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Furthermore, the results rest extensively on unrefereed and unformalized work recently released by OpenAI. \end{warningpage} \begin{authorsnote} With the exception of these notes, the entire document is machine-written. We have carefully supervised its preparation to ensure that the proofs are readable and that previous literature is appropriately cited. We are releasing this manuscript on Hexagon to disseminate the results as quickly as possible because they may be of interest to the mathematical community. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas. P.\ L.\ was partially supported by NSF grant DMS-2450004. \end{authorsnote} \begin{abstractpage} We study the three-dimensional Bose gas of hard spheres of diameter $a$ at density $\rho$ and temperature $T$, in the thermodynamic limit with $a$, $\rho$, and $T$ fixed. When the gas parameter $\rho a^3$ is sufficiently small and $00$ depends only on the container and the axis. In both results the derivative of the free energy is taken at finite volume, before the thermodynamic limit. The temperature $T_*$ is of order $(\rho a^3)^{1/3}/\log(1/(\rho a^3))$ times the critical temperature of the ideal Bose gas. We also bound the moment generating function of the winding vector of the Feynman--Kac paths by that of a centered Gaussian, at every density and positive temperature. Combining this upper bound with the first result, we determine the order of the winding fluctuations in the regime above. Further, for $0<\varepsilon\le1$, $\rho a^3\le(\varepsilon/100)^3$, and $T\ge(1+\varepsilon)4\pi\rho^{2/3}$, about $1.9(1+\varepsilon)$ times the critical temperature of the ideal Bose gas, the superfluid fraction in a periodic box tends to zero exponentially in its side length. By identities of Pollock and Ceperley and of Sindzingre, Klein, and Ceperley, we can express the two responses as multiples of the variances of the winding and of the signed projected area of the paths. We bound these variances from below with a variance inequality of Hammersley--Chapman--Robbins type, applied to random rewirings of the paths in a short imaginary-time interval along closed lattice routes. The rewirings adapt route comparisons developed in OpenAI's preprints on Bose--Einstein condensation in dilute Bose gases, and we extend the thermal estimates of the positive-temperature hard-sphere preprint in that series to higher temperatures. We average the rewirings over randomly translated closed lattice walks whose independent copies rarely approach each other. This makes the changed law close to the original law in chi-square divergence. Using conditional independence between disjoint imaginary-time intervals, we obtain bounds uniform in temperature. We also transfer the thermal estimates to arbitrary containers. \end{abstractpage} \tableofcontents \mainmatter \chapter{Introduction}\label{sec:intro} Below a temperature of about two kelvin, liquid helium-4 flows through narrow channels without measurable friction, and part of it does not follow the rotation of its container. A fluid with these properties is called a superfluid. The goal of this paper is to prove two equilibrium forms of superfluidity for a basic model of an interacting Bose gas: hard spheres in three dimensions, at a fixed small density and a fixed positive temperature, in the thermodynamic limit. Recall that identical particles are called bosons if their wave functions are symmetric under permutations of the particle coordinates. The thermodynamic limit means that the particle number and the volume tend to infinity while their ratio tends to a fixed density. We show that the superfluid fraction, defined through the free-energy cost of a twist of the boundary conditions, has a positive lower bound, and that the moment of inertia of the gas in a slowly rotating container is smaller than its classical value by a positive fraction. We prove both statements through Feynman's representation of the gas as a system of interacting Brownian loops. We will define two geometric quantities of these loops, their winding and their stochastic area, and prove lower bounds on their fluctuations. \section{Superfluidity and its equilibrium signatures}\label{sec:intro-superfluidity} In 1938, Kapitza \cite{Kapitza1938} and Allen and Misener \cite{AllenMisener1938} independently reported that liquid helium-4, cooled below its transition temperature into the superfluid phase, flows through thin slits and capillaries with almost no viscosity. In the same year, London proposed that the transition is a form of Bose--Einstein condensation, in which a positive fraction of the particles occupies a single one-particle state \cite{London1938}. Einstein had predicted this condensation for an ideal (noninteracting) gas of bosons \cite{Einstein1925}, on the basis of the quantum statistics of Bose \cite{Bose1924}. Tisza then suggested that the liquid behaves as a mixture of two components, a normal one that carries viscosity and a superfluid one that does not \cite{Tisza1938}. Landau developed a hydrodynamics of two interpenetrating fluids in which the normal component is a gas of collective disturbances of the liquid \cite{Landau1941,Landau1941a}. Andronikashvili measured the normal fraction with closely spaced disks that oscillate about an axis \cite{Andronikashvili1946}. Viscosity makes the normal component move with the disks, and the effective moment of inertia decreases as the liquid is cooled. The two components describe different responses of the same liquid, not two permanently labeled populations of atoms. Bogoliubov showed that in a weakly interacting Bose gas, the energy of an excitation is approximately proportional to its momentum at small momentum, as for sound waves \cite{Bogoliubov1947}. Feynman gave an atomic theory of the transition and of the two-fluid model \cite{Feynman1953a,Feynman1953,Feynman1954,Feynman1955}. We refer to \cite{Leggett1999,Leggett2006,PitaevskiiStringari2016} for modern accounts. Motion without friction concerns the dynamics of the fluid, and a circulating current can last for a long time without being an equilibrium state. We study two equilibrium criteria for superfluidity: the responses of the equilibrium state to a twist and to a slow rotation, two perturbations whose effect on a classical fluid in equilibrium is trivial. We describe them for $N$ particles of mass $m$ at temperature $T$ in a region of side length $L$, and we write $\rho$ for the number density and $\hbar$ for the reduced Planck constant; Boltzmann's constant is set equal to one. For a Hamiltonian $H$ (the energy operator) at temperature $T$, we define the free energy by $F=-T\log Z$, where $Z=\operatorname{Tr}e^{-H/T}$ is the partition function. The first perturbation is a twist. Let the particles move on the torus $\Lambda_L=(\mathbb R/L\mathbb Z)^3$, and require the many-body wave function $\psi$ to satisfy \begin{equation}\label{eq:intro-twisted-bc} \psi(x_1,\dots,x_j+Le_1,\dots,x_N)=e^{\mathrm i\theta}\psi(x_1,\dots,x_N)\qquad(1\le j\le N), \end{equation} where $e_1=(1,0,0)$ and the real number $\theta$ is the twist angle. Equivalently, the particles obey periodic boundary conditions and feel a constant vector potential of magnitude $\hbar\theta/L$ in the direction $e_1$. Let $F(\theta)$ denote the free energy with the boundary condition \eqref{eq:intro-twisted-bc}. In classical statistical mechanics we can remove a constant vector potential by shifting the momentum variables. Then $F$ does not depend on $\theta$. In a superfluid, a gradient of the phase of the wave function corresponds to a velocity of the superfluid component. The twist imposes the phase gradient $\theta/L$. The superfluid component then moves with velocity $v_{\mathrm s}=\hbar\theta/(mL)$ relative to the box while the normal component stays at rest, and the free energy increases by the kinetic energy of this motion: \begin{equation}\label{eq:intro-helicity} F(\theta)-F(0)=\tfrac12\, m\rho_{\mathrm s}L^3v_{\mathrm s}^2+o(\theta^2). \end{equation} Here $\rho_{\mathrm s}$ is the number density of the superfluid component. Fisher, Barber, and Jasnow defined the helicity modulus through the second-order free-energy response to a slowly varying phase, and related it to the superfluid density in this way \cite{FisherBarberJasnow1973}. Solving \eqref{eq:intro-helicity} for $\rho_{\mathrm s}$, we obtain the superfluid fraction \begin{equation}\label{eq:intro-fs-physical} f_{\mathrm s}=\frac{\rho_{\mathrm s}}{\rho}=\frac{mL^2}{\hbar^2N}F''(0). \end{equation} The second perturbation is a slow rotation. Place the particles in a container and rotate it with a small angular velocity $\omega$ about an axis with direction $n$. In the frame of the container, the Hamiltonian $H$ becomes $H-\omega\mathcal L_n$, where $\mathcal L_n$ is the total angular momentum about the axis; its formula for our model is given in \cref{sec:intro-results}. Let $F^{\mathrm{rot}}(\omega)$ be the free energy of $H-\omega\mathcal L_n$. The moment of inertia of the equilibrium state is $I=-\partial_\omega^2F^{\mathrm{rot}}(0)$. A classical fluid in equilibrium rotates rigidly with its container, and for a classical system $I$ equals \begin{equation}\label{eq:intro-Icl} I_{\mathrm{cl}}=m\Big\langle\sum_{j=1}^N r_j^2\Big\rangle, \end{equation} where $r_j$ is the distance of particle $j$ from the axis and $\langle\cdot\rangle$ is the equilibrium average. In an axisymmetric vessel and in the absence of vortices (circulation around line defects), the superfluid component can remain at rest under slow rotation. In this case we have $Ia$ for all $t\in[0,\beta]$ and all $i\ne j$, where on the torus $|x_i(t)-x_j(t)|$ denotes the torus distance. In a container we use the free Euclidean heat kernel and free Euclidean bridges, and $\mathcal C$ also requires the paths to stay inside the container. The Feynman--Kac formula expresses the bosonic partition function as \begin{equation}\label{eq:intro-FK} Z=\operatorname{Tr}e^{-\beta H}=\frac1{N!}\sum_{\pi\in S_N}\int\prod_{j=1}^Np_\beta(x_j,x_{\pi(j)})\,\mathbb P_{X,\pi X}(\mathcal C)\,\mathrm dX, \end{equation} where $X=(x_1,\dots,x_N)$ and $\pi X=(x_{\pi(1)},\dots,x_{\pi(N)})$; see \cref{prop:id-fk,lem:id-ck} for the representation and the trace normalization, and \cite{Feynman1953,Ginibre1965a,Ginibre1965b,Ginibre1971,Simon2005} for background. The sum over permutations expresses the symmetry of bosonic wave functions. After normalization, the integrand in \eqref{eq:intro-FK} defines a probability measure $\mu$ on permutations and paths, which we call the Bose path measure. Since the path of particle $j$ ends where the path of particle $\pi(j)$ begins, the $N$ paths join into closed loops; a cycle of $\pi$ of length $k$ gives one loop of duration $k\beta$. In this way the Bose gas becomes a gas of Brownian loops with a hard-core interaction at equal times. On the torus, we lift each loop to $\mathbb R^3$. After one full turn, its displacement is $L$ times an integer vector, which we call its winding vector. Let $W\in\mathbb Z^3$ be the sum of the winding vectors of all loops. Pollock and Ceperley observed that the twist \eqref{eq:intro-twisted-bc} multiplies the weight of each configuration by the phase $e^{\mathrm i\theta W_1}$, the phase acquired by closed loops in a constant vector potential, so that $Z(\theta)/Z(0)=\mathbb E_\mu e^{\mathrm i\theta W_1}$ \cite{PollockCeperley1987}. Since $\mathbb E_\mu W_1=0$ by symmetry, differentiating twice at $\theta=0$ yields $F''(0)=T\operatorname{Var}_\mu(W_1)$. In our units, \eqref{eq:intro-fs-physical} becomes \begin{equation}\label{eq:intro-PC} f_{\mathrm s}=\frac{TL^2}{2N}\operatorname{Var}_\mu(W_1). \end{equation} This is the winding-number formula of Pollock and Ceperley \cite{PollockCeperley1987}. For rotation, the Hamiltonian $H-\omega\mathcal L_n$ differs from the Hamiltonian of particles in a uniform magnetic field along $n$ by a centrifugal term proportional to $\omega^2\sum_jr_j^2$, which accounts for $I_{\mathrm{cl}}$. The magnetic field multiplies the weight of each configuration by $e^{\mathrm i\omega\mathcal A_n}$, where the stochastic area $\mathcal A_n$ is the sum over loops of the signed areas of their projections onto the plane orthogonal to $n$. We obtain \begin{equation}\label{eq:intro-SKC} I_{\mathrm{cl}}-I=T\operatorname{Var}_\mu(\mathcal A_n), \end{equation} which is the area formula of Sindzingre, Klein, and Ceperley \cite{SindzingreKleinCeperley1989}; see Ceperley's review \cite{Ceperley1995} for both formulas and their use in path-integral Monte Carlo simulations. Superfluidity, in the two senses above, is the statement that the winding (or the area) of an interacting system of Brownian loops has macroscopic fluctuations. By \eqref{eq:intro-PC}, the superfluid fraction has a positive lower bound in the thermodynamic limit if and only if the normalized variance $\operatorname{Var}_\mu(W_1)/(\beta\rho L)$ has a positive lower limit. In a container of linear size $L$, the classical moment of inertia is at most a constant times $NL^2$. By \eqref{eq:intro-SKC}, the nonclassical rotational inertia fraction then has a positive lower limit whenever $\operatorname{Var}_\mu(\mathcal A_n)/(\beta\rho L^5)$ does. Bose statistics is essential here. For noninteracting distinguishable particles, only the identity permutation occurs, each loop has duration $\beta$, and the probability that a single loop winds around the torus is of order $e^{-L^2/(4\beta)}$; the superfluid fraction then tends to zero. At fixed inverse temperature, long permutation cycles provide a natural mechanism for winding on the scale of the container, since a cycle through $k$ particles is a loop of duration $k\beta$. Feynman related the superfluid transition to the appearance of such cycles \cite{Feynman1953}; see also \cite[Section~III.F]{Ceperley1995}. The present paper proves that, for the dilute hard-sphere gas at low positive temperature, the winding and the area have fluctuations of the required size. \section{The model and the main results}\label{sec:intro-results} We use units in which $\hbar=1$, the particle mass is $m=1/2$, and Boltzmann's constant is one, so that $\hbar^2/(2m)=1$ and temperature is measured in units of energy. The dictionary to general $\hbar$ and $m$ is given in \cref{rem:intro-units}. We fix the diameter $a>0$ of the hard spheres, which is the smallest allowed distance between two particle centers. It is also the scattering length of the interaction: the zero-energy scattering solution outside the excluded ball is $1-a/|x|$, and its coefficient $a$ is called the scattering length. Throughout, $T>0$ is the temperature and $\beta=1/T$. \paragraph{The gas on the torus.} For $N\ge1$ and $L>0$, let $d_L$ be the Euclidean distance on the torus $\Lambda_L=(\mathbb R/L\mathbb Z)^3$, and let \[ \mathcal X_{N,L}=\{X=(x_1,\dots,x_N)\in\Lambda_L^N:d_L(x_i,x_j)>a\text{ for all }i\ne j\}. \] We consider only $N$ and $L$ for which $\mathcal X_{N,L}$ is nonempty; along the sequences in our theorems, this holds for all large $L$. The Hamiltonian $H_{N,L}$ is the self-adjoint operator associated with the quadratic form $\psi\mapsto\sum_{j=1}^N\int_{\mathcal X_{N,L}}|\nabla_j\psi|^2$ on the symmetric functions in $H^1_0(\mathcal X_{N,L})$. In other words, $H_{N,L}$ is the Dirichlet realization of $-\sum_j\Delta_j$ on bosonic wave functions that vanish when two particles touch. For $\theta\in\mathbb R$, the twisted Hamiltonian $H_{N,L}(\theta)$ is defined by the same form domain and the differential expression $\sum_j(-\mathrm i\nabla_j+\theta e_1/L)^2$. It is unitarily equivalent to $-\sum_j\Delta_j$ with the boundary condition \eqref{eq:intro-twisted-bc}. The twisted free energy is \[ F_{N,L,T}(\theta)=-T\log\operatorname{Tr}e^{-H_{N,L}(\theta)/T}, \] where the trace is over symmetric functions. We define the superfluid fraction at finite volume by \begin{equation}\label{eq:intro-fs} f_{\mathrm s}(N,L,T)=\frac{L^2}{2N}\,\partial_\theta^2F_{N,L,T}(0). \end{equation} In \cref{sec:identities}, we show that $F_{N,L,T}$ is twice continuously differentiable, that \eqref{eq:intro-PC} holds with the winding vector $W$ of the Bose path measure $\mu$ on $\Lambda_L$, and that $0\le f_{\mathrm s}(N,L,T)\le1$. \paragraph{The gas in a container.} Let $\Omega\subset\mathbb R^3$ be a nonempty bounded open set, the container, and let $|\Omega|>0$ be its volume. For $L>0$, consider $N$ hard spheres with centers in the dilated container $L\Omega$, and let \[ \mathcal X^\Omega_{N,L}=\{X\in(L\Omega)^N:|x_i-x_j|>a\text{ for all }i\ne j\}. \] As on the torus, we consider only $N$ and $L$ for which $\mathcal X^\Omega_{N,L}$ is nonempty. The Hamiltonian $H^\Omega_{N,L}$ is the Dirichlet realization of $-\sum_j\Delta_j$ on the symmetric functions in $H^1_0(\mathcal X^\Omega_{N,L})$; the wave functions vanish when two particles touch and when a particle touches the wall. Fix a unit vector $n\in\mathbb R^3$ and a point $x_0\in\mathbb R^3$, and consider rotations about the axis $\{Lx_0+sn:s\in\mathbb R\}$. The angular momentum about this axis is \[ \mathcal L_n=\sum_{j=1}^Nn\cdot\bigl((x_j-Lx_0)\times(-\mathrm i\nabla_j)\bigr). \] The operator $H^\Omega_{N,L}-\omega\mathcal L_n$ is defined through its closed quadratic form on the form domain of $H^\Omega_{N,L}$; the first-order term is infinitesimally form bounded because the container is bounded. For $\omega\in\mathbb R$, let $F^{\mathrm{rot}}_n(\omega)=-T\log\operatorname{Tr}e^{-(H^\Omega_{N,L}-\omega\mathcal L_n)/T}$, again with the trace over symmetric functions. We define the moment of inertia and the classical moment of inertia by \begin{equation}\label{eq:intro-inertia} I_n=-\partial_\omega^2F^{\mathrm{rot}}_n(0),\qquad I_{\mathrm{cl},n}=\frac12\Big\langle\sum_{j=1}^Nr_j^2\Big\rangle, \end{equation} where $r_j$ is the distance from $x_j$ to the axis and $\langle\cdot\rangle$ denotes expectation in the normalized state $e^{-H^\Omega_{N,L}/T}/\operatorname{Tr}e^{-H^\Omega_{N,L}/T}$, the Gibbs state at temperature $T$. The number $1-I_n/I_{\mathrm{cl},n}$ is the nonclassical rotational inertia fraction. The circle in the integral below denotes the Stratonovich convention, obtained by midpoint evaluation in stochastic Riemann sums. In \cref{sec:identities}, we show that $I_n$ is well defined and that \begin{equation}\label{eq:intro-area-identity} I_{\mathrm{cl},n}-I_n=T\operatorname{Var}_\mu(\mathcal A_n),\qquad \mathcal A=\frac12\sum_{j=1}^N\int_0^\beta x_j(t)\times\circ\,\mathrm dx_j(t),\qquad \mathcal A_n=n\cdot\mathcal A, \end{equation} where $\mu$ is now the Bose path measure in the container, and $x_j(t)$ is the path of particle $j$. We call $\mathcal A\in\mathbb R^3$ the stochastic area vector. Because the paths join into closed loops, $\mathcal A$ does not depend on the choice of origin. In particular, the deficit $I_{\mathrm{cl},n}-I_n$ is the same for all parallel axes. The identity \eqref{eq:intro-area-identity} implies $I_n\le I_{\mathrm{cl},n}$. Since $F^{\mathrm{rot}}_n$ is concave (\cref{prop:id-rotation}), we also have $I_n\ge0$. We do not assume any symmetry of $\Omega$ or any regularity of its boundary. \paragraph{Main results.} Let $\alpha_0\ge2$ be the absolute constant fixed in Step~1 of \cref{sec:stiff-parameters}, large enough for the scaled setting of \cite[Section~2]{OpenAI2026BEC} and for \cref{app:temperature,app:walls}; see \cref{sec:thermal-setting}. Let $\kappa_{\mathrm T}$ be the numerical constant fixed in \cref{app:temperature}. For $a,\rho>0$ with $\rho a^3<\alpha_0^3$, we set $D_\rho=\alpha_0^{3/2}(\rho a^3)^{-1/2}$ and define \begin{equation}\label{eq:intro-Tstar} T_*(a,\rho)=\frac{\kappa_{\mathrm T}\,\rho a}{\log D_\rho}. \end{equation} Under the change of units of \cite[Section 10]{OpenAI2026BEC}, every temperature in the open interval $(0,T_*(a,\rho))$ maps for all large $L$ into the scaled temperature window $00$ and $c_1>0$ such that the following holds. Let $a>0$ and $\rho>0$ satisfy $\rho a^3\le c_0$, and let $00$ and all sufficiently large $L$ in any such sequence. For comparison, the ideal Bose gas of density $\rho$ condenses below the temperature $T_{\mathrm c}(\rho)=4\pi(\rho/\zeta(3/2))^{2/3}$, where $\zeta$ is the Riemann zeta function, and \begin{equation}\label{eq:intro-Tc-ratio} \frac{T_*(a,\rho)}{T_{\mathrm c}(\rho)}=\frac{\kappa_{\mathrm T}\,\zeta(3/2)^{2/3}}{4\pi}\,\frac{(\rho a^3)^{1/3}}{\log D_\rho}. \end{equation} Numerical simulations place the transition of the dilute hard-sphere gas close to $T_{\mathrm c}$ \cite{GruterCeperleyLaloe1997,PilatiGiorginiProkofev2008}. Seiringer and Ueltschi proved an upper bound on the critical temperature for Bose--Einstein condensation, by proving exponential decay, with spatial separation, of the kernel of the one-particle density matrix above an explicit temperature \cite{SeiringerUeltschi2009}. The normal regime of \cref{thm:main-normal} below starts at $(1+\varepsilon)\zeta(3/2)^{2/3}T_{\mathrm c}\approx1.897(1+\varepsilon)T_{\mathrm c}$, whereas their bound for the absence of condensation is much closer to $T_{\mathrm c}$, namely $T_{\mathrm c}(1+5.09(\rho a^3)^{1/6}(1+o(1)))$ as $\rho a^3\to0$, for a different quantity, in the grand-canonical ensemble on domains with Dirichlet boundary conditions \cite{SeiringerUeltschi2009}. The interval $(0,T_*)$ is a small fraction, of order $(\rho a^3)^{1/3}/\log(1/(\rho a^3))$, of the expected range of superfluidity. It lies a logarithmic factor below the energy scale $\rho a$ of the excitations of the gas. For momentum $p\in\mathbb R^3$, writing $p^2=|p|^2$, Bogoliubov's excitation spectrum in our units is $\epsilon(p)=(p^2(p^2+16\pi a\rho))^{1/2}$ \cite{Bogoliubov1947}. It is approximately linear for $p^2\ll a\rho$, where the excitations are phonons, that is, low-momentum collective excitations, and approximately quadratic for $p^2\gg a\rho$; the crossover energy is of order $\rho a$. We have $T_*/(\rho a)=\kappa_{\mathrm T}/\log D_\rho$. In this regime the theories of Landau and Bogoliubov predict that the normal fraction $1-f_{\mathrm s}$ is very small \cite{Landau1941,Bogoliubov1947,PitaevskiiStringari2016}. Our lower bound $c_1(\rho a^3)^{3/8}$ is far from this prediction. \Cref{thm:main-stiffness} implies that the gas is superfluid in the sense of \eqref{eq:intro-helicity}, but it does not determine its expected near-unit value. Our second result concerns rotation. The covariance matrix $\operatorname{Cov}_\mu(\mathcal A)$ of the stochastic area vector has smallest eigenvalue $\lambda_{\min}(\operatorname{Cov}_\mu(\mathcal A))=\inf_{|n|=1}\operatorname{Var}_\mu(\mathcal A_n)$. For $x_0\in\mathbb R^3$, let $R_\Omega(x_0)=\sup_{y\in\Omega}|y-x_0|$. \begin{theorem}\label{thm:main-ncri} Let $\Omega\subset\mathbb R^3$ be a nonempty bounded open set. There exist constants $c_0>0$ and $c_1>0$, depending only on $\Omega$, such that the following holds. Let $a>0$ and $\rho>0$ satisfy $\rho a^3\le c_0$, let $00$ and all sufficiently large $L$. By \eqref{eq:intro-area-identity}, the inertia deficit $I_{\mathrm{cl},n}-I_n$ equals $T\,n\cdot\operatorname{Cov}_\mu(\mathcal A)\,n$ for every axis with direction $n$. Then \eqref{eq:intro-area-cov} bounds the deficit from below for all axes at once. For a general direction $n$, the proof rotates the coordinates so that $n$ becomes a coordinate direction; no symmetry of $\Omega$ is used. The constant in \eqref{eq:intro-ncri} depends on $\Omega$ and on the axis point $x_0$, and not on $n$. Combining \eqref{eq:intro-area-cov}, the identity \eqref{eq:intro-area-identity}, and the inequality $I_{\mathrm{cl},n}\le NL^2R_\Omega(x_0)^2/2$, we obtain the bound \eqref{eq:intro-ncri}. When $\Omega=(-1/2,1/2)^3$ and $x_0=0$, the symmetries of the cube imply that $I_n$ and $I_{\mathrm{cl},n}$ do not depend on $n$, at every finite volume. In this case the nonclassical rotational inertia fraction is the same for all axes through the center of the cube. Our third result bounds the moment generating function of the winding vector by that of a centered Gaussian (a sub-Gaussian bound), without any restriction on the density or the temperature. Let $\mu$ be the Bose path measure on $\Lambda_L$ and $W$ its winding vector, as in \eqref{eq:intro-PC}. \begin{theorem}\label{thm:main-winding} Let $N\ge1$, $L>0$, $a>0$, and $\beta>0$, and suppose that $\mathcal X_{N,L}$ is nonempty. Then for every $t\in\mathbb R^3$, \[ \mathbb E_\mu\exp(t\cdot W)\le\exp\bigl(\beta N|t|^2/L^2\bigr). \] \end{theorem} Expanding both sides to second order in $t$, we find that $\operatorname{Var}_\mu(u\cdot W)\le2\beta N|u|^2/L^2$ for $u\in\mathbb R^3$; by \eqref{eq:intro-PC}, this is the upper bound $f_{\mathrm s}\le1$. Combining this with \cref{thm:main-stiffness}, we obtain the order of magnitude of the winding vector. \begin{corollary}\label{cor:main-winding-scaling} Assume the hypotheses of \cref{thm:main-stiffness}, let $c_1$ be the constant there, and set $\widehat W=W/(\beta\rho L)^{1/2}$. Then $\mathbb E_\mu\exp(t\cdot\widehat W)\le\exp\bigl(N|t|^2/(\rho L^3)\bigr)$ for every $t\in\mathbb R^3$, and $\sup_{N,L}\mathbb E_\mu|\widehat W|^p<\infty$ for every $p\ge1$, where the supremum is over $N$ and $L$ in any sequence with $N/L^3\to\rho$. Further, every subsequential limit in distribution $\widehat W_\infty$ of $\widehat W$ along such a sequence satisfies $\mathbb E\widehat W_\infty=0$ and \[ 2c_1(\rho a^3)^{3/8}\,\mathrm{Id}\le\operatorname{Cov}(\widehat W_\infty)\le2\,\mathrm{Id}. \] \end{corollary} The area satisfies a two-sided bound of the same kind. Since $I_n\ge0$, we have $\operatorname{Var}_\mu(\mathcal A_n)\le\beta I_{\mathrm{cl},n}$ by \eqref{eq:intro-area-identity}, and $\beta I_{\mathrm{cl},n}\le(1+o(1))|\Omega|R_\Omega(x_0)^2\beta\rho L^5/2$. Then, by \cref{thm:main-ncri}, we find that the area variance has order of magnitude $\beta\rho L^5$. We do not prove the analog of \cref{cor:main-winding-scaling} for the area, because we have no uniform bound on its higher moments. \begin{theorem}\label{thm:main-normal} Let $\varepsilon\in(0,1]$, let $a,\rho>0$ satisfy $\rho a^3\le(\varepsilon/100)^3$, and consider a sequence with $N,L\to\infty$ and $N/L^3\to\rho$. Then, for all sufficiently large $L$ in the sequence and all $T\ge(1+\varepsilon)4\pi\rho^{2/3}$, \begin{equation}\label{eq:intro-normal} f_{\mathrm s}(N,L,T)\le57\Bigl(\frac{L^2T}{4\pi}\Bigr)^{3/4}\exp\Bigl(-\frac L4\sqrt{T\log(1+\varepsilon)}\Bigr), \end{equation} and $\mathbb P_\mu(W\neq0)\le190(L^2T/4\pi)^{3/2}\exp\bigl(-\frac L2\sqrt{T\log(1+\varepsilon)}\bigr)$. In particular, $f_{\mathrm s}(N,L,T)\to0$ and $\operatorname{Var}_\mu(W_1)\to0$, uniformly in $T\ge(1+\varepsilon)4\pi\rho^{2/3}$. \end{theorem} With $\varepsilon=1$ and $\rho a^3\le\min\{c_0,10^{-6}\}$, \cref{thm:main-stiffness,thm:main-normal} give a superfluid interval $(0,T_*(a,\rho))$ and a normal interval $[8\pi\rho^{2/3},\infty)$, which are disjoint; we prove nothing about the temperatures in between. We conclude this section with several remarks on the scope of these results. \begin{remark}\label{rem:intro-limits} In \cref{thm:main-stiffness,thm:main-ncri}, the derivatives in $\theta$ and $\omega$ are taken at $\theta=0$ and $\omega=0$ at each finite $N$ and $L$, and the thermodynamic limit is taken afterward. We do not prove that $f_{\mathrm s}(N,L,T)$ or $I_n/I_{\mathrm{cl},n}$ converges, and we make no statement about the free energy at a fixed nonzero twist or angular velocity. A twist of angle $\theta$ changes the free energy by $f_{\mathrm s}N\theta^2/L^2+o(\theta^2)$, which is of order $L$, while the free energy itself is of order $L^3$. For this reason, asymptotic formulas for the free energy per unit volume, such as those discussed in \cref{sec:intro-previous}, do not by themselves control this curvature; one needs estimates that resolve the twist-dependent part of the free energy, which is smaller than the volume, together with control of its derivatives. Both of our quantities are responses of the equilibrium state to static perturbations. Fisher, Barber, and Jasnow emphasized that a nonzero equilibrium helicity modulus does not by itself establish the dynamical stability of finite superflow \cite[p.~1113]{FisherBarberJasnow1973}. Prokof'ev and Svistunov discuss how such definitions relate to dynamical ones \cite{ProkofevSvistunov2000}. \end{remark} \begin{remark}\label{rem:intro-uniform} The proofs give the following finite-volume form of \cref{thm:main-stiffness}, uniform in the density and the temperature. Let $c_0$ and $c_1$ be the constants of that theorem, let $a>0$, and let $0<\rho_1\le\rho_2$ with $\rho_2a^3\le2c_0$. There exists $L_0$ such that $f_{\mathrm s}(N,L,T)\ge c_1(\rho_La^3)^{3/8}$, where $\rho_L=N/L^3$, for all $L\ge L_0$, all $N$ with $\rho_L\in[\rho_1,\rho_2]$, and all $0=Stealth,font=\small] \foreach \k in {0,...,5} { \draw[gray!60] (\k,0) rectangle (\k+1,2); \node at (\k+0.5,-0.35) {$v_{\k}$}; } \draw[gray!60,dashed] (6,0) rectangle (6.6,2); \node at (6.3,-0.35) {$v_0$}; \foreach \k in {0,...,5} { \fill (\k+0.3,1.45) circle (0.05); \draw (\k+0.72,0.55) circle (0.06); \draw[thick,gray] (\k+0.3,1.45) .. controls (\k+0.45,1.1) and (\k+0.5,0.8) .. (\k+0.66,0.6); } \foreach \k in {0,...,4} { \draw[->,thick,blue!60!black,densely dashed] (\k+0.3,1.45) .. controls (\k+0.9,1.6) and (\k+1.4,0.9) .. (\k+1.66,0.62); } \draw[->,thick,blue!60!black,densely dashed] (5.3,1.45) .. controls (5.9,1.6) and (6.3,1.0) .. (6.55,0.7); \node[align=left,anchor=west] at (0,2.45) {$\bullet$ initial point $Y_i$ \quad $\circ$ terminal point $Z_{\sigma(i)}$ \quad solid: old pairing \quad dashed: new pairing}; \end{tikzpicture} \caption{The surgery along a route $\gamma=(v_0,\dots,v_5)$ that winds once around the torus in the first coordinate direction; the dashed cell is $v_0$ again. In each cell one retained path is chosen. Its initial point is reconnected to the terminal point of the path chosen in the next cell, and the chosen paths are resampled from their exact conditional law. All other paths, and all endpoints, are unchanged. The changes of the displacements add up to $K\,\mathrm{hom}(\gamma)$, so in the example shown the first coordinate of the winding vector increases by one.} \label{fig:ov-surgery} \end{figure} \section{Why the chi-square divergence is small}\label{sec:ov-chi-small} Fix a group $a$ and a labeled route of this group, that is, a route together with the paths of the group chosen along it, and write $\pi$ for the corresponding cyclic reassignment and $S$ for the chosen rows. The accepted surgery along it is a subprobability kernel whose outgoing mass at a matching $\sigma$ is the probability $R_\pi(\sigma)$ of these choices. Its incoming density at a configuration reached by the surgery is explicit (\cref{lem:surgery-reverse}): \[ c_\pi=R_\pi(\sigma\pi^{-1}) \frac{Z_S(\sigma\pi^{-1};\omega_{S^c})}{Z_S(\sigma;\omega_{S^c})}, \] the probability of the choices at the matching before the surgery, multiplied by the conditional partition function of the chosen paths at that matching, and divided by their conditional partition function at the matching after the surgery. After the completion by the identity, the likelihood of the surgeries of group $a$ for a skeleton $\Gamma$ is $L_{a,\Gamma}=1+\sum_\pi(c_\pi-R_\pi)$, where the sum is over the labeled routes of group $a$. The kernel $Q$ is the mixture of the kernels $Q^{(a)}$ of the $m$ groups, each with weight $1/m$, so the density $L_\Gamma$ in \eqref{eq:ov-ingster} is $m^{-1}\sum_aL_{a,\Gamma}$, and $\chi^2(\mu Q\,\|\,\mu)$ is the average over the $m^2$ pairs of groups $(a,b)$ of $\E_{\Gamma,\Gamma'}\int(L_{a,\Gamma}-1)(L_{b,\Gamma'}-1)\dd\mu$. For the $m$ diagonal terms, $a=b$, and the $m(m-1)$ cross terms, $a\neq b$, we must bound the sums, over pairs of labeled routes of groups $a$ and $b$ built from two independent skeletons, of \begin{equation}\label{eq:ov-four} \int(c_\pi-R_\pi)(c_{\pi'}-R_{\pi'})\dd\mu . \end{equation} Each such term involves four matchings of the retained paths: the current one, the two matchings before each of the two surgeries, and the matching before both; the last enters through the product $c_\pi c_{\pi'}$. First let $a=b$, and suppose that the four matchings produce the same pair of routes from the two skeletons. If, in addition, no cell of one route is within a fixed distance $R$ of the other, then the hard-core interaction between the two sets of chosen paths vanishes, the normalizing constants factor, neither surgery changes the choices available to the other, and the four products in \eqref{eq:ov-four} cancel exactly (\cref{lem:surgery-exact}). When the routes do come within distance $R$ of each other, we only need an upper bound. Each nearby chosen path costs a factor $(1-\eta_{\mathrm{loc}})^{-1}$, where $\eta_{\mathrm{loc}}$ bounds the probability that a resampled path collides with one of the other chosen paths near it, and a path chosen by both routes costs a fixed factor $G$. The first cost is small because $\eta_{\mathrm{loc}}$ tends to zero as $D\to\infty$, by the collision estimates of \cite{OpenAI2026BEC}. The second is rare, because each route chooses among at least $m_*$ paths of the group in each cell, and two routes choose the same path with probability at most $1/m_*$. By \cref{lem:surgery-four,lem:surgery-labels}, when the routes agree in the four matchings, we find that a pair of routes contributes zero if $J_R(\gamma,\gamma')=0$ and at most $Ce^{\eps J_R(\gamma,\gamma')}$ otherwise, where $J_R(\gamma,\gamma')$ counts pairs of cells of the two routes at distance at most $R$ and $\eps$ is small. Since $e^{x}\le C(e^{2x}-1)$ for $x\ge\eps$, if the routes did not depend on the matching we would obtain \begin{equation}\label{eq:ov-encounter} \chi^2(\mu Q^{(a)}\,\|\,\mu)\le C\,\E\bigl[e^{2\eps J_R(\gamma,\gamma')}-1\bigr]. \end{equation} \Cref{sec:ov-routes} explains the additional term needed when they do. Now let $a\neq b$. The two surgeries move rows of different groups, and a move of group $b$ changes none of the counts, bad sets, rates, and route indicators of group $a$. After reindexing the matchings, three of the four products in \eqref{eq:ov-four} have the same integral, and the comparison of the four normalizing constants bounds the signed sum from above by a loss confined to the encounters of the two routes, which vanishes if $J_R(\gamma,\gamma')=0$ (\cref{lem:surgery-cross}). This is an upper bound on an integrated signed sum, not a pointwise cancellation: the hard core couples paths of different groups, and a move of one group changes the conditional law of the paths of another. Averaged over the skeletons and the configuration, each cross term is at most $C_\times\eta_{\mathrm{loc}}/K$. This comparison adapts two arguments of \cite{OpenAI2026HS,OpenAI2026B}. The first is the cancellation of conditional normalizing constants away from the overlap of two changes \cite[Lemma~8.3]{OpenAI2026HS}, reproduced in \cite[Lemma~9.3]{OpenAI2026BEC}. The second is the comparison of four endpoint assignments, with the costs concentrated on shared paths and their predecessors, in \cite[Section~10]{OpenAI2026B}. There is one essential difference. In \cite{OpenAI2026HS,OpenAI2026B,OpenAI2026BEC}, a bound of order one on the second moment of a density suffices, while here we need the divergence to be small. We must keep the subtracted outgoing rates in \eqref{eq:ov-four} until we have used the cancellation for route pairs without encounters. This is why we work with the signed expansion rather than with positive bounds alone. \section{Random routes}\label{sec:ov-routes} The bound \eqref{eq:ov-encounter} is useful only if two routes built from independent skeletons rarely meet and, when they do meet, separate quickly. A route built from a simple random walk would not do: two independent walks in the two transverse directions return to each other with probability of order $k^{-1}$ after $k$ steps, which is not summable. We use instead the dyadic random skeleton of \cite[Section~8.1]{OpenAI2026B}, also used in \cite{OpenAI2026HS,OpenAI2026BEC}. It advances by one cell in the first coordinate direction at each step, and its transverse displacement over $k$ steps has a conditional density bounded by $Ck^{-3/2}$, given its past. The exponent $3/2$ exceeds one, so that the expected number of future encounters with a fixed opposing route is bounded. This is the mechanism of unpredictable paths, due to Benjamini, Pemantle, and Peres \cite{BenjaminiPemantlePeres1998} and H\"aggstr\"om and Mossel \cite{HaggstromMossel1998}. To close the loop we read the transverse displacement forward and then backward, and to make encounters rare we translate the loop by a uniform transverse vector. By an ordered-moment argument, we obtain \begin{equation}\label{eq:ov-excess} \E e^{\kappa J_R(\Gamma,\Gamma')}\le1+C/K \end{equation} for two independent skeletons $\Gamma,\Gamma'$ (\cref{prop:routes-nominal}). The excess over one, rather than a bounded exponential moment, is the new requirement here; it uses the random translation in an essential way. The skeleton ignores the bad cells. We replace each visit of the skeleton to a connected cluster of bad cells by a detour along the outer boundary of the cluster, which consists of good cells and is connected by a theorem of Kesten and Tim\'ar \cite{Kesten1986,Timar2013}. The joint bound $p^{|A|}$ makes large clusters rare, so the detoured route is accepted with probability at least $1/2$ and has few additional encounters. These detours are the construction of \cite[Section~7]{OpenAI2026HS} and \cite[Section~8]{OpenAI2026BEC} (\cref{prop:routes-router}). There is an apparent circularity here, since the bad cells depend on the matching of the retained paths, which the surgery changes. Precisely, a cell is bad for a group when it contains fewer than $m_*$ retained paths of the group that can be moved, and it is vulnerable when it contains fewer than $m_*+2$ of them. The four matchings in \eqref{eq:ov-four} may produce different routes from the same skeleton, and the exact cancellation described above can fail. The way out is that a surgery changes the number of usable paths only in the cells it visits, and only by a bounded amount. All four bad sets lie in the vulnerable set of a common matching, and the routes can differ only if a component of the vulnerable set is approached by both skeletons. Let $A_*$ be the total number of cells in such components. If $A_*=0$, the route indicators agree in all four matchings. Contributions of route pairs without encounters then cancel exactly, and contributions with encounters are bounded by the positive estimate above. In general, the number of possible pairs of routes is at most $4^{A_*}$ (\cref{lem:routes-stability}). This follows the stability and path-list argument of \cite[Lemmas~10.2--10.3]{OpenAI2026B}. Averaging over the environment and using the joint bound for the vulnerable set, we obtain $\E[e^{2\eps J_R(\Gamma,\Gamma')+2c_{\mathrm{rec}}A_*}]-1\le C/K$ for a constant $c_{\mathrm{rec}}$ (\cref{cor:routes-final}), which bounds the divergence of the surgeries of one group by $C_*/K$. Combining the diagonal and the cross terms, \cref{thm:surgery} bounds $\chi^2(\mu Q\,\|\,\mu)$ by $C_*/(mK)+C_\times\eta_{\mathrm{loc}}/K$. The one-opponent collision bound of \cref{lem:slab-insertion} applies to the normalized one-path trial law before conditioning on avoidance of the other retained paths. The resulting union bound over all relevant retained paths is at most $1/2$ for the choice of $m$ in \eqref{eq:stiff-groups}, with $m$ comparable to $D^{1/4}$. Since $\eta_{\mathrm{loc}}$ is of order $D^{-1/4}$, we obtain \eqref{eq:ov-chi}. \section{Rotation}\label{sec:ov-rotation} For \cref{thm:main-ncri} we replace the winding loops by polygons of large area. Inside the container we place a cube of side $H$ comparable to $K$, at distance comparable to $K$ from the walls, and in it a diamond-shaped lattice polygon of side of order $H$, with the same dyadic transverse fluctuations and a uniform random translation. Detours are made separately on each of its four sides, so that the area of the routed polygon remains at least $cH^2$. The polygon is not planar, since its transverse fluctuations also move it out of the plane, but its projection onto the plane orthogonal to the axis has signed area of order $H^2$. The surgery along a polygon $\gamma$ changes the component of the stochastic area along the axis by this projected area, up to two errors. The chords between neighboring endpoints contribute an error that is bounded pathwise by a constant times the length of $\gamma$. The stochastic areas of the original and resampled Brownian bridges are not controlled pathwise, since a confined Brownian path can sweep a large signed area through small loops. We control them in mean, averaging over both the original chosen paths and their replacements, conditionally on everything else (\cref{prop:ncri-gain}). The mean gain of the surgeries of each group is at least $cH^2$, and so is that of their mixture, whose $\chi^2$ divergence is at most $C_\sharp D^{-1/4}/H$. Combining \eqref{eq:ov-hcr} with the slab decomposition, we obtain $\Var(n\cdot\mathcal A)\ge c\,n_\beta D^{1/4}K^5$. Since $\Icl\le CNK^2$, we conclude from the identity \eqref{eq:ov-skc} that $1-I/\Icl$ has the claimed lower bound. Rotating the container reduces every axis direction to the vertical one. The thermal estimates of \cite{OpenAI2026BEC} are proved on the torus, and in \cref{app:temperature} we extend them to higher temperatures. In \cref{app:walls} we prove them for cells at a fixed distance from the walls of an arbitrary bounded open container. The routes never approach the walls, so the remaining arguments are the same as on the torus. \section{The order of choices}\label{sec:ov-constants} The constants are chosen in the following order (\cref{sec:proof-stiffness}). We first fix the spatial constants of the thermal estimates and the temperature coefficient $\kappa_{\mathrm T}$, then the geometry of the routes and the encounter coefficients, including the exponent $\theta_\times$ for routes of different groups, then the upper bound $p_{\max}$ on the base for bad cells. From the thermal estimates we then obtain the slab length $\delta$ and the local constants, including the factor $G$. Next we choose the number $M$ of retained paths per cell in each group, which makes the cost of shared paths small, and then a density threshold $D_{\min}$, which makes the collision probability $\eta_{\mathrm{loc}}$ small, and, for each $D\ge D_{\min}$, the number $m(D)$ of groups. The constant $C_\sharp$ depends only on the choices made before $D_{\min}$. Only after these choices do we let $K$ tend to infinity. In particular, no constant depends on the volume or on the temperature, so we have the uniformity stated in \cref{thm:main-stiffness,thm:main-ncri}. \part{Exact identities}\label{part:identities} \chapter{The Bose path measure and exact identities}\label{sec:identities} This section collects the exact response identities. All statements are at finite volume and require no dilution assumption; the configuration domain is assumed nonempty. All derivatives are taken at finite volume. In \cref{sec:id-model}, we define the hard-core Hamiltonians on the torus and in a bounded container, the twisted and rotating free energies, the superfluid fraction, and the moment of inertia. \Cref{sec:id-fk} recalls the Feynman--Kac representation, defines the Bose path measure, and introduces the winding vector and the stochastic area. \Cref{sec:id-subgaussian} bounds the moment generating function of the winding vector by that of a centered Gaussian and records the consequences for its scaling. In \cref{sec:id-winding}, we express the superfluid fraction through the variance of the winding vector and through the fluctuations of the total momentum. In \cref{sec:id-rotation}, we prove that the moment of inertia is nonnegative and that its deficit from the classical value is the variance of the stochastic area. Finally, \cref{sec:id-slabs} decomposes imaginary time into slabs, proves that the slabs are conditionally independent given the configurations at the cut times, and deduces the variance inequality used to amplify single-slab estimates. \section{The model}\label{sec:id-model} We use the units of \cref{sec:intro-results,rem:intro-units}: $\hbar=1$, $m=1/2$, and Boltzmann's constant one. The one-particle kinetic energy is $-\Delta$, the temperature is $T>0$, and $\beta=1/T$. \paragraph{Containers and configuration domains.} The torus of side $L>0$ is $\Lambda_L=(\R/L\mathbb Z)^3$ with the distance $d_L(x,y)=\min_{k\in\mathbb Z^3}\abs{x-y+Lk}$. A container is a nonempty bounded open set $U\subset\R^3$, equipped with the Euclidean distance. Fix an exclusion distance $a>0$ and $N\ge1$. The configuration domains are \begin{equation}\label{eq:id-domains} \begin{aligned} \mathcal X_{N,L}&=\{X\in\Lambda_L^N: d_L(x_i,x_j)>a\text{ for all }i\neq j\},\\ \mathcal X_N(U)&=\{X\in U^N:\abs{x_i-x_j}>a\text{ for all }i\neq j\}, \end{aligned} \end{equation} where $X=(x_1,\dots,x_N)$. With the periodic distance, $\mathcal X_N(\Lambda_L)=\mathcal X_{N,L}$. In applications the container is $U=L\Omega$ in physical units or $U=K\Omega$ in scaled units; in particular, $\mathcal X_N(L\Omega)=\mathcal X^\Omega_{N,L}$ in the notation of \cref{sec:intro}. We write $\mathcal X_N$ for either $\mathcal X_{N,L}$ or $\mathcal X_N(U)$ when a statement applies to both, and $\mathbb X$ for the corresponding one-particle space, which is $\Lambda_L$ on the torus and $\R^3$ for a container. The set $\mathcal X_N$ is open, and it is invariant under the permutations $(\pi X)_i=x_{\pi(i)}$, $\pi\in S_N$. We assume throughout that $\mathcal X_N$ is nonempty. \paragraph{Hamiltonians.} Let $L^2_{\mathrm{sym}}(\mathcal X_N)$ be the space of square-integrable functions on $\mathcal X_N$ that are invariant under all permutations of the particle coordinates. Let $H^1_0(\mathcal X_N)$ be the closure in $H^1$ of the smooth functions with compact support in $\mathcal X_N$ (on the torus, smooth functions on $\Lambda_L^N$ with compact support in $\mathcal X_{N,L}$), and set $\mathcal Q_N=H^1_0(\mathcal X_N)\cap L^2_{\mathrm{sym}}(\mathcal X_N)$. The form \begin{equation}\label{eq:id-form} q[\psi]=\sum_{i=1}^N\int_{\mathcal X_N}\abs{\nabla_i\psi}^2\dd X,\qquad \psi\in\mathcal Q_N, \end{equation} is closed and nonnegative. We denote the associated self-adjoint operator on $L^2_{\mathrm{sym}}(\mathcal X_N)$ by $H_{N,L}$ on the torus and by $H_N(U)$ for a container, and write $H$ for either. It is the Dirichlet realization of $-\sum_i\Delta_i$ on symmetric functions: the boundary condition is imposed on the collision set $\{\abs{x_i-x_j}=a\}$ and, for a container, on the walls $\partial U$. The same form on $H^1_0(\mathcal X_N)$, without the symmetry constraint, defines the labeled operator $H^{\mathrm{lab}}$ on $L^2(\mathcal X_N)$. Since $\mathcal X_N$ is invariant under permutations, $H^{\mathrm{lab}}$ commutes with them, and $H$ is the restriction of $H^{\mathrm{lab}}$ to the reducing subspace $L^2_{\mathrm{sym}}(\mathcal X_N)$. \begin{lemma}\label{lem:id-trace} The operator $H$ has compact resolvent, and for every $t>0$ and $k\ge0$ we have $\tr(H^k e^{-tH})<\infty$. The partition function $Z=\tr e^{-\beta H}$ satisfies $00$. \end{proof} We write $\langle B\rangle=Z^{-1}\tr(Be^{-\beta H})$ for the Gibbs expectation of a bounded operator $B$. For a bounded measurable function $f$ on $\mathcal X_N$ that is invariant under permutations, $\langle f\rangle$ denotes the expectation of the corresponding multiplication operator. \paragraph{Twisted boundary conditions.} On the torus, let $\vartheta\in\R^3$. The form \begin{equation}\label{eq:id-twisted-form} q_\vartheta[\psi]=\sum_{i=1}^N\int_{\mathcal X_{N,L}}\abs{(-\mathrm i\nabla_i+\vartheta/L)\psi}^2\dd X,\qquad \psi\in\mathcal Q_N, \end{equation} is closed and bounded below, since $q_\vartheta\le 2q+2N\abs{\vartheta}^2L^{-2}$ and $q\le 2q_\vartheta+2N\abs{\vartheta}^2L^{-2}$ in the sense of forms. We denote the associated self-adjoint operator by $H_{N,L}(\vartheta)$, and for $\theta\in\R$ we write $H_{N,L}(\theta)=H_{N,L}(\theta e_1)$, so that \[ H_{N,L}(\theta)=\sum_{i=1}^N\bigl(-\mathrm i\nabla_i+\theta e_1/L\bigr)^2. \] The twist convention is that of \cref{sec:intro-results}: $H_{N,L}(\theta)$ is unitarily equivalent to $-\sum_i\Delta_i$ with phase $e^{\mathrm i\theta}$ when one particle is translated by $Le_1$. By the min--max comparison of $q_\vartheta$ with $q$ and \cref{lem:id-trace}, the operator $e^{-tH_{N,L}(\vartheta)}$ is trace class for every $t>0$. The twisted free energy is \begin{equation}\label{eq:id-twisted-free-energy} F_{N,L,T}(\vartheta)=-T\log\tr e^{-\beta H_{N,L}(\vartheta)},\qquad F_{N,L,T}(\theta)=F_{N,L,T}(\theta e_1). \end{equation} \begin{definition}\label{def:id-superfluid-fraction} The superfluid fraction of the canonical hard-core Bose gas on $\Lambda_L$ is \begin{equation}\label{eq:id-fs} f_{\mathrm s}(N,L,T)=\frac{L^2}{2N}\,\partial_\theta^2F_{N,L,T}(\theta)\Big|_{\theta=0}. \end{equation} \end{definition} By \cref{lem:id-blocks}(3), $\theta\mapsto F_{N,L,T}(\theta)$ is twice continuously differentiable. The derivative in \eqref{eq:id-fs} exists. The physical interpretation and the order of limits are given in \cref{sec:intro-results,rem:intro-units}. \paragraph{Rotation.} Let $U$ be a container, let $n\in\R^3$ be a unit vector, and let $x_0\in\R^3$. The axis of rotation is the line through $x_0$ with direction $n$. We write $y_i=x_i-x_0$ and $R_0=\sup_{x\in U}\abs{x-x_0}<\infty$. For $\psi\in\mathcal Q_N$, define the components of the angular momentum about $x_0$ by \begin{equation}\label{eq:id-angular-momentum} \mathcal L_b\psi=\sum_{i=1}^N e_b\cdot\bigl(y_i\times(-\mathrm i\nabla_i)\bigr)\psi =-\mathrm i\sum_{i=1}^N(e_b\times y_i)\cdot\nabla_i\psi,\qquad b=1,2,3, \end{equation} and set $\mathcal L=(\mathcal L_1,\mathcal L_2,\mathcal L_3)$ and $\mathcal L_n=n\cdot\mathcal L$. For $\bm\omega\in\R^3$, the operator $\bm\omega\cdot\mathcal L$ is given by \eqref{eq:id-angular-momentum} with $e_b$ replaced by $\bm\omega$. \begin{lemma}\label{lem:id-angular} Let $\bm\omega\in\R^3$. For all $\phi,\psi\in\mathcal Q_N$, we have $\bm\omega\cdot\mathcal L\psi\in L^2_{\mathrm{sym}}(\mathcal X_N(U))$, \begin{equation}\label{eq:id-angular-bound} \norm{\bm\omega\cdot\mathcal L\psi}^2\le\abs{\bm\omega}^2NR_0^2\,q[\psi],\qquad \langle\phi,\bm\omega\cdot\mathcal L\psi\rangle=\langle\bm\omega\cdot\mathcal L\phi,\psi\rangle. \end{equation} The form $q[\psi]-\langle\psi,\bm\omega\cdot\mathcal L\psi\rangle$ on $\mathcal Q_N$ is real, closed, and bounded below. Its self-adjoint operator $H-\bm\omega\cdot\mathcal L$ satisfies $\tr e^{-t(H-\bm\omega\cdot\mathcal L)}<\infty$ for every $t>0$. \end{lemma} \begin{proof} The vector field $v(x)=\bm\omega\times(x-x_0)$ satisfies $\abs{v}\le\abs{\bm\omega}R_0$ on $U$ and $\operatorname{div}v=0$. By the Cauchy--Schwarz inequality in $\R^{3N}$, we have pointwise \[ \Bigl\lvert\sum_iv(x_i)\cdot\nabla_i\psi\Bigr\rvert^2\le\Bigl(\sum_i\abs{v(x_i)}^2\Bigr)\sum_i\abs{\nabla_i\psi}^2\le\abs{\bm\omega}^2NR_0^2\sum_i\abs{\nabla_i\psi}^2, \] and integrating, we obtain the first inequality in \eqref{eq:id-angular-bound}. Since the differential expression is invariant under permutations of the particles, it maps symmetric functions to symmetric functions. For smooth $\phi,\psi$ with compact support in $\mathcal X_N(U)$, we obtain the symmetry relation in \eqref{eq:id-angular-bound} by integration by parts, using $\operatorname{div}v=0$. Since both sides are continuous on $\mathcal Q_N$ with the $H^1$ norm, the relation extends to $\mathcal Q_N$; in particular $\langle\psi,\bm\omega\cdot\mathcal L\psi\rangle$ is real. The first inequality in \eqref{eq:id-angular-bound} implies \begin{equation}\label{eq:id-klmn} \abs{\langle\psi,\bm\omega\cdot\mathcal L\psi\rangle}\le\tfrac12 q[\psi]+\tfrac12\abs{\bm\omega}^2NR_0^2\norm{\psi}^2, \end{equation} so the perturbation has relative form bound less than one. By the theorem on relatively form-bounded perturbations of closed quadratic forms \cite[Theorem~X.17]{ReedSimon1975}, we conclude that the perturbed form is closed and bounded below on $\mathcal Q_N$. By \eqref{eq:id-klmn} and the min--max principle, the $k$-th eigenvalue of $H-\bm\omega\cdot\mathcal L$ is at least half the $k$-th eigenvalue of $H$ minus $\abs{\bm\omega}^2NR_0^2/2$, and the trace bound follows from \cref{lem:id-trace}. \end{proof} The rotating-frame free energy is \begin{equation}\label{eq:id-rot-free-energy} F^{\mathrm{rot}}(\bm\omega)=-T\log\tr e^{-\beta(H-\bm\omega\cdot\mathcal L)},\qquad \bm\omega\in\R^3, \end{equation} and for the fixed axis we write $F^{\mathrm{rot}}(\omega)=F^{\mathrm{rot}}(\omega n)$, $\omega\in\R$. Both depend on $x_0$ through $\mathcal L$. \begin{definition}\label{def:id-inertia} The moment of inertia tensor about $x_0$ and the moment of inertia about the axis $(n,x_0)$ are \begin{equation}\label{eq:id-inertia} I_{bc}(x_0)=-\partial_{\omega_b}\partial_{\omega_c}F^{\mathrm{rot}}(\bm\omega)\Big|_{\bm\omega=0},\qquad I(n,x_0)=-\partial_\omega^2F^{\mathrm{rot}}(\omega n)\Big|_{\omega=0}=\sum_{b,c}n_bn_cI_{bc}(x_0). \end{equation} The classical moment of inertia tensor and the classical moment of inertia are \begin{equation}\label{eq:id-classical-inertia} I_{\mathrm{cl},bc}(x_0)=m\Bigl\langle\sum_{i=1}^N\bigl(\abs{y_i}^2\delta_{bc}-y_{i,b}y_{i,c}\bigr)\Bigr\rangle,\qquad I_{\mathrm{cl}}(n,x_0)=m\Bigl\langle\sum_{i=1}^N\abs{n\times y_i}^2\Bigr\rangle, \end{equation} with $m=1/2$ and $y_i=x_i-x_0$. \end{definition} The function $F^{\mathrm{rot}}$ is twice continuously differentiable by \cref{prop:id-rotation}, which justifies \eqref{eq:id-inertia}; the second identity in \eqref{eq:id-inertia} is the chain rule. The identity $\abs{y}^2-(n\cdot y)^2=\abs{n\times y}^2$ shows that $I_{\mathrm{cl}}(n,x_0)=\sum_{b,c}n_bn_cI_{\mathrm{cl},bc}(x_0)$. The ratio $1-I(n,x_0)/I_{\mathrm{cl}}(n,x_0)$ is the nonclassical rotational inertia fraction. For $U=L\Omega$, the axis point in this definition is the physical point $Lx_0$ used in \cref{sec:intro-results}. \section{The Feynman--Kac representation and the Bose path measure}\label{sec:id-fk} \paragraph{Brownian bridges.