lemma.116}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:lem:positive-mass@cref\endcsname{{[lemma][5][]4.5}{[1][42][]42}}
\expandafter\gdef\csname r@ch01:eq:three-measures\endcsname{{4.7}{43}{The three measures}{equation.118}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:eq:three-measures@cref\endcsname{{[equation][7][4]4.7}{[1][43][]43}}
\expandafter\gdef\csname r@ch01:eq:complete-white-kernel\endcsname{{4.8}{44}{The conditional fluctuation kernel}{equation.121}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:eq:complete-white-kernel@cref\endcsname{{[equation][8][4]4.8}{[1][43][]44}}
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\expandafter\gdef\csname r@ch01:eq:tensor-norm@cref\endcsname{{[equation][9][4]4.9}{[1][45][]45}}
\expandafter\gdef\csname r@ch01:eq:scalar-root\endcsname{{4.10}{45}{}{equation.126}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:eq:scalar-root@cref\endcsname{{[equation][10][4]4.10}{[1][45][]45}}
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\expandafter\gdef\csname r@ch01:eq:regional-norm@cref\endcsname{{[equation][11][4]4.11}{[1][46][]46}}
\expandafter\gdef\csname r@ch01:rem:factorial\endcsname{{4.7}{47}{}{remark.128}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:rem:factorial@cref\endcsname{{[remark][7][]4.7}{[1][47][]47}}
\expandafter\gdef\csname r@ch01:sec:polymers\endcsname{{4.5}{48}{Polymers and exponential moments of blocked actions}{section.131}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:sec:polymers@cref\endcsname{{[section][5][4]4.5}{[1][48][]48}}
\expandafter\gdef\csname r@ch01:sec:flow\endcsname{{4.6}{50}{Heat semigroup, Wilson flow, and flow data}{section.135}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch01:def:enhancement\endcsname{{4.8}{50}{}{definition.136}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:def:enhancement@cref\endcsname{{[definition][8][]4.8}{[1][50][]50}}
\expandafter\gdef\csname r@ch01:sec:current\endcsname{{4.7}{51}{The heat-smoothed circulation}{section.139}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch01:def:heat-current\endcsname{{4.9}{52}{}{definition.140}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:def:heat-current@cref\endcsname{{[definition][9][]4.9}{[1][51][]52}}
\expandafter\gdef\csname r@ch01:sec:orbits\endcsname{{4.8}{52}{Gauge orbits and the trajectory space}{section.141}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch01:def:orbit-space\endcsname{{4.10}{52}{}{definition.142}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch01:def:orbit-space@cref\endcsname{{[definition][10][]4.10}{[1][52][]52}}
\expandafter\gdef\csname r@ch01:sec:weak\endcsname{{4.9}{53}{Observables near commuting holonomies}{section.143}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch01:def:toron-observables\endcsname{{4.11}{53}{}{definition.144}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch01:eq:normal-covariance\endcsname{{4.12}{54}{}{equation.145}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch04:chapter@cref\endcsname{{[chapter][7][]7}{[1][121][]121}}
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\expandafter\gdef\csname r@ch04:eq:patch-logs\endcsname{{7.7}{129}{Energy estimates and physical curvature}{equation.322}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch04:eq:patch-logs@cref\endcsname{{[equation][7][7]7.7}{[1][129][]129}}
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\expandafter\gdef\csname r@ch04:eq:curl-near-contraction\endcsname{{7.9}{130}{Energy estimates and physical curvature}{equation.324}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch04:eq:curl-near-contraction@cref\endcsname{{[equation][9][7]7.9}{[1][130][]130}}
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\expandafter\gdef\csname r@ch04:eq:scaled-recurrence\endcsname{{7.11}{131}{Energy estimates and physical curvature}{equation.326}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch04:eq:scaled-recurrence@cref\endcsname{{[equation][11][7]7.11}{[1][131][]131}}
\expandafter\gdef\csname r@ch04:sec:cells\endcsname{{7.3}{132}{Dependence sets and reflected cell partitions}{section.327}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch04:sec:cells@cref\endcsname{{[section][3][7]7.3}{[1][132][]132}}
\expandafter\gdef\csname r@ch04:fig:cells\endcsname{{7.1}{132}{Left: the blocks of side $m_j$ (thin squares) around the vertices of the level-$j$ lattice, in a plane spanned by two coordinate directions. The patch of the coarse link $c$ consists of its two blocks, and the patch of the coarse plaquette $P$ consists of the four blocks at its corners (thick outlines). Right: cell boundaries of two of the eight shifted tilings, with shifts $(0,0)$ (solid) and $(1,0)$ (dashed) in the two directions shown; each cell is the union of the blocks of $2\times 2\times 2$ level-$j$ vertices. The thick plaquette crosses a cell boundary of the first tiling, but lies in a cell of the second}{figure.caption.328}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch04:eq:block-nesting@cref\endcsname{{[equation][12][7]7.12}{[1][134][]134}}
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\expandafter\gdef\csname r@ch04:eq:bounded-cell-tests@cref\endcsname{{[equation][13][7]7.13}{[1][135][]135}}
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\expandafter\gdef\csname r@ch04:sec:gauge@cref\endcsname{{[section][4][7]7.4}{[1][136][]136}}
\expandafter\gdef\csname r@ch04:lem:quantitative-fillings\endcsname{{7.5}{136}{}{lemma.337}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch04:lem:quantitative-fillings@cref\endcsname{{[lemma][5][]7.5}{[1][136][]136}}
\expandafter\gdef\csname r@ch04:eq:chord-comparison\endcsname{{7.14}{137}{Lipschitz extensions and periodic gauge fixing}{equation.341}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch04:eq:chord-comparison@cref\endcsname{{[equation][14][7]7.14}{[1][137][]137}}
\expandafter\gdef\csname r@ch04:eq:stereo-bounds\endcsname{{7.15}{138}{Lipschitz extensions and periodic gauge fixing}{equation.342}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch04:lem:cell-gluing@cref\endcsname{{[lemma][6][]7.6}{[1][139][]139}}
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\expandafter\gdef\csname r@ch04:eq:frame-hypothesis@cref\endcsname{{[equation][16][7]7.16}{[1][139][]139}}
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\expandafter\gdef\csname r@ch04:thm:small-curvature-admission\endcsname{{7.7}{142}{}{theorem.348}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch04:thm:small-curvature-admission@cref\endcsname{{[theorem][7][7]7.7}{[1][142][]142}}
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\expandafter\gdef\csname r@ch04:eq:curvature-radius@cref\endcsname{{[equation][19][7]7.19}{[1][143][]143}}
\expandafter\gdef\csname r@ch05:chapter\endcsname{{8}{146}{Linearized averaging and effective quadratic forms}{chapter.354}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:chapter@cref\endcsname{{[chapter][8][]8}{[1][146][]146}}
\expandafter\gdef\csname r@ch05:sec:linear\endcsname{{8.1}{147}{Linearized block averaging}{section.355}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:sec:linear@cref\endcsname{{[section][1][8]8.1}{[1][147][]147}}
\expandafter\gdef\csname r@ch05:fig:paths\endcsname{{8.1}{148}{A two-dimensional slice of two neighboring blocks for $M=3$. The thick line is the reference path $C_c$ of the coarse link $c=(v,v+Me_1)$. The arrows show the translated path with endpoint connectors $F_{c,x}$: the coordinate path $\sigma _{v,x}$, the segment $[x,x+Me_1]$, and the reversed coordinate path $\sigma _{v+Me_1,x+Me_1}^{-1}$. At the identity the derivatives of the two connectors produce the gradient term in \eqref {ch05:eq:one-step-endpoint}; in the gauge-corrected average of \cref {ch05:sec:corrected} they are canceled by gauge transformations at $v$ and $v+Me_1$}{figure.caption.358}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch05:eq:one-step-endpoint@cref\endcsname{{[equation][1][8]8.1}{[1][149][]149}}
\expandafter\gdef\csname r@ch05:sec:maxwell\endcsname{{8.2}{150}{The effective Maxwell quadratic form}{section.365}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:sec:maxwell@cref\endcsname{{[section][2][8]8.2}{[1][150][]150}}
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\expandafter\gdef\csname r@ch05:def:flat-symbols@cref\endcsname{{[definition][2][]8.2}{[1][150][]150}}
\expandafter\gdef\csname r@ch05:eq:holomorphic-alias\endcsname{{8.2}{151}{}{equation.367}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:eq:holomorphic-alias@cref\endcsname{{[equation][2][8]8.2}{[1][151][]151}}
\expandafter\gdef\csname r@ch05:thm:flat-precision\endcsname{{8.3}{151}{}{theorem.368}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:thm:flat-precision@cref\endcsname{{[theorem][3][8]8.3}{[1][151][]151}}
\expandafter\gdef\csname r@ch05:rem:constant-modes\endcsname{{8.4}{153}{}{remark.373}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:rem:constant-modes@cref\endcsname{{[remark][4][]8.4}{[1][153][]153}}
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\expandafter\gdef\csname r@ch05:rem:infinite-depth@cref\endcsname{{[remark][5][]8.5}{[1][153][]153}}
\expandafter\gdef\csname r@ch05:eq:curl-symbol\endcsname{{8.3}{154}{Step 1: Fourier conventions and the covariance}{equation.376}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:eq:curl-symbol@cref\endcsname{{[equation][3][8]8.3}{[1][154][]154}}
\expandafter\gdef\csname r@ch05:eq:beta-bound\endcsname{{8.4}{156}{Step 4: Uniform bounds for the continued alias sums}{equation.380}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch05:sec:ward\endcsname{{8.3}{160}{Gradient annihilation and spatial inversion}{section.386}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:sec:ward@cref\endcsname{{[section][3][8]8.3}{[1][160][]160}}
\expandafter\gdef\csname r@ch05:lem:ward-parity\endcsname{{8.6}{161}{}{lemma.388}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:lem:ward-parity@cref\endcsname{{[lemma][6][]8.6}{[1][161][]161}}
\expandafter\gdef\csname r@ch05:rem:curl-factor\endcsname{{8.7}{163}{}{remark.392}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:rem:curl-factor@cref\endcsname{{[remark][7][]8.7}{[1][163][]163}}
\expandafter\gdef\csname r@ch05:sec:corrected\endcsname{{8.4}{163}{The gauge-corrected block average}{section.393}{3d-ym-torus-volume-i.pdf}}
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\expandafter\gdef\csname r@ch05:lem:word-bounds\endcsname{{8.9}{165}{}{lemma.395}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:lem:word-bounds@cref\endcsname{{[lemma][9][]8.9}{[1][164][]165}}
\expandafter\gdef\csname r@ch05:lem:modified-average\endcsname{{8.10}{166}{}{lemma.399}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:lem:modified-average@cref\endcsname{{[lemma][10][]8.10}{[1][166][]166}}
\expandafter\gdef\csname r@ch05:eq:curvature-sum\endcsname{{8.6}{168}{The gauge-corrected block average}{equation.403}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:eq:curvature-sum@cref\endcsname{{[equation][6][8]8.6}{[1][168][]168}}
\expandafter\gdef\csname r@ch05:eq:ordinary-differential\endcsname{{8.7}{168}{Step 1: The one-step differential and its endpoint terms}{equation.405}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:eq:ordinary-differential@cref\endcsname{{[equation][7][8]8.7}{[1][168][]168}}
\expandafter\gdef\csname r@ch05:eq:bootstrap-choice\endcsname{{8.8}{170}{Step 3: Uniform iteration}{equation.408}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:eq:bootstrap-choice@cref\endcsname{{[equation][8][8]8.8}{[1][170][]170}}
