% 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 IV
\expandafter\gdef\csname r@part:flow\endcsname{{VII}{2}{\contentsname \@mkboth {\MakeUppercase \contentsname }{\MakeUppercase \contentsname }}{part.2}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@part:flow@cref\endcsname{{[part][7][]VII}{[1][2][]2}}
\expandafter\gdef\csname r@ch21:chapter\endcsname{{24}{2}{Local Wilson flow and continuum solutions}{chapter.3}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:chapter@cref\endcsname{{[chapter][24][]24}{[1][2][]2}}
\expandafter\gdef\csname r@ch21:eq:intro-equation\endcsname{{24.1}{2}{Local Wilson flow and continuum solutions}{equation.4}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:intro-equation@cref\endcsname{{[equation][1][24]24.1}{[1][2][]2}}
\expandafter\gdef\csname r@ch21:eq:intro-decomposition\endcsname{{24.2}{2}{Local Wilson flow and continuum solutions}{equation.5}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:intro-decomposition@cref\endcsname{{[equation][2][24]24.2}{[1][2][]2}}
\expandafter\gdef\csname r@ch21:sec:equation\endcsname{{24.1}{3}{The local equation for the Wilson flow}{section.6}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:sec:equation@cref\endcsname{{[section][1][24]24.1}{[1][3][]3}}
\expandafter\gdef\csname r@ch21:eq:c1a-norm\endcsname{{24.3}{3}{Notation}{equation.8}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:c1a-norm@cref\endcsname{{[equation][3][24]24.3}{[1][3][]3}}
\expandafter\gdef\csname r@ch21:lem:stencil-analytic\endcsname{{24.1}{4}{}{lemma.10}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:lem:stencil-analytic@cref\endcsname{{[lemma][1][]24.1}{[1][4][]4}}
\expandafter\gdef\csname r@ch21:lem:wilson-local-algebra\endcsname{{24.2}{6}{}{lemma.17}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:lem:wilson-local-algebra@cref\endcsname{{[lemma][2][]24.2}{[1][6][]6}}
\expandafter\gdef\csname r@ch21:eq:Na-form\endcsname{{24.4}{7}{}{equation.22}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:Na-form@cref\endcsname{{[equation][4][24]24.4}{[1][7][]7}}
\expandafter\gdef\csname r@ch21:eq:curl-adjoint\endcsname{{24.5}{8}{Step 1: Hodge identities}{equation.26}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:curl-adjoint@cref\endcsname{{[equation][5][24]24.5}{[1][8][]8}}
\expandafter\gdef\csname r@ch21:eq:wilson-coordinate-force\endcsname{{24.6}{9}{Step 3: the group gradient in logarithmic coordinates}{equation.29}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:wilson-coordinate-force@cref\endcsname{{[equation][6][24]24.6}{[1][9][]9}}
\expandafter\gdef\csname r@ch21:eq:physical-cubic-series\endcsname{{24.7}{11}{Step 6: the remainder}{equation.33}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:physical-cubic-series@cref\endcsname{{[equation][7][24]24.7}{[1][11][]11}}
\expandafter\gdef\csname r@ch21:eq:lattice-damping-generator\endcsname{{24.8}{12}{The equation}{equation.35}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:lattice-damping-generator@cref\endcsname{{[equation][8][24]24.8}{[1][12][]12}}
\expandafter\gdef\csname r@ch21:sec:data\endcsname{{24.2}{12}{Linear and quadratic flow data}{section.36}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:sec:data@cref\endcsname{{[section][2][24]24.2}{[1][12][]12}}
\expandafter\gdef\csname r@ch21:eq:flow-norms\endcsname{{24.9}{12}{Linear and quadratic flow data}{equation.37}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:flow-norms@cref\endcsname{{[equation][9][24]24.9}{[1][12][]12}}
\expandafter\gdef\csname r@ch21:def:two-grade-datum\endcsname{{24.3}{12}{}{definition.38}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:def:two-grade-datum@cref\endcsname{{[definition][3][]24.3}{[1][12][]12}}
\expandafter\gdef\csname r@ch21:eq:full-equation\endcsname{{24.10}{13}{}{equation.39}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:full-equation@cref\endcsname{{[equation][10][24]24.10}{[1][12][]13}}
\expandafter\gdef\csname r@ch21:eq:comparison-metric\endcsname{{24.11}{13}{}{equation.40}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:eq:comparison-metric@cref\endcsname{{[equation][11][24]24.11}{[1][13][]13}}
\expandafter\gdef\csname r@ch21:sec:local\endcsname{{24.3}{14}{Local existence for the Wilson flow}{section.41}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:sec:local@cref\endcsname{{[section][3][24]24.3}{[1][14][]14}}
\expandafter\gdef\csname r@ch21:lem:source-convolution\endcsname{{24.4}{14}{}{lemma.42}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch21:lem:source-convolution@cref\endcsname{{[lemma][4][]24.4}{[1][14][]14}}
\expandafter\gdef\csname r@ch21:eq:source-convolution\endcsname{{24.12}{14}{}{equation.43}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch22:thm:root-matrix-modulus@cref\endcsname{{[theorem][10][25]25.10}{[1][65][]65}}
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\expandafter\gdef\csname r@ch22:cor:bounded-input-trace-stability@cref\endcsname{{[corollary][11][]25.11}{[1][67][]67}}
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\expandafter\gdef\csname r@ch22:eq:bounded-enhancement-trace-stability@cref\endcsname{{[equation][23][25]25.23}{[1][67][]67}}
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\expandafter\gdef\csname r@ch22:thm:real-flow-variations@cref\endcsname{{[theorem][13][25]25.13}{[1][70][]70}}
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\expandafter\gdef\csname r@ch22:eq:integrable-variation-profile@cref\endcsname{{[equation][33][25]25.33}{[1][74][]74}}