} Let $p_t(x)=(4\pi t)^{-3/2}\exp(-\abs{x}^2/(4t))$ be the heat kernel of $\Delta$ on $\R^3$. Brownian motion with generator $\Delta$ has independent coordinates, each of variance $2s$ at time $s$. For $x,y\in\R^3$ and $t>0$, let $\mathbb W^t_{x,y}$ be the finite measure on $C([0,t];\R^3)$ with total mass $p_t(y-x)$ whose normalization is the law of the Brownian bridge from $x$ to $y$ of duration $t$. Every continuous path $\gamma\colon[0,t]\to\Lambda_L$ has a continuous lift $\tilde\gamma$ to $\R^3$, unique up to a translation by $L\mathbb Z^3$, and we call \[ d(\gamma)=\tilde\gamma(t)-\tilde\gamma(0)\in\R^3 \] the lifted displacement of $\gamma$. For $x,y\in\Lambda_L$ with representatives $\tilde x,\tilde y\in\R^3$, the torus bridge measure is \[ \mathbb W^t_{x,y}=\sum_{k\in\mathbb Z^3}(\mathrm{pr}_L)_*\,\mathbb W^t_{\tilde x,\tilde y+Lk}, \] where $\mathrm{pr}_L\colon\R^3\to\Lambda_L$ is the projection; it does not depend on the choice of representatives, and the paths in the $k$-th term are exactly those with $d(\gamma)=\tilde y+Lk-\tilde x$. For configurations $X,Y\in\mathbb X^N$, we set $\mathbb W^t_{X,Y}=\bigotimes_{i=1}^N\mathbb W^t_{x_i,y_i}$, a measure on paths $X(\cdot)=(x_1(\cdot),\dots,x_N(\cdot))$ in $C([0,t];\mathbb X^N)$. We use two standard properties of these measures. First, they are invariant under time reversal, in the sense that the image of $\mathbb W^t_{X,Y}$ under $X(\cdot)\mapsto X(t-\cdot)$ is $\mathbb W^t_{Y,X}$. Second, they satisfy the concatenation identity \begin{equation}\label{eq:id-concatenation} \mathbb W^{s+t}_{X,Z}=\int_{\mathbb X^N}\mathbb W^s_{X,Y}\circledast\mathbb W^t_{Y,Z}\dd Y,\qquad s,t>0, \end{equation} where $\circledast$ denotes the image of the product measure under concatenation of paths at time $s$. Identity \eqref{eq:id-concatenation} is the Markov property of Brownian motion. For $t>0$, let $\mathcal E_t=\{X(\cdot): X(s)\in\mathcal X_N\text{ for all }s\in[0,t]\}$ be the event that the configuration path stays in $\mathcal X_N$; on this event no two particles come within distance $a$ and, for a container, no particle leaves $U$. For $X,Y\in\mathcal X_N$, the killed kernel is \begin{equation}\label{eq:id-killed-kernel} K_t(X,Y)=\mathbb W^t_{X,Y}(\mathcal E_t),\qquad K^{\mathrm s}_t(X,Y)=\frac1{N!}\sum_{\pi\in S_N}K_t(X,\pi Y). \end{equation} \begin{proposition}\label{prop:id-fk} Let $t>0$. \begin{enumerate} \item The operator $e^{-tH^{\mathrm{lab}}}$ on $L^2(\mathcal X_N)$ is an integral operator with kernel $K_t$. \item For a container $U$, let $\mathsf a(x)=Ax+c$ be an affine real vector field with $\operatorname{tr}A=0$, let $V$ be a bounded measurable real function on $\mathcal X_N(U)$ satisfying $V(\pi X)=V(X)$ for every $\pi\in S_N$ (almost everywhere in $X$), and let $H^{\mathrm{lab}}_{\mathsf a,V}$ be the operator associated with the closed form $\sum_i\norm{(-\mathrm i\nabla_i-\mathsf a(x_i))\psi}^2+\langle\psi,V\psi\rangle$ on $H^1_0(\mathcal X_N(U))$. Then $e^{-tH^{\mathrm{lab}}_{\mathsf a,V}}$ has the kernel \begin{equation}\label{eq:id-fk-magnetic} \int_{\mathcal E_t}\exp\Bigl(-\mathrm i\sum_{i=1}^N\int_0^t\mathsf a(x_i(s))\cdot\circ\dd x_i(s)-\int_0^tV(X(s))\dd s\Bigr)\,\mathbb W^t_{X,Y}(\dd X(\cdot)), \end{equation} where $\circ$ denotes the Stratonovich integral, which agrees with the It\^o integral because $\operatorname{div}\mathsf a=0$. \end{enumerate} The same statements hold for the restrictions of these operators to $L^2_{\mathrm{sym}}(\mathcal X_N)$, with the kernels symmetrized in the second variable as in \eqref{eq:id-killed-kernel}. \end{proposition} The path-integral description of the Bose gas through permutation cycles of Brownian paths goes back to Feynman \cite{Feynman1953}. Ginibre studied reduced density matrices of quantum gases, including hard-core potentials \cite{Ginibre1965a,Ginibre1965b}; see also his account of functional integration \cite{Ginibre1971}. For general background on functional integration, see \cite[Sections~6 and~15]{Simon2005}. For arbitrary open domains, see \cite[Proposition~2.9 and Theorem~6.1]{BroderixHundertmarkLeschke2000}, with time rescaled from generator $\Delta/2$ to $\Delta$. For a container, let $\mathcal O_m\uparrow\mathcal X_N$ be an increasing smooth relatively compact exhaustion. After extension by zero, $H^1_0(\mathcal O_m)\subset H^1_0(\mathcal X_N)$, and their union is dense in $H^1_0(\mathcal X_N)$ in the form norm. Then the Dirichlet semigroups on $\mathcal O_m$, extended by zero, converge strongly to the semigroup on $\mathcal X_N$. The same conclusion holds for the magnetic forms, since the vector potential is bounded on the configuration domain and the scalar potential is bounded. A continuous path whose image on $[0,t]$ lies in $\mathcal X_N$ lies in some $\mathcal O_m$. The survival indicators increase to $\mathbf1_{\mathcal E_t}$. The nonnegative path integrals converge monotonically; the magnetic ones converge by dominated convergence, since their integrands have modulus at most $e^{t\|V_-\|_\infty}$. For the torus case of (1), see \cite[Proposition~4.4 in arXiv version~4]{BeiGuneysu2021}, applied to the $H^1_0$ realization on the open configuration domain; conditioning on the endpoint gives the kernel above. The sign conventions for the phases in part (2) play no role below, since all the partition functions we consider are real. \begin{lemma}\label{lem:id-ck} The kernels have the following properties. \begin{enumerate} \item For all $X,Y\in\mathcal X_N$ and $\pi\in S_N$, we have $K_t(X,Y)=K_t(Y,X)\ge0$ and $K_t(\pi X,\pi Y)=K_t(X,Y)$. The kernel $K^{\mathrm s}_t$ is symmetric and invariant under permutations of either argument. \item For $s,t>0$ and $X,Z\in\mathcal X_N$, \begin{equation}\label{eq:id-ck} \int_{\mathcal X_N}K_s(X,Y)K_t(Y,Z)\dd Y=K_{s+t}(X,Z),\qquad \int_{\mathcal X_N}K^{\mathrm s}_s(X,Y)K^{\mathrm s}_t(Y,Z)\dd Y=K^{\mathrm s}_{s+t}(X,Z). \end{equation} \item We have \begin{equation}\label{eq:id-trace-formula} Z=\tr e^{-\beta H}=\int_{\mathcal X_N}K^{\mathrm s}_\beta(X,X)\dd X=\frac1{N!}\sum_{\pi\in S_N}\int_{\mathcal X_N}K_\beta(X,\pi X)\dd X, \end{equation} and $\langle f\rangle=Z^{-1}\int_{\mathcal X_N}f(X)K^{\mathrm s}_\beta(X,X)\dd X$ for every bounded measurable symmetric function $f$. The analogous formula holds for the operators of \cref{prop:id-fk}(2). \end{enumerate} \end{lemma} \begin{proof} Part (1) follows from the time-reversal invariance of the bridge measures, the invariance of $\mathcal X_N$ under permutations, and the definition of $K^{\mathrm s}_t$. For part (2), we apply \eqref{eq:id-concatenation} and note that a concatenated path lies in $\mathcal E_{s+t}$ if and only if both pieces lie in $\mathcal E_s$ and $\mathcal E_t$, respectively; the requirement $Y\in\mathcal X_N$ is part of this event. This proves the first identity. For the second, we insert the definition of $K^{\mathrm s}$, change variables $Y\mapsto\pi^{-1}Y$ in the term indexed by $(\pi,\sigma)$, and use part (1) and the first identity; the double sum over $S_N\times S_N$ then reduces to $N!$ copies of the sum defining $K^{\mathrm s}_{s+t}$. For part (3), let $P_{\mathrm{sym}}$ be the orthogonal projection of $L^2(\mathcal X_N)$ onto $L^2_{\mathrm{sym}}(\mathcal X_N)$. It commutes with $H^{\mathrm{lab}}$, and by \cref{prop:id-fk}(1) the operator $A=e^{-\beta H^{\mathrm{lab}}/2}P_{\mathrm{sym}}$ has the kernel $K^{\mathrm s}_{\beta/2}$. It is self-adjoint and Hilbert--Schmidt by \cref{lem:id-trace}, and $\tr_{L^2_{\mathrm{sym}}}e^{-\beta H}=\tr A^2=\iint\abs{K^{\mathrm s}_{\beta/2}(X,Y)}^2\dd X\dd Y$. By part (1) and \eqref{eq:id-ck}, the last integral equals $\int K^{\mathrm s}_\beta(X,X)\dd X$. The formula for $\langle f\rangle$ follows in the same way from $\tr(fe^{-\beta H})=\tr(Afe^{-\beta H^{\mathrm{lab}}/2}P_{\mathrm{sym}})$. For the operators of \cref{prop:id-fk}(2), write $k_t$ for the kernel \eqref{eq:id-fk-magnetic} and $k_t^{\mathrm s}$ for its symmetrization. Apply \cite[Theorem~6.1]{BroderixHundertmarkLeschke2000} to the globally defined affine configuration-space vector field $(\mathsf a(x_1),\ldots,\mathsf a(x_N))$, with scalar potential $V/2$ extended by zero and time $2t$. It gives joint continuity of $k_t$ on $\mathcal X_N(U)^2$. The bound $\abs{k_t(X,Y)}\le e^{t\norm{V_-}_\infty}(4\pi t)^{-3N/2}$ and the finite volume of $\mathcal X_N(U)$ show that the convolution of two symmetrized kernels is continuous. Thus the semigroup identity, initially valid almost everywhere on $\mathcal X_N(U)^2$, holds everywhere. Self-adjointness likewise gives $k_t^{\mathrm s}(Y,X)=\overline{k_t^{\mathrm s}(X,Y)}$ everywhere. In particular, \[ k_\beta^{\mathrm s}(X,X)=\int_{\mathcal X_N(U)}\abs{k_{\beta/2}^{\mathrm s}(X,Y)}^2\dd Y. \] Since the vector field and the scalar potential are bounded on the configuration domain, their form satisfies $q_{\mathsf a,V}[\psi]\ge\tfrac12q[\psi]-C\norm\psi^2$. The min--max argument of \cref{lem:id-trace} therefore makes the half-time symmetrized semigroup Hilbert--Schmidt. The same argument as above proves both trace formulas. \end{proof} \begin{definition}\label{def:id-path-measure} We denote the Bose path measure by $\mu$, or by $\mu_{N,L,\beta}$ on the torus and $\mu_{N,U,\beta}$ in a container, suppressing the fixed exclusion distance. It is the probability measure on $S_N\times C([0,\beta];\mathbb X^N)$ given by \begin{equation}\label{eq:id-path-measure} \mu(\{\pi\}\times\dd X(\cdot))=\frac1{Z\,N!}\int_{\mathcal X_N}\mathbf 1_{\mathcal E_\beta}(X(\cdot))\,\mathbb W^\beta_{X_0,\pi X_0}(\dd X(\cdot))\dd X_0. \end{equation} We write $\E$, $\Var$, and $\Cov$ for expectation, variance, and covariance under $\mu$. \end{definition} By \eqref{eq:id-killed-kernel} and \eqref{eq:id-trace-formula}, $\mu$ has total mass one, and its normalizing constant is $Z=\int_{\mathcal X_N}K^{\mathrm s}_\beta(X,X)\dd X$. This expression is positive and finite without reference to $H$: $K_\beta(X,\pi X)$ is bounded by $\prod_i\sup_z\sum_{k\in\mathbb Z^3}p_\beta(z+Lk)$ on the torus and by $(4\pi\beta)^{-3N/2}$ in a container, $\mathcal X_N$ has finite volume, and, since $\mathcal X_N$ is open, the bridge from $X\in\mathcal X_N$ to itself stays in $\mathcal X_N$ with positive probability. The proof of \cref{thm:id-subgaussian} uses only this description of $\mu$. Under $\mu$, the random path satisfies $X(0)=X_0$ and $x_i(\beta)=x_{\pi(i)}(0)$ for every $i$. If $(i_1,\dots,i_k)$ is a cycle of $\pi$, with $\pi(i_j)=i_{j+1}$ and $i_{k+1}=i_1$, then the concatenation of $x_{i_1}(\cdot),\dots,x_{i_k}(\cdot)$ is a continuous closed loop of duration $k\beta$. We call these loops the worldlines. They are closed by the endpoint permutation alone; no segments are added to close them. The labels are bookkeeping: all observables below are functions of the unlabeled collection of worldlines. \begin{lemma}\label{lem:id-marginal} For every $\tau\in[0,\beta]$ and every bounded measurable function $f$ on $\mathcal X_N$ that is invariant under permutations, we have $\E f(X(\tau))=\langle f\rangle$. \end{lemma} \begin{proof} For $\tau=0$, the claim follows from \eqref{eq:id-path-measure} and \cref{lem:id-ck}(3), and for $\tau=\beta$ it follows from the case $\tau=0$ because $X(\beta)=\pi X(0)$ and $f$ is symmetric. For $0<\tau<\beta$, we split the bridge measure at time $\tau$ using \eqref{eq:id-concatenation}, as in the proof of \cref{lem:id-ck}(2). This gives \begin{align*} \E f(X(\tau))&=\frac1{Z\,N!}\sum_\pi\iint K_\tau(X,Y)f(Y)K_{\beta-\tau}(Y,\pi X)\dd X\dd Y\\ &=\frac1Z\iint K^{\mathrm s}_{\beta-\tau}(Y,X)K_\tau(X,Y)f(Y)\dd X\dd Y. \end{align*} Using $K_\tau(\pi^{-1}X',Y)=K_\tau(X',\pi Y)$ and \eqref{eq:id-ck}, the integral over $X$ equals $K^{\mathrm s}_\beta(Y,Y)$, and \cref{lem:id-ck}(3) completes the proof. \end{proof} \paragraph{Winding vector and stochastic area.} On the torus, the winding vector of $(\pi,X(\cdot))$ is \begin{equation}\label{eq:id-winding} W=\frac1L\sum_{i=1}^Nd(x_i(\cdot))\in\R^3. \end{equation} Let $\tilde x_i\in[0,L)^3$ be representatives of $x_i(0)$. Since $x_i(\beta)=x_{\pi(i)}(0)$, we have $d(x_i(\cdot))=\tilde x_{\pi(i)}-\tilde x_i+Lk_i$ for some $k_i\in\mathbb Z^3$. Summing over $i$, we find that $W=\sum_ik_i\in\mathbb Z^3$. The vector $W$ is the total homology class of the worldlines. We write $W=(W_1,W_2,W_3)$. For a container, the stochastic area vector of $(\pi,X(\cdot))$ is \begin{equation}\label{eq:id-area} \mathcal A=\frac12\sum_{i=1}^N\int_0^\beta(x_i(s)-x_0)\times\circ\dd x_i(s). \end{equation} For a path of any duration $s>0$ and reference point $a$, we use the vector limit of the dyadic left-point sums $\frac12\sum_{k=0}^{2^\ell-1}(\omega(ks2^{-\ell})-a)\times(\omega((k+1)s2^{-\ell})-\omega(ks2^{-\ell}))$, set to zero where that vector limit does not exist. This is a Borel function of the path and agrees $\mathbb W^s_{y,z}$-almost everywhere, for every $y,z$, with the It\^o and Stratonovich area integrals, whose coordinate integrands and integrators have zero mutual quadratic covariation. To justify the asserted almost-sure limit, let $A_\ell$ be the dyadic sum under a normalized bridge of duration $s$. Given the level-$\ell$ vertices, each midpoint of an interval with endpoints $P,Q$ and duration $h$ is $(P+Q)/2+\xi$, with independent centered Gaussian $\xi$ of covariance $(h/2)I$. Its contribution to $A_{\ell+1}-A_\ell$ is $\tfrac12\xi\times(Q-P)$. Thus $(A_\ell)$ is a martingale and $\E\abs{A_{\ell+1}-A_\ell}^2\le C_{s,y,z}2^{-\ell}$. The summable increment variances give convergence almost surely and in $L^2$; convergence in probability of the left-point sums to the It\^o integral identifies the limit. All stochastic areas below use this convention, and configuration areas are sums of the single-path areas. We write $\mathcal A_n=n\cdot\mathcal A$. Replacing $x_0$ by $x_0'$ changes the right side of \eqref{eq:id-area} by \[ -\frac12(x_0'-x_0)\times\sum_{i=1}^N\bigl(x_i(\beta)-x_i(0)\bigr)=-\frac12(x_0'-x_0)\times\sum_{i=1}^N\bigl(x_{\pi(i)}(0)-x_i(0)\bigr)=0, \] so $\mathcal A$ does not depend on $x_0$. Here we used the fact that the worldlines are closed; the area of an individual path $x_i(\cdot)$, or of a piece of the worldlines on a subinterval of $[0,\beta]$, depends on the origin in general. The vector $\mathcal A$ is the stochastic area estimator of Sindzingre, Klein, and Ceperley \cite{SindzingreKleinCeperley1989}. \begin{lemma}\label{lem:id-moments} For every nonnegative measurable function $G$ of $(\pi,X(\cdot))$, we have \begin{equation}\label{eq:id-domination} Z\,\E G\le\frac1{N!}\sum_{\pi\in S_N}\int_{\mathcal X_N}\int G(\pi,X(\cdot))\,\mathbb W^\beta_{X_0,\pi X_0}(\dd X(\cdot))\dd X_0. \end{equation} On the torus, $\E e^{r\abs{W}}<\infty$ for every $r\ge0$. For a container, $\E\abs{\mathcal A}^2<\infty$. \end{lemma} \begin{proof} Inequality \eqref{eq:id-domination} follows from \eqref{eq:id-path-measure} by dropping the indicator of $\mathcal E_\beta$. On the torus, we use $e^{r\abs W}\le\prod_ie^{r\abs{d(x_i(\cdot))}/L}$ and decompose each torus bridge measure into its lifts as above. On the term in which $d(x_i(\cdot))=d_i$, with $d_i=\tilde x_{\pi(i)}+Lk_i-\tilde x_i$, the integrand is constant, and the mass of that term is $\prod_ip_\beta(d_i)$. Since $r\abs{d}/L\le\abs{d}^2/(8\beta)+2\beta r^2/L^2$, we have \[ p_\beta(d)e^{r\abs d/L}\le e^{2\beta r^2/L^2}(4\pi\beta)^{-3/2}e^{-\abs d^2/(8\beta)}, \] and $\sup_{z\in\R^3}\sum_{k\in\mathbb Z^3}e^{-\abs{z+Lk}^2/(8\beta)}<\infty$. After inserting these bounds into \eqref{eq:id-domination}, the integral over $X_0\in[0,L)^{3N}$ is finite. For a container, write $\mathcal A=\sum_i\mathcal A^{(i)}$, where $\mathcal A^{(i)}$ is the contribution of particle $i$ in \eqref{eq:id-area}, so that $\abs{\mathcal A}^2\le N\sum_i\abs{\mathcal A^{(i)}}^2$. Under $\mathbb W^\beta_{X,Y}$, the paths of different particles are independent, and the normalized law of the path of particle $i$ is that of $s\mapsto x_i+s\beta^{-1}(y_i-x_i)+b(s)$, where $b$ is a Brownian bridge from $0$ to $0$ of duration $\beta$. Expanding \eqref{eq:id-area} in terms of $b$, we find that $\mathcal A^{(i)}$ is the sum of a deterministic vector, linear functionals of $b$, and the stochastic area $(1/2)\int_0^\beta b(s)\times\circ\dd b(s)$. Each of these terms has a finite second moment, bounded uniformly for $x_i,y_i\in\overline U$. Since $p_\beta\le(4\pi\beta)^{-3/2}$ and $U$ is bounded, we conclude from \eqref{eq:id-domination} that $\E\abs{\mathcal A}^2<\infty$. \end{proof} \usedin{the square-integrability assertions in \cref{prop:id-slabs,prop:var-one-slab}.} \section{An exact sub-Gaussian bound for the winding vector}\label{sec:id-subgaussian} In this section and the next, we work on the torus $\Lambda_L$. For $w\in\R^3$ and $X\in\Lambda_L^N$, we write $\tau_wX=(x_1+w,\dots,x_N+w)$. The map $\tau_w$ preserves $\mathcal X_{N,L}$ and Lebesgue measure and commutes with permutations, and $K_s(\tau_wX,\tau_wY)=K_s(X,Y)$ for $s>0$, since the torus bridge measures and the events $\mathcal E_s$ are invariant under a common translation of all particles. The identity below is a path-space form of the common-translation relation behind the winding and momentum formulas of Pollock and Ceperley \cite{PollockCeperley1987,Ceperley1995}. The normalized overlap $\Phi(v)/Z$ is the characteristic function of the total momentum and equals the center-of-mass density-matrix kernel divided by the density in \cite[Theorem~III.1, Eqs.~(52)--(60) in arXiv version~2]{Suto2015Momentum}. We prove the winding moment-generating-function identity directly by shifting the bridges. \begin{theorem}\label{thm:id-subgaussian} Let $N\ge1$, $L>0$, $\beta>0$, and $a>0$ be such that $\mathcal X_{N,L}$ is nonempty. For $v\in\R^3$, let \begin{equation}\label{eq:id-overlap} \Phi(v)=\int_{\mathcal X_{N,L}}K^{\mathrm s}_\beta(X,\tau_{-v}X)\dd X. \end{equation} Then $0\le \Phi(v)\le \Phi(0)=Z$ for every $v\in\R^3$, with $\Phi(v)=Z$ if $v\in L\mathbb Z^3$, and for every $t\in\R^3$, \begin{equation}\label{eq:id-mgf-identity} \E e^{t\cdot W}=\exp\Bigl(\frac{\beta N\abs t^2}{L^2}\Bigr)\frac{\Phi(2\beta t/L)}{Z}. \end{equation} In particular, \begin{equation}\label{eq:id-mgf} \E e^{t\cdot W}\le\exp\Bigl(\frac{\beta N\abs t^2}{L^2}\Bigr), \end{equation} with equality for $t\in(L^2/2\beta)\mathbb Z^3$. The law of $W$ is invariant under $W\mapsto SW$ for every signed permutation matrix $S$; in particular $\E W=0$, $\Cov(W)=\Var(W_1)\,\mathrm{Id}$, and \begin{equation}\label{eq:id-var-bound} \Var(u\cdot W)\le\frac{2\beta N}{L^2}\abs u^2,\qquad u\in\R^3. \end{equation} \end{theorem} \begin{proof} Let $t\in\R^3$ and $v=2\beta t/L$. Fix $\pi\in S_N$ and $X_0\in\mathcal X_{N,L}$, and let $\tilde x_1,\dots,\tilde x_N\in\R^3$ be representatives of the coordinates of $X_0$. For $k=(k_1,\dots,k_N)\in(\mathbb Z^3)^N$, put $d_i=\tilde x_{\pi(i)}+Lk_i-\tilde x_i$ and $\mathbb W_k=\bigotimes_i(\mathrm{pr}_L)_*\mathbb W^\beta_{\tilde x_i,\tilde x_i+d_i}$. By the definition of the torus bridge measures, $\mathbb W^\beta_{X_0,\pi X_0}=\sum_k\mathbb W_k$, and the paths in the term indexed by $k$ satisfy $d(x_i(\cdot))=d_i$. Since $\sum_i(\tilde x_{\pi(i)}-\tilde x_i)=0$, we have $W=\sum_ik_i$ and $t\cdot W=L^{-1}\sum_it\cdot d_i$ on these paths. Completing the square in the exponent of $p_\beta$ gives \[ p_\beta(d)e^{t\cdot d/L}=e^{\beta\abs t^2/L^2}p_\beta(d-v),\qquad d\in\R^3. \] The normalized Brownian bridge from $\tilde x$ to $\tilde x+d$ of duration $\beta$ is the law of $s\mapsto\tilde x+(s/\beta)d+b(s)$, where $b$ is a Brownian bridge from $0$ to $0$. Hence the shear $\omega(s)\mapsto\omega(s)-(s/\beta)v$ maps it onto the normalized bridge from $\tilde x$ to $\tilde x+d-v$. Since $\mathbb W^\beta_{\tilde x,\tilde x+d}$ has mass $p_\beta(d)$, the last display shows that the shear maps $e^{t\cdot d/L}\,\mathbb W^\beta_{\tilde x,\tilde x+d}$ onto $e^{\beta\abs t^2/L^2}\,\mathbb W^\beta_{\tilde x,\tilde x+d-v}$. We apply this shear to every particle, on $\R^3$ and then on $\Lambda_L$, where it maps $X(s)$ to $\tau_{-sv/\beta}X(s)$. It translates all particles by the same vector at each time, and therefore it preserves $\mathcal E_\beta$. Hence \[ \int\mathbf 1_{\mathcal E_\beta}e^{t\cdot W}\dd\mathbb W_k=e^{\beta N\abs t^2/L^2}\,\mathbb W'_k(\mathcal E_\beta),\qquad \mathbb W'_k=\bigotimes_i(\mathrm{pr}_L)_*\mathbb W^\beta_{\tilde x_i,\tilde x_i+d_i-v}. \] As $k$ ranges over $(\mathbb Z^3)^N$, the endpoints $\tilde x_i+d_i-v=\tilde x_{\pi(i)}-v+Lk_i$ range over all lifts of $x_{\pi(i)}-v$, which is the $i$-th coordinate of $\pi\tau_{-v}X_0$. Summing over $k$, we find \[ \int_{\mathcal E_\beta}e^{t\cdot W}\,\mathbb W^\beta_{X_0,\pi X_0}(\dd X(\cdot))=e^{\beta N\abs t^2/L^2}K_\beta(X_0,\pi\tau_{-v}X_0). \] We sum over $\pi$, divide by $Z\,N!$, and integrate over $X_0$; all integrands are nonnegative. By \eqref{eq:id-path-measure} and \eqref{eq:id-killed-kernel}, this gives \eqref{eq:id-mgf-identity}. Next, let $A=K^{\mathrm s}_{\beta/2}\ge0$. Since $\tau_w$ commutes with permutations, $K^{\mathrm s}_s(\tau_wX,\tau_wY)=K^{\mathrm s}_s(X,Y)$. By \cref{lem:id-ck}(1)--(2) and this invariance, \[ \Phi(v)=\iint A(X,Y)A(Y,\tau_{-v}X)\dd Y\dd X=\iint A(X,Y)A(X,\tau_vY)\dd Y\dd X. \] Since $Y\mapsto\tau_vY$ preserves $\mathcal X_{N,L}$ and Lebesgue measure, the Cauchy--Schwarz inequality in $L^2(\dd X\dd Y)$ gives $0\le \Phi(v)\le\iint A(X,Y)^2\dd X\dd Y$. By \cref{lem:id-ck}(1)--(2), the last integral equals $\int K^{\mathrm s}_\beta(X,X)\dd X=\Phi(0)$, which is $Z$ by the description of $\mu$ after \cref{def:id-path-measure}. If $v\in L\mathbb Z^3$, then $\tau_{-v}$ is the identity of $\Lambda_L^N$, and $\Phi(v)=\Phi(0)$. Together with \eqref{eq:id-mgf-identity}, this proves \eqref{eq:id-mgf} and the equality case. Let $S$ be a $3\times3$ signed permutation matrix. The map $x\mapsto Sx$ is an isometry of $\Lambda_L$, it maps Brownian bridges to Brownian bridges, and it preserves $\mathcal X_{N,L}$. Then the measure $\mu$ is invariant under $(\pi,X(\cdot))\mapsto(\pi,SX(\cdot))$, and this map sends $W$ to $SW$. With $S=-\mathrm{Id}$, and since $W$ is integrable by \cref{lem:id-moments}, we obtain $\E W=0$. Further, $\Cov(W)=S\Cov(W)S^{\mathsf T}$ for all such $S$. Invariance under the diagonal sign changes forces the off-diagonal entries of $\Cov(W)$ to vanish, and invariance under coordinate permutations makes the diagonal entries equal. Finally, let $u\in\R^3$. By \cref{lem:id-moments}, the function $r\mapsto\E e^{ru\cdot W}$ is smooth, with derivatives computed under the expectation. By \eqref{eq:id-mgf}, it is bounded by $e^{\beta N\abs u^2r^2/L^2}$. The two functions are equal to one and have derivative zero at $r=0$, since $\E W=0$. Their difference is nonpositive, so its second derivative at $r=0$ is nonpositive, which gives $\E(u\cdot W)^2\le2\beta N\abs u^2/L^2$. This is \eqref{eq:id-var-bound}. \end{proof} \usedin{\cref{thm:main-winding}, \cref{cor:main-winding-scaling}, and \cref{lem:normal-response}; see \cref{sec:stiff-winding}.} The bound \eqref{eq:id-mgf} states that every linear functional $u\cdot W$ with $\abs u=1$ is sub-Gaussian with variance proxy \begin{equation}\label{eq:id-sigma} \sigma^2=\frac{2\beta N}{L^2}=2\beta\rho_LL,\qquad \rho_L=\frac N{L^3}. \end{equation} Here a variance proxy means that $\E e^{t u\cdot W}\le e^{\sigma^2t^2/2}$ for all real $t$; it bounds the variance from above without asserting equality. By \eqref{eq:id-pc}, $\sigma^2$ is also the value of $\Var(W_1)$ that corresponds to $f_{\mathrm s}=1$. We record the standard consequences of \eqref{eq:id-mgf}. \begin{corollary}\label{cor:id-tails} Let $u\in\R^3$ be a unit vector. For all $r\ge0$ and $p>0$, \begin{equation}\label{eq:id-tails} \PP(\abs{u\cdot W}\ge r)\le2e^{-r^2/(2\sigma^2)},\qquad \E\abs{u\cdot W}^p\le p\,(2\sigma^2)^{p/2}\,\Gamma(p/2). \end{equation} In particular $\E(u\cdot W)^4\le16\sigma^4$. If $\Var(W_1)\ge v>0$, then \begin{equation}\label{eq:id-pz} \PP\bigl(\abs{u\cdot W}\ge(v/2)^{1/2}\bigr)\ge\frac{v^2}{64\sigma^4}. \end{equation} \end{corollary} \begin{proof} For $s>0$, Markov's inequality and \eqref{eq:id-mgf} imply that $\PP(u\cdot W\ge r)\le e^{-sr+\sigma^2s^2/2}$, and the choice $s=r/\sigma^2$ gives $e^{-r^2/(2\sigma^2)}$. The same bound holds for $-u\cdot W$, which proves the first inequality in \eqref{eq:id-tails}. Then \[ \E\abs{u\cdot W}^p=\int_0^\infty pr^{p-1}\PP(\abs{u\cdot W}>r)\dd r\le2p\int_0^\infty r^{p-1}e^{-r^2/(2\sigma^2)}\dd r=p\,(2\sigma^2)^{p/2}\Gamma(p/2). \] For \eqref{eq:id-pz}, set $Y=(u\cdot W)^2$. By \cref{thm:id-subgaussian}, we have $\E Y=\Var(W_1)\ge v$. Since $\E[Y\mathbf 1_{\{Y0$. Then for all $k$ and all unit vectors $u$, \begin{equation}\label{eq:id-scale-probability} \PP\bigl(\abs{u\cdot W^{(k)}}\ge(cL_k/2)^{1/2}\bigr)\ge\frac{c^2}{256\lambda^2}, \end{equation} and every subsequential weak limit $X_\infty$ satisfies $\Cov(X_\infty)=v_\infty\mathrm{Id}$ with $c\le v_\infty\le2\lambda$. \end{corollary} The scope of these fluctuation bounds is discussed after \cref{cor:main-winding-scaling}; the hypotheses here do not imply a unique limiting law or a central limit theorem. \begin{proof} Let $\sigma_k^2=2\beta_kN_k/L_k^2$, so that $\sigma_k^2\le2\lambda L_k$. Then \eqref{eq:id-mgf} implies that $\E e^{t\cdot X_k}\le\exp(\sigma_k^2\abs t^2/(2L_k))\le e^{\lambda\abs t^2}$, and the tail bound in \eqref{eq:id-scaled-mgf} follows from \eqref{eq:id-tails}. The tail bound implies that the laws are tight. Since $\E e^{2t\cdot X_k}\le e^{4\lambda\abs t^2}$, the variables $e^{t\cdot X_k}$ are uniformly integrable, and $\E e^{t\cdot X_k}$ converges to $\E e^{t\cdot X_\infty}$ along a weakly convergent subsequence. This proves the bound on the limit. The law of $X_k$ is symmetric by \cref{thm:id-subgaussian}, and so is any weak limit. Under the variance assumption, \eqref{eq:id-pz} with $v=cL_k$ and $\sigma_k^4\le4\lambda^2L_k^2$ implies \eqref{eq:id-scale-probability}. By \cref{thm:id-subgaussian}, $\Cov(X_k)=v_k\mathrm{Id}$ with $v_k=\Var(W^{(k)}_1)/L_k$, and $c\le v_k\le\sigma_k^2/L_k\le2\lambda$ by \eqref{eq:id-var-bound}. By \eqref{eq:id-tails}, the fourth moments of $X_k$ are bounded, and the second moments converge along a weakly convergent subsequence. Then $\Cov(X_\infty)$ is the limit of $v_k\mathrm{Id}$ along this subsequence, and its value lies in $[c,2\lambda]$. \end{proof} \section{Winding and the superfluid fraction}\label{sec:id-winding} In this section, we abbreviate $H=H_{N,L}$, $H(\vartheta)=H_{N,L}(\vartheta)$, and $Z(\vartheta)=\tr e^{-\beta H(\vartheta)}$, so that $Z=Z(0)$. Pollock and Ceperley related the superfluid density to winding fluctuations and momentum-density correlations, through the representation $Z(\vartheta)/Z=\E e^{\mathrm i\vartheta\cdot W}$ of the twisted partition function \cite{PollockCeperley1987,Ceperley1995}. We verify the second-order formulas \eqref{eq:id-pc} and \eqref{eq:id-momentum-identity} for the present hard-core model from the untwisted Feynman--Kac representation and \cref{thm:id-subgaussian}, without using this representation. For $z\in\R^3$, let $(\mathcal T_z\psi)(X)=\psi(\tau_{-z}X)$ on $L^2_{\mathrm{sym}}(\mathcal X_{N,L})$. For an eigenvalue $\lambda$ of $H$, let $E_\lambda$ be its eigenspace, which is finite-dimensional by \cref{lem:id-trace}. \begin{lemma}\label{lem:id-blocks} The following statements hold. \begin{enumerate} \item For every $z\in\R^3$, the operator $\mathcal T_z$ is unitary, commutes with $H$, and maps every eigenspace $E_\lambda$ onto itself. \item For every eigenvalue $\lambda$, there are commuting Hermitian operators $P_{\lambda,1},P_{\lambda,2},P_{\lambda,3}$ on $E_\lambda$ such that $\mathcal T_z|_{E_\lambda}=e^{-\mathrm iz\cdot P_\lambda}$ for $z\in\R^3$, and $P_{\lambda,b}\psi=-\mathrm i\sum_j\partial_{x_{j,b}}\psi$ for $\psi\in E_\lambda$. Moreover $\sum_b\norm{P_{\lambda,b}\psi}^2\le N\lambda\norm\psi^2$. \item For every $\vartheta\in\R^3$, the operator $H(\vartheta)$ maps $E_\lambda$ into itself and acts there as $\lambda+2L^{-1}\vartheta\cdot P_\lambda+NL^{-2}\abs\vartheta^2$. Consequently \begin{equation}\label{eq:id-block-trace} Z(\vartheta)=e^{-\beta N\abs\vartheta^2/L^2}\sum_\lambda e^{-\beta\lambda}\tr e^{-2\beta\vartheta\cdot P_\lambda/L}, \end{equation} where the series converges uniformly on compact sets together with all partial derivatives of order at most two, and $F_{N,L,T}$ is twice continuously differentiable on $\R^3$. \end{enumerate} \end{lemma} \begin{proof} For part (1), the map $\tau_{-z}$ preserves $\mathcal X_{N,L}$ and Lebesgue measure and commutes with permutations, so $\mathcal T_z$ is unitary. It maps smooth functions with compact support in $\mathcal X_{N,L}$ to such functions and preserves the $H^1$ norm and the form $q$. Hence it preserves $\mathcal Q_N$, and $q(\varphi,\mathcal T_z\psi)=q(\mathcal T_{-z}\varphi,\psi)$ for $\varphi,\psi\in\mathcal Q_N$. For $\psi$ in the domain of $H$, this gives $q(\varphi,\mathcal T_z\psi)=\langle\mathcal T_{-z}\varphi,H\psi\rangle=\langle\varphi,\mathcal T_zH\psi\rangle$ for every $\varphi\in\mathcal Q_N$. Then $\mathcal T_z\psi$ lies in the domain of $H$, and $H\mathcal T_z\psi=\mathcal T_zH\psi$. In particular $\mathcal T_z$ maps $E_\lambda$ into itself, and since $\mathcal T_{-z}$ does too, onto itself. For part (2), the translations are strongly continuous, so $z\mapsto\mathcal T_z|_{E_\lambda}$ is a continuous unitary representation of $\R^3$ on the finite-dimensional space $E_\lambda$. For each $b$, the continuous one-parameter group $s\mapsto\mathcal T_{se_b}|_{E_\lambda}$ of unitary matrices is $e^{-\mathrm isP_{\lambda,b}}$ for a Hermitian matrix $P_{\lambda,b}$. These matrices commute, since the groups do, and $\mathcal T_z|_{E_\lambda}=e^{-\mathrm iz\cdot P_\lambda}$. Let $\psi\in E_\lambda$. After extension by zero, $\psi\in H^1(\Lambda_L^N)$, and the difference quotients $s^{-1}(\mathcal T_{se_b}\psi-\psi)$ converge in $L^2$ to $-\sum_j\partial_{x_{j,b}}\psi$. Their limit is also $-\mathrm iP_{\lambda,b}\psi$, which proves the formula for $P_{\lambda,b}$. The pointwise bound $\abs{\sum_j\nabla_j\psi}^2\le N\sum_j\abs{\nabla_j\psi}^2$ gives $\sum_b\norm{P_{\lambda,b}\psi}^2\le Nq[\psi]=N\lambda\norm\psi^2$. For part (3), let $\psi\in E_\lambda$ and $\varphi\in\mathcal Q_N$. By integration by parts for smooth functions with compact support in $\mathcal X_{N,L}$, and by density, $\langle\varphi,-\mathrm i\sum_j\partial_{x_{j,b}}\psi\rangle=\langle-\mathrm i\sum_j\partial_{x_{j,b}}\varphi,\psi\rangle$. Expanding the square in \eqref{eq:id-twisted-form} and using part (2), we find \[ q_\vartheta(\varphi,\psi)=q(\varphi,\psi)+\frac2L\langle\varphi,\vartheta\cdot P_\lambda\psi\rangle+\frac{N\abs\vartheta^2}{L^2}\langle\varphi,\psi\rangle=\Bigl\langle\varphi,\Bigl(\lambda+\frac2L\vartheta\cdot P_\lambda+\frac{N\abs\vartheta^2}{L^2}\Bigr)\psi\Bigr\rangle. \] Hence $\psi$ lies in the domain of $H(\vartheta)$, and $H(\vartheta)$ acts on $E_\lambda$ as stated. Let $B_\lambda(\vartheta)$ be this Hermitian operator on $E_\lambda$. Since $H$ has compact resolvent, $L^2_{\mathrm{sym}}(\mathcal X_{N,L})$ is the orthogonal sum of the spaces $E_\lambda$. The orthogonal sum of the operators $B_\lambda(\vartheta)$ is self-adjoint, and its restriction to finite sums of eigenvectors of $H$ is a core. This restriction is contained in $H(\vartheta)$. Since $H(\vartheta)$ is closed, it also contains the closure of the restriction, which is the orthogonal sum. A self-adjoint operator has no proper symmetric extension, so the orthogonal sum is $H(\vartheta)$. This gives \eqref{eq:id-block-trace}, since $e^{-\beta B_\lambda(\vartheta)}=e^{-\beta\lambda-\beta N\abs\vartheta^2/L^2}e^{-2\beta\vartheta\cdot P_\lambda/L}$. Let $r>0$, and let $\gamma$ be a multi-index with $\abs\gamma\le2$. By part (2) and the Cauchy--Schwarz inequality, $\norm{\vartheta\cdot P_\lambda}\le\abs\vartheta\sqrt{N\lambda}$ for $\vartheta\in\R^3$. Since the matrices $P_{\lambda,b}$ commute, we obtain for $\abs\vartheta\le r$ \[ \Bigl\lvert\partial_\vartheta^\gamma\tr e^{-2\beta\vartheta\cdot P_\lambda/L}\Bigr\rvert\le\dim E_\lambda\Bigl(\frac{2\beta\sqrt{N\lambda}}L\Bigr)^{\abs\gamma}e^{2\beta r\sqrt{N\lambda}/L}\le\dim E_\lambda\Bigl(\frac{2\beta\sqrt{N\lambda}}L\Bigr)^{\abs\gamma}e^{\beta\lambda/2+2\beta Nr^2/L^2}. \] Since $\sum_\lambda(1+\lambda)\dim E_\lambda\,e^{-\beta\lambda/2}=\tr\bigl((1+H)e^{-\beta H/2}\bigr)<\infty$ by \cref{lem:id-trace}, the series in \eqref{eq:id-block-trace} and its partial derivatives of order at most two converge uniformly on compact subsets of $\R^3$. Hence $Z$ is twice continuously differentiable, and since $Z(\vartheta)>0$, so is $F_{N,L,T}=-T\log Z$. \end{proof} \begin{proposition}\label{prop:id-winding} Let $\langle P_1^2\rangle=Z^{-1}\sum_\lambda e^{-\beta\lambda}\tr P_{\lambda,1}^2$, with $P_\lambda$ as in \cref{lem:id-blocks}. Then \begin{equation}\label{eq:id-var-identity} \Var(W_1)=\frac{2\beta N}{L^2}-\frac{4\beta^2}{L^2}\langle P_1^2\rangle, \end{equation} \begin{equation}\label{eq:id-pc} f_{\mathrm s}(N,L,T)=\frac{TL^2}{2N}\Var(W_1), \end{equation} \begin{equation}\label{eq:id-momentum-identity} f_{\mathrm s}(N,L,T)=1-\frac{2}{NT}\langle P_1^2\rangle, \end{equation} and $0\le f_{\mathrm s}(N,L,T)\le1$. \end{proposition} Since $\langle P_1\rangle=Z^{-1}\sum_\lambda e^{-\beta\lambda}\tr P_{\lambda,1}=0$ (see the proof), $\langle P_1^2\rangle$ is the variance of the first component of the total momentum in the Gibbs state, and \eqref{eq:id-momentum-identity} reads $f_{\mathrm s}=1-2\Var(P_1)/(NT)$. In physical units, it states that the normal fraction $1-f_{\mathrm s}$ equals $\Var(P_{1})/(mNT)$, where $P_1$ is the physical total momentum. This is the total-momentum fluctuation formula for the normal fraction; see \cite{PollockCeperley1987}. \begin{proof} Let $v\in\R^3$. As in the proof of \cref{lem:id-ck}(3), let $A=e^{-\beta H^{\mathrm{lab}}/2}P_{\mathrm{sym}}$, and extend $\mathcal T_{-v}$ to $L^2(\mathcal X_{N,L})$ by the same formula. Since $(\mathcal T_{-v}\psi)(Y)=\psi(\tau_vY)$ and $\tau_{-v}$ preserves $\mathcal X_{N,L}$ and Lebesgue measure, \cref{prop:id-fk}(1) shows that the Hilbert--Schmidt operators $A$ and $A\mathcal T_{-v}$ have the kernels $K^{\mathrm s}_{\beta/2}(X,Y)$ and $K^{\mathrm s}_{\beta/2}(X,\tau_{-v}Y)$. Since $\mathcal T_{-v}$ commutes with $P_{\mathrm{sym}}$, \cref{lem:id-ck}(2) gives \[ \tr\bigl(e^{-\beta H}\mathcal T_{-v}\bigr)=\tr\bigl(A\cdot A\mathcal T_{-v}\bigr)=\iint K^{\mathrm s}_{\beta/2}(X,Y)K^{\mathrm s}_{\beta/2}(Y,\tau_{-v}X)\dd Y\dd X=\Phi(v), \] with $\Phi$ as in \eqref{eq:id-overlap}. On the other hand, \cref{lem:id-blocks}(1)--(2) give $\tr(e^{-\beta H}\mathcal T_{-v})=\sum_\lambda e^{-\beta\lambda}\tr e^{\mathrm iv\cdot P_\lambda}$. With $t=se_1$ and $v=2\beta se_1/L$, \eqref{eq:id-mgf-identity} becomes \[ Z\,\E e^{sW_1}=e^{\beta Ns^2/L^2}\sum_\lambda e^{-\beta\lambda}\tr e^{2\mathrm i\beta sP_{\lambda,1}/L},\qquad s\in\R. \] We differentiate twice at $s=0$. On the left, this is justified by \cref{lem:id-moments}. On the right, the $j$-th derivative of the term indexed by $\lambda$ is bounded by $\dim E_\lambda\,e^{-\beta\lambda}(2\beta\sqrt{N\lambda}/L)^j$ by \cref{lem:id-blocks}(2), and these bounds are summable by \cref{lem:id-trace}. The first derivatives give $\E W_1=(2\mathrm i\beta/L)\langle P_1\rangle$, where $\langle P_1\rangle=Z^{-1}\sum_\lambda e^{-\beta\lambda}\tr P_{\lambda,1}$. Both $\E W_1$ and $\langle P_1\rangle$ are real, so $\langle P_1\rangle=0$. The second derivatives then give $\E W_1^2=2\beta NL^{-2}-4\beta^2L^{-2}\langle P_1^2\rangle$, which is \eqref{eq:id-var-identity}, since $\E W_1=0$. On the operator side, \eqref{eq:id-block-trace} with $\vartheta=\theta e_1$ may be differentiated term by term by \cref{lem:id-blocks}(3). Since $\langle P_1\rangle=0$, the second derivative of $\log Z(\theta e_1)$ at $\theta=0$ equals $-2\beta NL^{-2}+4\beta^2L^{-2}\langle P_1^2\rangle$. Multiplying by $-T$, we obtain $\partial_\theta^2F_{N,L,T}(0)=2NL^{-2}-4\beta L^{-2}\langle P_1^2\rangle=T\Var(W_1)$, and \eqref{eq:id-pc} follows from \eqref{eq:id-fs}. Then \eqref{eq:id-momentum-identity} follows from \eqref{eq:id-pc} and \eqref{eq:id-var-identity}. Finally, $f_{\mathrm s}\ge0$ by \eqref{eq:id-pc}, and $f_{\mathrm s}\le1$ by \eqref{eq:id-momentum-identity}, since $\tr P_{\lambda,1}^2\ge0$. \end{proof} \begin{remark}\label{rem:id-units} In general units, with mass $m$, the reduced Planck constant $\hbar$, and $\kappa=\hbar^2/(2m)$, the Brownian motions have generator $\kappa\Delta$ on the time interval $[0,\beta]$, and the identities of this section read \[ f_{\mathrm s}=\frac{mL^2}{\hbar^2N}F''_{N,L,T}(0)=\frac{mL^2}{\beta\hbar^2N}\Var(W_1)=1-\frac{\beta}{mN}\Var(\hbar P_1),\qquad \Var(u\cdot W)\le\frac{2\beta\kappa N}{L^2}\abs u^2. \] The physical interpretation of the twist response is given in \cref{sec:intro-superfluidity}; the conversion to general units is recorded in \cref{rem:intro-units}. For the rotational identities of \cref{sec:id-rotation}, the factor $T$ in front of $\Cov(\mathcal A)$ becomes $4m^2/(\beta\hbar^2)$. \end{remark} \section{Rotation and the stochastic area}\label{sec:id-rotation} In this section, $U$ is a container, $n$ is a unit vector, $x_0\in\R^3$, and $H=H_N(U)$. We do not assume that $U$ has any rotational symmetry. For a container, the area functional and the angular momentum are related by the magnetic completion of the square. Let $\bm\omega\in\R^3$, and let $\mathsf R(X)$ be the symmetric matrix with entries $\mathsf R_{bc}(X)=m\sum_i(\abs{y_i}^2\delta_{bc}-y_{i,b}y_{i,c})$, where $y_i=x_i-x_0$ and $m=1/2$, so that $I_{\mathrm{cl},bc}(x_0)=\langle\mathsf R_{bc}\rangle$. Expanding the square and using \cref{lem:id-angular}, we find for $\psi\in\mathcal Q_N$ that \begin{equation}\label{eq:id-magnetic-completion} \sum_{i=1}^N\Bigl\lVert\bigl(-\mathrm i\nabla_i-\tfrac12\bm\omega\times y_i\bigr)\psi\Bigr\rVert^2=q[\psi]-\langle\psi,\bm\omega\cdot\mathcal L\psi\rangle+\tfrac12\bigl\langle\psi,(\bm\omega\cdot\mathsf R\bm\omega)\psi\bigr\rangle, \end{equation} because $\abs{\bm\omega\times y}^2=\abs{\bm\omega}^2\abs y^2-(\bm\omega\cdot y)^2$. We denote the operator associated with the left side of \eqref{eq:id-magnetic-completion} by $H^{\mathrm{mag}}(\bm\omega)$, so that $H^{\mathrm{mag}}(\bm\omega)=H-\bm\omega\cdot\mathcal L+(1/2)\bm\omega\cdot\mathsf R\bm\omega$. It is the Hamiltonian of the particles in the vector potential $\bm\omega\times(x-x_0)/2$, which corresponds to the uniform magnetic field $\bm\omega$. We write $G(\bm\omega)=-T\log\tr e^{-\beta H^{\mathrm{mag}}(\bm\omega)}$. \begin{proposition}\label{prop:id-rotation} Let $U$ be a container, $n$ a unit vector, and $x_0\in\R^3$. Then the following statements hold. \begin{enumerate} \item For every $\bm\omega\in\R^3$, \begin{equation}\label{eq:id-rot-path} \begin{aligned} \frac{\tr e^{-\beta(H-\bm\omega\cdot\mathcal L)}}{Z}&=\E\exp\Bigl(-\mathrm i\bm\omega\cdot\mathcal A+\frac12\int_0^\beta\bm\omega\cdot\mathsf R(X(s))\bm\omega\dd s\Bigr),\\ \frac{\tr e^{-\beta H^{\mathrm{mag}}(\bm\omega)}}{Z}&=\E e^{-\mathrm i\bm\omega\cdot\mathcal A}. \end{aligned} \end{equation} The functions $F^{\mathrm{rot}}$ and $G$ are twice continuously differentiable on $\R^3$. \item The function $F^{\mathrm{rot}}$ is concave on $\R^3$. In particular, $I(x_0)\ge0$ in the sense of symmetric matrices, and $I(n,x_0)\ge0$. \item We have $\E\mathcal A=0$ and \begin{equation}\label{eq:id-area-formula} \begin{aligned} I_{\mathrm{cl},bc}(x_0)-I_{bc}(x_0)&=\partial_{\omega_b}\partial_{\omega_c}G(0)=T\Cov(\mathcal A)_{bc},\\ I_{\mathrm{cl}}(n,x_0)-I(n,x_0)&=T\Var(\mathcal A_n). \end{aligned} \end{equation} In particular, $I_{\mathrm{cl}}-I$ does not depend on $x_0$. \item We have $0\le I(n,x_0)\le I_{\mathrm{cl}}(n,x_0)$, and $0\le I(x_0)\le I_{\mathrm{cl}}(x_0)$ in the sense of symmetric matrices. Further, $\tr e^{-\beta H^{\mathrm{mag}}(\bm\omega)}\le Z$, so that $G(\bm\omega)\ge G(0)$, for every $\bm\omega\in\R^3$. \end{enumerate} \end{proposition} Part (2) is the standard concavity of a free energy in a parameter that enters the Hamiltonian linearly. Formula \eqref{eq:id-area-formula} is the stochastic area estimator of the moment of inertia introduced by Sindzingre, Klein, and Ceperley \cite{SindzingreKleinCeperley1989}; the proof below verifies it, with its constant, for the hard-core gas with hard walls. The inequality $\tr e^{-\beta H^{\mathrm{mag}}(\bm\omega)}\le Z$ is a diamagnetic trace bound, proved below from the Feynman--Kac representation; see \cite{Simon1976} for ground-state diamagnetism of spinless bosons and \cite[Theorems~2.3--2.5]{AvronHerbstSimon1978} for magnetic semigroup domination. \begin{proof} For part (1), by \cref{lem:id-angular} and \eqref{eq:id-magnetic-completion}, we have $H-\bm\omega\cdot\mathcal L=H^{\mathrm{mag}}(\bm\omega)+V_{\bm\omega}$ with the bounded symmetric potential $V_{\bm\omega}(X)=-(1/2)\bm\omega\cdot\mathsf R(X)\bm\omega$. We apply \cref{prop:id-fk}(2) with the affine, divergence-free vector field $\mathsf a(x)=\bm\omega\times(x-x_0)/2$. Since $(\bm\omega\times y)\cdot\circ\dd y=\bm\omega\cdot(y\times\circ\dd y)$, the phase in \eqref{eq:id-fk-magnetic} is \[ -\frac{\mathrm i}2\sum_{i=1}^N\int_0^\beta(\bm\omega\times y_i(s))\cdot\circ\dd y_i(s)=-\mathrm i\bm\omega\cdot\mathcal A. \] Inserting this into the trace formula of \cref{lem:id-ck}(3) and comparing with \eqref{eq:id-path-measure} yields \eqref{eq:id-rot-path}. Let $g(\bm\omega)$ denote the right side of the first identity in \eqref{eq:id-rot-path}, and write $M=\int_0^\beta\mathsf R(X(s))\dd s$. The matrix $M$ is positive semidefinite and bounded by $\beta mNR_0^2$ in operator norm, and $\E\abs{\mathcal A}^2<\infty$ by \cref{lem:id-moments}. The second derivatives in $\bm\omega$ of the integrand are bounded by $((\abs{\mathcal A}+\norm M\abs{\bm\omega})^2+\norm M)e^{\norm M\abs{\bm\omega}^2/2}$, which is integrable uniformly for $\bm\omega$ in compact sets. Then $g$ is twice continuously differentiable, and we may differentiate under the expectation. The function $g$ is positive, since it is a ratio of traces of positive operators, and $F^{\mathrm{rot}}=F^{\mathrm{rot}}(0)-T\log g$ is twice continuously differentiable. The same argument applies to $G$. For part (3), we first note that $\partial_{\omega_b}g(0)=-\mathrm i\E\mathcal A_b$. This number is real because $g$ is real, and $\E\mathcal A_b$ is real, so $\E\mathcal A_b=0$. Then \[ \partial_{\omega_b}\partial_{\omega_c}\log g(0)=\partial_{\omega_b}\partial_{\omega_c}g(0)=-\E[\mathcal A_b\mathcal A_c]+\E M_{bc}. \] By \cref{lem:id-marginal} and Fubini's theorem, $\E M_{bc}=\beta\langle\mathsf R_{bc}\rangle=\beta I_{\mathrm{cl},bc}(x_0)$. Multiplying by $-T$ gives $\partial_{\omega_b}\partial_{\omega_c}F^{\mathrm{rot}}(0)=T\Cov(\mathcal A)_{bc}-I_{\mathrm{cl},bc}(x_0)$, which is the first claim of \eqref{eq:id-area-formula} after recalling \eqref{eq:id-inertia}. The same computation applied to the second identity in \eqref{eq:id-rot-path} yields $\partial_{\omega_b}\partial_{\omega_c}G(0)=T\Cov(\mathcal A)_{bc}$. Contracting with $n$ gives the second identity in \eqref{eq:id-area-formula}. The vector $\mathcal A$ does not depend on $x_0$, as shown below \eqref{eq:id-area}, and neither does $I_{\mathrm{cl}}-I$. For part (2), let $\sigma=\sum_k\lambda_k\lvert f_k\rangle\langle f_k\rvert$ be a symmetric density matrix of finite kinetic energy: $f_k\in\mathcal Q_N$ when $\lambda_k>0$, and $\sum_k\lambda_kq[f_k]<\infty$. By \eqref{eq:id-angular-bound}, $\abs{\langle f_k,\bm\omega\cdot\mathcal Lf_k\rangle}\le\abs{\bm\omega}\sqrt NR_0\,q[f_k]^{1/2}$, and $\sum_k\lambda_kq[f_k]^{1/2}\le(\sum_k\lambda_kq[f_k])^{1/2}$, so the series $\sum_k\lambda_k\langle f_k,\bm\omega\cdot\mathcal Lf_k\rangle$ converges absolutely. We use the Gibbs variational principle \cite[Lemma~3.1]{FrankLieb2012} for the self-adjoint operators of the forms, with energies given by the forms. For $H$, it gives $S(\sigma)\le\beta\sum_k\lambda_kq[f_k]+\log Z<\infty$, where $S(\sigma)=-\tr\sigma\log\sigma$. For $H-\bm\omega\cdot\mathcal L$, it gives \[ F^{\mathrm{rot}}(\bm\omega)\le\mathcal F_{\bm\omega}(\sigma)=\sum_k\lambda_k\bigl(q[f_k]-\langle f_k,\bm\omega\cdot\mathcal Lf_k\rangle\bigr)-TS(\sigma),\qquad\bm\omega\in\R^3, \] with equality for the Gibbs state $\Gamma_{\bm\omega}$ of $H-\bm\omega\cdot\mathcal L$. This state has finite kinetic energy. Indeed, by \cref{lem:id-angular}, $H-\bm\omega\cdot\mathcal L$ has an orthonormal basis of eigenvectors $\phi_k\in\mathcal Q_N$, with eigenvalues $E_k(\bm\omega)$ that are bounded below and satisfy $\sum_k\abs{E_k(\bm\omega)}e^{-\beta E_k(\bm\omega)}<\infty$, and \eqref{eq:id-klmn} gives $q[\phi_k]\le2E_k(\bm\omega)+\abs{\bm\omega}^2NR_0^2$. Thus $F^{\mathrm{rot}}(\bm\omega)$ is the minimum of $\mathcal F_{\bm\omega}(\sigma)$ over the symmetric density matrices of finite kinetic energy, a set that does not depend on $\bm\omega$. Each function $\bm\omega\mapsto\mathcal F_{\bm\omega}(\sigma)$ is affine, and $F^{\mathrm{rot}}$, as their infimum, is concave. By part (1), it is twice continuously differentiable, and its Hessian at $\bm\omega=0$ is negative semidefinite. By \eqref{eq:id-inertia}, this means that $I(x_0)\ge0$, and then $I(n,x_0)\ge0$. For part (4), the inequality $I\ge0$ follows from part (2), and $I\le I_{\mathrm{cl}}$ follows from part (3), since covariance matrices are positive semidefinite. By the second identity in \eqref{eq:id-rot-path}, we have $\tr e^{-\beta H^{\mathrm{mag}}(\bm\omega)}\le Z\,\E\abs{e^{-\mathrm i\bm\omega\cdot\mathcal A}}=Z$. \end{proof} \section{Imaginary-time slabs}\label{sec:id-slabs} In this section, $\mathcal X_N$ is either $\mathcal X_{N,L}$ or $\mathcal X_N(U)$. Fix a slab length $\delta\in(0,\beta]$, and let \[ n_\beta=\lfloor\beta/\delta\rfloor\ge1,\qquad r_\beta=\beta-n_\beta\delta\in[0,\delta). \] We cut the interval $[0,\beta]$ at the times $\tau_j=j\delta$, $0\le j\le n_\beta$. This produces $n_\beta$ full slabs $[\tau_{j-1},\tau_j]$ of length $\delta$ and, if $r_\beta>0$, a remainder slab $[n_\beta\delta,\beta]$ of length $r_\beta$. Let $J$ be the number of slabs, so that $J=n_\beta$ if $r_\beta=0$ and $J=n_\beta+1$ otherwise, and let $s_1,\dots,s_J$ be their lengths. \paragraph{The marked slab measure.} The construction uses the semigroup and Bose symmetrization identities of \cref{lem:id-ck}; for functional-integration background, see \cite{Ginibre1971,Simon2005}. Its one-slab marginal is obtained from \cite[Proposition~5.1]{OpenAI2026BEC} by integrating out the auxiliary configuration variable; we give the multiple-cut representation needed here. The slab decomposition is described on an extended space. Its coordinates are cut configurations $Y_0,\dots,Y_{J-1}\in\mathcal X_N$, with the convention $Y_J=Y_0$, and, for each slab $1\le j\le J$, a permutation $\pi_j\in S_N$ and a configuration path $X^{(j)}(\cdot)\in C([0,s_j];\mathbb X^N)$. We define the measure \begin{equation}\label{eq:id-marked} \bar\mu=\frac1{Z\,(N!)^J}\prod_{j=1}^J\Bigl(\mathbf 1_{\mathcal E_{s_j}}(X^{(j)}(\cdot))\,\mathbb W^{s_j}_{Y_{j-1},\pi_jY_j}(\dd X^{(j)}(\cdot))\Bigr)\prod_{j=0}^{J-1}\dd Y_j, \end{equation} summed over $(\pi_1,\dots,\pi_J)$ with counting measure. Under $\bar\mu$, the path of slab $j$ runs from the cut $Y_{j-1}$ to the cut $\pi_jY_j$, so that particle $i$ of slab $j$ ends at particle $\pi_j(i)$ of the cut $Y_j$. The concatenation map $\mathrm{Cat}$ sends the slab data to a pair $(\pi,X(\cdot))$ as follows. Set $\sigma_0=\mathrm{id}$ and $\sigma_j=\pi_j\circ\dots\circ\pi_1$, and define \begin{equation}\label{eq:id-concatenation-map} x_i(\tau)=X^{(j)}_{\sigma_{j-1}(i)}(\tau-\tau_{j-1})\quad\text{for }\tau\in[\tau_{j-1},\tau_{j-1}+s_j],\qquad \pi=\sigma_J, \end{equation} where $\tau_J=\beta$. Then particle $i$ follows the path of slab $1$ that starts at the $i$-th particle of $Y_0$, and at every cut it continues with the path of the next slab that starts where it arrived. Let $\mathcal F_{\mathrm{cut}}$ be the $\sigma$-algebra generated by $Y_0,\dots,Y_{J-1}$. \begin{proposition}\label{prop:id-slabs} The following statements hold. \begin{enumerate} \item The measure $\bar\mu$ is a probability measure. The cut configurations $(Y_0,\dots,Y_{J-1})\in\mathcal X_N^J$ have the density \begin{equation}\label{eq:id-cut-density} \rho_{\mathrm{cut}}(Y_0,\dots,Y_{J-1})=\frac1Z\prod_{j=1}^JK^{\mathrm s}_{s_j}(Y_{j-1},Y_j). \end{equation} \item Conditionally on $\mathcal F_{\mathrm{cut}}$, the slab data $(\pi_j,X^{(j)}(\cdot))$, $1\le j\le J$, are independent, and the conditional law of $(\pi_j,X^{(j)}(\cdot))$ is \begin{equation}\label{eq:id-slab-law} \nu_{s_j}(\cdot\mid Y_{j-1},Y_j)=\frac1{N!\,K^{\mathrm s}_{s_j}(Y_{j-1},Y_j)}\sum_{\pi_j\in S_N}\mathbf 1_{\mathcal E_{s_j}}\,\mathbb W^{s_j}_{Y_{j-1},\pi_jY_j}, \end{equation} which depends only on $s_j$, $Y_{j-1}$, and $Y_j$. This formula is used where $K^{\mathrm s}_{s_j}(Y_{j-1},Y_j)>0$; elsewhere choose an arbitrary probability law. The exceptional cut tuples have $\rho_{\mathrm{cut}}$-measure zero. \item The image of $\bar\mu$ under $\mathrm{Cat}$ is the Bose path measure $\mu$. \item If $\beta>\delta$, then for every $1\le j\le n_\beta$, the joint law of $(Y_{j-1},Y_j,\pi_j,X^{(j)}(\cdot))$ is \begin{equation}\label{eq:id-slab-marginal} \frac1{Z\,N!}\,K^{\mathrm s}_{\beta-\delta}(Y_j,Y_{j-1})\,\mathbf 1_{\mathcal E_\delta}(X^{(j)}(\cdot))\,\mathbb W^\delta_{Y_{j-1},\pi_jY_j}(\dd X^{(j)}(\cdot))\dd Y_{j-1}\dd Y_j, \end{equation} which does not depend on $j$. \item On the torus, let $O^W_j=L^{-1}\sum_id(X^{(j)}_i(\cdot))\in\R^3$. For a container, let $O^{\mathcal A}_j=(1/2)\sum_i\int_0^{s_j}(X^{(j)}_i(s)-x_0)\times\circ\dd X^{(j)}_i(s)$. Then $\bar\mu$-almost surely, \begin{equation}\label{eq:id-additivity} W\circ\mathrm{Cat}=\sum_{j=1}^JO^W_j,\qquad \mathcal A\circ\mathrm{Cat}=\sum_{j=1}^JO^{\mathcal A}_j, \end{equation} and every $O^W_j$ and $O^{\mathcal A}_j$ has finite second moment under $\bar\mu$. \end{enumerate} \end{proposition} \begin{proof} For fixed cuts, the total mass of the slab-$j$ factor in \eqref{eq:id-marked}, summed over $\pi_j$, is $N!\,K^{\mathrm s}_{s_j}(Y_{j-1},Y_j)$ by \eqref{eq:id-killed-kernel}. Integrating out the slab data in \eqref{eq:id-marked} gives \eqref{eq:id-cut-density}, and the product form of \eqref{eq:id-marked} implies part (2). Integrating \eqref{eq:id-cut-density} over $Y_1,\dots,Y_{J-1}$ with \eqref{eq:id-ck} leaves $Z^{-1}K^{\mathrm s}_\beta(Y_0,Y_0)$, whose integral is one by \eqref{eq:id-trace-formula}. This proves part (1). For part (3), fix $(\pi_1,\dots,\pi_J)$ and push the corresponding term of \eqref{eq:id-marked} forward under $\mathrm{Cat}$. Consider first the two slabs $j=1,2$. By \eqref{eq:id-concatenation-map}, particle $i$ follows $X^{(1)}_i$ from $y_{0,i}$ to $y_{1,\pi_1(i)}$ and then $X^{(2)}_{\pi_1(i)}$ from $y_{1,\pi_1(i)}$ to $y_{2,\sigma_2(i)}$. The change of variables $z_i=y_{1,\pi_1(i)}$ preserves Lebesgue measure on $\mathcal X_N$, and \eqref{eq:id-concatenation} applied to each particle shows that integrating over $Y_1$ produces the bridge measure of duration $s_1+s_2$ from $Y_0$ to $\sigma_2Y_2$. Since $\mathcal X_N$ is invariant under permutations, the configuration path of slab $2$ stays in $\mathcal X_N$ if and only if its relabeling by $\pi_1$ does; the killing indicators combine into $\mathbf 1_{\mathcal E_{s_1+s_2}}$ of the concatenated path. Repeating this for the remaining cuts, we conclude that the image of the term indexed by $(\pi_1,\dots,\pi_J)$ is \[ \frac1{Z\,(N!)^J}\,\mathbf 1_{\mathcal E_\beta}(X(\cdot))\,\mathbb W^\beta_{Y_0,\sigma_JY_0}(\dd X(\cdot))\dd Y_0 \] at $\pi=\sigma_J$. For each $\pi\in S_N$, exactly $(N!)^{J-1}$ tuples $(\pi_1,\dots,\pi_J)$ satisfy $\pi_J\circ\dots\circ\pi_1=\pi$, since $\pi_1,\dots,\pi_{J-1}$ are arbitrary and $\pi_J$ is then determined. Summing over the tuples gives \eqref{eq:id-path-measure}. For part (4), note that $\beta>\delta$ implies $J\ge2$. In \eqref{eq:id-marked}, we integrate out the slab data of all slabs other than $j$, which produces the factors $K^{\mathrm s}_{s_i}(Y_{i-1},Y_i)$ for $i\neq j$, and then the cuts other than $Y_{j-1}$ and $Y_j$. By \eqref{eq:id-ck}, applied around the cycle of cuts from $Y_j$ back to $Y_{j-1}$, the product of these kernels integrates to $K^{\mathrm s}_{\beta-\delta}(Y_j,Y_{j-1})$. The result is \eqref{eq:id-slab-marginal}. For part (5), the lifted displacement is additive under concatenation of paths, and for each $j$ the map $i\mapsto\sigma_{j-1}(i)$ is a bijection. Then \[ \sum_{i=1}^Nd(x_i(\cdot))=\sum_{i=1}^N\sum_{j=1}^Jd\bigl(X^{(j)}_{\sigma_{j-1}(i)}(\cdot)\bigr)=\sum_{j=1}^J\sum_{i=1}^Nd(X^{(j)}_i(\cdot)). \] The Stratonovich integral in \eqref{eq:id-area} is additive over the subintervals $[\tau_{j-1},\tau_{j-1}+s_j]$, and the same reindexing proves the second identity in \eqref{eq:id-additivity}. For the second moments, we argue as in the proof of \cref{lem:id-moments}. If $J=1$, then $O^W_1=W\circ\mathrm{Cat}$ and $O^{\mathcal A}_1=\mathcal A\circ\mathrm{Cat}$, and the claim is \cref{lem:id-moments}. If $J\ge2$, then the law of slab $j$ with its cuts is given by the analog of \eqref{eq:id-slab-marginal} with $\delta$ replaced by $s_j$. In this formula, $K^{\mathrm s}_{\beta-s_j}$ is bounded by the supremum of the free heat kernel at time $\beta-s_j>0$, which is finite, and dropping the killing indicator leaves a free Gaussian integral of the same type as in \cref{lem:id-moments}, which is finite. \end{proof} \usedin{\cref{lem:thermal-slab,prop:var-one-slab}.} We record three comments on \cref{prop:id-slabs}. First, the labels of the cuts in \eqref{eq:id-marked} are assigned independently at each cut: the density \eqref{eq:id-cut-density}, which contains a symmetrized kernel for every slab, is the law of the cut configurations after an independent uniform relabeling of the particles at each cut. It is not the joint law of $X(\tau_0),\dots,X(\tau_{J-1})$ under $\mu$ with the labels followed continuously from time $0$; in that labeling, the internal factors would be the labeled kernels $K_{s_j}$ and only the closing factor would be symmetrized. Part (3) shows that the relabeled description has the correct image, and the observables $W$ and $\mathcal A$ do not depend on the labels. Second, conditioning on $\mathcal F_{\mathrm{cut}}$ fixes the cut configurations but not the permutations $\pi_j$, the cycle structure of $\pi$, or the winding vector. Third, $O^W_j$ is not an integer in general: with representatives $\tilde y$ in a fundamental cell, $O^W_j$ equals $L^{-1}\sum_i(\tilde y_{j,i}-\tilde y_{j-1,i})$ plus an integer vector, and the noninteger part is a function of the cuts. Similarly, replacing $x_0$ by $x_0'$ in $O^{\mathcal A}_j$ adds $-(1/2)(x_0'-x_0)\times\sum_i(y_{j,i}-y_{j-1,i})$, a function of the cuts that cancels in the sum \eqref{eq:id-additivity}. Neither change affects conditional variances given $\mathcal F_{\mathrm{cut}}$. Using conditional independence, additivity, and the law of total variance, we obtain the following bound. The kernel form used below is proved in \cref{lem:var-slabs,prop:var-one-slab}. \begin{lemma}\label{lem:id-slab-variance} Let $g$ be a real measurable function of $(Y,Y',\pi',X'(\cdot))\in\mathcal X_N\times\mathcal X_N\times S_N\times C([0,\delta];\mathbb X^N)$, and set $O_j=g(Y_{j-1},Y_j,\pi_j,X^{(j)}(\cdot))$ for $1\le j\le n_\beta$. Let $O_{\mathrm{rem}}$ be a function of the cuts and, if $r_\beta>0$, of the remainder slab. Suppose that $O_1,\dots,O_{n_\beta}$ and $O_{\mathrm{rem}}$ have finite second moments under $\bar\mu$, and let $O=\sum_{j=1}^{n_\beta}O_j+O_{\mathrm{rem}}$. Then \begin{equation}\label{eq:id-slab-variance} \Var_{\bar\mu}(O)\ge\sum_{j=1}^{n_\beta}\E_{\bar\mu}\Var_{\bar\mu}(O_j\mid\mathcal F_{\mathrm{cut}})=n_\beta\,\E_{\bar\mu}\Var_{\bar\mu}(O_1\mid Y_0,Y_1), \end{equation} where $\Var_{\bar\mu}(O_1\mid Y_0,Y_1)$ is the variance of $g(Y_0,Y_1,\cdot)$ under $\nu_\delta(\cdot\mid Y_0,Y_1)$. \end{lemma} \begin{proof} By \cref{prop:id-slabs}(2), conditional variances add given $\mathcal F_{\mathrm{cut}}$. Dropping the nonnegative remainder variance and applying the law of total variance, we obtain the inequality in \eqref{eq:id-slab-variance}. Each full-slab conditional law depends only on its two cuts. If $n_\beta\ge2$, then $\beta>\delta$, and \cref{prop:id-slabs}(4) implies that the expected conditional variance is the same for every full slab. For $n_\beta=1$, the equality is immediate. \end{proof} For winding and area, the additivity and square integrability required here are given in \cref{prop:id-slabs}(5). Their one-slab kernel bounds are combined in \cref{prop:var-one-slab}. \chapter{A normal phase at high temperature}\label{sec:normal} \Cref{lem:normal-ratio} adapts the insertion argument in the proof of \cite[Proposition~3.5]{OpenAI2026BEC}, which combines the normalized insertion orbital of \cite[Lemma~3.4]{OpenAI2026BEC} and \cite[Lemma~3.3]{OpenAI2026HS}, a refinement of the nearest-neighbor construction of \cite[Section~3.2 in arXiv version~1]{LiebSeiringerYngvason2000}, with a Fourier frame. That proof applies Parseval's identity to all frequencies and then restricts the sum to low ones; we keep all frequencies. Canonical ratio estimates for integrable repulsive pair potentials are proved in \cite[Lemma~3.4 in arXiv version~3]{AdamsCollevecchioKonig2011}; their bound contains the integral of the pair potential, which is infinite for a hard core. \Cref{lem:normal-cycles} is the inequality form of Ueltschi's canonical cycle formula \cite[Eq.