\expandafter\gdef\csname r@ch05:sec:columns\endcsname{{8.5}{172}{Volume-normalized column sums and derivative bounds}{section.411}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:sec:columns@cref\endcsname{{[section][5][8]8.5}{[1][172][]172}}
\expandafter\gdef\csname r@ch05:lem:physical-jets\endcsname{{8.11}{172}{}{lemma.412}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:lem:physical-jets@cref\endcsname{{[lemma][11][]8.11}{[1][172][]172}}
\expandafter\gdef\csname r@ch05:eq:error-sum\endcsname{{8.9}{174}{Step 2: Products and the ordered telescope}{equation.415}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:eq:error-sum@cref\endcsname{{[equation][9][8]8.9}{[1][174][]174}}
\expandafter\gdef\csname r@ch05:eq:one-step-higher-jets\endcsname{{8.10}{175}{Step 4: Absolute bounds for higher derivatives of one step}{equation.418}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:eq:one-step-higher-jets@cref\endcsname{{[equation][10][8]8.10}{[1][174][]175}}
\expandafter\gdef\csname r@ch05:eq:partition-chain\endcsname{{8.11}{175}{Step 5: Induction over the order}{equation.420}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch05:eq:partition-chain@cref\endcsname{{[equation][11][8]8.11}{[1][175][]175}}
\expandafter\gdef\csname r@part:background\endcsname{{III}{178}{Volume-normalized column sums and derivative bounds}{part.422}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@part:background@cref\endcsname{{[part][3][]III}{[1][178][]178}}
\expandafter\gdef\csname r@ch06:chapter\endcsname{{9}{178}{Constraint coordinates and local gauge fixing}{chapter.423}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:chapter@cref\endcsname{{[chapter][9][]9}{[1][178][]178}}
\expandafter\gdef\csname r@ch06:sec:spaces\endcsname{{9.1}{179}{Lattices, scalar products, and norms}{section.424}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:sec:spaces@cref\endcsname{{[section][1][9]9.1}{[1][179][]179}}
\expandafter\gdef\csname r@ch06:lem:physical-schur\endcsname{{9.1}{180}{}{lemma.425}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:lem:physical-schur@cref\endcsname{{[lemma][1][]9.1}{[1][180][]180}}
\expandafter\gdef\csname r@ch06:sec:covers\endcsname{{9.2}{180}{Transfer of periodic kernels through finite covers}{section.426}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:sec:covers@cref\endcsname{{[section][2][9]9.2}{[1][180][]180}}
\expandafter\gdef\csname r@ch06:lem:periodic-kernel-descent\endcsname{{9.2}{181}{}{lemma.427}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:lem:periodic-kernel-descent@cref\endcsname{{[lemma][2][]9.2}{[1][181][]181}}
\expandafter\gdef\csname r@ch06:eq:cover-pairing\endcsname{{9.1}{182}{Transfer of periodic kernels through finite covers}{equation.431}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:cover-pairing@cref\endcsname{{[equation][1][9]9.1}{[1][182][]182}}
\expandafter\gdef\csname r@ch06:eq:cover-scalar-subspace\endcsname{{9.2}{183}{Transfer of periodic kernels through finite covers}{equation.432}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:cover-scalar-subspace@cref\endcsname{{[equation][2][9]9.2}{[1][183][]183}}
\expandafter\gdef\csname r@ch06:eq:cover-projector\endcsname{{9.3}{183}{Transfer of periodic kernels through finite covers}{equation.433}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:cover-projector@cref\endcsname{{[equation][3][9]9.3}{[1][183][]183}}
\expandafter\gdef\csname r@ch06:eq:cover-weighted-row\endcsname{{9.4}{184}{Transfer of periodic kernels through finite covers}{equation.434}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:cover-weighted-row@cref\endcsname{{[equation][4][9]9.4}{[1][184][]184}}
\expandafter\gdef\csname r@ch06:tab:balaban\endcsname{{9.1}{185}{Linear estimates of Bałaban used in this section, at the identity connection, and their relation to the operators defined here}{table.caption.436}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:tab:balaban@cref\endcsname{{[table][1][9]9.1}{[1][185][]185}}
\expandafter\gdef\csname r@ch06:sec:extension\endcsname{{9.3}{186}{The constrained linear extension}{section.437}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:sec:extension@cref\endcsname{{[section][3][9]9.3}{[1][185][]186}}
\expandafter\gdef\csname r@ch06:thm:matched-flat-extension\endcsname{{9.3}{186}{}{theorem.438}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:thm:matched-flat-extension@cref\endcsname{{[theorem][3][9]9.3}{[1][185][]186}}
\expandafter\gdef\csname r@ch06:eq:centered-physical-average\endcsname{{9.5}{187}{Step 1: The averages}{equation.444}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:centered-physical-average@cref\endcsname{{[equation][5][9]9.5}{[1][187][]187}}
\expandafter\gdef\csname r@ch06:eq:translated-projector\endcsname{{9.6}{187}{Step 1: The averages}{equation.445}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:translated-projector@cref\endcsname{{[equation][6][9]9.6}{[1][187][]187}}
\expandafter\gdef\csname r@ch06:eq:precision-descent\endcsname{{9.7}{188}{Step 3: Coercivity on the torus of side $n$}{equation.448}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:precision-descent@cref\endcsname{{[equation][7][9]9.7}{[1][187][]188}}
\expandafter\gdef\csname r@ch06:eq:extension-cover-kernel\endcsname{{9.8}{189}{Step 5: Kernel bounds}{equation.451}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:extension-cover-kernel@cref\endcsname{{[equation][8][9]9.8}{[1][189][]189}}
\expandafter\gdef\csname r@ch06:eq:exponential-sums\endcsname{{9.9}{189}{Step 5: Kernel bounds}{equation.452}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:exponential-sums@cref\endcsname{{[equation][9][9]9.9}{[1][189][]189}}
\expandafter\gdef\csname r@ch06:sec:plaquette\endcsname{{9.4}{190}{Plaquette expansions}{section.454}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:sec:plaquette@cref\endcsname{{[section][4][9]9.4}{[1][189][]190}}
\expandafter\gdef\csname r@ch06:lem:plaquette-taylor\endcsname{{9.4}{190}{}{lemma.455}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:lem:plaquette-taylor@cref\endcsname{{[lemma][4][]9.4}{[1][190][]190}}
\expandafter\gdef\csname r@ch06:eq:face-value\endcsname{{9.10}{190}{}{equation.456}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:face-value@cref\endcsname{{[equation][10][9]9.10}{[1][190][]190}}
\expandafter\gdef\csname r@ch06:eq:face-first\endcsname{{9.11}{190}{}{equation.457}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:face-first@cref\endcsname{{[equation][11][9]9.11}{[1][190][]190}}
\expandafter\gdef\csname r@ch06:eq:face-second\endcsname{{9.12}{190}{}{equation.458}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:face-second@cref\endcsname{{[equation][12][9]9.12}{[1][190][]190}}
\expandafter\gdef\csname r@ch06:sec:fiber\endcsname{{9.5}{191}{Local coordinates on constraint fibers}{section.459}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:sec:fiber@cref\endcsname{{[section][5][9]9.5}{[1][190][]191}}
\expandafter\gdef\csname r@ch06:lem:analytic-contraction\endcsname{{9.5}{191}{}{lemma.460}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:lem:analytic-contraction@cref\endcsname{{[lemma][5][]9.5}{[1][190][]191}}
\expandafter\gdef\csname r@ch06:thm:nonlinear-section\endcsname{{9.6}{191}{}{theorem.461}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:thm:nonlinear-section@cref\endcsname{{[theorem][6][9]9.6}{[1][191][]191}}
\expandafter\gdef\csname r@ch06:eq:average-remainder-derivative\endcsname{{9.13}{193}{Step 1: The remainder}{equation.467}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:average-remainder-derivative@cref\endcsname{{[equation][13][9]9.13}{[1][193][]193}}
\expandafter\gdef\csname r@ch06:eq:coarse-remainder\endcsname{{9.14}{193}{Step 2: The coarse remainder}{equation.469}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:coarse-remainder@cref\endcsname{{[equation][14][9]9.14}{[1][193][]193}}
\expandafter\gdef\csname r@ch06:eq:q-partitions\endcsname{{9.15}{194}{Step 4: Derivatives of higher order of the right inverse}{equation.472}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:q-partitions@cref\endcsname{{[equation][15][9]9.15}{[1][194][]194}}
\expandafter\gdef\csname r@ch06:eq:D-quadratic\endcsname{{9.16}{195}{Step 5: The coordinates $\Xi _d$}{equation.474}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:D-quadratic@cref\endcsname{{[equation][16][9]9.16}{[1][195][]195}}
\expandafter\gdef\csname r@ch06:eq:D-prime\endcsname{{9.17}{195}{Step 5: The coordinates $\Xi _d$}{equation.475}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:D-prime@cref\endcsname{{[equation][17][9]9.17}{[1][195][]195}}
\expandafter\gdef\csname r@ch06:eq:D-partitions\endcsname{{9.18}{195}{Step 5: The coordinates $\Xi _d$}{equation.476}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:D-partitions@cref\endcsname{{[equation][18][9]9.18}{[1][195][]195}}
\expandafter\gdef\csname r@ch06:sec:hessian\endcsname{{9.6}{196}{The constrained Wilson Hessian}{section.478}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:sec:hessian@cref\endcsname{{[section][6][9]9.6}{[1][196][]196}}
\expandafter\gdef\csname r@ch06:thm:constrained-convexity\endcsname{{9.7}{196}{}{theorem.479}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:thm:constrained-convexity@cref\endcsname{{[theorem][7][9]9.7}{[1][196][]196}}
\expandafter\gdef\csname r@ch06:eq:local-wilson-hessian\endcsname{{9.19}{197}{Step 1: The Hessian of the Wilson action}{equation.485}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:local-wilson-hessian@cref\endcsname{{[equation][19][9]9.19}{[1][197][]197}}
\expandafter\gdef\csname r@ch06:eq:curl-chart-image\endcsname{{9.20}{198}{Step 3: The size of the chart}{equation.488}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:curl-chart-image@cref\endcsname{{[equation][20][9]9.20}{[1][198][]198}}
\expandafter\gdef\csname r@ch06:eq:tangent-correction\endcsname{{9.21}{198}{Step 4: The term with the tangent map}{equation.490}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:tangent-correction@cref\endcsname{{[equation][21][9]9.21}{[1][198][]198}}
\expandafter\gdef\csname r@ch06:eq:first-hessian-term\endcsname{{9.22}{198}{Step 4: The term with the tangent map}{equation.491}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:first-hessian-term@cref\endcsname{{[equation][22][9]9.22}{[1][198][]198}}
\expandafter\gdef\csname r@ch06:eq:acceleration-l1\endcsname{{9.23}{199}{Step 5: The term with the second derivative of the coordinates}{equation.493}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:acceleration-l1@cref\endcsname{{[equation][23][9]9.23}{[1][198][]199}}
\expandafter\gdef\csname r@ch06:eq:acceleration-force\endcsname{{9.24}{199}{Step 5: The term with the second derivative of the coordinates}{equation.494}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:acceleration-force@cref\endcsname{{[equation][24][9]9.24}{[1][198][]199}}
\expandafter\gdef\csname r@ch06:sec:scalar\endcsname{{9.7}{199}{A scalar constrained inverse including constant modes}{section.496}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:sec:scalar@cref\endcsname{{[section][7][9]9.7}{[1][199][]199}}
\expandafter\gdef\csname r@ch06:lem:scalar-gauge-inverse\endcsname{{9.8}{200}{}{lemma.497}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:lem:scalar-gauge-inverse@cref\endcsname{{[lemma][8][]9.8}{[1][199][]200}}
\expandafter\gdef\csname r@ch06:eq:LB-identity\endcsname{{9.25}{201}{Step 1: Algebra}{equation.504}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:LB-identity@cref\endcsname{{[equation][25][9]9.25}{[1][201][]201}}