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\expandafter\gdef\csname r@ch22:eq:early-generator-modulus@cref\endcsname{{[equation][34][25]25.34}{[1][74][]74}}
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\expandafter\gdef\csname r@ch22:eq:linear-divergence-generator@cref\endcsname{{[equation][35][25]25.35}{[1][75][]75}}
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\expandafter\gdef\csname r@ch22:eq:gauge-two-variations@cref\endcsname{{[equation][36][25]25.36}{[1][75][]75}}
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\expandafter\gdef\csname r@ch22:eq:endpoint-near-smooth-compact@cref\endcsname{{[equation][37][25]25.37}{[1][75][]75}}
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\expandafter\gdef\csname r@ch22:eq:accumulated-root-gauge@cref\endcsname{{[equation][38][25]25.38}{[1][76][]76}}
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\expandafter\gdef\csname r@ch22:eq:matrix-product-variations@cref\endcsname{{[equation][39][25]25.39}{[1][76][]77}}
\expandafter\gdef\csname r@ch22:eq:matrix-in-initial-frame\endcsname{{25.40}{77}{Parameter derivatives in a fixed initial gauge}{equation.190}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch22:eq:matrix-in-initial-frame@cref\endcsname{{[equation][40][25]25.40}{[1][77][]77}}
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\expandafter\gdef\csname r@ch22:eq:twice-differentiated-residual@cref\endcsname{{[equation][41][25]25.41}{[1][78][]78}}
\expandafter\gdef\csname r@ch22:eq:C2-smooth-family-convergence\endcsname{{25.42}{79}{Parameter derivatives in a fixed initial gauge}{equation.192}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch22:eq:C2-smooth-family-convergence@cref\endcsname{{[equation][42][25]25.42}{[1][79][]79}}
\expandafter\gdef\csname r@ch22:eq:L1-generator-convergence\endcsname{{25.43}{79}{Parameter derivatives in a fixed initial gauge}{equation.193}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch22:eq:L1-generator-convergence@cref\endcsname{{[equation][43][25]25.43}{[1][79][]79}}
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\expandafter\gdef\csname r@ch22:eq:root-ODE-consistency@cref\endcsname{{[equation][44][25]25.44}{[1][80][]80}}
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\expandafter\gdef\csname r@ch22:sec:modified@cref\endcsname{{[section][8][25]25.8}{[1][81][]81}}
\expandafter\gdef\csname r@ch22:lem:alternative-smooth-reference\endcsname{{25.14}{81}{}{lemma.196}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch22:lem:alternative-smooth-reference@cref\endcsname{{[lemma][14][]25.14}{[1][81][]81}}
\expandafter\gdef\csname r@ch22:eq:alternative-reference-bound\endcsname{{25.45}{82}{}{equation.199}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch22:eq:alternative-reference-bound@cref\endcsname{{[equation][45][25]25.45}{[1][82][]82}}
\expandafter\gdef\csname r@ch22:lem:alternative-uniform-handoff\endcsname{{25.15}{83}{}{lemma.200}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch22:lem:alternative-uniform-handoff@cref\endcsname{{[lemma][15][]25.15}{[1][83][]83}}
\expandafter\gdef\csname r@ch22:eq:alternative-uniform-loops\endcsname{{25.46}{83}{}{equation.201}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch22:eq:alternative-uniform-loops@cref\endcsname{{[equation][46][25]25.46}{[1][83][]83}}
\expandafter\gdef\csname r@part:stochastic\endcsname{{VIII}{86}{Flows with the modified plaquette potential}{part.202}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@part:stochastic@cref\endcsname{{[part][8][]VIII}{[1][86][]86}}
\expandafter\gdef\csname r@ch23:chapter\endcsname{{26}{86}{Concentration of reconstruction fluctuations}{chapter.203}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:chapter@cref\endcsname{{[chapter][26][]26}{[1][86][]86}}
\expandafter\gdef\csname r@ch23:sec:setup\endcsname{{26.1}{88}{Setting and the exact decomposition of an increment}{section.206}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:sec:setup@cref\endcsname{{[section][1][26]26.1}{[1][88][]88}}
\expandafter\gdef\csname r@ch23:lem:kernel-reserve\endcsname{{26.1}{91}{}{lemma.213}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:kernel-reserve@cref\endcsname{{[lemma][1][]26.1}{[1][90][]91}}
\expandafter\gdef\csname r@ch23:eq:implemented-gates\endcsname{{26.1}{91}{The conditional law and the Taylor coefficients}{equation.214}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:implemented-gates@cref\endcsname{{[equation][1][26]26.1}{[1][91][]91}}
\expandafter\gdef\csname r@ch23:tab:four-terms\endcsname{{26.1}{93}{The four terms of the increment $d_j=x_j-x_{j-1}$ in \cref {ch23:lem:four-piece-increment}}{table.caption.216}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:tab:four-terms@cref\endcsname{{[table][1][26]26.1}{[1][92][]93}}
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\expandafter\gdef\csname r@ch23:eq:taylor-at-zero@cref\endcsname{{[equation][7][26]26.7}{[1][94][]94}}
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\expandafter\gdef\csname r@ch23:eq:taylor-remainder@cref\endcsname{{[equation][9][26]26.9}{[1][95][]95}}
\expandafter\gdef\csname r@ch23:rem:base-point\endcsname{{26.3}{97}{}{remark.236}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:rem:base-point@cref\endcsname{{[remark][3][]26.3}{[1][96][]97}}
\expandafter\gdef\csname r@ch23:sec:deterministic\endcsname{{26.2}{97}{Deterministic estimates for heat kernels and coefficients}{section.237}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:sec:deterministic@cref\endcsname{{[section][2][26]26.2}{[1][97][]97}}