~(2.8) in arXiv version~2]{Ueltschi2006a}. The Gaussian winding-sector weights of \cref{lem:normal-sectors} appear for negative chemical potential in the proof of \cite[Theorem~2 in arXiv version~2]{Ueltschi2006a}, which establishes the absence of infinite cycles and of off-diagonal long-range order; for the canonical free gas, cycle winding weights are computed in \cite[equations~(4)--(7) in the preprint version]{HolzmannKrauth1999}. Sütő related cycle percolation to condensation in the ideal and mean-field gases \cite{Suto1993,Suto2002}. Seiringer and Ueltschi bounded the density in terms of the activity and deduced the decay of off-diagonal correlations above an explicit temperature close to $T_{\mathrm c}$, in the grand-canonical ensemble on domains with Dirichlet boundary conditions \cite{SeiringerUeltschi2009}. The mechanism, suppression of long cycles by a fugacity below one, is that of the grand-canonical results of Ueltschi and of Seiringer and Ueltschi. Throughout this chapter, we work on the torus $\Lambda_L$, write $V=L^3$, fix $a,\beta>0$, and set $\lambda_\beta=(4\pi\beta)^{1/2}$. For $j\ge1$ such that the domain $\mathcal X_{j,L}$ of \eqref{eq:id-domains} is nonempty, let $Z_j=\tr e^{-\beta H_{j,L}}$ be the partition function of $j$ particles at the same $L$, $a$, and $\beta$. We regard $\mathcal X_{0,L}$ as a point and set $Z_0=1$. We write $q_j$ and $\mathcal Q_j$ for the form \eqref{eq:id-form} and its domain for $j$ particles, and $H^{\mathrm{lab}}_j$ for the labeled operator. Let $N\ge1$ be such that $\mathcal X_{N,L}$ is nonempty, and let $\rho_L=N/V$ as in \eqref{eq:id-sigma}. We set $\epsilon_{\mathrm{ex}}=36\pi\rho_La^3$ and, if $\epsilon_{\mathrm{ex}}<1$, \begin{equation}\label{eq:normal-fugacity} \mathfrak z=\frac{\rho_L\lambda_\beta^3}{1-\epsilon_{\mathrm{ex}}}\exp\Bigl(\frac{152\pi}{3}\,\frac{\beta\rho_La}{1-\epsilon_{\mathrm{ex}}}\Bigr). \end{equation} If $\mathfrak z<1$, we set $\gamma_*=(\log(1/\mathfrak z)/(2\beta))^{1/2}$. The Bose path measure $\mu=\mu_{N,L,\beta}$ and the winding vector $W$ are those of \cref{def:id-path-measure} and \eqref{eq:id-winding}, and $\E$, $\PP$, and $\Var$ refer to $\mu$. \section{A canonical insertion bound}\label{sec:normal-insertion} \begin{lemma}\label{lem:normal-ratio} Let $j\ge1$ be such that $\mathcal X_{j-1,L}$ is nonempty, and suppose that $\epsilon_j=36\pi(j-1)a^3/V<1$. Then $\mathcal X_{j,L}$ is nonempty, and \begin{equation}\label{eq:normal-ratio} Z_{j-1}\le\frac{j\lambda_\beta^3}{V(1-\epsilon_j)}\exp\Bigl(\frac{152\pi}{3}\,\frac{\beta(j-1)a}{(1-\epsilon_j)V}\Bigr)Z_j. \end{equation} Consequently, if $\epsilon_{\mathrm{ex}}<1$, then $Z_{N-m}\le\mathfrak z^mZ_N$ for $0\le m\le N$. \end{lemma} \begin{proof} The consequence follows by applying \eqref{eq:normal-ratio} for $j=N-m+1,\dots,N$; the domains $\mathcal X_{j-1,L}$ are nonempty because $\mathcal X_{N,L}$ is. The factor in front of $Z_j$ in \eqref{eq:normal-ratio} is increasing in $j/V$, in $(j-1)/V$, and in $\epsilon_j$. For $j\le N$, we have $(j-1)/V\le j/V\le\rho_L$ and $\epsilon_j\le\epsilon_{\mathrm{ex}}<1$, so this factor is at most $\mathfrak z$. We prove \eqref{eq:normal-ratio}. Write points of $\Lambda_L^j$ as $(X,y)$, with $X=(x_1,\dots,x_{j-1})$ and $y=x_j$. For $j=1$, the variable $X$ is absent, $\epsilon_1=0$, and the orbital below is the constant $V^{-1/2}$; the steps below apply with these conventions. \emph{Step 1: the orbital.} Let $h(r)=\min\{1,(r-2a)_+/a\}$ for $r\ge0$. It vanishes on $[0,2a]$, equals one on $[3a,\infty)$, and is $a^{-1}$-Lipschitz. For $X\in\Lambda_L^{j-1}$ and $y\in\Lambda_L$, let \[ b(X,y)=V^{-1/2}\min_{i0$, so $\varphi$ is bounded and Lipschitz on $\Lambda_L^{j-1}\times\Lambda_L$. If $u$ is Lipschitz on $\Lambda_L^{j-1}\times\Lambda_L$, then $X\mapsto\int u(X,y)\dd y$ is Lipschitz, with gradient $\int\nabla_Xu(X,y)\dd y$ almost everywhere. To see this, we pair with derivatives of a smooth function of $X$, integrate by parts in $X$ for fixed $y$, and use Fubini's theorem; by Rademacher's theorem and Fubini's theorem, for almost every $y$ the gradient of $u(\cdot,y)$ agrees almost everywhere with $\nabla_Xu(\cdot,y)$. Since $\int\varphi_X^2\dd y=1$ for every $X$, applying this to $u=\varphi^2$ and $u=b^2$ gives, for almost every $X$, \[ \int\varphi_X\nabla_{x_i}\varphi_X\dd y=0,\qquad\nabla_{x_i}\mathfrak m(X)=2\int b\,\nabla_{x_i}b\dd y. \] Then $\nabla_{x_i}\varphi_X=\mathfrak m^{-1/2}\bigl(\nabla_{x_i}b-\varphi_X\int\varphi_X\nabla_{x_i}b\dd y\bigr)$. Thus $\mathfrak m^{1/2}\nabla_{x_i}\varphi_X$ is the component of $\nabla_{x_i}b$ orthogonal to the unit vector $\varphi_X$ in $L^2(\dd y)$, and $\int\abs{\nabla_{x_i}\varphi_X}^2\dd y\le\mathfrak m^{-1}\int\abs{\nabla_{x_i}b}^2\dd y$. Also $\nabla_y\varphi_X=\mathfrak m^{-1/2}\nabla_yb$. By the shell volume of Step 1 and \eqref{eq:normal-normalizer}, we obtain for almost every $X$ \[ \begin{aligned} G(X)&:=\int\Bigl(\abs{\nabla_y\varphi_X}^2+\sum_{ia$; by mollification, the function lies in $H^1_0(\mathcal X_{j,L})$. For $f\in\mathcal Q_{j-1}$, we approximate $f$ in $H^1$ by such functions. Multiplication by the bounded Lipschitz function $\varphi_X(y)e^{\mathrm ip\cdot y}$ is bounded in $H^1$, and $H^1_0(\mathcal X_{j,L})$ is closed. Since $f_k$ and $\varphi_X$ are symmetric in $X$, this proves the claim for every $p$. For $X\in\mathcal X_{j-1,L}$, the function $\varphi_X$ vanishes at every $y$ with $(X,y)\notin\mathcal X_{j,L}$, so $\norm{\psi_{k,p}}=1$ since $\int\varphi_X^2\dd y=1$. Almost everywhere, \[ \nabla_{x_i}\psi_{k,p}=(\varphi_X\nabla_if_k+f_k\nabla_{x_i}\varphi_X)e^{\mathrm ip\cdot y},\qquad\nabla_y\psi_{k,p}=f_k(\nabla_y\varphi_X+\mathrm ip\varphi_X)e^{\mathrm ip\cdot y}. \] Since $\varphi_X$ is real, $\abs{\nabla_y\psi_{k,p}}^2=\abs{f_k}^2(\abs{\nabla_y\varphi_X}^2+\abs p^2\varphi_X^2)$. We expand $\abs{\nabla_{x_i}\psi_{k,p}}^2$ and integrate first in $y$. By Step 2, the cross terms vanish, and \begin{equation}\label{eq:normal-trial} q_j[\psi_{k,p}]=q_{j-1}[f_k]+\abs p^2+\int\abs{f_k(X)}^2G(X)\dd X\le E_k+\abs p^2+G_*. \end{equation} \emph{Step 4: a frame bound for all frequencies.} Let $u\in\operatorname{ran}P'$, extended by zero to $\Lambda_L^j$, and let $g_p(X)=\int\varphi_X(y)e^{-\mathrm ip\cdot y}u(X,y)\dd y$. Then $g_p$ is symmetric, it vanishes outside $\mathcal X_{j-1,L}$, and $\abs{g_p(X)}^2\le\int\abs{u(X,y)}^2\dd y$. Hence $g_p\in L^2_{\mathrm{sym}}(\mathcal X_{j-1,L})$, and $\langle\psi_{k,p},u\rangle=\langle f_k,g_p\rangle$. Parseval's identity for the basis $(f_k)$, Parseval's identity in $L^2(\Lambda_L)$ for the orthonormal basis $(V^{-1/2}e^{\mathrm ip\cdot y})_p$, Tonelli's theorem, and the bound on $\varphi_X$ in Step 1 give \begin{equation}\label{eq:normal-frame} \sum_{k,p}\abs{\langle\psi_{k,p},u\rangle}^2=\int\sum_p\abs{g_p(X)}^2\dd X=V\iint\varphi_X(y)^2\abs{u(X,y)}^2\dd y\dd X\le\frac{\norm u^2}{1-\epsilon_j}. \end{equation} \emph{Step 5: partition functions.} Let $Z_{\mathrm{dist}}=\tr e^{-\beta H_{\mathrm{dist}}}$, and let $(e_n)$ be an orthonormal basis of $\operatorname{ran}P'$. Applying \eqref{eq:normal-frame} to $u=e^{-\beta H_{\mathrm{dist}}/2}e_n$ and summing over $n$, we obtain by Tonelli's theorem \[ \sum_{k,p}\langle\psi_{k,p},e^{-\beta H_{\mathrm{dist}}}\psi_{k,p}\rangle=\sum_n\sum_{k,p}\abs{\langle\psi_{k,p},e^{-\beta H_{\mathrm{dist}}/2}e_n\rangle}^2\le\frac{Z_{\mathrm{dist}}}{1-\epsilon_j}. \] For a unit vector $\psi$ in the form domain of $H_{\mathrm{dist}}$, Jensen's inequality for the spectral measure of $\psi$ gives $\langle\psi,e^{-\beta H_{\mathrm{dist}}}\psi\rangle\ge e^{-\beta q_j[\psi]}$. We apply this to each $\psi_{k,p}$ and use \eqref{eq:normal-trial} and the Poisson summation formula. This gives \[ \frac{Z_{\mathrm{dist}}}{1-\epsilon_j}\ge e^{-\beta G_*}\sum_ke^{-\beta E_k}\sum_pe^{-\beta\abs p^2}=e^{-\beta G_*}Z_{j-1}\,\frac V{\lambda_\beta^3}\sum_{n\in\mathbb Z^3}e^{-L^2\abs n^2/(4\beta)}\ge e^{-\beta G_*}Z_{j-1}\,\frac V{\lambda_\beta^3}. \] It remains to compare $Z_{\mathrm{dist}}$ with $Z_j$. The proof of \cref{lem:id-ck}(3) applies with $P_{\mathrm{sym}}$ replaced by $P'$. By \cref{prop:id-fk}(1), the operator $e^{-\beta H^{\mathrm{lab}}_j/2}P'$ has the kernel $((j-1)!)^{-1}\sum_{\sigma}K_{\beta/2}((X,y),(\sigma X',y'))$. It is Hilbert--Schmidt by the min--max argument in the proof of \cref{lem:id-trace}, which does not use the symmetry. Then \cref{lem:id-ck}(1) and~(2) give \[ Z_{\mathrm{dist}}=\frac1{(j-1)!}\sum_{\sigma\in S_{j-1}}\int_{\mathcal X_{j,L}}K_\beta\bigl((X,y),(\sigma X,y)\bigr)\dd X\dd y. \] In the formula of \cref{lem:id-ck}(3) for $Z_j$, the kernel $K_\beta$ is nonnegative by \cref{lem:id-ck}(1). Retaining only the $(j-1)!$ permutations that fix the label $j$, we obtain $Z_j\ge Z_{\mathrm{dist}}/j$. Together with the previous display, this proves \eqref{eq:normal-ratio}. \end{proof} \section{Cycles and winding sectors}\label{sec:normal-cycles} Let $(\pi,X(\cdot))\in S_N\times C([0,\beta];\Lambda_L^N)$ satisfy $x_i(\beta)=x_{\pi(i)}(0)$ for every $i$; this holds $\mu$-almost surely. We write $\mathrm{cyc}(\pi)$ for the set of cycles of $\pi$. For a cycle $c=(i_1,\dots,i_m)$ of $\pi$ with $i_1=\min c$ and $\pi(i_l)=i_{l+1}$, where $i_{m+1}=i_1$, we write $\abs c=m$, and we define the worldline $\omega_c\in C([0,m\beta];\Lambda_L)$ by $\omega_c(s)=x_{i_l}(s-(l-1)\beta)$ for $s\in[(l-1)\beta,l\beta]$. It is a closed loop, and we set $w_c=d(\omega_c)/L$. \begin{lemma}\label{lem:normal-cycles} For each $m\ge1$, let $F_m\colon C([0,m\beta];\Lambda_L)\to[0,\infty]$ be measurable. Then \[ \E\sum_{c\in\mathrm{cyc}(\pi)}F_{\abs c}(\omega_c)\le\sum_{m=1}^N\frac{Z_{N-m}}{mZ_N}\int_{\Lambda_L}\int F_m\dd\mathbb W^{m\beta}_{x,x}\dd x. \] \end{lemma} \begin{proof} By \eqref{eq:id-path-measure}, \[ Z_N\,\E\sum_{c\in\mathrm{cyc}(\pi)}F_{\abs c}(\omega_c)=\frac1{N!}\sum_{\pi\in S_N}\sum_{c\in\mathrm{cyc}(\pi)}\int_{\Lambda_L^N}\int\mathbf 1_{\mathcal E_\beta}F_{\abs c}(\omega_c)\dd\mathbb W^\beta_{X_0,\pi X_0}\dd X_0, \] where we may integrate over $\Lambda_L^N$ because $\mathbf 1_{\mathcal E_\beta}$ vanishes unless $X_0=X(0)\in\mathcal X_{N,L}$. The pairs $(\pi,c)$ with $\abs c=m$ correspond bijectively to the triples $(S,c,\pi')$, where $S=\supp c$ has $m$ elements, $c$ is a cyclic permutation of $S$, and $\pi'$ is a permutation of the complement $S^{\mathrm c}$; there are $\binom Nm(m-1)!$ choices of $(S,c)$. Fix such a triple. We bound $\mathbf 1_{\mathcal E_\beta}$ by the indicator of the event that the paths with labels in $S^{\mathrm c}$ stay in $\mathcal X_{N-m,L}$; this drops every exclusion that involves a label in $S$. The measure $\mathbb W^\beta_{X_0,\pi X_0}$ is a product over the particles, and by Tonelli's theorem the integral is at most the product of two factors. The first is the integral of $F_m(\omega_c)$ over the initial positions and paths with labels in $S$. By $m-1$ applications of \eqref{eq:id-concatenation} for one particle, it equals $\int_{\Lambda_L}\int F_m\dd\mathbb W^{m\beta}_{x,x}\dd x$. The second factor, summed over $\pi'$, equals $(N-m)!\,Z_{N-m}$ by \cref{lem:id-ck}(3) for $N-m$ particles, and it equals $1=0!\,Z_0$ if $m=N$. Since $\binom Nm(m-1)!\,(N-m)!/N!=1/m$, the lemma follows. \end{proof} \begin{lemma}\label{lem:normal-sectors} \leavevmode \begin{enumerate}[label=\textup{(\roman*)}] \item For $t>0$ and $k\in\mathbb Z^3$, we have $\int_{\Lambda_L}\mathbb W^t_{x,x}(\{\omega:d(\omega)=Lk\})\dd x=Vp_t(Lk)$. \item For every $c\in\mathrm{cyc}(\pi)$, we have $w_c\in\mathbb Z^3$, and $W=\sum_{c\in\mathrm{cyc}(\pi)}w_c$. \end{enumerate} \end{lemma} \begin{proof} By the definition of the torus bridge measure in \cref{sec:id-fk}, $\mathbb W^t_{x,x}$ is the sum over $k'\in\mathbb Z^3$ of the images of $\mathbb W^t_{\tilde x,\tilde x+Lk'}$, and the $k'$-th term is carried by $\{d(\omega)=Lk'\}$ and has mass $p_t(Lk')$. Integrating over $x$, we obtain (i). The lifted displacement is additive under concatenation, so $d(\omega_c)=\sum_{i\in c}d(x_i(\cdot))$, and $d(\omega_c)\in L\mathbb Z^3$ because $\omega_c$ is closed. Summing over the cycles and using \eqref{eq:id-winding}, we obtain $LW=\sum_id(x_i(\cdot))=\sum_cd(\omega_c)$, which proves (ii). \end{proof} \section{The normal-phase estimates}\label{sec:normal-proof} \begin{lemma}\label{lem:normal-response} Let $N\ge1$ and $L,a,\beta>0$ be such that $\mathcal X_{N,L}$ is nonempty. Then \[ \Var(W_1)\le\frac{8\beta N}{L^2}\,\PP(W_1\neq0)^{1/2},\qquad\text{equivalently}\qquad f_{\mathrm s}(N,L,T)\le4\,\PP(W_1\neq0)^{1/2}. \] \end{lemma} \begin{proof} Let $\sigma^2=2\beta N/L^2$ as in \eqref{eq:id-sigma}. By \cref{thm:id-subgaussian}, we have $\E W_1=0$, and by \cref{cor:id-tails}, we have $\E W_1^4\le16\sigma^4$. By the Cauchy--Schwarz inequality, \[ \Var(W_1)=\E\bigl[W_1^2\mathbf 1_{\{W_1\neq0\}}\bigr]\le(\E W_1^4)^{1/2}\,\PP(W_1\neq0)^{1/2}\le4\sigma^2\,\PP(W_1\neq0)^{1/2}. \] By \eqref{eq:id-pc}, $f_{\mathrm s}(N,L,T)=\Var(W_1)/\sigma^2$, which gives the equivalent form. \end{proof} \begin{theorem}\label{thm:normal-finite} Let $N\ge1$ and $L,a,\beta>0$ be such that $\mathcal X_{N,L}$ is nonempty, and suppose that $\epsilon_{\mathrm{ex}}<1$, $\mathfrak z<1$, and $\gamma_*L\ge1$. Then the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item We have \[ \PP(W\neq0)\le\E\,\#\{c\in\mathrm{cyc}(\pi):w_c\neq0\}\le190\Bigl(\frac L{\lambda_\beta}\Bigr)^3e^{-\gamma_*L}=:P_*. \] \item We have $0\le f_{\mathrm s}(N,L,T)\le4\,\PP(W_1\neq0)^{1/2}\le4P_*^{1/2}$ and $\Var(W_1)\le(8\beta N/L^2)P_*^{1/2}$. \end{enumerate} \end{theorem} \begin{proof} If $W\neq0$, then $w_c\neq0$ for some cycle $c$ by \cref{lem:normal-sectors}(ii), which proves the first inequality in (a). We apply \cref{lem:normal-cycles} with $F_m(\omega)=\mathbf 1_{\{d(\omega)\neq0\}}$, and then \cref{lem:normal-sectors}(i) and \cref{lem:normal-ratio}. Since $Vp_{m\beta}(Lk)=(L/\lambda_\beta)^3m^{-3/2}e^{-L^2\abs k^2/(4m\beta)}$ and all terms are nonnegative, we obtain \[ \E\,\#\{c:w_c\neq0\}\le\sum_{m=1}^N\frac{\mathfrak z^m}m\sum_{k\neq0}Vp_{m\beta}(Lk)\le\Bigl(\frac L{\lambda_\beta}\Bigr)^3\sum_{m\ge1}\frac{\mathfrak z^m}{m^{5/2}}\sum_{k\in\mathbb Z^3\setminus\{0\}}e^{-L^2\abs k^2/(4m\beta)}. \] By the inequality between the arithmetic and geometric means, \[ \frac m2\log\frac1{\mathfrak z}+\frac{L^2\abs k^2}{4m\beta}\ge L\abs k\Bigl(\frac{\log(1/\mathfrak z)}{2\beta}\Bigr)^{1/2}=\gamma_*L\abs k, \] and therefore $\mathfrak z^me^{-L^2\abs k^2/(4m\beta)}\le\mathfrak z^{m/2}e^{-\gamma_*L\abs k}$. Now $\sum_{m\ge1}m^{-5/2}\mathfrak z^{m/2}\le\zeta(5/2)<1.342$. There are $24n^2+2\le26n^2$ points $k\in\mathbb Z^3$ with $\abs k_\infty=n\ge1$, and $\abs k\ge\abs k_\infty$. Since $\sum_{n\ge1}n^2q^n=q(1+q)/(1-q)^3$ for $q=e^{-\gamma_*L}\le e^{-1}$, we have \[ \sum_{k\neq0}e^{-\gamma_*L\abs k}\le26\,\frac{1+e^{-1}}{(1-e^{-1})^3}\,e^{-\gamma_*L}\le140.9\,e^{-\gamma_*L}. \] Since $1.342\cdot140.9<190$, this proves (a). By \cref{prop:id-winding}, we have $f_{\mathrm s}(N,L,T)\ge0$. The remaining bounds in (b) follow from \cref{lem:normal-response}, the inequality $\PP(W_1\neq0)\le\PP(W\neq0)$, and (a). \end{proof} \begin{proof}[Proof of \cref{thm:main-normal}] Let $\varepsilon\in(0,1]$ and $a,\rho>0$ with $g=\rho a^3\le(\varepsilon/100)^3$, and set $T_{\mathrm{hi}}=(1+\varepsilon)4\pi\rho^{2/3}$ and $\eta=\log(1+\varepsilon)$. Consider a sequence of $N$ and $L$ with $L\to\infty$ and $\rho_L=N/L^3\to\rho$. Choose $L_0$ such that, for all $L\ge L_0$ in the sequence, \[ \rho_L\le(1+\varepsilon)^{1/2}\rho\qquad\text{and}\qquad L\sqrt{T_{\mathrm{hi}}}\ge6\eta^{-1/2}. \] Neither condition depends on $T$. Let $L\ge L_0$ and $T\ge T_{\mathrm{hi}}$. Since $\rho(4\pi/T_{\mathrm{hi}})^{3/2}=(1+\varepsilon)^{-3/2}$, we have \[ \rho_L\lambda_\beta^3\le(1+\varepsilon)^{-1},\qquad36\pi\rho_La^3\le36\pi\sqrt2\,g<160g\le1.6\cdot10^{-4}\varepsilon^3. \] In particular $\epsilon_j\le36\pi\rho_La^3<1$ for $1\le j\le N$, and \cref{lem:normal-ratio}, applied for $j=1,\dots,N$, shows that $\mathcal X_{N,L}$ is nonempty. Since $\beta\rho_La=(\rho_L\lambda_\beta^3)^{2/3}(\rho_La^3)^{1/3}/(4\pi)\le2^{1/6}g^{1/3}/(4\pi)$, the exponent in \eqref{eq:normal-fugacity} satisfies \[ \frac{152\pi}3\,\frac{\beta\rho_La}{1-\epsilon_{\mathrm{ex}}}\le\frac{38}3\,\frac{2^{1/6}}{1-1.6\cdot10^{-4}}\,g^{1/3}\le14.3\,g^{1/3}\le0.143\,\varepsilon. \] We use $(1-\epsilon_{\mathrm{ex}})^{-1}\le1+0.00017\varepsilon$, $e^{0.143\varepsilon}\le1+0.143e^{0.143}\varepsilon\le1+0.165\varepsilon$, and $(1+\varepsilon)^{1/2}\ge1+(\sqrt2-1)\varepsilon$ for $\varepsilon\in[0,1]$. We obtain \[ \mathfrak z\le(1+\varepsilon)^{-1}(1+0.00017\varepsilon)(1+0.165\varepsilon)\le(1+\varepsilon)^{-1}(1+0.17\varepsilon)\le(1+\varepsilon)^{-1/2}. \] Hence $\mathfrak z<1$, $\gamma_*\ge\frac12(T\eta)^{1/2}$, and $\gamma_*L\ge\frac12L(T_{\mathrm{hi}}\eta)^{1/2}\ge3$. Since $(L/\lambda_\beta)^3=(L^2T/4\pi)^{3/2}$ and $4\sqrt{190}<57$, \cref{thm:normal-finite} gives \eqref{eq:intro-normal} and the bound on $\PP(W\neq0)$, for all $L\ge L_0$ in the sequence and all $T\ge T_{\mathrm{hi}}$. For the uniformity, let $b=\frac12\eta^{1/2}$ and $s=L\sqrt T$. The bounds just proved are $57(4\pi)^{-3/4}s^{3/2}e^{-bs/2}$ for $f_{\mathrm s}$ and $190(4\pi)^{-3/2}s^3e^{-bs}$ for $\PP(W\neq0)$. Both decrease in $s$ for $s\ge3/b$, and $L\sqrt T\ge L\sqrt{T_{\mathrm{hi}}}\ge3/b$ for $L\ge L_0$. Therefore their suprema over $T\ge T_{\mathrm{hi}}$ are attained at $T=T_{\mathrm{hi}}$, and these tend to zero as $L\to\infty$. By \eqref{eq:id-pc}, \[ \Var(W_1)=\frac{2\rho_LL}{T}f_{\mathrm s}(N,L,T)\le114(4\pi)^{-3/4}\rho_LL^{5/2}T^{-1/4}e^{-bL\sqrt T/2}, \] and the right side decreases in $T$. At $T=T_{\mathrm{hi}}$, it tends to zero as $L\to\infty$, since $\rho_L$ is bounded. This proves the uniform convergence. \end{proof} \part{Surgery in one slab}\label{part:surgery} \chapter{A variance inequality for surgeries}\label{sec:variance} We first complete accepted moves by the identity and prove the Hammersley--Chapman--Robbins variance inequality. We then combine it with conditional independence of the time slabs. In \cref{prop:var-one-slab}, we reduce each response bound to the mean gain and chi-square divergence of a kernel acting on one slab. \section{Completed kernels and the chi-square divergence}\label{sec:var-hcr} Throughout this section, $(\mathsf X,\mathcal F)$ is a standard Borel space and $\mu$ is a probability measure on $\mathsf X$. A subprobability kernel on $\mathsf X$ is a map $q\colon\mathsf X\times\mathcal F\to[0,1]$ such that $q(x,\cdot)$ is a measure of total mass at most one for every $x\in\mathsf X$, and $x\mapsto q(x,B)$ is measurable for every $B\in\mathcal F$. It is a probability kernel if every $q(x,\cdot)$ is a probability measure. We write $\mu q$ for the measure $B\mapsto\int q(x,B)\,\mu(\dd x)$, and $qf(x)=\int f(y)\,q(x,\dd y)$ for measurable $f\ge0$. For probability measures $\nu$ and $\mu$ on $\mathsf X$, the chi-square divergence of $\nu$ from $\mu$ is \[ \chi^2(\nu\,\|\,\mu)=\int\Bigl(\frac{\dd\nu}{\dd\mu}-1\Bigr)^2\dd\mu \] if $\nu$ is absolutely continuous with respect to $\mu$, and $\chi^2(\nu\,\|\,\mu)=\infty$ otherwise. We write $\langle f,g\rangle_\mu=\int fg\dd\mu$. \begin{definition}\label{def:var-completion} Let $q$ be a subprobability kernel on $\mathsf X$, and set $\bar q(x)=q(x,\mathsf X)$. The completion of $q$ is the probability kernel \[ Q(x,\dd y)=q(x,\dd y)+\bigl(1-\bar q(x)\bigr)\delta_x(\dd y). \] We call $\bar q$ the outgoing mass of $q$. If $\mu q$ is absolutely continuous with respect to $\mu$, we call its density $c=\dd(\mu q)/\dd\mu$ the incoming density of $q$. \end{definition} A sample of $\mu Q$ is obtained from a sample $x$ of $\mu$ by moving according to $q(x,\cdot)$ with probability $\bar q(x)$, and by leaving $x$ unchanged otherwise. In our application, $q$ describes the accepted surgeries, $\bar q(x)$ is the probability that a surgery is accepted at $x$, and a rejected surgery leaves the configuration unchanged. The kernel $q$ is not normalized by its acceptance probability. \begin{lemma}\label{lem:var-hcr} Let $q$ be a subprobability kernel on $\mathsf X$ with outgoing mass $\bar q$ and completion $Q$, and suppose that $\mu q$ is absolutely continuous with respect to $\mu$, with incoming density $c$. \begin{enumerate} \item The measure $\mu Q$ is absolutely continuous with respect to $\mu$, and \begin{equation}\label{eq:var-likelihood} L=\frac{\dd(\mu Q)}{\dd\mu}=1-\bar q+c,\qquad \chi^2(\mu Q\,\|\,\mu)=\int(c-\bar q)^2\dd\mu. \end{equation} \item Let $O\in L^2(\mu)$, and suppose that $L\in L^2(\mu)$. Then all integrals in \eqref{eq:var-gain} converge absolutely, \begin{equation}\label{eq:var-gain} \E_{\mu Q}O-\E_\mu O=\iint\bigl(O(y)-O(x)\bigr)\,q(x,\dd y)\,\mu(\dd x)=\int(O-\E_\mu O)(L-1)\dd\mu, \end{equation} and \begin{equation}\label{eq:var-hcr} \bigl(\E_{\mu Q}O-\E_\mu O\bigr)^2\le\Var_\mu(O)\,\chi^2(\mu Q\,\|\,\mu). \end{equation} In particular, if $\chi^2(\mu Q\,\|\,\mu)=0$, then $\E_{\mu Q}O=\E_\mu O$. \end{enumerate} \end{lemma} \begin{proof} For $B\in\mathcal F$, we have \[ \mu Q(B)=\mu q(B)+\int_B(1-\bar q)\dd\mu=\int_B(c+1-\bar q)\dd\mu, \] which proves the first identity in \eqref{eq:var-likelihood}. The second follows from $L-1=c-\bar q$. For part (2), we note that $0\le c\le L$, since $0\le\bar q\le1$ and $c\ge0$; in particular, $c\in L^2(\mu)$. By the Cauchy--Schwarz inequality, we have \[ \iint\abs{O(y)}\,q(x,\dd y)\,\mu(\dd x)=\int\abs O\,c\dd\mu\le\norm O_{L^2(\mu)}\norm c_{L^2(\mu)}<\infty, \] and $\iint\abs{O(x)}\,q(x,\dd y)\,\mu(\dd x)=\int\abs O\,\bar q\dd\mu\le\norm O_{L^1(\mu)}<\infty$. In the same way, $\int\abs O\dd(\mu Q)=\int\abs OL\dd\mu<\infty$. Then the double integral in \eqref{eq:var-gain} converges absolutely, and it equals $\int O(c-\bar q)\dd\mu=\int O(L-1)\dd\mu$, which is also $\E_{\mu Q}O-\E_\mu O$. Since $\int(L-1)\dd\mu=0$, we may replace $O$ by $O-\E_\mu O$ in the last integral, which proves \eqref{eq:var-gain}. The Cauchy--Schwarz inequality applied to the last expression in \eqref{eq:var-gain} implies \eqref{eq:var-hcr}. \end{proof} \usedin{\cref{lem:var-slabs}, \cref{sec:surgery}, and the gain computations in \cref{sec:proof-stiffness,sec:proof-ncri}.} Inequality \eqref{eq:var-hcr} is the Cauchy--Schwarz inequality for the functions $O-\E_\mu O$ and $L-1$ in $L^2(\mu)$. In this form, which compares the means of one observable under two probability measures, it is due to Hammersley \cite[p.~197, equation~(31)]{Hammersley1950} and to Chapman and Robbins \cite[Section~2, equations~(3)--(5)]{ChapmanRobbins1951}, who used it to bound the variance of an estimator without regularity assumptions; see also Barankin \cite[Theorem~4]{Barankin1949}. The middle expression in \eqref{eq:var-gain} shows that the gain is computed from the accepted moves alone: a rejected move does not change $O$. The integrability assumption $L\in L^2(\mu)$ is needed; for a general positive measure on pairs of states, the gain can fail to be defined even when both marginals have the same density. The kernels of \cref{sec:surgery} are averages over an auxiliary random parameter, the skeleton of a route, which is sampled independently of the state. The next lemma expresses the divergence of such an average through two independent copies of the parameter. \begin{lemma}\label{lem:var-mixture} Let $(\mathfrak G,\mathcal G)$ be a standard Borel space with a probability measure $\mathfrak p$. For every $\Gamma\in\mathfrak G$, let $Q_\Gamma$ be a probability kernel on $\mathsf X$, such that $(\Gamma,x)\mapsto Q_\Gamma(x,B)$ is measurable for every $B\in\mathcal F$. Suppose that, for $\mathfrak p$-almost every $\Gamma$, the measure $\mu Q_\Gamma$ is absolutely continuous with respect to $\mu$, with a nonnegative density $L_\Gamma(x)$ that is jointly measurable in $(\Gamma,x)$. Let $Q=\int Q_\Gamma\,\mathfrak p(\dd\Gamma)$. Then $\mu Q$ is absolutely continuous with respect to $\mu$, with density $L=\int L_\Gamma\,\mathfrak p(\dd\Gamma)$, and \begin{equation}\label{eq:var-mixture} 1+\chi^2(\mu Q\,\|\,\mu)=\iint\langle L_\Gamma,L_{\Gamma'}\rangle_\mu\,\mathfrak p(\dd\Gamma)\,\mathfrak p(\dd\Gamma'), \end{equation} where both sides may be infinite. If moreover $\int\norm{L_\Gamma}_{L^2(\mu)}^2\,\mathfrak p(\dd\Gamma)<\infty$, then \begin{equation}\label{eq:var-mixture-signed} \chi^2(\mu Q\,\|\,\mu)=\iint\langle L_\Gamma-1,L_{\Gamma'}-1\rangle_\mu\,\mathfrak p(\dd\Gamma)\,\mathfrak p(\dd\Gamma'), \end{equation} and the integrand in \eqref{eq:var-mixture-signed} is integrable with respect to $\mathfrak p\otimes\mathfrak p$. \end{lemma} \begin{proof} For $B\in\mathcal F$, we have by Tonelli's theorem \[ \mu Q(B)=\int\mu Q_\Gamma(B)\,\mathfrak p(\dd\Gamma)=\iint_BL_\Gamma\dd\mu\,\mathfrak p(\dd\Gamma)=\int_BL\dd\mu, \] which shows that $L$ is the density of $\mu Q$. Another application of Tonelli's theorem yields \[ \int L^2\dd\mu=\int\Bigl(\int L_\Gamma\,\mathfrak p(\dd\Gamma)\Bigr)\Bigl(\int L_{\Gamma'}\,\mathfrak p(\dd\Gamma')\Bigr)\dd\mu=\iint\langle L_\Gamma,L_{\Gamma'}\rangle_\mu\,\mathfrak p(\dd\Gamma)\,\mathfrak p(\dd\Gamma'). \] Here we used the fact that the integrands are nonnegative. If $L\in L^2(\mu)$, then $\chi^2(\mu Q\,\|\,\mu)=\int L^2\dd\mu-1$, because $\int L\dd\mu=1$. If $L\notin L^2(\mu)$, both sides of \eqref{eq:var-mixture} are infinite. This proves \eqref{eq:var-mixture}. Under the additional assumption, by the Cauchy--Schwarz inequality, we have $\abs{\langle L_\Gamma-1,L_{\Gamma'}-1\rangle_\mu}\le(1+\norm{L_\Gamma}_{L^2(\mu)})(1+\norm{L_{\Gamma'}}_{L^2(\mu)})$, and the right side is integrable with respect to $\mathfrak p\otimes\mathfrak p$. Since $\int L_\Gamma\dd\mu=1$ for $\mathfrak p$-almost every $\Gamma$, we have $\langle L_\Gamma-1,L_{\Gamma'}-1\rangle_\mu=\langle L_\Gamma,L_{\Gamma'}\rangle_\mu-1$. Integrating this identity and using \eqref{eq:var-mixture} proves \eqref{eq:var-mixture-signed}. \end{proof} \usedin{\cref{thm:surgery}.} Identity \eqref{eq:var-mixture} is the elementary independent-copy expansion of the second moment of an averaged likelihood. Mixture likelihoods in minimax hypothesis testing are discussed in \cite[Section~1.1, equations~(1.1)--(1.4) in the preprint version]{Ingster1997}. The independent-copy expansion in the Gaussian model is given in \cite[Section~3.3.1, equation~(3.66), pp.~97--98]{IngsterSuslina2003}; see also \cite{Ingster1982} for related minimax testing background. Explicit forms of the independent-path expansion appear in \cite[Section~2.1.2, equation~(2.6) in arXiv version~2]{AriasCastroCandesHelgasonZeitouni2008}, \cite[proof of Lemma~2.4, equation~(2.8) in arXiv version~1]{BergerPeres2013}, and \cite[proof of Proposition~2.1, equation~(2.3) in arXiv version~1]{PeresSly2014}. Its Bose-gas applications in \cite[Section~9]{OpenAI2026B} and \cite[Section~9]{OpenAI2026BEC} average over random skeletons, as we do here. It is important that the parameter is averaged out in $Q$. If $\Gamma$ were kept as a coordinate of the state, the divergence would be $\iint(L_\Gamma-1)^2\dd\mu\,\mathfrak p(\dd\Gamma)$, which corresponds to two copies of the same route, and the decay that comes from the rarity of encounters between two independent routes would be lost. The kernels of \cref{sec:surgery} also preserve part of the state, namely the exposed data of a slab, and the slab kernels of \cref{sec:var-slabs} preserve the cut configurations. The next lemma shows that such kernels can be analyzed on each fiber. \begin{lemma}\label{lem:var-fibers} Let $(\mathsf Y,\mathcal Y)$ be a standard Borel space, let $\Phi\colon\mathsf X\to\mathsf Y$ be measurable, write $\mu_\Phi$ for the image of $\mu$ under $\Phi$, and let $(\mu_y)_{y\in\mathsf Y}$ be a disintegration of $\mu$ along $\Phi$: a probability kernel from $\mathsf Y$ to $\mathsf X$ such that $\mu_y(\Phi^{-1}(y))=1$ for $\mu_\Phi$-almost every $y$ and $\mu=\int\mu_y\,\mu_\Phi(\dd y)$. Let $Q$ be a probability kernel on $\mathsf X$ such that, for $\mu$-almost every $x$, we have $\Phi(x')=\Phi(x)$ for $Q(x,\cdot)$-almost every $x'$. Suppose that $\mu Q$ is absolutely continuous with respect to $\mu$, with density $L$. Then, for $\mu_\Phi$-almost every $y$, the measure $\mu_yQ$ is absolutely continuous with respect to $\mu_y$ with density $L$, and $\int L\dd\mu_y=1$. Moreover, \begin{equation}\label{eq:var-fibers} \chi^2(\mu Q\,\|\,\mu)=\int\chi^2(\mu_yQ\,\|\,\mu_y)\,\mu_\Phi(\dd y). \end{equation} \end{lemma} \begin{proof} Let $f$ be a bounded measurable function on $\mathsf X$, and let $\varphi$ be a bounded measurable function on $\mathsf Y$. Since $Q$ preserves $\Phi$, we have \[ \begin{split} \int\varphi(\Phi(x))f(x)L(x)\,\mu(\dd x)&=\iint\varphi(\Phi(x'))f(x')\,Q(x,\dd x')\,\mu(\dd x)\\ &=\int\varphi(\Phi(x))\,Qf(x)\,\mu(\dd x). \end{split} \] Disintegrating both sides along $\Phi$ gives \[ \int\varphi(y)\Bigl(\int fL\dd\mu_y\Bigr)\mu_\Phi(\dd y)=\int\varphi(y)\Bigl(\int Qf\dd\mu_y\Bigr)\mu_\Phi(\dd y). \] Since $\varphi$ is arbitrary, $\int fL\dd\mu_y=\int f\dd(\mu_yQ)$ for $\mu_\Phi$-almost every $y$, for each fixed $f$. Since $\mathsf X$ is standard Borel, $\mathcal F$ is generated by a countable collection of sets that is closed under finite intersections and contains $\mathsf X$. Applying the last identity to the indicators of these sets, we conclude that the finite measures $L\mu_y$ and $\mu_yQ$ agree on this collection for $\mu_\Phi$-almost every $y$, and they are equal. Taking $f=1$ yields $\int L\dd\mu_y=1$. Finally, \eqref{eq:var-fibers} follows from $\int(L-1)^2\dd\mu=\int\bigl(\int(L-1)^2\dd\mu_y\bigr)\mu_\Phi(\dd y)$. \end{proof} \usedin{\cref{prop:var-one-slab}.} \section{Reduction to one slab}\label{sec:var-slabs} The next lemma combines the law of total variance with \cref{lem:var-hcr}. It applies to an observable that is a sum of contributions of conditionally independent pieces, and to a kernel that changes one piece at a time. \begin{lemma}\label{lem:var-slabs} Let $C,S_1,\dots,S_m,R$ be random elements of standard Borel spaces, defined on a probability space with probability measure $\PP$. Suppose that, conditionally on $C$, the variables $S_1,\dots,S_m,R$ are independent, and let \[ O=\sum_{j=1}^mO_j(C,S_j)+O_R(C,R),\qquad O_1(C,S_1),\dots,O_m(C,S_m),O_R(C,R)\in L^2(\PP). \] Let $\nu_{j,c}$ be a regular conditional law of $S_j$ given $C=c$. For each $j$, let $Q_{j,c}$ be a probability kernel on the space of $S_j$ such that $(c,s)\mapsto Q_{j,c}(s,B)$ is jointly measurable for every measurable set $B$, and suppose that, for almost every value $c$ of $C$, the measure $\nu_{j,c}Q_{j,c}$ is absolutely continuous with respect to $\nu_{j,c}$, with a density $L_j(c,\cdot)$ that is jointly measurable in its two arguments. Set \[ h_j=\E\int\bigl(L_j(C,s)-1\bigr)^2\,\nu_{j,C}(\dd s),\qquad g_j=\E\Bigl[\int O_j(C,s')\,Q_{j,C}(S_j,\dd s')-O_j(C,S_j)\Bigr], \] and suppose that $h_j<\infty$ for every $j$. Then $g_j$ is well defined, $h_j=0$ implies $g_j=0$, and, with the convention $0/0=0$, \begin{equation}\label{eq:var-slabs} \Var(O)\ge\sum_{j=1}^m\E\Var(O_j\mid C)\ge\sum_{j=1}^m\frac{g_j^2}{h_j}. \end{equation} In particular, if $g_j\ge g>0$ and $h_j\le h$ for every $j$, then $\Var(O)\ge mg^2/h$. \end{lemma} \begin{proof} By the law of total variance, \[ \Var(O)=\E\Var(O\mid C)+\Var\bigl(\E[O\mid C]\bigr)\ge\E\Var(O\mid C). \] Conditionally on $C$, the variables $O_1(C,S_1),\dots,O_m(C,S_m),O_R(C,R)$ are independent and have finite second moments almost surely. Then $\Var(O\mid C)=\sum_j\Var(O_j\mid C)+\Var(O_R\mid C)\ge\sum_j\Var(O_j\mid C)$, which proves the first inequality in \eqref{eq:var-slabs}. Fix $j$, and write $\bar O_j(c)=\int O_j(c,s)\,\nu_{j,c}(\dd s)$ for the conditional mean of $O_j$. Since $\int L_j(c,s)\,\nu_{j,c}(\dd s)=1$ for almost every $c$, we have $\E\int L_j(C,s)^2\,\nu_{j,C}(\dd s)=1+h_j<\infty$. By the Cauchy--Schwarz inequality, \[ \begin{split} \E\iint\abs{O_j(C,s')}\,Q_{j,C}(s,\dd s')\,\nu_{j,C}(\dd s)&=\E\int\abs{O_j(C,s)}L_j(C,s)\,\nu_{j,C}(\dd s)\\ &\le\norm{O_j}_{L^2(\PP)}(1+h_j)^{1/2}, \end{split} \] so $g_j$ is well defined. Conditioning on $C$, we obtain \[ g_j=\E\int O_j(C,s)\bigl(L_j(C,s)-1\bigr)\nu_{j,C}(\dd s)=\E\int\bigl(O_j(C,s)-\bar O_j(C)\bigr)\bigl(L_j(C,s)-1\bigr)\nu_{j,C}(\dd s), \] where in the second equality we used $\int(L_j(c,s)-1)\,\nu_{j,c}(\dd s)=0$. The Cauchy--Schwarz inequality applied to the joint law of $C$ and $s$ gives $g_j^2\le h_j\,\E\Var(O_j\mid C)$. If $h_j=0$, this forces $g_j=0$. This proves the second inequality in \eqref{eq:var-slabs}. \end{proof} \usedin{\cref{prop:var-one-slab}.} We record two comments on \cref{lem:var-slabs}. First, extend $Q_{j,c}$ to a kernel $Q_j$ on the joint state by keeping $C$, the variables $S_k$ with $k\neq j$, and $R$ fixed. Since $S_j$ is conditionally independent of the other variables given $C$, the density of $\PP Q_j$ with respect to $\PP$ is $L_j(C,S_j)$, and we have $h_j=\chi^2(\PP Q_j\,\|\,\PP)$. Second, $g_j$ and $h_j$ are averages over $C$; neither needs to be bounded on every fiber of $C$, and the conditional gain may even be negative on some fibers. This matters for us, because the thermal estimates of \cref{sec:thermal} are statements about the unconditional Gibbs law. On the other hand, conditioning on $C$ must not fix the variable that the kernel changes. In our application, $C$ consists of the cut configurations, and conditioning on them does not fix the endpoint matchings of the slabs, which the surgeries of \cref{sec:surgery} change. We now apply \cref{lem:var-slabs} to the slab decomposition of \cref{sec:id-slabs}. We use the notation of \cref{prop:id-slabs}: $\delta\in(0,\beta]$ is the slab length, $n_\beta=\lfloor\beta/\delta\rfloor$, the cut configurations are $Y_0,\dots,Y_{J-1}$, and the data of slab $j$ are $(\pi_j,X^{(j)}(\cdot))$. A one-slab variable is a quadruple $s=(Y,Y',\pi',X'(\cdot))$ with $Y,Y'\in\mathcal X_N$, $\pi'\in S_N$, and $X'(\cdot)\in C([0,\delta];\mathbb X^N)$, and we call $(Y,Y')$ its cuts. If $\beta>\delta$, we let $\mu_\delta$ be the law \eqref{eq:id-slab-marginal}, which by \cref{prop:id-slabs}(4) is the law of $(Y_{j-1},Y_j,\pi_j,X^{(j)}(\cdot))$ for every $1\le j\le n_\beta$. The surgery of \cref{sec:surgery} acts on a slab that carries additional random marks. For this reason, we allow a probability kernel $\mathfrak m$ from one-slab variables to a standard Borel space $\mathsf M$ of marks, and we let \[ \mu^{\mathfrak m}_\delta(\dd s,\dd\theta)=\mu_\delta(\dd s)\,\mathfrak m(s,\dd\theta) \] be the law of a marked one-slab variable $(s,\theta)$. For a unit vector $u\in\R^3$ on the torus, and for a unit vector $n\in\R^3$ in a container, let \[ \varphi^W_u(s)=L^{-1}\sum_{i=1}^Nu\cdot d(X'_i(\cdot)),\qquad \varphi^{\mathcal A}_n(s)=\frac12\sum_{i=1}^N\int_0^\delta n\cdot\bigl((X'_i(t)-x_0)\times\circ\dd X'_i(t)\bigr), \] so that $\varphi^W_u(Y_{j-1},Y_j,\pi_j,X^{(j)}(\cdot))=u\cdot O^W_j$ and $\varphi^{\mathcal A}_n(Y_{j-1},Y_j,\pi_j,X^{(j)}(\cdot))=n\cdot O^{\mathcal A}_j$ in the notation of \cref{prop:id-slabs}(5). \begin{proposition}\label{prop:var-one-slab} Let $0<\delta<\beta$ and $n_\beta=\lfloor\beta/\delta\rfloor$. Let $\varphi$ be $\varphi^W_u$ on the torus or $\varphi^{\mathcal A}_n$ in a container, and let $O$ be $u\cdot W$, respectively $\mathcal A_n=n\cdot\mathcal A$. Let $\mathfrak m$ be a marking kernel as above, and let $Q$ be a probability kernel on the space of marked one-slab variables such that, for $\mu^{\mathfrak m}_\delta$-almost every $(s,\theta)$, the measure $Q((s,\theta),\cdot)$ is concentrated on marked one-slab variables with the same cuts as $s$. Suppose that $\mu^{\mathfrak m}_\delta Q$ is absolutely continuous with respect to $\mu^{\mathfrak m}_\delta$, and that, for some $g>0$ and $h<\infty$, \[ \E_{\mu^{\mathfrak m}_\delta Q}\varphi-\E_{\mu^{\mathfrak m}_\delta}\varphi\ge g,\qquad \chi^2\bigl(\mu^{\mathfrak m}_\delta Q\,\big\|\,\mu^{\mathfrak m}_\delta\bigr)\le h, \] where $\varphi$ is regarded as a function of $(s,\theta)$ that does not depend on $\theta$. Then \begin{equation}\label{eq:var-one-slab} \Var_\mu(O)\ge\frac{n_\beta\,g^2}{h}. \end{equation} \end{proposition} \begin{proof} We extend the marked slab measure $\bar\mu$ of \eqref{eq:id-marked} by marks $\Theta_1,\dots,\Theta_{n_\beta}$ that are sampled, given all cuts and all slab data, independently with laws $\mathfrak m((Y_{j-1},Y_j,\pi_j,X^{(j)}(\cdot)),\cdot)$. Let $\PP$ be the extended measure. We apply \cref{lem:var-slabs} with $m=n_\beta$, with $C=(Y_0,\dots,Y_{J-1})$, with $S_j=(\pi_j,X^{(j)}(\cdot),\Theta_j)$ for $1\le j\le n_\beta$, and with $R$ the data of the remainder slab if $\beta-n_\beta\delta>0$ (and a constant otherwise). By \cref{prop:id-slabs}(2) and the construction of the marks, $S_1,\dots,S_{n_\beta},R$ are conditionally independent given $C$, and the conditional law of $S_j$ given $C$ is $\nu^{\mathfrak m}(\cdot\mid Y_{j-1},Y_j)$, where \[ \nu^{\mathfrak m}(\dd\pi',\dd X',\dd\theta\mid Y,Y')=\nu_\delta(\dd\pi',\dd X'\mid Y,Y')\,\mathfrak m\bigl((Y,Y',\pi',X'),\dd\theta\bigr). \] By \cref{prop:id-slabs}(5), we have $O\circ\mathrm{Cat}=\sum_{j=1}^{n_\beta}O_j+O_R$ almost surely, where $O_j=\varphi(Y_{j-1},Y_j,\pi_j,X^{(j)}(\cdot))$ and $O_R$ is the contribution of the remainder slab, and all these terms are in $L^2(\PP)$. Since the marks do not change the law of the slab data, \cref{prop:id-slabs}(3) implies that $\Var_{\PP}(O\circ\mathrm{Cat})=\Var_\mu(O)$. For each $j$, let $Q_{j,C}$ be the kernel that acts on $S_j$ as $Q$ acts on the marked one-slab variable $(Y_{j-1},Y_j,S_j)$; this is well defined because $Q$ preserves the cuts. By \cref{prop:id-slabs}(4), the pair $(Y_{j-1},Y_j)$ has the law of the cuts under $\mu_\delta$, and by \eqref{eq:id-slab-law} and \eqref{eq:id-slab-marginal}, the measures $\delta_{(Y,Y')}\otimes\nu^{\mathfrak m}(\cdot\mid Y,Y')$ form a disintegration of $\mu^{\mathfrak m}_\delta$ along its cuts. Let $L$ be the density of $\mu^{\mathfrak m}_\delta Q$ with respect to $\mu^{\mathfrak m}_\delta$. By \cref{lem:var-fibers}, applied with $\Phi$ the map to the cuts, for almost every $(Y,Y')$ the density of $\nu^{\mathfrak m}(\cdot\mid Y,Y')Q$ with respect to $\nu^{\mathfrak m}(\cdot\mid Y,Y')$ is the restriction of $L$, and the quantity $h_j$ of \cref{lem:var-slabs} equals $\chi^2(\mu^{\mathfrak m}_\delta Q\,\|\,\mu^{\mathfrak m}_\delta)\le h$. In the same way, $g_j=\E_{\mu^{\mathfrak m}_\delta Q}\varphi-\E_{\mu^{\mathfrak m}_\delta}\varphi\ge g$. Then \cref{lem:var-slabs} implies \eqref{eq:var-one-slab}. \end{proof} \usedin{\cref{sec:proof-stiffness} (proof of \cref{thm:main-stiffness}) and \cref{sec:proof-ncri} (proof of \cref{thm:main-ncri}).} For winding, \cref{lem:stiff-gain} computes the gain used with \cref{prop:var-one-slab}. For area, the conditional mean gain is proved in \cref{prop:ncri-gain}. \chapter{Random routes with few nearby pairs}\label{sec:routes} We construct two families of random closed walks: winding walks on the torus and polygons with large projected area inside a cube. A deterministic detour construction turns either walk into a simple cycle avoiding a sparse set of bad cells. We bound encounters between independent walks, control changes of the resulting cycles when the bad set changes locally, and average these estimates over the random bad set. The final bound, \cref{cor:routes-final}, is the lattice input to \cref{thm:surgery}. We give the source attributions at the individual constructions below. \section{Notation}\label{sec:routes-notation} \paragraph{Cells and distances.} Throughout this section $\mathbb L$ denotes either $\mathbb Z^3$ or the discrete torus $\mathbb L_K=(\mathbb Z/K\mathbb Z)^3$ with an integer $K\ge3$; its elements are called cells. We write $\mathrm{pr}\colon\mathbb Z^3\to\mathbb L_K$ for the projection, and a lift of a cell or set of cells of $\mathbb L_K$ is a cell or set of cells of $\mathbb Z^3$ that $\mathrm{pr}$ maps onto it. The maximum distance $d_\infty$ is $\abs{x-y}_\infty$ on $\mathbb Z^3$ and $\min_{k\in\mathbb Z^3}\abs{\tilde x-\tilde y+Kk}_\infty$ on $\mathbb L_K$, for lifts $\tilde x,\tilde y$ of $x,y$. For sets $F,F'$ we write $d_\infty(x,F)=\min_{y\in F}d_\infty(x,y)$ and $d_\infty(F,F')=\min_{y\in F}d_\infty(y,F')$, a minimum over the empty set being $+\infty$, and $\mathcal N_r(F)=\{x\in\mathbb L:d_\infty(x,F)\le r\}$ for $r\ge0$; thus $\mathcal N_r(\varnothing)=\varnothing$. The level of a cell is its first coordinate, in $\mathbb Z$ or in $\mathbb Z/K\mathbb Z$. \paragraph{Adjacency and components.} Two cells are nearest neighbors if they differ by one of $\pm e_1,\pm e_2,\pm e_3$ (on $\mathbb L_K$, if they have lifts that do). For an integer $r\ge1$, two distinct cells $x,y$ are range-$r$ adjacent if $d_\infty(x,y)\le r$. Range-one adjacency is called $*$-adjacency, and $*$-adjacent cells are called $*$-neighbors; every cell has at most $26$ of them. A set $F\subset\mathbb L$ is range-$r$ connected if any two of its cells are joined by a chain of cells of $F$ in which consecutive cells are range-$r$ adjacent. The range-$r$ components of a set are its maximal range-$r$ connected subsets, and its $*$-components are its range-one components. A range-$r$ connected set of $m$ cells has $d_\infty$-diameter at most $r(m-1)$. For $F\subset\mathbb Z^3$, the coordinate span of $F$ is the largest of the lengths of its three coordinate projections, and it is at most $r(m-1)$ for such a set. \paragraph{Walks and routes.} A walk is a finite sequence $P=(z_0,\dots,z_{\abs P})$ of cells in which consecutive cells are nearest neighbors. The integer $\abs P\ge0$ is its length, and the pairs $(z_k,z_{k+1})$ are its steps. It is closed if $z_{\abs P}=z_0$. The occurrence list of a closed walk is $(z_0,\dots,z_{\abs P-1})$; the closing cell is not repeated. A sum over $z\in P$ runs over the occurrence list and counts repeated cells with multiplicity, while $d_\infty(x,P)$, $\mathcal N_r(P)$, and the phrase ``cell of $P$'' refer to the set of cells of $P$. A closed walk is simple if $z_0,\dots,z_{\abs P-1}$ are distinct. A route is either a simple closed walk of length at least three, represented by starting at its smallest cell in lexicographic order and keeping its orientation (on $\mathbb L_K$, the order of the representatives in $\{0,\dots,K-1\}^3$), or the empty route $\dagger$, which stands for a rejection and has no cells. \paragraph{Encounter statistics.} For closed walks or routes $P,P'$, an integer $R\ge0$, and an integer $R_c\ge1$, the encounter count and the exponentially weighted distance sum are \begin{equation}\label{eq:routes-J} J_R(P,P')=\sum_{z\in P}\sum_{z'\in P'}\mathbf 1\{d_\infty(z,z')\le R\},\qquad S_{R_c}(P,P')=\sum_{z\in P}e^{-d_\infty(z,P')/R_c}. \end{equation} Both count occurrences, and both vanish if $P$ or $P'$ is $\dagger$; for two routes they are sums over cells. In applications $R$ is the encounter radius and $R_c$ the component range, which is the length scale of the exponentially weighted distance sum because that sum arises from the expansion over components in \cref{lem:routes-environment}. \paragraph{Homology and area.} Let $P$ be a closed walk on $\mathbb L_K$. Since $K\ge3$, each step $(z_k,z_{k+1})$ has a unique displacement $\delta_k\in\{\pm e_1,\pm e_2,\pm e_3\}$ such that $\mathrm{pr}(\tilde z+\delta_k)=z_{k+1}$ for every lift $\tilde z$ of $z_k$. The sum of the displacements lies in $K\mathbb Z^3$, and we define the homology vector and the first homology by \[ \mathrm{hom}(P)=K^{-1}\sum_{k=0}^{\abs P-1}\delta_k\in\mathbb Z^3,\qquad \mathrm{hom}_1(P)=e_1\cdot\mathrm{hom}(P). \] Equivalently, the lift of $P$ that starts at a lift $\tilde z_0$ of $z_0$ ends at $\tilde z_0+K\,\mathrm{hom}(P)$. For a closed walk $P$ in $\mathbb Z^3$, write $x(z)$ and $y(z)$ for the first two coordinates of a cell $z$, and define the projected area \begin{equation}\label{eq:routes-area} \operatorname{area}(P)=\sum_{k=0}^{\abs P-1}x(z_k)\bigl(y(z_{k+1})-y(z_k)\bigr)=-\sum_{k=0}^{\abs P-1}y(z_k)\bigl(x(z_{k+1})-x(z_k)\bigr). \end{equation} The two sums agree because at most one of $x$ and $y$ changes in a step, so that $x(z_k)(y(z_{k+1})-y(z_k))+y(z_k)(x(z_{k+1})-x(z_k))$ is the increment of $xy$ in the $k$-th step, and these increments sum to zero. The quantity $\operatorname{area}(P)$ is the line integral of $x\,\mathrm dy$ along the projection of $P$ to the plane of the first two coordinates. It is invariant under translations, changes sign under reversal of orientation, and for a simple closed curve in that plane equals the enclosed area, positive for counterclockwise orientation. \paragraph{Cycle decomposition.} Every closed walk $P$ is decomposed into simple closed walks by the following fixed rule. If $\abs P=0$, the decomposition is empty. Otherwise let $j\ge1$ be the smallest index such that $z_j=z_i$ for some $i0$, and for $K\ge3$ let $X_K$ equal $(\log K)/(2\vartheta)$ with probability $1/(K\log K)$ and zero otherwise. Then $\E X_K=1/(2\vartheta K)$ and $\E e^{2\vartheta X_K}=1+(K-1)/(K\log K)\le2$, but \[ \E e^{\vartheta X_K}-1=\frac{K^{1/2}-1}{K\log K}, \] which is much larger than $1/K$. The random root must therefore enter the ordered-moment argument itself, as in the proof of \cref{prop:routes-nominal}. \end{remark} \paragraph{The dyadic velocity array.} Let $c_{\rm vel}=(1-2^{-1/4})/2$, so that $c_{\rm vel}\sum_{l\ge0}2^{-l/4}=1/2$. A dyadic array is a family $(U_{l,k})_{l,k\ge0}$ of independent random variables, each uniform on $[-1,1]$. Its velocities and displacements are \begin{equation}\label{eq:routes-velocity} v_i=c_{\rm vel}\sum_{l\ge0}2^{-l/4}U_{l,\lfloor(i-1)/2^l\rfloor}\quad(i\ge1),\qquad Z_s=\sum_{i=1}^sv_i\quad(s\ge0). \end{equation} The series converges absolutely for every sample, $\abs{v_i}\le1/2$, and $\abs{Z_s-Z_{s'}}\le\abs{s-s'}/2$. The variable $U_{l,k}$ is attached to the dyadic block $I_{l,k}=\{k2^l+1,\dots,(k+1)2^l\}$: it enters $v_i$ if and only if $i\in I_{l,k}$. For $u\ge0$, the variables revealed by time $u$ are the $U_{l,k}$ with $I_{l,k}\cap\{1,\dots,u\}\neq\varnothing$. They determine $Z_1,\dots,Z_u$, because $v_i$ involves only variables whose blocks contain $i$. \begin{lemma}\label{lem:routes-dyadic} Let $U^{(2)}$ and $U^{(3)}$ be independent dyadic arrays with displacements $Z^{(2)}_s$ and $Z^{(3)}_s$, and write $Z_s=(Z^{(2)}_s,Z^{(3)}_s)$. Let $u\in\mathbb Z_{\ge0}$, let $\mathcal G$ be a $\sigma$-algebra independent of the two arrays, and let $\mathcal F$ be generated by $\mathcal G$ and by the variables of both arrays revealed by time $u$. Set $c_{\rm d}=4^{3/4}/(2c_{\rm vel})$. Then for every integer $k\ge1$ the conditional law of $Z_{u+k}$ given $\mathcal F$ has a density bounded by $c_{\rm d}^2k^{-3/2}$. Moreover, for every $w\in\mathbb Z^2$, we have \begin{equation}\label{eq:routes-point-Z} \PP\bigl(\lfloor Z_{u+k}/8\rfloor=w\bigm|\mathcal F\bigr)\le64\,c_{\rm d}^2\,k^{-3/2}, \end{equation} and for every integer $K\ge k$, we have \begin{equation}\label{eq:routes-point-torus} \PP\bigl(\lfloor Z_{u+k}\rfloor\in w+K\mathbb Z^2\bigm|\mathcal F\bigr)\le4\,c_{\rm d}^2\,k^{-3/2}. \end{equation} Here the integer part is taken in each coordinate. \end{lemma} \begin{proof} We first find an unrevealed block inside $\{u+1,\dots,u+k\}$. If $k=1$, take $l=0$ and $m=u$, so that $I_{l,m}=\{u+1\}$. If $k\ge2$, take $l$ with $k/4<2^l\le k/2$ and let $m$ be the smallest integer with $m2^l\ge u$. Then $m2^l1$ by the same inequality. The event $\lfloor Z^{(j)}_{u+k}/8\rfloor=w_j$ is the event that $Z^{(j)}_{u+k}$ lies in an interval of length $8$, and its conditional probability is at most $8c_{\rm d}k^{-3/4}$. Taking the product over the two coordinates yields \eqref{eq:routes-point-Z}. For \eqref{eq:routes-point-torus}, the event $\lfloor Z^{(j)}_{u+k}\rfloor\in w_j+K\mathbb Z$ is the event that $Z^{(j)}_{u+k}$ lies in the union of the intervals $[w_j+Kt,w_j+Kt+1)$, $t\in\mathbb Z$. Since $2^l\le k\le K$ and $2c_{\rm vel}<1$, we have $\lambda\le k^{3/4}\le K$. Points of two of these intervals whose indices $t$ differ by at least two are more than $2K-1\ge K$ apart, so an interval of length $\lambda\le K$ meets at most two of them. The conditional probability is at most $2c_{\rm d}k^{-3/4}$. Taking the product over the two coordinates gives \eqref{eq:routes-point-torus}. \end{proof} The velocity array \eqref{eq:routes-velocity} is the one of \cite[Section~8.1]{OpenAI2026B}, reproduced in \cite[Equation~(8.4)]{OpenAI2026BEC}, and the density bound is proved in the course of \cite[Lemma~8.2]{OpenAI2026BEC} for the first half of a folded skeleton; we have added the two lattice consequences. The mechanism of independent aligned blocks on all dyadic scales goes back to H\"aggstr\"om and Mossel \cite[Section~3, Proposition~3.1]{HaggstromMossel1998}. The use of an unpredictable path, whose future position has small conditional point probabilities, to control intersections with an independent path is due to Benjamini, Pemantle, and Peres \cite[Theorem~1.3 and Lemma~3.1]{BenjaminiPemantlePeres1998}; see Hoffman \cite{Hoffman1998} for the limits of predictability profiles of nearest-neighbor processes. Stopping and reflecting an unpredictable path is used as a symmetrization device in \cite[Section~6, Lemma~6.1]{AbbeMassoulieMontanariSlySrivastava2018} and in \cite[Section~2, proof of Theorem~1.3]{GarbanSpencer2022}; the latter's Lemma~2.5 bounds the second moment of the resulting estimator. The lattice constructions below instead fold the transverse displacements at deterministic midpoints; their geometry and encounter bounds are proved here. \usedin{\cref{prop:routes-nominal}.} \section{Two random skeletons}\label{sec:routes-skeletons} A skeleton is a random closed walk built from a finite family of independent random variables, which we call its seed. Two skeletons are independent if their seeds are independent. We use $\Gamma$ for the full random input and for the walk it determines; statistics of $\Gamma$ depend only on the walk. In \cref{sec:surgery}, the seeds are drawn independently of the gas. \begin{definition}\label{def:routes-loop} Let $K\ge20$ be an integer. The seed of a winding skeleton consists of three independent components: a root $\varrho$ uniform on $(\mathbb Z/K\mathbb Z)^2$, and two dyadic arrays $U^{(2)},U^{(3)}$ with displacements $Z_s=(Z^{(2)}_s,Z^{(3)}_s)$. Identify $\varrho$ with its representative in $\{0,\dots,K-1\}^2$. For $0\le s\le K$, the lifted base vertices are \begin{equation}\label{eq:routes-loop-base} \tilde b_s=\bigl(s,\ \varrho+\lfloor Z_{\min(s,K-s)}\rfloor\bigr)\in\mathbb Z^3, \end{equation} so that $\tilde b_K=\tilde b_0+Ke_1$. For $0\le s0$. Since these coordinates are integers, they are at most $-2$ and at least $1$, respectively. The cells created at a level by a refined step have second coordinates between those of the two base vertices of the step. For $2\le q\le2n-2$, the cells of the lower branch at level $q$ have second coordinate at most $-2$, and those of the upper branch at least $1$. At level $1$, the lower branch, whose step starts at $o_0$, has second coordinates at most $0$, and the upper branch at least $1$; level $2n-1$ is symmetric. At level $0$ the lower branch consists of $o_0$ alone, and at level $2n$ the upper branch consists of $o_2$ alone. We conclude that $\Gamma$ is simple and that it passes through the anchors in order. For the area, note that in each refined step the only step changing the first coordinate is the first one, which starts at the base vertex. By the second formula in \eqref{eq:routes-area} and translation invariance, we have \[ \operatorname{area}(\Gamma)=\sum_{s=0}^{n-1}\Bigl[\bigl(2s-d^{(1,2)}_s\bigr)+\bigl(2n-2s-d^{(2,2)}_s\bigr)+\bigl(2s+d^{(3,2)}_s\bigr)+\bigl(2n-2s+d^{(4,2)}_s\bigr)\Bigr]. \] The terms without $d$ sum to $4n^2$. Since $\abs{d^{(\iota,2)}_s}\le\min(s,n-s)/16+1$ and $\sum_{s=0}^{n-1}\min(s,n-s)\le n^2/4$, the remaining terms are at least $-4(n^2/64+n)$. This proves the first bound in \eqref{eq:routes-polygon-area}. For the margin, the bounds on $d^{(\iota,j)}_s$ show that all base vertices, and hence all cells of $\Gamma-\varrho$, lie in $[0,2n]\times[-2n-1,2n+1]\times[-n/32-1,n/32+1]$. Since $2n+1\le K_{\rm in}/32+1$ and $\varrho-x_{\rm in}\in[K_{\rm in}/3,K_{\rm in}/2]^3$, every coordinate of $z-x_{\rm in}$, for a cell $z$ of $\Gamma$, lies in $[29K_{\rm in}/96-1,\,17K_{\rm in}/32+1]$. A cell $v\notin Q_{\rm in}$ has a coordinate of $v-x_{\rm in}$ below $0$ or at least $K_{\rm in}$. The distance $d_\infty(z,v)$ is at least $\min\{29K_{\rm in}/96,\,15K_{\rm in}/32-1\}\ge K_{\rm in}/4$. \end{proof} \begin{proof}[Proof of \cref{prop:routes-nominal}\textup{(c)}] Fix $R\ge0$ and $R_c\ge1$; in this proof $C$ denotes a constant depending only on $R$ and $R_c$, which may change from line to line. In both cases we construct an integer $\bar u\ge0$, a filtration $(\mathcal F_u)_{0\le u\le\bar u}$, and adapted random variables $X_u,X'_u\in[0,1]$ such that, with $\bar K=K$ for winding skeletons and $\bar K=K_{\rm in}$ for area skeletons, \begin{equation}\label{eq:routes-three} \begin{gathered} J_R(\Gamma,\Gamma')\le C_1\sum_{u=0}^{\bar u}X_u,\qquad S_{R_c}(\Gamma,\Gamma')\le C_1\sum_{u=0}^{\bar u}X'_u,\\ \E[X_{u'}\mid\mathcal F_u]+\E[X'_{u'}\mid\mathcal F_u]\le C(u'-u)^{-3/2}\quad(0\le u2r$ be integers, and let $C\subset\mathbb L_K$ be range-$r$ connected with $m$ cells, where $rm2r$, if $x,y\in\mathbb L_K$ are range-$r$ adjacent and $\tilde x$ is a lift of $x$, then $y$ has a unique lift at $d_\infty$-distance at most $r$ from $\tilde x$. Lifting the tree edge by edge from a lift of one of its cells produces a range-$r$ connected set $\tilde C\subset\mathrm{pr}^{-1}(C)$ of $m$ cells, with coordinate span at most $r(m-1)r(m-1)$. Hence the translates are the range-$r$ components of $\mathrm{pr}^{-1}(C)$. For the second, let $\tilde x\in\mathrm{pr}^{-1}(B)$ be $*$-adjacent to a cell $\tilde y$ of a lift $\tilde C$. Since $K\ge3$, the cells $\mathrm{pr}(\tilde x)$ and $\mathrm{pr}(\tilde y)$ are distinct and $*$-adjacent, so $\mathrm{pr}(\tilde x)\in C$ by maximality of $C$. Hence $\tilde x\in\mathrm{pr}^{-1}(C)$ is range-one adjacent to $\tilde y$, and $\tilde x\in\tilde C$. \begin{lemma}\label{lem:routes-animals} Let $r\ge1$ and $m\ge1$ be integers. On $\mathbb Z^3$ and on $\mathbb L_K$, the number of range-$r$ connected sets of $m$ cells containing a given cell is at most $(2r+1)^{6(m-1)}$. If $B$ is a random field with $\PP(F\subset B)\le p^{\abs F}$ for every finite $F\subset\mathbb L$, then for every cell $x$, \begin{equation}\label{eq:routes-animal-tail} \PP\bigl(x\in B\text{ and the range-$r$ component of $B$ containing $x$ has at least $m$ cells}\bigr)\le\bigl((2r+1)^6p\bigr)^m. \end{equation} \end{lemma} \begin{proof} Assign to each such set a spanning tree for range-$r$ adjacency, rooted at the given cell, and traverse it depth first. The traversal has $2(m-1)$ steps, each given by one of fewer than $(2r+1)^3$ displacements, and the visited cells recover the set. This proves the first bound. If the range-$r$ component of $B$ containing $x$ has at least $m$ cells, then growing a spanning tree from $x$ produces a range-$r$ connected set $F\ni x$ of exactly $m$ cells with $F\subset B$. The union bound and the first statement give at most $(2r+1)^{6(m-1)}p^m$, which implies \eqref{eq:routes-animal-tail}. \end{proof} This counting bound and the depth-first argument are recorded in \cite[Section~8.2]{OpenAI2026BEC}. Such estimates are instances of the classical Peierls counting principle \cite[pp.