\expandafter\gdef\csname r@ch06:eq:scalar-cover-identities\endcsname{{9.26}{202}{Step 3: Transfer to the torus of side $n$}{equation.507}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:scalar-cover-identities@cref\endcsname{{[equation][26][9]9.26}{[1][202][]202}}
\expandafter\gdef\csname r@ch06:eq:penalty-normalization\endcsname{{9.27}{203}{Step 4: Conversion to penalty one}{equation.509}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:penalty-normalization@cref\endcsname{{[equation][27][9]9.27}{[1][203][]203}}
\expandafter\gdef\csname r@ch06:sec:gauge\endcsname{{9.8}{203}{Local gauge fixing with prescribed coarse gauge}{section.511}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:sec:gauge@cref\endcsname{{[section][8][9]9.8}{[1][203][]203}}
\expandafter\gdef\csname r@ch06:thm:projected-landau-slice\endcsname{{9.9}{204}{}{theorem.512}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:thm:projected-landau-slice@cref\endcsname{{[theorem][9][9]9.9}{[1][204][]204}}
\expandafter\gdef\csname r@ch06:eq:gauge-word-local\endcsname{{9.28}{205}{Step 1: The gauge transformation on one link}{equation.518}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:gauge-word-local@cref\endcsname{{[equation][28][9]9.28}{[1][205][]205}}
\expandafter\gdef\csname r@ch06:eq:gauge-word-remainder\endcsname{{9.29}{205}{Step 1: The gauge transformation on one link}{equation.519}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:gauge-word-remainder@cref\endcsname{{[equation][29][9]9.29}{[1][205][]205}}
\expandafter\gdef\csname r@ch06:eq:gauge-div-exact\endcsname{{9.30}{206}{Step 2: The divergence}{equation.521}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:gauge-div-exact@cref\endcsname{{[equation][30][9]9.30}{[1][206][]206}}
\expandafter\gdef\csname r@ch06:eq:gauge-terminal-linearization\endcsname{{9.31}{207}{Step 3: The composed coarse gauge transformation}{equation.523}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:gauge-terminal-linearization@cref\endcsname{{[equation][31][9]9.31}{[1][207][]207}}
\expandafter\gdef\csname r@ch06:eq:gauge-fixed-point\endcsname{{9.32}{207}{Step 4: Solving the two equations}{equation.525}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:gauge-fixed-point@cref\endcsname{{[equation][32][9]9.32}{[1][207][]207}}
\expandafter\gdef\csname r@ch06:eq:gauge-curl-comparison\endcsname{{9.33}{208}{Step 6: Real data and the chart}{equation.528}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:gauge-curl-comparison@cref\endcsname{{[equation][33][9]9.33}{[1][208][]208}}
\expandafter\gdef\csname r@ch06:sec:trial\endcsname{{9.9}{209}{A trial configuration with prescribed coarse field}{section.529}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:sec:trial@cref\endcsname{{[section][9][9]9.9}{[1][209][]209}}
\expandafter\gdef\csname r@ch06:cor:trial-configuration\endcsname{{9.10}{209}{}{corollary.530}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:cor:trial-configuration@cref\endcsname{{[corollary][10][]9.10}{[1][209][]209}}
\expandafter\gdef\csname r@ch06:eq:section-curvature\endcsname{{9.34}{210}{Step 3: The coordinates}{equation.538}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:section-curvature@cref\endcsname{{[equation][34][9]9.34}{[1][210][]210}}
\expandafter\gdef\csname r@ch06:eq:one-shell-sides\endcsname{{9.35}{211}{Step 3: The coordinates}{equation.539}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:one-shell-sides@cref\endcsname{{[equation][35][9]9.35}{[1][210][]211}}
\expandafter\gdef\csname r@ch06:eq:trial-plaquette-action\endcsname{{9.36}{211}{Step 4: The action}{equation.541}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:trial-plaquette-action@cref\endcsname{{[equation][36][9]9.36}{[1][211][]211}}
\expandafter\gdef\csname r@ch06:eq:trial-action-bound\endcsname{{9.37}{211}{Step 4: The action}{equation.542}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch06:eq:trial-action-bound@cref\endcsname{{[equation][37][9]9.37}{[1][211][]211}}
\expandafter\gdef\csname r@ch07:chapter\endcsname{{10}{212}{Classical background fields and analytic estimates}{chapter.543}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:chapter@cref\endcsname{{[chapter][10][]10}{[1][212][]212}}
\expandafter\gdef\csname r@ch07:sec:conventions\endcsname{{10.1}{213}{Rescaled fields, scalar products, and norms}{section.544}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:sec:conventions@cref\endcsname{{[section][1][10]10.1}{[1][213][]213}}
\expandafter\gdef\csname r@ch07:def:physical-classical-data\endcsname{{10.1}{214}{}{definition.547}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:def:physical-classical-data@cref\endcsname{{[definition][1][]10.1}{[1][214][]214}}
\expandafter\gdef\csname r@ch07:sec:green\endcsname{{10.2}{217}{The constrained Green operator}{section.554}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:sec:green@cref\endcsname{{[section][2][10]10.2}{[1][217][]217}}
\expandafter\gdef\csname r@ch07:lem:green-rows\endcsname{{10.2}{217}{}{lemma.555}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:lem:green-rows@cref\endcsname{{[lemma][2][]10.2}{[1][217][]217}}
\expandafter\gdef\csname r@ch07:eq:green-rows\endcsname{{10.1}{217}{}{equation.556}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:green-rows@cref\endcsname{{[equation][1][10]10.1}{[1][217][]217}}
\expandafter\gdef\csname r@ch07:eq:cover-green-block\endcsname{{10.2}{219}{Step 4: Ba{\l }aban's estimate on the cover}{equation.561}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:cover-green-block@cref\endcsname{{[equation][2][10]10.2}{[1][219][]219}}
\expandafter\gdef\csname r@ch07:eq:cover-green-row\endcsname{{10.3}{219}{Step 5: absolute rows on the cover}{equation.563}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:cover-green-row@cref\endcsname{{[equation][3][10]10.3}{[1][219][]219}}
\expandafter\gdef\csname r@ch07:eq:cover-inner-product\endcsname{{10.4}{220}{Step 6: restriction to deck-invariant fields}{equation.565}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:cover-inner-product@cref\endcsname{{[equation][4][10]10.4}{[1][220][]220}}
\expandafter\gdef\csname r@ch07:eq:projector-descent\endcsname{{10.5}{220}{Step 6: restriction to deck-invariant fields}{equation.566}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:projector-descent@cref\endcsname{{[equation][5][10]10.5}{[1][220][]220}}
\expandafter\gdef\csname r@ch07:eq:covariance-descent\endcsname{{10.6}{220}{Step 6: restriction to deck-invariant fields}{equation.567}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:covariance-descent@cref\endcsname{{[equation][6][10]10.6}{[1][220][]220}}
\expandafter\gdef\csname r@ch07:eq:covariance-image-sum\endcsname{{10.7}{221}{Step 6: restriction to deck-invariant fields}{equation.568}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:covariance-image-sum@cref\endcsname{{[equation][7][10]10.7}{[1][221][]221}}
\expandafter\gdef\csname r@ch07:sec:block-maps\endcsname{{10.3}{221}{Analytic block maps and constraint coordinates}{section.570}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:sec:block-maps@cref\endcsname{{[section][3][10]10.3}{[1][221][]221}}
\expandafter\gdef\csname r@ch07:lem:composition-majorant\endcsname{{10.3}{222}{}{lemma.571}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:lem:composition-majorant@cref\endcsname{{[lemma][3][]10.3}{[1][221][]222}}
\expandafter\gdef\csname r@ch07:eq:positive-recurrence\endcsname{{10.8}{223}{Analytic block maps and constraint coordinates}{equation.576}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:positive-recurrence@cref\endcsname{{[equation][8][10]10.8}{[1][222][]223}}
\expandafter\gdef\csname r@ch07:lem:word-majorant\endcsname{{10.4}{223}{}{lemma.577}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:lem:word-majorant@cref\endcsname{{[lemma][4][]10.4}{[1][223][]223}}
\expandafter\gdef\csname r@ch07:eq:modified-word\endcsname{{10.9}{224}{}{equation.578}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:modified-word@cref\endcsname{{[equation][9][10]10.9}{[1][223][]224}}
\expandafter\gdef\csname r@ch07:eq:D-majorant\endcsname{{10.10}{224}{}{equation.582}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:D-majorant@cref\endcsname{{[equation][10][10]10.10}{[1][224][]224}}
\expandafter\gdef\csname r@ch07:eq:accumulated-gauge\endcsname{{10.11}{224}{}{equation.584}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:accumulated-gauge@cref\endcsname{{[equation][11][10]10.11}{[1][224][]224}}
\expandafter\gdef\csname r@ch07:eq:positive-Q\endcsname{{10.12}{225}{Step 2: the linear part at zero}{equation.587}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:positive-Q@cref\endcsname{{[equation][12][10]10.12}{[1][225][]225}}
\expandafter\gdef\csname r@ch07:eq:word-step\endcsname{{10.13}{225}{Step 3: bounds for one step}{equation.589}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:word-step@cref\endcsname{{[equation][13][10]10.13}{[1][225][]225}}
\expandafter\gdef\csname r@ch07:eq:ordinary-gauge-covariance\endcsname{{10.14}{227}{Step 7: gauge covariance of the block averaging map}{equation.594}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:ordinary-gauge-covariance@cref\endcsname{{[equation][14][10]10.14}{[1][227][]227}}
\expandafter\gdef\csname r@ch07:sec:euler\endcsname{{10.4}{227}{The Euler equation in constraint coordinates}{section.596}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:sec:euler@cref\endcsname{{[section][4][10]10.4}{[1][227][]227}}
\expandafter\gdef\csname r@ch07:lem:wilson-euler\endcsname{{10.5}{228}{}{lemma.597}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:lem:wilson-euler@cref\endcsname{{[lemma][5][]10.5}{[1][228][]228}}
\expandafter\gdef\csname r@ch07:sec:background\endcsname{{10.5}{231}{The classical background field and local minimality}{section.609}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:sec:background@cref\endcsname{{[section][5][10]10.5}{[1][231][]231}}
\expandafter\gdef\csname r@ch07:thm:regular-minimum\endcsname{{10.6}{231}{}{theorem.611}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:thm:regular-minimum@cref\endcsname{{[theorem][6][10]10.6}{[1][231][]231}}
\expandafter\gdef\csname r@ch07:eq:min-contraction\endcsname{{10.15}{233}{Step 1: contraction}{equation.620}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:min-contraction@cref\endcsname{{[equation][15][10]10.15}{[1][233][]233}}
\expandafter\gdef\csname r@ch07:eq:ambient-min-pair\endcsname{{10.16}{234}{Step 2: factorial bounds for the background field}{equation.622}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:ambient-min-pair@cref\endcsname{{[equation][16][10]10.16}{[1][234][]234}}
\expandafter\gdef\csname r@ch07:eq:min-all-order-row\endcsname{{10.17}{235}{Step 2: factorial bounds for the background field}{equation.623}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:min-all-order-row@cref\endcsname{{[equation][17][10]10.17}{[1][235][]235}}
\expandafter\gdef\csname r@ch07:eq:row-to-physical-column\endcsname{{10.18}{235}{Step 3: from rows to columns}{equation.625}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:row-to-physical-column@cref\endcsname{{[equation][18][10]10.18}{[1][235][]235}}
\expandafter\gdef\csname r@ch07:eq:envelope\endcsname{{10.19}{236}{Step 5: the effective classical action}{equation.628}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:envelope@cref\endcsname{{[equation][19][10]10.19}{[1][236][]236}}
\expandafter\gdef\csname r@ch07:eq:action-majorant\endcsname{{10.20}{237}{Step 6: coefficient bounds for the action}{equation.630}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:action-majorant@cref\endcsname{{[equation][20][10]10.20}{[1][237][]237}}