\expandafter\gdef\csname r@ch23:eq:convolution-profile\endcsname{{26.10}{97}{Standing conventions}{equation.239}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:convolution-profile@cref\endcsname{{[equation][10][26]26.10}{[1][97][]97}}
\expandafter\gdef\csname r@ch23:eq:background-radius\endcsname{{26.11}{97}{Standing conventions}{equation.240}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:background-radius@cref\endcsname{{[equation][11][26]26.11}{[1][97][]97}}
\expandafter\gdef\csname r@ch23:lem:physical-heat\endcsname{{26.4}{98}{}{lemma.242}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:physical-heat@cref\endcsname{{[lemma][4][]26.4}{[1][98][]98}}
\expandafter\gdef\csname r@ch23:lem:pair-schur\endcsname{{26.5}{99}{}{lemma.244}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:pair-schur@cref\endcsname{{[lemma][5][]26.5}{[1][99][]99}}
\expandafter\gdef\csname r@ch23:tab:placement\endcsname{{26.2}{100}{Placement of differences in the proof of \cref {ch23:lem:marked-calculus}}{table.caption.246}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:tab:placement@cref\endcsname{{[table][2][26]26.2}{[1][100][]100}}
\expandafter\gdef\csname r@ch23:lem:marked-calculus\endcsname{{26.6}{100}{}{lemma.247}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:marked-calculus@cref\endcsname{{[lemma][6][]26.6}{[1][100][]100}}
\expandafter\gdef\csname r@ch23:eq:background-differences\endcsname{{26.12}{101}{}{equation.248}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:background-differences@cref\endcsname{{[equation][12][26]26.12}{[1][101][]101}}
\expandafter\gdef\csname r@ch23:eq:profile-bounds\endcsname{{26.13}{102}{Step 5: the convolution integrals}{equation.254}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:profile-bounds@cref\endcsname{{[equation][13][26]26.13}{[1][102][]102}}
\expandafter\gdef\csname r@ch23:lem:coefficient-cover\endcsname{{26.7}{103}{}{lemma.256}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:coefficient-cover@cref\endcsname{{[lemma][7][]26.7}{[1][102][]103}}
\expandafter\gdef\csname r@ch23:eq:normalized-cover\endcsname{{26.14}{103}{}{equation.257}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:normalized-cover@cref\endcsname{{[equation][14][26]26.14}{[1][103][]103}}
\expandafter\gdef\csname r@ch23:eq:cover-source-derivatives\endcsname{{26.15}{103}{Covering numbers of coefficient rows}{equation.258}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:cover-source-derivatives@cref\endcsname{{[equation][15][26]26.15}{[1][103][]103}}
\expandafter\gdef\csname r@ch23:eq:normalized-modulus\endcsname{{26.16}{104}{Covering numbers of coefficient rows}{equation.259}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:eq:normalized-modulus@cref\endcsname{{[equation][16][26]26.16}{[1][104][]104}}
\expandafter\gdef\csname r@ch23:sec:maxima\endcsname{{26.3}{105}{Concentration and maximal inequalities}{section.260}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:sec:maxima@cref\endcsname{{[section][3][26]26.3}{[1][105][]105}}
\expandafter\gdef\csname r@ch23:lem:coefficient-chaining\endcsname{{26.8}{105}{}{lemma.261}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:coefficient-chaining@cref\endcsname{{[lemma][8][]26.8}{[1][105][]105}}
\expandafter\gdef\csname r@ch23:lem:input-maxima\endcsname{{26.9}{106}{}{lemma.262}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:lem:input-maxima@cref\endcsname{{[lemma][9][]26.9}{[1][106][]106}}
\expandafter\gdef\csname r@ch23:sec:linear\endcsname{{26.4}{109}{The linear fluctuation and the mixed Duhamel terms}{section.273}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:sec:linear@cref\endcsname{{[section][4][26]26.4}{[1][109][]109}}
\expandafter\gdef\csname r@ch23:thm:fresh-and-mixed-rows\endcsname{{26.10}{110}{}{theorem.274}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:thm:fresh-and-mixed-rows@cref\endcsname{{[theorem][10][26]26.10}{[1][109][]110}}
\expandafter\gdef\csname r@ch23:sec:residual\endcsname{{26.5}{114}{The conditionally centered nonlinear residual}{section.289}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:sec:residual@cref\endcsname{{[section][5][26]26.5}{[1][114][]114}}
\expandafter\gdef\csname r@ch23:thm:centered-residual\endcsname{{26.11}{114}{}{theorem.290}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:thm:centered-residual@cref\endcsname{{[theorem][11][26]26.11}{[1][114][]114}}
\expandafter\gdef\csname r@ch23:sec:square\endcsname{{26.6}{118}{The quadratic Duhamel term of the linear fluctuation}{section.300}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:sec:square@cref\endcsname{{[section][6][26]26.6}{[1][118][]118}}
\expandafter\gdef\csname r@ch23:thm:fresh-square\endcsname{{26.12}{118}{}{theorem.301}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:thm:fresh-square@cref\endcsname{{[theorem][12][26]26.12}{[1][118][]118}}
\expandafter\gdef\csname r@ch23:rem:centering-scope\endcsname{{26.13}{124}{}{remark.315}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch23:rem:centering-scope@cref\endcsname{{[remark][13][]26.13}{[1][124][]124}}
\expandafter\gdef\csname r@ch24:chapter\endcsname{{27}{125}{Conditional means and propagated fluctuations}{chapter.316}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:chapter@cref\endcsname{{[chapter][27][]27}{[1][125][]125}}