~479--480]{Peierls1936}; see also \cite[Section~1.4, pp.~16--18]{Grimmett1999}. An early tree-encoding exponential bound for plane animals is given in \cite[pp.~852--853, Lemma~3]{Klarner1967}. Only the joint bound on prescribed sets enters; no independence between cells is assumed. \begin{lemma}\label{lem:routes-area} Every closed walk $P$ in $\mathbb Z^3$ satisfies $\abs{\operatorname{area}(P)}\le\abs P^2/16$. If $\operatorname{area}(P)>0$, then some walk $c$ in the cycle decomposition of $P$ satisfies \[ \operatorname{area}(c)\ge\frac{16\operatorname{area}(P)^2}{\abs P^2}. \] \end{lemma} \begin{proof} Let $t_x=\sum_k\abs{x(z_{k+1})-x(z_k)}$ and $t_y=\sum_k\abs{y(z_{k+1})-y(z_k)}$ be the total variations of the first two coordinates along $P$, so that $t_x+t_y\le\abs P$. Since $P$ is closed, the values of $x$ along $P$ lie in an interval of length at most $t_x/2$; let $\bar x$ be its midpoint. Since the increments of $y$ along $P$ sum to zero, we obtain from \eqref{eq:routes-area} \[ \abs{\operatorname{area}(P)}=\Bigl\lvert\sum_k\bigl(x(z_k)-\bar x\bigr)\bigl(y(z_{k+1})-y(z_k)\bigr)\Bigr\rvert\le\frac{t_xt_y}4\le\frac{(t_x+t_y)^2}{16}\le\frac{\abs P^2}{16}. \] For the second statement, let $c_1,\dots,c_J$ be the cycle decomposition of $P$ and let $A_{\max}=\max_i\operatorname{area}(c_i)$, which is positive since the areas sum to $\operatorname{area}(P)>0$. By additivity and the first statement applied to each $c_i$, we have \[ \operatorname{area}(P)\le\sum_{i:\operatorname{area}(c_i)>0}\operatorname{area}(c_i)\le A_{\max}^{1/2}\sum_{i:\operatorname{area}(c_i)>0}\operatorname{area}(c_i)^{1/2}\le A_{\max}^{1/2}\sum_i\frac{\abs{c_i}}4=\frac{A_{\max}^{1/2}\abs P}4, \] which is the claim. \end{proof} \usedin{\cref{prop:routes-router} (the area bound after cycle selection).} \section{Routing maps}\label{sec:routes-routing} A routing map turns a skeleton and a field into a route. In \cref{sec:surgery}, the routing map of one surgery is evaluated at fields obtained from the current field by the other surgery, and \cref{lem:routes-stability} compares the outputs. This requires locality in a precise sense, which we formulate through the components of the field that meet the skeleton. \begin{definition}\label{def:routes-local} Let $\Gamma$ be a closed walk on $\mathbb L$. For a field $B\subset\mathbb L$, let $\mathcal C(B,\Gamma)$ be the set of $*$-components of $B$ that contain a cell of $\Gamma$, and let \[ \mathcal Q(B,\Gamma)=\Gamma\cup\mathcal N_1\Bigl(\bigcup_{C\in\mathcal C(B,\Gamma)}C\Bigr). \] A routing map for $\Gamma$ assigns to every field $B$ either $\dagger$ or a route $\mathcal R(B,\Gamma)$. (For the skeletons of \cref{sec:routes-skeletons}, the lift and the anchors are determined by $\Gamma$.) The routing map is component-local if the following hold for all fields $B,B'$. \begin{enumerate}[label=\textup{(\alph*)}] \item If $\mathcal C(B,\Gamma)=\mathcal C(B',\Gamma)$, then $\mathcal R(B,\Gamma)=\mathcal R(B',\Gamma)$. \item If $\mathcal R(B,\Gamma)\neq\dagger$, every cell of $\mathcal R(B,\Gamma)$ is a cell of $\Gamma$ or lies within $d_\infty$-distance one of a component in $\mathcal C(B,\Gamma)$. \end{enumerate} \end{definition} \begin{lemma}\label{lem:routes-explored} Let $\Gamma$ be a closed walk on $\mathbb L$, and let $B,B'\subset\mathbb L$ be fields that agree on $\mathcal Q(B,\Gamma)$. Then $\mathcal C(B',\Gamma)=\mathcal C(B,\Gamma)$ and $\mathcal Q(B',\Gamma)=\mathcal Q(B,\Gamma)$. In particular, $\mathcal R(B',\Gamma)=\mathcal R(B,\Gamma)$ for every component-local routing map for $\Gamma$. \end{lemma} \begin{proof} Let $C\in\mathcal C(B,\Gamma)$. Its cells lie in $\mathcal Q(B,\Gamma)\cap B$, hence in $B'$. The cells at $d_\infty$-distance one from $C$ lie in $\mathcal Q(B,\Gamma)$, and by maximality of $C$ they lie outside $B$, hence outside $B'$. So $C$ is a $*$-component of $B'$, and it contains a cell of $\Gamma$. Conversely, let $C'\in\mathcal C(B',\Gamma)$, and let $z$ be a cell of $C'$ and of $\Gamma$. Then $z\in B'\cap\mathcal Q(B,\Gamma)=B\cap\mathcal Q(B,\Gamma)$, so $z$ lies in some $C\in\mathcal C(B,\Gamma)$. That $C$ is a $*$-component of $B'$ containing $z$, so $C'=C$. This proves $\mathcal C(B',\Gamma)=\mathcal C(B,\Gamma)$, which implies $\mathcal Q(B',\Gamma)=\mathcal Q(B,\Gamma)$, and the last assertion is \cref{def:routes-local}(a). \end{proof} \begin{definition}\label{def:routes-routing} \label{def:routes-router-torus} \label{def:routes-router-box} The two routing maps are defined as follows. In both, a walk produced in the last step is represented as a route. \emph{Winding routing map.} Let $K\ge20$ and let $\Gamma$ be a winding skeleton, with lift $\tilde\Gamma$ from $\tilde b_0$ to $\tilde b_0+Ke_1$. For a field $B\subset\mathbb L_K$, the route $\mathcal R(B,\Gamma)$ is computed in four steps. \begin{enumerate} \item \emph{Size rule.} Reject if some $C\in\mathcal C(B,\Gamma)$ has at least $\lfloor K/20\rfloor$ cells. Otherwise every $C\in\mathcal C(B,\Gamma)$ has fewer than $K/20$ cells, and by \cref{lem:routes-lifts} its lifts are finite $*$-components of $\mathrm{pr}^{-1}(B)$ mapped bijectively onto it. \item \emph{Fill rule.} Reject if $b_0=\mathrm{pr}(\tilde b_0)$ belongs to $\mathrm{pr}(\operatorname{fill}(\tilde C))$ for a lift $\tilde C$ of a component in $\mathcal C(B,\Gamma)$; this set does not depend on the choice of the lift. \item \emph{Detours.} Set $P=\tilde\Gamma$. As long as $P$ has a cell in $\mathrm{pr}^{-1}(B)$, let $k$ be the smallest index with $P_k\in\mathrm{pr}^{-1}(B)$, let $\tilde C$ be the lift of a component in $\mathcal C(B,\Gamma)$ that contains $P_k$, and let $k'$ be the largest index with $P_{k'}\in\tilde C$. Replace the part $(P_{k-1},\dots,P_{k'+1})$ of $P$ by the shortest nearest-neighbor path from $P_{k-1}$ to $P_{k'+1}$ in $\partial^{\rm ext}_*\tilde C$ that comes first in lexicographic order. \item \emph{Selection.} Decompose the closed walk $\mathrm{pr}(P)$ on $\mathbb L_K$ by the cycle decomposition, and output the first walk in it with positive first homology. \end{enumerate} \emph{Area routing map.} Let $K_{\rm in}\ge2^{14}$, let $n=\lfloor K_{\rm in}/64\rfloor$, and let $\Gamma$ be an area skeleton with anchors $o_0,\dots,o_3$. For a field $B\subset\mathbb Z^3$, the route $\mathcal R(B,\Gamma)$ is computed in four steps. \begin{enumerate} \item \emph{Budget rule.} Reject if $\sum_{C\in\mathcal C(B,\Gamma)}\abs C^2>n$, in particular if some $C\in\mathcal C(B,\Gamma)$ is infinite. \item \emph{Anchor rule.} Reject if one of $o_0,o_1,o_2,o_3$ belongs to $\operatorname{fill}(C)$ for a component $C\in\mathcal C(B,\Gamma)$. \item \emph{Detours segment by segment.} For each $\iota\in\{1,2,3,4\}$ separately, let $P$ be segment $\iota$ of $\Gamma$, from $o_{\iota-1}$ to $o_\iota$. As long as $P$ has a bad cell, let $k$ be the smallest index with $P_k\in B$, let $C$ be the component in $\mathcal C(B,\Gamma)$ that contains $P_k$, and let $k'$ be the largest index with $P_{k'}\in C$. Replace $(P_{k-1},\dots,P_{k'+1})$ by the shortest nearest-neighbor path from $P_{k-1}$ to $P_{k'+1}$ in $\partial^{\rm ext}_*C$ that comes first in lexicographic order. Concatenate the four resulting walks into a closed walk $P$ through $o_0,\dots,o_3$. \item \emph{Selection.} Decompose $P$ by the cycle decomposition. Reject if no walk in it has positive area; otherwise output the first walk in the decomposition whose area is the largest. \end{enumerate} \end{definition} \begin{figure}[tb] \centering \begin{tikzpicture}[scale=1.0] \draw[thin] (0,0) -- (0.75,-0.65) -- (1.5,-1.6) -- (2.25,-2.2) -- (3,-3); \draw[thin] (3,-3) -- (3.75,-2.35) -- (4.0,-2.033); \draw[thin,dashed] (4.0,-2.033) -- (4.5,-1.4) -- (5.0,-1.0); \draw[thin] (5.0,-1.0) -- (5.25,-0.8) -- (6,0); \draw[thin] (6,0) -- (5.25,0.85) -- (4.5,1.4) -- (3.75,2.3) -- (3,3); \draw[thin] (3,3) -- (2.25,2.15) -- (1.5,1.55) -- (0.75,0.7) -- (0,0); \draw[->,thin] (1.0,-0.967) -- (1.12,-1.12); \draw[->,thin] (5.6,0.45) -- (5.45,0.62); \fill[gray!45] (4.5,-1.45) circle (0.3); \node[font=\small] at (4.5,-1.45) {$C$}; \draw[line width=1.4pt] (4.0,-2.033) -- (4.55,-2.0) -- (4.95,-1.6) -- (5.0,-1.0); \fill (0,0) circle (0.06); \node[left] at (0,0) {$o_0$}; \fill (3,-3) circle (0.06); \node[below] at (3,-3) {$o_1$}; \fill (6,0) circle (0.06); \node[right] at (6,0) {$o_2$}; \fill (3,3) circle (0.06); \node[above] at (3,3) {$o_3$}; \end{tikzpicture} \caption{Schematic projection of an area skeleton (thin) to the plane of the first two coordinates, traversed counterclockwise through the anchors $o_0,\dots,o_3$. The second segment meets a bad $*$-component $C$ (gray). The area routing map replaces the part of this segment between its first and last visits to $C$ (dashed) by a path in the exterior shell of $C$ (thick). The other segments and the four anchors are unchanged. Perturbations, the third coordinate, and the lattice are not drawn to scale.} \label{fig:routes-diamond} \end{figure} The detours follow the exterior shell of a component from the cell immediately before the first visit of the walk to that component to the cell immediately after its last visit, as in \cite[Section~7.3]{OpenAI2026HS}, reproduced in \cite[Section~8.3]{OpenAI2026BEC}; related detours appear in \cite[Lemma~8.2]{OpenAI2026B}. In these sources the size and fill tests are applied to the whole environment before the path is drawn \cite[Equation~(8.16)]{OpenAI2026BEC}. Here they are applied only to components met by the skeleton, which makes the maps component-local, and the loop erasure of an open path is replaced by the cycle decomposition, which preserves homology and area. The size rule gives finite lifts, and the fill rule keeps the endpoints of the lifted walk outside the fills of the detoured components. The area map is anchored: each segment is corrected separately and its endpoints never move. A single first-to-last detour along the whole closed skeleton could remove the entire polygon, for instance when a small bad component near $o_0$ is met at the beginning and at the end of the walk. The bound on the sum of squared component sizes controls the area lost in the detours, since a detour around $C$ changes the area by at most a constant times $\abs C^2$, and it keeps the components that are used close to the skeleton. \begin{proposition}\label{prop:routes-router} Let $p_0=1/1458$. There is an absolute constant $C_{\rm acc}$ such that the following hold. \begin{enumerate}[label=\textup{(\alph*)}] \item Let $K\ge20$ and let $\Gamma$ be a winding skeleton. The winding routing map is a component-local routing map for $\Gamma$. For every field $B\subset\mathbb L_K$, the route $\gamma=\mathcal R(B,\Gamma)$ is either $\dagger$ or a good route with $\mathrm{hom}_1(\gamma)\ge1$, every cell of which is a cell of $\Gamma$ or the projection of a cell of $\partial^{\rm ext}_*\tilde C$ for a lift $\tilde C$ of a $*$-component of $B$ containing a cell of $\Gamma$. \item Let $K_{\rm in}\ge2^{14}$, let $n=\lfloor K_{\rm in}/64\rfloor$, and let $\Gamma$ be an area skeleton with inner cube $Q_{\rm in}$. The area routing map is a component-local routing map for $\Gamma$. For every field $B\subset\mathbb Z^3$, the route $\gamma=\mathcal R(B,\Gamma)$ is either $\dagger$ or a good route with \begin{equation}\label{eq:routes-area-output} \abs\gamma\le124\,n,\qquad\operatorname{area}(\gamma)\ge\frac{n^2}{250}, \end{equation} every cell of which is a cell of $\Gamma$ or a cell of $\partial^{\rm ext}_*C$ for a $*$-component $C$ of $B$ containing a cell of $\Gamma$. Every cell of $\gamma$ lies in $\mathcal N_{\lfloor\sqrt n\rfloor+1}(\Gamma)\subset Q_{\rm in}$. \item Let $\Gamma$ be a random winding skeleton with $K\ge20$, or a random area skeleton with $K_{\rm in}\ge2^{14}$. Let $B$ be a random field, independent of the seed of $\Gamma$, such that $\PP(F\subset B)\le p^{\abs F}$ for every finite $F\subset\mathbb L$, where $0\le p\le p_0$. Then $\PP(\mathcal R(B,\Gamma)=\dagger)\le C_{\rm acc}\,p$. \end{enumerate} \end{proposition} \begin{proof} \emph{Component-locality.} In both maps, the size and budget rules, the fill and anchor rules, and the exterior shells, with their lifts in the winding case, are functions of $\Gamma$ and $\mathcal C(B,\Gamma)$, and the selection steps read only the walk $P$. In the detour steps, every cell of the current walk $P$ is a cell of $\Gamma$ (in the winding case, of $\tilde\Gamma$) or a cell of the exterior shell of some $C\in\mathcal C(B,\Gamma)$ (in the winding case, of a lift of such a $C$). A cell of $\partial^{\rm ext}_*C$ is good, since otherwise it would be $*$-adjacent to $C$ and lie in $C$. In the winding case, a cell $\tilde x$ of $\partial^{\rm ext}_*\tilde C$ satisfies $\mathrm{pr}(\tilde x)\notin C$, since otherwise $\tilde x$ would be a cell of $\mathrm{pr}^{-1}(C)$ that is $*$-adjacent to $\tilde C$, hence a cell of $\tilde C$ by \cref{lem:routes-lifts}. So $\mathrm{pr}(\tilde x)$ is a $*$-neighbor of $C$ outside $C$, and it is good. A cell of $\Gamma$ is bad exactly when it lies in a component in $\mathcal C(B,\Gamma)$. Hence a cell of $P$ lies in $B$ (in the winding case, in $\mathrm{pr}^{-1}(B)$) if and only if it lies in a component in $\mathcal C(B,\Gamma)$ (in the winding case, in a lift of one). Every test in the detour steps, and the component used in each replacement, is therefore decided by $\mathcal C(B,\Gamma)$. By induction over the replacements, two fields $B,B'$ with $\mathcal C(B,\Gamma)=\mathcal C(B',\Gamma)$ produce the same walks and the same route. This is \cref{def:routes-local}(a). \emph{Proof of (a).} Suppose that neither the size rule nor the fill rule rejects. The cell $b_0$ is good, since otherwise it would belong to a component $C\in\mathcal C(B,\Gamma)$ and to $C\subset\mathrm{pr}(\operatorname{fill}(\tilde C))$, and the fill rule would reject. We claim that throughout the detour step, $P$ is a walk from $\tilde b_0$ to $\tilde b_0+Ke_1$ whose cells are cells of $\tilde\Gamma$ or good cells, so that its cells in $\mathrm{pr}^{-1}(B)$ are cells of $\tilde\Gamma$. This holds initially. Let $k$, $\tilde C$, and $k'$ be as in the definition; the lift $\tilde C$ exists because $\mathrm{pr}(P_k)$ is a bad cell of $\Gamma$, whose $*$-component lies in $\mathcal C(B,\Gamma)$. Since the endpoints of $P$ are good, $1\le k\le k'<\abs P$. The cells $P_0,\dots,P_{k-1}$ are good. They avoid $\tilde C$ and join $\tilde b_0$ to $P_{k-1}$. Since $\mathrm{pr}(\tilde b_0)=b_0\notin\mathrm{pr}(\operatorname{fill}(\tilde C))$, the cell $\tilde b_0$ lies in the exterior of $\tilde C$, and so does $P_{k-1}$. By the choice of $k'$, the cells after $P_{k'}$ avoid $\tilde C$ and join $P_{k'+1}$ to $\tilde b_0+Ke_1$, which also lies in the exterior since it projects to $b_0$. Both $P_{k-1}$ and $P_{k'+1}$ are nearest neighbors of cells of $\tilde C$. They lie in $\partial^{\rm ext}_*\tilde C$. By \cref{lem:routes-shell,lem:routes-lifts}, they are joined by a nearest-neighbor path in $\partial^{\rm ext}_*\tilde C$, and all its cells are good. The replacement removes every visit of $P$ to $\tilde C$ and adds only good cells. This proves the claim, and shows that the number of visits of $P$ to $\mathrm{pr}^{-1}(B)$ decreases strictly at each replacement. Then the detour step ends with a walk $P$ from $\tilde b_0$ to $\tilde b_0+Ke_1$ consisting of good cells. The closed walk $\mathrm{pr}(P)$ has homology $e_1$, because its lift $P$ ends at $\tilde b_0+Ke_1$. By additivity, the first homologies of the walks in its cycle decomposition sum to one. Then one of them is positive, and the selection step does not reject. The selected walk $\gamma$ is simple, consists of good cells, and has length at least $K\ge3$, because a closed walk with nonzero first homology has at least $K$ steps in the first coordinate. Its cells are cells of $\Gamma$ or projections of shell cells of lifts of components in $\mathcal C(B,\Gamma)$, and the latter lie within distance one of the corresponding component. This proves (a) and \cref{def:routes-local}(b). \emph{Proof of (b).} On acceptance, $\sum_{C\in\mathcal C(B,\Gamma)}\abs C^2\le n$. Each $C\in\mathcal C(B,\Gamma)$ is $*$-connected, contains a cell of $\Gamma$, and has at most $\lfloor\sqrt n\rfloor$ cells, so $C$ and its exterior shell lie in $\mathcal N_{\lfloor\sqrt n\rfloor}(\Gamma)$. All cells of an accepted route lie in $\mathcal N_{\lfloor\sqrt n\rfloor+1}(\Gamma)$. Since $n\le K_{\rm in}/64$ and $K_{\rm in}\ge2^{14}$, we have $\sqrt n+1\le\sqrt{K_{\rm in}}/8+1\frac{n^2}{250}. \] It is simple, its cells are good, and its length is at least four since its area is positive. Its cells are cells of $\Gamma$ or shell cells of components in $\mathcal C(B,\Gamma)$. This proves (b) and \cref{def:routes-local}(b). \emph{Proof of (c).} Fix the seed of $\Gamma$. Since $B$ is independent of the seed, the prescribed-set bound holds conditionally on it, and we apply \cref{lem:routes-animals} with $r=1$, for which $(2r+1)^6=729$; note that $729p\le1/2$. In the winding case, the size rule rejects only if some $*$-connected set of $\lfloor K/20\rfloor$ bad cells contains an occurrence of $\Gamma$, which has probability at most $3K(729p)^{\lfloor K/20\rfloor}$. If the fill rule rejects because of a component $C\in\mathcal C(B,\Gamma)$ with $m$ cells, then a lift of $b_0$ lies in the bounding box of a lift of $C$, whose coordinate span is at most $m-1$ by \cref{lem:routes-lifts}. Every cell of $C$ is then within distance $m-1$ of $b_0$, so the $*$-component of $B$ containing one of the at most $(2m-1)^3$ cells within distance $m-1$ of $b_0$ has at least $m$ cells. Summing over $m$, we obtain \begin{align*} \PP(\mathcal R(B,\Gamma)=\dagger)&\le3K(729p)^{\lfloor K/20\rfloor}+\sum_{m\ge1}(2m-1)^3(729p)^m\\ &\le729p\Bigl(3K\,2^{1-\lfloor K/20\rfloor}+\sum_{m\ge1}(2m-1)^32^{1-m}\Bigr), \end{align*} and the bracket is bounded by an absolute constant for $K\ge20$. In the area case, if the budget rule rejects, then the sum of $\abs C^2$ over the $*$-components $C$ of $B$ containing a cell of $\Gamma$ exceeds $n$. Each such component contains an occurrence of $\Gamma$. Writing $C(z)$ for the $*$-component of $B$ containing $z$ when $z\in B$, we have by Markov's inequality and \cref{lem:routes-animals} \begin{align*} \PP(\text{budget rule rejects})&\le\frac1n\sum_{z\in\Gamma}\E\bigl[\mathbf 1\{z\in B\}\abs{C(z)}^2\bigr]\\ &\le\frac{\abs\Gamma}n\sum_{m\ge1}m^2(729p)^m\le20\cdot729p\sum_{m\ge1}m^22^{1-m}. \end{align*} Here we used $\E Y^2=\sum_{m\ge1}(2m-1)\PP(Y\ge m)$ for $Y=\mathbf 1\{z\in B\}\abs{C(z)}$, which is almost surely finite by \eqref{eq:routes-animal-tail}, and $\abs\Gamma\le20n$. The anchor rule is bounded as the fill rule in the winding case, for each of the four anchors. This proves (c). \end{proof} \usedin{\cref{thm:surgery} (acceptance and component-locality), \cref{lem:stiff-gain} (first homology), \cref{prop:ncri-gain} (length and area), and \cref{sec:proof-ncri} (containment in the inner cube).} \section{Stability of routes under local changes of the field}\label{sec:routes-stability} In \cref{sec:surgery}, the route of one surgery is evaluated at a field obtained from the current one by the other surgery. Such fields differ from a baseline field only within distance one of the other route, and all of them lie in a common vulnerable set. The next lemma shows that, for component-local routing maps, these changes matter only through the components of the vulnerable set that are approached by both skeletons. \begin{lemma}\label{lem:routes-stability} Let $\mathbb L$ be $\mathbb Z^3$ or $\mathbb L_K$, let $\Gamma,\Gamma'$ be closed walks of positive length on $\mathbb L$, and let $\mathcal R(\cdot,\Gamma)$ and $\mathcal R(\cdot,\Gamma')$ be component-local routing maps for $\Gamma$ and $\Gamma'$. Let $R\ge0$ and $R_c\ge R+2$ be integers, let $V\subset\mathbb L$ be finite, and let $B_0\subset V$. Let $\mathscr H$ be the set of range-$R_c$ components $H$ of $V$ such that $d_\infty(H,\Gamma)\le R_c$ and $d_\infty(H,\Gamma')\le R_c$, and set \[ U_*=\bigcup_{H\in\mathscr H}H,\qquad A_*=\abs{U_*}. \] For $D\subset U_*$ put $B[D]=(B_0\setminus U_*)\cup D$, and define the route lists \[ \mathscr L=\{\mathcal R(B[D],\Gamma):D\subset U_*\},\qquad\mathscr L'=\{\mathcal R(B[D],\Gamma'):D\subset U_*\}. \] \begin{enumerate}[label=\textup{(\roman*)}] \item Let $B^1,B^2\subset V$, let $\gamma=\mathcal R(B^1,\Gamma)$ and $\gamma'=\mathcal R(B^2,\Gamma')$, and suppose that \begin{equation}\label{eq:routes-local-change} B^1\triangle B_0\subset\mathcal N_1(\gamma'),\qquad B^2\triangle B_0\subset\mathcal N_1(\gamma). \end{equation} Then $\mathcal R(B[B^1\cap U_*],\Gamma)=\gamma$ and $\mathcal R(B[B^2\cap U_*],\Gamma')=\gamma'$. In particular $(\gamma,\gamma')\in\mathscr L\times\mathscr L'$, and at most $4^{A_*}$ ordered pairs $(\gamma,\gamma')$ arise in this way. \item Let $\gamma,\gamma'$ be routes, possibly $\dagger$, and let $B^1,B^2\subset V$ satisfy \eqref{eq:routes-local-change} for this pair. If $A_*=0$, then \begin{equation}\label{eq:routes-product-identity} \mathbf 1\{\mathcal R(B^1,\Gamma)=\gamma\}\,\mathbf 1\{\mathcal R(B^2,\Gamma')=\gamma'\}=\mathbf 1\{\mathcal R(B_0,\Gamma)=\gamma\}\,\mathbf 1\{\mathcal R(B_0,\Gamma')=\gamma'\}. \end{equation} \item For all $B^1,B^2\subset V$, the routes $\gamma=\mathcal R(B^1,\Gamma)$ and $\gamma'=\mathcal R(B^2,\Gamma')$ satisfy \begin{equation}\label{eq:routes-charge} J_R(\gamma,\gamma')\le J_R(\Gamma,\Gamma')+C_{R,1}A_*,\qquad C_{R,1}=54(2R+1)^3. \end{equation} In particular \eqref{eq:routes-charge} holds for the pairs in \textup{(i)} and for every pair in $\mathscr L\times\mathscr L'$. \end{enumerate} \end{lemma} \begin{proof} We first prove the following localization claim. Let $B,B'\subset V$, let $x\in\mathcal Q(B,\Gamma)\cap V$, and suppose that $d_\infty(x,\mathcal R(B',\Gamma'))\le1$. Then $x\in U_*$. The same holds with the roles of $\Gamma$ and $\Gamma'$ interchanged. To prove the claim, let $H$ be the range-$R_c$ component of $V$ containing $x$. Either $x$ is a cell of $\Gamma$, or $x$ lies within distance one of some $C\in\mathcal C(B,\Gamma)$. In the second case, $C\subset B\subset V$ is $*$-connected and contains a cell of $\Gamma$, so it lies in a range-$R_c$ component of $V$ that contains a cell of $\Gamma$, and this component is $H$, since $x\in V$ and $d_\infty(x,C)\le1\le R_c$. In both cases $H$ contains a cell of $\Gamma$. The route $\mathcal R(B',\Gamma')$ is not $\dagger$, since $\mathcal N_1(\dagger)=\varnothing$; let $y'$ be one of its cells with $d_\infty(x,y')\le1$. By \cref{def:routes-local}(b) for $B'$, either $y'$ is a cell of $\Gamma'$, and then $d_\infty(H,\Gamma')\le1\le R_c$; or $y'$ lies within distance one of a $*$-component $C'$ of $B'$ containing a cell of $\Gamma'$. In the latter case $C'\subset V$ is $*$-connected, so it lies in a single range-$R_c$ component $H'$ of $V$, which contains a cell of $\Gamma'$. Since $d_\infty(x,C')\le2\le R_c$, we have $H'=H$. In both cases $H\in\mathscr H$, and therefore $x\in U_*$. We prove (i). Let $B'=B[B^1\cap U_*]$. The fields $B'$ and $B^1$ agree on $U_*$, and $B'$ agrees with $B_0$ outside $U_*$. If $B'$ and $B^1$ differed at a cell $x\in\mathcal Q(B^1,\Gamma)$, then $x\notin U_*$ and $x\in B^1\triangle B_0\subset V\cap\mathcal N_1(\gamma')$. The claim, applied with $B=B^1$ and $B'=B^2$, would then give $x\in U_*$, a contradiction. So $B'$ and $B^1$ agree on $\mathcal Q(B^1,\Gamma)$, and \cref{lem:routes-explored} gives $\mathcal R(B',\Gamma)=\mathcal R(B^1,\Gamma)=\gamma$. The argument for $\Gamma'$ is the same, with the roles of $\Gamma$ and $\Gamma'$ interchanged. Since $B^1\cap U_*$ and $B^2\cap U_*$ are subsets of $U_*$, we have $\gamma\in\mathscr L$ and $\gamma'\in\mathscr L'$, and each list has at most $2^{A_*}$ elements. We prove (ii). Since $A_*=0$, we have $U_*=\varnothing$ and $B[\varnothing]=B_0$. If the left side of \eqref{eq:routes-product-identity} is one, then $\gamma$ and $\gamma'$ are the routes computed on $B^1$ and $B^2$, and by (i) we have $\mathcal R(B_0,\Gamma)=\gamma$ and $\mathcal R(B_0,\Gamma')=\gamma'$, so the right side is one. Conversely, suppose that the right side is one, so that $\gamma=\mathcal R(B_0,\Gamma)$ and $\gamma'=\mathcal R(B_0,\Gamma')$. Let $x\in\mathcal Q(B_0,\Gamma)$ with $x\in B^1\triangle B_0$. Then $x\in V\cap\mathcal N_1(\gamma')$, and the claim with $B=B'=B_0$ gives $x\in U_*=\varnothing$, which is impossible. Then $B^1$ and $B_0$ agree on $\mathcal Q(B_0,\Gamma)$, and $\mathcal R(B^1,\Gamma)=\gamma$ by \cref{lem:routes-explored}. In the same way $\mathcal R(B^2,\Gamma')=\gamma'$, and the left side is one. In all other cases both sides vanish. The converse direction is what identifies all four products of route indicators in the signed sum of \cref{sec:surgery} with their baseline values. We prove (iii). If $\gamma$ or $\gamma'$ is $\dagger$, the left side vanishes. Otherwise both are simple, and $J_R(\gamma,\gamma')$ counts the pairs of cells $(x,x')$ with $x\in\gamma$, $x'\in\gamma'$, and $d_\infty(x,x')\le R$. For pairs in which $x$ is a cell of $\Gamma$ and $x'$ a cell of $\Gamma'$, choose an occurrence of $x$ in $\Gamma$ and one of $x'$ in $\Gamma'$. Distinct pairs give distinct pairs of occurrences, and these are counted in $J_R(\Gamma,\Gamma')$. Suppose now that $x$ is not a cell of $\Gamma$. By \cref{def:routes-local}(b), $x$ lies within distance one of a $*$-component $C$ of $B^1$ containing a cell of $\Gamma$, and $C$ lies in a range-$R_c$ component $H$ of $V$ containing a cell of $\Gamma$. If $x'$ is a cell of $\Gamma'$, then $d_\infty(H,\Gamma')\le d_\infty(C,x')\le R+1\le R_c$. Otherwise $x'$ lies within distance one of a $*$-component $C'$ of $B^2$ containing a cell of $\Gamma'$, which lies in a range-$R_c$ component $H'$ of $V$ containing a cell of $\Gamma'$; then $d_\infty(C,C')\le R+2\le R_c$, so $H'=H$. In both cases $H\in\mathscr H$, and $x\in\mathcal N_1(U_*)$. Symmetrically, if $x'$ is not a cell of $\Gamma'$, then $x'\in\mathcal N_1(U_*)$. Since $\abs{\mathcal N_1(U_*)}\le27A_*$, and at most $(2R+1)^3$ cells of a simple route lie within distance $R$ of a given cell, the pairs with $x\in\mathcal N_1(U_*)$ number at most $27(2R+1)^3A_*$, and so do the pairs with $x'\in\mathcal N_1(U_*)$. This proves \eqref{eq:routes-charge}. The routes in $\mathscr L$ and $\mathscr L'$ are computed from the fields $B[D]\subset V$. Then \eqref{eq:routes-charge} applies to them. \end{proof} \begin{remark}\label{rem:routes-stability-sources} In \cite[Lemmas~10.2--10.3]{OpenAI2026B}, all unusable sets arising from the permitted replacements lie in one enlarged set; the resulting pairs of paths are counted by an exponential in the total size of components approached by both paths. There are three differences. First, our routes are closed cycles whose rejection rules read the components met by the skeleton, so the localization is formulated through the set $\mathcal Q(B,\Gamma)$ of \cref{lem:routes-explored}; \cref{def:routes-local}(a) replaces the property in \cite[Lemma~8.2]{OpenAI2026B} that bad components not hit by an accepted skeleton can be removed without changing the path. Second, the inclusion of the perturbed fields in $V$ and the location \eqref{eq:routes-local-change} of their differences are hypotheses here; in \cref{sec:surgery} they follow from row-local accessibility, with $B_0$ the bad set and $V$ the vulnerable set of the reindexed base matching. Third, part (ii), the identity in both directions including rejection, is needed because \cref{thm:surgery} bounds a signed sum, whereas \cite{OpenAI2026B} bounds a positive second moment, for which (i) and (iii) suffice. The fields $B[D]$ need not arise from any matching, which is why the routing maps are defined on every field; the lists count routes, not the rows selected along them. \end{remark} \usedin{\cref{thm:surgery}.} \section{Averaging over the environment}\label{sec:routes-environment} \begin{lemma}\label{lem:routes-environment} Let $\mathbb L$ be $\mathbb Z^3$ or $\mathbb L_K$, let $R_c\ge1$ be an integer, and let $c_{\mathrm{rec}}\ge0$ and $p\ge0$ satisfy $q=(2R_c+1)^6pe^{c_{\mathrm{rec}}}\le e^{-2}$. Let $\Gamma,\Gamma'$ be random closed walks on $\mathbb L$, each of positive length almost surely, and let $V$ be a random finite subset of $\mathbb L$, independent of $(\Gamma,\Gamma')$, such that $\PP(F\subset V)\le p^{\abs F}$ for every finite $F\subset\mathbb L$. Define $A_*$ from $V$, $\Gamma$, $\Gamma'$, and $R_c$ as in \cref{lem:routes-stability}. Then almost surely \begin{equation}\label{eq:routes-environment} \E\bigl[e^{c_{\mathrm{rec}}A_*}\bigm|\Gamma,\Gamma'\bigr]\le\exp\bigl(\zeta(p,c_{\mathrm{rec}})\,S_{R_c}(\Gamma,\Gamma')\bigr),\qquad\zeta(p,c_{\mathrm{rec}})=\frac{e(2R_c+1)^3\sqrt q}{1-\sqrt q}. \end{equation} In particular $\zeta(p,c_{\mathrm{rec}})\to0$ as $p\downarrow0$, for fixed $R_c$ and $c_{\mathrm{rec}}$. \end{lemma} \begin{proof} Since $V$ is independent of $(\Gamma,\Gamma')$, it suffices to prove \eqref{eq:routes-environment} for fixed closed walks $\Gamma,\Gamma'$, with the expectation taken over $V$. Call a set $F\subset\mathbb L$ admissible if it is finite, nonempty, range-$R_c$ connected, and satisfies $d_\infty(F,\Gamma)\le R_c$ and $d_\infty(F,\Gamma')\le R_c$. The elements of $\mathscr H$ are pairwise disjoint admissible sets contained in $V$, and $A_*=\sum_{H\in\mathscr H}\abs H$. Expanding the product, we have \[ e^{c_{\mathrm{rec}}A_*}=\prod_{H\in\mathscr H}\bigl(1+(e^{c_{\mathrm{rec}}\abs H}-1)\bigr)=\sum_{\mathscr F\subset\mathscr H}\prod_{H\in\mathscr F}\bigl(e^{c_{\mathrm{rec}}\abs H}-1\bigr)\le\sum_{\mathscr F}\mathbf 1\Bigl\{\bigcup_{F\in\mathscr F}F\subset V\Bigr\}\prod_{F\in\mathscr F}e^{c_{\mathrm{rec}}\abs F}, \] where on the right $\mathscr F$ ranges over the finite collections of pairwise disjoint admissible sets. In the last step we dropped maximality, since every subcollection of $\mathscr H$ is such a collection, and used $e^{c_{\mathrm{rec}}\abs H}-1\le e^{c_{\mathrm{rec}}\abs H}$. By disjointness the union has $\sum_{F\in\mathscr F}\abs F$ cells. Taking expectations, which is legitimate for the nonnegative countable sum by Tonelli's theorem, and applying the prescribed-set bound, we obtain \[ \E e^{c_{\mathrm{rec}}A_*}\le\sum_{\mathscr F}\prod_{F\in\mathscr F}\bigl(pe^{c_{\mathrm{rec}}}\bigr)^{\abs F}\le\prod_F\bigl(1+(pe^{c_{\mathrm{rec}}})^{\abs F}\bigr)\le\exp\Bigl(\sum_F(pe^{c_{\mathrm{rec}}})^{\abs F}\Bigr). \] In the second inequality we dropped disjointness, and the product and the last sum run over all admissible sets $F$. Let $F$ be admissible with $m$ cells. Let $z$ be an occurrence of $\Gamma$ and $x\in F$ a cell such that $d_\infty(z,x)\le R_c$. Since $F$ has diameter at most $R_c(m-1)$ and lies within distance $R_c$ of $\Gamma'$, we have $d_\infty(z,\Gamma')\le R_c+R_c(m-1)+R_c=R_c(m+1)$. Assigning $F$ to the pair $(z,x)$, and using \cref{lem:routes-animals} for the number of range-$R_c$ connected sets of $m$ cells containing $x$, we find \begin{align*} \sum_F(pe^{c_{\mathrm{rec}}})^{\abs F}&\le\sum_{z\in\Gamma}\sum_{x\in\mathcal N_{R_c}(z)}\sum_{m\ge1}(2R_c+1)^{6(m-1)}(pe^{c_{\mathrm{rec}}})^m\,\mathbf 1\{d_\infty(z,\Gamma')\le R_c(m+1)\}\\ &\le(2R_c+1)^3\sum_{z\in\Gamma}\sum_{m\ge1}q^m\,\mathbf 1\{d_\infty(z,\Gamma')\le R_c(m+1)\}. \end{align*} Counting a set more than once only increases the bound. Let $d=d_\infty(z,\Gamma')$. If $d\le R_c(m+1)$, then $m\ge d/R_c-1$, and $q\le e^{-2}$ gives $q^m=q^{m/2}q^{m/2}\le q^{m/2}e^{-m}\le e\,q^{m/2}e^{-d/R_c}$. Summing over $m\ge1$ gives $\sum_mq^m\mathbf 1\{d\le R_c(m+1)\}\le e\sqrt q(1-\sqrt q)^{-1}e^{-d/R_c}$. Inserting this bound into the previous display shows that the exponent is at most $\zeta(p,c_{\mathrm{rec}})S_{R_c}(\Gamma,\Gamma')$, which proves \eqref{eq:routes-environment}. On $\mathbb L_K$, admissible sets may wrap around the torus; the argument uses only the periodic distance and the counting bound of \cref{lem:routes-animals}, both valid there. \end{proof} This is the expansion over components in the proof of \cite[Proposition~7.1]{OpenAI2026HS} (\cite[Section~7.4]{OpenAI2026HS}), reproduced in \cite[Section~8.4, Equations~(8.18)--(8.20)]{OpenAI2026BEC}; see also \cite[Section~10.5]{OpenAI2026B}. There it is applied to bad components approached by two paths in a common environment; here it is applied to the vulnerable set, with an exponent $c_{\mathrm{rec}}$ that in \cref{thm:surgery} includes the term $\log4$ from the list count in \cref{lem:routes-stability}. Only the prescribed-set bound under the law of $V$ enters, with both walks fixed; no independence between cells and no conditioning on explored cells is used. \usedin{\cref{cor:routes-final}.} \begin{corollary}\label{cor:routes-final} Let $R\ge0$ and $R_c\ge1$ be integers, and let $\kappa,\xi,C$ be the constants of \cref{prop:routes-nominal}\textup{(c)}. Let $\Gamma,\Gamma'$ be independent winding skeletons with $K\ge20$, respectively independent area skeletons with a common inner cube and $K_{\rm in}\ge2^{14}$. Let $V$ be a random finite subset of $\mathbb L_K$, respectively of $\mathbb Z^3$, independent of the two seeds, such that $\PP(F\subset V)\le p^{\abs F}$ for every finite $F$. Let $\eps\ge0$ and $c_{\mathrm{rec}}\ge0$ satisfy \[ 2\eps\le\kappa,\qquad(2R_c+1)^6pe^{c_{\mathrm{rec}}}\le e^{-2},\qquad2\zeta(p,c_{\mathrm{rec}})\le\xi, \] with $\zeta$ as in \cref{lem:routes-environment}, and define $A_*$ as in \cref{lem:routes-stability}. Then \begin{equation}\label{eq:routes-final} \E\bigl[e^{\eps J_R(\Gamma,\Gamma')+c_{\mathrm{rec}}A_*}\bigr]-1\le\frac CK,\qquad\text{respectively}\qquad\E\bigl[e^{\eps J_R(\Gamma,\Gamma')+c_{\mathrm{rec}}A_*}\bigr]-1\le\frac C{K_{\rm in}}. \end{equation} \end{corollary} \begin{proof} We treat the winding case; the area case is identical with $K_{\rm in}$ in place of $K$. Conditionally on the two seeds, $J_R(\Gamma,\Gamma')$ is fixed, and by \cref{lem:routes-environment} we have \[ \E\bigl[e^{\eps J_R(\Gamma,\Gamma')+c_{\mathrm{rec}}A_*}\bigm|\Gamma,\Gamma'\bigr]\le\exp\bigl(\eps J_R(\Gamma,\Gamma')+\zeta(p,c_{\mathrm{rec}})S_{R_c}(\Gamma,\Gamma')\bigr). \] By the Cauchy--Schwarz inequality, monotonicity of the exponential moments in their coefficients, and \cref{prop:routes-nominal}(c), we have \[ \E e^{\eps J_R+\zeta S_{R_c}}\le\bigl(\E e^{2\eps J_R}\bigr)^{1/2}\bigl(\E e^{2\zeta S_{R_c}}\bigr)^{1/2}\le\bigl(\E e^{\kappa J_R}\bigr)^{1/2}\bigl(\E e^{\xi S_{R_c}}\bigr)^{1/2}\le1+\frac CK, \] where $J_R$ and $S_{R_c}$ are evaluated at $(\Gamma,\Gamma')$. Taking expectations in the previous display proves \eqref{eq:routes-final}. \end{proof} In \cref{thm:surgery}, we apply \cref{cor:routes-final} with encounter coefficient $2\eps_{\max}$ and coefficient $2c_{\mathrm{rec}}$ of $A_*$ in the exponential. The parameter choices, including $4\eps_{\max}\le\kappa$, $(2R_c+1)^6pe^{2c_{\mathrm{rec}}}\le e^{-2}$, and $2\zeta(p,2c_{\mathrm{rec}})\le\xi$, are given in \cref{sec:stiff-parameters,sec:ncri-parameters}. \usedin{\cref{thm:surgery}.} \chapter{Matching systems and the surgery theorem}\label{sec:surgery} A row labels a path's initial endpoint, a slot labels a terminal endpoint, and a matching assigns one slot to each row. We call the data revealed before this assignment is resampled the exposed data. This section introduces matching systems, which include the law of one slab of the hard-sphere gas after suitable data have been exposed (\cref{sec:slab}). The rows are partitioned into $m$ groups with at most $M$ rows of each group in each cell, and a surgery moves rows of one group, chosen uniformly at random. Under hypotheses (H1)--(H5), \cref{thm:surgery} bounds the chi-square divergence of the completed surgery kernel, averaged over the group and over the random skeleton, by $C_*/(mK)+C_\times\eta_{\mathrm{loc}}/K$, where the collision error $\eta_{\mathrm{loc}}$ does not depend on $m$, and its acceptance probability from below by $1-Cp$. Paths belong to an abstract standard Borel space; the analytic inputs are the mass and insertion bounds (H2), (H3), and (H3$_1$), and the route properties (H5) come from \cref{sec:routes}. The applications in \cref{sec:proof-stiffness,sec:proof-ncri} combine this theorem with the slab verification of (H1)--(H4) and (H4$^+$). The proof follows the second-moment method of \cref{lem:var-mixture}. By \eqref{eq:var-mixture-signed}, the divergence of the surgeries of one group is an average over two independent skeletons $\Gamma,\Gamma'$ of the inner product $\langle L_\Gamma-1,L_{\Gamma'}-1\rangle$ of two centered likelihoods. In \cref{sec:surgery-move}, we compute $L_\Gamma$ and write this inner product as a sum, over pairs of labeled routes, of four terms. Three of them are given by exact identities (\cref{lem:surgery-easy}). For the fourth, we compare the conditional normalizing constants of four matchings (\cref{lem:surgery-four}) in \cref{sec:surgery-four}. In \cref{sec:surgery-labels}, we sum over the selected rows (\cref{lem:surgery-labels}). In \cref{sec:surgery-theorem}, we assemble the four terms. When the two routes have no encounters and their skeletons approach no common vulnerable component, route stability identifies their indicators, and the four terms cancel exactly (\cref{lem:surgery-exact}). Otherwise, only the two positive terms are kept; they are bounded by the cost of the encounters, and, when the skeletons approach a common vulnerable component, the routes are counted by route lists. Finally, an expansion over clusters of vulnerable cells averages over the configuration. The average over groups also produces inner products of the likelihoods of two different groups. A move of one group changes none of the counts, bad sets, and rates of another group, so three of the four terms coincide, and the comparison of four matchings bounds the signed sum from above by a loss confined to the encounters of the two routes (\cref{lem:surgery-cross}). Most ingredients adapt arguments of \cite{OpenAI2026HS,OpenAI2026B,OpenAI2026BEC}. The cyclic shift of endpoints along a route, combined with exact conditional resampling, is the construction of \cite[Section~9]{OpenAI2026B}, where the routes are open; ours are closed, and the surgery changes the endpoint matching of a closed slab. The cancellation of conditional normalizing constants away from the places where two routes come close is the argument of \cite[Lemma~8.3]{OpenAI2026HS}, reproduced as \cite[Lemma~9.3]{OpenAI2026BEC}, which we adapt to changes of the endpoint matching; \cref{rem:surgery-bec} explains why that lemma cannot be applied verbatim. The comparison of four endpoint states, including rows shared by the two routes and their neighbors on a route, adapts \cite[Section~10]{OpenAI2026B}. The summation over selected rows adapts \cite[Equation~(9.16)]{OpenAI2026BEC} and \cite[Section~10.5]{OpenAI2026B}, and the route lists that control routes depending on the configuration adapt \cite[Lemmas~10.2 and~10.3]{OpenAI2026B}. The signed estimate is proved in \cref{thm:surgery}; see \cref{rem:surgery-signed} for its role in the variance bound. \section{Matching systems}\label{sec:surgery-systems} Throughout this section, $\Lat$ is the cell lattice of \cref{sec:routes}, that is, $(\Z/K\Z)^3$ on the torus and $\Z^3$ for a container, with the maximum distance $d_\infty$. Two cells are face neighbors if they differ by $\pm e_k$ for some $k\in\{1,2,3\}$; these are the nearest neighbors of \cref{sec:routes-notation}. For $P\subset\Lat$ and $r\ge0$, we write $\mathcal N_r(P)=\{x\in\Lat:d_\infty(x,P)\le r\}$. As in \cref{sec:routes-notation}, a route is either a simple closed walk $\gamma=(v_0,\dots,v_{r-1},v_r=v_0)$ of length $r\ge3$, or the empty route $\dagger$, which stands for a rejection. The cells $v_0,\dots,v_{r-1}$ of a route are distinct, consecutive cells $v_k$ and $v_{k+1}$ are face neighbors for $0\le k\le r-1$, and the orientation is part of the route. We identify $\gamma$ with its set of cells when we write $v\in\gamma$, $\abs\gamma$, or $d_\infty(x,\gamma)$, and $\mathcal N_r(\dagger)=\varnothing$. For finite sequences $P,P'$ of cells and $R\ge0$, the encounter count of \cref{sec:routes} is \[ J_R(P,P')=\sum_{z\in P}\sum_{w\in P'}\ind\{d_\infty(z,w)\le R\}, \] where occurrences are counted with multiplicity, and $J_R(\dagger,P')=J_R(P,\dagger)=0$. For a finite measure $\varrho$, we write $\varrho(1)$ for its total mass. \begin{definition}\label{def:surgery-system} A matching system consists of the following data. \begin{enumerate} \item A probability measure $\nu$ on a standard Borel space $\mathsf E$. Its elements $E$ are called exposed data. \item For each $E\in\mathsf E$, finite sets $I=I(E)$ of rows and $J=J(E)$ of slots with $\abs I=\abs J$, an anchor map $\operatorname{anch}=\operatorname{anch}_E\colon I\to\Lat$, and a partition $I=I^{(1)}\sqcup\dots\sqcup I^{(m)}$ of the rows into groups, where the integer $m\ge1$ does not depend on $E$. We write $I_v=\{i\in I:\operatorname{anch}(i)=v\}$ and $I^{(a)}_v=I_v\cap I^{(a)}$. There exists an integer $M\ge1$ such that $\abs{I^{(a)}_v}\le M$ for all $E$, $v$, and $a$; in particular, $\abs{I_v}\le mM$. The sets $I(E)$ and $J(E)$ are subsets of a fixed countable set, and they, the anchor map, and the groups depend measurably on $E$. \item A standard Borel space $\mathsf P$ of paths. For each $E$, each $i\in I$, and each $j\in J$, a finite measure $K^E_{ij}$ on $\mathsf P$ and a measurable set $\mathcal P^E_{ij}\subset\mathsf P$ with $K^E_{ij}(\mathsf P\setminus\mathcal P^E_{ij})=0$. For distinct rows $i,k\in I$, a measurable function $h^E_{ik}\colon\mathsf P\times\mathsf P\to\{0,1\}$, the pair check, with $h^E_{ki}(\omega',\omega)=h^E_{ik}(\omega,\omega')$. The maps $(E,\omega)\mapsto\ind\{\omega\in\mathcal P^E_{ij}\}$ and $(E,\omega,\omega^{\prime})\mapsto h^E_{ik}(\omega,\omega^{\prime})$ are Borel measurable on their index domains, and $E\mapsto K^E_{ij}(B)$ is measurable for every Borel set $B\subset\mathsf P$. \item Numbers $A^E_{ij}\in\{0,1\}$ for $i\in I$ and $j\in J$, the accessibility indicators, depending measurably on $E$. \item A finite set $\Lambda_{\mathrm{act}}\subset\Lat$ of active cells, which may depend on $K$ but does not depend on $E$, and an integer $m_*\ge4$. \end{enumerate} A matching is a bijection $\tau\colon I\to J$. For a matching $\tau$, let \begin{equation}\label{eq:surgery-measure} M^E_\tau(\dd\omega)=\prod_{i\in I}K^E_{i\tau(i)}(\dd\omega_i)\prod_{\{i,k\}}h^E_{ik}(\omega_i,\omega_k),\qquad Z_\tau(E)=M^E_\tau(1),\qquad \Xi(E)=\sum_\tau Z_\tau(E), \end{equation} where the second product is over unordered pairs of distinct rows, and the sum is over all matchings. We assume that $0<\Xi(E)<\infty$ for $\nu$-almost every $E$. The matching law is the probability measure defined, for $E$ in this full-measure set, by \[ \mu(\dd E,\{\tau\},\dd\omega)=\nu(\dd E)\,\Xi(E)^{-1}M^E_\tau(\dd\omega) \] on the state space $\mathsf X$ of triples $(E,\tau,\omega)$ with $E\in\mathsf E$, $\tau$ a matching, and $\omega\in\mathsf P^{I(E)}$. For a matching $\tau$, a group $a$, and a cell $v$, the count of accessible rows of group $a$, the bad set, and the vulnerable set of group $a$ are \[ \begin{gathered} m^{(a)}_v(\tau)=\sum_{i\in I^{(a)}_v}A^E_{i\tau(i)},\\ B^{(a)}_\tau=\{v\in\Lambda_{\mathrm{act}}:m^{(a)}_v(\tau)R_{\mathrm{int}}$. \item (Mass bounds for crossed pairings.) $z_-\le K^E_{il}(1)\le z_+$ for every crossed pairing $(i,l)$. \item (Insertion.) For every matching $\tau$ and row $i$ such that $(i,\tau(i))$ is a crossed pairing, every set $I'\subset I\setminus\{i\}$, and all paths $\omega_k\in\mathcal P^E_{k\tau(k)}$, $k\in I'$, \[ \int\prod_{k\in I'}h^E_{ik}(\omega_i,\omega_k)\,K^E_{i\tau(i)}(\dd\omega_i)\ge(1-\eta_{\mathrm{glob}})\,K^E_{i\tau(i)}(1). \] \item[(H3$_1$)] (One opponent.) For every matching $\tau$ and row $i$ such that $(i,\tau(i))$ is a crossed pairing, every row $k\neq i$, and every path $\omega_k\in\mathcal P^E_{k\tau(k)}$, \[ K^E_{i\tau(i)}\bigl(\{\omega:h^E_{ik}(\omega,\omega_k)=0\}\bigr)\le\eta_1\,K^E_{i\tau(i)}(1). \] \item (Rare vulnerable cells.) $\PP_\mu(\mathcal F\subset V^{(a)}_\sigma)\le p^{\abs{\mathcal F}}$ for every group $a$ and every finite set $\mathcal F\subset\Lat$. \item[(H4$^+$)] (Rare vulnerable cells of two groups.) $\PP_\mu(\mathcal F\subset V^{(a)}_\sigma\cup V^{(b)}_\sigma)\le p_\times^{\abs{\mathcal F}}$ for all groups $a\neq b$ and every finite set $\mathcal F\subset\Lat$. \item (Routes.) There exist a standard Borel space $\mathfrak G$ with a probability measure $\mathfrak p$, whose elements are called skeletons, and a routing map $\mathcal R$ that assigns to every set $B\subset\Lambda_{\mathrm{act}}$ and every $\Gamma\in\mathfrak G$ either $\dagger$ or a route $\mathcal R(B,\Gamma)$, measurably in $\Gamma$. Every skeleton $\Gamma$ determines, measurably in $\Gamma$, a finite sequence of cells, which we also denote by $\Gamma$. Fix integers $R\ge\max\{R_{\mathrm{int}},1\}$ and $R_{\mathrm c}\ge R+2$. For $V\subset\Lambda_{\mathrm{act}}$ and $\Gamma,\Gamma'\in\mathfrak G$, let $A_*(V;\Gamma,\Gamma')$ be the total number of cells in those components $H$ of $V$, for the adjacency $00$, $p_0>0$, $C_{\mathrm{acc}}$, and $C_{\mathrm{ch}}$, and, for each $c\ge0$, numbers $p_{\mathrm{env}}(c)>0$ and $C_{\mathrm{fin}}(c)<\infty$, none of which depends on $K$, such that the following hold for $K\ge K_0$. \begin{enumerate}[label=(\alph*)] \item Every route $\mathcal R(B,\Gamma)$ has all its cells in $\Lambda_{\mathrm{act}}\setminus B$. \item If $B$ is a random subset of $\Lambda_{\mathrm{act}}$ that is independent of $\Gamma$, where $\Gamma$ has law $\mathfrak p$, and $\PP(\mathcal F\subset B)\le p^{\abs{\mathcal F}}$ for every finite $\mathcal F\subset\Lat$, where $p\le p_0$, then $\PP(\mathcal R(B,\Gamma)=\dagger)\le C_{\mathrm{acc}}p$. \item Let $\Gamma,\Gamma'\in\mathfrak G$, let $V\subset\Lambda_{\mathrm{act}}$ be finite, let $B_0\subset V$, and write $A_*=A_*(V;\Gamma,\Gamma')$ and $\mathscr L=\mathscr L(V,B_0;\Gamma,\Gamma')$. Then: \begin{enumerate}[label=(\roman*)] \item $\mathscr L$ has at most $4^{A_*}$ elements; \item if $A_*=0$, then for every pair of routes $(\gamma,\gamma')$ and all $B^1,B^2\subset V$ with $B^1\triangle B_0\subset\mathcal N_1(\gamma')$ and $B^2\triangle B_0\subset\mathcal N_1(\gamma)$, \[ \ind\{\mathcal R(B^1,\Gamma)=\gamma\}\,\ind\{\mathcal R(B^2,\Gamma')=\gamma'\}=\ind\{\mathcal R(B_0,\Gamma)=\gamma\}\,\ind\{\mathcal R(B_0,\Gamma')=\gamma'\}; \] \item[(iii$^+$)] for all $B^1,B^2\subset V$, the routes $\gamma=\mathcal R(B^1,\Gamma)$ and $\gamma'=\mathcal R(B^2,\Gamma')$ satisfy $J_R(\gamma,\gamma')\le J_R(\Gamma,\Gamma')+C_{\mathrm{ch}}A_*$; in particular, every pair of routes in $\mathscr L$ satisfies this bound. \end{enumerate} \item Let $c\ge0$, $p\le p_{\mathrm{env}}(c)$, and $0\le\eps'\le\kappa/2$. Let $\Gamma,\Gamma'$ be independent with law $\mathfrak p$, and let $V$ be a random subset of $\Lambda_{\mathrm{act}}$ that is independent of $(\Gamma,\Gamma')$ and satisfies $\PP(\mathcal F\subset V)\le p^{\abs{\mathcal F}}$ for every finite $\mathcal F\subset\Lat$. Then \[ \E\exp\bigl(\eps'J_R(\Gamma,\Gamma')+cA_*(V;\Gamma,\Gamma')\bigr)\le1+\frac{C_{\mathrm{fin}}(c)}K. \] \end{enumerate} \end{enumerate} \end{definition} The properties in (H5) are proved in \cref{sec:routes} for the skeletons and routing maps used there: the winding skeletons of \cref{def:routes-loop} with the winding routing map of \cref{def:routes-routing} on the torus, and the area skeletons of \cref{def:routes-polygon} with the area routing map of \cref{def:routes-routing} in a container. Properties (a) and (b) are part of \cref{prop:routes-router}. Property (c) is \cref{lem:routes-stability}, applied with perturbation radius one, and part (iii$^+$) is its part (iii), which is stated for all $B^1,B^2\subset V$; the condition $R_{\mathrm c}\ge R+2$ is the condition on the component range there. Property (d) combines \cref{lem:routes-environment} with the encounter bounds of \cref{prop:routes-nominal}, as in \cref{cor:routes-final}, whose hypotheses $2\eps'\le\kappa$, $(2R_{\mathrm c}+1)^6pe^{c}\le e^{-2}$, and $2\zeta(p,c)\le\xi$ hold for $\eps'\le\kappa/2$ and $p\le p_{\mathrm{env}}(c)$ if $p_{\mathrm{env}}(c)$ is chosen small enough; in a container, that corollary gives the bound $C/K_{\mathrm{in}}$ for the side $K_{\mathrm{in}}$ of the inner cube. In this application the scale $K$ in (H5) and \cref{thm:surgery} is read as $K_{\mathrm{in}}$ (\cref{sec:proof-ncri}): the proof uses this scale only in the threshold $K\ge K_0$ and in (H5)(d), so the conclusion holds with $K_{\mathrm{in}}$ in place of $K$ and with the same comparison constants. In (c), the sets $B^1,B^2$ are arbitrary subsets of $V$; they need not be bad sets of actual matchings. Part (ii) compares the indicators of a prescribed pair of routes in both directions, which is needed to identify all four terms of the signed expansion below. We record some comments on the hypotheses. Let $C_{\mathrm{int}}=(2R_{\mathrm{int}}+1)^3$. By (H1) and $\abs{I_v}\le mM$, every row has nontrivial checks with at most $C_{\mathrm{int}}mM$ other rows. Hence (H3$_1$) and a union bound imply (H3) with $\eta_{\mathrm{glob}}=C_{\mathrm{int}}mM\eta_1$, provided that this number is at most $1/2$; in \cref{sec:slab}, this is how (H3) is verified. We set $\eta_{\mathrm{loc}}=4C_{\mathrm{int}}\eta_1$; this local error, used in \cref{lem:surgery-four}, does not involve the number $mM$ of rows per cell. Hypothesis (H2) concerns row-slot pairings that need not occur together in one matching: the rows $i$ and $k$ may coincide, the slots $j$ and $l$ may coincide, and the two accessibility conditions may hold in different matchings. This form is needed to bound masses in the four matchings of \cref{lem:surgery-sectors}. In (H3), the other rows need only be individually permitted; they need not be accessible, and they need not pass their checks with each other. Since $I'$ is arbitrary, the bound persists when rows are deleted. In (H3$_1$), the opponent $k$ may belong to any group. Hypotheses (H4) and (H4$^+$) are statements about the joint law $\mu$, including the exposed data; they are not assumed conditionally on $E$. Since $B^{(a)}_\sigma\subset V^{(a)}_\sigma$, they also hold for the bad sets. The next lemma is the form in which (H3) is used. Fix $E$, a matching $\tau$, and a set $O\subset I$ of rows with paths $\omega_O=(\omega_k)_{k\in O}$ that are individually permitted for $\tau$. For $i\notin O$, we absorb the checks with $O$ into the path measure of $i$ and write \[ \widehat K^\tau_i(\dd\omega_i)=K_{i\tau(i)}(\dd\omega_i)\prod_{k\in O}h_{ik}(\omega_i,\omega_k),\qquad \widehat z^\tau_i=\widehat K^\tau_i(1). \] For disjoint sets $W,X\subset I\setminus O$ and paths $\omega_X$ that are individually permitted for $\tau$, let \[ Z_W(\tau;\omega_X)=\int\prod_{i\in W}\widehat K^\tau_i(\dd\omega_i)\prod_{\{i,k\}\subset W}h_{ik}(\omega_i,\omega_k)\prod_{i\in W,\,k\in X}h_{ik}(\omega_i,\omega_k), \] with $Z_\varnothing(\tau;\omega_X)=1$. In $Z_W(\tau;\omega_X)$, the rows of $W$ are integrated, the rows of $O\cup X$ are fixed opponents, and all other rows are deleted together with their checks. It depends on $\tau$ only through its restriction to $W$. With $O=I\setminus S$ and $X=\varnothing$, we recover $Z_S(\tau;\omega_{S^c})$ from \eqref{eq:surgery-conditional}. \begin{lemma}\label{lem:surgery-insertion} Assume (H3) for $E$, and let $\tau$, $O$, $\omega_O$, $X$, and $\omega_X$ be as above. Let $W_0\subset W\subset I\setminus(O\cup X)$. \begin{enumerate} \item We have \begin{equation}\label{eq:surgery-insertion-upper} Z_W(\tau;\omega_X)\le\Bigl(\prod_{i\in W\setminus W_0}\widehat z^\tau_i\Bigr)Z_{W_0}(\tau;\omega_X). \end{equation} If moreover $(i,\tau(i))$ is a crossed pairing for every $i\in W\setminus W_0$, then $(1-\eta_{\mathrm{glob}})K_{i\tau(i)}(1)\le\widehat z^\tau_i\le K_{i\tau(i)}(1)$ for these $i$, and \begin{equation}\label{eq:surgery-insertion-lower} Z_W(\tau;\omega_X)\ge(1-\eta_{\mathrm{glob}})^{\abs{W\setminus W_0}}\Bigl(\prod_{i\in W\setminus W_0}\widehat z^\tau_i\Bigr)Z_{W_0}(\tau;\omega_X). \end{equation} \item Assume moreover (H1) and (H3$_1$) for $E$, and that $(i,\tau(i))$ is a crossed pairing for every $i\in W\setminus W_0$. Let $N\ge0$ be an integer such that every $i\in W\setminus W_0$ has at most $N$ rows $k\in(W\cup X)\setminus\{i\}$ with $d_\infty(\operatorname{anch}(i),\operatorname{anch}(k))\le R_{\mathrm{int}}$, and suppose that $2N\eta_1\le1$. Then \[ Z_W(\tau;\omega_X)\ge(1-2N\eta_1)^{\abs{W\setminus W_0}}\Bigl(\prod_{i\in W\setminus W_0}\widehat z^\tau_i\Bigr)Z_{W_0}(\tau;\omega_X). \] \end{enumerate} \end{lemma} \begin{proof} The upper bound \eqref{eq:surgery-insertion-upper} follows by bounding the checks incident to the rows of $W\setminus W_0$ by one. For the lower bound, the inequality $\widehat z^\tau_i\le K_{i\tau(i)}(1)$ holds because the checks are at most one, and (H3) with $I'=O$ gives $\widehat z^\tau_i\ge(1-\eta_{\mathrm{glob}})K_{i\tau(i)}(1)$. We prove \eqref{eq:surgery-insertion-lower} by induction on $\abs{W\setminus W_0}$, the case $W=W_0$ being trivial. Let $i\in W\setminus W_0$ and $W_1=W\setminus\{i\}$. Integrating $\omega_i$ last, we have \[ Z_W(\tau;\omega_X)=\int\Lambda_{W_1}(\dd\omega_{W_1})\int\widehat K^\tau_i(\dd\omega_i)\prod_{k\in W_1\cup X}h_{ik}(\omega_i,\omega_k), \] where the measure \[ \Lambda_{W_1}(\dd\omega_{W_1})=\prod_{k\in W_1}\widehat K^\tau_k(\dd\omega_k)\prod_{\{k,l\}\subset W_1}h_{kl}(\omega_k,\omega_l)\prod_{k\in W_1,\,l\in X}h_{kl}(\omega_k,\omega_l) \] has total mass $Z_{W_1}(\tau;\omega_X)$. The measure $\Lambda_{W_1}$ is carried by paths that are individually permitted for $\tau$. For such $\omega_{W_1}$, by (H3) with $I'=O\cup W_1\cup X$, we have \[ \begin{split} \int\widehat K^\tau_i(\dd\omega_i)\prod_{k\in W_1\cup X}h_{ik}(\omega_i,\omega_k)&=\int K_{i\tau(i)}(\dd\omega_i)\prod_{k\in O\cup W_1\cup X}h_{ik}(\omega_i,\omega_k)\\ &\ge(1-\eta_{\mathrm{glob}})K_{i\tau(i)}(1)\ge(1-\eta_{\mathrm{glob}})\widehat z^\tau_i. \end{split} \] We conclude that $Z_W(\tau;\omega_X)\ge(1-\eta_{\mathrm{glob}})\widehat z^\tau_iZ_{W_1}(\tau;\omega_X)$, and the induction hypothesis applied to $W_1$ completes the proof of part (1). For part (2), we use the same induction and the same representation of $Z_W(\tau;\omega_X)$. Fix paths $\omega_{W_1}$ that are individually permitted for $\tau$. Since the checks take values in $\{0,1\}$, we have $1-\prod_kh_{ik}\le\sum_k(1-h_{ik})$, and since $\widehat K^\tau_i\le K_{i\tau(i)}$, \[ \int\widehat K^\tau_i(\dd\omega_i)\prod_{k\in W_1\cup X}h_{ik}(\omega_i,\omega_k)\ge\widehat z^\tau_i-\sum_{k\in W_1\cup X}K_{i\tau(i)}\bigl(\{\omega:h_{ik}(\omega,\omega_k)=0\}\bigr). \] By (H1), only the rows $k$ with $d_\infty(\operatorname{anch}(i),\operatorname{anch}(k))\le R_{\mathrm{int}}$ contribute, and there are at most $N$ of them. For each of them, $k\neq i$ and $\omega_k\in\mathcal P_{k\tau(k)}$, so (H3$_1$) bounds its term by $\eta_1K_{i\tau(i)}(1)$. By part (1), $K_{i\tau(i)}(1)\le\widehat z^\tau_i/(1-\eta_{\mathrm{glob}})\le2\widehat z^\tau_i$. Hence the left side is at least $(1-2N\eta_1)\widehat z^\tau_i$, and $Z_W(\tau;\omega_X)\ge(1-2N\eta_1)\widehat z^\tau_iZ_{W_1}(\tau;\omega_X)$. Since $W_1\cup X\subset W\cup X$, the assumption on $N$ holds for $W_1$, and the induction hypothesis applied to $W_1$ completes the proof. The rows of $O$ enter only through $\widehat z^\tau_i$, and they are not counted in $N$. \end{proof} \usedin{\cref{lem:surgery-reverse} and \cref{lem:surgery-four}.