\expandafter\gdef\csname r@ch07:sec:symmetries\endcsname{{10.6}{238}{Gauge extensions and symmetries of the effective action}{section.632}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:sec:symmetries@cref\endcsname{{[section][6][10]10.6}{[1][238][]238}}
\expandafter\gdef\csname r@ch07:lem:criticality-transfer\endcsname{{10.7}{238}{}{lemma.633}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:lem:criticality-transfer@cref\endcsname{{[lemma][7][]10.7}{[1][238][]238}}
\expandafter\gdef\csname r@ch07:lem:classical-symmetries\endcsname{{10.8}{239}{}{lemma.637}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:lem:classical-symmetries@cref\endcsname{{[lemma][8][]10.8}{[1][239][]239}}
\expandafter\gdef\csname r@ch07:sec:nested\endcsname{{10.7}{243}{One averaging step and nested minimization}{section.647}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:sec:nested@cref\endcsname{{[section][7][10]10.7}{[1][243][]243}}
\expandafter\gdef\csname r@ch07:lem:shell-branch\endcsname{{10.9}{244}{}{lemma.649}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:lem:shell-branch@cref\endcsname{{[lemma][9][]10.9}{[1][244][]244}}
\expandafter\gdef\csname r@ch07:eq:forest-gauge\endcsname{{10.21}{244}{Step 1: the block axial gauge}{equation.654}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:forest-gauge@cref\endcsname{{[equation][21][10]10.21}{[1][244][]244}}
\expandafter\gdef\csname r@ch07:thm:bellman\endcsname{{10.10}{246}{}{theorem.660}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:thm:bellman@cref\endcsname{{[theorem][10][10]10.10}{[1][246][]246}}
\expandafter\gdef\csname r@ch07:eq:flat-value-hessian\endcsname{{10.22}{248}{Step 1: the Hessian of the action at the identity}{equation.668}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:flat-value-hessian@cref\endcsname{{[equation][22][10]10.22}{[1][248][]248}}
\expandafter\gdef\csname r@ch07:eq:shell-full-hessian\endcsname{{10.23}{249}{Step 3: the Hessian away from the identity}{equation.671}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:shell-full-hessian@cref\endcsname{{[equation][23][10]10.23}{[1][249][]249}}
\expandafter\gdef\csname r@ch07:eq:shell-hessian-perturbation\endcsname{{10.24}{249}{Step 3: the Hessian away from the identity}{equation.672}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:shell-hessian-perturbation@cref\endcsname{{[equation][24][10]10.24}{[1][249][]249}}
\expandafter\gdef\csname r@ch07:sec:intermediate\endcsname{{10.8}{252}{Analytic estimates at intermediate scales}{section.679}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:sec:intermediate@cref\endcsname{{[section][8][10]10.8}{[1][252][]252}}
\expandafter\gdef\csname r@ch07:thm:all-root-classical\endcsname{{10.11}{252}{}{theorem.680}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:thm:all-root-classical@cref\endcsname{{[theorem][11][10]10.11}{[1][252][]252}}
\expandafter\gdef\csname r@ch07:eq:exact-jet-word\endcsname{{10.25}{254}{Step 1: one step on field and difference variables}{equation.687}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:exact-jet-word@cref\endcsname{{[equation][25][10]10.25}{[1][254][]254}}
\expandafter\gdef\csname r@ch07:eq:jet-power-ledger\endcsname{{10.26}{254}{Step 2: the mesh powers}{equation.689}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:jet-power-ledger@cref\endcsname{{[equation][26][10]10.26}{[1][254][]254}}
\expandafter\gdef\csname r@ch07:eq:joint-word-tensors\endcsname{{10.27}{254}{Step 2: the mesh powers}{equation.690}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:joint-word-tensors@cref\endcsname{{[equation][27][10]10.27}{[1][254][]254}}
\expandafter\gdef\csname r@ch07:eq:composed-joint-word\endcsname{{10.28}{255}{Step 4: composition of the steps}{equation.693}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch07:eq:composed-joint-word@cref\endcsname{{[equation][28][10]10.28}{[1][255][]255}}
\expandafter\gdef\csname r@ch08:chapter\endcsname{{11}{258}{Classical continuum limits and contraction estimates}{chapter.698}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:chapter@cref\endcsname{{[chapter][11][]11}{[1][258][]258}}
\expandafter\gdef\csname r@ch08:sec:setting\endcsname{{11.1}{259}{Setting and conventions}{section.703}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:sec:setting@cref\endcsname{{[section][1][11]11.1}{[1][259][]259}}
\expandafter\gdef\csname r@ch08:eq:gauge-corrected-relation\endcsname{{11.1}{261}{Block averages and gauge-corrected block averages}{equation.712}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:gauge-corrected-relation@cref\endcsname{{[equation][1][11]11.1}{[1][260][]261}}
\expandafter\gdef\csname r@ch08:eq:aligned-series-convention\endcsname{{11.2}{261}{Majorant series}{equation.714}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:aligned-series-convention@cref\endcsname{{[equation][2][11]11.2}{[1][261][]261}}
\expandafter\gdef\csname r@ch08:lem:majorant-composition\endcsname{{11.1}{261}{}{lemma.715}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:lem:majorant-composition@cref\endcsname{{[lemma][1][]11.1}{[1][261][]261}}
\expandafter\gdef\csname r@ch08:eq:partition-count\endcsname{{11.3}{262}{}{equation.716}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:partition-count@cref\endcsname{{[equation][3][11]11.3}{[1][261][]262}}
\expandafter\gdef\csname r@ch08:sec:refinement\endcsname{{11.2}{262}{Classical actions at a fixed coarse lattice}{section.717}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:sec:refinement@cref\endcsname{{[section][2][11]11.2}{[1][262][]262}}
\expandafter\gdef\csname r@ch08:thm:classical-cutoff\endcsname{{11.2}{262}{}{theorem.718}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:thm:classical-cutoff@cref\endcsname{{[theorem][2][11]11.2}{[1][262][]262}}
\expandafter\gdef\csname r@ch08:lem:one-step\endcsname{{11.3}{264}{}{lemma.722}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:lem:one-step@cref\endcsname{{[lemma][3][]11.3}{[1][263][]264}}
\expandafter\gdef\csname r@ch08:eq:one-step-energy\endcsname{{11.4}{264}{}{equation.723}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:one-step-energy@cref\endcsname{{[equation][4][11]11.4}{[1][264][]264}}
\expandafter\gdef\csname r@ch08:eq:one-step-potential\endcsname{{11.5}{264}{}{equation.724}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:one-step-potential@cref\endcsname{{[equation][5][11]11.5}{[1][264][]264}}
\expandafter\gdef\csname r@ch08:lem:analytic-compactness\endcsname{{11.4}{264}{}{lemma.725}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:lem:analytic-compactness@cref\endcsname{{[lemma][4][]11.4}{[1][264][]264}}
\expandafter\gdef\csname r@ch08:eq:value-ceiling\endcsname{{11.6}{266}{Step 2: the monotone limit}{equation.731}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:value-ceiling@cref\endcsname{{[equation][6][11]11.6}{[1][266][]266}}
\expandafter\gdef\csname r@ch08:eq:root-integral\endcsname{{11.7}{269}{Step 4: finite reconstruction maps}{equation.734}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:root-integral@cref\endcsname{{[equation][7][11]11.7}{[1][269][]269}}
\expandafter\gdef\csname r@ch08:rem:complex-domain\endcsname{{11.5}{270}{}{remark.735}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:rem:complex-domain@cref\endcsname{{[remark][5][]11.5}{[1][270][]270}}
\expandafter\gdef\csname r@ch08:sec:overlap\endcsname{{11.3}{270}{Gauge invariance on overlapping charts}{section.736}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:sec:overlap@cref\endcsname{{[section][3][11]11.3}{[1][270][]270}}
\expandafter\gdef\csname r@ch08:thm:compact-overlap\endcsname{{11.6}{270}{}{theorem.737}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:thm:compact-overlap@cref\endcsname{{[theorem][6][11]11.6}{[1][270][]270}}
\expandafter\gdef\csname r@ch08:rem:quaternion-picture\endcsname{{11.7}{271}{}{remark.741}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:rem:quaternion-picture@cref\endcsname{{[remark][7][]11.7}{[1][271][]271}}
\expandafter\gdef\csname r@ch08:eq:quaternion-cardinal\endcsname{{11.8}{271}{Step 2: the interpolation}{equation.744}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:quaternion-cardinal@cref\endcsname{{[equation][8][11]11.8}{[1][271][]271}}
\expandafter\gdef\csname r@ch08:eq:compact-lift-derivatives\endcsname{{11.9}{272}{Step 3: derivative bounds and symmetries}{equation.746}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:compact-lift-derivatives@cref\endcsname{{[equation][9][11]11.9}{[1][272][]272}}
\expandafter\gdef\csname r@ch08:eq:large-gauge-word\endcsname{{11.10}{273}{Step 5: the effect on a fine potential}{equation.749}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:large-gauge-word@cref\endcsname{{[equation][10][11]11.10}{[1][273][]273}}
\expandafter\gdef\csname r@ch08:eq:large-gauge-c1\endcsname{{11.11}{273}{Step 5: the effect on a fine potential}{equation.750}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:large-gauge-c1@cref\endcsname{{[equation][11][11]11.11}{[1][273][]273}}
\expandafter\gdef\csname r@ch08:sec:alignment\endcsname{{11.4}{275}{Gauge alignment of the successive constrained minimizers}{section.753}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:sec:alignment@cref\endcsname{{[section][4][11]11.4}{[1][275][]275}}
\expandafter\gdef\csname r@ch08:lem:successive-minimizer-orbit\endcsname{{11.8}{275}{}{lemma.755}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:lem:successive-minimizer-orbit@cref\endcsname{{[lemma][8][]11.8}{[1][275][]275}}
\expandafter\gdef\csname r@ch08:lem:aligned-zero-white\endcsname{{11.9}{277}{}{lemma.758}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:lem:aligned-zero-white@cref\endcsname{{[lemma][9][]11.9}{[1][277][]277}}
\expandafter\gdef\csname r@ch08:tab:constants\endcsname{{11.1}{279}{Order of the constants in \cref {ch08:chapter}}{table.caption.765}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:tab:constants@cref\endcsname{{[table][1][11]11.1}{[1][278][]279}}
\expandafter\gdef\csname r@ch08:eq:remaining-gauge\endcsname{{11.12}{279}{Step 2: the accumulated gauge transformation}{equation.768}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:remaining-gauge@cref\endcsname{{[equation][12][11]11.12}{[1][279][]279}}
\expandafter\gdef\csname r@ch08:eq:local-gauge-series\endcsname{{11.13}{279}{Step 2: the accumulated gauge transformation}{equation.769}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:local-gauge-series@cref\endcsname{{[equation][13][11]11.13}{[1][279][]279}}
\expandafter\gdef\csname r@ch08:eq:terminal-gauge-inverse\endcsname{{11.14}{280}{Step 2: the accumulated gauge transformation}{equation.770}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:terminal-gauge-inverse@cref\endcsname{{[equation][14][11]11.14}{[1][280][]280}}
\expandafter\gdef\csname r@ch08:eq:volume-exponential\endcsname{{11.15}{280}{Step 3: from rows to columns}{equation.772}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:volume-exponential@cref\endcsname{{[equation][15][11]11.15}{[1][280][]280}}
\expandafter\gdef\csname r@ch08:eq:positive-alignment\endcsname{{11.16}{281}{Step 4: the interpolated gauge transformation and the exact constraint}{equation.774}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:positive-alignment@cref\endcsname{{[equation][16][11]11.16}{[1][280][]281}}
\expandafter\gdef\csname r@ch08:eq:removable-alignment\endcsname{{11.17}{281}{Step 5: removal of the inverse mesh}{equation.776}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:removable-alignment@cref\endcsname{{[equation][17][11]11.17}{[1][281][]281}}