\expandafter\gdef\csname r@ch24:tab:constants\endcsname{{27.1}{127}{Dependence of constants in \cref {ch24:chapter}. No constant depends on $K$, on the scale indices $i$ and $j$, on the number of preceding averaging steps, on $a/\delta $, or on the values of the coarser fields}{table.caption.321}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:tab:constants@cref\endcsname{{[table][1][27]27.1}{[1][126][]127}}
\expandafter\gdef\csname r@ch24:sec:defect\endcsname{{27.1}{127}{Volume-normalized column sums and the interpolation defect}{section.322}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch24:lem:signed-heat-defect@cref\endcsname{{[lemma][1][]27.1}{[1][127][]127}}
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\expandafter\gdef\csname r@ch24:sec:drifts@cref\endcsname{{[section][2][27]27.2}{[1][132][]133}}
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\expandafter\gdef\csname r@ch24:lem:geometric-convolution@cref\endcsname{{[lemma][2][]27.2}{[1][133][]133}}
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\expandafter\gdef\csname r@ch24:eq:uniform-white-covariance@cref\endcsname{{[equation][8][27]27.8}{[1][134][]134}}
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\expandafter\gdef\csname r@ch24:eq:uniform-linear-coefficient@cref\endcsname{{[equation][9][27]27.9}{[1][134][]134}}
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\expandafter\gdef\csname r@ch24:lem:quadratic-mean@cref\endcsname{{[lemma][4][]27.4}{[1][136][]136}}
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\expandafter\gdef\csname r@ch24:eq:unweighted-quantum@cref\endcsname{{[equation][12][27]27.12}{[1][136][]136}}
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\expandafter\gdef\csname r@ch24:eq:unweighted-classical@cref\endcsname{{[equation][13][27]27.13}{[1][136][]136}}
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\expandafter\gdef\csname r@ch24:eq:nuclear@cref\endcsname{{[equation][14][27]27.14}{[1][136][]136}}
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\expandafter\gdef\csname r@ch24:eq:B-mean-bounded@cref\endcsname{{[equation][16][27]27.16}{[1][137][]137}}
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\expandafter\gdef\csname r@ch24:thm:ancestor-gram-rows@cref\endcsname{{[theorem][5][27]27.5}{[1][138][]138}}
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\expandafter\gdef\csname r@ch24:eq:source-coefficients@cref\endcsname{{[equation][17][27]27.17}{[1][140][]140}}
\expandafter\gdef\csname r@ch24:eq:value-source-white\endcsname{{27.18}{140}{Step 1: sources and late rows}{equation.371}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:value-source-white@cref\endcsname{{[equation][18][27]27.18}{[1][140][]140}}
\expandafter\gdef\csname r@ch24:eq:slot-gram\endcsname{{27.19}{140}{Step 1: sources and late rows}{equation.372}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:slot-gram@cref\endcsname{{[equation][19][27]27.19}{[1][140][]140}}
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\expandafter\gdef\csname r@ch24:eq:late-source-row@cref\endcsname{{[equation][20][27]27.20}{[1][140][]140}}
\expandafter\gdef\csname r@ch24:eq:J-formula\endcsname{{27.21}{141}{Step 2: the outer heat semigroup at small times}{equation.375}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:J-formula@cref\endcsname{{[equation][21][27]27.21}{[1][141][]141}}
\expandafter\gdef\csname r@ch24:eq:retained-mesh-log\endcsname{{27.22}{141}{Step 4: quadratic coefficients and weights}{equation.378}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:retained-mesh-log@cref\endcsname{{[equation][22][27]27.22}{[1][141][]141}}
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\expandafter\gdef\csname r@ch24:eq:crude-source@cref\endcsname{{[equation][23][27]27.23}{[1][142][]142}}
\expandafter\gdef\csname r@ch24:eq:weighted-mean-mesh\endcsname{{27.24}{142}{Step 6: the drifts}{equation.382}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:weighted-mean-mesh@cref\endcsname{{[equation][24][27]27.24}{[1][142][]142}}
\expandafter\gdef\csname r@ch24:eq:coarse-caps\endcsname{{27.25}{143}{Step 7: bounds on the coarser fields and summability}{equation.384}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:coarse-caps@cref\endcsname{{[equation][25][27]27.25}{[1][143][]143}}
\expandafter\gdef\csname r@ch24:eq:mean-majorant\endcsname{{27.26}{143}{Step 7: bounds on the coarser fields and summability}{equation.385}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:mean-majorant@cref\endcsname{{[equation][26][27]27.26}{[1][143][]143}}
\expandafter\gdef\csname r@ch24:sec:ordered\endcsname{{27.3}{143}{Maximal estimates and ultraviolet tails of the ordered terms}{section.386}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:sec:ordered@cref\endcsname{{[section][3][27]27.3}{[1][143][]143}}
\expandafter\gdef\csname r@ch24:thm:ordered-rough-return\endcsname{{27.6}{144}{}{theorem.387}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:thm:ordered-rough-return@cref\endcsname{{[theorem][6][27]27.6}{[1][143][]144}}
\expandafter\gdef\csname r@ch24:eq:sample-cover-bounds\endcsname{{27.27}{145}{Step 2: a deterministic net}{equation.393}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:sample-cover-bounds@cref\endcsname{{[equation][27][27]27.27}{[1][145][]145}}
\expandafter\gdef\csname r@ch24:eq:microscopic-cover-ratio\endcsname{{27.28}{145}{Step 2: a deterministic net}{equation.394}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:microscopic-cover-ratio@cref\endcsname{{[equation][28][27]27.28}{[1][145][]145}}
\expandafter\gdef\csname r@ch24:eq:net-cardinality\endcsname{{27.29}{146}{Step 2: a deterministic net}{equation.395}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:net-cardinality@cref\endcsname{{[equation][29][27]27.29}{[1][146][]146}}