} For the lower bound in part (1), we used the fact that the absorbed measure $\widehat K^\tau_i$ still passes the remaining checks with probability at least $1-\eta_{\mathrm{glob}}$ relative to its own mass: the numerator passes the outside and the remaining checks together with probability at least $1-\eta_{\mathrm{glob}}$ relative to $K_{i\tau(i)}(1)$, and the mass $\widehat z^\tau_i$ is at most $K_{i\tau(i)}(1)$. \section{Surgeries}\label{sec:surgery-move} Let $\gamma=(v_0,\dots,v_{r-1},v_r=v_0)$ be a route other than $\dagger$. A labeling of $\gamma$ is a family $(i_v)_{v\in\gamma}$ of rows with $i_v\in I_v$ for every $v\in\gamma$. The permutation of the labeled route is the permutation $\pi$ of $I$ with $\pi(i_{v_k})=i_{v_{k+1}}$ for $0\le k\le r-1$, where $v_r=v_0$, and $\pi(i)=i$ for all other rows. Its support $S=\supp\pi=\{i_v:v\in\gamma\}$ is the set of selected rows. It contains exactly one row in each cell of $\gamma$ and no other row, and $\pi$ is a cyclic permutation of $S$ since $r\ge3$. The labeled route is determined by $\pi$, and we identify the two. We write $\gamma(\pi)$ for the route of $\pi$, and $\Pi_E$ for the finite set of permutations of labeled routes whose cells lie in $\Lambda_{\mathrm{act}}$. For a group $a$, we write $\Pi^{(a)}_E$ for the set of $\pi\in\Pi_E$ whose selected rows belong to $I^{(a)}$, that is, with $i_v\in I^{(a)}_v$ for every $v\in\gamma(\pi)$. These sets are disjoint. Permutations act on the right: $(\sigma\pi)(i)=\sigma(\pi(i))$. In the matching $\sigma\pi$, the selected row of $v_k$ is paired with $\sigma(i_{v_{k+1}})$, the slot of the selected row of the next cell along $\gamma$, and all other rows keep their slots. \begin{lemma}\label{lem:surgery-sectors} Assume (H2) for $E$. Let $\sigma$ be a matching, and let $\pi,\pi'\in\Pi_E$ have selected rows $S,S'$. Set $U=S\cup S'$ and \[ \sigma_1=\sigma\pi^{-1},\qquad \sigma_2=\sigma\pi'^{-1},\qquad \sigma_{12}=\sigma\pi^{-1}\pi'^{-1}. \] \begin{enumerate} \item If $A_{i\sigma(i)}=1$ for every $i\in S$, then $(i,\sigma(i))$ and $(i,\sigma\pi(i))$ are crossed pairings for every $i\in S$. \item If $A_{i\sigma_1(i)}=1$ for every $i\in S$ and $A_{i\sigma_2(i)}=1$ for every $i\in S'$, then $(i,\tau(i))$ is a crossed pairing for every $i\in U$ and every $\tau\in\{\sigma,\sigma_1,\sigma_2,\sigma_{12}\}$. \end{enumerate} In both cases, $z_-\le K_{i\tau(i)}(1)\le z_+$ for the indicated pairings, by (H2). \end{lemma} \begin{proof} For part (1), let $i\in S$. The pairing $(i,\sigma(i))$ is accessible, hence crossed. Since $\sigma\pi(i)=\sigma(\pi(i))$, where $\pi(i)$ is the selected row of the next cell of the route, which is a face neighbor of $\operatorname{anch}(i)$, and since $A_{i\sigma(i)}=A_{\pi(i)\sigma(\pi(i))}=1$, the pairing $(i,\sigma\pi(i))$ is crossed, with $k=\pi(i)$ and $j=\sigma(i)$ in the definition. For part (2), we consider the four matchings in turn. For $i\in S$, we have $\sigma(i)=\sigma_1(\pi(i))$ and $A_{i\sigma_1(i)}=A_{\pi(i)\sigma_1(\pi(i))}=1$, and the anchors of $i$ and $\pi(i)$ are face neighbors; this implies that $(i,\sigma(i))$ is crossed. For $i\in S'\setminus S$, the same argument with $\sigma_2$ and $\pi'$ applies. This treats $\tau=\sigma$. For $\tau=\sigma_1$ and $i\in S$, the pairing $(i,\sigma_1(i))$ is accessible; for $i\in S'\setminus S$, we have $\sigma_1(i)=\sigma(i)$, which was treated. The case $\tau=\sigma_2$ is symmetric. For $\tau=\sigma_{12}=\sigma_1\pi'^{-1}$ and $i\in S\setminus S'$, we have $\sigma_{12}(i)=\sigma_1(i)$. For $i\in S'$, let $k=\pi'^{-1}(i)$, the selected row of the cell preceding $\operatorname{anch}(i)$ on the route of $\pi'$, so that $\sigma_{12}(i)=\sigma_1(k)$. If $k\notin S$, then $\sigma_1(k)=\sigma(k)=\sigma_2(i)$, which was treated. If $k\in S$, then $A_{i\sigma_2(i)}=A_{k\sigma_1(k)}=1$, and $\operatorname{anch}(i)$ and $\operatorname{anch}(k)$ are face neighbors, which implies that $(i,\sigma_1(k))$ is crossed. This includes the case $k\in S\cap S'$ and the case $i\in S\cap S'$. \end{proof} \usedin{\cref{lem:surgery-reverse} and \cref{lem:surgery-four}.} \begin{definition}\label{def:surgery-move} Let a matching system satisfy (H1)--(H5). Fix a Borel set $\mathsf E_0\subset\mathsf E$ with $\nu(\mathsf E_0)=1$ on which $0<\Xi(E)<\infty$ and (H1)--(H3) and (H3$_1$) hold, with their full universal path quantifiers. Such a set exists: after partitioning by the countably many finite index sets and matchings, failures of these hypotheses are projections of Borel sets of finite tuples of paths, hence analytic. They are null in the completion of $\nu$ and are contained in a Borel $\nu$-null set; take its complement, also intersected with $\{0<\Xi<\infty\}$. For a group $a$, $\pi\in\Pi^{(a)}_E$ with route $\gamma$ and labeling $(i_v)_{v\in\gamma}$, a matching $\tau$, and a skeleton $\Gamma\in\mathfrak G$, the selection rate is \[ R_\pi(\tau\mid\Gamma)=\ind\{\mathcal R(B^{(a)}_\tau,\Gamma)=\gamma\}\,r_\pi(\tau),\qquad r_\pi(\tau)=\prod_{v\in\gamma}\frac{A_{i_v\tau(i_v)}}{m^{(a)}_v(\tau)}, \] where a factor with $A_{i_v\tau(i_v)}=0$ is zero; if $A_{i_v\tau(i_v)}=1$, then $m^{(a)}_v(\tau)\ge1$. For $E\in\mathsf E_0$, we set $q_{\pi,\Gamma}(x,\cdot)=0$ whenever $R_\pi(\sigma\mid\Gamma)=0$, before forming any quotient. Every zero-rate contribution below is understood in this sense. When the rate is positive, the surgery along $\pi$ maps a state $x=(E,\sigma,\omega)$ to the state $(E,\sigma\pi,\omega')$, where $\omega'_i=\omega_i$ for $i\notin S=\supp\pi$, and $\omega'_S$ is a sample of \[ P_\pi(\dd\omega'_S\mid x)=\frac{M^S_{\sigma\pi}(\dd\omega'_S;\omega_{S^c})}{Z_S(\sigma\pi;\omega_{S^c})}. \] The denominator is positive by \cref{lem:surgery-reverse}(1). The surgery kernel of group $a$ with skeleton $\Gamma$ is the subprobability kernel \[ q^{(a)}_\Gamma(x,\cdot)=\sum_{\pi\in\Pi^{(a)}_E}q_{\pi,\Gamma}(x,\cdot),\qquad q_{\pi,\Gamma}(x,\cdot)=R_\pi(\sigma\mid\Gamma)\,\delta_E\otimes\delta_{\sigma\pi}\otimes\delta_{\omega_{S^c}}\otimes P_\pi(\cdot\mid x), \] where the formula for $q_{\pi,\Gamma}$ is used only when $R_\pi(\sigma\mid\Gamma)>0$, and $Q^{(a)}_\Gamma$ is the completion of $q^{(a)}_\Gamma$. On the exceptional exposed fibers $E\notin\mathsf E_0$, we set $q_{\pi,\Gamma}(x,\cdot)=q^{(a)}_\Gamma(x,\cdot)=0$ and $Q^{(a)}_\Gamma(x,\cdot)=\delta_x$. The surgery kernel of group $a$ is $Q^{(a)}=\int Q^{(a)}_\Gamma\,\mathfrak p(\dd\Gamma)$, and the surgery kernel is \[ Q=\frac1m\sum_{a=1}^mQ^{(a)}. \] \end{definition} For $E\in\mathsf E_0$, the surgery kernel acts as follows. Choose a group $a$ uniformly at random and a skeleton $\Gamma$ with law $\mathfrak p$, independently of each other and of the state, and let $\gamma=\mathcal R(B^{(a)}_\sigma,\Gamma)$. If $\gamma=\dagger$, do nothing. Otherwise, choose independently in each cell $v$ of $\gamma$ a row uniformly among the $m^{(a)}_v(\sigma)$ rows $i\in I^{(a)}_v$ with $A_{i\sigma(i)}=1$, let $\pi$ be the permutation of the resulting labeled route, and perform the surgery along $\pi$. Indeed, by (H5)(a), every cell of $\gamma$ lies in $\Lambda_{\mathrm{act}}\setminus B^{(a)}_\sigma$. Then $m^{(a)}_v(\sigma)\ge m_*$ for $v\in\gamma$, and the sum of $r_\pi(\sigma)$ over all $\pi\in\Pi^{(a)}_E$ with $\gamma(\pi)=\gamma$ is $\prod_{v\in\gamma}\sum_{i\in I^{(a)}_v}A_{i\sigma(i)}/m^{(a)}_v(\sigma)=1$. In particular, the outgoing mass of $q^{(a)}_\Gamma$ is $\ind\{\mathcal R(B^{(a)}_\sigma,\Gamma)\neq\dagger\}$. Since completion commutes with averaging, $Q^{(a)}$ is the completion of $q^{(a)}=\int q^{(a)}_\Gamma\,\mathfrak p(\dd\Gamma)$, whose outgoing mass is the acceptance probability $\mathfrak p(\{\Gamma:\mathcal R(B^{(a)}_\sigma,\Gamma)\neq\dagger\})$ of group $a$, and $Q$ is the completion of $q=m^{-1}\sum_aq^{(a)}$. The rates depend on $(E,\sigma,\Gamma)$ only, and not on the paths. The surgery preserves $E$, and the new paths of the selected rows are sampled from their conditional law under $\mu_E$ given the matching $\sigma\pi$ and all other paths, including those of the other groups. By \cref{lem:surgery-reverse}(1) below, the normalizer $Z_S(\sigma\pi;\omega_{S^c})$ is positive whenever $R_\pi(\sigma\mid\Gamma)>0$. This construction adapts \cite[Sections~9.2 and~9.3]{OpenAI2026B}. There, one eligible slot is chosen uniformly in each cell of an open path of cells, the starting points of the chosen paths are shifted cyclically along the chosen tuple, and the chosen paths are resampled from their conditional interacting law; the change opens one path of a closed configuration. Here the route is closed, and the surgery maps a closed slab with one endpoint matching to a closed slab with another. \begin{lemma}\label{lem:surgery-reverse} Fix $E\in\mathsf E_0$. Fix $\Gamma\in\mathfrak G$, a group $a$, and $\pi\in\Pi^{(a)}_E$ with selected rows $S$. \begin{enumerate} \item If $R_\pi(\sigma\mid\Gamma)>0$ and $\omega$ is individually permitted for $\sigma$, then \[ Z_S(\sigma\pi;\omega_{S^c})\ge\prod_{i\in S}(1-\eta_{\mathrm{glob}})^2K_{i\sigma\pi(i)}(1)\ge\bigl((1-\eta_{\mathrm{glob}})^2z_-\bigr)^{\abs S}>0. \] \item The measure $\mu_Eq_{\pi,\Gamma}$ is absolutely continuous with respect to $\mu_E$, with density \begin{equation}\label{eq:surgery-reverse} c_\pi(\sigma,\omega\mid\Gamma)=R_\pi(\sigma\pi^{-1}\mid\Gamma)\,\frac{Z_S(\sigma\pi^{-1};\omega_{S^c})}{Z_S(\sigma;\omega_{S^c})}, \end{equation} where the right side is zero if $R_\pi(\sigma\pi^{-1}\mid\Gamma)=0$, and $Z_S(\sigma;\omega_{S^c})>0$ for $\mu_E$-almost every $(\sigma,\omega)$. \item The measure $\mu_EQ^{(a)}_\Gamma$ is absolutely continuous with respect to $\mu_E$, with density \begin{equation}\label{eq:surgery-likelihood} L_{a,\Gamma}=1+\sum_{\pi\in\Pi^{(a)}_E}\bigl(c_\pi(\cdot\mid\Gamma)-R_\pi(\cdot\mid\Gamma)\bigr). \end{equation} \end{enumerate} \end{lemma} \begin{proof} For part (1), if $R_\pi(\sigma\mid\Gamma)>0$, then $A_{i\sigma(i)}=1$ for every $i\in S$, and by \cref{lem:surgery-sectors}(1), the pairing $(i,\sigma\pi(i))$ is crossed for every $i\in S$. The paths $\omega_{S^c}$ are individually permitted for $\sigma\pi$, since $\sigma\pi$ and $\sigma$ agree off $S$. We apply \cref{lem:surgery-insertion}(1) with $\tau=\sigma\pi$, $O=I\setminus S$, $X=\varnothing$, $W=S$, and $W_0=\varnothing$. Since $Z_S(\sigma\pi;\omega_{S^c})=Z_S(\sigma\pi;\varnothing)$ in the notation of that lemma, this implies the first inequality, and (H2) implies the second. For part (2), let $f\ge0$ be measurable. By \eqref{eq:surgery-split} for the matching $\sigma$, and since $P_\pi(\cdot\mid x)$ does not depend on $\omega_S$, we may integrate out $\omega_S$ to obtain \[ \begin{split} \int f\dd(\mu_Eq_{\pi,\Gamma})&=\frac1{\Xi}\sum_\sigma R_\pi(\sigma\mid\Gamma)\int N^{S^c}_\sigma(\dd\omega_{S^c})\,\frac{Z_S(\sigma;\omega_{S^c})}{Z_S(\sigma\pi;\omega_{S^c})}\\ &\qquad\times\int f(\sigma\pi,\omega_{S^c},\omega'_S)\,M^S_{\sigma\pi}(\dd\omega'_S;\omega_{S^c}). \end{split} \] By part (1), the divisor is positive for $N^{S^c}_\sigma$-almost every $\omega_{S^c}$ whenever $R_\pi(\sigma\mid\Gamma)>0$. We substitute $\tau=\sigma\pi$. Since $\sigma$ and $\tau$ agree off $S$, we have $N^{S^c}_\sigma=N^{S^c}_\tau$, and by \eqref{eq:surgery-split} for $\tau$, we have \[ \begin{split} \int f\dd(\mu_Eq_{\pi,\Gamma})&=\frac1{\Xi}\sum_\tau\int M_\tau(\dd\omega)\,R_\pi(\tau\pi^{-1}\mid\Gamma)\,\frac{Z_S(\tau\pi^{-1};\omega_{S^c})}{Z_S(\tau;\omega_{S^c})}\,f(\tau,\omega)\\ &=\int fc_\pi(\cdot\mid\Gamma)\dd\mu_E. \end{split} \] By part (1), applied to the matching $\tau\pi^{-1}$, the divisor $Z_S(\tau;\omega_{S^c})$ is positive whenever $R_\pi(\tau\pi^{-1}\mid\Gamma)>0$, for $N^{S^c}_\tau$-almost every $\omega_{S^c}$. Moreover, the set $\{Z_S(\sigma;\omega_{S^c})=0\}$ is a $\mu_E$-null set, since by \eqref{eq:surgery-split} its measure is $\Xi^{-1}\sum_\sigma\int N^{S^c}_\sigma(\dd\omega_{S^c})\ind\{Z_S(\sigma;\omega_{S^c})=0\}Z_S(\sigma;\omega_{S^c})=0$. This proves part (2). Part (3) follows from part (2) and \cref{lem:var-hcr}(1), since the outgoing mass of $q^{(a)}_\Gamma$ is $\sum_{\pi\in\Pi^{(a)}_E}R_\pi(\cdot\mid\Gamma)$ and its incoming density is $\sum_{\pi\in\Pi^{(a)}_E}c_\pi(\cdot\mid\Gamma)$. \end{proof} \usedin{\cref{lem:surgery-easy}, \cref{lem:surgery-four}, \cref{lem:surgery-cross}, and \cref{thm:surgery}.} Formula \eqref{eq:surgery-reverse} has the structure of the density of the opened slab in \cite[Equation~(26)]{OpenAI2026B} and of the exact likelihood \cite[Equation~(9.10)]{OpenAI2026BEC}: the normalizer of the source matching appears in the numerator and that of the target matching in the denominator. Its derivation uses only the product form \eqref{eq:surgery-measure} of the matching law, with no assumption on the distribution of the matching. Since the rates do not depend on the paths, three of the four inner products in the expansion of $\langle L_{a,\Gamma}-1,L_{b,\Gamma'}-1\rangle$ are given by exact identities. \begin{lemma}\label{lem:surgery-easy} Fix $E$ as in \cref{lem:surgery-reverse}. For groups $a$ and $b$, $\Gamma,\Gamma'\in\mathfrak G$, $\pi\in\Pi^{(a)}_E$, and $\pi'\in\Pi^{(b)}_E$, \begin{align*} \bigl\langle c_\pi(\cdot\mid\Gamma),R_{\pi'}(\cdot\mid\Gamma')\bigr\rangle_{\mu_E}&=\E_{\mu_E}\bigl[R_\pi(\sigma\mid\Gamma)\,R_{\pi'}(\sigma\pi\mid\Gamma')\bigr],\\ \bigl\langle R_\pi(\cdot\mid\Gamma),c_{\pi'}(\cdot\mid\Gamma')\bigr\rangle_{\mu_E}&=\E_{\mu_E}\bigl[R_\pi(\sigma\pi'\mid\Gamma)\,R_{\pi'}(\sigma\mid\Gamma')\bigr],\\ \bigl\langle R_\pi(\cdot\mid\Gamma),R_{\pi'}(\cdot\mid\Gamma')\bigr\rangle_{\mu_E}&=\E_{\mu_E}\bigl[R_\pi(\sigma\mid\Gamma)\,R_{\pi'}(\sigma\mid\Gamma')\bigr]. \end{align*} \end{lemma} \begin{proof} By \cref{lem:surgery-reverse}(2), $\langle c_\pi(\cdot\mid\Gamma),f\rangle_{\mu_E}=\int q_{\pi,\Gamma}f\dd\mu_E$ for measurable $f\ge0$. Since the function $f(\sigma,\omega)=R_{\pi'}(\sigma\mid\Gamma')$ does not depend on $\omega$, we have $q_{\pi,\Gamma}f(\sigma,\omega)=R_\pi(\sigma\mid\Gamma)R_{\pi'}(\sigma\pi\mid\Gamma')$. This proves the first identity. The second is the first with the roles of $(\pi,\Gamma)$ and $(\pi',\Gamma')$ exchanged, and the third is the definition of the inner product. \end{proof} \section{Comparison of four matchings}\label{sec:surgery-four} It remains to bound the inner product of two incoming densities. In this section, we fix $E$ as in \cref{lem:surgery-reverse} such that (H1) holds, a matching $\sigma$, and $\pi,\pi'\in\Pi_E$ with routes $\gamma,\gamma'$ and selected rows $S,S'$. We set $U=S\cup S'$ and define $\sigma_1,\sigma_2,\sigma_{12}$ as in \cref{lem:surgery-sectors}. \begin{center} \begin{tabular}{lll} \toprule Matching & Assignment & Role\\ \midrule $\sigma$ & $\sigma$ & Current matching\\ $\sigma_1$ & $\sigma\pi^{-1}$ & Pre-surgery matching for $\pi$\\ $\sigma_2$ & $\sigma\pi'^{-1}$ & Pre-surgery matching for $\pi'$\\ $\sigma_{12}$ & $\sigma\pi^{-1}\pi'^{-1}$ & Composed matching\\ \bottomrule \end{tabular} \end{center} At a state with matching $\sigma$, the product $c_\pi(\cdot\mid\Gamma)c_{\pi'}(\cdot\mid\Gamma')$ contains the rates $R_\pi(\sigma_1\mid\Gamma)$ and $R_{\pi'}(\sigma_2\mid\Gamma')$, which vanish unless \begin{equation}\label{eq:surgery-source} A_{i\sigma_1(i)}=1\quad(i\in S),\qquad A_{i\sigma_2(i)}=1\quad(i\in S'). \end{equation} We fix paths $\omega_{U^c}$ that are individually permitted for $\sigma$ and for the other three matchings, which agree with $\sigma$ outside $U$. We use the notation of \cref{lem:surgery-insertion} with $O=U^c$, so that $\widehat K^\tau_i$ and $\widehat z^\tau_i$, for $i\in U$, contain the checks with the rows outside $U$. In this notation, $Z_S(\tau;\omega_{S^c})=Z_S(\tau;\omega_{S'\setminus S})$ and $Z_{S'}(\tau;\omega_{S'^c})=Z_{S'}(\tau;\omega_{S\setminus S'})$, the measure $M^U_\sigma(\dd\omega_U;\omega_{U^c})$ of \eqref{eq:surgery-conditional} equals $\prod_{i\in U}\widehat K^\sigma_i(\dd\omega_i)\prod_{\{i,k\}\subset U}h_{ik}(\omega_i,\omega_k)$, and $Z_U(\tau;\omega_{U^c})=Z_U(\tau;\varnothing)$. Let $R\ge\max\{R_{\mathrm{int}},1\}$. We call a row $i\in S$ near if $d_\infty(\operatorname{anch}(i),\gamma')\le R$, and a row $i\in S'$ near if $d_\infty(\operatorname{anch}(i),\gamma)\le R$; other rows of $S$ and $S'$ are far. Let $C_S,F_S$ be the near and far rows of $S$, let $C_{S'},F_{S'}$ be those of $S'$, and let $C_U=C_S\cup C_{S'}=U\setminus(F_S\cup F_{S'})$. These sets have the following properties. \begin{enumerate}[label=(N\arabic*)] \item Every row of $S\cap S'$ is anchored at a cell of $\gamma\cap\gamma'$, so it lies in $C_S\cap C_{S'}$. In particular, $F_S\subset S\setminus S'$, $F_{S'}\subset S'\setminus S$, $F_S\cap F_{S'}=\varnothing$, $C_U\cap S=C_S$, and $C_U\cap S'=C_{S'}$. \item Each near row of $S$ is the selected row of a cell $v\in\gamma$ with $d_\infty(v,\gamma')\le R$, and there are at most $J_R(\gamma,\gamma')$ such cells; the same holds for $S'$. This implies that $\abs{C_S}+\abs{C_{S'}}\le2J_R(\gamma,\gamma')$ and $\abs{C_U}\le2J_R(\gamma,\gamma')$. \item A far row of $S$ is anchored at distance more than $R\ge R_{\mathrm{int}}$ from every row of $S'$. By (H1), it has no nontrivial check with any row of $S'$. The same holds with $S$ and $S'$ exchanged. \item $\sigma_{12}=\sigma_1$ on $S\setminus S'$, and $\sigma_{12}=\sigma_2$ on $F_{S'}$. \end{enumerate} To prove (N4), we note that $\sigma_{12}(i)=\sigma_1(\pi'^{-1}(i))$, which equals $\sigma_1(i)$ if $i\notin S'$. If $i\in F_{S'}$, then $k=\pi'^{-1}(i)$ is the selected row of the cell preceding $\operatorname{anch}(i)$ on $\gamma'$, which is at distance one from $\operatorname{anch}(i)$. Since $d_\infty(\operatorname{anch}(i),\gamma)>R\ge1$, we have $k\notin S$ and $\sigma_{12}(i)=\sigma_1(k)=\sigma(k)=\sigma_2(i)$. \begin{lemma}\label{lem:surgery-four} Assume (H1)--(H3) and (H3$_1$) for $E$ and $\eta_{\mathrm{loc}}\le1/2$, and let $R\ge\max\{R_{\mathrm{int}},1\}$. In the setting above, assume \eqref{eq:surgery-source}. Then all divisors in the following integral are positive, and \begin{equation}\label{eq:surgery-four} \int\frac{Z_S(\sigma_1;\omega_{S^c})}{Z_S(\sigma;\omega_{S^c})}\,\frac{Z_{S'}(\sigma_2;\omega_{S'^c})}{Z_{S'}(\sigma;\omega_{S'^c})}\,M^U_\sigma(\dd\omega_U;\omega_{U^c})\le G^{\abs{S\cap S'}}(1-\eta_{\mathrm{loc}})^{-4J_R(\gamma,\gamma')}\,Z_U(\sigma_{12};\omega_{U^c}), \end{equation} where $G=(2z_+/z_-)^3$. Further, if $\pi\in\Pi^{(a)}_E$ and $\pi'\in\Pi^{(b)}_E$ for groups $a$ and $b$, then for all $\Gamma,\Gamma'\in\mathfrak G$, \begin{equation}\label{eq:surgery-cc} \bigl\langle c_\pi(\cdot\mid\Gamma),c_{\pi'}(\cdot\mid\Gamma')\bigr\rangle_{\mu_E}\le G^{\abs{S\cap S'}}(1-\eta_{\mathrm{loc}})^{-4J_R(\gamma,\gamma')}\,\E_{\mu_E}\bigl[R_\pi(\sigma\pi'\mid\Gamma)\,R_{\pi'}(\sigma\pi'\pi\pi'^{-1}\mid\Gamma')\bigr]. \end{equation} \end{lemma} \begin{proof} We proceed in five steps. \emph{Step 1: masses.} By \eqref{eq:surgery-source} and \cref{lem:surgery-sectors}(2), every pairing $(i,\tau(i))$ with $i\in U$ and $\tau\in\{\sigma,\sigma_1,\sigma_2,\sigma_{12}\}$ is crossed. By \cref{lem:surgery-insertion}(1) and (H2), we have \begin{equation}\label{eq:surgery-zhat} (1-\eta_{\mathrm{glob}})z_-\le\widehat z^\tau_i\le z_+,\qquad\text{so that}\qquad \frac{\widehat z^\tau_i}{\widehat z^{\tau'}_i}\le\frac{2z_+}{z_-}=G^{1/3} \end{equation} for all such $i$ and $\tau,\tau'$, where we used $\eta_{\mathrm{glob}}\le1/2$. Recall that $\widehat z^\tau_i$ depends on $\tau$ only through $\tau(i)$. Since $S$ and $S'$ contain exactly one row in each cell of $\gamma$ and of $\gamma'$, respectively, $U$ contains at most two rows in each cell. A cell has at most $C_{\mathrm{int}}$ cells within $d_\infty$-distance $R_{\mathrm{int}}$, including itself. Hence every row of $U$ has at most $N=2C_{\mathrm{int}}-1$ other rows of $U$ within $d_\infty$-distance $R_{\mathrm{int}}$, and $2N\eta_1\le\eta_{\mathrm{loc}}\le1/2$. \emph{Step 2: normalizers of $S$ and $S'$.} Let $\tau\in\{\sigma,\sigma_1\}$. The paths $\omega_{S'\setminus S}$ under $M^U_\sigma$ are individually permitted for $\sigma$ and for $\sigma_1$, which agrees with $\sigma$ on $S'\setminus S$. By \cref{lem:surgery-insertion} with $W=S$, $W_0=F_S$, and $X=S'\setminus S$, we have \begin{equation}\label{eq:surgery-near} (1-\eta_{\mathrm{loc}})^{\abs{C_S}}\Bigl(\prod_{i\in C_S}\widehat z^\tau_i\Bigr)Z_{F_S}(\tau;\omega_{S'\setminus S})\le Z_S(\tau;\omega_{S'\setminus S})\le\Bigl(\prod_{i\in C_S}\widehat z^\tau_i\Bigr)Z_{F_S}(\tau;\omega_{S'\setminus S}). \end{equation} The upper bound is part (1) of that lemma. The lower bound is part (2) with the number $N$ of Step 1, which applies since $W\cup X=U$ and the pairings are crossed by Step 1, together with $1-2N\eta_1\ge1-\eta_{\mathrm{loc}}$. By (N3), the checks between $F_S$ and $S'\setminus S$ are identically one, and we have $Z_{F_S}(\tau;\omega_{S'\setminus S})=Z_{F_S}(\tau;\varnothing)$, which we abbreviate as $Z_{F_S}(\tau)$. It does not depend on $\omega_{S'\setminus S}$, and it depends on $\tau$ only through its restriction to $F_S$. By \cref{lem:surgery-insertion}(1) with $W=F_S$ and $W_0=\varnothing$, it is positive. In the same way, for $\tau\in\{\sigma,\sigma_2\}$, \eqref{eq:surgery-near} holds with $S,S'$ exchanged, and $Z_{F_{S'}}(\tau)>0$. In particular, the divisors in \eqref{eq:surgery-four} are positive. \emph{Step 3: upper bound.} We insert the upper bounds from \eqref{eq:surgery-near} for the numerators in \eqref{eq:surgery-four} and the lower bounds for the denominators. These bounds hold pointwise in $\omega_U$, and the remaining integral of $M^U_\sigma$ is $Z_U(\sigma;\varnothing)$. Then the left side of \eqref{eq:surgery-four} is at most \[ (1-\eta_{\mathrm{loc}})^{-\abs{C_S}-\abs{C_{S'}}}\prod_{i\in C_S}\frac{\widehat z^{\sigma_1}_i}{\widehat z^\sigma_i}\prod_{i\in C_{S'}}\frac{\widehat z^{\sigma_2}_i}{\widehat z^\sigma_i}\cdot\frac{Z_{F_S}(\sigma_1)\,Z_{F_{S'}}(\sigma_2)}{Z_{F_S}(\sigma)\,Z_{F_{S'}}(\sigma)}\cdot Z_U(\sigma;\varnothing). \] By \eqref{eq:surgery-insertion-upper} with $W=U$ and $W_0=F_S\cup F_{S'}$, and since by (N3) there are no checks between $F_S$ and $F_{S'}$, we have \[ Z_U(\sigma;\varnothing)\le\Bigl(\prod_{i\in C_U}\widehat z^\sigma_i\Bigr)Z_{F_S\cup F_{S'}}(\sigma;\varnothing)=\Bigl(\prod_{i\in C_U}\widehat z^\sigma_i\Bigr)Z_{F_S}(\sigma)\,Z_{F_{S'}}(\sigma). \] Canceling the far normalizers of $\sigma$, we find that the left side of \eqref{eq:surgery-four} is at most \begin{equation}\label{eq:surgery-upper} (1-\eta_{\mathrm{loc}})^{-\abs{C_S}-\abs{C_{S'}}}\Bigl(\prod_{i\in C_U}\widehat z^\sigma_i\Bigr)\prod_{i\in C_S}\frac{\widehat z^{\sigma_1}_i}{\widehat z^\sigma_i}\prod_{i\in C_{S'}}\frac{\widehat z^{\sigma_2}_i}{\widehat z^\sigma_i}\cdot Z_{F_S}(\sigma_1)\,Z_{F_{S'}}(\sigma_2). \end{equation} \emph{Step 4: the composed matching.} By \cref{lem:surgery-insertion}(2) with $\tau=\sigma_{12}$, $W=U$, $W_0=F_S\cup F_{S'}$, $X=\varnothing$, and the number $N$ of Step 1, which applies by Step 1, we have \[ \begin{split} Z_U(\sigma_{12};\varnothing)&\ge(1-\eta_{\mathrm{loc}})^{\abs{C_U}}\Bigl(\prod_{i\in C_U}\widehat z^{\sigma_{12}}_i\Bigr)Z_{F_S}(\sigma_{12})\,Z_{F_{S'}}(\sigma_{12})\\ &=(1-\eta_{\mathrm{loc}})^{\abs{C_U}}\Bigl(\prod_{i\in C_U}\widehat z^{\sigma_{12}}_i\Bigr)Z_{F_S}(\sigma_1)\,Z_{F_{S'}}(\sigma_2). \end{split} \] In the first step, we used (N3) to factorize the far normalizer, and in the second step, we used (N4) and $F_S\subset S\setminus S'$. Combining this with \eqref{eq:surgery-upper}, and using (N2), we conclude that the left side of \eqref{eq:surgery-four} is at most \[ (1-\eta_{\mathrm{loc}})^{-\abs{C_S}-\abs{C_{S'}}-\abs{C_U}}\,\mathcal Q\,Z_U(\sigma_{12};\varnothing)\le(1-\eta_{\mathrm{loc}})^{-4J_R(\gamma,\gamma')}\,\mathcal Q\,Z_U(\sigma_{12};\varnothing), \] where \[ \mathcal Q=\prod_{i\in C_U}\mathcal Q_i,\qquad \mathcal Q_i=\frac{\widehat z^\sigma_i}{\widehat z^{\sigma_{12}}_i}\Bigl(\frac{\widehat z^{\sigma_1}_i}{\widehat z^\sigma_i}\Bigr)^{\ind\{i\in S\}}\Bigl(\frac{\widehat z^{\sigma_2}_i}{\widehat z^\sigma_i}\Bigr)^{\ind\{i\in S'\}}. \] Here we used $C_U\cap S=C_S$ and $C_U\cap S'=C_{S'}$ from (N1). \emph{Step 5: the scalar ratio.} We show that $\mathcal Q\le G^{\abs{S\cap S'}}$, which proves \eqref{eq:surgery-four}. Let $i\in C_U$. If $i\in S\setminus S'$, then $\mathcal Q_i=\widehat z^{\sigma_1}_i/\widehat z^{\sigma_{12}}_i=1$ by (N4). If $i\in S'\setminus S$ and $k=\pi'^{-1}(i)\notin S$, then $\sigma_{12}(i)=\sigma_1(k)=\sigma(k)=\sigma_2(i)$, and we have $\mathcal Q_i=\widehat z^{\sigma_2}_i/\widehat z^{\sigma_{12}}_i=1$. If $i\in S'\setminus S$ and $k=\pi'^{-1}(i)\in S$, then $k\in S\cap S'$ and $i\in\pi'(S\cap S')$, and $\mathcal Q_i\le2z_+/z_-$ by \eqref{eq:surgery-zhat}. If $i\in S\cap S'$, then $\mathcal Q_i=\widehat z^{\sigma_1}_i\widehat z^{\sigma_2}_i/(\widehat z^\sigma_i\widehat z^{\sigma_{12}}_i)\le(2z_+/z_-)^2$. The number of rows in the third case is at most $\abs{\pi'(S\cap S')}=\abs{S\cap S'}$. Since $2z_+/z_-\ge1$, we conclude that $\mathcal Q\le(2z_+/z_-)^{3\abs{S\cap S'}}=G^{\abs{S\cap S'}}$. \emph{Consequence.} Let $\pi\in\Pi^{(a)}_E$ and $\pi'\in\Pi^{(b)}_E$, and write $F=(1-\eta_{\mathrm{loc}})^{-4J_R(\gamma,\gamma')}$. By \eqref{eq:surgery-reverse}, \[ \begin{split} \bigl\langle c_\pi(\cdot\mid\Gamma),c_{\pi'}(\cdot\mid\Gamma')\bigr\rangle_{\mu_E}&=\frac1\Xi\sum_\sigma R_\pi(\sigma_1\mid\Gamma)R_{\pi'}(\sigma_2\mid\Gamma')\int N^{U^c}_\sigma(\dd\omega_{U^c})\\ &\qquad\times\int\frac{Z_S(\sigma_1;\omega_{S^c})}{Z_S(\sigma;\omega_{S^c})}\,\frac{Z_{S'}(\sigma_2;\omega_{S'^c})}{Z_{S'}(\sigma;\omega_{S'^c})}\,M^U_\sigma(\dd\omega_U;\omega_{U^c}), \end{split} \] where we used \eqref{eq:surgery-split} with $U$ in place of $S$ and that the rates do not depend on $\omega$. If the product of the two rates is positive, then \eqref{eq:surgery-source} holds, and \eqref{eq:surgery-four} applies for $N^{U^c}_\sigma$-almost every $\omega_{U^c}$. Since $\sigma_{12}$ agrees with $\sigma$ outside $U$, we have $N^{U^c}_\sigma=N^{U^c}_{\sigma_{12}}$, and by \eqref{eq:surgery-split}, \[ \int N^{U^c}_{\sigma_{12}}(\dd\omega_{U^c})\,Z_U(\sigma_{12};\omega_{U^c})=Z_{\sigma_{12}}(E). \] We obtain \[ \bigl\langle c_\pi(\cdot\mid\Gamma),c_{\pi'}(\cdot\mid\Gamma')\bigr\rangle_{\mu_E}\le G^{\abs{S\cap S'}}F\,\frac1\Xi\sum_\sigma Z_{\sigma_{12}}(E)\,R_\pi(\sigma_1\mid\Gamma)R_{\pi'}(\sigma_2\mid\Gamma'). \] We have obtained the normalizing mass of the composed matching. We now sum using that matching as the base variable. The conjugated permutation in the second rate follows from the order of composition. The map $\sigma\mapsto\tau=\sigma\pi^{-1}\pi'^{-1}$ is a bijection of the set of matchings, with inverse $\sigma=\tau\pi'\pi$. In terms of $\tau$, we have $\sigma_1=\tau\pi'$ and $\sigma_2=\tau\pi'\pi\pi'^{-1}$. Reindexing the sum by $\tau$ and using $Z_\tau(E)/\Xi(E)=\mu_E(\{\tau\}\times\mathsf P^I)$ proves \eqref{eq:surgery-cc}. \end{proof} The second rate in \eqref{eq:surgery-cc} is evaluated at $\sigma\pi'\pi\pi'^{-1}$, which differs from $\sigma\pi$ when the two labeled routes share a row: two cyclic permutations with a common row need not commute. When $J_R(\gamma,\gamma')=0$, the comparison is an identity, and the ordinary rates $r_\pi,r_{\pi'}$ are unchanged by the other surgery; for two moves of the same group, the route indicators still require the stability argument. \begin{lemma}\label{lem:surgery-exact} Assume (H1)--(H3) for $E$, and let $R\ge\max\{R_{\mathrm{int}},1\}$. Let $\pi\in\Pi^{(a)}_E$ and $\pi'\in\Pi^{(b)}_E$ for groups $a$ and $b$, with routes $\gamma,\gamma'$ and selected rows $S,S'$, and suppose that $J_R(\gamma,\gamma')=0$. \begin{enumerate} \item The sets $S$ and $S'$ are disjoint, the permutations $\pi$ and $\pi'$ commute, so that $\pi'\pi\pi'^{-1}=\pi$, and $h_{ik}\equiv1$ for all $i\in S$ and $k\in S'$. \item For every matching $\sigma$, we have $r_\pi(\sigma\pi')=r_\pi(\sigma)$ and $r_{\pi'}(\sigma\pi)=r_{\pi'}(\sigma)$. \item In the setting of \cref{lem:surgery-four}, the inequality \eqref{eq:surgery-four} holds with equality. Consequently, for all $\Gamma,\Gamma'\in\mathfrak G$, \begin{equation}\label{eq:surgery-exact} \bigl\langle c_\pi(\cdot\mid\Gamma),c_{\pi'}(\cdot\mid\Gamma')\bigr\rangle_{\mu_E}=\E_{\mu_E}\bigl[R_\pi(\sigma\pi'\mid\Gamma)\,R_{\pi'}(\sigma\pi\mid\Gamma')\bigr]. \end{equation} \end{enumerate} \end{lemma} \begin{proof} For part (1), every cell of $\gamma$ is at $d_\infty$-distance more than $R$ from every cell of $\gamma'$, since $J_R(\gamma,\gamma')=0$. In particular, $\gamma$ and $\gamma'$ have no common cell. Each selected row is anchored at a cell of its route, so $S\cap S'=\varnothing$, and permutations with disjoint supports commute. Since $R\ge R_{\mathrm{int}}$, (H1) implies that $h_{ik}\equiv1$ for $i\in S$ and $k\in S'$. For part (2), the matching $\sigma\pi'$ differs from $\sigma$ only at the rows of $S'$, which are anchored at cells of $\gamma'$. No cell of $\gamma$ contains such a row, and no selected row of $\pi$ lies in $S'$. Since accessibility is row-local, we have $m^{(a)}_v(\sigma\pi')=m^{(a)}_v(\sigma)$ and $A_{i_v\sigma\pi'(i_v)}=A_{i_v\sigma(i_v)}$ for every $v\in\gamma$, and $r_\pi(\sigma\pi')=r_\pi(\sigma)$. The second identity follows in the same way. For part (3), we use the notation of \cref{lem:surgery-insertion} with $O=U^c$. By part (1), there are no checks between $S$ and $S'$. Then $Z_S(\tau;\omega_{S^c})=Z_S(\tau;\varnothing)$ and $Z_{S'}(\tau;\omega_{S'^c})=Z_{S'}(\tau;\varnothing)$ do not depend on $\omega_U$, and $M^U_\sigma(\dd\omega_U;\omega_{U^c})$ is the product of the measures $\prod_{i\in S}\widehat K^\sigma_i(\dd\omega_i)\prod_{\{i,k\}\subset S}h_{ik}(\omega_i,\omega_k)$ and $\prod_{i\in S'}\widehat K^\sigma_i(\dd\omega_i)\prod_{\{i,k\}\subset S'}h_{ik}(\omega_i,\omega_k)$, whose masses are $Z_S(\sigma;\varnothing)$ and $Z_{S'}(\sigma;\varnothing)$. These masses are positive by Step 2 of the proof of \cref{lem:surgery-four}. Integrating, we find that the left side of \eqref{eq:surgery-four} equals $Z_S(\sigma_1;\varnothing)\,Z_{S'}(\sigma_2;\varnothing)$. For $i\in S$, we have $\pi'^{-1}(i)=i$ and $\sigma_{12}(i)=\sigma_1(i)$; for $i\in S'$, the row $\pi'^{-1}(i)$ lies in $S'$ and not in $S$, and $\sigma_{12}(i)=\sigma_1(\pi'^{-1}(i))=\sigma(\pi'^{-1}(i))=\sigma_2(i)$. Since there are no checks between $S$ and $S'$, we obtain $Z_U(\sigma_{12};\varnothing)=Z_S(\sigma_1;\varnothing)\,Z_{S'}(\sigma_2;\varnothing)$. The right side of \eqref{eq:surgery-four} is $Z_U(\sigma_{12};\omega_{U^c})=Z_U(\sigma_{12};\varnothing)$, because $S\cap S'=\varnothing$ and $J_R(\gamma,\gamma')=0$, which proves the equality in \eqref{eq:surgery-four}. The derivation of \eqref{eq:surgery-cc} from \eqref{eq:surgery-four} in the proof of \cref{lem:surgery-four} uses no other inequality. Then \eqref{eq:surgery-cc} also holds with equality, with right side $\E_{\mu_E}[R_\pi(\sigma\pi'\mid\Gamma)R_{\pi'}(\sigma\pi'\pi\pi'^{-1}\mid\Gamma')]$. By part (1), $\pi'\pi\pi'^{-1}=\pi$, which gives \eqref{eq:surgery-exact}. \end{proof} Part (3) is the cancellation of conditional normalizing constants of \cite[Lemma~8.3]{OpenAI2026HS}, reproduced as \cite[Lemma~9.3]{OpenAI2026BEC}, in its simplest form: when no check joins the two changes, the normalizers factor and cancel exactly. Together with part (2), it shows that a pair of labeled routes whose routes do not meet contributes exactly zero to the signed expansion, once the route indicators in the four terms agree; this is used in Step 4 of the proof of \cref{thm:surgery} and in \cref{lem:surgery-cross}. \begin{remark}\label{rem:surgery-bec} The proof of \cref{lem:surgery-four} follows the proof of \cite[Lemma~8.3]{OpenAI2026HS}, reproduced as \cite[Lemma~9.3]{OpenAI2026BEC}. Rows far from the other route are separated from the near rows by deletion and insertion, the normalizers of the far rows cancel, and the cost of the insertions is confined to the near rows. The comparison of four states with a defect supported on the shared rows and their neighbors on a route adapts \cite[Section~10.1]{OpenAI2026B}, where the selected slots are partitioned into the shared set $\mathsf h$ and the set $\mathsf d=\mathsf h\cup\operatorname{pred}(\mathsf h)$; our set $(S\cap S')\cup\pi'(S\cap S')$ plays the role of $\mathsf d$. The fixed-reference-law comparison in \cite[Lemma~9.3]{OpenAI2026BEC} cannot be applied directly when endpoints are reassigned. With the matching fixed, a path ending at a different slot is singular with respect to its former bridge law. If slots are instead included in the path coordinates, enforcing their distinctness does not satisfy the near-one insertion bound. We compare the four matching normalizers directly in \cref{lem:surgery-four}. The crossed mass hypothesis (H2) controls their individual path masses, and the composed normalizer permits the reindexing back to the matching law. \end{remark} \section{Label averaging}\label{sec:surgery-labels} By \cref{lem:surgery-easy,lem:surgery-four}, the four inner products for a pair of labeled routes are bounded by expectations of explicit functions of the matching. We now sum these functions over the labelings of a fixed pair of routes by rows of one group $a$. The next lemma describes how the bad sets of group $a$ of the matchings that appear differ from $B^{(a)}_\sigma$. \begin{lemma}\label{lem:surgery-fields} Let $\sigma$ be a matching, let $a$ be a group, let $\pi,\pi'\in\Pi^{(a)}_E$ have routes $\gamma,\gamma'$ and selected rows $S,S'$, and let $\tilde\pi=\pi'\pi\pi'^{-1}$. For $\tau\in\{\sigma\pi,\sigma\pi',\sigma\tilde\pi\}$, every cell contains at most two rows $i$ with $\tau(i)\neq\sigma(i)$, and these rows belong to group $a$; in particular, $\abs{m^{(a)}_v(\tau)-m^{(a)}_v(\sigma)}\le2$ for every cell $v$, and $B^{(a)}_\tau\subset V^{(a)}_\sigma$. Further, \begin{equation}\label{eq:surgery-fields} B^{(a)}_{\sigma\pi}\triangle B^{(a)}_\sigma\subset\gamma,\qquad B^{(a)}_{\sigma\pi'}\triangle B^{(a)}_\sigma\subset\gamma',\qquad B^{(a)}_{\sigma\tilde\pi}\triangle B^{(a)}_\sigma\subset\mathcal N_1(\gamma). \end{equation} \end{lemma} \begin{proof} The matching $\sigma\pi$ changes the slots of exactly the rows of $S$, one in each cell of $\gamma$, and $\sigma\pi'$ those of $S'$. The permutation $\tilde\pi$ moves a row $i$ if and only if $\pi'^{-1}(i)\in S$; its support is $\pi'(S)=(S\setminus S')\cup\pi'(S\cap S')$. The set $S\setminus S'$ has at most one row in each cell, and these cells lie on $\gamma$. The set $\pi'(S\cap S')\subset S'$ has at most one row in each cell, and the row $\pi'(k)$, for $k\in S\cap S'$, is anchored at the cell following $\operatorname{anch}(k)\in\gamma$ on $\gamma'$, which is a face neighbor of a cell of $\gamma$. The rows of $S$ and $S'$ belong to group $a$. Since $A_{i\tau(i)}$ depends only on $i$ and $\tau(i)$, we have $m^{(a)}_v(\tau)-m^{(a)}_v(\sigma)=\sum_{i\in I^{(a)}_v}(A_{i\tau(i)}-A_{i\sigma(i)})$, and only rows with $\tau(i)\neq\sigma(i)$ contribute. This proves the bound on the counts and \eqref{eq:surgery-fields}. If $v\in B^{(a)}_\tau$, then $v\in\Lambda_{\mathrm{act}}$ and $m^{(a)}_v(\sigma)\le m^{(a)}_v(\tau)+2{\raggedright\arraybackslash}p{0.40\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}p{0.28\textwidth}@{}} \toprule Object & In \cite{OpenAI2026BEC} & Here\\ \midrule symmetric killed heat kernel; partition function & $B^{\rm sym}_s$; $Z_N$ & $K^{\mathrm s}_s$ (\cref{sec:id-fk}); $\tr e^{-\beta H}$\\ purified slab law & $\mathcal R$ & $\PP^{\mathrm{pur}}$ \eqref{eq:thermal-pur}\\ close-count fraction; separation constant & $\theta$; $b$ & $\theta_0$; $b_0$\\ close count at a cell $v$ & $C_v(X)$ & $N^{\mathrm{cl}}_v(X)$\\ scaled oscillation on a mesh interval & $A_k(\omega)$ & $\osc_k(\omega)$\\ regularity exponent & $\gamma=1/12$ & written out\\ trial laws & $Q^u_{y,z}$, $Q^B_{y,z}$, $Q^0_{y,z}$, $Q^u_i$ & $P^u_{y,z}$, $P^B_{y,z}$, $P^0_{y,z}$, $P^u_i$\\ \bottomrule \end{tabular} \end{table} In a quoted statement, $c,C>0$ denote constants independent of $K$, $D$, and $T$ in the ranges under consideration; they may depend on parameters fixed earlier, including the window for $\alpha$ \cite[Section~2]{OpenAI2026BEC}. \section{The scaled setting}\label{sec:thermal-setting} Let $K\ge100$ be an integer, let $\Lambda_K=(\R/K\mathbb Z)^3$ with the torus distance $d$, and for $N\ge1$ set \begin{equation}\label{eq:thermal-window} V=K^3,\enspace D=N/V,\enspace r=\alpha/D,\enspace \alpha\in[\alpha_0/2,2\alpha_0],\enspace 00$ sufficiently small, as in \cite[Proposition~4.5]{OpenAI2026BEC}; then $\theta_0>0$ with $3\theta_0<\ell/2$; then $b_0\in(0,1)$ sufficiently small, as in \cite[Proposition~4.6]{OpenAI2026BEC}. Set \begin{equation}\label{eq:thermal-closecount} s_0=b_0D^{-1/3},\qquad N^{\mathrm{cl}}_v(X)=\#\{i: d(x_i,v)\le5,\ d_i(X)\le s_0\}, \end{equation} and increase $D$ so that $r\delta$; since $T\le\tau_0/\log D\le1/4$, this holds. The time $t$ is fixed before the lower threshold on $D$. Let $\nu_N$ be the measure of $N$ independent torus Brownian motions with generator $\Delta$ on $[0,\delta]$, with Lebesgue initial measure, and write $X(s)=(\omega_1(s),\dots,\omega_N(s))$. For $W'\in\Lambda_K^N$, the purified slab law of \cite[(5.1)]{OpenAI2026BEC} is \begin{equation}\label{eq:thermal-pur} \PP^{\mathrm{pur}}(\dd\omega,\dd W')=\frac{K^{\mathrm s}_{\beta/2-t}(W',X(0))\,K^{\mathrm s}_{\beta/2-t}(X(\delta),W')}{\tr e^{-\beta H}}\,\mathbf 1_{\mathcal E_\delta}(\omega)\,\nu_N(\dd\omega)\dd W', \end{equation} where $\mathbf 1_{\mathcal E_\delta}(\omega)=1$ if and only if $d(\omega_i(s),\omega_k(s))>r$ for all $s\in[0,\delta]$ and $i0$ there are $H_1$ and $\bar t\in(0,1]$, and for each $t\in(0,\bar t]$ there is $D_0$, such that, for every set $A$ of distinct cell centers, \begin{equation}\label{eq:thermal-certificates} \PP^{\mathrm{pur}}\{\text{no site of $A$ has a certificate}\}\le p_1^{|A|} \end{equation} whenever $K\ge K_1$, $D\ge D_0$, $\alpha\in[\alpha_0/2,2\alpha_0]$, and $0s_0$ and $d(z,c(2t))>s_0$. For either $P=P^B_{y,z}$, with any target cell $B$, or $P=P^0_{y,z}$, one has \begin{equation}\label{eq:thermal-single} P(\mathcal C_v\cap\mathcal H(c))\le Ctr+C\frac{rD^{1/4}}{\sqrt{h_0}}\le C\alpha\bigl(tD^{-1}+D^{-1/4}\bigr). \end{equation} Thus this probability tends to zero with $D$ for each fixed $t$. \end{lemma} \trilogynote{Adapted in:}{the proof of \cref{lem:slab-insertion}, for the supports and endpoints of \cref{sec:slab}; the statement itself is not applied.} The next statement is \cite[Remark~6.4]{OpenAI2026BEC}, without its last sentence, which refers to later uses in \cite{OpenAI2026BEC}. \begin{remark}\label{rem:thermal-midpoint} The midpoint in $P^B_{y,z}$ is averaged over a unit cell. An estimate uniform in a prescribed midpoint is false: if $u=c(t)$, the trial collides at time $t$. Restricting a uniform midpoint to a measurable set of volume at least $a_0>0$ multiplies an upper bound for a nonnegative event by at most $a_0^{-1}$. Conditioning a trial on an event of probability at least $a_0$ has the same cost. \end{remark} In \cref{sec:slab} the midpoint is never prescribed: every collision bound integrates the full bridge law, and the only normalization is a division by a lower bound on a total mass. \usedin{the integrated bridge estimates of \cref{sec:slab-mass,sec:slab-insertion}; no estimate at a prescribed midpoint is asserted.} \section{Useful labels under the local conditions}\label{sec:thermal-useful} Label $i$ refers to the path $\omega_i$ of the sample. Following \cite[Section~7.1]{OpenAI2026BEC}, a label $i$ is \emph{eligible at $v$ in the sense of} \cite{OpenAI2026BEC} if \begin{enumerate}[label=\textup{(\roman*)}] \item $\omega_i(0)\in B_v$ and $d(\omega_i(2t),v)\le3$; \item $d(\omega_i(0),\omega_k(0))>s_0$ and $d(\omega_i(2t),\omega_k(2t))>s_0$ for every $k\ne i$; \item its trajectory is regular and remains in the ball of radius $10$ about $v$. \end{enumerate} For such a label and $u$ in a target cell, write $P^u_i=P^u_{\omega_i(0),\omega_i(2t)}$, with all lifts near $v$. For a collection $F$ of fixed paths with labels other than $i$, let $C_i(F)$ be the event that the trial is regular, stays within distance $10$ of $v$, and avoids every path in $F$ at distance $r$, that is, stays at distance greater than $r$ at every time. The next statement reproduces \cite[Lemma~7.1]{OpenAI2026BEC}, retaining its terminology. \begin{lemma}\label{lem:thermal-useful} At a cell with a certificate there are at least $\ell D/2$ eligible labels. After $t$ has been chosen sufficiently small and $D$ sufficiently large, at least nine tenths of these labels satisfy the following stronger property: for every target cell $B_w$, \begin{equation}\label{eq:thermal-useful} \bigl|\bigl\{u\in B_w:P^u_i\bigl(C_i(F_i^{\rm all})\bigr)\ge0.9\bigr\}\bigr|\ge0.99. \end{equation} Here $F_i^{\rm all}$ consists of all other trajectories in the sampled configuration, and $\abs{\cdot}$ denotes spatial volume. \end{lemma} We call an eligible label with property \eqref{eq:thermal-useful} \emph{useful}. The requirement on $t$ depends on $H_1$, through \cite[Corollary~6.5]{OpenAI2026BEC}, which is why the reduction of $t$ permitted in \cref{prop:thermal-certificates} is needed. Since every path that starts in $B_v$ visits the ball of radius $12$ about $v$, \cref{def:thermal-certificate} implies that at most $H_1D$ paths start in $B_v$ at a cell satisfying the local conditions. By \cref{lem:thermal-useful}, we have at least $9\ell D/20$ useful labels there, a fraction at least $9\ell/(20H_1)$ of the paths starting in $B_v$. \usedin{\cref{prop:slab-vulnerable}.} \section{The container}\label{sec:thermal-container} Let $\Omega\subset\R^3$ be a fixed nonempty bounded open set, and for an integer $K\ge1$ let $U=K\Omega$, $V=\abs U$, $D=N/V$, $r=\alpha/D$, with $\alpha\in[\alpha_0/2,2\alpha_0]$ and $012$, a cell $B_v$ is \emph{interior} if $v\in U$ and $\dist(v,\R^3\setminus U)\ge R_0$. Local conditions, regular paths, eligible and useful labels, the trial laws $P^u_i$, and the events $C_i(F)$ are defined as in \cref{sec:thermal-certificates,sec:thermal-useful}, with Euclidean distances; the trial laws consist of two independent unrestricted Euclidean bridges. \begin{proposition}\label{prop:thermal-container} The constants $\alpha_0$, $\ell$, $\theta_0$, $b_0$, and $C_{\rm reg}$ of \cref{sec:thermal-setting} can be chosen so that the following holds. Fix $R_0>12$. For every $p_1>0$, one can choose $H_1$ large, then $t\in(0,1]$ sufficiently small, and then $D_0$ sufficiently large, and there exists $K_\Omega<\infty$, depending only on $\Omega$ and $R_0$, such that, whenever $K\ge K_\Omega$, $D\ge D_0$, $N=DV\in\mathbb N$, $\alpha\in[\alpha_0/2,2\alpha_0]$, and $0\delta$, and with $V=\abs U$ in a container. We write $d$ for the torus distance or the Euclidean distance, $\mathbb L$ for the cell lattice $\mathbb L_K$ or $\mathbb Z^3$, and $\mathcal N(v)=\{v,v\pm e_1,v\pm e_2,v\pm e_3\}$ for a cell $v$; the cells $B_w$, $w\in\mathcal N(v)$, are the seven target cells of $v$. \paragraph{Active cells.} On the torus, every cell is active: $\Lambda_{\mathrm{act}}=\mathbb L_K$. In a container, set $R_{\mathrm{act}}=64$ and \begin{equation}\label{eq:slab-active} \Lambda_{\mathrm{act}}=\{v\in\mathbb Z^3: v\in U,\ \dist(v,\R^3\setminus U)\ge R_{\mathrm{act}}\}. \end{equation} These are the interior cells of \cref{sec:thermal-container} for $R_0=R_{\mathrm{act}}$. In both settings, $\Lambda_{\mathrm{act}}$ is finite: the torus has $K^3$ cells, and the active cells in a container belong to the bounded set $U=K\Omega$. Rows are retained only at active cells, and the bad and vulnerable sets of \cref{sec:slab-mass} are subsets of $\Lambda_{\mathrm{act}}$. The depth $64$ ensures that permitted paths of rows at active cells stay in $U$ (\cref{lem:slab-range}). That accepted routes lie in $\Lambda_{\mathrm{act}}$ is part of (H5) and is proved in \cref{sec:proof-ncri}. \paragraph{The slab.} Fix $t\in(0,1]$ and set $\delta=2t$. We write $(Y,Z,\sigma,\omega)$ for a sample of the one-slab law $\PP$ of \eqref{eq:thermal-one-slab}; the path $\omega_i$ runs from $Y_i$ to $Z_{\sigma(i)}$. By \cref{prop:id-slabs}(4), $\PP$ is the law of every full slab of the Bose path measure, and by \cref{lem:thermal-slab} the results of \cref{sec:thermal} apply to events of the path family under $\PP$. \paragraph{Standing assumptions.} Unless stated otherwise, we assume \begin{equation}\label{eq:slab-standing} D>2,\qquad\text{\eqref{eq:thermal-scales} holds with }C=60,\qquad\text{and }K\ge2000\text{ on the torus}. \end{equation} The first two conditions hold for all $D$ larger than a threshold depending only on $t$, $\alpha_0$, and $b_0$. Under them, $r<1$. \paragraph{Local lifts.} On the torus, for $K\ge2000$ the projection $\R^3\to\Lambda_K$ is an isometry from every Euclidean ball of radius $400$ onto its image, since two points of such a ball are at distance at most $800s_0$ for every $k\ne i$, and $d(Z_{\sigma(i)},Z_l)>s_0$ for every $l\ne\sigma(i)$; \item $\omega_i\in\mathcal C^-_{Y_i}\cap\mathcal C^+_{Z_{\sigma(i)}}$. \end{enumerate} \end{definition} Note that condition (E1) depends only on $Y$, on $Z$, and on the slot $\sigma(i)$; condition (E2) consists of a test on the first half of $\omega_i$ that refers to its initial point and a test on the second half that refers to its terminal point. \begin{lemma}\label{lem:slab-eligible} Assume \eqref{eq:slab-standing}. \begin{enumerate}[label=\textup{(\alph*)}] \item Let $\omega$ be a path on $[0,2t]$ with $\omega(0)\in B_v$. If $\omega$ is regular and its lift stays within distance $10$ of $\tilde v$, then $\omega\in\mathcal C^-_{\omega(0)}\cap\mathcal C^+_{\omega(2t)}$. In particular, every label that is eligible at $v$ in the sense of \cite{OpenAI2026BEC} (\cref{sec:thermal-useful}) is an eligible row. \item If $\omega\in\mathcal C^-_y\cap\mathcal C^+_z$, then $\abs{\tilde\omega(2t)-\tilde\omega(0)}\le40$ and $\abs{\tilde\omega(s)-\tilde\omega(0)}\le60$ for all $s\in[0,2t]$. Moreover, $\osc_k(\omega)\le D^{1/12}$ for all $1\le k\le2m_t$, and the diameters of $\tilde\omega$ on the two windows of length $h_0$ are at most $s_0/8$. \item For all $y,z$, the indicator of $\mathcal C^-_y$ is a function of the restriction of the path to $[0,t]$, and the indicator of $\mathcal C^+_z$ is a function of its restriction to $[t,2t]$. \end{enumerate} \end{lemma} \begin{proof} For (a), every point of the lift is within $10+\sqrt3/2<20$ of $\tilde\omega(0)$, since $\abs{\tilde\omega(0)-\tilde v}\le\sqrt3/2$, and within $20$ of $\tilde\omega(2t)$, since both lie within $10$ of $\tilde v$. The maximum test in (ii) is part of \eqref{eq:thermal-regular}. Since the summands in \eqref{eq:thermal-regular} are nonnegative, each half sum is at most the full sum, and we have \[ \frac1{m_t}\sum_{k\in\text{half}}(1+\osc_k(\omega))^3\le\frac2{2m_t}\sum_{k=1}^{2m_t}(1+\osc_k(\omega))^3\le2C_{\rm reg}. \] The window conditions are those of \cref{def:thermal-regular}. A label eligible at $v$ in the sense of \cite{OpenAI2026BEC} satisfies these hypotheses by its condition (iii), and condition (ii) there is (E1), since $Z_{\sigma(i)}=\omega_i(2t)$. For (b), the point $\tilde\omega(t)$ is within $20$ of both $\tilde\omega(0)$ and $\tilde\omega(2t)$, which gives the first bound. For $s\le t$ we have $\abs{\tilde\omega(s)-\tilde\omega(0)}\le20$, and for $s\ge t$ we have $\abs{\tilde\omega(s)-\tilde\omega(0)}\le\abs{\tilde\omega(s)-\tilde\omega(2t)}+40\le60$. The remaining assertions are part of \cref{def:slab-half}. Part (c) holds because the mesh intervals and the window used in each half event lie in the corresponding half of $[0,2t]$. \end{proof} \paragraph{Ordered lists.} For $v\in\Lambda_{\mathrm{act}}$, let $\mathcal I^{\rm el}_v$ be the set of eligible rows $i$ with $Y_i\in B_v$, and let $n_v=\abs{\mathcal I^{\rm el}_v}$. Fix integers $m,M\ge1$. Conditionally on the sample $(Y,Z,\sigma,\omega)$, choose independently for every $v\in\Lambda_{\mathrm{act}}$ a list $L_v$ as follows. If $n_v\ge mM$, then $L_v$ is a uniformly random ordered list of $mM$ distinct elements of $\mathcal I^{\rm el}_v$, each of the $(n_v)_{mM}=n_v!/(n_v-mM)!$ such lists having probability $1/(n_v)_{mM}$. If $n_vr$ for all $s\in[0,2t]$ and $k\notin I$ that, in a container, stay in $U$. For $i\in I$ and $j\in J$, define the finite measure \begin{equation}\label{eq:slab-kernel} K^E_{ij}(\dd\omega)=\mathbf 1_{\mathcal C^-_{Y_i}\cap\mathcal C^+_{Z_j}\cap\mathrm{Av}_F}(\omega)\,\mathbb W^{2t}_{Y_i,Z_j}(\dd\omega), \end{equation} and let $\mathcal P^E_{ij}$ be the set of paths from $Y_i$ to $Z_j$ in $\mathcal C^-_{Y_i}\cap\mathcal C^+_{Z_j}\cap\mathrm{Av}_F$. The measure $K^E_{ij}$ is carried by $\mathcal P^E_{ij}$, and we call the elements of $\mathcal P^E_{ij}$ the \emph{permitted paths} of the pairing $(i,j)$. Let $R_{\rm loc}=61$, $R_{\rm int}=123$, and $C_{\rm int}=(2R_{\rm int}+1)^3$. For distinct $i,k\in I$, set \[ h_{ik}(\omega,\omega')=\mathbf 1\{d(\omega(s),\omega'(s))>r\text{ for all }s\in[0,2t]\}\quad\text{if }d_\infty(\operatorname{anch}(i),\operatorname{anch}(k))\le R_{\rm int}, \] and $h_{ik}\equiv1$ otherwise. For a bijection $\sigma'\colon I\to J$, we define \begin{equation}\label{eq:slab-matching-measure} \mathcal M^E_{\sigma'}(\dd\omega_I)=\prod_{i\in I}K^E_{i\sigma'(i)}(\dd\omega_i)\prod_{iR_{\rm int}$, then $d(\omega(s),\omega'(s))>r$ for all $s$, all $\omega\in\mathcal P^E_{ij}$, and all $\omega'\in\mathcal P^E_{kl}$. For each $i\in I$, at most $C_{\rm int}mM$ rows $k\in I$ satisfy $d_\infty(\operatorname{anch}(i),\operatorname{anch}(k))\le R_{\rm int}$. In a container, every path in $\mathcal C^-_{Y_i}\cap\mathcal C^+_{Z_j}$ stays in $U$. These assertions do not depend on the matching. \end{lemma} \begin{proof} Let $v=\operatorname{anch}(i)$. By \cref{lem:slab-eligible}(b), a permitted path stays within $60$ of $Y_i$. Since $\abs{Y_i-\tilde v}\le\sqrt3/2$, the first assertion follows. If $d_\infty(v,v')>R_{\rm int}$, then some coordinate of $v-v'$ has absolute value at least $124$. Then $d(v,v')\ge124$, and two paths within $61$ of $v$ and of $v'$ are at distance at least $2>r$ at all times. There are at most $(2R_{\rm int}+1)^3$ cells $w$ with $d_\infty(v,w)\le R_{\rm int}$, and each carries at most $mM$ retained rows. In a container, $\dist(v,\R^3\setminus U)\ge64>61$ for $v\in\Lambda_{\mathrm{act}}$. \end{proof} By \cref{lem:slab-range}, replacing the avoidance indicator by one for rows with distant anchors does not change $\mathcal M^E_{\sigma'}$: on permitted paths both are equal to one. \usedin{\cref{prop:slab-law,lem:slab-insertion}, and \cref{cor:slab-system}, where it gives (H1).