\expandafter\gdef\csname r@ch08:eq:alignment-zero-mesh\endcsname{{11.18}{281}{Step 5: removal of the inverse mesh}{equation.777}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:alignment-zero-mesh@cref\endcsname{{[equation][18][11]11.18}{[1][281][]281}}
\expandafter\gdef\csname r@ch08:eq:aligned-positive-composition\endcsname{{11.19}{282}{Step 6: one majorant for all bounds}{equation.779}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:aligned-positive-composition@cref\endcsname{{[equation][19][11]11.19}{[1][282][]282}}
\expandafter\gdef\csname r@ch08:eq:aligned-positive-majorant\endcsname{{11.20}{282}{Step 6: one majorant for all bounds}{equation.780}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:aligned-positive-majorant@cref\endcsname{{[equation][20][11]11.20}{[1][282][]282}}
\expandafter\gdef\csname r@ch08:eq:forest-order\endcsname{{11.21}{283}{Step 7: the block axial gauge transformation for fixed $J$}{equation.782}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:forest-order@cref\endcsname{{[equation][21][11]11.21}{[1][283][]283}}
\expandafter\gdef\csname r@ch08:eq:physical-schur-blocks\endcsname{{11.22}{284}{Step 8: identification with the zero-fluctuation reconstruction}{equation.784}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:physical-schur-blocks@cref\endcsname{{[equation][22][11]11.22}{[1][284][]284}}
\expandafter\gdef\csname r@ch08:sec:contraction\endcsname{{11.5}{285}{Contraction of the interaction under the classical background field}{section.785}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:sec:contraction@cref\endcsname{{[section][5][11]11.5}{[1][285][]285}}
\expandafter\gdef\csname r@ch08:thm:principal-contraction\endcsname{{11.10}{286}{}{theorem.786}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:thm:principal-contraction@cref\endcsname{{[theorem][10][11]11.10}{[1][286][]286}}
\expandafter\gdef\csname r@ch08:lem:two-differences\endcsname{{11.11}{288}{}{lemma.791}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:lem:two-differences@cref\endcsname{{[lemma][11][]11.11}{[1][288][]288}}
\expandafter\gdef\csname r@ch08:eq:small-two-jets\endcsname{{11.23}{288}{Contraction of the interaction under the classical background field}{equation.792}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:small-two-jets@cref\endcsname{{[equation][23][11]11.23}{[1][288][]288}}
\expandafter\gdef\csname r@ch08:eq:laurent-division\endcsname{{11.24}{289}{Contraction of the interaction under the classical background field}{equation.793}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:laurent-division@cref\endcsname{{[equation][24][11]11.24}{[1][289][]289}}
\expandafter\gdef\csname r@ch08:eq:primitive-bound\endcsname{{11.25}{290}{Step 1: the quadratic term}{equation.795}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:primitive-bound@cref\endcsname{{[equation][25][11]11.25}{[1][290][]290}}
\expandafter\gdef\csname r@ch08:eq:cubic-force\endcsname{{11.26}{290}{Step 2: the cubic term}{equation.797}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:cubic-force@cref\endcsname{{[equation][26][11]11.26}{[1][290][]290}}
\expandafter\gdef\csname r@ch08:eq:path-difference\endcsname{{11.27}{290}{Step 2: the cubic term}{equation.798}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:path-difference@cref\endcsname{{[equation][27][11]11.27}{[1][290][]290}}
\expandafter\gdef\csname r@ch08:eq:jet-series\endcsname{{11.28}{291}{Step 3: majorants of the background field and power counting}{equation.800}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:jet-series@cref\endcsname{{[equation][28][11]11.28}{[1][291][]291}}
\expandafter\gdef\csname r@ch08:tab:powers\endcsname{{11.2}{292}{Powers of $\delta =M^{-J}$ in the proof of \cref {ch08:thm:principal-contraction}}{table.caption.801}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:tab:powers@cref\endcsname{{[table][2][11]11.2}{[1][291][]292}}
\expandafter\gdef\csname r@ch08:eq:principal-series\endcsname{{11.29}{292}{Step 6: terms of degree at least four}{equation.805}{3d-ym-torus-volume-i.pdf}}
\expandafter\gdef\csname r@ch08:eq:principal-series@cref\endcsname{{[equation][29][11]11.29}{[1][292][]292}}
% 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}}
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\expandafter\gdef\csname r@ch12:lem:backward-law@cref\endcsname{{[lemma][10][]15.10}{[1][144][]144}}
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\expandafter\gdef\csname r@ch12:rem:joint-haar-realization@cref\endcsname{{[remark][11][]15.11}{[1][147][]147}}
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\expandafter\gdef\csname r@ch12:lem:terminal-selector-tv@cref\endcsname{{[lemma][12][]15.12}{[1][147][]147}}
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\expandafter\gdef\csname r@ch12:thm:replacement-family@cref\endcsname{{[theorem][13][15]15.13}{[1][148][]148}}
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\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}}
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\expandafter\gdef\csname r@ch12:thm:representative-prefixes@cref\endcsname{{[theorem][14][15]15.14}{[1][155][]155}}
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\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}}
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\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}}
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\expandafter\gdef\csname r@ch12:sec:amplitude@cref\endcsname{{[section][7][15]15.7}{[1][157][]157}}
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\expandafter\gdef\csname r@ch12:lem:deterministic-prolongation-reserve@cref\endcsname{{[lemma][16][]15.16}{[1][157][]158}}
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\expandafter\gdef\csname r@part:reconstruction@cref\endcsname{{[part][5][]V}{[1][170][]170}}
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\expandafter\gdef\csname r@ch13:eq:cell-derivatives\endcsname{{16.6}{176}{The continuum kernel and the cell integrals}{equation.529}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:cell-derivatives@cref\endcsname{{[equation][6][16]16.6}{[1][176][]176}}
\expandafter\gdef\csname r@ch13:sec:cardinal\endcsname{{16.2}{177}{Cardinal interpolation and volume-normalized columns}{section.530}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:sec:cardinal@cref\endcsname{{[section][2][16]16.2}{[1][177][]177}}
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\expandafter\gdef\csname r@ch13:eq:cardinal-map@cref\endcsname{{[equation][7][16]16.7}{[1][177][]177}}
\expandafter\gdef\csname r@ch13:lem:cardinal-columns\endcsname{{16.3}{177}{}{lemma.532}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:lem:cardinal-columns@cref\endcsname{{[lemma][3][]16.3}{[1][177][]177}}
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\expandafter\gdef\csname r@ch13:eq:coefficient-hypotheses@cref\endcsname{{[equation][8][16]16.8}{[1][178][]178}}
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\expandafter\gdef\csname r@ch13:eq:difference-average@cref\endcsname{{[equation][10][16]16.10}{[1][179][]179}}
\expandafter\gdef\csname r@ch13:eq:exact-cardinal-mass\endcsname{{16.11}{180}{The basis functions and their masses}{equation.540}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:exact-cardinal-mass@cref\endcsname{{[equation][11][16]16.11}{[1][180][]180}}
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\expandafter\gdef\csname r@ch13:eq:interpolated-schur@cref\endcsname{{[equation][12][16]16.12}{[1][180][]180}}
\expandafter\gdef\csname r@ch13:eq:late-rows\endcsname{{16.13}{181}{Row sums and volume-normalized column sums}{equation.544}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:late-rows@cref\endcsname{{[equation][13][16]16.13}{[1][180][]181}}
\expandafter\gdef\csname r@ch13:eq:physical-schur\endcsname{{16.14}{181}{Row sums and volume-normalized column sums}{equation.545}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:physical-schur@cref\endcsname{{[equation][14][16]16.14}{[1][181][]181}}
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\expandafter\gdef\csname r@ch13:sec:chaining@cref\endcsname{{[section][3][16]16.3}{[1][182][]182}}
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\expandafter\gdef\csname r@ch13:lem:coefficient-metric-chaining@cref\endcsname{{[lemma][4][]16.4}{[1][182][]182}}
\expandafter\gdef\csname r@ch13:sec:concentration\endcsname{{16.4}{185}{Concentration of conditionally centered reconstructions}{section.550}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:sec:concentration@cref\endcsname{{[section][4][16]16.4}{[1][185][]185}}
\expandafter\gdef\csname r@ch13:eq:two-support-kernels\endcsname{{16.15}{185}{The conditional fluctuation kernel}{equation.553}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:two-support-kernels@cref\endcsname{{[equation][15][16]16.15}{[1][185][]185}}
\expandafter\gdef\csname r@ch13:lem:reconstruction-domain\endcsname{{16.5}{187}{}{lemma.556}{3d-ym-torus-volume-ii.pdf}}
\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}}
\expandafter\gdef\csname r@ch13:eq:actual-shell-gates\endcsname{{16.17}{187}{}{equation.562}{3d-ym-torus-volume-ii.pdf}}
\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}}
\expandafter\gdef\csname r@ch13:eq:shell-product-inequalities\endcsname{{16.18}{191}{Step 1: the kernel and its convexity}{equation.566}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:shell-product-inequalities@cref\endcsname{{[equation][18][16]16.18}{[1][191][]191}}
\expandafter\gdef\csname r@ch13:eq:shell-counting-jacobian\endcsname{{16.19}{191}{Step 2: the Jacobian}{equation.568}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:shell-counting-jacobian@cref\endcsname{{[equation][19][16]16.19}{[1][191][]191}}
\expandafter\gdef\csname r@ch13:eq:shell-jacobian\endcsname{{16.20}{192}{Step 2: the Jacobian}{equation.569}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:shell-jacobian@cref\endcsname{{[equation][20][16]16.20}{[1][192][]192}}
\expandafter\gdef\csname r@ch13:eq:global-array-lipschitz\endcsname{{16.21}{192}{Step 2: the Jacobian}{equation.570}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:global-array-lipschitz@cref\endcsname{{[equation][21][16]16.21}{[1][192][]192}}
\expandafter\gdef\csname r@ch13:eq:embedded-heat\endcsname{{16.22}{192}{Step 3: a fixed time}{equation.572}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:embedded-heat@cref\endcsname{{[equation][22][16]16.22}{[1][192][]192}}
\expandafter\gdef\csname r@ch13:eq:shell-sigma\endcsname{{16.23}{192}{Step 3: a fixed time}{equation.573}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:shell-sigma@cref\endcsname{{[equation][23][16]16.23}{[1][192][]192}}
\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}}
\expandafter\gdef\csname r@ch13:eq:shell-metric@cref\endcsname{{[equation][24][16]16.24}{[1][192][]192}}
\expandafter\gdef\csname r@ch13:eq:fixed-time-max\endcsname{{16.25}{193}{Step 3: a fixed time}{equation.575}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:fixed-time-max@cref\endcsname{{[equation][25][16]16.25}{[1][193][]193}}
\expandafter\gdef\csname r@ch13:eq:cell-max\endcsname{{16.26}{193}{Step 4: the supremum over times}{equation.577}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:cell-max@cref\endcsname{{[equation][26][16]16.26}{[1][193][]193}}
\expandafter\gdef\csname r@ch13:eq:dyadic-heat\endcsname{{16.27}{193}{Step 4: the supremum over times}{equation.578}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:dyadic-heat@cref\endcsname{{[equation][27][16]16.27}{[1][193][]193}}