\expandafter\gdef\csname r@ch24:eq:ordered-telescope\endcsname{{27.30}{147}{Step 4: the three orders and the telescope}{equation.398}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:ordered-telescope@cref\endcsname{{[equation][30][27]27.30}{[1][147][]147}}
\expandafter\gdef\csname r@ch24:sec:predictable\endcsname{{27.4}{148}{The predictable term on the finest lattice}{section.400}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:sec:predictable@cref\endcsname{{[section][4][27]27.4}{[1][148][]148}}
\expandafter\gdef\csname r@ch24:thm:full-smoother-return\endcsname{{27.7}{148}{}{theorem.401}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:thm:full-smoother-return@cref\endcsname{{[theorem][7][27]27.7}{[1][148][]148}}
\expandafter\gdef\csname r@ch24:eq:static-coefficient\endcsname{{27.31}{150}{Step 1: the propagated quadratic coefficient}{equation.408}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:static-coefficient@cref\endcsname{{[equation][31][27]27.31}{[1][150][]150}}
\expandafter\gdef\csname r@ch24:eq:static-micro\endcsname{{27.32}{151}{Step 3: the centered term}{equation.411}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:static-micro@cref\endcsname{{[equation][32][27]27.32}{[1][150][]151}}
\expandafter\gdef\csname r@ch24:eq:static-telescope\endcsname{{27.33}{151}{Step 4: the drifts and the coarsest term}{equation.413}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch24:eq:uniform-static-mean\endcsname{{27.34}{152}{Step 4: the drifts and the coarsest term}{equation.414}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:uniform-static-mean@cref\endcsname{{[equation][34][27]27.34}{[1][152][]152}}
\expandafter\gdef\csname r@ch24:eq:bounded-smoother\endcsname{{27.35}{152}{Step 5: the first difference, including the smallest times}{equation.416}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:bounded-smoother@cref\endcsname{{[equation][35][27]27.35}{[1][152][]152}}
\expandafter\gdef\csname r@ch24:eq:static-rate-convolution\endcsname{{27.36}{153}{Step 6: the rate of the quadratic part}{equation.418}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:static-rate-convolution@cref\endcsname{{[equation][36][27]27.36}{[1][152][]153}}
\expandafter\gdef\csname r@ch24:eq:quadratic-smoother-rate\endcsname{{27.37}{153}{Step 6: the rate of the quadratic part}{equation.419}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:quadratic-smoother-rate@cref\endcsname{{[equation][37][27]27.37}{[1][153][]153}}
\expandafter\gdef\csname r@ch24:eq:mean-score-covariance\endcsname{{27.38}{154}{Step 7: the cubic term and the conditional mean}{equation.421}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:mean-score-covariance@cref\endcsname{{[equation][38][27]27.38}{[1][154][]154}}
\expandafter\gdef\csname r@ch24:eq:mean-explicit-target-derivative\endcsname{{27.39}{154}{Step 7: the cubic term and the conditional mean}{equation.422}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:mean-explicit-target-derivative@cref\endcsname{{[equation][39][27]27.39}{[1][154][]154}}
\expandafter\gdef\csname r@ch24:eq:uniform-reconstructed-mean\endcsname{{27.40}{155}{Step 7: the cubic term and the conditional mean}{equation.423}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:uniform-reconstructed-mean@cref\endcsname{{[equation][40][27]27.40}{[1][155][]155}}
\expandafter\gdef\csname r@ch24:eq:physical-mean\endcsname{{27.41}{155}{Step 7: the cubic term and the conditional mean}{equation.424}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:eq:physical-mean@cref\endcsname{{[equation][41][27]27.41}{[1][155][]155}}
\expandafter\gdef\csname r@ch24:rem:product-application\endcsname{{27.8}{155}{}{remark.426}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch24:rem:product-application@cref\endcsname{{[remark][8][]27.8}{[1][155][]155}}
\expandafter\gdef\csname r@ch25:chapter\endcsname{{28}{156}{Linear and quadratic flow data and convergence of Wilson loops}{chapter.427}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:chapter@cref\endcsname{{[chapter][28][]28}{[1][156][]156}}
\expandafter\gdef\csname r@ch25:sec:setting\endcsname{{28.1}{158}{Setting and main results}{section.431}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:sec:setting@cref\endcsname{{[section][1][28]28.1}{[1][157][]158}}
\expandafter\gdef\csname r@ch25:thm:selected-loop-limit\endcsname{{28.1}{160}{}{theorem.438}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:thm:selected-loop-limit@cref\endcsname{{[theorem][1][28]28.1}{[1][160][]160}}
\expandafter\gdef\csname r@ch25:thm:original-loop-stitching\endcsname{{28.2}{160}{}{theorem.439}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:thm:original-loop-stitching@cref\endcsname{{[theorem][2][28]28.2}{[1][160][]160}}
\expandafter\gdef\csname r@ch25:sec:deterministic\endcsname{{28.2}{161}{Deterministic estimates}{section.440}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:sec:deterministic@cref\endcsname{{[section][2][28]28.2}{[1][161][]161}}
\expandafter\gdef\csname r@ch25:lem:elementary-estimates\endcsname{{28.3}{161}{}{lemma.442}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:lem:elementary-estimates@cref\endcsname{{[lemma][3][]28.3}{[1][161][]161}}
\expandafter\gdef\csname r@ch25:eq:cardinal-elementary\endcsname{{28.1}{161}{}{equation.443}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:cardinal-elementary@cref\endcsname{{[equation][1][28]28.1}{[1][161][]161}}