} \begin{proposition}\label{prop:slab-law} Assume \eqref{eq:slab-standing}. For almost every value of $E$ under $\PP$, we have $0<\Xi(E)<\infty$, and the conditional law of $(\sigma_I,\omega_I)$ given $E$ is \begin{equation}\label{eq:slab-law} \mu_E(\{\sigma'\}\times\dd\omega_I)=\Xi(E)^{-1}\mathcal M^E_{\sigma'}(\dd\omega_I),\qquad \sigma'\in\mathrm{Bij}(I,J). \end{equation} In particular, the joint law of $(E,\sigma_I,\omega_I)$ under $\PP$ is a matching system in the sense of \cref{def:surgery-system}, with exposed data $E$, rows $I=\bigsqcup_vI_v$ partitioned into the groups $I^{(1)},\dots,I^{(m)}$ with $\abs{I^{(a)}_v}\le M$, anchors $\operatorname{anch}(i)$, slots $J$, kernels $K^E_{ij}$, permitted sets $\mathcal P^E_{ij}$, pair checks $h_{ik}$, and active region $\Lambda_{\mathrm{act}}$. \end{proposition} \begin{proof} We first show that the exposed quantities that enter the density are the same for every matching of the retained rows. Fix a value of $E$ such that $\varepsilon_i=1$ for all $i\in I$; other values have probability zero, since the lists contain only eligible rows. Let $\sigma'\colon I\to J$ be a bijection, let $\omega'_i\in\mathcal P^E_{i\sigma'(i)}$ for $i\in I$, and let $\mathbf S'$ be the sample with the cut configurations $Y,Z$, the matching equal to $\sigma'$ on $I$ and to the exposed slots on $I^c$, the paths $\omega'_I$, and the frozen paths $F$. Then the following hold. \begin{enumerate}[label=\textup{(\arabic*)}] \item Every retained row $i$ is eligible in $\mathbf S'$. Indeed, $Y_i$ is separated from the other initial positions because $i$ was eligible in the original sample and this condition depends on $Y$ only. The slot $\sigma'(i)\in J$ is the slot of some retained row in the original sample; since that row was eligible, $Z_{\sigma'(i)}$ is separated from the other terminal positions. Condition (E2) holds because $\omega'_i\in\mathcal P^E_{i\sigma'(i)}$. \item Every row $k\notin I$ has the same eligibility indicator in $\mathbf S'$ as in the original sample, because this indicator depends only on $Y$, $Z$, $\sigma(k)$, and $\omega_k$, which are exposed. \item By (1) and (2), the eligible sets $\mathcal I^{\rm el}_v$ and the counts $n_v$ of $\mathbf S'$ are those of the original sample, and the lists $L$ have the same conditional probability $\prod_vs_v(L_v)$, where $s_v(L_v)=\mathbf 1\{n_v\ge mM\}/(n_v)_{mM}$ if $L_v\ne\varnothing$ and $s_v(\varnothing)=\mathbf 1\{n_vs_0\ \forall k\ne i\}\prod_{j\in J}\mathbf 1\{d(Z_j,Z_l)>s_0\ \forall l\ne j\}, \] which does not depend on $\sigma'$. By (3), the sampling factors are functions of $E$. Collecting the factors that depend only on $E$ into a function $G(E)\ge0$, and using \eqref{eq:slab-kernel}, the definition of $h_{ik}$, and \cref{lem:slab-range} for the pairs of retained paths, we factor the integrand as \[ c_NK^{\mathrm s}_{\beta-\delta}(Z,Y)\,G(E)\,\Phi(E)\prod_{k\notin I}\mathbb W^{2t}_{Y_k,Z_{\varphi(k)}}(\dd\omega_k)\times\Psi(E,\sigma',\omega_I)\,\mathcal M^E_{\sigma'}(\dd\omega_I). \] Let $\nu^{\rm ref}$ be the measure on exposed data that, for each choice of $L$, $\varphi$, and $\varepsilon$, equals $c_NK^{\mathrm s}_{\beta-\delta}(Z,Y)G(E)\dd Y\dd Z\prod_{k\notin I}\mathbb W^{2t}_{Y_k,Z_{\varphi(k)}}(\dd\omega_k)$. Summing over the finitely many choices gives \begin{equation}\label{eq:slab-disintegration} \E[\Phi(E)\Psi(E,\sigma_I,\omega_I)]=\int\Phi(E)\sum_{\sigma'\in\mathrm{Bij}(I,J)}\int\Psi(E,\sigma',\omega_I)\,\mathcal M^E_{\sigma'}(\dd\omega_I)\,\nu^{\rm ref}(\dd E). \end{equation} With $\Psi=1$, \eqref{eq:slab-disintegration} implies that the law of $E$ is $\Xi(E)\,\nu^{\rm ref}(\dd E)$. In particular, the set $\{\Xi=0\}$ has probability $\int_{\{\Xi=0\}}\Xi\,\dd\nu^{\rm ref}=0$. Since each $\mathcal M^E_{\sigma'}(1)$ is at most a product of the finite masses $\mathbb W^{2t}_{Y_i,Z_j}(1)$, we have $\Xi(E)<\infty$. Then \eqref{eq:slab-disintegration} states that \eqref{eq:slab-law} is a version of the conditional law of $(\sigma_I,\omega_I)$ given $E$. The exposed data take values in a finite disjoint union, over the choices of $L$, $\varphi$, and $\varepsilon$, of products of Euclidean spaces and path spaces, which is standard Borel. Since bridge measures depend measurably on their endpoints and the events in \eqref{eq:slab-kernel} are Borel sets of continuous paths, the maps $E\mapsto K^E_{ij}(A)$, $\mathcal M^E_{\sigma'}$, and $\Xi$ are measurable. Each retained row uses the path space $C([0,2t];\mathbb X)$ for all of its pairings, and the checks $h_{ik}$ do not depend on the matching. This is the structure required in \cref{def:surgery-system}; accessibility and the threshold $m_*$ are defined in \cref{sec:slab-mass}. \end{proof} \begin{remark}\label{rem:slab-nonuniform} The retained matching is not uniformly distributed given $E$: its conditional probabilities are $\mathcal M^E_{\sigma'}(1)/\Xi(E)$. For example, $K^E_{ij}=0$ when the lifted distance from $Y_i$ to $Z_j$ exceeds $40$, by \cref{lem:slab-eligible}(b), and every matching pairing such $i$ and $j$ has probability zero, while the matching of the sample has positive mass almost surely. If $\sigma_I$ is exposed as well, the conditional law is the product of the normalized enlarged kernels $K^E_{i\sigma_I(i)}$, conditioned on mutual avoidance. This has the same disintegration form as \cite[Lemma~7.2]{OpenAI2026BEC}, whose individual cutoffs and endpoint class are different. We cannot expose $\sigma_I$, since the surgeries of \cref{sec:surgery} change it. Apart from the eligibility and allocation marks whose fiber invariance was proved above, $E$ imposes no further condition on the retained paths. Neither these local conditions nor route acceptance is exposed. Steps (1)--(3) of the proof, the invariance of eligibility and of the sampling factors on the matching fiber, are the reason for imposing eligibility separately on the two halves of a path. \end{remark} \usedin{\cref{cor:slab-system}; the form \eqref{eq:slab-law} is the matching law of \cref{def:surgery-system} used in \cref{sec:surgery}.} \section{Accessibility and masses}\label{sec:slab-mass} Fix a value of $E$. For $i\in I_v$, $j\in J$, and $u\in\R^3$ with $\abs{u-\tilde v}\le400$ (on the torus, a point of the lifted ball about $\tilde v$), let $p^-_i(u)$ be the probability that a Brownian bridge of duration $t$ from $Y_i$ to $u$, with law $\widehat{\mathbb W}^{t,\mathrm E}_{\tilde Y_i,u}$, satisfies conditions (i)--(iii) of \cref{def:slab-half} for $\mathcal C^-_{Y_i}$, avoids every frozen path at distance $r$ during $[0,t]$, and, in a container, stays in $U$. Let $p^+_j(u)$ be the probability that a Brownian bridge of duration $t$ from $u$ to $Z_j$, with law $\widehat{\mathbb W}^{t,\mathrm E}_{u,\tilde Z_j}$ and run on the time interval $[t,2t]$, satisfies the conditions of \cref{def:slab-half} for $\mathcal C^+_{Z_j}$, avoids every frozen path at distance $r$ during $[t,2t]$, and, in a container, stays in $U$. On the torus, $Z_j$ is lifted to the lift nearest to $u$; if that lift is at distance more than $20$ from $u$, then $p^+_j(u)=0$. \begin{definition}\label{def:slab-access} For $i\in I_v$ and $j\in J$, set $A_{ij}=1$ if, for every $w\in\mathcal N(v)$, \begin{equation}\label{eq:slab-access} \bigl|\{u\in B_w:p^-_i(u)\,p^+_j(u)\ge0.9\}\bigr|\ge0.99, \end{equation} and $A_{ij}=0$ otherwise. For a bijection $\sigma'\colon I\to J$, a group $a$, and a cell $v$, let $m^{(a)}_v(\sigma')=\sum_{i\in I^{(a)}_v}A_{i\sigma'(i)}$. For an integer $m_*\ge4$, the \emph{bad set} and the \emph{vulnerable set} of the group $a$ are \[ B^{(a)}_{\sigma'}=\{v\in\Lambda_{\mathrm{act}}:m^{(a)}_v(\sigma')0$ requires a path from $u$ to $Z_l$ that stays within $20$ of $Z_l$. Since $p_t$ is radially decreasing, both heat kernels in \eqref{eq:slab-midpoint} are at least $p_t(20e_1)=g_*$ on $S_1\cap S_2$. Restricting the integral to $S_1\cap S_2$ yields $K^E_{il}(1)\ge0.98\,(0.9)^2g_*^2$. For the upper bound, the confinement of \cref{lem:slab-eligible}(b) selects a unique Euclidean lift on the torus. We first pass to that Euclidean bridge restriction and then drop all restrictions to obtain $K^E_{ij}(1)\le p_{2t}(Z_j-Y_i)\le p_{2t}(0)=z_+$, with the selected lift of $Z_j$ on the torus. If no lift is permitted, the mass is zero. \end{proof} The constants $z_\pm$ depend only on $t$. \Cref{lem:slab-mass} bounds the mass of the crossed pairing $(i,l)$; it does not assert that this pairing is accessible. The intersection of two success sets of volume $0.99$ is used in \cite[Section~7.4]{OpenAI2026BEC}, after Proposition~7.4 there, for two accessible labels sharing a midpoint target set, following \cite[Proposition~6.4]{OpenAI2026HS}. Here it bounds the unnormalized mass of a pairing whose two halves come from different accessible pairings. \usedin{\cref{cor:slab-system}, where it gives (H2); the lower bound $z_-$ is also used in \cref{lem:slab-insertion} and in the conditional area moments of \cref{sec:proof-ncri}.} \section{Insertion}\label{sec:slab-insertion} \begin{lemma}\label{lem:slab-insertion} Assume \eqref{eq:slab-standing}. There exists $C_t<\infty$, depending only on $t$ and $\alpha_0$, with the following property. Let $E$ be a value of the exposed data satisfying $d(Y_i,Y_k)>s_0$ for every $i\in I$ and $k\ne i$, and $d(Z_j,Z_l)>s_0$ for every $j\in J$ and $l\ne j$. Every value produced by the selection procedure satisfies these conditions, since retained rows are eligible and $J=\sigma(I)$. Let $i\in I$ and $j\in J$ satisfy $K^E_{ij}(1)\ge z_-$, let $k\in I\setminus\{i\}$ and $j'\in J\setminus\{j\}$, and let $\omega'\in\mathcal P^E_{kj'}$. Then \begin{equation}\label{eq:slab-one-opponent} K^E_{ij}\bigl(\{\omega:h_{ik}(\omega,\omega')=0\}\bigr)\le\eta_1K^E_{ij}(1),\qquad \eta_1=C_tD^{-1/4}. \end{equation} Further, for every bijection $\sigma'\colon I\to J$ with $K^E_{i\sigma'(i)}(1)\ge z_-$, every $I'\subset I\setminus\{i\}$, and every choice of $\omega_k\in\mathcal P^E_{k\sigma'(k)}$ for $k\in I'$, \begin{equation}\label{eq:slab-insertion} \int\prod_{k\in I'}h_{ik}(\omega,\omega_k)\,K^E_{i\sigma'(i)}(\dd\omega)\ge(1-\eta_{\mathrm{glob}})\,K^E_{i\sigma'(i)}(1),\qquad \eta_{\mathrm{glob}}=C_{\rm int}mMC_tD^{-1/4}. \end{equation} \end{lemma} The opponents $\omega_k$ in \eqref{eq:slab-one-opponent} and \eqref{eq:slab-insertion} are only required to be permitted for their own pairings. They need not be accessible, they need not belong to the group of $i$, they need not avoid one another, and the frozen paths remain imposed through the kernels. Deleting rows from $I'$ or checks from the product only increases the left side. By \cref{lem:slab-mass}, every crossed pairing $(i,\sigma'(i))$ of (H2) has mass at least $z_-$. Hence \eqref{eq:slab-insertion} is hypothesis (H3) with collision error $\eta_{\mathrm{glob}}$ whenever $\eta_{\mathrm{glob}}\le1/2$, and \eqref{eq:slab-one-opponent} is hypothesis (H3$_1$) with $\eta_1$, since distinct rows have distinct slots under a bijection. \begin{proof} If $d_\infty(\operatorname{anch}(i),\operatorname{anch}(k))>R_{\rm int}$, then $h_{ik}\equiv1$ and there is nothing to prove. Otherwise all paths involved stay within distance $400$ of $\tilde v$, $v=\operatorname{anch}(i)$, by \cref{lem:slab-range}, and we pass to the Euclidean bridge restriction in this chart. We write $y=Y_i$ and $z=Z_j$ in these Euclidean coordinates. Since $K^E_{ij}\le\mathbf 1_{\mathcal C^-_y\cap\mathcal C^+_z}\mathbb W^{2t,\mathrm E}_{y,z}$, the restrictions to $\mathrm{Av}_F$ only decrease the left side of \eqref{eq:slab-one-opponent}. It suffices to prove \begin{equation}\label{eq:slab-collision-mass} \mathbb W^{2t,\mathrm E}_{y,z}\bigl(\mathcal C^-_y\cap\mathcal C^+_z\cap\mathcal H(\omega')\bigr)\le C\alpha_0t^{-3/2}D^{-1/4} \end{equation} with a numerical constant $C$, where $\mathcal H(\omega')$ is the collision event of \cref{sec:thermal-bridges}. Then \eqref{eq:slab-one-opponent} follows with $C_t=C\alpha_0t^{-3/2}z_-^{-1}$, after division by $K^E_{ij}(1)\ge z_-$. We first collect the properties of the opponent. Since $\omega'\in\mathcal C^-_{Y_k}\cap\mathcal C^+_{Z_{j'}}$, \cref{lem:slab-eligible}(b) implies that $\osc_m(\omega')\le D^{1/12}$ for all $m$, and the diameters of $\omega'$ on the two windows of length $h_0$ are at most $s_0/8$. Its initial point is $Y_k$, and $d(y,Y_k)>s_0$ because the retained row $i$ is eligible. Its terminal point is $Z_{j'}$, and $d(z,Z_{j'})>s_0$ because $j$ is the slot of an eligible retained row in the original sample. Note that $j'\ne j$ is used here: the trial and the opponent have distinct terminal slots. \emph{The endpoint windows.} Let $\omega\in\mathcal C^-_y$ and $s\in[0,h_0]$. Then \[ d(\omega(s),\omega'(s))\ge d(y,Y_k)-d(\omega(s),y)-d(\omega'(s),Y_k)>s_0-\frac{s_0}8-\frac{s_0}8=\frac{3s_0}4>r, \] where in the last inequality we used \eqref{eq:thermal-scales}. The same holds on $[2t-h_0,2t]$. It follows that on $\mathcal C^-_y\cap\mathcal C^+_z\cap\mathcal H(\omega')$ a collision occurs during a mesh interval $[(m-1)\tau,m\tau]$ that is not contained in $[0,h_0]\cup[2t-h_0,2t]$. \emph{One mesh interval in the first half.} Let $m\le m_t$ with $m\tau>h_0$, and set $s=(m-1)\tau$; then $s>h_0-\tau\ge h_0/2$, since $\tau\le h_0/2$. By the concatenation identity, under $\mathbb W^{2t,\mathrm E}_{y,z}$ the position $X_s=\omega(s)$ has the unnormalized density $p_s(x-y)p_{2t-s}(z-x)$, and conditionally on $X_s=x$ the path $(\omega(s+s'))_{0\le s'\le2t-s}$ is a Brownian bridge from $x$ to $z$ of duration $t'=2t-s\ge t$. We write it as \[ \omega(s+s')=x+\frac{s'}{t'}(z-x)+b(s'),\qquad 0\le s'\le t', \] where $b$ is a Brownian bridge from $0$ to $0$ of duration $t'$, whose law does not depend on $x$. Let $\Upsilon=r^{-1}\sup_{0\le s'\le\tau}\abs{b(s')}$. Writing $b(s')=\mathfrak B(s')-(s'/t')\mathfrak B(t')$ for a Brownian motion $\mathfrak B$ of generator $\Delta$, Doob's maximal inequality and Gaussian moments give \[ \E\sup_{s'\le\tau}\abs{\mathfrak B(s')}^3\le\Bigl(\frac32\Bigr)^3\E\abs{\mathfrak B(\tau)}^3\le C\tau^{3/2},\qquad \E\Bigl(\frac\tau{t'}\abs{\mathfrak B(t')}\Bigr)^3\le C\tau^3t'^{-3/2}\le C\tau^{3/2}, \] where we used $\tau\le t'$ in the last step. Since $\tau\le r^2$, we obtain $\E\Upsilon^3\le C_\Upsilon$ with a numerical constant $C_\Upsilon$. On $\mathcal C^-_y\cap\mathcal C^+_z$, \cref{lem:slab-eligible}(b) implies that $\abs{z-X_s}\le60$, and the drift term satisfies $s'\abs{z-X_s}/t'\le60\tau/t\le r$ for $s'\le\tau$, by \eqref{eq:thermal-scales} with $C=60$. If, moreover, $d(\omega(s+s'),\omega'(s+s'))\le r$ for some $s'\in[0,\tau]$, then \[ \begin{aligned} \abs{X_s-\omega'(s)}&\le\abs{X_s-\omega(s+s')}+\abs{\omega(s+s')-\omega'(s+s')}+\abs{\omega'(s+s')-\omega'(s)}\\ &\le(r+r\Upsilon)+r+r\osc_m(\omega')=r\bigl(2+\osc_m(\omega')+\Upsilon\bigr). \end{aligned} \] This is a necessary condition for a collision during the $m$th interval on $\mathcal C^-_y\cap\mathcal C^+_z$. We now discard the cutoffs. Since $\Upsilon$ is independent of $X_s$, $p_s\le(4\pi s)^{-3/2}$, and $p_{2t-s}\le(4\pi t)^{-3/2}$, the unnormalized mass of this necessary condition is at most \[ \frac{4\pi}3\,\frac{r^3}{(4\pi s)^{3/2}(4\pi t)^{3/2}}\,\E\bigl(2+\osc_m(\omega')+\Upsilon\bigr)^3\le C\,\frac{r^3D^{1/4}}{t^{3/2}s^{3/2}}. \] In the last inequality, we used $(x+y)^3\le4x^3+4y^3$, $\osc_m(\omega')\le D^{1/12}$, and $\E\Upsilon^3\le C_\Upsilon$. \emph{Summing over the intervals.} The starting times $s=(m-1)\tau\ge h_0/2$ are multiples of $\tau$. With $k_0=\lceil h_0/(2\tau)\rceil\ge1$, we have \[ \sum_{k\ge k_0}(k\tau)^{-3/2}\le\tau^{-3/2}\bigl(k_0^{-3/2}+2k_0^{-1/2}\bigr)\le3\tau^{-3/2}k_0^{-1/2}\le\frac{3\sqrt2}{\tau\sqrt{h_0}}\le\frac{6\sqrt2}{r^2\sqrt{h_0}}, \] where we used $\tau\ge r^2/2$. The collisions during the first half contribute at most $Ct^{-3/2}rD^{1/4}h_0^{-1/2}=Ct^{-3/2}\alpha D^{-1/4}\le2C\alpha_0t^{-3/2}D^{-1/4}$, since $r=\alpha/D$ and $h_0=D^{-1}$. \emph{The second half.} The time reversal $\omega\mapsto\omega(2t-\cdot)$ maps $\mathbb W^{2t,\mathrm E}_{y,z}$ to $\mathbb W^{2t,\mathrm E}_{z,y}$ (\cref{sec:id-fk}). It maps $\mathcal C^+_z$ to $\mathcal C^-_z$ and $\mathcal C^-_y$ to $\mathcal C^+_y$, because the two halves of \cref{def:slab-half} are mirror images, and it maps the mesh intervals of $[t,2t]$ to those of $[0,t]$. The reversed opponent has the same oscillation and window bounds, and its endpoints are separated from $z$ and $y$ by more than $s_0$. Collisions during the second half become collisions during the first half of the reversed paths, and by the preceding argument, we obtain the same bound. This proves \eqref{eq:slab-collision-mass}. For \eqref{eq:slab-insertion}, we have $1-\prod_{k\in I'}h_{ik}\le\sum_{k\in I'}(1-h_{ik})$. Only rows $k$ with $d_\infty(\operatorname{anch}(i),\operatorname{anch}(k))\le R_{\rm int}$ contribute, and there are fewer than $C_{\rm int}mM$ of them by \cref{lem:slab-range}. For each of them $\sigma'(k)\ne\sigma'(i)$, since $\sigma'$ is a bijection, and $\omega_k\in\mathcal P^E_{k\sigma'(k)}$. Applying \eqref{eq:slab-one-opponent} with $j=\sigma'(i)$ and $j'=\sigma'(k)$ to each term and summing yields \eqref{eq:slab-insertion}. \end{proof} The normalization of the integrated bridge estimates is as in \cref{rem:thermal-midpoint}, with divisor $K^E_{ij}(1)\ge z_-$. We adapt the proof of \cref{lem:thermal-single} for the whole bridge $P^0_{y,z}$, with three changes: permitted paths are confined to balls of radius $20$ about their two endpoints instead of a ball of radius $10$ about the home cell, and the drift bound uses the distance $60$; the opponent is any permitted path of another retained row with a different terminal slot; and the normalization is by the mass bound of \cref{lem:slab-mass} instead of an accessibility probability as in \cite[Proposition~7.4]{OpenAI2026BEC}. Only the maximum test on the opponent's oscillations is used, which is why the bound is $C_tD^{-1/4}$ rather than $C(tr+rD^{1/4}h_0^{-1/2})$. \usedin{\cref{cor:slab-system}, where it gives (H3) and (H3$_1$); the bound $(1-\eta_{\mathrm{glob}})z_-\ge z_-/2$ on the one-row normalizing masses after imposing the fixed opponents is also used in \cref{sec:proof-ncri}.} \section{Vulnerable cells}\label{sec:slab-vulnerable} We use the following form of the multiplicative Chernoff bound for sampling without replacement. \begin{lemma}\label{lem:slab-hypergeometric} Let a set of $n$ elements contain at least $fn$ marked elements, where $00$, applying this with $\phi(x)=e^{-\lambda x}$ and Markov's inequality, we obtain \[ \PP(X0$, $t\in(0,1]$, and $D_0$ be such that the useful-label conclusion holds for these $H_1$, $t$, and $D\ge D_0$: on the torus by \cref{lem:thermal-useful}, and in a container by the conclusion derived from \cref{prop:walls-bridges} in the proof of \cref{thm:walls-main}(ii). Set $q_{\mathrm u}=9\ell/(20H_1)$, let $M$ be an integer with $q_{\mathrm u}M\ge16$ and $p_1+2e^{-q_{\mathrm u}M/8}\le1$, and let $m_*=\lfloor q_{\mathrm u}M/4\rfloor$. Assume \eqref{eq:slab-standing}, $D\ge D_0$, $m\ge1$, $mM\le\ell D/2$, $\eta_{\mathrm{glob}}\le1/2$, $\alpha\in[\alpha_0/2,2\alpha_0]$, $\beta>\delta$, in a container $K\ge K_\Omega$, and that $\PP$ satisfies the joint bound \eqref{eq:thermal-joint} with base $p_1$ (for sets of interior cells in a container). Then $m_*\ge4$, $m_*+2\delta$, let $m\ge1$ be an integer with $mM\le\ell D/2$ and $\eta_{\mathrm{glob}}=C_{\rm int}mMC_tD^{-1/4}\le1/2$, let $K\ge2000$ on the torus and $K\ge K_\Omega$ in a container, and suppose that $\PP$ satisfies the joint bound \eqref{eq:thermal-joint} with base $p_1$ (for sets of interior cells in a container). Then the law of $(E,\sigma_I,\omega_I)$ under $\PP$ is a matching system in the sense of \cref{def:surgery-system}, with $m$ groups of at most $M$ rows per cell, active region $\Lambda_{\mathrm{act}}$, accessibility $A_{ij}$, and threshold $m_*\ge4$, and it satisfies \begin{enumerate} \item[(H1)] with $R_{\rm loc}=61$, $R_{\rm int}=123$, and at most $C_{\rm int}mM=247^3mM$ rows with a nontrivial check with a given row; \item[(H2)] with $z_-=0.98\,(0.9)^2g_*^2$ and $z_+=(8\pi t)^{-3/2}$, where $g_*=(4\pi t)^{-3/2}e^{-100/t}$; \item[(H3)] with collision error $\eta_{\mathrm{glob}}$; \item[(H3$_1$)] with $\eta_1=C_tD^{-1/4}$; \item[(H4)] for every group, with $p=p_1+e^{-q_{\mathrm u}M/8}$; \item[(H4$^+$)] with $p_\times=p_1+2e^{-q_{\mathrm u}M/8}\le2p$. \end{enumerate} Moreover $M/m_*\le8/q_{\mathrm u}$. The constants $R_{\rm loc}$, $R_{\rm int}$, and $C_{\rm int}$ are numerical, and $z_\pm$, $q_{\mathrm u}$, and $C_t$ depend only on $H_1$, $t$, and the constants of \cref{sec:thermal-setting}. None of them depends on $M$, $m$, $D$, $T$, $K$, or the exposed data. \end{corollary} \begin{proof} Let $D_1\ge\max\{D_0,2M/\ell\}$ be large enough that \eqref{eq:slab-standing} holds for $D\ge D_1$, apart from the condition on $K$. By \cref{prop:slab-law}, we have a matching system with the groups $I^{(1)},\dots,I^{(m)}$. Hypothesis (H1) follows from \cref{lem:slab-range}, and \cref{lem:slab-mass} implies (H2). By \cref{lem:slab-insertion}, we have (H3), since $\eta_{\mathrm{glob}}\le1/2$, and (H3$_1$): for a matching $\tau$, a crossed pairing $(i,\tau(i))$, and a row $k\ne i$ of any group, $K^E_{i\tau(i)}(1)\ge z_-$ by \cref{lem:slab-mass}, and $\tau(k)\ne\tau(i)$, so that \eqref{eq:slab-one-opponent} applies with $j=\tau(i)$, $j'=\tau(k)$, and every $\omega'\in\mathcal P^E_{k\tau(k)}$. Finally, \cref{prop:slab-vulnerable} implies (H4), (H4$^+$), and the bound on $M/m_*$. \end{proof} The full order of parameter choices is given in \cref{sec:stiff-parameters}, with the permission to reduce $t$ stated in \cref{rem:thermal-order}. Hypothesis (H5), including the containment of accepted routes in $\Lambda_{\mathrm{act}}$ in a container, is verified in \cref{sec:proof-stiffness,sec:proof-ncri}. \usedin{\cref{sec:proof-stiffness,sec:proof-ncri}, together with \cref{thm:surgery}.} \chapter{Proof of the theorem on the superfluid fraction}\label{sec:proof-stiffness} We apply \cref{thm:surgery} to the slab matching system, compute the exact winding gain, and use \cref{prop:var-one-slab} to obtain the scaled lower bound. After fixing the parameters, we return to physical units and prove \cref{thm:main-stiffness}; the final subsection proves \cref{cor:main-winding-scaling}. \section{The scaled setting}\label{sec:stiff-setting} In \cref{sec:stiff-setting,sec:stiff-gain,sec:stiff-parameters,sec:stiff-scaled-proof}, we work in the scaled units of \cref{sec:thermal-setting}. Here $K\ge100$ is an integer, $\Lambda_K=(\R/K\Z)^3$, $\Lat_K=(\Z/K\Z)^3$ is the cell lattice with cells $B_v=v+[-1/2,1/2)^3$, and $N$, $D$, $r$, $\alpha$, and $T$ satisfy \eqref{eq:thermal-window}, with the temperature-window condition in \eqref{eq:thermal-window} replaced by $\beta>\delta$. We apply the results of \cref{sec:identities} to $N$ particles on $\Lambda_K$ with exclusion distance $r$; the side $L$ and the exclusion distance $a$ of \cref{sec:identities} are then $K$ and $r$. We write $\mu$ for the Bose path measure, $W$ for its winding vector, and $\fs(N,K,T)$ for the superfluid fraction \eqref{eq:id-fs}. Since $N=DK^3$, the identity \eqref{eq:id-pc} reads \begin{equation}\label{eq:stiff-pc} \fs(N,K,T)=\frac{TK^2}{2N}\Var_\mu(W_1)=\frac{T}{2DK}\Var_\mu(W_1). \end{equation} \paragraph{Slabs and marks.} Let $t\in(0,1]$ and the group size $M$ be fixed as in \cref{sec:stiff-parameters}, let $m=m(D)$ be the number of groups in \eqref{eq:stiff-groups}, and set $\delta=2t$ and $n_\beta=\lfloor\beta/\delta\rfloor\ge1$. We use the slab decomposition of \cref{prop:id-slabs} with slab length $\delta$, and we mark each full slab by the ordered lists of \cref{sec:slab-rows}, of $mM$ rows divided into $m$ groups. The lists of a slab are chosen using that slab and its two cuts only, and they define a marking kernel $\mathfrak m$ as in \cref{prop:var-one-slab}. We write $\mu^{\mathfrak m}_\delta$ for the resulting marked one-slab law. In the notation of \cref{sec:slab}, it is the joint law $\PP$ of the slab sample $(Y,Z,\sigma,\omega)$ and the lists, where $Y$ and $Z$ are the two cuts of the slab and $\omega_i$ runs from $Y_i$ to $Z_{\sigma(i)}$. By \cref{prop:slab-law}, the marked one-slab variable and the triple $(E,\sigma_I,\omega_I)$ of exposed data, retained matching, and retained paths are measurable functions of each other, and we identify the two descriptions. As in \cref{sec:slab}, we write $\sigma$ for the retained matching $\sigma_I$ when no confusion can arise. For a unit vector $u$, the function $\varphi^W_u$ of \cref{sec:var-slabs} is, in the present notation, \begin{equation}\label{eq:stiff-phi} \varphi^W_u=\frac1K\sum_{k=1}^Nu\cdot d(\omega_k), \end{equation} where $d(\omega_k)$ is the lifted displacement of the path $\omega_k$ (\cref{sec:id-fk}). By \cref{prop:id-slabs}(4) and (5), it is the contribution of any full slab to $u\cdot W$, and it is square integrable under $\mu^{\mathfrak m}_\delta$. \paragraph{The surgery kernel.} Let $\Gamma$ be a winding skeleton of \cref{def:routes-loop}, sampled independently of the slab, and let $\mathcal R$ be the torus routing map of \cref{def:routes-router-torus}. We use them in the kernels of \cref{def:surgery-move}. For each group $a$, averaging the subprobability kernel of the group over the random skeleton gives a subprobability kernel $q^{(a)}$, whose outgoing mass $\bar q^{(a)}$ is the probability, over the random skeleton, that the routing map accepts the bad set $B^{(a)}_\sigma$. The surgery kernel $Q=m^{-1}\sum_aQ^{(a)}$, where $Q^{(a)}$ is the completion of $q^{(a)}$ (\cref{def:var-completion}), is the completion of $q=m^{-1}\sum_aq^{(a)}$, and it preserves the exposed data $E$, including the two cuts. For an accepted labeled route of the group $a$, the selected rows $i_k$ belong to $I^{(a)}$, $\pi(i_k)=i_{k+1}$, and the matching changes from $\sigma$ to $\sigma\pi$. We compute the resulting winding change below. The following proposition is the scaled form of \cref{thm:main-stiffness}. Its constants are fixed in \cref{sec:stiff-parameters}. \begin{proposition}\label{prop:stiff-scaled} Let $\alpha_0$, $p_1$, $\delta$, $D_{\min}$, and $C_\sharp$ be the constants fixed in \cref{sec:stiff-parameters}, and let $K_0$ be the threshold defined there, which depends only on the constants fixed there. Let $K\ge K_0$ and $N\ge1$ with $D=N/K^3\ge D_{\min}$, let $\alpha\in[\alpha_0/2,2\alpha_0]$ and $r=\alpha/D$, and let $\beta>\delta$. Suppose that the one-slab law of $\mu$ satisfies the joint bound \eqref{eq:thermal-joint} with base $p_1$. Then \begin{equation}\label{eq:stiff-scaled} \Var_\mu(W_1)\ge\frac{n_\beta KD^{1/4}}{4C_\sharp},\qquad \fs(N,K,T)\ge\frac{Tn_\beta}{8C_\sharp D^{3/4}}\ge\frac1{16C_\sharp\delta D^{3/4}}. \end{equation} \end{proposition} By \cref{prop:thermal-certificates}, the joint bound holds whenever $K\ge K_1$, $D\ge D_{\min}$, $\alpha\in[\alpha_0/2,2\alpha_0]$, and $00$ and $C_h,C_s<\infty$ such that two independent winding skeletons $\Gamma$, $\Gamma'$ satisfy $\E e^{\kappa J_R(\Gamma,\Gamma')}\le1+C_h/K$ and $\E e^{\xi S_{R_c}(\Gamma,\Gamma')}\le1+C_s/K$ for all sufficiently large $K$. Let $C_{\rm acc}$ be the acceptance constant of \cref{prop:routes-router}, $C_{\rm ch}$ the constant in the encounter bound of \cref{lem:routes-stability}, and $\zeta(p,c)$ the coefficient of \cref{lem:routes-environment}. The polygon skeletons of \cref{sec:ncri-setting} have constants of the same kind, and we use the smaller exponential coefficients and the larger multiplicative constants of the two families. We fix $\eps_{\max}\in(0,\kappa/4]$, and then the coefficient $c_{\mathrm{rec}}>\log4+C_{\rm ch}\eps_{\max}+1$ of $A_*$ in the exponential of \cref{thm:surgery}. We also fix the cross-group exponent and coefficient of \cref{lem:surgery-cross}, \[ \theta_\times=\min\{\kappa/2,1/\max(1,C_{\rm ch})\},\qquad c_\times=\theta_\times C_{\rm ch}, \] and the cross-group constant $C_\times=8C_{\mathrm{fin}}(c_\times)/\theta_\times$ of \cref{thm:surgery}, where $C_{\mathrm{fin}}(c_\times)$ is the constant of (H5)(d), the larger of its values for the two skeleton families. These choices are made before the thermal constants, and the base for bad cells in Step 4 does not depend on $t$. \item \emph{Base for bad cells.} Let $p_{\max}\in(0,1)$ be such that, for every $p\in(0,p_{\max}]$, we have $p\le p_0$, with $p_0$ as in \cref{prop:routes-router}, $C_{\rm acc}p\le1/2$, the expansion of \cref{lem:routes-environment} applies with exponent $2c_{\mathrm{rec}}$, and $2\zeta(p,2c_{\mathrm{rec}})\le\xi$. Require moreover that $(2R_c+1)^6(2p_{\max})e^{c_\times}\le e^{-2}$ and $2\zeta(2p_{\max},c_\times)\le\xi$, so that \cref{cor:routes-final} applies with the coefficient $c_\times$ of $A_*$ for every base at most $2p_{\max}$; that is, we may take $p_{\mathrm{env}}(c_\times)=2p_{\max}$ in (H5)(d). Fix $p_1\in(0,p_{\max}/2)$. \item \emph{Thermal constants.} We apply \cref{prop:thermal-certificates}, \cref{lem:thermal-useful}, and \cref{prop:thermal-container} with base $p_1$ and $R_0=64$, and we fix a common $H_1$, then a common $t\in(0,1]$, as permitted by \cref{rem:thermal-order}, together with the density thresholds $D_0$. The larger of the two values of $H_1$ is allowed: increasing $H_1$ weakens the certificate conditions in \cref{def:thermal-certificate}, and then $t$ may be decreased to meet the useful-label requirements for this common value. Set $\delta=2t$, the useful fraction $q_{\mathrm u}=9\ell/(20H_1)$ of \cref{prop:slab-vulnerable}, the masses $z_\pm$ of \cref{lem:slab-mass}, $G=(2z_+/z_-)^3$, and the constant $C_t$ of \cref{lem:slab-insertion}. No later step changes $t$. \item \emph{Group size $M$.} Set $\lambda_0=8/q_{\mathrm u}$, and let $M$ be an integer such that $q_{\mathrm u}M\ge16$, $e^{-q_{\mathrm u}M/8}2$ of \eqref{eq:slab-standing}, and at least $(2C_{\rm int}MC_t)^4$. Let $D_{\min}$ moreover be so large that \[ 32C_{\rm int}C_tD_{\min}^{-1/4}\le\min\{\eps_{\max}/2,\theta_\times\},\qquad D_{\min}^{3/4}\ge\frac1{\ell C_{\rm int}C_t},\qquad \frac{\tau_0}{\log D_{\min}}\le\frac14. \] For $D\ge D_{\min}$, the number of groups is \begin{equation}\label{eq:stiff-groups} m(D)=\Bigl\lfloor\frac{D^{1/4}}{2C_{\rm int}MC_t}\Bigr\rfloor. \end{equation} Since $D\ge(2C_{\rm int}MC_t)^4$, the argument of the floor is at least one, and $\lfloor x\rfloor\ge x/2$ for $x\ge1$ gives \[ \frac{D^{1/4}}{4C_{\rm int}MC_t}\le m(D)\le\frac{D^{1/4}}{2C_{\rm int}MC_t}. \] Hence $m(D)\ge1$, the global collision error $\eta_{\mathrm{glob}}=C_{\rm int}m(D)MC_tD^{-1/4}$ of \cref{lem:slab-insertion} satisfies $\eta_{\mathrm{glob}}\le1/2$, and $m(D)M\le D^{1/4}/(2C_{\rm int}C_t)\le\ell D/2$. With $\eta_1=C_tD^{-1/4}$ and $\eta_{\mathrm{loc}}=4C_{\rm int}\eta_1$, we have $8\eta_{\mathrm{loc}}\le\min\{\eps_{\max}/2,\theta_\times\}$, and therefore $\eps=(3+8G\lambda_0^2)/m_*+8\eta_{\mathrm{loc}}<\eps_{\max}$. Since $G$ was defined using only $\eta_{\mathrm{glob}}\le1/2$, increasing $D_{\min}$ changes neither $G$ nor $M$. \item \emph{Comparison constants.} With the constants of Steps 1--7, \cref{thm:surgery} provides the constant $C_*$ in its bound on the chi-square divergence; for the polygon skeletons of \cref{sec:ncri-setting} it provides a constant of the same kind, and we let $C_*$ be the larger of the two. Set \[ C_\sharp=4C_{\rm int}C_t\bigl(MC_*+C_\times\bigr). \] The constants $C_*$, $C_\times$, and $C_\sharp$ depend only on the constants of Steps 1--6, and not on $K$, on $D\ge D_{\min}$, on $m$, on $\alpha$, or on $T$. Finally, we increase $D_{\min}$, if necessary, so that $8C_\sharp\delta D_{\min}^{3/4}\ge1$; since $C_\sharp$ does not depend on $D_{\min}$, this changes no constant fixed before, and the requirements of Step 7 persist. The constants of the area estimate are fixed at this step as well (\cref{sec:ncri-parameters}). \item \emph{Physical parameters, then the volume.} For \cref{thm:main-stiffness}, fix $a$ and $0<\rho_1\le\rho_2$ with $\rho_2a^3\le2c_0$, where $c_0=\alpha_0^3/(16D_{\min}^2)$. Only after that do we let $N$ and $L$, and with them $K$, tend to infinity. \end{enumerate} \paragraph{Volume thresholds.} Let $K_0$ be the smallest integer $K_0\ge2000$ such that, for all $K\ge K_0$, the excess bounds of Step 3 for the winding skeletons, the acceptance bound of \cref{prop:routes-router} for the torus routing map, and the volume condition of \cref{thm:surgery} hold. These are lattice conditions, which depend only on the constants of Steps 1--8. No condition involves $D$, $\alpha$, $T$, or the number of groups, and none relates $\beta$ to $K$. \section{Proof of the scaled bound}\label{sec:stiff-scaled-proof} \begin{proof}[Proof of \cref{prop:stiff-scaled}] Let $K$, $N$, $\alpha$, and $\beta$ be as in the statement, and let $m=m(D)$ be the number of groups \eqref{eq:stiff-groups}. By assumption, $\beta>\delta$. We first verify the hypotheses of \cref{thm:surgery} for one slab. By Step 7, we have $m\ge1$, $mM\le\ell D/2$, and $\eta_{\mathrm{glob}}\le1/2$, and $D\ge D_{\min}$ is at least the threshold $D_1$ of \cref{cor:slab-system}; moreover $K\ge K_0\ge2000$. Since the one-slab law satisfies the joint bound \eqref{eq:thermal-joint} with base $p_1$, \cref{cor:slab-system} shows that the marked one-slab law $\mu^{\mathfrak m}_\delta$ is the law of a matching system with $m$ groups of at most $M$ rows per cell that satisfies (H1) with the radii of Step 2, (H2) with the masses $z_\pm$ of Step 5, (H3) with the collision error $\eta_{\mathrm{glob}}\le1/2$, (H3$_1$) with $\eta_1=C_tD^{-1/4}$, (H4) with the base $p0$, we write $\fs(N,L,T;a)$ for the superfluid fraction of \cref{def:id-superfluid-fraction}. \begin{lemma}\label{lem:stiff-dilation} Let $N\ge1$ and $L,a,s>0$, and set $K=L/s$. The configuration domains of $N$ particles on $\Lambda_L$ with exclusion distance $a$ and on $\Lambda_K$ with exclusion distance $a/s$ are either both empty or both nonempty. In the second case, $F_{N,L,T}(\theta)=s^{-2}F_{N,K,s^2T}(\theta)$ for every $\theta\in\R$, where the free energies are computed with the exclusion distances $a$ and $a/s$, respectively, and \begin{equation}\label{eq:stiff-dilation} \fs(N,L,T;a)=\fs(N,K,s^2T;a/s). \end{equation} \end{lemma} \begin{proof} The map $X\mapsto X/s$ is a bijection from $\Lambda_L^N$ onto $\Lambda_K^N$, and $d_L(x,y)=s\,d_K(x/s,y/s)$. It maps the configuration domain with exclusion distance $a$ onto the one with exclusion distance $a/s$, which proves the first claim. Define $(U\psi)(X)=s^{3N/2}\psi(sX)$ for $X\in\Lambda_K^N$. Then $U$ is unitary from $L^2(\Lambda_L^N)$ onto $L^2(\Lambda_K^N)$, it preserves symmetry, and it maps smooth functions with compact support in the first domain onto those in the second. It also maps $\mathcal Q_N$ onto $\mathcal Q_N$. For smooth $\psi$, note that $\nabla_i(U\psi)(X)=s^{3N/2}s(\nabla_i\psi)(sX)$ and $\theta e_1/K=s\,\theta e_1/L$. Then \[ \sum_{i=1}^N\int\abs{(-\mathrm i\nabla_i+\theta e_1/K)U\psi}^2\dd X=s^2\sum_{i=1}^N\int\abs{(-\mathrm i\nabla_i+\theta e_1/L)\psi}^2\dd X, \] where we changed variables $X\mapsto sX$ on the right, with Jacobian $s^{3N}$. By density, this identity holds on $\mathcal Q_N$. The forms \eqref{eq:id-twisted-form} then satisfy $q^{(L,a)}_\theta=s^{-2}q^{(K,a/s)}_\theta\circ U$, which implies that $H_{N,L}(\theta)=s^{-2}U^{-1}H_{N,K}(\theta)U$, with the indicated exclusion distances. It follows that $\tr e^{-H_{N,L}(\theta)/T}=\tr e^{-H_{N,K}(\theta)/(s^2T)}$, and taking logarithms, we obtain the relation between the free energies. Differentiating twice at $\theta=0$ and using \eqref{eq:id-fs} yields \[ \fs(N,L,T;a)=\frac{L^2}{2N}\,s^{-2}\,\partial_\theta^2F_{N,K,s^2T}(0)=\frac{K^2}{2N}\,\partial_\theta^2F_{N,K,s^2T}(0)=\fs(N,K,s^2T;a/s). \] This completes the proof. \end{proof} \usedin{the proof of \cref{thm:main-stiffness}.} Equivalently, the winding vector is invariant under the dilation of paths $x(\tau)\mapsto s^{-1}x(s^2\tau)$, and $TL^2=(s^2T)K^2$. \begin{proof}[Proof of \cref{thm:main-stiffness}] Let $\alpha_0$, $p_1$, $\tau_0$, $D_{\min}$, $\delta$, $C_\sharp$, and $K_0$ be the constants fixed in \cref{sec:stiff-parameters}, and let $K_1$ be the threshold of \cref{prop:thermal-certificates}; they are absolute. We prove the theorem with the constants $c_0=\alpha_0^3/(16D_{\min}^2)$ and $c_1=1/(16C_\sharp\delta\alpha_0^{9/8})$. For $a,\rho>0$, put \[ s_\rho=\Bigl(\frac{\alpha_0}{a\rho}\Bigr)^{1/2},\qquad D_\rho=\rho s_\rho^3=\frac{\alpha_0^{3/2}}{(\rho a^3)^{1/2}}. \] By \eqref{eq:intro-Tstar}, we have $T_*(a,\rho)=\tau_0/(s_\rho^2\log D_\rho)$. We first prove the finite-volume form stated in \cref{rem:intro-uniform}. Let $a>0$ and $0<\rho_1\le\rho_2$ with $\rho_2a^3\le2c_0$. We claim that there exists $L_0$ such that $\fs(N,L,T)\ge c_1(\rho_La^3)^{3/8}$, where $\rho_L=N/L^3$, for all $L\ge L_0$, all $N$ with $\rho_L\in[\rho_1,\rho_2]$, and all $01$, both factors in $T_K\log D$ are nonnegative, and \[ T_K\log D\le s_{\rho_L}^2T\log D_{\rho_L}=\tau_0\frac T{T_*(a,\rho_L)}\le\tau_0. \] \item $\beta_K=1/T_K\ge\log D_{\min}/\tau_0\ge4>\delta$, since $\tau_0/\log D_{\min}\le1/4$ and $\delta=2t\le2$. \end{itemize} By \cref{prop:thermal-certificates}, in the form \eqref{eq:thermal-joint}, the one-slab law of $N$ particles on $\Lambda_K$ with exclusion distance $r$ and temperature $T_K$ satisfies the joint bound with base $p_1$; the threshold $K_1$ depends on neither $D$ nor $T_K$ (\cref{rem:thermal-volume}). By \cref{lem:stiff-dilation} and \cref{prop:stiff-scaled}, we have \[ \fs(N,L,T;a)=\fs(N,K,T_K;r)\ge\frac{D^{-3/4}}{16C_\sharp\delta}\ge\frac{D_{\rho_L}^{-3/4}}{16C_\sharp\delta}=\frac{(\rho_La^3)^{3/8}}{16C_\sharp\delta\alpha_0^{9/8}}=c_1(\rho_La^3)^{3/8}, \] which proves the claim. Now let $a,\rho>0$ satisfy $\rho a^3\le c_0$, let $00,\qquad \beta=T^{-1}>\delta. \end{equation} Here $\delta=2t$, with $t$ fixed in \cref{sec:stiff-parameters}, and $D$ is the number of particles per unit volume in scaled coordinates. It is not $N$ divided by the number of cells that meet $U$, which can be much larger than $\abs U$ when $\Omega$ is irregular. With this convention, $D$ is the scaled counterpart of the finite-volume density $\rho_L=N/(L^3\abs\Omega)$, which converges to the density $\rho$ in \cref{thm:main-ncri}. We apply the results of \cref{sec:identities} to $N$ particles in the container $U$ with exclusion distance $r$, and we write $\mu$ for the Bose path measure and $\mathcal A$ for the stochastic area vector \eqref{eq:id-area}. By \cref{prop:id-rotation}(3) and (4), for every unit vector $n$ and every $x_0\in\R^3$, \begin{equation}\label{eq:ncri-identity} \Icl(n,Kx_0)-I(n,Kx_0)=T\Var_\mu(\mathcal A_n),\qquad 0\le I(n,Kx_0)\le\Icl(n,Kx_0), \end{equation} where the axis passes through $Kx_0$ with direction $n$; the matrix form is $\Icl(Kx_0)-I(Kx_0)=T\Cov_\mu(\mathcal A)$. No symmetry of $\Omega$ is used in \eqref{eq:ncri-identity}. \paragraph{An inscribed cube.} Since $\Omega$ is open and nonempty, we may fix $x_*\in\Omega$ and $R_{\mathrm{in}}>0$ such that the closed ball $\overline B(x_*,2R_{\mathrm{in}})$ is contained in $\Omega$. For $K$ with $R_{\mathrm{in}}K\ge20$, set \[ H=\Bigl\lfloor\frac{2R_{\mathrm{in}}K}{\sqrt3}\Bigr\rfloor-2, \] let $v_{\mathfrak Q}\in\Z^3$ be a lattice point nearest to $Kx_*-(H-1)(1,1,1)/2$, and let $\mathfrak Q_K$ be the union of the cells $B_v$ with $v\in v_{\mathfrak Q}+\{0,\dots,H-1\}^3$. The set $\mathfrak Q_K=v_{\mathfrak Q}+[-1/2,H-1/2)^3$ is a cube of side $H$ made of $H^3$ cells. \begin{lemma}\label{lem:ncri-cube} If $R_{\mathrm{in}}K\ge20$, then $R_{\mathrm{in}}K\le H\le2R_{\mathrm{in}}K$, the cube $\mathfrak Q_K$ is contained in the ball $B(Kx_*,R_{\mathrm{in}}K)$, and every point of $\mathfrak Q_K$ has distance at least $R_{\mathrm{in}}K$ from $\R^3\setminus U$. \end{lemma} \begin{proof} The upper bound on $H$ is immediate. Since $2/\sqrt3-1>0.15$ and $R_{\mathrm{in}}K\ge20$, we have $H\ge2R_{\mathrm{in}}K/\sqrt3-3\ge R_{\mathrm{in}}K+0.15R_{\mathrm{in}}K-3\ge R_{\mathrm{in}}K$. The center $c=v_{\mathfrak Q}+(H-1)(1,1,1)/2$ of $\mathfrak Q_K$ satisfies $\abs{c-Kx_*}\le\sqrt3/2$ by the choice of $v_{\mathfrak Q}$, and every point of $\mathfrak Q_K$ is within distance $\sqrt3H/2$ of $c$. Then every point of $\mathfrak Q_K$ is within distance $\sqrt3(H+1)/2\le R_{\mathrm{in}}K-\sqrt3/2$ of $Kx_*$, where we used $H+1\le2R_{\mathrm{in}}K/\sqrt3-1$. The closed ball $\overline B(Kx_*,2R_{\mathrm{in}}K)=K\overline B(x_*,2R_{\mathrm{in}})$ is contained in $U$, which gives the last claim. \end{proof} \usedin{\cref{lem:ncri-containment} and the proof of \cref{prop:ncri-scaled}.} \paragraph{Active cells and the matching system.} We use the matching system of \cref{sec:slab} for the gas in $U$. By \eqref{eq:slab-active}, the active cells are \[ \Lambda_{\mathrm{act}}=\{v\in\Z^3:v\in U,\ \dist(v,\R^3\setminus U)\ge64\}, \] a finite set, since $U$ is bounded. We recall the conventions of \cref{sec:slab} that matter in a container. Rows are retained only at active cells. All other slab paths, in particular all paths that start near the walls, are frozen and exposed, and the wall condition is imposed on them. The bad sets $B^{(a)}_\sigma$ and the vulnerable sets $V^{(a)}_\sigma$ of the groups are subsets of $\Lambda_{\mathrm{act}}$; a prescribed set of cells that is not contained in $\Lambda_{\mathrm{act}}$ is never contained in $V^{(a)}_\sigma\cup V^{(b)}_\sigma$ for groups $a,b$, and the inactive cells do not affect (H4) and (H4$^+$). The exposed data name the terminal positions as slots, and they do not contain the assignment of the retained rows to the retained slots (\cref{prop:slab-law}); exposing the terminal position of each retained row would fix the matching that the surgery changes. Restricting the retained rows to the deterministic set $\Lambda_{\mathrm{act}}$ does not affect the proof of \cref{prop:slab-law}, since the absence of rows at inactive cells is not a random event. By \cref{lem:slab-range}, every permitted path of a retained row stays in $U$, within distance $61$ of the center of its anchor; the walls do not restrict retained rows in any pairing, and the kernels $K^E_{ij}$ of \eqref{eq:slab-kernel} are restrictions of Euclidean bridge measures $\mathbb W^{2t}_{Y_i,Z_j}$. By \cref{cor:slab-system}, for $K\ge K_\Omega$ and $D\ge D_{\min}$, and whenever the one-slab law of the gas in $U$ satisfies the joint bound \eqref{eq:thermal-joint} with base $p_1$ for sets of interior cells, the marked one-slab law $\mu^{\mathfrak m}_\delta$ is the law of a grouped matching system with active region $\Lambda_{\mathrm{act}}$ and the number of groups of \eqref{eq:stiff-groups}, which satisfies (H1)--(H4), (H3$_1$), and (H4$^+$), with the same constants as on the torus. Here $K_\Omega$ is the threshold of \cref{prop:thermal-container} with $R_0=64$, the interior cells are the active cells, and $\mu^{\mathfrak m}_\delta$ is defined as in \cref{sec:stiff-setting}. The cells outside $\Lambda_{\mathrm{act}}$ never belong to a bad set, but they carry no rows, and a surgery cannot use them. Declaring them bad instead would violate (H4). \Cref{lem:ncri-containment} below shows that accepted routes never reach them. \paragraph{Routes.} For points $y_0,\dots,y_{m-1}\in\R^3$, with indices modulo $m$, we write \begin{equation}\label{eq:ncri-polygon-area} \operatorname{ar}(y_0,\dots,y_{m-1})=\frac12\sum_{k=0}^{m-1}y_k\times y_{k+1} \end{equation} for the area vector of the closed polygon with vertices $y_0,\dots,y_{m-1}$. It is invariant under translations $y_k\mapsto y_k+c$, since $\sum_kc\times(y_{k+1}-y_k)=0$. The number $e_3\cdot\operatorname{ar}(y_0,\dots,y_{m-1})$ depends only on the projections of the $y_k$ onto the plane orthogonal to $e_3$; it is the signed area enclosed by the projected polygon, counted positively for counterclockwise orientation. For an oriented cycle of cells $\gamma=(v_0,\dots,v_{\abs\gamma-1})$, we write $\operatorname{ar}(\gamma)=\operatorname{ar}(v_0,\dots,v_{\abs\gamma-1})$, using the cell centers $v_k\in\Z^3$. Let $n_H=\lfloor H/64\rfloor$, the scale denoted $n$ in \cref{def:routes-polygon}. Let $\Gamma$ be a polygonal skeleton of that definition with $K_{\rm in}=H$ and inner cell cube $Q_{\rm in}=v_{\mathfrak Q}+\{0,\dots,H-1\}^3$, the cell-index set of $\mathfrak Q_K$. Let $\mathcal R$ be the area routing map of \cref{def:routes-router-box}. \begin{lemma}\label{lem:ncri-containment} Assume that $R_{\mathrm{in}}K\ge64$ and $H\ge2^{14}$. For every polygonal skeleton $\Gamma$ and every field $B\subset\mathbb Z^3$, every cell of an accepted output of the area routing map lies in $\mathcal N_{\lfloor\sqrt{n_H}\rfloor+1}(\Gamma)\subset Q_{\rm in}\subset\Lambda_{\mathrm{act}}$. \end{lemma} \begin{proof} By \cref{lem:ncri-cube}, every cell center in $Q_{\rm in}$ has distance at least $R_{\mathrm{in}}K\ge64$ from $\mathbb R^3\setminus U$. In particular, $Q_{\rm in}\subset\Lambda_{\mathrm{act}}$. The conclusion follows from \cref{prop:routes-router}(b), applied with $K_{\rm in}=H$. \end{proof} \usedin{the verification of (H5) below.} \paragraph{The surgery theorem in the container.} By \cref{prop:routes-nominal}, applied to the polygon law in a cube of side $H$, two independent polygonal skeletons satisfy the excess bounds of Step 3 in \cref{sec:stiff-parameters} with $K$ replaced by $H$, for all sufficiently large $H$. By \cref{prop:routes-router}, if $n_H\ge256$, then for a random set of bad cells that is independent of the random skeleton and satisfies the bound of (H4), the area routing map rejects with probability at most $C_{\rm acc}p$; and every accepted output of the routing map is an oriented nearest-neighbor cycle $\gamma$ of distinct cells outside the bad set, with $\abs\gamma\le124n_H$ and $e_3\cdot\operatorname{ar}(\gamma)\ge n_H^2/250$. By \cref{lem:routes-stability}, (H5)(c) holds, and by \cref{lem:ncri-containment} the routes lie in $\Lambda_{\mathrm{act}}$, as (H5)(a) requires. We conclude that (H5) holds, with the excess bounds stated in terms of $H$. Let $q$ be the subprobability kernel of \cref{def:surgery-move}, built from the polygonal skeletons and the area routing map and averaged over the random skeleton and over the groups, and let $Q$ be its completion. It acts on one slab as described in \cref{sec:stiff-setting}, with $\Lat_K$ replaced by $\Lambda_{\mathrm{act}}$. By \cref{thm:surgery}, with the constants of \cref{sec:stiff-parameters} and the excess bounds in terms of $H$, and by the bounds on the number of groups and on $\eta_{\rm loc}$ used for \eqref{eq:stiff-chi}, the measure $\mu^{\mathfrak m}_\delta Q$ is absolutely continuous with respect to $\mu^{\mathfrak m}_\delta$ with a square-integrable density, and \begin{equation}\label{eq:ncri-chi} \chi^2\bigl(\mu^{\mathfrak m}_\delta Q\,\big\|\,\mu^{\mathfrak m}_\delta\bigr)\le\frac{C_\sharp D^{-1/4}}H,\qquad \int\bar q\dd\mu^{\mathfrak m}_\delta\ge1-C_{\rm acc}p\ge\frac12, \end{equation} whenever $K$ exceeds the volume thresholds collected in \cref{sec:ncri-parameters} and the joint bound holds. \section{Rotating the container}\label{sec:ncri-rotation} The routes of \cref{sec:ncri-setting} produce area in the direction $e_3$. For another direction, we rotate the container. We write $\mathrm{SO}(3)$ for the group of rotations of $\R^3$, and $\mu_U$ for the Bose path measure of $N$ particles in a container $U$ with exclusion distance $r$. \begin{lemma}\label{lem:ncri-rotation} Let $\Theta\in\mathrm{SO}(3)$. The Bose path measure $\mu_{\Theta U}$ is the image of $\mu_U$ under the map $(\pi,X(\cdot))\mapsto(\pi,\Theta X(\cdot))$, where $\Theta X(\cdot)=(\Theta x_1(\cdot),\dots,\Theta x_N(\cdot))$, and the stochastic area satisfies $\mathcal A(\Theta X(\cdot))=\Theta\mathcal A(X(\cdot))$. In particular, $\Cov_{\mu_{\Theta U}}(\mathcal A)=\Theta\Cov_{\mu_U}(\mathcal A)\Theta^{\mathsf T}$, and $\Var_{\mu_U}(n\cdot\mathcal A)=\Var_{\mu_{\Theta U}}((\Theta n)\cdot\mathcal A)$ for every unit vector $n$. \end{lemma} \begin{proof} Since $\abs{\Theta x-\Theta y}=\abs{x-y}$, the map $X\mapsto\Theta X$ sends $\mathcal X_N(U)$ onto $\mathcal X_N(\Theta U)$, preserves Lebesgue measure, and commutes with permutations of the particles. The heat kernel $p_t$ is rotation invariant, and the image of the bridge measure $\mathbb W^t_{x,y}$ under $\omega\mapsto\Theta\omega$ is $\mathbb W^t_{\Theta x,\Theta y}$; this is the rotation invariance of Brownian motion with generator $\Delta$. A configuration path stays in $\mathcal X_N(U)$ if and only if its image stays in $\mathcal X_N(\Theta U)$. Inserting these facts into \eqref{eq:id-trace-formula} and \eqref{eq:id-path-measure}, we conclude that the two partition functions are equal and that $\mu_{\Theta U}$ is the image of $\mu_U$. For $\Theta\in\mathrm{SO}(3)$, recall that $\Theta a\times\Theta b=\Theta(a\times b)$ for all $a,b\in\R^3$. Each coordinate of the Stratonovich integral in \eqref{eq:id-area} is a bilinear expression in the coordinates of the integrand and of the integrator, and this identity carries over to the integrals. We obtain $\mathcal A(\Theta X(\cdot))=\Theta\mathcal A(X(\cdot))$, computed with the reference points $\Theta x_0$ and $x_0$; recall that $\mathcal A$ does not depend on the reference point. The remaining claims follow. \end{proof} \usedin{the proof of \cref{prop:ncri-scaled}.} If $\overline B(x_*,2R_{\mathrm{in}})\subset\Omega$, then $\Theta\Omega$ is a nonempty bounded open set with $\overline B(\Theta x_*,2R_{\mathrm{in}})\subset\Theta\Omega$ and $\abs{\Theta\Omega}=\abs\Omega$. The construction of \cref{sec:ncri-setting} applies to $\Theta\Omega$, with the point $\Theta x_*$, the same radius $R_{\mathrm{in}}$, and the same values of $H$, $N$, and $D$. Only the cell lattice $\mathbb Z^3$ is not rotated. The volume thresholds do not depend on $\Theta$: conditions (K1) and (K3) of \cref{sec:ncri-parameters} involve only $R_{\mathrm{in}}$ and $H$, and for the threshold of \cref{prop:thermal-container} we use $g_{\Theta\Omega}=g_\Omega\circ\Theta^{-1}$ in \eqref{eq:walls-g-omega}, so that $K_{\Theta\Omega}=K_\Omega$ by \cref{rem:walls-constants}(2). \section{Stochastic area of a Brownian bridge}\label{sec:ncri-bridge} For a path $\omega$ on $[0,\delta]$ and $y\in\R^3$, we write \[ \mathcal B_y(\omega)=\frac12\int_0^\delta e_3\cdot\bigl((\omega(s)-y)\times\circ\dd\omega(s)\bigr) \] for the stochastic area of $\omega$ about $y$ in the direction $e_3$, defined almost everywhere under the bridge measures, as in \eqref{eq:id-area}. If $\omega(0)=y$ and $x'\in\R^3$, then, since $\omega-x'=(\omega-y)+(y-x')$ and $\int_0^\delta\circ\dd\omega=\omega(\delta)-y$, \begin{equation}\label{eq:ncri-area-split} \frac12\int_0^\delta e_3\cdot\bigl((\omega(s)-x')\times\circ\dd\omega(s)\bigr)=\frac12e_3\cdot\bigl((y-x')\times(\omega(\delta)-y)\bigr)+\mathcal B_y(\omega). \end{equation} For the stochastic area of planar Brownian motion and its conditional characteristic function given the endpoint, see L\'evy \cite[Sections~1.2--1.3]{Levy1951} and Yor \cite[Equation~(2)]{Yor1980}. We need only the second moment of the area of a Brownian bridge. The proof below obtains it from the decomposition of Brownian motion into a bridge and its independent endpoint and from It\^o's isometry \cite[Theorems~2.1(I) and~2.2]{Ito1944}; see also \cite[Chapter~I, Section~3, and Theorem~IV.2.2]{RevuzYor1999}. \begin{lemma}\label{lem:ncri-bridge-area} Let $\delta>0$ and $y,d\in\R^3$, and let $\omega$ be a Brownian bridge with generator $\Delta$ from $y$ to $y+d$ of duration $\delta$, with law $\widehat{\mathbb W}^\delta_{y,y+d}=\mathbb W^\delta_{y,y+d}/p_\delta(d)$. Then \begin{equation}\label{eq:ncri-bridge-moments} \E\mathcal B_y(\omega)=0,\qquad \E\mathcal B_y(\omega)^2=\frac{\delta^2}3+\frac{\delta\abs{d_\perp}^2}6, \end{equation} where $d_\perp=(d_1,d_2,0)$. Further, for $R\ge0$ and $\abs d\le R$, \begin{equation}\label{eq:ncri-bridge-unnormalized} \int\abs{\mathcal B_y}\dd\mathbb W^\delta_{y,y+d}\le(4\pi\delta)^{-3/2}\Bigl(\frac{\delta^2}3+\frac{\delta R^2}6\Bigr)^{1/2}. \end{equation} \end{lemma} \begin{proof} We write $\omega(s)=y+sd/\delta+b(s)$, where $b$ is a Brownian bridge from $0$ to $0$ of duration $\delta$, with $\E b_a(s)b_c(s')=2\,\ind\{a=c\}(\min(s,s')-ss'/\delta)$ for the coordinates $a,c\in\{1,2,3\}$. Since the coordinates of $b$ are independent, the Stratonovich integrals of one coordinate against another coincide with the It\^o integrals. Since $d\times d=0$, we have \[ (\omega(s)-y)\times\circ\dd\omega(s)=\frac s\delta\,d\times\dd b(s)+\frac1\delta\,b(s)\times d\dd s+b(s)\times\circ\dd b(s). \] Integration by parts gives $\int_0^\delta s\dd b(s)=-\int_0^\delta b(s)\dd s$, because $b(\delta)=0$. With $J=\int_0^\delta b(s)\dd s$, we obtain \begin{equation}\label{eq:ncri-bridge-split} \mathcal B_y(\omega)=A(b)+\frac1\delta\,e_3\cdot(J\times d),\qquad A(b)=\frac12\int_0^\delta e_3\cdot\bigl(b(s)\times\circ\dd b(s)\bigr). \end{equation} The law of $b$ is invariant under $b\mapsto-b$, which leaves $A(b)$ unchanged and changes the sign of $J$. This implies that $\E J=0$ and that $A(b)$ and $e_3\cdot(J\times d)$ are uncorrelated. The law of $b$ is also invariant under the reflection of its first coordinate, which changes the sign of $A(b)$. Then $\E A(b)=0$. This proves the first identity in \eqref{eq:ncri-bridge-moments}. The coordinates of $J$ are independent centered Gaussian variables with variance \[ 2\int_0^\delta\int_0^\delta\Bigl(\min(s,s')-\frac{ss'}\delta\Bigr)\dd s\dd s'=2\Bigl(\frac{\delta^3}3-\frac{\delta^3}4\Bigr)=\frac{\delta^3}6. \] The variance of $\delta^{-1}e_3\cdot(J\times d)=\delta^{-1}(J_1d_2-J_2d_1)$ is $\delta\abs{d_\perp}^2/6$. To compute the variance of $A(b)$, let $\mathfrak B$ be a Brownian motion with generator $\Delta$ started at $0$, and let $\mathfrak U=\mathfrak B(\delta)$. The process $\mathfrak B(s)-s\mathfrak U/\delta$ is a Brownian bridge from $0$ to $0$ of duration $\delta$, independent of $\mathfrak U$, and we may take $b(s)=\mathfrak B(s)-s\mathfrak U/\delta$. Applying \eqref{eq:ncri-bridge-split} with $y=0$ to the path $\mathfrak B(s)=s\mathfrak U/\delta+b(s)$, whose computation above is pathwise, yields $A(\mathfrak B)=A(b)+\delta^{-1}e_3\cdot(J\times\mathfrak U)$. By It\^o's isometry, since $\dd\langle\mathfrak B_a\rangle_s=2\dd s$ and $\E\mathfrak B_a(s)^2=2s$, \[ \E A(\mathfrak B)^2=\frac14\int_0^\delta\E\bigl(\mathfrak B_1(s)^2+\mathfrak B_2(s)^2\bigr)\,2\dd s=\frac12\int_0^\delta4s\dd s=\delta^2. \] Since $\mathfrak U$ is centered and independent of $b$, the variables $A(b)$ and $\delta^{-1}e_3\cdot(J\times\mathfrak U)$ are uncorrelated, and the variance of the latter is \[ \delta^{-2}\bigl(\E J_1^2\,\E\mathfrak U_2^2+\E J_2^2\,\E\mathfrak U_1^2\bigr)=\delta^{-2}\cdot2\cdot\frac{\delta^3}6\cdot2\delta=\frac{2\delta^2}3. \] We obtain $\E A(b)^2=\delta^2-2\delta^2/3=\delta^2/3$. Together with \eqref{eq:ncri-bridge-split} and the computation for $J$, this proves the second identity in \eqref{eq:ncri-bridge-moments}. Finally, $\mathbb W^\delta_{y,y+d}$ has total mass $p_\delta(d)\le p_\delta(0)=(4\pi\delta)^{-3/2}$. By the Cauchy--Schwarz inequality, we have $\int\abs{\mathcal B_y}\dd\mathbb W^\delta_{y,y+d}=p_\delta(d)\E\abs{\mathcal B_y(\omega)}\le p_\delta(0)(\E\mathcal B_y(\omega)^2)^{1/2}$. Since $\abs{d_\perp}\le\abs d\le R$, this proves \eqref{eq:ncri-bridge-unnormalized}. \end{proof} \usedin{\cref{lem:ncri-conditional-area}.} The stochastic area of a confined Brownian path is not bounded pathwise: a path that stays in a small ball can sweep a large signed area by winding many times. For this reason, we control the area gain of a surgery through first moments under the exact conditional laws of the matching system. Set \begin{equation}\label{eq:ncri-CB} M_{\mathcal B}=(4\pi\delta)^{-3/2}\Bigl(\frac{\delta^2}3+\frac{1600\,\delta}6\Bigr)^{1/2},\qquad C_{\mathcal B}=\frac{2M_{\mathcal B}}{z_-}, \end{equation} where $z_-$ is the constant of \cref{lem:slab-mass}. \begin{lemma}\label{lem:ncri-conditional-area} Consider the matching system of one slab in $U$, and assume that its global collision error satisfies $\eta_{\rm glob}\le1/2$. Let $E$ be a value of the exposed data satisfying the endpoint separation conditions of \cref{lem:slab-insertion}, let $\tau\colon I\to J$ be a bijection, and let $i\in I$ satisfy $K^E_{i\tau(i)}(1)\ge z_-$. Let $\omega_k\in\mathcal P^E_{k\tau(k)}$ for $k\in I\setminus\{i\}$, and let $\nu$ be the probability measure proportional to $\prod_{k\ne i}h_{ik}(\omega,\omega_k)\,K^E_{i\tau(i)}(\dd\omega)$. Then \begin{equation}\label{eq:ncri-conditional-area} \int\abs{\mathcal B_{Y_i}(\omega)}\,\nu(\dd\omega)\le C_{\mathcal B}. \end{equation} \end{lemma} \begin{proof} By \eqref{eq:slab-insertion} with $I'=I\setminus\{i\}$, the normalizing constant of $\nu$ is at least $(1-\eta_{\rm glob})K^E_{i\tau(i)}(1)\ge z_-/2>0$. By \eqref{eq:slab-kernel}, $K^E_{i\tau(i)}$ is bounded by the bridge measure $\mathbb W^{2t}_{Y_i,Z_{\tau(i)}}$, and the checks $h_{ik}$ are at most one. Since $K^E_{i\tau(i)}(1)>0$, there exists a permitted path from $Y_i$ to $Z_{\tau(i)}$, and \cref{lem:slab-eligible}(b) implies that $\abs{Z_{\tau(i)}-Y_i}\le40$. By \eqref{eq:ncri-bridge-unnormalized} with $R=40$, we have \[ \int\abs{\mathcal B_{Y_i}(\omega)}\prod_{k\ne i}h_{ik}(\omega,\omega_k)\,K^E_{i\tau(i)}(\dd\omega)\le\int\abs{\mathcal B_{Y_i}}\dd\mathbb W^{2t}_{Y_i,Z_{\tau(i)}}\le M_{\mathcal B}. \] Dividing by the normalizing constant gives \eqref{eq:ncri-conditional-area}. \end{proof} \usedin{the proof of \cref{prop:ncri-gain}.