\expandafter\gdef\csname r@ch13:eq:weighted-sigma\endcsname{{16.28}{193}{Step 4: the supremum over times}{equation.579}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:weighted-sigma@cref\endcsname{{[equation][28][16]16.28}{[1][193][]193}}
\expandafter\gdef\csname r@ch13:eq:source-cell-weights\endcsname{{16.29}{194}{Step 6: deterministic sources and generating functions}{equation.582}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:source-cell-weights@cref\endcsname{{[equation][29][16]16.29}{[1][194][]194}}
\expandafter\gdef\csname r@ch13:sec:convolution\endcsname{{16.5}{195}{Weighted heat convolutions}{section.584}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:sec:convolution@cref\endcsname{{[section][5][16]16.5}{[1][195][]195}}
\expandafter\gdef\csname r@ch13:lem:weighted-convolution\endcsname{{16.8}{196}{}{lemma.585}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:lem:weighted-convolution@cref\endcsname{{[lemma][8][]16.8}{[1][195][]196}}
\expandafter\gdef\csname r@ch13:eq:short-convolution\endcsname{{16.30}{196}{Weighted heat convolutions}{equation.586}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:short-convolution@cref\endcsname{{[equation][30][16]16.30}{[1][196][]196}}
\expandafter\gdef\csname r@ch13:eq:weighted-powers\endcsname{{16.31}{197}{Weighted heat convolutions}{equation.587}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:weighted-powers@cref\endcsname{{[equation][31][16]16.31}{[1][197][]197}}
\expandafter\gdef\csname r@ch13:sec:consistency\endcsname{{16.6}{197}{Heat consistency for finite-scale reconstructions}{section.588}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:sec:consistency@cref\endcsname{{[section][6][16]16.6}{[1][197][]197}}
\expandafter\gdef\csname r@ch13:lem:fixed-prefix-heat\endcsname{{16.9}{198}{}{lemma.589}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:lem:fixed-prefix-heat@cref\endcsname{{[lemma][9][]16.9}{[1][197][]198}}
\expandafter\gdef\csname r@ch13:rem:interpolant-derivative\endcsname{{16.10}{198}{}{remark.590}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:rem:interpolant-derivative@cref\endcsname{{[remark][10][]16.10}{[1][198][]198}}
\expandafter\gdef\csname r@ch13:eq:prefix-smooth-bounds\endcsname{{16.32}{198}{Heat consistency for finite-scale reconstructions}{equation.591}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:prefix-smooth-bounds@cref\endcsname{{[equation][32][16]16.32}{[1][198][]198}}
\expandafter\gdef\csname r@ch13:eq:generator-error\endcsname{{16.33}{199}{Heat consistency for finite-scale reconstructions}{equation.592}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:generator-error@cref\endcsname{{[equation][33][16]16.33}{[1][199][]199}}
\expandafter\gdef\csname r@ch13:eq:heat-duhamel\endcsname{{16.34}{199}{Heat consistency for finite-scale reconstructions}{equation.593}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:heat-duhamel@cref\endcsname{{[equation][34][16]16.34}{[1][199][]199}}
\expandafter\gdef\csname r@ch13:eq:differentiated-data\endcsname{{16.35}{199}{Heat consistency for finite-scale reconstructions}{equation.594}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:differentiated-data@cref\endcsname{{[equation][35][16]16.35}{[1][199][]199}}
\expandafter\gdef\csname r@ch13:eq:sampling-interpolation\endcsname{{16.36}{199}{Heat consistency for finite-scale reconstructions}{equation.595}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch13:eq:sampling-interpolation@cref\endcsname{{[equation][36][16]16.36}{[1][199][]199}}
\expandafter\gdef\csname r@ch14:chapter\endcsname{{17}{201}{Backward reconstruction and conditional quadratic moments}{chapter.596}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:chapter@cref\endcsname{{[chapter][17][]17}{[1][201][]201}}
\expandafter\gdef\csname r@ch14:eq:increment-decomposition\endcsname{{17.1}{201}{The purpose of the three estimates}{equation.598}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:increment-decomposition@cref\endcsname{{[equation][1][17]17.1}{[1][201][]201}}
\expandafter\gdef\csname r@ch14:sec:setting\endcsname{{17.1}{202}{Setting}{section.600}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:sec:setting@cref\endcsname{{[section][1][17]17.1}{[1][202][]202}}
\expandafter\gdef\csname r@ch14:eq:backward-chain\endcsname{{17.2}{203}{The coarse-to-fine reconstruction}{equation.603}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:backward-chain@cref\endcsname{{[equation][2][17]17.2}{[1][203][]203}}
\expandafter\gdef\csname r@ch14:eq:heat-norm\endcsname{{17.3}{203}{Norms}{equation.606}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:heat-norm@cref\endcsname{{[equation][3][17]17.3}{[1][203][]203}}
\expandafter\gdef\csname r@ch14:eq:bounded-heat\endcsname{{17.4}{203}{Norms}{equation.607}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:bounded-heat@cref\endcsname{{[equation][4][17]17.4}{[1][203][]203}}
\expandafter\gdef\csname r@ch14:sec:means\endcsname{{17.2}{204}{Conditional means along the reconstruction}{section.609}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:sec:means@cref\endcsname{{[section][2][17]17.2}{[1][204][]204}}
\expandafter\gdef\csname r@ch14:lem:backward-caps-and-means\endcsname{{17.1}{204}{}{lemma.610}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:lem:backward-caps-and-means@cref\endcsname{{[lemma][1][]17.1}{[1][204][]204}}
\expandafter\gdef\csname r@ch14:eq:cap-constants\endcsname{{17.5}{206}{Step 1: the reconstruction amplitude bound}{equation.612}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:cap-constants@cref\endcsname{{[equation][5][17]17.5}{[1][206][]206}}
\expandafter\gdef\csname r@ch14:eq:p-affine\endcsname{{17.6}{206}{Step 1: the reconstruction amplitude bound}{equation.613}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:p-affine@cref\endcsname{{[equation][6][17]17.6}{[1][206][]206}}
\expandafter\gdef\csname r@ch14:eq:original-second-moment\endcsname{{17.7}{207}{Step 3: the conditional mean in the gauge of the reconstruction maps}{equation.616}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch14:eq:geometric-entropy\endcsname{{17.8}{208}{Step 4: the centered part}{equation.618}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:geometric-entropy@cref\endcsname{{[equation][8][17]17.8}{[1][208][]208}}
\expandafter\gdef\csname r@ch14:sec:signed\endcsname{{17.3}{209}{Column cancellation and heat-kernel estimates}{section.619}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:sec:signed@cref\endcsname{{[section][3][17]17.3}{[1][208][]209}}
\expandafter\gdef\csname r@ch14:lem:signed-physical-columns\endcsname{{17.2}{209}{}{lemma.620}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:lem:signed-physical-columns@cref\endcsname{{[lemma][2][]17.2}{[1][209][]209}}
\expandafter\gdef\csname r@ch14:eq:center-units\endcsname{{17.9}{210}{Step 1: the expansion and its units}{equation.622}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:center-units@cref\endcsname{{[equation][9][17]17.9}{[1][210][]210}}
\expandafter\gdef\csname r@ch14:eq:column-shell\endcsname{{17.10}{211}{Step 3: localization and the first moment}{equation.625}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:column-shell@cref\endcsname{{[equation][10][17]17.10}{[1][211][]211}}
\expandafter\gdef\csname r@ch14:eq:cube-sup\endcsname{{17.11}{211}{Step 4: heat-kernel cancellation in the continuum}{equation.627}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:cube-sup@cref\endcsname{{[equation][11][17]17.11}{[1][211][]211}}
\expandafter\gdef\csname r@ch14:eq:refinement-ratios\endcsname{{17.12}{212}{Step 5: the finer lattice}{equation.629}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:refinement-ratios@cref\endcsname{{[equation][12][17]17.12}{[1][212][]212}}
\expandafter\gdef\csname r@ch14:eq:cardinal-signed-mass\endcsname{{17.13}{212}{Step 5: the finer lattice}{equation.630}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:cardinal-signed-mass@cref\endcsname{{[equation][13][17]17.13}{[1][212][]212}}
\expandafter\gdef\csname r@ch14:eq:discrete-defect-value\endcsname{{17.14}{213}{Step 5: the finer lattice}{equation.631}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:discrete-defect-value@cref\endcsname{{[equation][14][17]17.14}{[1][213][]213}}
\expandafter\gdef\csname r@ch14:sec:quadratic\endcsname{{17.4}{213}{Conditional quadratic moments}{section.632}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:sec:quadratic@cref\endcsname{{[section][4][17]17.4}{[1][213][]213}}
\expandafter\gdef\csname r@ch14:eq:weighted-quadratic-norm\endcsname{{17.15}{214}{Quadratic maps and their norms}{equation.634}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:weighted-quadratic-norm@cref\endcsname{{[equation][15][17]17.15}{[1][214][]214}}
\expandafter\gdef\csname r@ch14:eq:quadratic-norm\endcsname{{17.16}{214}{Quadratic maps and their norms}{equation.635}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:quadratic-norm@cref\endcsname{{[equation][16][17]17.16}{[1][214][]214}}
\expandafter\gdef\csname r@ch14:lem:quadratic-pullback\endcsname{{17.3}{215}{}{lemma.637}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:lem:quadratic-pullback@cref\endcsname{{[lemma][3][]17.3}{[1][215][]215}}
\expandafter\gdef\csname r@ch14:eq:tangent-rows-columns\endcsname{{17.17}{215}{}{equation.638}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:tangent-rows-columns@cref\endcsname{{[equation][17][17]17.17}{[1][215][]215}}
\expandafter\gdef\csname r@ch14:thm:quadratic-ancestor-return\endcsname{{17.4}{215}{}{theorem.639}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:thm:quadratic-ancestor-return@cref\endcsname{{[theorem][4][17]17.4}{[1][215][]215}}
\expandafter\gdef\csname r@ch14:eq:white-data\endcsname{{17.18}{218}{Step 1: the conditional expectation of a quadratic term}{equation.646}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:white-data@cref\endcsname{{[equation][18][17]17.18}{[1][217][]218}}
\expandafter\gdef\csname r@ch14:eq:score-identity\endcsname{{17.19}{218}{Step 1: the conditional expectation of a quadratic term}{equation.647}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:score-identity@cref\endcsname{{[equation][19][17]17.19}{[1][218][]218}}
\expandafter\gdef\csname r@ch14:eq:true-mean-expansion\endcsname{{17.20}{218}{Step 1: the conditional expectation of a quadratic term}{equation.648}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:true-mean-expansion@cref\endcsname{{[equation][20][17]17.20}{[1][218][]218}}
\expandafter\gdef\csname r@ch14:eq:coefficient-response\endcsname{{17.21}{218}{Step 1: the conditional expectation of a quadratic term}{equation.649}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch14:eq:coefficient-response@cref\endcsname{{[equation][21][17]17.21}{[1][218][]218}}
\expandafter\gdef\csname r@ch14:eq:covariance-response\endcsname{{17.22}{219}{Step 1: the conditional expectation of a quadratic term}{equation.650}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:root-fraction@cref\endcsname{{[equation][2][18]18.2}{[1][232][]232}}
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\expandafter\gdef\csname r@ch15:eq:coefficient-resolvent@cref\endcsname{{[equation][5][18]18.5}{[1][234][]234}}
\expandafter\gdef\csname r@ch15:eq:schur-lower\endcsname{{18.6}{235}{Coefficient norms with boundary decay}{equation.688}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:schur-lower@cref\endcsname{{[equation][6][18]18.6}{[1][235][]235}}