\expandafter\gdef\csname r@ch25:eq:cardinal-sampling\endcsname{{28.2}{162}{}{equation.444}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:cardinal-sampling@cref\endcsname{{[equation][2][28]28.2}{[1][161][]162}}
\expandafter\gdef\csname r@ch25:eq:coarse-heat-short\endcsname{{28.3}{162}{}{equation.445}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:coarse-heat-short@cref\endcsname{{[equation][3][28]28.3}{[1][161][]162}}
\expandafter\gdef\csname r@ch25:eq:geometric-sums\endcsname{{28.4}{162}{}{equation.446}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:geometric-sums@cref\endcsname{{[equation][4][28]28.4}{[1][161][]162}}
\expandafter\gdef\csname r@ch25:eq:cutoff-linear\endcsname{{28.5}{162}{Cardinal interpolation}{equation.447}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:cutoff-linear@cref\endcsname{{[equation][5][28]28.5}{[1][162][]162}}
\expandafter\gdef\csname r@ch25:lem:fixed-smooth-prefix\endcsname{{28.4}{162}{}{lemma.449}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:lem:fixed-smooth-prefix@cref\endcsname{{[lemma][4][]28.4}{[1][162][]162}}
\expandafter\gdef\csname r@ch25:sec:ordered\endcsname{{28.3}{164}{The four-term decomposition and the ordered bilinear terms}{section.450}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:sec:ordered@cref\endcsname{{[section][3][28]28.3}{[1][164][]164}}
\expandafter\gdef\csname r@ch25:eq:quadratic-increment\endcsname{{28.6}{164}{The four-term decomposition and the ordered bilinear terms}{equation.451}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:quadratic-increment@cref\endcsname{{[equation][6][28]28.6}{[1][164][]164}}
\expandafter\gdef\csname r@ch25:lem:ordered-product-inventory\endcsname{{28.5}{165}{}{lemma.452}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:lem:ordered-product-inventory@cref\endcsname{{[lemma][5][]28.5}{[1][165][]165}}
\expandafter\gdef\csname r@ch25:eq:coarse-ordered-product\endcsname{{28.7}{168}{Step 5: the product inequality}{equation.464}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:coarse-ordered-product@cref\endcsname{{[equation][7][28]28.7}{[1][168][]168}}
\expandafter\gdef\csname r@ch25:eq:rough-two-responses\endcsname{{28.8}{168}{Step 6: the sequence $W_j$}{equation.466}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:rough-two-responses@cref\endcsname{{[equation][8][28]28.8}{[1][168][]168}}
\expandafter\gdef\csname r@ch25:eq:marked-background-identification\endcsname{{28.9}{169}{Step 8: the products of $s_j$ and $r_j$}{equation.469}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:marked-background-identification@cref\endcsname{{[equation][9][28]28.9}{[1][169][]169}}
\expandafter\gdef\csname r@ch25:eq:marked-background-envelopes\endcsname{{28.10}{169}{Step 8: the products of $s_j$ and $r_j$}{equation.470}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:marked-background-envelopes@cref\endcsname{{[equation][10][28]28.10}{[1][169][]169}}
\expandafter\gdef\csname r@ch25:eq:marked-cells\endcsname{{28.11}{170}{Step 8: the products of $s_j$ and $r_j$}{equation.471}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:marked-cells@cref\endcsname{{[equation][11][28]28.11}{[1][170][]170}}
\expandafter\gdef\csname r@ch25:eq:factor-powers\endcsname{{28.12}{170}{Step 9: summation}{equation.473}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:eq:factor-powers@cref\endcsname{{[equation][12][28]28.12}{[1][170][]170}}
\expandafter\gdef\csname r@ch25:sec:truncation\endcsname{{28.4}{171}{Truncation estimates}{section.474}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch25:sec:truncation@cref\endcsname{{[section][4][28]28.4}{[1][171][]171}}
\expandafter\gdef\csname r@ch25:thm:all-prefix-first-grade\endcsname{{28.6}{171}{}{theorem.475}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch26:eq:matching-energy-domination@cref\endcsname{{[equation][19][29]29.19}{[1][214][]214}}
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\expandafter\gdef\csname r@ch26:sec:unconditioning@cref\endcsname{{[section][5][29]29.5}{[1][215][]216}}
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\expandafter\gdef\csname r@ch26:thm:orbit-limit@cref\endcsname{{[theorem][20][29]29.20}{[1][215][]216}}
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\expandafter\gdef\csname r@ch26:sec:coupling-bound@cref\endcsname{{[section][6][29]29.6}{[1][217][]217}}
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\expandafter\gdef\csname r@ch26:thm:finite-ceiling-matching@cref\endcsname{{[theorem][21][29]29.21}{[1][217][]217}}
\expandafter\gdef\csname r@ch26:lem:vanishing-area-selected-passage\endcsname{{29.22}{218}{}{lemma.622}{3d-ym-torus-volume-iv.pdf}}
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\expandafter\gdef\csname r@ch26:thm:matched-paths@cref\endcsname{{[theorem][23][29]29.23}{[1][219][]219}}
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\expandafter\gdef\csname r@ch26:thm:finite-ceiling-tier-one@cref\endcsname{{[theorem][24][29]29.24}{[1][220][]220}}
\expandafter\gdef\csname r@ch26:thm:tier-one\endcsname{{29.25}{221}{}{theorem.639}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:tier-one@cref\endcsname{{[theorem][25][29]29.25}{[1][221][]221}}
\expandafter\gdef\csname r@ch26:cor:common-ladder-family\endcsname{{29.26}{222}{}{corollary.640}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:cor:common-ladder-family@cref\endcsname{{[corollary][26][]29.26}{[1][222][]222}}
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\expandafter\gdef\csname r@ch26:eq:common-original-cutoffs@cref\endcsname{{[equation][20][29]29.20}{[1][223][]223}}