} No independence between the area and the avoidance constraints is used: the constraints are dropped in the numerator and controlled in the denominator. The bound requires the insertion estimate for every permitted configuration of the other rows, including the rows of the other groups, and not only on average; this is the form of (H3), with the global error $\eta_{\rm glob}$, proved in \cref{lem:slab-insertion}. \section{The mean area gain of a surgery}\label{sec:ncri-gain} The change of the stochastic area under a surgery consists of a deterministic part, which is computed from the endpoints of the selected paths, and the change of the stochastic areas of the selected paths about their initial points. The deterministic part is the area of the polygon of initial points, up to an error from the chords between neighboring endpoints. \begin{lemma}\label{lem:ncri-chord} Let $m\ge2$, let $Y_0,\dots,Y_{m-1},Z_0,\dots,Z_{m-1},x'\in\R^3$, with indices modulo $m$, and let $d_k=Z_k-Y_k$. Then \begin{equation}\label{eq:ncri-chord} \frac12\sum_{k=0}^{m-1}(Y_k-x')\times(Z_{k+1}-Z_k)=\operatorname{ar}(Y_0,\dots,Y_{m-1})+\frac12\sum_{k=0}^{m-1}(Y_{k-1}-Y_k)\times d_k. \end{equation} If $Y_k=V_k+e_k$ for $V_k,e_k\in\R^3$, then \begin{equation}\label{eq:ncri-center} \operatorname{ar}(Y_0,\dots,Y_{m-1})-\operatorname{ar}(V_0,\dots,V_{m-1})=\frac12\sum_{k=0}^{m-1}\bigl(e_k\times(V_{k+1}-V_{k-1})+e_k\times e_{k+1}\bigr). \end{equation} \end{lemma} \begin{proof} Since $\sum_k(Z_{k+1}-Z_k)=0$, the terms with $x'$ cancel. Next, note that $Z_{k+1}-Z_k=Y_{k+1}-Y_k+d_{k+1}-d_k$ and $Y_k\times Y_k=0$. Expanding gives \[ \sum_kY_k\times(Z_{k+1}-Z_k)=\sum_kY_k\times Y_{k+1}+\sum_kY_k\times d_{k+1}-\sum_kY_k\times d_k. \] Reindexing the second sum on the right as $\sum_kY_{k-1}\times d_k$ proves \eqref{eq:ncri-chord}. For \eqref{eq:ncri-center}, we expand $Y_k\times Y_{k+1}=V_k\times V_{k+1}+V_k\times e_{k+1}+e_k\times V_{k+1}+e_k\times e_{k+1}$ and use $\sum_kV_k\times e_{k+1}=\sum_kV_{k-1}\times e_k=-\sum_ke_k\times V_{k-1}$. \end{proof} \usedin{the proof of \cref{prop:ncri-gain}.} In the rest of this section, we assume the setting of \cref{sec:ncri-setting}, with $K$ exceeding the volume thresholds of \cref{sec:ncri-parameters} and the one-slab law satisfying the joint bound, so that \eqref{eq:ncri-chi} holds and \cref{lem:ncri-containment} applies. Fix $x'\in\R^3$, and let $\varphi=\varphi^{\mathcal A}_{e_3}$ be the function of \cref{sec:var-slabs} with the reference point $x'$; in the notation of \cref{sec:slab}, \[ \varphi=\frac12\sum_{k=1}^N\int_0^\delta e_3\cdot\bigl((\omega_k(s)-x')\times\circ\dd\omega_k(s)\bigr), \] the sum running over all paths of the slab. By \cref{prop:id-slabs}(5), $\varphi$ is square integrable under $\mu^{\mathfrak m}_\delta$. Let \begin{equation}\label{eq:ncri-Cerr} C_{\mathrm{err}}=57+2C_{\mathcal B}. \end{equation} \begin{proposition}\label{prop:ncri-gain} Let $a$ be a group, and let $q^{(a)}$ be the subprobability kernel of group $a$ in \cref{def:surgery-move}, averaged over the random skeleton. The double integral below converges absolutely, and \begin{equation}\label{eq:ncri-gain} \iint\bigl(\varphi(y)-\varphi(x)\bigr)\,q^{(a)}(x,\dd y)\,\mu^{\mathfrak m}_\delta(\dd x)\ge\E\Bigl[\ind\{\mathcal R(B^{(a)}_\sigma,\Gamma)\text{ accepts}\}\bigl(e_3\cdot\operatorname{ar}(\gamma)-C_{\mathrm{err}}\abs\gamma\bigr)\Bigr], \end{equation} where the expectation on the right is over $(E,\sigma,\omega_I)$ with law $\mu^{\mathfrak m}_\delta$ and an independent polygonal skeleton $\Gamma$, and $\gamma=\mathcal R(B^{(a)}_\sigma,\Gamma)$. \end{proposition} \begin{proof} Since $q$ is the average of the group kernels, $q^{(a)}$ is at most the number of groups times $q$. The absolute convergence then follows from \cref{lem:var-hcr}(2), applied to $q$, since $\varphi$ and the density of $\mu^{\mathfrak m}_\delta Q$ are square integrable. All fiber computations below are on $E\in\mathsf E_0$ satisfying the endpoint separation conditions of \cref{lem:slab-insertion}, a set of full measure for the exposed data. For fixed $\Gamma$, the kernel $q^{(a)}_\Gamma$ is a finite sum over labeled routes as in \cref{def:surgery-move}, with one row of group $a$ in each cell. For $\gamma=(v_0,\dots,v_{\abs\gamma-1})$ and a labeling $i_k\in I^{(a)}_{v_k}$, the rate is \[ R_\pi(\sigma\,|\,\Gamma)=\ind\{\mathcal R(B^{(a)}_\sigma,\Gamma)=\gamma\}\prod_{k}\frac{A_{i_k\sigma(i_k)}}{m^{(a)}_{v_k}(\sigma)}, \] where $\pi$ is the permutation with $\pi(i_k)=i_{k+1}$ that fixes all other rows (\cref{def:surgery-move}). The rate depends only on $E$, $\sigma$, and $\Gamma$. Let $S=\{i_k\}$, and let $P^{\rm tgt}$ denote the exact target conditional law of \cref{def:surgery-move}, given $E$, $\sigma\pi$, and $\omega_{I\setminus S}$. For an accepted route, the rates of its labelings sum to one. Fix $\Gamma$, a state $x=(E,\sigma,\omega_I)$ with $\omega_i\in\mathcal P^E_{i\sigma(i)}$ for all $i\in I$, and a labeled route with $R_\pi(\sigma\,|\,\Gamma)>0$. We write $Y_k=Y_{i_k}$, $Z_k=Z_{\sigma(i_k)}$, and $d_k=Z_k-Y_k$. The move replaces the path $\omega_{i_k}$, from $Y_k$ to $Z_k$, by a path $\omega'_{i_k}$ from $Y_k$ to $Z_{\sigma\pi(i_k)}=Z_{k+1}$, and leaves all other paths of the slab unchanged. By \eqref{eq:ncri-area-split}, applied to $\omega_{i_k}$ and to $\omega'_{i_k}$, and by \eqref{eq:ncri-chord}, the new state $y$ satisfies \begin{equation}\label{eq:ncri-change} \varphi(y)-\varphi(x)=e_3\cdot\operatorname{ar}(\gamma)+\mathcal E_1+\mathcal E_2+\sum_k\bigl(\mathcal B_{Y_k}(\omega'_{i_k})-\mathcal B_{Y_k}(\omega_{i_k})\bigr), \end{equation} where \[ \mathcal E_1=\frac12\sum_ke_3\cdot\bigl((Y_{k-1}-Y_k)\times d_k\bigr),\qquad \mathcal E_2=e_3\cdot\bigl(\operatorname{ar}(Y_0,\dots,Y_{\abs\gamma-1})-\operatorname{ar}(\gamma)\bigr). \] We bound the two errors. We write $e_k=Y_k-v_k$. Since $Y_k\in B_{v_k}$ and neighboring cell centers are at distance one, we have $\abs{e_k}\le\sqrt3/2$ and $\abs{Y_{k-1}-Y_k}\le1+\sqrt3$. The path $\omega_{i_k}$ is permitted, and \cref{lem:slab-eligible}(b) implies that $\abs{d_k}\le40$. We obtain $\abs{\mathcal E_1}\le20(1+\sqrt3)\abs\gamma$. By \eqref{eq:ncri-center} with $V_k=v_k$, and by $\abs{v_{k+1}-v_{k-1}}\le2$, we have $\abs{\mathcal E_2}\le(\sqrt3+3/4)\abs\gamma/2\le5\abs\gamma/4$. Since $20(1+\sqrt3)+5/4\le57$, we conclude from \eqref{eq:ncri-change} that \begin{equation}\label{eq:ncri-pointwise} \varphi(y)-\varphi(x)\ge e_3\cdot\operatorname{ar}(\gamma)-57\abs\gamma-\sum_k\bigl(\abs{\mathcal B_{Y_k}(\omega'_{i_k})}+\abs{\mathcal B_{Y_k}(\omega_{i_k})}\bigr). \end{equation} We next bound the conditional means of the stochastic areas, starting with the target paths. Under $P^{\rm tgt}$, conditionally on $\omega'_{S\setminus\{i_k\}}$, the path $\omega'_{i_k}$ has the law $\nu$ of \cref{lem:ncri-conditional-area} with $\tau=\sigma\pi$, $i=i_k$, and the opponents $\omega'_l$, $l\in S\setminus\{i_k\}$, and $\omega_l$, $l\notin S$, which are permitted for their pairings under $\sigma\pi$. Its pairing $(i_k,\sigma(i_{k+1}))$ has mass at least $z_-$ by \cref{lem:slab-mass}, applied to the rows $i_k$ and $i_{k+1}$, whose anchors $v_k$ and $v_{k+1}$ are face neighbors, and to $A_{i_k\sigma(i_k)}=A_{i_{k+1}\sigma(i_{k+1})}=1$. Then \cref{lem:ncri-conditional-area} implies that $\int\abs{\mathcal B_{Y_k}(\omega'_{i_k})}\,P^{\rm tgt}(\dd\omega'_S)\le C_{\mathcal B}$. We now turn to the source paths. By \eqref{eq:slab-law}, under $\mu^{\mathfrak m}_\delta$ and conditionally on $(E,\sigma,\omega_{I\setminus\{i_k\}})$, the path $\omega_{i_k}$ has the law $\nu$ of \cref{lem:ncri-conditional-area} with $\tau=\sigma$ and $i=i_k$. If $R_\pi(\sigma\,|\,\Gamma)>0$, then $A_{i_k\sigma(i_k)}=1$, and by \cref{lem:slab-mass}, applied with equal rows and equal slots, we have $K^E_{i_k\sigma(i_k)}(1)\ge z_-$. Since $R_\pi(\sigma\,|\,\Gamma)$ is a function of $(E,\sigma)$ for fixed $\Gamma$, by \cref{lem:ncri-conditional-area}, we have \[ \begin{aligned} \int R_\pi(\sigma\,|\,\Gamma)\abs{\mathcal B_{Y_k}(\omega_{i_k})}\dd\mu^{\mathfrak m}_\delta&=\int R_\pi(\sigma\,|\,\Gamma)\,\E\bigl[\abs{\mathcal B_{Y_k}(\omega_{i_k})}\bigm|E,\sigma,\omega_{I\setminus\{i_k\}}\bigr]\dd\mu^{\mathfrak m}_\delta\\ &\le C_{\mathcal B}\int R_\pi(\sigma\,|\,\Gamma)\dd\mu^{\mathfrak m}_\delta. \end{aligned} \] This is the point where we use that the rates do not read the paths that are integrated. We now multiply \eqref{eq:ncri-pointwise} by $R_\pi(\sigma\,|\,\Gamma)$, integrate over $P^{\rm tgt}$ and over $\mu^{\mathfrak m}_\delta$, sum over the labeled routes, and average over $\Gamma$. All terms are integrable: $\operatorname{ar}(\gamma)$ and $\abs\gamma$ are bounded at fixed $K$, because $\Lambda_{\mathrm{act}}$ is finite, and the stochastic areas were just bounded. Since the rates of the labelings of an accepted route sum to one, the two bounds above give \eqref{eq:ncri-gain} with $C_{\mathrm{err}}=57+2C_{\mathcal B}$. \end{proof} \usedin{\cref{cor:ncri-slab-gain}.} In \cref{prop:ncri-gain}, the endpoint terms of \eqref{eq:ncri-area-split} reproduce the area of the polygon of initial points through the chord identity \eqref{eq:ncri-chord}, and the reference point $x'$ drops out, in agreement with the independence of $\mathcal A$ from the reference point. The stochastic areas of the selected paths enter only through their conditional first moments, which is where the exact resampling of the target paths and the independence of the rates from the path values are used. \begin{corollary}\label{cor:ncri-slab-gain} Suppose, in addition, that $H\ge128$ and $n_H\ge\max\{256,62000\,C_{\mathrm{err}}\}$. Then \begin{equation}\label{eq:ncri-slab-gain} \E_{\mu^{\mathfrak m}_\delta Q}\varphi-\E_{\mu^{\mathfrak m}_\delta}\varphi\ge\frac{n_H^2}{1000}\ge c_{\mathrm g}H^2,\qquad c_{\mathrm g}=2^{-14}\cdot10^{-3}. \end{equation} \end{corollary} \begin{proof} By \cref{lem:var-hcr}(2), the left side of \eqref{eq:ncri-slab-gain} equals $\iint(\varphi(y)-\varphi(x))\,q(x,\dd y)\,\mu^{\mathfrak m}_\delta(\dd x)$, which is the average over the groups $a$ of the left sides of \eqref{eq:ncri-gain}. By \cref{prop:routes-router}, every accepted route satisfies $e_3\cdot\operatorname{ar}(\gamma)\ge n_H^2/250$ and $\abs\gamma\le124n_H$. Then \[ e_3\cdot\operatorname{ar}(\gamma)-C_{\mathrm{err}}\abs\gamma\ge\frac{n_H^2}{250}-124\,C_{\mathrm{err}}n_H\ge\frac{n_H^2}{500}, \] where in the last inequality we used $n_H\ge62000\,C_{\mathrm{err}}$. By \cref{prop:ncri-gain}, the left side of \eqref{eq:ncri-gain} is at least $n_H^2/500$ times the acceptance probability $\int\bar q^{(a)}\dd\mu^{\mathfrak m}_\delta$ of group $a$. The average of these acceptance probabilities over the groups is $\int\bar q\dd\mu^{\mathfrak m}_\delta\ge1/2$, by \eqref{eq:ncri-chi}. Hence the gain is at least $n_H^2/1000$. Finally, $n_H\ge H/64-1\ge H/128$ since $H\ge128$, and $(H/128)^2/1000=c_{\mathrm g}H^2$. \end{proof} \usedin{the proof of \cref{prop:ncri-scaled}.} \section{Choice of the parameters}\label{sec:ncri-parameters} We use the constants of \cref{sec:stiff-parameters}. At the steps listed there, the choices also meet the following requirements of the container. \begin{itemize} \item \emph{Step 1.} The spatial constants satisfy the requirements of \cref{prop:thermal-container}. \item \emph{Step 2.} The active cells have depth $64$, as in \eqref{eq:slab-active}. The polygonal skeleton law of \cref{def:routes-polygon} and the area routing map of \cref{def:routes-router-box} are fixed. The point $x_*$ and the radius $R_{\mathrm{in}}$ are fixed properties of $\Omega$; they enter only the constant $c^\sharp_\Omega$ below and the volume thresholds. \item \emph{Steps 3 and 4.} The skeleton constants, including $\theta_\times$ and $C_\times$, serve both skeleton families, and the area routing map accepts with probability at least $1-C_{\rm acc}p\ge1/2$. \item \emph{Steps 5 and 7.} The thermal constants $H_1$ and $t$ are chosen for \cref{prop:thermal-container} with base $p_1$ and $R_0=64$ as well, and $D_{\min}$ is at least the threshold of \cref{cor:slab-system} in a container, which includes the threshold $D_0$ of \cref{prop:thermal-container}. \item \emph{Step 8.} The area constants are $M_{\mathcal B}$ and $C_{\mathcal B}=2M_{\mathcal B}/z_-$ of \eqref{eq:ncri-CB}, $C_{\mathrm{err}}=57+2C_{\mathcal B}$ of \eqref{eq:ncri-Cerr}, $c_{\mathrm g}=2^{-14}\cdot10^{-3}$, and \begin{equation}\label{eq:ncri-cOmega} c^\sharp_\Omega=\frac{c_{\mathrm g}^2R_{\mathrm{in}}^5}{2C_\sharp\delta}. \end{equation} The constant $C_{\mathcal B}$ uses only the bound $\eta_{\rm glob}\le1/2$. It does not depend on $D\ge D_{\min}$. These constants depend on $t$, $C_\sharp$, and $R_{\mathrm{in}}$ only. They enter the volume thresholds and the final constants, and none of Steps 1--7. \end{itemize} \paragraph{Volume thresholds.} Let $K_0$ be the smallest integer such that, for every $\Theta\in\mathrm{SO}(3)$ and every $K\ge K_0$, the following hold for the container $\Theta\Omega$: \begin{enumerate}[label=\textup{(K\arabic*)}] \item $R_{\mathrm{in}}K\ge64$, $H\ge2^{14}$, and $n_H\ge\max\{256,62000\,C_{\mathrm{err}}\}$; \item $K\ge K_\Omega$, the threshold of \cref{prop:thermal-container} with $R_0=64$; \item the excess bounds of \cref{prop:routes-nominal} for the polygon law in a cube of side $H$, the acceptance bound of \cref{prop:routes-router} for the area routing map, and the volume condition of \cref{thm:surgery}. \end{enumerate} Conditions (K1) hold for all large $K$, since $H\ge R_{\mathrm{in}}K$ and $n_H=\lfloor H/64\rfloor\ge H/64-1\ge R_{\mathrm{in}}K/64-1$ by \cref{lem:ncri-cube}. The threshold $K_0$ depends only on $\Omega$, through $R_{\mathrm{in}}$ and $K_\Omega$, and on the constants of \cref{sec:stiff-parameters}; it does not depend on $\Theta$, $D$, $\alpha$, or $T$, and no condition relates $\beta$ to $K$. \section{Proof of the theorem}\label{sec:ncri-proof} We first prove the lower bound on the area variance in a fixed direction, in scaled units. \begin{proposition}\label{prop:ncri-scaled} Let $\Omega$, $x_*$, and $R_{\mathrm{in}}$ be as in \cref{sec:ncri-setting}, let $c^\sharp_\Omega$ be the constant \eqref{eq:ncri-cOmega}, and let $K_0$ be the threshold of \cref{sec:ncri-parameters}. Let $K\ge K_0$ and $N\ge1$ with $D=N/(K^3\abs\Omega)\ge D_{\min}$, let $\alpha\in[\alpha_0/2,2\alpha_0]$ and $r=\alpha/D$, and let $\beta>\delta$. Let $n$ be a unit vector and $\Theta\in\mathrm{SO}(3)$ with $\Theta n=e_3$, and suppose that the one-slab law of $N$ particles in $K\Theta\Omega$ with exclusion distance $r$ satisfies the joint bound \eqref{eq:thermal-joint} with base $p_1$ for sets of interior cells. Then the Bose path measure $\mu$ of $N$ particles in $U=K\Omega$ satisfies \begin{equation}\label{eq:ncri-var-lower} \Var_\mu(n\cdot\mathcal A)\ge c^\sharp_\Omega\,D^{1/4}\beta K^5. \end{equation} \end{proposition} \begin{proof} By \cref{lem:ncri-rotation}, $\Var_{\mu_U}(n\cdot\mathcal A)=\Var_{\mu_{\Theta U}}(e_3\cdot\mathcal A)$, and $\Theta U=K(\Theta\Omega)$. By the comment after \cref{lem:ncri-rotation}, the results of \cref{sec:ncri-setting,sec:ncri-gain} apply to the container $\Theta U$, whose one-slab law satisfies the joint bound by assumption, with the point $\Theta x_*$ and the same $R_{\mathrm{in}}$, $H$, $N$, and $D$. By assumption $\beta>\delta$. The completed surgery kernel $Q$ preserves the exposed data, and in particular the cuts, and by \eqref{eq:ncri-chi} it satisfies $\chi^2(\mu^{\mathfrak m}_\delta Q\,\|\,\mu^{\mathfrak m}_\delta)\le C_\sharp D^{-1/4}/H$. By \cref{cor:ncri-slab-gain}, its gain for $\varphi^{\mathcal A}_{e_3}$ is at least $c_{\mathrm g}H^2$. By \cref{prop:var-one-slab}, applied in the container $\Theta U$ with the unit vector $e_3$, $g=c_{\mathrm g}H^2$, and $h=C_\sharp D^{-1/4}/H$, we obtain \[ \Var_{\mu_{\Theta U}}(e_3\cdot\mathcal A)\ge\frac{n_\beta\,c_{\mathrm g}^2H^4}{C_\sharp D^{-1/4}/H}=\frac{n_\beta\,c_{\mathrm g}^2H^5D^{1/4}}{C_\sharp}. \] Since $\beta>\delta$, we have $n_\beta=\lfloor\beta/\delta\rfloor\ge\beta/(2\delta)$, and $H\ge R_{\mathrm{in}}K$ by \cref{lem:ncri-cube}. The right side is at least $c_{\mathrm g}^2R_{\mathrm{in}}^5D^{1/4}\beta K^5/(2C_\sharp\delta)=c^\sharp_\Omega D^{1/4}\beta K^5$. \end{proof} \usedin{the proof of \cref{thm:main-ncri}.} By \cref{prop:thermal-container}, applied to $\Theta\Omega$ with $K_{\Theta\Omega}=K_\Omega$ (\cref{sec:ncri-rotation}), the joint bound holds for every $\Theta$ whenever $D\ge D_{\min}$, $00$, $K=L/s$, and $x_0\in\R^3$. Consider $N$ particles in $L\Omega$ with exclusion distance $a$ at temperature $T$, and $N$ particles in $K\Omega$ with exclusion distance $a/s$ at temperature $s^2T$. Their configuration domains are both empty or both nonempty. In the second case, the inertia tensors of \cref{def:id-inertia} satisfy \[ I_{bc}(Lx_0)=s^2I_{bc}(Kx_0),\qquad I_{\mathrm{cl},bc}(Lx_0)=s^2I_{\mathrm{cl},bc}(Kx_0), \] where the left sides refer to the first gas and the right sides to the second, and $\Cov(\mathcal A)$ for the first gas is $s^4$ times $\Cov(\mathcal A)$ for the second. In particular, $1-I(n,Lx_0)/\Icl(n,Lx_0)=1-I(n,Kx_0)/\Icl(n,Kx_0)$ for every unit vector $n$. \end{lemma} \begin{proof} The first claim and the properties of the unitary map $U$ are as in the proof of \cref{lem:stiff-dilation}, with $\Lambda_L$ and $\Lambda_K$ replaced by $L\Omega$ and $K\Omega$; in particular $q^{(L\Omega,a)}=s^{-2}q^{(K\Omega,a/s)}\circ U$, so that $H^{(L\Omega,a)}=s^{-2}U^{-1}H^{(K\Omega,a/s)}U$. Under the change of variables $x=s\tilde x$, we have $x-Lx_0=s(\tilde x-Kx_0)$ and $\nabla_x=s^{-1}\nabla_{\tilde x}$, and the angular momentum \eqref{eq:id-angular-momentum} about $Lx_0$ is conjugated by $U$ to the angular momentum about $Kx_0$. By \cref{lem:id-angular}, the operators $H-\bm\omega\cdot\mathcal L$ are defined by their forms on the respective domains, and the last two facts give \[ \tr e^{-(H^{(L\Omega,a)}-\bm\omega\cdot\mathcal L)/T}=\tr e^{-(H^{(K\Omega,a/s)}-s^2\bm\omega\cdot\mathcal L)/(s^2T)}. \] The rotating-frame free energies \eqref{eq:id-rot-free-energy} then satisfy $F^{\mathrm{rot}}(\bm\omega)=s^{-2}F^{\mathrm{rot}}(s^2\bm\omega)$, with the first gas on the left and the second on the right. Differentiating twice at $\bm\omega=0$ gives $I_{bc}(Lx_0)=s^2I_{bc}(Kx_0)$. For a bounded measurable symmetric function $f$, the operator $UfU^{-1}$ is multiplication by $f(s\,\cdot)$, and the Gibbs expectation of $f$ in the first gas equals that of $f(s\,\cdot)$ in the second. The matrix function $\mathsf R$ of \cref{sec:id-rotation} with reference point $Lx_0$, evaluated at $sX$, is $s^2$ times the one with reference point $Kx_0$, evaluated at $X$. This gives $I_{\mathrm{cl},bc}(Lx_0)=s^2I_{\mathrm{cl},bc}(Kx_0)$. By \eqref{eq:id-area-formula} in both gases, $T\Cov(\mathcal A)$ for the first gas equals $s^2(\Icl-I)$ for the second, which is $s^2\cdot s^2T\Cov(\mathcal A)$ for the second. Dividing by $T$ gives the claim. \end{proof} \usedin{the proof of \cref{thm:main-ncri}.} \begin{proof}[Proof of \cref{thm:main-ncri}] Let $\alpha_0$, $D_{\min}$, $\delta$, $\tau_0$, and $C_\sharp$ be the constants of \cref{sec:stiff-parameters}, and let $c^\sharp_\Omega$ be the constant \eqref{eq:ncri-cOmega}. We prove the theorem with the constants $c_0=\alpha_0^3/(16D_{\min}^2)$ and $c_1=c^\sharp_\Omega/\alpha_0^{9/8}$. The first is absolute; the second depends on $\Omega$ only through $R_{\mathrm{in}}$. We first prove the following finite-volume bounds. Let $a>0$ and $0<\rho_1\le\rho_2$ with $\rho_2a^3\le2c_0$. There exists $L_0$ such that, for all $L\ge L_0$, all $N$ with $\rho_L=N/(L^3\abs\Omega)\in[\rho_1,\rho_2]$, all $00$, let $s_\rho=(\alpha_0/(a\rho))^{1/2}$ and $D_\rho=\rho s_\rho^3=\alpha_0^{3/2}(\rho a^3)^{-1/2}$, as in the proof of \cref{thm:main-stiffness}. By \eqref{eq:intro-Tstar}, $T_*(a,\rho)=\tau_0/(s_\rho^2\log D_\rho)$, and since $\rho_2a^3\le2c_0$, we have $D_\rho\ge D_{\rho_2}\ge\sqrt8\,D_{\min}$ for $\rho\le\rho_2$. For $N$, $L$, and $T$ as above, let \[ K=\Bigl\lceil\Bigl(\frac{aN}{\alpha_0\abs\Omega L}\Bigr)^{1/2}\Bigr\rceil=\Bigl\lceil\frac L{s_{\rho_L}}\Bigr\rceil,\qquad s=\frac LK,\qquad D=\frac N{K^3\abs\Omega}=\rho_Ls^3, \] \[ r=\frac as,\qquad \alpha=Dr=\rho_Las^2,\qquad T_K=s^2T,\qquad \beta_K=\frac1{T_K}. \] Since $K-10$, \[ T_K\log D\le s_{\rho_L}^2T\log D_{\rho_L}=\tau_0\,\frac T{T_*(a,\rho_L)}\le\tau_0, \] and $\beta_K\ge\log D/\tau_0\ge\log D_{\min}/\tau_0\ge4>\delta$, since $\tau_0/\log D_{\min}\le1/4$ by Step~7 of \cref{sec:stiff-parameters} and $\delta=2t\le2$. Let $K_0$ be the threshold of \cref{sec:ncri-parameters}, and let $L_0=\max\{K_0,8\}\,s_{\rho_1}$. For $L\ge L_0$, we have $K\ge L/s_{\rho_L}\ge L/s_{\rho_1}\ge\max\{K_0,8\}$, since $\rho_L\ge\rho_1$. We denote by $\Cov_{\rm sc}$ and $\Var_{\rm sc}$ the covariance and variance under the Bose path measure of $N$ particles in $K\Omega$ with exclusion distance $r$ at temperature $T_K$. \emph{All directions in scaled units.} Let $L\ge L_0$. By the comment after \cref{prop:ncri-scaled}, the joint bound holds for every $\Theta\in\mathrm{SO}(3)$, since $D\ge D_{\min}$, $00$. The function $f$ vanishes only on the Lebesgue null set of configurations with all particles on the axis, which implies that $\Icl(n,Lx_0)>0$. Moreover, $\Icl(n,Lx_0)\le NL^2R_\Omega(x_0)^2/2$, since $\abs{x-Lx_0}\le LR_\Omega(x_0)$ for $x\in L\Omega$. By \eqref{eq:id-area-formula}, \[ 1-\frac{I(n,Lx_0)}{\Icl(n,Lx_0)}=\frac{T\Var_\mu(\mathcal A_n)}{\Icl(n,Lx_0)}\ge\frac{2\lambda_{\min}(\Cov_\mu(\mathcal A))}{\beta NL^2R_\Omega(x_0)^2}=\frac{\lambda_{\min}(\Cov_\mu(\mathcal A))}{\beta\rho_LL^5}\cdot\frac2{\abs\Omega R_\Omega(x_0)^2}, \] and the first finite-volume bound gives the second. \emph{The limits.} Let $a,\rho>0$ satisfy $\rho a^3\le c_0$, let $00$, and by \eqref{eq:id-area-formula}, $I(0)=(w-Tv)\,\mathrm{Id}$. It follows that $I(n,0)$, $\Icl(n,0)$, and the fraction $1-I(n,0)/\Icl(n,0)$ do not depend on $n$, at every finite volume, and $\lambda_{\min}(\Cov_\mu(\mathcal A))=\Var_\mu(\mathcal A_3)$. In this case the proof of \cref{thm:main-ncri} needs only the direction $e_3$, and \cref{lem:ncri-rotation} is not used. \end{remark} \begin{remark}\label{rem:ncri-scope} The constant $c_1$ depends on $\Omega$ through $R_{\mathrm{in}}^5$, and the fraction bound \eqref{eq:intro-ncri} also involves $\abs\Omega$ and $R_\Omega(x_0)$. The thermal constants are independent of $\Omega$; the volume thresholds may depend on its shape. For the interpretation of the rotational response and its dependence on the walls, see \cref{rem:intro-boundary}. Unlike for winding in \cref{cor:main-winding-scaling}, no higher-moment or distributional-limit bound for the normalized area is proved here. \end{remark} \appendix \crefname{appendix}{Appendix}{Appendices}\Crefname{appendix}{Appendix}{Appendices} \providecommand{\osc}{\operatorname{osc}} \chapter{Thermal estimates in a container}\label[appendix]{app:walls} In this appendix, we prove the estimates on the hard-sphere Gibbs measure that \cref{sec:slab,sec:proof-ncri} require for a gas in a container with hard walls. The container is the dilation of a fixed nonempty bounded open set $\Omega\subset\R^3$. We assume no regularity of $\partial\Omega$, no connectivity, and nothing about the equilibrium density profile. The main result, \cref{thm:walls-main}, is \cref{prop:thermal-container}: the analog in the container of the joint bound for failure of the local conditions \cite[Proposition~5.4]{OpenAI2026BEC} and of the supply of useful labels \cite[Lemma~7.1]{OpenAI2026BEC}, for cells at a fixed distance from the walls. The proof follows the chain of estimates in \cite[Sections~3--7]{OpenAI2026BEC}, much of which is reproduced there from \cite{OpenAI2026HS,OpenAI2026B}. The largest changes are a lower bound on the free energy depending only on the volume and an insertion orbital satisfying the wall condition: a smooth orbital supported in the container and equal, before normalization, to one on a set of volume at least half the container volume. We also adapt the close-neighbor estimate, the entropy bound, the shortage argument, and the marked-path trace comparison. \Cref{tab:walls-status} states which proofs are reproduced, modified, or restricted to interior cells. Upper bounds on local particle numbers hold at every unit cell, including cells cut by the walls; lower bounds require cells whose closures lie in the container. Auxiliary marked particles may leave the physical container, as explained in \cref{sec:walls-marked}. The domain arguments in \cref{sec:walls-domains} apply to arbitrary open sets. \section{Setting and statement}\label{sec:walls-setting} \paragraph{The scaled container.} We use the scaled container and parameter ranges of \cref{sec:thermal-container}, with \begin{equation}\label{eq:walls-parameters} \begin{gathered} U=K\Omega,\qquad V=\abs U,\qquad D=N/V,\qquad r=\alpha/D,\qquad\beta=T^{-1},\\ \alpha\in[\alpha_0/2,2\alpha_0],\qquad00$ denote constants that may change from line to line. Unless stated otherwise, they depend only on $\alpha_0$ and on the constants $\ell,\theta_0,b_0,C_{\rm reg}$ fixed below, and never on $\Omega$, $K$, $N$, $D$, $\alpha$, or $T$ in the ranges under consideration. A constant called numerical does not depend on $\alpha_0$ either. \paragraph{Hamiltonians.} For an open set $G\subset\R^d$, we write $H^1_0(G)$ for the closure in $H^1(\R^d)$ of the smooth functions with compact support in $G$, and we identify its elements with their extensions by zero. We use only this definition, and never a description of $H^1_0(G)$ as the set of $H^1$ functions vanishing outside $G$, which fails for some irregular boundaries. For an open set $U'\subset\R^3$ and $j\ge1$, let \[ \mathcal G_j(U')=\{X=(x_1,\dots,x_j)\in(U')^j:\abs{x_i-x_k}>r\text{ for }i\neq k\},\qquad \mathcal G_j=\mathcal G_j(U). \] Let $H_j^{\mathrm{lab}}$ be the operator on $L^2(\mathcal G_j)$ associated with the closed form $q_j[f]=\sum_{i=1}^j\int\abs{\nabla_if}^2$ on $H^1_0(\mathcal G_j)$, and let $H_j$ be its restriction to the symmetric subspace. In the notation of \cref{sec:id-model}, $\mathcal G_N$ is the configuration domain $\mathcal X_N(U)$ and $H=H_N$ is the container Hamiltonian, with the exclusion distance $r$. For a bounded measurable function $W\ge0$ on $\R^3$, set \[ H_j^{(-W)}=H_j-\sum_{i=1}^jW(x_i),\qquad Z_j^{(-W)}=\tr e^{-\beta H_j^{(-W)}},\qquad F_j^{(-W)}=-T\log Z_j^{(-W)}, \] where the trace is over the symmetric subspace. We omit the superscript when $W=0$, and set $Z_0=1$ and $F_0=0$; thus $Z_N=\tr e^{-\beta H}$. A symmetric trace-one state $\sigma=\sum_k\lambda_k\lvert f_k\rangle\langle f_k\rvert$ on $L^2(\mathcal G_j)$ has \emph{finite kinetic energy} if $f_k\in H^1_0(\mathcal G_j)$ whenever $\lambda_k>0$ and $\tr H_j\sigma=\sum_k\lambda_kq_j[f_k]<\infty$; energies are always form expectations. For such $\sigma$ we write $S(\sigma)=-\tr\sigma\log\sigma$ and \[ \mathcal F_j^{(-W)}(\sigma)=\tr H_j\sigma-\tr\Bigl(\sum_{i=1}^jW(x_i)\Bigr)\sigma-TS(\sigma). \] The Gibbs variational principle \cite[Lemma~3.1]{FrankLieb2012}, applied to $\beta H_j^{(-W)}$ with form energies, gives $\mathcal F_j^{(-W)}(\sigma)\ge F_j^{(-W)}$, with equality for the Gibbs state $\Gamma_j^{(-W)}=e^{-\beta H_j^{(-W)}}/Z_j^{(-W)}$ when $Z_j^{(-W)}>0$; this state has finite kinetic energy, because $\tr e^{-sH_j^{(-W)}}<\infty$ for every $s>0$ (\cref{lem:walls-domains}(5)). We write $\E_{j,-W}$ for expectation under the position law of $\Gamma_j^{(-W)}$, and \[ \mu(\dd X)=Z_N^{-1}K^{\mathrm s}_\beta(X,X)\dd X \] for the equilibrium position law of $N$ particles (\cref{lem:id-ck}). For a bounded measurable function $f$ on $\R^3$, let $n_f(X)=\sum_if(x_i)$, and let $\E_\sigma$ denote expectation under the position law of a state $\sigma$. \paragraph{Cells.} The cells are the half-open unit cubes $B_v=v+[-1/2,1/2)^3$, $v\in\Z^3$, and $n_v(X)=\#\{i:x_i\in B_v\}$. Since configurations lie in $U$, the count $n_v$ refers to the points in $B_v\cap U$. This set may be empty or of arbitrarily small volume, and we never divide by its volume. For $A\subset\Z^3$ and $x\in\R^3$, let $d(x,A)=\min_{v\in A}\abs{x-v}$, and let $\abs A$ be the number of elements of $A$. Fix $R_0>12$. As in \cref{sec:thermal-container}, a cell $B_v$ is \emph{interior} if $v\in U$ and $\dist(v,\R^3\setminus U)\ge R_0$. Then the closed ball $\overline B(v,12)$ and the closure of $B_v$ lie in $U$. Distinct interior cells are disjoint unit cubes in $U$. A set $A$ of interior cells satisfies $\abs A\le V$. \paragraph{Separation and the slab.} Let $s_0=b_0D^{-1/3}$. For a configuration $X$ of $j$ points, let $d_i(X)=\min_{k\neq i}\abs{x_i-x_k}$, with $d_i=+\infty$ if $j=1$, and let $N^{\mathrm{cl}}_v(X)=\#\{i:\abs{x_i-v}\le5,\ d_i(X)\le s_0\}$, as in \eqref{eq:thermal-closecount}. Fix $t\in(0,1]$ and set $\delta=2t<\beta$. Let $\nu_N$ be the measure of $N$ independent Euclidean Brownian motions with generator $\Delta$ on $[0,\delta]$ with Lebesgue initial measure, write $X(s)=(\omega_1(s),\dots,\omega_N(s))$, $Y=X(0)$, and $Z^\circ=X(\delta)$, and let $\mathcal E_\delta$ be the event that $X(s)\in\mathcal G_N$ for all $s\in[0,\delta]$. On this event no two paths come within distance $r$ and no path leaves $U$. The purified slab law is \begin{equation}\label{eq:walls-slab} \PP^{\mathrm{pur}}(\dd\omega,\dd W')=\frac1{Z_N}K^{\mathrm s}_{\beta/2-t}(W',Y)\,K^{\mathrm s}_{\beta/2-t}(Z^\circ,W')\,\ind_{\mathcal E_\delta}(\omega)\,\nu_N(\dd\omega)\dd W', \end{equation} where $K^{\mathrm s}_s$ are the symmetrized killed kernels \eqref{eq:id-killed-kernel} of $\mathcal G_N$, which vanish outside $\mathcal G_N\times\mathcal G_N$. This is \eqref{eq:thermal-pur} for the container. By \cref{lem:thermal-slab}, every event of the path family $\omega$ has the same probability under $\PP^{\mathrm{pur}}$ as under the one-slab law $\PP$ of \eqref{eq:thermal-one-slab} for $U$. \paragraph{Regular paths, local conditions, and useful labels.} We use \cref{def:thermal-regular,def:thermal-certificate,sec:thermal-useful} with Euclidean distances and diameters, including the definitions of eligible and useful labels and the unrestricted trial laws $P^u_{y,z}$. The mesh variables $m_t$ and $\tau$ and the oscillations $\osc_k$ are those of \cref{def:thermal-regular}; set $h_0=D^{-1}$. The mesh conditions are \begin{equation}\label{eq:walls-regular} \max_k\osc_k(\omega)\le D^{1/12},\qquad\frac1{2m_t}\sum_{k=1}^{2m_t}\bigl(1+\osc_k(\omega)\bigr)^3\le C_{\rm reg}, \end{equation} together with the two endpoint-window conditions in that definition. The local conditions quoted from \cite[Definition~5.2 and Proposition~5.4]{OpenAI2026BEC} are those of \cref{def:thermal-certificate}; their six failure tests are listed in \cref{sec:walls-completion}. We read the ball of radius $12$ there as $B(v,12)$; the proof also applies to $\overline B(v,12)$. For an eligible label $i$, usefulness means \begin{equation}\label{eq:walls-useful} \bigl\lvert\{u\in B_w:P^u_i(C_i(F^{\rm all}_i))\ge0.9\}\bigr\rvert\ge0.99,\qquad F^{\rm all}_i=\{\omega_k:k\neq i\}, \end{equation} for each target cell $B_w$ of $v$, with $P^u_i$ and $C_i$ as in \cref{sec:thermal-useful}. At an interior cell, $\overline B(v,10)\subset U$. A trial in $C_i(F^{\rm all}_i)$ stays in $U$. \paragraph{The shape threshold.} By inner regularity of Lebesgue measure, $\Omega$ contains a compact set of volume greater than $\abs\Omega/2$. We fix once and for all a smooth function $g_\Omega$ with compact support in $\Omega$, with $0\le g_\Omega\le1$, and equal to one on such a compact set. Then \begin{equation}\label{eq:walls-g-omega} \int g_\Omega^2\ge\frac{\abs\Omega}2,\qquad e_\Omega=\frac{\int\abs{\nabla g_\Omega}^2}{\int g_\Omega^2},\qquad K_\Omega=\max\bigl\{1,\abs\Omega^{-1/3},e_\Omega^{1/2}\bigr\}. \end{equation} We prove \cref{prop:thermal-container} in the following form. \begin{theorem}\label{thm:walls-main} With the parameters, ranges, and order of choices specified in \cref{prop:thermal-container}, in particular for $00$, $C^*_{\rm reg}$ and, for each $\theta_0>0$, a numerical $b^*(\theta_0)\in(0,1)$ such that the theorem holds whenever $\alpha_0\ge\alpha_0^*$, $0<\ell\le\ell^*$, $0<\theta_0<\ell/6$, $0{\raggedright\arraybackslash}p{0.27\textwidth}>{\raggedright\arraybackslash}p{0.47\textwidth}>{\raggedright\arraybackslash}p{0.17\textwidth}@{}} \caption{Status in the container of the statements of \cite{OpenAI2026BEC} used in the proof of \cref{thm:walls-main}.}\label{tab:walls-status}\\ \toprule Statement of \cite{OpenAI2026BEC} & Status in the container & Here\\ \midrule \endfirsthead \toprule Statement of \cite{OpenAI2026BEC} & Status in the container & Here\\ \midrule \endhead \bottomrule \endfoot Lemma 3.1, cube energy (from \cite[Lemma~3.1]{OpenAI2026HS}) & Same proof for the cube inequality. The cell inequality is applied to the zero extension on full cubes, summed over all cells, including cells cut by the walls. & \cref{lem:walls-local}\\ Lemma 3.2, close-neighbor energy (from \cite[Lemma~3.2]{OpenAI2026HS}) & Modified proof: translated Euclidean grid of side $8s$. & \cref{lem:walls-local}\\ Lemma 3.3, entropy (from \cite[Lemma~3.2]{OpenAI2026B}) & Modified proof: open sets of finite volume in $\R^3$; the term $V^{-1}$ disappears. & \cref{lem:walls-entropy}\\ Lemma 3.4, normalized insertion (from \cite[Lemma~3.3]{OpenAI2026HS}) & Same proof for orbitals with compact support in $U$. & \cref{lem:walls-insertion}\\ Proposition 3.5, lower bound $F_N\ge c\alpha N$ & Modified proof: new proof using only $\abs U$. & \cref{lem:walls-global}, \cref{cor:walls-global}\\ Proposition 3.5, monotonicity of $F_j/j$ (from \cite[Proposition~3.3]{OpenAI2026B}) & Same proof at fixed $U$, $r$, $T$, $D$. & \cref{prop:walls-increments}\\ Proposition 3.5, upper increments & Modified proof: orbital $w_K$ and Fourier frame on a cube containing $U$. & \cref{prop:walls-increments}\\ Lemma 4.1, two-color localization (from \cite[Lemma~4.1]{OpenAI2026B}) & Same proof; first color on $U\cap O$, second color on $U$. & \cref{lem:walls-localization}\\ Lemma 4.2, diagonal tilt (from \cite[Lemma~4.1]{OpenAI2026B}) & Same proof. & \cref{lem:walls-tilt}\\ Proposition 4.3, local penalty (from \cite[Lemma~3.6]{OpenAI2026HS}) & Same proof with Euclidean distances, for every finite set of cells; temperature uses as in \cref{prop:temp-spatial}. & \cref{prop:walls-penalty}\\ Corollary 4.4, density in a small field & Same proof, for every finite set of cells; temperature uses as in \cref{prop:temp-spatial}. & \cref{cor:walls-density}\\ Proposition 4.5, low counts (method of \cite[Section~4]{OpenAI2026B}) & Restricted to cells with closure in $U$; modified proof (filling inside $U$, $\abs A\le V$); temperature uses as in \cref{prop:temp-spatial}. & \cref{prop:walls-shortage}\\ Proposition 4.6, close counts (method of \cite[Section~4]{OpenAI2026B}; geometry from \cite[Section~3.5]{OpenAI2026HS}) & Same proof with Euclidean distances, for every finite set of cells; temperature uses as in \cref{prop:temp-spatial}. & \cref{prop:walls-close}\\ Proposition 5.1, purification & Modified proof: kernels include wall killing; the occupation identity is not used. & \cref{prop:walls-purification}\\ Definition 5.2, regular path (from \cite[Definition~4.3]{OpenAI2026HS}) & Same proof, with Euclidean diameters. & \eqref{eq:walls-regular}\\ Lemma 5.3, free cutoffs (from \cite[Lemma~4.4]{OpenAI2026HS}) & Same proof for Euclidean free laws. & \cref{lem:walls-cutoffs}\\ Proposition 5.4, [BEC]'s ``certificates'' & Restricted to interior cells; extended to the window as in \cref{prop:temp-certificates}, with the remainder $r^{\rm wall}_D$ of \cref{sec:walls-moment}. & \cref{thm:walls-main}(i)\\ Lemma 5.5, marked kernel (adapting \cite[Section~6]{OpenAI2026B}) & Modified proof: Euclidean neighborhoods, no lifts; marked paths without wall killing. & \cref{lem:walls-marked}, \cref{lem:walls-colormap}\\ Lemma 5.6, trace comparison (from \cite[Lemma~6.2]{OpenAI2026B}) & Same proof; hypotheses verified for the mixed color space. & \cref{lem:walls-sandwich}\\ Proposition 5.7, trajectory moment (adapting \cite[Proposition~6.1]{OpenAI2026B}) & Modified proof: unmarked physical particles in $U$, all-cell density bound, no term $V^{-1}$. & \cref{prop:walls-moment}\\ Lemma 6.1, Gaussian density (from \cite[Lemma~5.1]{OpenAI2026HS}) & Same proof. & \cref{prop:walls-bridges}\\ Lemma 6.2, averaged collision (from \cite[Proposition~5.2]{OpenAI2026HS}) & Same proof in $\R^3$; the paragraph on lifts is deleted. & \cref{prop:walls-bridges}\\ Lemma 6.3, single replacement (from \cite[Proposition~5.3]{OpenAI2026HS}) & Same proof in $\R^3$. & \cref{prop:walls-bridges}\\ Remark 6.4, midpoints (from \cite[Remark~5.4]{OpenAI2026HS}) & Same proof. & \cref{prop:walls-bridges}\\ Corollary 6.5, successful labels (from \cite[Corollary~5.5]{OpenAI2026HS}) & Same proof in $\R^3$. & \cref{prop:walls-bridges}\\ Lemma 7.1, useful labels (from \cite[Lemma~6.1]{OpenAI2026HS}) & Restricted to interior cells; same proof. & \cref{thm:walls-main}(ii)\\ \end{longtable} } No auxiliary torus is used. In \cite{OpenAI2026BEC}, several neighborhoods of sets of cells are allowed to be the whole torus; in $\R^3$ they are bounded, and this case does not occur. The only place where periodic functions enter is Parseval's identity on a cube containing $U$, in the proof of \cref{prop:walls-increments}. \section{Open-domain preliminaries}\label{sec:walls-domains} The following facts hold for arbitrary open sets. They are used without comment in the rest of the appendix. \begin{lemma}\label{lem:walls-domains} Let $G\subset\R^d$ be open. \begin{enumerate} \item If $f\in H^1_0(G)$, then $f=0$ and $\nabla f=0$ almost everywhere on $\R^d\setminus G$. \item Let $m\colon\R^d\to\mathbb C$ be bounded and Lipschitz, and let $G'\subset G$ be open with $\supp m\cap G\subset G'$. Then $f\mapsto mf$ is bounded from $H^1_0(G)$ to $H^1_0(G')$, and $\nabla(mf)=m\nabla f+f\nabla m$ almost everywhere. If $G_1\subset\R^{d_1}$ and $G_2\subset\R^{d_2}$ are open, $f_1\in H^1_0(G_1)$, and $f_2\in H^1_0(G_2)$, then $f_1\otimes f_2\in H^1_0(G_1\times G_2)$. \item Let $G_1\subset\R^{d_1}$ and $G_2\subset\R^{d_2}$ be open with $G\subset G_1\times G_2$, and let $\nabla_1$ be the gradient in the first $d_1$ variables. Let $\rho=\sum_k\lambda_k\lvert f_k\rangle\langle f_k\rvert$ be a positive trace-class operator on $L^2(G)$ with $\lambda_k\ge0$, $f_k\in H^1_0(G)$, and $\sum_k\lambda_k\norm{\nabla_1f_k}^2<\infty$. Then the partial trace $\rho_1$ of $\rho$ over the last $d_2$ variables is a positive trace-class operator on $L^2(G_1)$, its eigenvectors with nonzero eigenvalues lie in $H^1_0(G_1)$, and its kinetic energy $\sum_l\mu_l\norm{\nabla h_l}^2$, computed in an eigenbasis $\rho_1=\sum_l\mu_l\lvert h_l\rangle\langle h_l\rvert$, equals $\sum_k\lambda_k\norm{\nabla_1f_k}^2$. \item Let $W$ be a bounded measurable real function on $G$, and let $h$ be the operator of the closed form $\int\abs{\nabla f}^2-\int W\abs f^2$ on $H^1_0(G)$. For $s>0$, the operator $e^{-sh}$ has the nonnegative kernel \[ \int\ind\{\gamma(\tau)\in G\text{ for all }\tau\in[0,s]\}\exp\Bigl(\int_0^sW(\gamma(\tau))\dd\tau\Bigr)\,\mathbb W^s_{x,y}(\dd\gamma), \] where $\mathbb W^s_{x,y}$ is the Brownian bridge measure of \cref{sec:id-fk}, on $\R^d$, of total mass $(4\pi s)^{-d/2}e^{-\abs{y-x}^2/(4s)}$. \item If $d=3$ and $\abs G<\infty$, then $\tr e^{s\Delta_G}\le\abs G(4\pi s)^{-3/2}$ for $s>0$, where $\Delta_G$ is the Dirichlet Laplacian of $G$. If $G$ is bounded, the operator $h$ of part (4) has compact resolvent and $\tr e^{-sh}<\infty$ for $s>0$. \end{enumerate} \end{lemma} \begin{proof} For part (1), approximants vanish outside $G$ together with their gradients, and both properties pass to limits in $L^2$. For part (2), let $f\in C^\infty_c(G)$. Then $mf$ is Lipschitz, and its support lies in the compact set $\supp f\cap\supp m$, which is contained in $G\cap\supp m\subset G'$. Mollification shows that Lipschitz functions with compact support in $G'$ belong to $H^1_0(G')$. The product rule for a Lipschitz function and a Sobolev function gives $\norm{mf}_{H^1}\le C_m\norm f_{H^1}$, and since $H^1_0(G')$ is closed, approximation proves the first assertion for every $f\in H^1_0(G)$. Products of approximants of $f_1$ and $f_2$ are smooth, have compact support in $G_1\times G_2$, and converge to $f_1\otimes f_2$ in $H^1$. Part (3) is the domain argument of \cite[Section~3.2]{OpenAI2026BEC}. Let $(e_m)$ be an orthonormal basis of $L^2(G_2)$, and for $f\in H^1_0(G)$ set $f_m(x)=\int f(x,y)\overline{e_m(y)}\dd y$. If $f\in C^\infty_c(G)$, then $f_m$ is smooth with support in the projection of $\supp f$, which is a compact subset of $G_1$, and Parseval's identity in $y$ gives $\sum_m\norm{f_m}^2=\norm f^2$ and $\sum_m\norm{\nabla f_m}^2=\norm{\nabla_1f}^2$. Both identities extend to $f\in H^1_0(G)$ by approximation, and $f_m\in H^1_0(G_1)$ because $H^1_0(G_1)$ is closed. Then $\rho_1=\sum_{k,m}\lambda_k\lvert f_{k,m}\rangle\langle f_{k,m}\rvert$ with $\sum_{k,m}\lambda_k\norm{f_{k,m}}^2_{H^1}<\infty$. If $\rho_1h=\mu h$ with $\mu>0$ and $\norm h=1$, then $h=\mu^{-1}\sum_{k,m}\lambda_k\langle f_{k,m},h\rangle f_{k,m}$. By the Cauchy--Schwarz inequality, $\sum_{k,m}\lambda_k\abs{\langle f_{k,m},h\rangle}\norm{f_{k,m}}_{H^1}<\infty$. This series converges in $H^1$, and $h\in H^1_0(G_1)$. With $A$ the Dirichlet Laplacian of $G_1$ with the sign reversed, both expressions for the kinetic energy equal $\tr(A^{1/2}\rho_1A^{1/2})$. For part (4), the case $W=0$ with $G$ a configuration domain is \cref{prop:id-fk}(1). Its proof exhausts $G$ by relatively compact open subsets with smooth boundaries, applies the Feynman--Kac formula with killing \cite[Proposition~2.9]{BroderixHundertmarkLeschke2000} on each of them, and passes to the limit by strong convergence of the semigroups and monotone convergence of the killed path integrals. The only property of $G$ that it uses is that a continuous path which stays in $G$ on $[0,s]$ has compact image in $G$, and therefore stays in one of the exhausting subsets. The argument applies to every open set $G$ and every bounded $W$; it is the open-domain argument of \cite[Section~2.1]{OpenAI2026BEC} and \cite[Section~4]{OpenAI2026HS}. Kernel identities hold almost everywhere, and traces are evaluated after a split at a positive time, as in \cref{lem:id-ck}(3). For part (5), let $p^G_s$ be the kernel of $e^{s\Delta_G}$, and $p_s$ the free heat kernel. By part (4), $0\le p^G_s(x,y)\le p_s(y-x)$ almost everywhere. Since $e^{s\Delta_G}=(e^{s\Delta_G/2})^2$, \[ \tr e^{s\Delta_G}=\iint p^G_{s/2}(x,y)^2\dd x\dd y\le\int_G\int_{\R^3}p_{s/2}(y-x)^2\dd y\dd x=\abs Gp_s(0)=\abs G(4\pi s)^{-3/2}. \] If $G$ is bounded, extension by zero maps $H^1_0(G)$ isometrically into $H^1_0(Q)$ for an open cube $Q\supset G$, and the min--max principle compares the eigenvalues of $h+\norm W_\infty$ with those of the Dirichlet Laplacian on $Q$, as in the proof of \cref{lem:id-trace}. \end{proof} The configuration domains $\mathcal G_j$ are bounded open sets. By \cref{lem:walls-domains}(5), $H^{(-W)}_j$ has compact resolvent and $Z_j^{(-W)}<\infty$ for every $j$ and every bounded measurable $W\ge0$. That $Z_j^{(-W)}>0$ for $j\le2N$ is proved in \cref{prop:walls-increments}. \begin{lemma}\label{lem:walls-entropy} Let $O\subset\R^3$ be open with $\abs O<\infty$, let $\varsigma\in(0,1]$, and let $j\ge0$. For every symmetric trace-one state $\sigma$ on $L^2(O^j)$ of finite kinetic energy, \begin{equation}\label{eq:walls-entropy} \varsigma\tr\Bigl(\sum_{i=1}^j-\Delta_{O,i}\Bigr)\sigma-TS(\sigma)\ge-Tj-C_\varsigma T^{5/2}\abs O, \end{equation} where $\Delta_{O,i}$ is the Dirichlet Laplacian of $O$ in the $i$th coordinate. In particular $S(\sigma)<\infty$. The same bound holds for symmetric states of finite kinetic energy on $L^2(\mathcal G_j(O'))$ for every open set $O'\subset O$. \end{lemma} \begin{proof} This is \cite[Lemma~3.3]{OpenAI2026BEC}, the ideal Bose entropy bound of \cite[Lemma~3.2]{OpenAI2026B}, with the periodic heat-kernel bound replaced by \cref{lem:walls-domains}(5). Let $h=-\varsigma\Delta_O$. For $00$ and every partition of $\R^3$ into the cubes $Q$ of a grid $z+l\Z^3+[0,l)^3$, $z\in\R^3$, \[ q_I[F]\ge c_{\rm cube}\,rl^{-3}\,\E_F\sum_Qn^I_Q(n^I_Q-1),\qquad c_{\rm cube}=\frac1{64\sqrt3}. \] In particular $q_I[F]\ge c_{\rm cube}\,r\,\E_F\sum_{v\in\Z^3}n^I_v(n^I_v-1)$. \item If $0l^3/2$ and \emph{sparse} otherwise. The dense cubes are disjoint, and each contains a part of $U$ of volume greater than $l^3/2$. Their number $M_{\mathrm d}$ satisfies $M_{\mathrm d}\le2V/l^3$, where $V=\abs U$. For a configuration $X$, let $m(X)$ be the number of its points in dense cubes. We first bound the kinetic energy of the points in sparse cubes. Let $Q$ be a cube of side $l$, and let $g\in H^1(Q)$ vanish almost everywhere on a measurable set $E\subset Q$ with $\abs E\ge\abs Q/2$. Let $g_Q$ be the mean of $g$ over $Q$. Then $\abs E\abs{g_Q}^2=\int_E\abs{g-g_Q}^2\le\int_Q\abs{g-g_Q}^2$, and by the Poincar\'e inequality for the Neumann Laplacian of $Q$, whose smallest nonzero eigenvalue is $\pi^2/l^2$, we have \[ \int_Q\abs g^2=\int_Q\abs{g-g_Q}^2+\abs Q\abs{g_Q}^2\le3\int_Q\abs{g-g_Q}^2\le\frac{3l^2}{\pi^2}\int_Q\abs{\nabla g}^2. \] For almost every choice of the coordinates other than $x_i$, the function $x_i\mapsto F(X)$ lies in $H^1(\R^3)$, its gradient is $\nabla_iF$, and it vanishes almost everywhere outside $U$ by \cref{lem:walls-domains}(1). In a sparse cube $Q$, the set $Q\setminus U$ has volume at least $l^3/2$. We apply the last display in every sparse cube, and sum over the cubes and over $i$. Integrating over the other coordinates yields \begin{equation}\label{eq:walls-sparse} q_N[F]\ge\frac{\pi^2}{3l^2}\,\E_F(N-m)=c_{\mathrm P}\,\alpha\,\E_F(N-m),\qquad c_{\mathrm P}=\frac{\pi^2}3. \end{equation} If $M_{\mathrm d}=0$, then $m=0$, and \eqref{eq:walls-sparse} implies the lemma. Suppose that $M_{\mathrm d}>0$. By \cref{lem:walls-local}(1) with the grid of side $l$, keeping only the dense cubes, and by the Cauchy--Schwarz inequality $\sum_{Q\text{ dense}}n_Q^2\ge m^2/M_{\mathrm d}$, we have \begin{equation}\label{eq:walls-dense} q_N[F]\ge c_{\rm cube}\,rl^{-3}\,\E_F\sum_{Q\text{ dense}}n_Q(n_Q-1)\ge c_{\rm cube}\,rl^{-3}\,\E_F\Bigl(\frac{l^3m^2}{2V}-m\Bigr). \end{equation} We average \eqref{eq:walls-sparse} and \eqref{eq:walls-dense}, so that each kinetic energy is used once, and use $m\le N$. This gives \[ q_N[F]\ge\E_F\Bigl[\frac{c_{\rm cube}r}{4V}m^2+\frac{c_{\mathrm P}\alpha}2(N-m)\Bigr]-\frac{c_{\rm cube}r}{2l^3}N. \] Since $r/V=\alpha/N$ and $c_{\mathrm P}/2\ge c_{\rm cube}/4$, the expectation is at least $(c_{\rm cube}/4)\alpha N\,\E_F[(m/N)^2+1-m/N]\ge(3c_{\rm cube}/16)\alpha N$, where we used $x^2+1-x\ge3/4$. The last term equals $(c_{\rm cube}/2)\alpha^{5/2}N/D$, because $rl^{-3}=\alpha^{5/2}/D$. \end{proof} \begin{corollary}\label{cor:walls-global} There exists $D_1$, depending only on $\alpha_0$, such that the following holds for every bounded open set $U$, every $N$ with $D=N/\abs U\ge D_1$, every $\alpha\in[\alpha_0/2,2\alpha_0]$, and every $00$, the Gibbs variational principle gives $F_N=\mathcal F_N(\Gamma_N)\ge c_*\alpha N$; if $Z_N=0$, then $F_N=+\infty$. \end{proof} \section{Insertion, the Fourier frame, and particle-number comparisons}\label{sec:walls-insertion} \begin{lemma}\label{lem:walls-insertion} Let $w$ be a real smooth function with compact support in $U$ and $\norm w_2=1$. Let $n\ge0$, let $F\in H^1_0(\mathcal G_n)$, and suppose that on the support of $F$ at most $k$ old positions lie within distance $2r$ of $\supp w$. If $C\norm w_\infty^2kr^3\le1/4$, there exists $\widetilde F\in H^1_0(\mathcal G_{n+1})$ of the form $\widetilde F(X,y)=F(X)\varphi_X(y)$, where $\varphi_X$ is real, $\abs{\varphi_X}\le2\abs w$, $\varphi_X$ is a symmetric function of $X$, and, for $X$ in the support of $F$, \begin{equation}\label{eq:walls-insertion-orbital} \begin{gathered} \int\varphi_X^2\dd y=1,\qquad\int\varphi_X\nabla_{x_i}\varphi_X\dd y=0,\\ \int\Bigl(\abs{\nabla_y\varphi_X}^2+\sum_{i=1}^n\abs{\nabla_{x_i}\varphi_X}^2\Bigr)\dd y\le C\bigl(\norm{\nabla w}_2^2+\norm w_\infty^2kr\bigr). \end{gathered} \end{equation} In particular $\norm{\widetilde F}=\norm F$, $\widetilde F$ has the same old-coordinate position marginal as $F$, and \[ q_{n+1}[\widetilde F]\le q_n[F]+C\bigl(\norm{\nabla w}_2^2+\norm w_\infty^2kr\bigr)\norm F^2. \] \end{lemma} \begin{proof} This is \cite[Lemma~3.4]{OpenAI2026BEC}, the configurationwise normalized insertion of \cite[Lemma~3.3]{OpenAI2026HS}, which refines the nearest-neighbor trial construction of \cite[Section~3.2 in arXiv version~1]{LiebSeiringerYngvason2000}. Its proof applies with Euclidean distances, and we recall the construction because it is used below. Let $h_r\colon[0,\infty)\to[0,1]$ be nondecreasing, equal to zero on $[0,1.1r]$ and to one on $[2r,\infty)$, with Lipschitz constant at most $C/r$. For $X\in(\R^3)^n$ and $y\in\R^3$, let \[ f_X(y)=w(y)\min_{i\le n}h_r(\abs{y-x_i}),\qquad\mathfrak m(X)=\int f_X^2\dd y,\qquad\varphi_X=\frac{f_X}{\max\{\mathfrak m(X)^{1/2},1/2\}}, \] with the empty minimum equal to one. On the support of $F$, the hypothesis implies that $1-\mathfrak m(X)\le\norm w_\infty^2k\abs{B(0,2r)}\le1/4$. The proof of \eqref{eq:walls-insertion-orbital} and of the energy bound in \cite{OpenAI2026BEC} uses only these definitions and Euclidean geometry, and applies after the stated substitutions. It remains to verify that $\widetilde F\in H^1_0(\mathcal G_{n+1})$; this is where the walls enter. The function $(X,y)\mapsto\varphi_X(y)$ is bounded and Lipschitz, and $\int\abs{\nabla_X\varphi_X(y)}^2\dd y$ is bounded uniformly in $X$. The map $F\mapsto F\varphi$ is bounded from $H^1(\R^{3n})$ to $H^1(\R^{3n+3})$. If $F'\in C^\infty_c(\mathcal G_n)$, then $F'\varphi$ is Lipschitz and supported in $\{(X,y):X\in\supp F',\ y\in\supp w,\ \abs{y-x_i}\ge1.1r\text{ for all }i\}$. This set is compact and contained in $\mathcal G_{n+1}$: on $\supp F'$ the mutual distances of the old points are bounded below by a number larger than $r$, the new point lies in the compact set $\supp w\subset U$, and it has distance at least $1.1r>r$ from the old points. We have $F'\varphi\in H^1_0(\mathcal G_{n+1})$, and approximation of $F$ proves the claim. The Dirichlet condition at the walls for the new coordinate comes from the compact support of $w$ in $U$, and for the old coordinates from $F$. \end{proof} We now fix the insertion orbital. With $g_\Omega$ and $e_\Omega$ as in \eqref{eq:walls-g-omega}, let \begin{equation}\label{eq:walls-wK} w_K(x)=\frac{K^{-3/2}g_\Omega(x/K)}{\norm{g_\Omega}_2}. \end{equation} Then $w_K$ is real and smooth with compact support in $U$, $\norm{w_K}_2=1$, $\norm{w_K}_\infty^2\le K^{-3}(\abs\Omega/2)^{-1}=2/V$, and $\norm{\nabla w_K}_2^2=e_\Omega K^{-2}\le1$ for $K\ge K_\Omega$. This is the only place where a threshold depending on the shape of $\Omega$ enters. An orbital supported in a single ball inside $U$ would have a supremum controlled by the volume of that ball rather than by $V$; and a constant orbital, as in \cite{OpenAI2026BEC}, does not satisfy the Dirichlet condition at the walls. \begin{proposition}\label{prop:walls-increments} There exist a numerical constant $c_*>0$ and a threshold $D_1$, depending only on $\alpha_0$, such that the following holds whenever $K\ge K_\Omega$, $D\ge D_1$, $\alpha\in[\alpha_0/2,2\alpha_0]$, and $00$, and the constants $C_{\rm ins}\ge C'$ and $C_f\ge C$ of \cref{app:temperature} are chosen to dominate its constants. Since $j/V\le2D$, the logarithm is at most $\log(2C_fD)\le2\log D$ for $D\ge2C_f$. Since $T\le\tau_0/\log D$ with $\tau_0=\kappa_{\mathrm T}\alpha_0$, $\kappa_{\mathrm T}\le1/16$, and $\alpha\ge\alpha_0/2$, the last term is at most $2\kappa_{\mathrm T}\alpha_0\le\alpha_0/8\le\alpha/4$. Since $\alpha\ge1$, we obtain part (4) with $C_{\rm inc}=C_{\rm ins}+5/4$, as in \cref{app:temperature}. The constant $C_{\rm inc}$ is uniform in $W$, including its size, in $T$, and in the shape of $\Omega$, once $K\ge K_\Omega$. \end{proof} \section{Localization and diagonal tilting}\label{sec:walls-localization} \begin{lemma}\label{lem:walls-localization} Let $p,q\colon\R^3\to\R$ be Lipschitz with $p^2+q^2=1$, and let $O\subset\R^3$ be a bounded open set that contains the compact set $\supp p$. Let $0\le J\le N$, and let $\sigma$ be a symmetric trace-one state on $L^2(\mathcal G_J)$ of finite kinetic energy. Introduce an auxiliary label with two values by the isometry $f\mapsto(pf,qf)$ on each coordinate, and call the resulting state $\sigma_{\rm loc}$. If $P_j$ projects onto the subspace with exactly $j$ labels of the first type, replace $\sigma_{\rm loc}$ by $\sum_jP_j\sigma_{\rm loc}P_j$; this removes terms between distinct particle-count subspaces. We call the two label types colors. Let $(\varpi_j)$ be the distribution of $j$, and let $\sigma^{\mathrm{in}}_j$ and $\sigma^{\mathrm{out}}_j$ be the conditional marginal states of the two colors. Then $\sigma^{\mathrm{in}}_j$ is a symmetric state of finite kinetic energy on $L^2(\mathcal G_j(U\cap O))$, $\sigma^{\mathrm{out}}_j$ is one on $L^2(\mathcal G_{J-j})$ whose eigenvectors vanish unless every coordinate lies in $\supp q$, and for every bounded measurable $W\ge0$, \begin{equation}\label{eq:walls-two-color} \mathcal F^{(-W)}_J(\sigma)+\E_\sigma n_{\abs{\nabla p}^2+\abs{\nabla q}^2}\ge\sum_j\varpi_j\bigl[\mathcal F^{(-W)}_j(\sigma^{\mathrm{in}}_j)+\mathcal F^{(-W)}_{J-j}(\sigma^{\mathrm{out}}_j)\bigr]-T\mathcal S(\varpi), \end{equation} where $\mathcal F_j^{(-W)}(\sigma^{\mathrm{in}}_j)$ uses the Dirichlet form of $\mathcal G_j(U\cap O)$ and $\mathcal S(\varpi)=-\sum_j\varpi_j\log\varpi_j$. Moreover, \begin{equation}\label{eq:walls-color-entropy} \sum_jj\varpi_j=\E_\sigma n_{p^2},\qquad\mathcal S(\varpi)\le-\log(1-e^{-1})+\E_\sigma n_{p^2}. \end{equation} Conditional on the positions, the particles receive the first color independently with probabilities $p(x_i)^2$, and for every nonnegative measurable symmetric function $g$, $\sum_j\varpi_j\E_{\sigma^{\mathrm{in}}_j}g$ is the expectation of $g$ evaluated at the first-color points. \end{lemma} \begin{proof} This is \cite[Lemma~4.1]{OpenAI2026BEC}, the two-color localization of \cite[Lemma~4.1]{OpenAI2026B}; the geometric step localizes the many-particle kinetic energy by a partition of unity, see \cite[Section~3, Lemma~3.1]{DeuchertSeiringerYngvason2019}. Its proof applies after the stated substitutions once the following domain statements are verified. In the sector of color assignments with first-color label set $S$, the localized vector is $\prod_{i\in S}p(x_i)\prod_{i\notin S}q(x_i)F(X)$, where $F$ is an eigenvector of $\sigma$. By \cref{lem:walls-domains}(2), applied with $G=\mathcal G_J$ and $G'=\{X\in\mathcal G_J:x_i\in O\text{ for }i\in S\}$, this vector lies in $H^1_0(G')$. After grouping the coordinates by color, $G'$ is contained in $\mathcal G_{\abs S}(U\cap O)\times\mathcal G_{J-\abs S}$; dropping the exclusions between the colors only enlarges the domain. The wall condition for both colors comes from $F$, and the Dirichlet condition on $\partial O$ for the first color comes from the compact support of $p$ in $O$. This order matters when $\supp p$ meets $\partial U$: the mask multiplies approximants of $F$, which already vanish near the walls. By \cref{lem:walls-domains}(3), the conditional marginals have finite kinetic energy, and their kinetic energies add up to the localized kinetic energy. Their entropies are finite by \cref{lem:walls-entropy}. Every second-color factor contains $q(x_i)$, which gives the statement on the support of $\sigma^{\mathrm{out}}_j$. \end{proof} \begin{lemma}\label{lem:walls-tilt} Let $G$ be a bounded real symmetric Lipschitz function of $N$ positions, $g=e^G$, $M_G=\E_\mu e^{2G}$, and $\sigma_g=M_G^{-1}g\Gamma_Ng$. The state $\sigma_g$ has finite kinetic energy, and \[ \mathcal F_N(\sigma_g)-F_N\le\E_{\sigma_g}\abs{\nabla G}^2+T\bigl(2\E_{\sigma_g}G-\log M_G\bigr),\qquad\abs{\nabla G}^2=\sum_i\abs{\nabla_iG}^2. \] \end{lemma} \begin{proof} This is \cite[Lemma~4.2]{OpenAI2026BEC}, the diagonal tilt of \cite[Lemma~4.1]{OpenAI2026B}, and its proof applies after the stated substitutions. The only domain property it uses is that multiplication by the bounded Lipschitz symmetric functions $g$ and $g^2$ preserves $H^1_0(\mathcal G_N)\cap L^2_{\mathrm{sym}}$, which is \cref{lem:walls-domains}(2) with $G'=G$; this allows a weak eigenfunction $\psi$ of $H_N$ to be tested against $g^2\psi$. Energies and entropies are finite by \cref{lem:walls-domains}(5) and \cref{lem:walls-entropy}, and the relative-entropy step uses monotonicity under discarding a finite classical label \cite[Lemmas~1 and~2]{Lindblad1975}; see also \cite[Theorem~1]{MullerHermesReeb2017} for monotonicity under positive trace-preserving maps on separable Hilbert spaces. No regularity of $\partial U$ and no preservation of operator domains is used. \end{proof} \section{The local penalty and the density in a small field}\label{sec:walls-penalty} For a finite nonempty set $A\subset\Z^3$ and a configuration $X$ of $J\le N$ points, let \[ f_A(X)=\sum_{i=1}^Je^{-d(x_i,A)},\qquad k_A(X)=\#\{i:d(x_i,A)\le6,\ d_i(X)\le2s_0\}. \] Here $A$ is any finite set of cells. Its cells may meet $\R^3\setminus U$, or lie outside $U$. \begin{proposition}\label{prop:walls-penalty} There exist numerical constants $c,C>0$, in particular independent of $b_0$, such that the following holds for $K\ge K_\Omega$ and $D$ sufficiently large depending on $b_0$ and $\alpha_0$. For every finite nonempty $A\subset\Z^3$, every $0\le J\le N$, every measurable $W$ with $0\le W\le1$, and every symmetric trace-one state $\sigma$ on $L^2(\mathcal G_J)$ of finite kinetic energy, \begin{equation}\label{eq:walls-penalty} \mathcal F^{(-W)}_J(\sigma)-F^{(-W)}_J\ge\alpha\bigl[\E_\sigma(f_A+cb_0^{-3}k_A)-CD\abs A\bigr]. \end{equation} \end{proposition} \begin{proof} We follow the proof of \cite[Proposition~4.3]{OpenAI2026BEC}, which combines the exponential localization of \cite[Lemma~3.6]{OpenAI2026HS} with the two-color localization. We give it in full, because the inside states now live in $U\cap O_h$, the outside states in $U$, and the cell energies run over all cells $B_v$, $v\in\Z^3$. We take $D$ so large that $r0$, and $c_{\rm sh}>0$ such that the following holds for $\alpha_0\ge\alpha_0^*$, $0<\ell\le\ell^*$, $K\ge K_\Omega$, and $D$ sufficiently large depending on $\alpha_0$. For every finite set $A\subset\Z^3$ of cells whose closures lie in $U$, in particular for every finite set of interior cells, \[ \mu\{n_v<\ell D\text{ for every }v\in A\}\le e^{-c_{\rm sh}D\abs A}. \] \end{proposition} \usedin{\cref{sec:walls-completion}.