\expandafter\gdef\csname r@ch15:sec:coordinates\endcsname{{18.3}{236}{Local coordinates and Haar disintegration on block regions}{section.689}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:sec:coordinates@cref\endcsname{{[section][3][18]18.3}{[1][235][]236}}
\expandafter\gdef\csname r@ch15:eq:balanced-nesting\endcsname{{18.7}{236}{Local coordinates and Haar disintegration on block regions}{equation.690}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:lem:local-endpoint-dependence\endcsname{{18.4}{236}{}{lemma.691}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:lem:local-endpoint-dependence@cref\endcsname{{[lemma][4][]18.4}{[1][236][]236}}
\expandafter\gdef\csname r@ch15:rem:cones-comparison\endcsname{{18.5}{237}{}{remark.692}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:rem:cones-comparison@cref\endcsname{{[remark][5][]18.5}{[1][237][]237}}
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\expandafter\gdef\csname r@ch15:def:internal-coordinates@cref\endcsname{{[definition][6][]18.6}{[1][237][]237}}
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\expandafter\gdef\csname r@ch15:eq:internal-lift\endcsname{{18.9}{238}{}{equation.700}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:internal-lift@cref\endcsname{{[equation][9][18]18.9}{[1][238][]238}}
\expandafter\gdef\csname r@ch15:eq:internal-reconstruction\endcsname{{18.10}{238}{}{equation.701}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:internal-reconstruction@cref\endcsname{{[equation][10][18]18.10}{[1][238][]238}}
\expandafter\gdef\csname r@ch15:eq:anchoring\endcsname{{18.11}{239}{}{equation.703}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:anchoring@cref\endcsname{{[equation][11][18]18.11}{[1][239][]239}}
\expandafter\gdef\csname r@ch15:lem:regional-coarea\endcsname{{18.7}{239}{}{lemma.704}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:lem:regional-coarea@cref\endcsname{{[lemma][7][]18.7}{[1][239][]239}}
\expandafter\gdef\csname r@ch15:sec:minimum-calculus\endcsname{{18.4}{243}{Derivatives of constrained minima}{section.705}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:lem:minimum-calculus\endcsname{{18.8}{243}{}{lemma.706}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:lem:minimum-calculus@cref\endcsname{{[lemma][8][]18.8}{[1][243][]243}}
\expandafter\gdef\csname r@ch15:eq:horizontal-orthogonality\endcsname{{18.12}{244}{Derivatives of constrained minima}{equation.707}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:sec:freezing@cref\endcsname{{[section][5][18]18.5}{[1][245][]245}}
\expandafter\gdef\csname r@ch15:lem:freezing\endcsname{{18.9}{245}{}{lemma.709}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:lem:freezing@cref\endcsname{{[lemma][9][]18.9}{[1][245][]245}}
\expandafter\gdef\csname r@ch15:eq:uniform-surviving-radius\endcsname{{18.13}{248}{Freezing variables}{equation.714}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:uniform-surviving-radius@cref\endcsname{{[equation][13][18]18.13}{[1][248][]248}}
\expandafter\gdef\csname r@ch15:sec:gap\endcsname{{18.6}{250}{Coercivity and analytic constrained minimizers}{section.715}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:patch-constraints@cref\endcsname{{[equation][14][18]18.14}{[1][250][]250}}
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\expandafter\gdef\csname r@ch15:lem:patch-coercivity@cref\endcsname{{[lemma][11][]18.11}{[1][250][]250}}
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\expandafter\gdef\csname r@ch15:eq:extraction\endcsname{{18.16}{250}{}{equation.720}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:extraction@cref\endcsname{{[equation][16][18]18.16}{[1][250][]250}}
\expandafter\gdef\csname r@ch15:eq:patch-inequality\endcsname{{18.17}{251}{}{equation.725}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:patch-inequality@cref\endcsname{{[equation][17][18]18.17}{[1][251][]251}}
\expandafter\gdef\csname r@ch15:eq:gradient-blocking\endcsname{{18.18}{251}{Coercivity and analytic constrained minimizers}{equation.726}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:gradient-blocking@cref\endcsname{{[equation][18][18]18.18}{[1][251][]251}}
\expandafter\gdef\csname r@ch15:eq:lift-identities\endcsname{{18.19}{251}{Coercivity and analytic constrained minimizers}{equation.727}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:lift-identities@cref\endcsname{{[equation][19][18]18.19}{[1][251][]251}}
\expandafter\gdef\csname r@ch15:thm:regional-gap-and-center\endcsname{{18.12}{254}{}{theorem.728}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:thm:regional-gap-and-center@cref\endcsname{{[theorem][12][18]18.12}{[1][254][]254}}
\expandafter\gdef\csname r@ch15:eq:flat-floor\endcsname{{18.20}{254}{}{equation.729}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:flat-floor@cref\endcsname{{[equation][20][18]18.20}{[1][254][]254}}
\expandafter\gdef\csname r@ch15:eq:raw-majorant\endcsname{{18.21}{256}{Step 1: local charts and comparison of embeddings}{equation.736}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:embedding-definitions\endcsname{{18.22}{257}{Step 1: local charts and comparison of embeddings}{equation.737}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:regional-flat-gap\endcsname{{18.24}{258}{Step 3: the lower bound on the Hessian at the identity}{equation.741}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:objective-tensors\endcsname{{18.25}{259}{Step 4: analytic products and the inverse of the free Hessian}{equation.743}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:nonflat-hessian\endcsname{{18.26}{259}{Step 4: analytic products and the inverse of the free Hessian}{equation.744}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:hessian-lipschitz\endcsname{{18.27}{259}{Step 4: analytic products and the inverse of the free Hessian}{equation.745}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:force-bound\endcsname{{18.28}{259}{Step 4: analytic products and the inverse of the free Hessian}{equation.746}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:radius-choices\endcsname{{18.29}{259}{Step 4: analytic products and the inverse of the free Hessian}{equation.747}{3d-ym-torus-volume-ii.pdf}}
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\expandafter\gdef\csname r@ch15:eq:center-majorant\endcsname{{18.30}{260}{Step 5: the constrained minimizer and its coefficients}{equation.749}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:center-majorant@cref\endcsname{{[equation][30][18]18.30}{[1][260][]260}}
\expandafter\gdef\csname r@ch15:eq:prefix-margins\endcsname{{18.31}{261}{Step 6: domains for the conditional minimizers}{equation.751}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:prefix-margins@cref\endcsname{{[equation][31][18]18.31}{[1][261][]261}}
\expandafter\gdef\csname r@ch15:sec:bare\endcsname{{18.7}{262}{The first block integration}{section.752}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:sec:bare@cref\endcsname{{[section][7][18]18.7}{[1][262][]262}}
\expandafter\gdef\csname r@ch15:def:bare-completion\endcsname{{18.13}{262}{}{definition.753}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:def:bare-completion@cref\endcsname{{[definition][13][]18.13}{[1][262][]262}}
\expandafter\gdef\csname r@ch15:eq:near-deep\endcsname{{18.32}{263}{}{equation.756}{3d-ym-torus-volume-ii.pdf}}
\expandafter\gdef\csname r@ch15:eq:near-deep@cref\endcsname{{[equation][32][18]18.32}{[1][262][]263}}
\expandafter\gdef\csname r@ch15:eq:crossing-completion\endcsname{{18.33}{263}{}{equation.759}{3d-ym-torus-volume-ii.pdf}}
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\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}}
\expandafter\gdef\csname r@ch16:sec:perturbation\endcsname{{19.7}{36}{Perturbations of the incoming action}{section.121}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:sec:perturbation@cref\endcsname{{[section][7][19]19.7}{[1][36][]36}}
\expandafter\gdef\csname r@ch16:lem:input-wall-distance\endcsname{{19.8}{36}{}{lemma.122}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:lem:input-wall-distance@cref\endcsname{{[lemma][8][]19.8}{[1][36][]36}}
\expandafter\gdef\csname r@ch16:eq:input-wall-comparison\endcsname{{19.35}{36}{}{equation.123}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:input-wall-comparison@cref\endcsname{{[equation][35][19]19.35}{[1][36][]36}}
\expandafter\gdef\csname r@ch16:eq:uniform-input-after-freeze\endcsname{{19.36}{37}{Perturbations of the incoming action}{equation.124}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:uniform-input-after-freeze@cref\endcsname{{[equation][36][19]19.36}{[1][37][]37}}
\expandafter\gdef\csname r@ch16:thm:controlled-perturbation\endcsname{{19.9}{37}{}{theorem.125}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:thm:controlled-perturbation@cref\endcsname{{[theorem][9][19]19.9}{[1][37][]37}}
\expandafter\gdef\csname r@ch16:eq:unit-joint-hessian\endcsname{{19.37}{41}{Step 3: the normalized objective and the perturbing gradient}{equation.136}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:unit-joint-hessian@cref\endcsname{{[equation][37][19]19.37}{[1][41][]41}}
\expandafter\gdef\csname r@ch16:eq:perturbing-force\endcsname{{19.38}{42}{Step 3: the normalized objective and the perturbing gradient}{equation.137}{3d-ym-torus-volume-iii.pdf}}
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\expandafter\gdef\csname r@ch16:eq:perturbing-hessian@cref\endcsname{{[equation][39][19]19.39}{[1][41][]42}}
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\expandafter\gdef\csname r@ch16:eq:perturbing-bounds@cref\endcsname{{[equation][40][19]19.40}{[1][42][]42}}
\expandafter\gdef\csname r@ch16:eq:composed-jet-majorant\endcsname{{19.41}{42}{Step 3: the normalized objective and the perturbing gradient}{equation.140}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:composed-jet-majorant@cref\endcsname{{[equation][41][19]19.41}{[1][42][]42}}
\expandafter\gdef\csname r@ch16:eq:network-transport\endcsname{{19.42}{43}{Step 4: transport of the incoming boundary dependence}{equation.142}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:network-transport@cref\endcsname{{[equation][42][19]19.42}{[1][43][]43}}
\expandafter\gdef\csname r@ch16:eq:joint-perturbed-displacement\endcsname{{19.43}{44}{Step 5: the joint stationary point}{equation.144}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:joint-perturbed-displacement@cref\endcsname{{[equation][43][19]19.43}{[1][44][]44}}
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\expandafter\gdef\csname r@ch16:eq:controlled-conditional-force@cref\endcsname{{[equation][45][19]19.45}{[1][44][]45}}
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\expandafter\gdef\csname r@ch16:eq:prefix-perturbation-gates@cref\endcsname{{[equation][46][19]19.46}{[1][45][]45}}
\expandafter\gdef\csname r@ch16:eq:conditional-perturbed-displacement\endcsname{{19.47}{45}{Step 6: initial segments in the original coordinates}{equation.149}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch16:eq:conditional-perturbed-displacement@cref\endcsname{{[equation][47][19]19.47}{[1][45][]45}}
\expandafter\gdef\csname r@ch16:eq:perturbed-center-response\endcsname{{19.48}{47}{Step 9: the differences carry the perturbation}{equation.153}{3d-ym-torus-volume-iii.pdf}}
\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}}
\expandafter\gdef\csname r@ch17:eq:wall-transport\endcsname{{20.3}{56}{Density sums and boundary transport}{equation.174}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:wall-transport@cref\endcsname{{[equation][3][20]20.3}{[1][56][]56}}
\expandafter\gdef\csname r@ch17:eq:boundary-sum\endcsname{{20.4}{56}{Density sums and boundary transport}{equation.175}{3d-ym-torus-volume-iii.pdf}}