\expandafter\gdef\csname r@ch26:thm:full-M-ladder\endcsname{{29.28}{223}{}{theorem.646}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:full-M-ladder@cref\endcsname{{[theorem][28][29]29.28}{[1][223][]223}}
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\expandafter\gdef\csname r@ch26:rem:scope-full-sequence@cref\endcsname{{[remark][29][]29.29}{[1][224][]224}}
\expandafter\gdef\csname r@ch26:chapter-properties\endcsname{{30}{227}{Further properties of the continuum measure}{chapter.659}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:chapter-properties@cref\endcsname{{[chapter][30][]30}{[1][227][]227}}
\expandafter\gdef\csname r@ch26:sec:properties-setting\endcsname{{30.1}{228}{Setting and notation}{section.667}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:properties-setting@cref\endcsname{{[section][1][30]30.1}{[1][228][]228}}
\expandafter\gdef\csname r@ch26:eq:properties-failure-bound\endcsname{{30.1}{229}{Conditioned measures}{equation.669}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:properties-failure-bound@cref\endcsname{{[equation][1][30]30.1}{[1][229][]229}}
\expandafter\gdef\csname r@ch26:eq:conditioning-inequality\endcsname{{30.2}{229}{Conditioned measures}{equation.670}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:conditioning-inequality@cref\endcsname{{[equation][2][30]30.2}{[1][229][]229}}
\expandafter\gdef\csname r@ch26:eq:properties-radius-tightness\endcsname{{30.3}{230}{Flow data and reference solutions}{equation.674}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:properties-radius-tightness@cref\endcsname{{[equation][3][30]30.3}{[1][230][]230}}
\expandafter\gdef\csname r@ch26:sec:dilation\endcsname{{30.2}{230}{Exact dilation}{section.675}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:dilation@cref\endcsname{{[section][2][30]30.2}{[1][230][]230}}
\expandafter\gdef\csname r@ch26:thm:exact-L-dilation\endcsname{{30.1}{230}{}{theorem.676}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:exact-L-dilation@cref\endcsname{{[theorem][1][30]30.1}{[1][230][]230}}
\expandafter\gdef\csname r@ch26:eq:dilated-orbit-law\endcsname{{30.4}{231}{}{equation.677}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:dilated-orbit-law@cref\endcsname{{[equation][4][30]30.4}{[1][230][]231}}
\expandafter\gdef\csname r@ch26:eq:dilated-energy\endcsname{{30.5}{231}{}{equation.678}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:dilated-energy@cref\endcsname{{[equation][5][30]30.5}{[1][231][]231}}
\expandafter\gdef\csname r@ch26:eq:dilated-holonomy\endcsname{{30.6}{232}{Step 2: the continuum dilation}{equation.681}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:dilated-holonomy@cref\endcsname{{[equation][6][30]30.6}{[1][232][]232}}
\expandafter\gdef\csname r@ch26:rem:dilation-scope\endcsname{{30.2}{233}{}{remark.684}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:rem:dilation-scope@cref\endcsname{{[remark][2][]30.2}{[1][232][]233}}
\expandafter\gdef\csname r@ch26:sec:paths\endcsname{{30.3}{233}{Energy and loop processes at positive times}{section.685}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:paths@cref\endcsname{{[section][3][30]30.3}{[1][233][]233}}
\expandafter\gdef\csname r@ch26:lem:positive-window-observable-modulus\endcsname{{30.3}{233}{}{lemma.686}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:lem:positive-window-observable-modulus@cref\endcsname{{[lemma][3][]30.3}{[1][233][]233}}
\expandafter\gdef\csname r@ch26:eq:positive-window-modulus\endcsname{{30.7}{233}{}{equation.687}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:positive-window-modulus@cref\endcsname{{[equation][7][30]30.7}{[1][233][]233}}
\expandafter\gdef\csname r@ch26:eq:positive-window-time-derivative\endcsname{{30.8}{235}{Step 3: the time derivative}{equation.691}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:positive-window-time-derivative@cref\endcsname{{[equation][8][30]30.8}{[1][235][]235}}
\expandafter\gdef\csname r@ch26:thm:actual-observable-paths\endcsname{{30.4}{236}{}{theorem.693}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:actual-observable-paths@cref\endcsname{{[theorem][4][30]30.4}{[1][235][]236}}
\expandafter\gdef\csname r@ch26:eq:selected-orbit-bounded-payment\endcsname{{30.9}{237}{Step 2: comparison of the limits for Borel tests}{equation.696}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:selected-orbit-bounded-payment@cref\endcsname{{[equation][9][30]30.9}{[1][236][]237}}
\expandafter\gdef\csname r@ch26:sec:sobolev\endcsname{{30.4}{238}{Convergence in the Sobolev topology}{section.699}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:sobolev@cref\endcsname{{[section][4][30]30.4}{[1][238][]238}}
\expandafter\gdef\csname r@ch26:thm:sobolev-image-limit\endcsname{{30.5}{238}{}{theorem.700}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:sobolev-image-limit@cref\endcsname{{[theorem][5][30]30.5}{[1][238][]238}}
\expandafter\gdef\csname r@ch26:sec:symmetry\endcsname{{30.5}{240}{Spatial isometries and center symmetry}{section.701}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:symmetry@cref\endcsname{{[section][5][30]30.5}{[1][239][]240}}
\expandafter\gdef\csname r@ch26:thm:spatial-center-symmetry\endcsname{{30.6}{240}{}{theorem.702}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:spatial-center-symmetry@cref\endcsname{{[theorem][6][30]30.6}{[1][240][]240}}
\expandafter\gdef\csname r@ch26:eq:center-word-sign\endcsname{{30.10}{241}{Step 1: lattice symmetries}{equation.704}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:center-word-sign@cref\endcsname{{[equation][10][30]30.10}{[1][241][]241}}
