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\expandafter\gdef\csname r@e:sigma-potential\endcsname{{5.22}{125}{The partition function}{equation.310}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@e:sigma-radial-evolution\endcsname{{6.17}{153}{Invariant superfunctions and the singlet state}{equation.380}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@e:sigma-radial-integral\endcsname{{6.6}{141}{}{equation.351}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@e:sigma-rep\endcsname{{5.15}{117}{The invariant integral and the Casimir operator}{equation.281}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@e:sigma-singlet-divergence\endcsname{{6.16}{152}{}{equation.379}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@e:sigma-singlet-operator\endcsname{{6.1}{136}{The sigma model of rank one}{equation.339}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@e:sigma-weight\endcsname{{5.3}{109}{The underlying manifold}{equation.252}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@f:intro-crossover\endcsname{{1.2}{23}{The variance per unit length $V(\ell )/\ell $ of \zcref {t:crossover} (solid line), computed by numerical quadrature with the supplementary script \texttt {crossover\_ratio.py}, with the Poisson value $1$ (dashed gray line) and the large-window asymptote $\pi ^{-3/2} \ell ^{-1/2}$ (dotted gray line)}{figure.caption.67}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@f:intro-profile\endcsname{{1.1}{7}{The kernel $J_{\Z }(i - j)$ of \eqref {e:intro-images} for $W = 10$ (solid line), which is a row of the variance profile $J$ far from the ends of the chain, up to exponentially small terms, and the sharp profile $(2r_0 + 1)^{-1} \1_{|i - j| \le r_0}$ with $r_0 = 24$ (dashed gray line). Both have total mass one and the second moment $2W^2 = r_0(r_0 + 1)/3 = 200$}{figure.caption.10}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@f:ov-structure\endcsname{{2.1}{33}{The logical structure of the proof. An arrow from one box to another means that the second uses results of the first, and only the main dependencies are drawn (for instance, \zcref {s:end} also uses results of \zcref {s:tube,s:cons}). \zcref [S]{s:rep} is used throughout Part IV, and \zcref {s:sigma,s:sigma-radial} throughout Part III, which uses from Part IV only the interface theorem. The a priori bounds $e^{o(W)}$ on partial chains (integrals over consecutive sites of the chain) make the errors $e^{-\gamma W}$, for some $\gamma > 0$, of the attachment of the ends negligible}{figure.caption.77}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@f:rep-chain\endcsname{{4.1}{84}{The open chain of \zcref {p:rep-exact}. Each site carries the weight $w_\Gamma $ of \eqref {e:rep-weight}, and each pair of neighboring sites the link kernel $k_{\tau _{\rm phys}}$ of \zcref {d:rep-chain}. In the pairing \eqref {e:rep-pairing}, the partial chain $f_m$ collects the last $m$ sites, the power $T_\Gamma ^{N - 2m}$ the middle $N - 2m$ sites, and $\ell _m$ the first $m$ sites together with the link from $S_m$ to $S_{m + 1}$}{figure.caption.187}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@f:rep-cycle\endcsname{{4.2}{86}{The Sch\"afer--Wegner cycle for $k = 1$, drawn for $E = 0$ and $P = 0$, so that $s_E = 1$ and $P = E \Id = \kappa _E \Id $. Here $X$ has the single entry $X_{12} = \bar X_{21}$. The plane of $X_{12}$ at height zero carries $\mathcal S_\flat $, and $\mathcal S$ is the graph of $f_0(X)_{11} = -\Im S^{\mathrm {BB}}_{11}$ over it, the upper sheet of the hyperboloid $f_0(X)_{11}^2 - |X_{12}|^2 = s_E^2$. The two dotted segments join points $R$ of $\mathcal S_\flat $ to the points $S(R)$ above them, one at $X = 0$, where $S(R)$ is the vertex at height $s_E$, and one at the labeled point $R$. The dashed circle $|X_{12}| = \eta $, drawn with $\eta = 0.3$, carries the poles on $\mathcal S_\flat $ of the example in \zcref {s:rep-cycle}, with $|r| = |X_{12}|$, and $\mathcal S$ stays away from it. The points of $\mathcal S$ shown have the BB blocks $\kappa _E - \iu s_E Q_X$ of \zcref {l:rep-cycle}\ref {i:rep-cycle-saddle}, the BB blocks of critical points of the weight of the chain, and the vertex gives $\kappa _E - \iu s_E \Lambda _{\mathrm B}$}{figure.caption.192}{lopatto-critical-bandwidth-volume-i.pdf}}
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\expandafter\gdef\csname r@f:sigma-domain\endcsname{{6.1}{157}{The domain $D = [1, \infty ) \times [-1, 1]$. The dashed lines are level lines of $d = B - F$, which vanishes only at the corner. The radial diffusion, with generator $X \Delta _D$, does not reach the boundary of $D$ at positive times (the edges are entrance boundaries), and it is killed at rate $z d$. The singlet diffusion, its Doob transform by $1/d$, is drawn toward the corner and is absorbed there without further killing. The singlet state $\Psi ^{X, z}_t$ is the expectation of the survival factor of the singlet diffusion}{figure.caption.387}{lopatto-critical-bandwidth-volume-i.pdf}}
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