\expandafter\gdef\csname r@ch17:eq:boundary-sum@cref\endcsname{{[equation][4][20]20.4}{[1][56][]56}}
\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}}
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\expandafter\gdef\csname r@ch19:lem:covered-history-admission@cref\endcsname{{[lemma][3][]22.3}{[1][184][]184}}
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\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 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}}
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\expandafter\gdef\csname r@ch29:eq:macroscopic-prefix-approximation@cref\endcsname{{[equation][29][32]32.29}{[1][61][]61}}
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\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:eq:short-distance-macroscopic-mixing@cref\endcsname{{[equation][33][32]32.33}{[1][62][]62}}
\expandafter\gdef\csname r@ch29:sec:orbit-borel-mixing-addendum\endcsname{{32.9}{63}{Asymptotic independence from bounded Borel functions}{section.186}{3d-ym-torus-volume-v.pdf}}
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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:sec:stopped@cref\endcsname{{[section][2][33]33.2}{[1][71][]71}}
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\expandafter\gdef\csname r@ch27:eq:common-coupling-gates@cref\endcsname{{[equation][2][33]33.2}{[1][72][]72}}
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\expandafter\gdef\csname r@ch27:sec:cone@cref\endcsname{{[section][3][33]33.3}{[1][72][]72}}
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\expandafter\gdef\csname r@ch27:eq:global-laplace@cref\endcsname{{[equation][3][33]33.3}{[1][76][]76}}
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\expandafter\gdef\csname r@ch27:cor:weighted-tube-density@cref\endcsname{{[corollary][4][]33.4}{[1][77][]77}}
\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}}
\expandafter\gdef\csname r@ch27:eq:tube-tail\endcsname{{33.5}{77}{}{equation.222}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:tube-tail@cref\endcsname{{[equation][5][33]33.5}{[1][77][]77}}
\expandafter\gdef\csname r@ch27:lem:pointwise-laplace\endcsname{{33.5}{77}{}{lemma.223}{3d-ym-torus-volume-v.pdf}}
\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}}
\expandafter\gdef\csname r@ch27:eq:pointwise-laplace\endcsname{{33.7}{79}{Laplace asymptotics in normal coordinates}{equation.229}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:pointwise-laplace@cref\endcsname{{[equation][7][33]33.7}{[1][79][]79}}
\expandafter\gdef\csname r@ch27:sec:central\endcsname{{33.5}{81}{Based coordinates near the trivial connection}{section.230}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:sec:central@cref\endcsname{{[section][5][33]33.5}{[1][81][]81}}
\expandafter\gdef\csname r@ch27:lem:residual-derivatives\endcsname{{33.6}{81}{}{lemma.231}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:lem:residual-derivatives@cref\endcsname{{[lemma][6][]33.6}{[1][81][]81}}
\expandafter\gdef\csname r@ch27:eq:residual-interior-derivatives\endcsname{{33.8}{81}{}{equation.232}{3d-ym-torus-volume-v.pdf}}
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\expandafter\gdef\csname r@ch27:eq:residual-central-derivatives\endcsname{{33.9}{82}{}{equation.233}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:residual-central-derivatives@cref\endcsname{{[equation][9][33]33.9}{[1][82][]82}}
\expandafter\gdef\csname r@ch27:eq:residual-chart-transfer\endcsname{{33.10}{82}{}{equation.234}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:residual-chart-transfer@cref\endcsname{{[equation][10][33]33.10}{[1][82][]82}}
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\expandafter\gdef\csname r@ch27:lem:local-flat-admission@cref\endcsname{{[lemma][7][]33.7}{[1][84][]84}}
\expandafter\gdef\csname r@ch27:eq:local-flat-margin\endcsname{{33.11}{84}{Based coordinates near the trivial connection}{equation.236}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:local-flat-margin@cref\endcsname{{[equation][11][33]33.11}{[1][84][]84}}
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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}}
\expandafter\gdef\csname r@ch27:sec:normalizer\endcsname{{33.6}{90}{Flat configurations and the partition function}{section.247}{3d-ym-torus-volume-v.pdf}}
\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}}
\expandafter\gdef\csname r@ch27:eq:central-bare-bound\endcsname{{33.14}{95}{Step 2: the neighborhood of the identity}{equation.258}{3d-ym-torus-volume-v.pdf}}
\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}}
\expandafter\gdef\csname r@ch27:sec:covariance\endcsname{{33.7}{97}{Linearized flowed cycle logarithms}{section.263}{3d-ym-torus-volume-v.pdf}}
\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}}
\expandafter\gdef\csname r@ch27:eq:charged-quadratic\endcsname{{33.16}{99}{Step 2: the Gaussian field}{equation.271}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:charged-quadratic@cref\endcsname{{[equation][16][33]33.16}{[1][99][]99}}
\expandafter\gdef\csname r@ch27:eq:charged-covariance\endcsname{{33.17}{100}{Step 2: the Gaussian field}{equation.272}{3d-ym-torus-volume-v.pdf}}
\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}}
\expandafter\gdef\csname r@ch27:eq:lattice-maxwell-exponent@cref\endcsname{{[equation][18][33]33.18}{[1][102][]102}}
\expandafter\gdef\csname r@ch27:eq:lattice-sine-bounds\endcsname{{33.19}{103}{Step 9: comparison with the continuum}{equation.281}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:lattice-sine-bounds@cref\endcsname{{[equation][19][33]33.19}{[1][103][]103}}
\expandafter\gdef\csname r@ch27:sec:terminal\endcsname{{33.8}{104}{The stopped measure near regular flat configurations}{section.283}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:sec:terminal@cref\endcsname{{[section][8][33]33.8}{[1][104][]104}}
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\expandafter\gdef\csname r@ch27:lem:terminal-gaussian-telescope@cref\endcsname{{[proposition][12][]33.12}{[1][104][]104}}
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\expandafter\gdef\csname r@ch27:eq:whole-flat-margin@cref\endcsname{{[equation][20][33]33.20}{[1][106][]106}}
\expandafter\gdef\csname r@ch27:eq:gaussian-dimension-count\endcsname{{33.21}{109}{Step 5: the count of variables}{equation.293}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:gaussian-dimension-count@cref\endcsname{{[equation][21][33]33.21}{[1][109][]109}}
\expandafter\gdef\csname r@ch27:eq:quadratic-telescope\endcsname{{33.22}{110}{Step 7: completion of squares}{equation.296}{3d-ym-torus-volume-v.pdf}}
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\expandafter\gdef\csname r@ch27:eq:quadratic-isometry\endcsname{{33.23}{110}{Step 8: Gaussian linear algebra}{equation.298}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:eq:quadratic-isometry@cref\endcsname{{[equation][23][33]33.23}{[1][110][]110}}
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\expandafter\gdef\csname r@ch27:rem:scope@cref\endcsname{{[remark][13][]33.13}{[1][111][]112}}
\expandafter\gdef\csname r@ch27:sec:abelian\endcsname{{33.9}{112}{An Abelian comparison measure}{section.302}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch27:sec:abelian@cref\endcsname{{[section][9][33]33.9}{[1][112][]112}}
\expandafter\gdef\csname r@ch27:lem:abelian-reference\endcsname{{33.14}{112}{}{lemma.303}{3d-ym-torus-volume-v.pdf}}
\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}}
\expandafter\gdef\csname r@ch30:chapter@cref\endcsname{{[chapter][34][]34}{[1][114][]114}}
\expandafter\gdef\csname r@ch30:tab:settings\endcsname{{34.1}{115}{The three probability settings of this section}{table.caption.307}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch30:tab:settings@cref\endcsname{{[table][1][34]34.1}{[1][114][]115}}
\expandafter\gdef\csname r@ch30:sec:setting\endcsname{{34.1}{116}{Setting and notation}{section.308}{3d-ym-torus-volume-v.pdf}}
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\expandafter\gdef\csname r@ch30:def:paired-history@cref\endcsname{{[definition][1][]34.1}{[1][116][]116}}
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\expandafter\gdef\csname r@ch30:eq:local-mean@cref\endcsname{{[equation][13][34]34.13}{[1][123][]123}}
\expandafter\gdef\csname r@ch30:eq:input-radius\endcsname{{34.14}{123}{}{equation.341}{3d-ym-torus-volume-v.pdf}}
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\expandafter\gdef\csname r@ch30:eq:sum1\endcsname{{34.15}{123}{}{equation.343}{3d-ym-torus-volume-v.pdf}}
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\expandafter\gdef\csname r@ch30:eq:sum2\endcsname{{34.16}{123}{}{equation.344}{3d-ym-torus-volume-v.pdf}}
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\expandafter\gdef\csname r@ch30:eq:sum-fixed@cref\endcsname{{[equation][2147483647][]{\ref {ch30:eq:sum2}a}}{[1][123][]123}}
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\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}}
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\expandafter\gdef\csname r@ch30:eq:support-cap@cref\endcsname{{[equation][20][34]34.20}{[1][129][]129}}
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\expandafter\gdef\csname r@ch30:eq:ancestor-normalization@cref\endcsname{{[equation][21][34]34.21}{[1][129][]129}}
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\expandafter\gdef\csname r@ch30:lem:zero-white-and-chaining@cref\endcsname{{[lemma][4][]34.4}{[1][130][]130}}
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\expandafter\gdef\csname r@ch30:eq:classical-adjacent@cref\endcsname{{[equation][22][34]34.22}{[1][130][]131}}
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\expandafter\gdef\csname r@ch30:eq:det-ancestor@cref\endcsname{{[equation][23][34]34.23}{[1][131][]131}}
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\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}}
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\expandafter\gdef\csname r@ch32:eq:physical-action@cref\endcsname{{[equation][4][36]36.4}{[1][225][]225}}
\expandafter\gdef\csname r@ch32:eq:attained-value\endcsname{{36.5}{225}{Step 1: the variational comparison}{equation.604}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:attained-value@cref\endcsname{{[equation][5][36]36.5}{[1][225][]225}}
\expandafter\gdef\csname r@ch32:eq:minimum-domain\endcsname{{36.6}{225}{Step 1: the variational comparison}{equation.605}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:minimum-domain@cref\endcsname{{[equation][6][36]36.6}{[1][225][]225}}
\expandafter\gdef\csname r@ch32:eq:trial-domain\endcsname{{36.7}{226}{Step 1: the variational comparison}{equation.606}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:trial-domain@cref\endcsname{{[equation][7][36]36.7}{[1][226][]226}}
\expandafter\gdef\csname r@ch32:eq:trial-comparison\endcsname{{36.8}{226}{Step 1: the variational comparison}{equation.607}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:trial-comparison@cref\endcsname{{[equation][8][36]36.8}{[1][226][]226}}
\expandafter\gdef\csname r@ch32:eq:exact-trial\endcsname{{36.9}{227}{Step 3: an exactly constrained trial and the upper bound}{equation.610}{3d-ym-torus-volume-v.pdf}}
\expandafter\gdef\csname r@ch32:eq:exact-trial@cref\endcsname{{[equation][9][36]36.9}{[1][227][]227}}
\expandafter\gdef\csname r@ch32:eq:quartic-upper\endcsname{{36.10}{227}{Step 3: an exactly constrained trial and the upper bound}{equation.611}{3d-ym-torus-volume-v.pdf}}
\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