\expandafter\gdef\csname r@ch26:eq:continuum-center-holonomy\endcsname{{30.11}{242}{Step 2: the continuum actions}{equation.706}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:continuum-center-holonomy@cref\endcsname{{[equation][11][30]30.11}{[1][242][]242}}
\expandafter\gdef\csname r@ch26:sec:properties-full-sequence\endcsname{{30.6}{243}{Collected conclusions and the full sequence}{section.709}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:properties-full-sequence@cref\endcsname{{[section][6][30]30.6}{[1][243][]243}}
\expandafter\gdef\csname r@ch26:cor:finite-ceiling-observable-upgrades\endcsname{{30.7}{243}{}{corollary.710}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:cor:finite-ceiling-observable-upgrades@cref\endcsname{{[corollary][7][]30.7}{[1][243][]243}}
\expandafter\gdef\csname r@ch26:sec:compact-coupling-addendum\endcsname{{30.7}{245}{Continuity and uniform convergence in the coupling}{section.711}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:compact-coupling-addendum@cref\endcsname{{[section][7][30]30.7}{[1][245][]245}}
\expandafter\gdef\csname r@ch26:lem:bounded-input-energy-comparison\endcsname{{30.8}{245}{}{lemma.712}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:lem:bounded-input-energy-comparison@cref\endcsname{{[lemma][8][]30.8}{[1][245][]245}}
\expandafter\gdef\csname r@ch26:lem:compact-coupling-smooth-prefix-approximation\endcsname{{30.9}{246}{}{lemma.713}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:lem:compact-coupling-smooth-prefix-approximation@cref\endcsname{{[lemma][9][]30.9}{[1][246][]246}}
\expandafter\gdef\csname r@ch26:eq:compact-coupling-smooth-prefix-error\endcsname{{30.12}{246}{}{equation.714}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:compact-coupling-smooth-prefix-error@cref\endcsname{{[equation][12][30]30.12}{[1][246][]246}}
\expandafter\gdef\csname r@ch26:thm:positive-coupling-uniform-UV\endcsname{{30.10}{247}{}{theorem.715}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:positive-coupling-uniform-UV@cref\endcsname{{[theorem][10][30]30.10}{[1][247][]247}}
\expandafter\gdef\csname r@ch26:rem:compact-coupling-scope\endcsname{{30.11}{248}{}{remark.720}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:rem:compact-coupling-scope@cref\endcsname{{[remark][11][]30.11}{[1][248][]248}}
\expandafter\gdef\csname r@ch26:eq:finite-ceiling-uniform-feasibility\endcsname{{30.13}{250}{Step 3: uniform feasibility at a fixed level}{equation.724}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:finite-ceiling-uniform-feasibility@cref\endcsname{{[equation][13][30]30.13}{[1][250][]250}}
\expandafter\gdef\csname r@ch26:eq:finite-ceiling-matching-energy\endcsname{{30.14}{251}{Step 4: the diagonal choice}{equation.726}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:finite-ceiling-matching-energy@cref\endcsname{{[equation][14][30]30.14}{[1][251][]251}}
\expandafter\gdef\csname r@ch26:cor:continuous-torus-side\endcsname{{30.12}{253}{}{corollary.729}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:cor:continuous-torus-side@cref\endcsname{{[corollary][12][]30.12}{[1][253][]253}}
\expandafter\gdef\csname r@ch26:sec:bounded-quartic-flow-addendum\endcsname{{30.8}{254}{Modified smoothing flows}{section.730}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:bounded-quartic-flow-addendum@cref\endcsname{{[section][8][30]30.8}{[1][254][]254}}
\expandafter\gdef\csname r@ch26:thm:bounded-quartic-flow-universality\endcsname{{30.13}{254}{}{theorem.731}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:bounded-quartic-flow-universality@cref\endcsname{{[theorem][13][30]30.13}{[1][254][]254}}
\expandafter\gdef\csname r@ch26:eq:alternative-original-comparison\endcsname{{30.15}{254}{}{equation.732}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:alternative-original-comparison@cref\endcsname{{[equation][15][30]30.15}{[1][254][]254}}
\expandafter\gdef\csname r@ch26:eq:alternative-orbit-fidelity\endcsname{{30.16}{255}{}{equation.733}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:alternative-orbit-fidelity@cref\endcsname{{[equation][16][30]30.16}{[1][255][]255}}
\expandafter\gdef\csname r@ch26:sec:alternative-observable-paths-addendum\endcsname{{30.9}{257}{Energy and loop processes for the modified flows}{section.737}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:sec:alternative-observable-paths-addendum@cref\endcsname{{[section][9][30]30.9}{[1][256][]257}}
\expandafter\gdef\csname r@ch26:lem:alternative-energy-handoff\endcsname{{30.14}{257}{}{lemma.738}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:lem:alternative-energy-handoff@cref\endcsname{{[lemma][14][]30.14}{[1][257][]257}}
\expandafter\gdef\csname r@ch26:thm:bounded-quartic-observable-paths\endcsname{{30.15}{258}{}{theorem.739}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:thm:bounded-quartic-observable-paths@cref\endcsname{{[theorem][15][30]30.15}{[1][258][]258}}
\expandafter\gdef\csname r@ch26:eq:alternative-energy-path-comparison\endcsname{{30.17}{258}{}{equation.740}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:eq:alternative-energy-path-comparison@cref\endcsname{{[equation][17][30]30.17}{[1][258][]258}}
\expandafter\gdef\csname r@ch26:rem:modified-flow-scope\endcsname{{30.16}{260}{}{remark.744}{3d-ym-torus-volume-iv.pdf}}
\expandafter\gdef\csname r@ch26:rem:modified-flow-scope@cref\endcsname{{[remark][16][]30.16}{[1][260][]260}}
\makeatother