} \begin{proof} We follow the proof of \cite[Proposition~4.5]{OpenAI2026BEC}; the filling construction is now carried out inside $U$. Let $A$ be nonempty. Since the cells of $A$ are disjoint unit cubes in $U$, we have $\abs A\le V$. For $x\in\R^3$ let $\abs x_\infty$ be the maximum norm. Define $p$ in each cell $v\in A$ by $p(x)=\cos\bigl((\pi/2)\min\{\max\{12(\abs{x-v}_\infty-1/4),0\},1\}\bigr)$, so that $p=1$ where $\abs{x-v}_\infty\le1/4$ and $p=0$ where $\abs{x-v}_\infty\ge1/3$, let $p=0$ outside the cells of $A$, and let $q=(1-p^2)^{1/2}$, the corresponding sine. Then $\abs{\nabla p}^2+\abs{\nabla q}^2\le C$. Let $O$ be the union of the open cubes $\{\abs{x-v}_\infty<1/2\}$, $v\in A$; it is a bounded open subset of $U$ containing $\supp p$. Let $\phi=\phi_0^2$, where $\phi_0$ is smooth, $0\le\phi_0\le1$, $\phi_0=1$ where $\abs{x-v}_\infty\le1/3$ for some $v\in A$, and $\supp\phi_0\subset O$. Then $\abs{\nabla\phi}^2\le C\phi$ and \begin{equation}\label{eq:walls-phi} p^2+\abs{\nabla p}^2+\abs{\nabla q}^2\le C\phi. \end{equation} \emph{The filling vector.} Fix a real smooth function $g_0$ supported in $(-1/8,1/8)^3$ with $\norm{g_0}_2=1$, and let $w=\abs A^{-1/2}\sum_{v\in A}g_0(\cdot-v)$. Then $w$ is smooth with compact support in $U$, $\norm w_2=1$, $\norm w_\infty^2\le C/\abs A$, and $\norm{\nabla w}_2^2\le C$. Let $\varepsilon_{\rm f}\in(0,1)$ be fixed below, and let $k=\lfloor\varepsilon_{\rm f}D\abs A\rfloor$. We apply \cref{lem:walls-insertion} with this $w$ successively $k$ times, starting from the zero-particle vector $1$. At every step, at most $k$ old points are present, and the smallness condition holds because $C\norm w_\infty^2kr^3\le C\varepsilon_{\rm f}Dr^3=C\varepsilon_{\rm f}\alpha^3/D^2\le1/4$ for large $D$. Since $kr/\abs A\le\varepsilon_{\rm f}\alpha$, the result is a normalized labeled vector $\Psi\in H^1_0(\mathcal G_k)$ with \begin{equation}\label{eq:walls-fill-cost} q_k[\Psi]\le Ck(1+rk/\abs A)\le Ck(1+\varepsilon_{\rm f}\alpha), \end{equation} which vanishes unless every coordinate lies in $\supp w$. Let $\Upsilon_k=\bigl((k!)^{-1}\sum_\pi\abs{\Psi\circ\pi}^2\bigr)^{1/2}$. As in the proof of \cref{prop:walls-increments}(1), $\Upsilon_k$ is a symmetric normalized vector in $H^1_0(\mathcal G_k)$ with $q_k[\Upsilon_k]\le q_k[\Psi]$, and it vanishes unless every coordinate lies in $\supp w$. \emph{The filling map.} The support of $q$ lies where $\abs{x-v}_\infty\ge1/4$ for every $v\in A$, and $\supp w$ lies where $\abs{x-v}_\infty\le1/8$ for some $v\in A$; the two sets have distance at least $1/8$. Let $\sigma$ be a symmetric state of finite kinetic energy on $L^2(\mathcal G_N)$, and let $\sigma^{\mathrm{out}}_j$ be the outside states of \cref{lem:walls-localization} for these masks and $J=N$. For a symmetric $(N-j)$-particle form vector $f$ that vanishes unless every coordinate lies in $\supp q$, let \[ \mathcal U_jf=\binom{N-j+k}k^{1/2}P_{\mathrm{sym}}(f\otimes\Upsilon_k). \] The terms in $P_{\mathrm{sym}}(f\otimes\Upsilon_k)$ that correspond to different choices of the $k$ coordinates in $\supp w$ have disjoint supports. Then $\mathcal U_j$ is an isometry, and the kinetic energy of $\mathcal U_jf$ is $q_{N-j}[f]\norm{\Upsilon_k}^2+q_k[\Upsilon_k]\norm f^2$. The vector $\mathcal U_jf$ is admissible. Indeed, $f\otimes\Upsilon_k\in H^1_0(\mathcal G_{N-j}\times\mathcal G_k)$ by \cref{lem:walls-domains}(2). Let $\theta_1,\theta_2$ be Lipschitz functions on $\R^3$ with $\theta_1=1$ on $\supp q$, $\theta_1=0$ where $\abs{x-v}_\infty\le3/16$ for some $v\in A$, $\theta_2=1$ on $\supp w$, and $\theta_2=0$ where $\abs{x-v}_\infty\ge5/32$ for every $v\in A$. Then $f\otimes\Upsilon_k$ is unchanged by multiplication with $m=\theta_1^{\otimes(N-j)}\otimes\theta_2^{\otimes k}$, and on $\supp m$ every old point has distance at least $1/32>r$ from every new point. Then $\supp m\cap(\mathcal G_{N-j}\times\mathcal G_k)\subset\mathcal G_{N-j+k}$, and by \cref{lem:walls-domains}(2), we have $f\otimes\Upsilon_k\in H^1_0(\mathcal G_{N-j+k})$. The walls are respected because both factors vanish outside $U$, and the new exclusions because of the separation of supports. The isometry preserves the nonzero eigenvalues of $\sigma^{\mathrm{out}}_j$, hence its entropy. The Gibbs variational principle and \eqref{eq:walls-fill-cost} give \[ \mathcal F_{N-j}(\sigma^{\mathrm{out}}_j)\ge F_{N-j+k}-Ck(1+\varepsilon_{\rm f}\alpha). \] Since $k\le\varepsilon_{\rm f}D\abs A\le\varepsilon_{\rm f}N$, all particle numbers below are at most $2N$. By \cref{prop:walls-increments}(3) at $N+k$ and then part (4) applied $j$ times, \begin{equation}\label{eq:walls-outside-fill} \mathcal F_{N-j}(\sigma^{\mathrm{out}}_j)\ge F_N+c_*\alpha k-C_{\rm inc}\alpha j-Ck(1+\varepsilon_{\rm f}\alpha). \end{equation} The constant $C$ in \eqref{eq:walls-outside-fill} depends only on $g_0$, which was fixed before $\alpha_0$. Let $\alpha_0^*=8C/c_*$ and $\varepsilon_{\rm f}=c_*/(4C)$. For $\alpha_0\ge\alpha_0^*$ and $\alpha\ge\alpha_0/2$, we have $C(1+\varepsilon_{\rm f}\alpha)\le c_*\alpha/2$. For $D\ge2/\varepsilon_{\rm f}$, we have $k\ge\varepsilon_{\rm f}D\abs A/2$. Then \eqref{eq:walls-outside-fill} implies that $\mathcal F_{N-j}(\sigma^{\mathrm{out}}_j)\ge F_N+(c_*\varepsilon_{\rm f}/4)\alpha D\abs A-C_{\rm inc}\alpha j$. \emph{The inside states.} The state $\sigma^{\mathrm{in}}_j$ lives on $\mathcal G_j(U\cap O)$, and $\abs O=\abs A$. By \cref{lem:walls-entropy} with the full kinetic energy, $\mathcal F_j(\sigma^{\mathrm{in}}_j)\ge-Tj-CT^{5/2}\abs A$. Inserting both bounds into \eqref{eq:walls-two-color}, and using \eqref{eq:walls-color-entropy} and \eqref{eq:walls-phi}, we conclude that for every symmetric state $\sigma$ of finite kinetic energy \begin{align*} \mathcal F_N(\sigma)-F_N&\ge\frac{c_*\varepsilon_{\rm f}}4\alpha D\abs A-C\abs A-(C_{\rm inc}\alpha+2T)\E_\sigma n_{p^2}-\E_\sigma n_{\abs{\nabla p}^2+\abs{\nabla q}^2}\\ &\ge c_1\alpha D\abs A-C_2\alpha\E_\sigma n_\phi, \end{align*} with $c_1=c_*\varepsilon_{\rm f}/8$ and a numerical $C_2$, once $D$ is large enough to absorb $C\abs A$. This is a free-energy inequality for arbitrary states; no density profile has been assumed. \emph{The tilt.} For $0\le\lambda\le1$, let $G=-\lambda n_\phi$, a bounded Lipschitz symmetric function, and $L(\lambda)=-\log\E_\mu e^{-2\lambda n_\phi}$. By \cref{lem:walls-tilt}, with $\abs{\nabla G}^2\le C\lambda^2n_\phi$ from $\abs{\nabla\phi}^2\le C\phi$ and with $2\E_{\sigma_g}G\le0$, we have $\mathcal F_N(\sigma_g)-F_N\le C\E_{\sigma_g}n_\phi+TL(\lambda)$. Comparing with the previous display for $\sigma=\sigma_g$, and using that the position law of $\sigma_g$ is $e^{2G}\mu/M_G$, we find \[ C_3\alpha L'(\lambda)+TL(\lambda)\ge c_1\alpha D\abs A,\qquad L'(\lambda)=2\E_{\sigma_g}n_\phi, \] with a numerical $C_3$. The differentiation is justified because $0\le n_\phi\le N$. Let $a_*=\min\{c_1/2,c_1/(2C_3)\}$. If $L(1)0$ there exists a numerical $b^*(\theta_0)\in(0,1)$ such that, for $00$ is numerical. \end{proposition} \usedin{\cref{sec:walls-completion}.} \begin{proof} The proof of \cite[Proposition~4.6]{OpenAI2026BEC} applies after the stated substitutions with Euclidean distances, \cref{prop:walls-penalty} in place of \cite[Proposition~4.3]{OpenAI2026BEC}, and \cref{lem:walls-tilt} in place of \cite[Lemma~4.2]{OpenAI2026BEC}. It tilts the Gibbs state by $\lambda m_A$, $0\le\lambda\le s_0$, where $m_A=\sum_iu_{\rm d}(d(x_i,A))u_{\rm n}(d_i/s_0)$ for squared Lipschitz cutoffs $u_{\rm d}$, $u_{\rm n}$ equal to one on $[0,5]$ and $[0,1]$, and zero on $[6,\infty)$ and $[2,\infty)$, respectively. Then $0\le m_A\le k_A$ and $\sum_{v\in A}N^{\mathrm{cl}}_v\le C_0m_A$, where $C_0$ bounds the number of lattice sites within distance $5$ of a point. We note the two points where the domain could have entered. First, the bound $\abs{\nabla m_A}^2\le Cs_0^{-2}m_A$ rests on the fact that a point is the nearest neighbor of a numerically bounded number of other points: if $x_k$ is the nearest neighbor of $x_i$ and of $x_j$, then $\abs{x_i-x_j}$ is the largest side of the triangle, so the angle at $x_k$ is at least $\pi/3$. This is Euclidean geometry of the physical points, and restricting them to $U$ changes none of it. Second, $\lambda m_A$ is a bounded Lipschitz symmetric function. Then \cref{lem:walls-tilt} applies. The temperature enters only through the term $2T\lambda\le1$ in the upper bound on the tilted free energy, since $T\le1$ and $\lambda\le s_0<1/2$. The conditions on $b_0$ in the proof, namely that $c\alpha b_0^{-3}$ dominate the constant in the upper bound on the tilted free energy and that $Cb_0^3\le\theta_0/C_0$, where $C_0$ bounds the number of points of $\Z^3$ within distance $5$ of a point, involve only numerical constants and are monotone in $b_0$. This gives $b^*(\theta_0)$. \end{proof} \section{The slab and free cutoff bounds}\label{sec:walls-slab} \begin{proposition}\label{prop:walls-purification} Let $0<\delta<\beta$. Then \eqref{eq:walls-slab} defines a probability measure. At every fixed time $s\in[0,\delta]$, the position marginal of $X(s)$ is $\mu$. The joint law of $X(t)$ and $W'$ has density $\Phi(X,W')^2$, where $\Phi=Z_N^{-1/2}K^{\mathrm s}_{\beta/2}$ is nonnegative, symmetric separately in both arguments, and has squared integral one. The marginal of the path family $\omega$ is $Z_N^{-1}K^{\mathrm s}_{\beta-\delta}(Z^\circ,Y)\ind_{\mathcal E_\delta}(\omega)\nu_N(\dd\omega)$, the law \eqref{eq:thermal-path-marginal} for the container. \end{proposition} \begin{proof} This is \cite[Proposition~5.1]{OpenAI2026BEC} without its occupation identity, with the killed kernels of $\mathcal G_N$, which include the walls. The proof there uses only the Feynman--Kac representation of the labeled killed kernel, which is \cref{prop:id-fk}(1) for the container; the facts that $P_{\mathrm{sym}}$ commutes with the labeled semigroup and $P_{\mathrm{sym}}^2=P_{\mathrm{sym}}$; and the semigroup and trace identities, which are \cref{lem:id-ck}. These hold for every open configuration domain, and the proof applies after the stated substitutions. The last statement is the first step of the proof of \cref{lem:thermal-slab}. \end{proof} \usedin{\cref{sec:walls-completion}.} \begin{lemma}\label{lem:walls-cutoffs} Fix $t\in(0,1]$. For every $q>0$, a Euclidean Brownian motion with generator $\Delta$ on $[0,2t]$ fails to be regular with probability at most $C_{q,t}D^{-q}$, uniformly in its starting point and in $\alpha\in[\alpha_0/2,2\alpha_0]$. The same holds for each of the tests restricted to one half of $[0,2t]$, namely its endpoint window, the maximum over its mesh intervals, and the average over its mesh intervals compared with $C_{\rm reg}$. The same assertions hold for a Euclidean Brownian bridge of duration $2t$, and for two independent Euclidean bridges of duration $t$ joined at a prescribed midpoint, when the prescribed points lie in a fixed bounded region. The constant $C_{\rm reg}$ can be chosen as one numerical constant for all these laws, and any larger value is allowed. If the three prescribed points lie in $B(v,3)$, the probability that the joined path leaves $\overline B(v,10)$ is at most $Ce^{-c/t}$ for $01\},\qquad E_{\rm irr}=\{\omega\text{ is not regular}\}, \] and for one of them, $E$, and $\lambda>0$, the marked moment \begin{equation}\label{eq:walls-marked-moment} \mathcal M=\E_{\PP^{\mathrm{pur}}}\exp\Bigl(\lambda\sum_{i=1}^NJ_A(\omega_i)\ind_E(\omega_i)\Bigr)=\E_{\PP^{\mathrm{pur}}}\prod_{i=1}^N\bigl[1+(e^{\lambda J_A(\omega_i)}-1)\ind_E(\omega_i)\bigr]. \end{equation} As in \cite[Section~5.3]{OpenAI2026BEC}, on $[0,t]$ mark the first exit from the unit ball about $\omega(0)$, then the first exit from the unit ball about that exit point, and so on up to time $t$; on $[t,\delta]$ make the same construction backward from $\omega(\delta)$. Let $n(\omega)$ be the total number of completed exits. Between consecutive marks the path stays in a unit ball, so on $\{n(\omega)=n\}$, \begin{equation}\label{eq:walls-bins} J_A(\omega)\le C(n+1),\qquad\abs{\omega(\delta)-\omega(0)}\le n+2, \end{equation} and if $J_A(\omega)>0$, both endpoints lie within distance $n+15$ of $A$, and the closed unit neighborhood of the segment $[\omega(0),\omega(\delta)]$ lies in $\{x:d(x,A)<32(n+1)\}$. Let \[ O_n=\{x\in\R^3:d(x,A)<32(n+1)+2\}, \] and let $p_n\colon\R^3\to[0,1]$ be Lipschitz with $p_n=1$ where $d(x,A)\le32(n+1)$, $p_n=0$ where $d(x,A)\ge32(n+1)+1$, and $\abs{\nabla p_n}\le1$. Then $O_n$ is a bounded open set containing $\supp p_n$, $\abs{O_n}\le C(n+1)^3\abs A$, and $O_n$ is covered by at most $C(n+1)^3\abs A$ cells. The sets $O_n$ are auxiliary: they need not be contained in $U$. Let \[ \mathcal K_n(y,z)=\int\bigl(e^{\lambda J_A(\omega)}-1\bigr)\ind_E(\omega)\ind\{n(\omega)=n\}\,\mathbb W^\delta_{y,z}(\dd\omega) \] be the free Euclidean kernel of duration $\delta$ with the marking weight and the restriction $n(\omega)=n$, without walls or exclusion. \begin{lemma}\label{lem:walls-marked} For $y,z\in\R^3$ and $n\ge0$, \[ \mathcal K_n(y,z)\le l_n\,p_n(y)p_n(z)\,[e^{3\delta\Delta_{O_n}}](y,z),\qquad l_n=C\lambda(n+1)e^{C\lambda(n+1)}e^{-cn^2/t}\epsilon_E^{1/3}, \] where $\Delta_{O_n}$ is the Dirichlet Laplacian of $O_n$, and $\epsilon_E=1$, $Ce^{-c/t}$, $C_tD^{-12}$ for $E=E_{\rm all},E_{\rm diam},E_{\rm irr}$. The constants for $E_{\rm all}$ and $E_{\rm diam}$ are uniform in $0\delta$, we have $\beta\mathfrak g_n/2>(m_0+n)/2\ge1$. The direct sum below is an auxiliary comparison space. Its first summand retains a physical particle, while the additional summand indexed by $n$ replaces a marked path by the heat-kernel bound of \cref{lem:walls-marked}. The factor $\varkappa_n$ and the energy shift $\mathfrak g_n$ satisfy $\varkappa_ne^{-\delta\mathfrak g_n}=l_n$. The one-particle space is \[ \mathfrak h_e=L^2(U)\oplus\bigoplus_{n=0}^{n_*}L^2(O_n). \] The first summand represents unmarked physical particles, and the others represent auxiliary types indexed by the exit count $n$. For a color assignment $\mathsf c=(\mathsf c_1,\dots,\mathsf c_N)\in\{\mathrm o,0,\dots,n_*\}^N$, let $S(\mathsf c)=\{i:\mathsf c_i\neq\mathrm o\}$ and \[ G_{\mathsf c}=\bigl\{X\in(\R^3)^N:(x_i)_{i\notin S(\mathsf c)}\in\mathcal G_{N-\abs{S(\mathsf c)}},\ x_i\in O_{\mathsf c_i}\text{ for }i\in S(\mathsf c)\bigr\}. \] On $\mathcal H_e^{\mathrm{lab}}=\bigoplus_{\mathsf c}L^2(G_{\mathsf c})$, let $\mathbb H^{\mathrm{lab}}$ act on $L^2(G_{\mathsf c})$ as the operator of the closed form \[ \sum_{i\notin S(\mathsf c)}\int\abs{\nabla_if}^2+\sum_{i\in S(\mathsf c)}\int\bigl(3\abs{\nabla_if}^2+\mathfrak g_{\mathsf c_i}\abs f^2\bigr)\qquad\text{on }H^1_0(G_{\mathsf c}). \] Unmarked physical particles remain in $U$ with their mutual hard-core exclusions. A particle marked by $n$ instead moves in $O_n$ with a Dirichlet condition on $\partial O_n$; it is subject neither to the physical walls $\partial U$ nor to pair exclusions. Simultaneous permutations of positions and colors commute with $\mathbb H^{\mathrm{lab}}$. Let $\mathcal H_e$ be the invariant subspace and $\mathbb H$ the restriction of $\mathbb H^{\mathrm{lab}}$ to it. For $f\in L^2(U)$, let $v_ef=(f,(\varkappa_n^{1/2}p_nf)_{n\le n_*})$, where $p_nf$ is regarded as an element of $L^2(O_n)$; let $V_{\mathrm{lab}}=v_e^{\otimes N}$ on $L^2(\mathcal G_N)$, so that its component in $L^2(G_{\mathsf c})$ is $\prod_{i\in S(\mathsf c)}\varkappa_{\mathsf c_i}^{1/2}p_{\mathsf c_i}(x_i)F(X)$, and let $V_e$ be its restriction to symmetric functions. Set $\mathcal T_{\mathrm{lab}}=V_{\mathrm{lab}}^*e^{-\delta\mathbb H^{\mathrm{lab}}}V_{\mathrm{lab}}$ and $\mathcal T=V_e^*e^{-\delta\mathbb H}V_e$. \begin{lemma}\label{lem:walls-colormap} \begin{enumerate} \item The map $V_{\mathrm{lab}}$ is bounded, maps $H^1_0(\mathcal G_N)$ into $\bigoplus_{\mathsf c}H^1_0(G_{\mathsf c})$, and intertwines permutations of the coordinates with simultaneous permutations of positions and colors. \item We have \begin{equation}\label{eq:walls-holder} \mathcal M_{n_*}\le\frac{\tr_{\mathrm{sym}}\bigl(e^{-(\beta-\delta)H_N}\mathcal T\bigr)}{Z_N}\le\Bigl(\frac{\tr\mathcal T^{\beta/\delta}}{Z_N}\Bigr)^{\delta/\beta}. \end{equation} \end{enumerate} \end{lemma} \begin{proof} For part (1), $\norm{V_{\mathrm{lab}}F}^2=\int\prod_i(1+\sum_n\varkappa_np_n(x_i)^2)\abs F^2\le(1+\sum_n\varkappa_n)^N\norm F^2$. Let $F\in H^1_0(\mathcal G_N)$ and let $\mathsf c$ be a color assignment with $S=S(\mathsf c)$. We apply \cref{lem:walls-domains}(2) with $G=\mathcal G_N$, the bounded Lipschitz function $m(X)=\prod_{i\in S}\varkappa_{\mathsf c_i}^{1/2}p_{\mathsf c_i}(x_i)$, and $G'=\{X\in\mathcal G_N:x_i\in O_{\mathsf c_i}\text{ for }i\in S\}$. Since $\supp p_n\subset O_n$, we have $\supp m\cap\mathcal G_N\subset G'$, and $mF\in H^1_0(G')$. Since $G'\subset G_{\mathsf c}$, extension by zero gives $mF\in H^1_0(G_{\mathsf c})$. The coordinates of the unmarked physical particles keep the walls and their mutual exclusions from $F$; the marked coordinates acquire the Dirichlet condition on $\partial O_{\mathsf c_i}$ from the compact support of $p_{\mathsf c_i}$. The image vanishes at further configurations, for example where a marked coordinate lies outside $U$; this is allowed, since the target domain does not require it. Marked colors can lose the wall condition while the input of the map remains admissible. The intertwining property is immediate from the coordinatewise definition. For part (2), expand the product in $\mathcal M_{n_*}$. The term indexed by a set $S$ of marked labels is \[ \frac1{Z_N}\int K^{\mathrm s}_{\beta-\delta}(Z^\circ,Y)\ind_{\mathcal E_\delta}(\omega)\prod_{i\in S}\bigl(e^{\lambda J_A(\omega_i)}-1\bigr)\ind_E(\omega_i)\ind\{n(\omega_i)\le n_*\}\,\nu_N(\dd\omega), \] after integrating out $W'$ by \cref{prop:walls-purification}. For every $i\in S$, we drop the wall killing of $\omega_i$ and every exclusion between $\omega_i$ and another path: $\ind_{\mathcal E_\delta}(\omega)$ is at most the indicator that $(\omega_i)_{i\notin S}$ stays in $\mathcal G_{N-\abs S}$. The unmarked physical paths keep all their walls and mutual exclusions. All factors, including $K^{\mathrm s}_{\beta-\delta}$, are nonnegative. The integral does not decrease. Integrating the paths with fixed endpoints, \cref{prop:id-fk}(1) for the domain $\mathcal G_{N-\abs S}$ and the definition of $\mathcal K_n$ show that the term is at most \[ \frac1{Z_N}\int K^{\mathrm s}_{\beta-\delta}(Z^\circ,Y)\,K^{(S^c)}_\delta(Y_{S^c},Z^\circ_{S^c})\prod_{i\in S}\sum_{n_i\le n_*}\mathcal K_{n_i}(y_i,z^\circ_i)\dd Y\dd Z^\circ, \] where $K^{(S^c)}_\delta$ is the labeled killed kernel of $\mathcal G_{N-\abs S}$. Let $V_{\mathrm{lab},\mathsf c}$ be the component of $V_{\mathrm{lab}}$ in $L^2(G_{\mathsf c})$. By \cref{lem:walls-marked} and $l_n=\varkappa_ne^{-\delta\mathfrak g_n}$, the integrand is at most $K^{\mathrm s}_{\beta-\delta}(Z^\circ,Y)$ times the sum, over the assignments $\mathsf c$ with $S(\mathsf c)=S$, of the kernel of $V_{\mathrm{lab},\mathsf c}^*e^{-\delta\mathbb H^{\mathrm{lab}}}V_{\mathrm{lab},\mathsf c}$, which equals \[ K^{(S^c)}_\delta(Y_{S^c},Z^\circ_{S^c})\prod_{i\in S}\varkappa_{\mathsf c_i}p_{\mathsf c_i}(y_i)p_{\mathsf c_i}(z^\circ_i)e^{-\delta\mathfrak g_{\mathsf c_i}}[e^{3\delta\Delta_{O_{\mathsf c_i}}}](y_i,z^\circ_i). \] Every choice of marked labels and exit counts appears exactly once in the direct sum over $\mathsf c$, and summing over $S$ bounds the tested slab kernel pointwise by the kernel of $\mathcal T_{\mathrm{lab}}$. This is a comparison of nonnegative kernels; it does not compare normalized laws. The rest of the proof is that of \cite[Section~5.4]{OpenAI2026BEC}: both $e^{-(\beta-\delta)H^{\mathrm{lab}}_N}$ and $\mathcal T_{\mathrm{lab}}$ commute with $P_{\mathrm{sym}}=(N!)^{-1}\sum_\pi U_\pi$, and the kernel of $e^{-(\beta-\delta)H^{\mathrm{lab}}_N}P_{\mathrm{sym}}$ is $K^{\mathrm s}_{\beta-\delta}$. Then \[ \int K^{\mathrm s}_{\beta-\delta}(Z^\circ,Y)\,K_{\mathcal T_{\mathrm{lab}}}(Y,Z^\circ)\dd Y\dd Z^\circ=\tr\bigl(e^{-(\beta-\delta)H^{\mathrm{lab}}_N}P_{\mathrm{sym}}\mathcal T_{\mathrm{lab}}\bigr)=\tr_{\mathrm{sym}}\bigl(e^{-(\beta-\delta)H_N}\mathcal T\bigr). \] The only permutation normalization is the one in $P_{\mathrm{sym}}$; no factorial enters the color expansion. This proves the first inequality of \eqref{eq:walls-holder}, and the second is H\"older's inequality for Schatten norms \cite[Theorem~2.8]{Simon2005b} with exponents $\beta/(\beta-\delta)$ and $\beta/\delta$. All traces are finite by \cref{lem:walls-domains}(5). \end{proof} \begin{remark}\label{rem:walls-survival} The marked particles must be allowed to leave $U$, because there is no uniform lower bound on the probability that a short bridge near a wall survives. For example, a Brownian bridge with generator $\Delta$ and duration $s$, from a point at distance $h$ from a plane back to the same point, stays on one side of the plane with probability $1-e^{-h^2/s}$, which tends to zero as $h\to0$. The lower bound in \cref{lem:walls-marked} concerns a tube contained in the auxiliary set $O_n$, which is not contained in $U$. \end{remark} \section{A trace comparison and the path moment bound}\label{sec:walls-moment} Let $\Sigma_{\mathrm c}(x)=\sum_{n\le n_*}\varkappa_np_n(x)^2$, $\Theta(x)=(1/2)\log(1+\Sigma_{\mathrm c}(x))$, and $\widehat\Theta(X)=\sum_i\Theta(x_i)$, with the same multiplier in every color. The map $\widetilde V_e=e^{-\widehat\Theta}V_e$ is an isometry, because the squared pointwise norm of its one-coordinate color vector is one. With $\mathcal Y=e^{\widehat\Theta}e^{-\delta\mathbb H}e^{\widehat\Theta}$, we have $\mathcal T=\widetilde V_e^*\mathcal Y\widetilde V_e$. The min--max principle for compressions of positive compact operators gives $\tr\mathcal T^q\le\tr\mathcal Y^q$ for every $q>0$. \begin{lemma}\label{lem:walls-sandwich} Let $W_0=2\Theta/\delta+3\abs{\nabla\Theta}^2$ and $\widetilde{\mathbb H}=\mathbb H-\sum_iW_0(x_i)$. For every real $q>0$, $\tr\mathcal Y^q\le\tr e^{-q\delta\widetilde{\mathbb H}}$. \end{lemma} \begin{proof} This is \cite[Lemma~5.6]{OpenAI2026BEC}, the trace estimate for $e^{\widehat\Theta}e^{-\delta\mathbb H}e^{\widehat\Theta}$ with a Lipschitz multiplier of \cite[Lemma~6.2]{OpenAI2026B}. Its proof uses the following properties of the colored operator, which we verify for the present domains. (i) There are finitely many color assignments, and in each, $\mathbb H^{\mathrm{lab}}$ is given by a closed form on $H^1_0(G_{\mathsf c})$ with kinetic coefficients $1$ or $3$ and nonnegative constant energy shifts $\mathfrak g_n$. Since $U$ and the sets $O_n$ are bounded, every $G_{\mathsf c}$ is a bounded open set, and \cref{lem:walls-domains}(5) shows that $\mathbb H$ and $\widetilde{\mathbb H}$ have compact resolvent and finite heat traces; $W_0$ is bounded. (ii) Multiplication by $e^{u\widehat\Theta}$, $\abs u\le1$, preserves each $H^1_0(G_{\mathsf c})$ and the simultaneous permutation symmetry, by \cref{lem:walls-domains}(2) with $G'=G$; this retains the walls and exclusions of the unmarked physical particles and the boundaries $\partial O_n$ of the marked colors. (iii) The product rule holds almost everywhere. With these properties, the proof in \cite{OpenAI2026BEC} applies after the stated substitutions. It pairs form-domain vectors with operator-domain vectors only, so no preservation of operator domains by the multipliers is needed, and its final integration against $q^2e^{qh}\dd h$ gives the bound for every real $q>0$, including $q=\beta/\delta$. \end{proof} \begin{proposition}\label{prop:walls-moment} Let $E$ be one of $E_{\rm all}$, $E_{\rm diam}$, $E_{\rm irr}$, and let $l_n,\varkappa_n,\mathfrak g_n$ be as in \cref{lem:walls-marked} and \eqref{eq:walls-gaps}. There is a numerical $C_0$ such that, if \begin{equation}\label{eq:walls-smallness} \frac{C_0}\delta\sum_{n\ge0}\varkappa_n\le1, \end{equation} then, under the hypotheses of \cref{prop:walls-penalty} and for every finite nonempty $A\subset\Z^3$, \begin{align*} \log\E_{\PP^{\mathrm{pur}}}\exp\Bigl(\lambda\sum_{i=1}^NJ_A(\omega_i)\ind_E(\omega_i)\Bigr)&\le CD\abs A\sum_{n\ge0}(n+1)^3\varkappa_n\\ &\quad+C\delta T^{5/2}\abs A\sum_{n\ge0}(n+1)^3e^{-\beta\mathfrak g_n/2}. \end{align*} The constants do not depend on $n_*$, $K$, $\Omega$, $D$, $T$, or $00$. By \eqref{eq:walls-eta-sum}, $\sum_n\varkappa_n\le C\lambda=Cc't$ with $C$ independent of $t$. Then \eqref{eq:walls-smallness} holds for all $t\in(0,1]$ once $c'$ is small, and the first term in \cref{prop:walls-moment} is at most $C_4\lambda D\abs{A'}$ with $C_4$ independent of $t$. If every cell of $A'$ has more than $H_1D$ visitors, then $\sum_iJ_{A'}(\omega_i)>H_1D\abs{A'}$. By the exponential Markov inequality and \cref{prop:walls-moment}, we have the bound \[ \exp\bigl\{[-\lambda(H_1-C_4)D+r^{\rm wall}_D]\abs{A'}\bigr\}. \] We choose $H_1=C_4+1$ now. For every subsequently fixed $t$, the base $\exp(-c'tD+r^{\rm wall}_D)$ is at most $p'$ for $D$ large. Reducing $t$ does not affect the choice of $H_1$. \emph{Visitors of large diameter.} We set $E=E_{\rm diam}$ and $\lambda=1$. Then $\epsilon_E^{1/3}\le Ce^{-c/t}$ and $\sum_n(n+1)^3\varkappa_n\le Ce^{-c/t}$. Fix $t$ so small that \eqref{eq:walls-smallness} holds and the first coefficient in \cref{prop:walls-moment} is at most $\theta_0D/2$ per cell. If every cell of $A'$ is visited by at least $\theta_0D$ paths of diameter greater than one, then $\sum_iJ_{A'}(\omega_i)\ind_E(\omega_i)\ge\theta_0D\abs{A'}$, and the probability is at most $\exp[-(\theta_0D/2-r^{\rm wall}_D)\abs{A'}]$. This reduction of $t$ is permitted, and any further reduction preserves the bound. \emph{Irregular visitors.} With $t$ now fixed, take $E=E_{\rm irr}$ and $\lambda=\log(2/p')$. Then $\epsilon_E^{1/3}=C_tD^{-4}$, and we have $\sum_n(n+1)^3\varkappa_n\le C_{\lambda,t}D^{-4}$, \eqref{eq:walls-smallness} holds for $D$ large, and the first term in \cref{prop:walls-moment} is at most $C_{\lambda,t}D^{-3}\abs{A'}$. If every cell of $A'$ is visited by an irregular path, then $\sum_iJ_{A'}(\omega_i)\ind_E(\omega_i)\ge\abs{A'}$, even if a single path visits many of these cells. The probability is at most $\exp[-(\lambda-C_{\lambda,t}D^{-3}-r^{\rm wall}_D)\abs{A'}]\le(p')^{\abs{A'}}$ for $D$ large. \emph{Order of choices.} The constants $\alpha_0,\ell,\theta_0,b_0,C_{\rm reg}$ are fixed first, then $\kappa_{\mathrm T}$ and $\tau_0=\kappa_{\mathrm T}\alpha_0$, then $C_{\rm inc}$ and $m_0$ of \cref{app:temperature}, and $c'$. Given $p_1$, we fix $p'$ with $6(p')^{1/6}\le p_1$, then $H_1$, then $t$, which may be reduced further for the requirement of \cref{sec:walls-bridges}, then $\lambda$ for the irregular test, and finally $D_0$, larger than all the thresholds above and than those of \cref{prop:walls-increments,prop:walls-penalty,prop:walls-shortage,prop:walls-close,prop:walls-bridges}. We also require \eqref{eq:thermal-scales}, $2s_0<1$, and $r<1/32$, which proves \cref{rem:walls-constants}(3). The shape of $\Omega$ enters only through $K\ge K_\Omega$, in \cref{prop:walls-increments}; $R_0$ enters only through the set of cells to which the theorem applies. No lower bound on $T$ other than positivity is used. \end{proof} \section{Euclidean bridge estimates and useful labels}\label{sec:walls-bridges} For $y,z\in\R^3$ and a cell $B$, let $P^B_{y,z}=\int_BP^u_{y,z}\dd u$, and let $P^0_{y,z}$ be the Euclidean Brownian bridge with generator $\Delta$ from $y$ to $z$ of duration $2t$. For $v\in\Z^3$, let $\mathcal C_v$ be the event that a path on $[0,2t]$ is regular and stays in $\overline B(v,10)$, and for a deterministic continuous path $c\colon[0,2t]\to\R^3$ let $\mathcal H(c)=\{\omega:\abs{\omega(s)-c(s)}\le r\text{ for some }s\in[0,2t]\}$. \begin{proposition}\label{prop:walls-bridges} Fix $t\in(0,1]$, and let $D$ be large enough that \eqref{eq:thermal-scales} holds. Then \cite[Lemmas~6.1, 6.2, and~6.3, Remark~6.4, and Corollary~6.5]{OpenAI2026BEC} hold for the Euclidean laws $P^u_{y,z}$, $P^B_{y,z}$, and $P^0_{y,z}$, with the same constants at every $v\in\Z^3$. The obstacle paths are deterministic Euclidean paths. In particular: \begin{enumerate} \item If $y\in B_v$, $\abs{z-v}\le3$, and $c$ is a deterministic regular path with $\abs{y-c(0)}>s_0$ and $\abs{z-c(2t)}>s_0$, then, for $P=P^B_{y,z}$ with any target cell $B$ of $v$, or $P=P^0_{y,z}$, \[ P(\mathcal C_v\cap\mathcal H(c))\le Ctr+C\frac{rD^{1/4}}{\sqrt{h_0}}\le C\alpha\bigl(tD^{-1}+D^{-1/4}\bigr). \] The midpoint is averaged as in \cref{rem:thermal-midpoint}; no uniform prescribed-midpoint estimate is used. \item Let $\mathfrak F$ be a collection of deterministic continuous paths on $[0,2t]$, at most $H_1D$ of which visit $B(v,12)$, every visiting path being regular. Let a list of $m\ge\ell D/2$ distinct paths $c_i\in\mathfrak F$ have endpoints $y_i=c_i(0)\in B_v$ and $z_i=c_i(2t)$ with $\abs{z_i-v}\le3$, such that $\abs{y_i-c(0)}>s_0$ and $\abs{z_i-c(2t)}>s_0$ for every $c\in\mathfrak F\setminus\{c_i\}$. For $t$ sufficiently small depending on $H_1$, and then $D$ sufficiently large depending on $t$, at least $9m/10$ of the listed labels have the following property: for every target cell $B$ of $v$, the set of $u\in B$ with $P^u_{y_i,z_i}(\mathcal C_v\cap\bigcap_{c\in\mathfrak F\setminus\{c_i\}}\mathcal H(c)^c)\ge9/10$ has volume at least $99/100$. \end{enumerate} The choices are uniform in $\mathfrak F$, in $v$, and in $\alpha\in[\alpha_0/2,2\alpha_0]$. \end{proposition} \usedin{\cref{thm:walls-main}(ii), and through its proof \cref{prop:slab-vulnerable}; part (1) is the Euclidean form of \cref{lem:thermal-single}, whose proof is adapted in \cref{lem:slab-insertion}.} \begin{proof} The statements of \cite[Section~6]{OpenAI2026BEC}, reproduced there from \cite[Section~5]{OpenAI2026HS}, concern free Euclidean Brownian bridges and deterministic obstacle paths. The torus enters their proofs only through the projection of the trial paths to $\Lambda_K$, and through the paragraph on lifts in the proof of \cite[Lemma~6.2]{OpenAI2026BEC}. In that paragraph, the lift of the obstacle near $v$ is selected for each mesh interval. In $\R^3$ we delete the projection and that paragraph; every other line applies unchanged, with \cref{lem:walls-cutoffs} in place of \cite[Lemma~5.3]{OpenAI2026BEC}. In the proof of \cite[Corollary~6.5]{OpenAI2026BEC}, a trial in $\overline B(v,10)$ can collide only with an obstacle that visits $B(v,12)$, since $r<1$. For each such obstacle, we first average the collision probability over the $m$ listed labels using \cite[Lemma~6.2]{OpenAI2026BEC}, obtaining $Ctr+CrD^{1/4}/(D\sqrt{h_0})$, omitting the obstacle's own label when present but retaining the divisor $m$. By the union bound over the at most $H_1D$ such obstacles, we bound the averaged failure probability by $Ce^{-c/t}+\varepsilon_D(t)+CH_1\alpha(t+D^{-1/4})$, with $\varepsilon_D(t)\to0$ as $D\to\infty$; this is the reason $t$ is chosen small depending on $H_1$. The container $U$ does not appear in any of these statements. \end{proof} \begin{proof}[Proof of \cref{thm:walls-main}(ii)] The proof of \cite[Lemma~7.1]{OpenAI2026BEC} applies after the stated substitutions at interior cells; we recall it. Let $v$ be an interior cell at which the sample satisfies the local conditions. Every path that starts in $B_v$ visits $B(v,12)$. At most $H_1D$ paths start in $B_v$. By item 1 of \cref{def:thermal-certificate}, at least $\ell D$ paths start in $B_v$; item 2 makes all of them regular. We discard the fewer than $\theta_0D$ of them with diameter greater than one. The remaining paths stay within distance $\sqrt3/2+1<2$ of $v$; in particular they stay in $\overline B(v,10)$ and end within distance $3$ of $v$. We discard also those whose initial point has another initial point within distance $s_0$, and those whose terminal point has another terminal point within distance $s_0$. These endpoints lie within distance $5$ of $v$, so by item 1 of \cref{def:thermal-certificate} each of the two losses is less than $\theta_0D$. The remaining labels are eligible, and their number is at least $(\ell-3\theta_0)D\ge\ell D/2$. We now apply \cref{prop:walls-bridges}(2) to the entire list of labels eligible at $v$, not only to the labels just counted, with $\mathfrak F=\{\omega_k:1\le k\le N\}$. Every eligible label has its initial point in $B_v$, its terminal point within distance $3$ of $v$, and both endpoints at distance greater than $s_0$ from the corresponding endpoints of all other paths. By item 2 of \cref{def:thermal-certificate}, at most $H_1D$ paths visit $B(v,12)$, and all of them are regular. For an eligible label $i$, the event in \cref{prop:walls-bridges}(2) is $C_i(F^{\rm all}_i)$. For $t$ small depending on $H_1$ and $D$ large depending on $t$, we conclude that at least nine tenths of the eligible labels are useful. Since $\overline B(v,10)\subset U$, a successful trial stays in $U$. The useful property needs no factor for survival inside the container. The requirement on $t$ is the further reduction of $t$ after $H_1$ is fixed, which is permitted by part (i). \end{proof} \chapter{Thermal estimates in a larger temperature window}\label[appendix]{app:temperature} In this appendix, we prove \cref{prop:thermal-certificates}: on the torus, the joint bound of \cite[Proposition~5.4]{OpenAI2026BEC} for failure of the local conditions holds uniformly for $00$ denote constants that may change from line to line; unless stated otherwise, they depend only on the spatial constants, and not on $K$, $N$, $D$, $\alpha$, $T$, or $t$. \paragraph{The window.} Let $\alpha_+=2\alpha_0$. We fix \begin{equation}\label{eq:temp-window} \kappa_{\mathrm T}=\frac1{16},\qquad\tau_0=\kappa_{\mathrm T}\alpha_0,\qquad00$ and every bounded measurable $W\ge0$, both on the torus and in every container with $V=\abs U$ and $K\ge K_\Omega$. Here $\log_+x=\max\{\log x,0\}$. On the torus, \eqref{eq:temp-inc-raw} is the bound obtained in the proof of \cite[Proposition~3.5]{OpenAI2026BEC} before the temperature is restricted (see the proof of \cref{prop:temp-free-energy}); in a container, it is the corresponding bound in the proof of \cref{prop:walls-increments}. Every threshold on $D$ below is at least $\max\{2C_f,e^{4\tau_0}\}$. Then $\log(2C_fD)\le2\log D$, and in the window \begin{gather} T\le\frac{\tau_0}{\log D}\le\frac14,\qquad\beta\ge4>\delta\quad\text{for every }\delta=2t\text{ with }t\in(0,1],\label{eq:temp-small}\\ T\log(2C_fD)\le2T\log D\le2\tau_0=\frac{\alpha_0}8\le\frac\alpha4.\label{eq:temp-log} \end{gather} We set \begin{equation}\label{eq:temp-constants} C_{\rm inc}=C_{\rm ins}+\frac54,\qquad m_0=\max\{2,4(C_{\rm inc}\alpha_++1)\}. \end{equation} For $j\le2N$ we have $j/V\le2D$, and \eqref{eq:temp-log} and $\alpha\ge1$ bound the right side of \eqref{eq:temp-inc-raw} by $C_{\rm inc}\alpha$ in the window. The constant $C_{\rm inc}$ is numerical, and $m_0$ depends only on $\alpha_0$. \paragraph{Order of choices.} The constants are chosen in the following order, and each depends only on those chosen before it: \begin{enumerate} \item the spatial constants $\alpha_0$, $\ell$, $\theta_0$, $b_0$, and $C_{\rm reg}$ of \cref{sec:thermal-setting}; \item $\kappa_{\mathrm T}=1/16$ and $\tau_0=\kappa_{\mathrm T}\alpha_0$; \item $C_{\rm inc}$, and then $m_0$, by \eqref{eq:temp-constants}; \item the prescribed base $p_1$, and a base $p'\in(0,1)$ for the six tests with $6(p')^{1/6}\le p_1$ (for $p_1\ge1$ there is nothing to prove); \item $H_1$; \item $\bar t\in(0,1]$, and then the time $t\in(0,\bar t]$, with $\delta=2t$; \item the fixed tilt $\lambda=\log(2/p')$ of the test for irregular visitors; \item the threshold $D_0=D_0(t)$; \item the volume threshold $K_1=100$ of \cref{lem:temp-volume}. \end{enumerate} In particular, $\kappa_{\mathrm T}$, $C_{\rm inc}$, and $m_0$ do not depend on $p_1$, $H_1$, or $t$. The thermal threshold $D_0(t)$ is distinct from the density threshold $D_{\min}$ of \cref{sec:stiff-parameters}, which is chosen later and also involves $M$ and the constants of the surgery. \section{Statements whose proofs hold at every temperature} The setting of \cite[Section~2]{OpenAI2026BEC} involves the temperature only through $\beta>0$: the killed heat kernels are nonnegative, heat traces are finite at every positive time, and the Gibbs variational principle holds for every $T>0$. In \cite{OpenAI2026BEC}, the standing range is $00$, or for every $\beta>\delta$, the table records the reason, and we use the statement without change. The other statements are extended in \cref{sec:temp-free-energy,sec:temp-spatial,sec:temp-moment,sec:temp-certificates}. The endpoint-window length $h_0=D^{-1}$, the mesh scales of \eqref{eq:thermal-mesh}, and the scale restrictions \eqref{eq:thermal-scales} are spatial; they are kept, and the midpoint stays averaged, as in \cref{rem:thermal-midpoint}. {\small \begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.27\textwidth}>{\raggedright\arraybackslash}p{0.47\textwidth}>{\raggedright\arraybackslash}p{0.17\textwidth}@{}} \caption{Use of the temperature in the statements of \cite{OpenAI2026BEC} used to prove \cite[Proposition~5.4 and Lemma~7.1]{OpenAI2026BEC}.}\label{tab:temp-status}\\ \toprule Statement of \cite{OpenAI2026BEC} & Use of the temperature & Here\\ \midrule \endfirsthead \toprule Statement of \cite{OpenAI2026BEC} & Use of the temperature & Here\\ \midrule \endhead \bottomrule \endfoot Lemma 3.1, cube energy & None: a kinetic inequality for hard-core functions. & \cref{prop:temp-free-energy,prop:temp-spatial}\\ Lemma 3.2, close-neighbor energy & None. & \cref{prop:temp-spatial}\\ Lemma 3.3, entropy & Proved for every $T>0$. Its error $-Tj-CT\abs O(V^{-1}+T^{3/2})$, including the term $V^{-1}$ from the zero mode of the torus, is kept at every use. & \cref{prop:temp-free-energy,prop:temp-spatial}\\ Lemma 3.4, normalized insertion & None: its smallness condition is geometric. & \cref{prop:temp-free-energy,prop:temp-spatial}\\ Proposition 3.5, particle-number bounds & Stated for $00$. The color entropy $-T\mathcal S(\varpi)$, with $\mathcal S(\varpi)\le C+\E_\sigma n_{p^2}$, is kept. Each sector of fixed color numbers is one tensor product, with no binomial factor. & \cref{prop:temp-spatial}\\ Lemma 4.2, diagonal tilt & Proved for every $T>0$. The term $T(2\E_{\sigma_g}G-\log M_G)$ is kept; the gradient term has no factor $\beta$. & \cref{prop:temp-spatial}\\ Proposition 4.3 and Corollary 4.4, local penalty and density in a field & Standing range; the proof uses Proposition 3.5 and $T\le1$. Extended. & \cref{prop:temp-spatial}\allowbreak(1), (2)\\ Propositions 4.5 and 4.6, low and close counts & Stated for $0\delta$, so that the outer times $\beta/2-t$ are positive. & \cref{prop:temp-moment,prop:temp-certificates}\\ Definition 5.2, regular path & None: $h_0=D^{-1}$ is an endpoint-window length on $[0,2t]$. & \cref{def:thermal-regular}\\ Lemma 5.3, free cutoffs & None: free paths and bridges of duration $2t$, or two legs of duration $t$. & \cref{prop:temp-moment}\\ Proposition 5.4, certificates & Stated for $00$, and used at $q=\beta/\delta$; the H\"older step before it has the exponents $\beta/(\beta-\delta)$ and $\beta/\delta$, and requires $\beta>\delta$. & \cref{prop:temp-moment}\\ Proposition 5.7, trajectory moment & Standing range; the proof uses $\beta\ge D$ in the choice of the color gaps and Proposition 3.5 for the removal of particles. Extended. & \cref{prop:temp-moment}\\ Lemmas 6.1--6.3, Remark 6.4, and Corollary 6.5, bridge estimates & None: deterministic estimates for free bridges of duration $t$ and $2t$; the scale restrictions relate $D$ to $t$, not to $\beta$. & \cref{lem:thermal-single,rem:thermal-midpoint,lem:thermal-useful}\\ Lemma 7.1, useful labels & None: deterministic on the event of a certificate. & \cref{lem:thermal-useful}\\ \end{longtable} } \section{Free energies and particle numbers}\label{sec:temp-free-energy} The next proposition extends \cite[Proposition~3.5]{OpenAI2026BEC}, which is stated uniformly in $K\ge100$ and $00$ and every symmetric trace-one $N$-particle state $\sigma$ of finite kinetic energy, \begin{equation}\label{eq:temp-global} \frac{\mathcal F_N(\sigma)}N\ge\frac{c_{\rm cube}}2\alpha\bigl(1-D^{-1}\bigr)-T-\frac{CT}N-\frac{CT^{5/2}}D. \end{equation} The last two terms are the error $CTV(V^{-1}+T^{3/2})$ of \cite[Lemma~3.3]{OpenAI2026BEC} divided by $N=DV$; the term $CT/N$ comes from the zero mode of the torus. In \cite{OpenAI2026BEC}, the negative terms are absorbed using $T\le D^{-1}$. In the window, since $N\ge100^3D$ and $\alpha\ge1$, \[ T+\frac{CT}N+\frac{CT^{5/2}}D\le\frac{\tau_0}{\log D}+\frac{C\tau_0}{N\log D}+\frac{C\tau_0^{5/2}}{D(\log D)^{5/2}}\le\frac{c_{\rm cube}}8\alpha \] once $D$ is large depending on $\alpha_0$, uniformly in $K\ge100$. For $D\ge4$, \eqref{eq:temp-global} then gives $\mathcal F_N(\sigma)\ge(c_{\rm cube}/4)\alpha N$, and the Gibbs variational principle gives part (2). For part (3), \cite{OpenAI2026BEC}, following \cite[Proposition~3.3]{OpenAI2026B}, shows that $F_j/j$ is nondecreasing for $1\le j\le2N$. The proof uses strong subadditivity of the entropies of the marginals of the Gibbs state \cite{LiebRuskai1973}, the fact that taking a marginal preserves the hard-core form domain and the kinetic energy per particle, and the variational principle. Writing $S_i$ for the entropy of the $i$-particle marginal of the $j$-particle Gibbs state, strong subadditivity gives $S_i/i\ge S_j/j$. Hence $-TS_i/i\le-TS_j/j$ for every $T>0$, which, together with preservation of kinetic energy per particle and the variational principle, proves the monotonicity in the window. With part (2), $F_{N+k}\ge(1+k/N)F_N\ge F_N+(c_{\rm cube}/4)\alpha k$. For part (4), the insertion-frame argument of \cite{OpenAI2026BEC} distinguishes the last label, inserts it with the normalized orbital of \cite[Lemma~3.4]{OpenAI2026BEC} for $w=V^{-1/2}$ and the plane waves $e^{\mathrm ip\cdot y}$, $p\in(2\pi/K)\Z^3$, $\abs p\le1$, bounds the frame operator of these insertions, applies Jensen's inequality in the spectral measure of each inserted vector, and returns to the symmetric partition function by positivity of the labeled kernel. These steps hold for every $\beta>0$ and every bounded measurable $W\ge0$: the old field terms are unchanged, and the new one is nonpositive. For $D$ large depending on $\alpha_0$, they give \eqref{eq:temp-inc-raw} with numerical constants. In \cite{OpenAI2026BEC}, the last term of \eqref{eq:temp-inc-raw} is absorbed using $T\le D^{-1}$. In the window, $j/V\le2D$ and \eqref{eq:temp-log} give $T\log_+(C_fj/V)\le T\log(2C_fD)\le\alpha/4$; hence, since $\alpha\ge1$, \[ F_j^{(-W)}-F_{j-1}^{(-W)}\le C_{\rm ins}\alpha+1+\frac\alpha4\le C_{\rm inc}\alpha. \] The constant does not depend on $W$, including its size. \end{proof} \section{Spatial estimates}\label{sec:temp-spatial} The next proposition extends \cite[Proposition~4.3 and Corollary~4.4]{OpenAI2026BEC}, which are proved in the standing range $00$, independent of $b_0$, and a threshold $D_2\ge D_1$ such that the following hold for every integer $K\ge100$, every $N$ with $D=N/V\ge D_2$, every $\alpha\in[\alpha_0/2,2\alpha_0]$, and every $00$ (\cref{tab:temp-status}), and \cite[Proposition~3.5]{OpenAI2026BEC}, for which we use \cref{prop:temp-free-energy}. The temperature enters only in the inequalities displayed below. We use $T\le1/4$ from \eqref{eq:temp-small}, and $\alpha\ge1$. \emph{Part (1).} For $h\ge0$, let $p_h$, $q_h$, and the inside region $O_h$ be as in the proof of \cite[Proposition~4.3]{OpenAI2026BEC}, so that $\abs{O_h}\le\min\{V,C(h+4)^3\abs A\}$. We apply \cite[Lemma~4.1]{OpenAI2026BEC} to $\sigma$, with the distribution $\varpi$ of the number of particles of the first color, its entropy $\mathcal S(\varpi)$, and the inside and outside states $\sigma^{\mathrm{in}}_j$ and $\sigma^{\mathrm{out}}_j$. Outside, \cref{prop:temp-free-energy}(4), applied $j$ times with the field $W$, gives \[ \mathcal F_{J-j}^{(-W)}(\sigma^{\mathrm{out}}_j)\ge F_{J-j}^{(-W)}\ge F_J^{(-W)}-C_{\rm inc}\alpha j. \] Inside, one third of the kinetic energy and \cite[Lemma~3.3]{OpenAI2026BEC} with $O=O_h$ pay for the entropy $TS(\sigma^{\mathrm{in}}_j)$, with the error $-Tj-CT\abs{O_h}(V^{-1}+T^{3/2})$; the field costs at most $j$; and the color entropy costs $T\mathcal S(\varpi)\le T(C+\E_\sigma n_{p_h^2})$. Since $\sum_jj\varpi_j=\E_\sigma n_{p_h^2}$, the losses that are linear in the inside number are at most \[ (C_{\rm inc}\alpha+2T+1)\,\E_\sigma n_{p_h^2}\le C\alpha\,\E_\sigma n_{p_h^2}, \] and the remaining entropy error is at most \[ CT\bigl[1+\abs{O_h}(V^{-1}+T^{3/2})\bigr]\le C(h+4)^3\abs A, \] since $T\le1$, $\abs{O_h}\le V$, and $\abs A\ge1$. These are the two inequalities of the proof of \cite[Proposition~4.3]{OpenAI2026BEC} that use the temperature: there, the first relies on \cite[Proposition~3.5]{OpenAI2026BEC} in its range, and the second uses $T\le1$. The term $V^{-1}$ is kept. The rest of that proof, namely the averaging over $h$ with the density $e^{-h}$, the cell and close-neighbor energies of \cite[Lemmas~3.1 and~3.2]{OpenAI2026BEC}, and the completion of the square, does not involve the temperature, and it gives part (1) with constants independent of $b_0$. \emph{Part (2).} The left side of part (1) vanishes for the Gibbs state $e^{-\beta H_J^{(-W)}}/Z_J^{(-W)}$, which exists by \cref{prop:temp-free-energy}(1). Dropping $k_A$ and using $\sum_{v\in A}n_v\le ef_A$, we obtain part (2), as in the proof of \cite[Corollary~4.4]{OpenAI2026BEC}. \emph{Part (3).} Let $A\neq\varnothing$, and let the masks $p$, $q$, the squared cutoff $\phi$ with $p^2+\abs{\nabla p}^2+\abs{\nabla q}^2\le C\phi$, the fraction $\eta$, the number $k=\lfloor\eta D\abs A\rfloor$, the symmetric filling vector $\psi_k$, and the filling map $\mathcal U_jf=\binom{N-j+k}k^{1/2}P_{\mathrm{sym}}(f\otimes\psi_k)$ be as in the proof of \cite[Proposition~4.5]{OpenAI2026BEC}. The map $\mathcal U_j$ is an isometry, because the terms for different choices of the $k$ added labels have disjoint supports; it preserves the nonzero spectrum, and hence the entropy, of the outside state. The binomial coefficient is its normalization, and no binomial term enters the free energy. Since $\abs A\le V$, we have $k\le\eta N$, so all particle numbers below are at most $2N$. \Cref{prop:temp-free-energy}(3) at $N+k$, and then \cref{prop:temp-free-energy}(4) applied $j$ times, give \[ \mathcal F_{N-j}(\sigma^{\mathrm{out}}_j)\ge F_{N-j+k}-Ck(1+rk/\abs A)\ge F_N+\frac{c_{\rm cube}}4\alpha k-C_{\rm inc}\alpha j-Ck(1+rk/\abs A). \] With $\alpha_0$ large and then $\eta$ small, as in \cite{OpenAI2026BEC}, and with $D$ large, the last term is at most $(c_{\rm cube}/8)\alpha k$, and $k\ge\eta D\abs A/2$. Inside, \cite[Lemma~3.3]{OpenAI2026BEC} with the full kinetic energy, on the union of the interiors of the cells of $A$, gives $\mathcal F_j(\sigma^{\mathrm{in}}_j)\ge-Tj-CT\abs A(V^{-1}+T^{3/2})$. We insert both bounds and the color entropy $T\mathcal S(\varpi)\le T(C+\E_\sigma n_{p^2})$ into \cite[Lemma~4.1]{OpenAI2026BEC}. For every symmetric trace-one $N$-particle state $\sigma$ of finite kinetic energy, we obtain \begin{align*} \mathcal F_N(\sigma)-F_N&\ge\frac{c_{\rm cube}\eta}{16}\alpha D\abs A-(C_{\rm inc}\alpha+2T)\E_\sigma n_{p^2}-\E_\sigma n_{\abs{\nabla p}^2+\abs{\nabla q}^2}\\ &\quad-CT-CT\abs A\bigl(V^{-1}+T^{3/2}\bigr)\\ &\ge c_{\rm p}\alpha D\abs A-C_{\rm p}\alpha\,\E_\sigma n_\phi, \end{align*} with $c_{\rm p}=c_{\rm cube}\eta/32$, using $T\le1$, $V\ge1$, $p^2+\abs{\nabla p}^2+\abs{\nabla q}^2\le C\phi$, and $D$ large to absorb $C\abs A$. For $0\le\lambda\le1$, let $G=-\lambda n_\phi$ and $L(\lambda)=-\log\E_\mu e^{-2\lambda n_\phi}$, and let $\sigma_g$ be the tilted state of \cite[Lemma~4.2]{OpenAI2026BEC}. That lemma, with $\abs{\nabla G}^2\le C\lambda^2n_\phi$ and $2\E_{\sigma_g}G\le0$, gives $\mathcal F_N(\sigma_g)-F_N\le C\E_{\sigma_g}n_\phi+TL(\lambda)$. With the previous display for $\sigma=\sigma_g$, we obtain \[ C_{\rm p}'\alpha\,\E_{\sigma_g}n_\phi+TL(\lambda)\ge c_{\rm p}\alpha D\abs A,\qquad L'(\lambda)=2\E_{\sigma_g}n_\phi, \] with $C_{\rm p}'=C_{\rm p}+C$. Let $a'=\min\{c_{\rm p}/2,c_{\rm p}/(2C_{\rm p}')\}$. Whenever $L(\lambda)\le a'D\abs A$, we have \[ TL(\lambda)\le a'D\abs A\le\frac{c_{\rm p}}2\alpha D\abs A, \] since $T\le1\le\alpha$. This is the only use of the temperature in the tilt, and the factor $D\abs A$ appears on both sides. Then $L'(\lambda)\ge2a'D\abs A$ wherever $L(\lambda)\le a'D\abs A$, and the argument of \cite{OpenAI2026BEC} gives $L(1)\ge a'D\abs A$ and the bound $\exp[-(a'-2\ell)D\abs A]$ for the event in part (3). The requirement $2\ell0$, every $E\in\{E_{\rm all},E_{\rm diam},E_{\rm irr}\}$, and every nonempty $A\subset\mathbb L_K$. If $(C_0/\delta)\sum_{n\ge0}\varkappa_n\le1$, then \begin{equation}\label{eq:temp-moment} \log\E_{\PP^{\mathrm{pur}}}\exp\Bigl(\lambda\sum_{i=1}^NJ_A(\omega_i)\ind_E(\omega_i)\Bigr)\le CD\abs A\sum_{n\ge0}(n+1)^3\varkappa_n+R_{D,K,\delta}\abs A, \end{equation} where \[ R_{D,K,\delta}=C\delta T\bigl(V^{-1}+T^{3/2}\bigr)\sum_{n\ge0}(n+1)^3e^{-(m_0+n)/(2\delta T)}. \] Moreover, uniformly in $t\in(0,1]$ and $K\ge100$, \begin{equation}\label{eq:temp-remainder} \begin{gathered} R_{D,K,\delta}\le C\Bigl[\frac{\tau_0}{V\log D}+\frac{\tau_0^{5/2}}{(\log D)^{5/2}}\Bigr]D^{-m_0/(4\tau_0)}\le r_D,\\ r_D=C\Bigl[\frac{\tau_0}{\log D}+\frac{\tau_0^{5/2}}{(\log D)^{5/2}}\Bigr]D^{-m_0/(4\tau_0)}, \end{gathered} \end{equation} and $r_D\to0$ as $D\to\infty$. \end{proposition} \usedin{\cref{prop:temp-certificates}.} \begin{proof} The proof of \cite[Proposition~5.7]{OpenAI2026BEC} bounds the moment, with the bins truncated at $n\le n_*$, by the colored comparison of \cite[Section~5.4]{OpenAI2026BEC}: the marked kernel of \cite[Lemma~5.5]{OpenAI2026BEC}, the color map, the pointwise domination of the tested slab kernel, the symmetric trace identity, the Schatten H\"older inequality with exponents $\beta/(\beta-\delta)$ and $\beta/\delta$, the compression inequality, and \cite[Lemma~5.6]{OpenAI2026BEC} at $q=\beta/\delta$. These steps hold for every $\beta>\delta$ (\cref{tab:temp-status}), and $\beta>\delta$ by \eqref{eq:temp-small}. As in \cite{OpenAI2026BEC}, the smallness condition gives a field $W_0$ with $0\le W_0\le(C/\delta)\sum_{n\le n_*}\varkappa_n\ind_{O_n}\le1$, and the comparison reduces the claim to a bound on $\log(\tr e^{-\beta\widetilde{\mathbb H}}/Z_N)$, where $\widetilde{\mathbb H}$ is the colored operator with $W_0$ subtracted. The temperature enters this bound at the following places. \emph{The color gaps.} In \cite{OpenAI2026BEC}, $m_0$ is chosen so that $\mathfrak g_n/2\ge C_{\rm inc}\alpha+1$ and $\beta\mathfrak g_n/2\ge1$, the latter using $\beta\ge D\ge1$. With \eqref{eq:temp-constants} and $\delta\le2$, \begin{equation}\label{eq:temp-gaps} \frac{\mathfrak g_n}2=\frac{m_0+n}{2\delta}\ge\frac{m_0}4\ge C_{\rm inc}\alpha_++1\ge C_{\rm inc}\alpha+1,\qquad\frac{\beta\mathfrak g_n}2>\frac{m_0+n}2\ge1, \end{equation} where the last inequalities use only $\beta>\delta$ and $m_0\ge2$. \emph{The original color.} For $0\le J\le N$ and $0\le u\le1$, we have $\partial_u\log Z^{(-uW_0)}_J=\beta\,\E_{J,-uW_0}\sum_iW_0(x_i)$. \Cref{prop:temp-spatial}(2), applied to covers of the sets $O_n$ by at most $C(n+1)^3\abs A$ cells, gives, after integration in $u$, \begin{equation}\label{eq:temp-field} \log\frac{Z^{(-W_0)}_J}{Z_J}\le\frac{\beta C}\delta D\abs A\sum_{n\le n_*}(n+1)^3\varkappa_n. \end{equation} \emph{Removal and the gaps, paid once.} By \cref{prop:temp-free-energy}(4) with $W=0$, applied $N-J$ times, $Z_J/Z_N\le e^{\beta C_{\rm inc}\alpha(N-J)}$. The enlarged symmetric space decomposes into sectors with $J$ original-color particles and $k_n$ particles of color $n$, where $J=N-\sum_nk_n$. Each sector is one normalized tensor product of symmetric species spaces, with no multinomial multiplicity, as shown in the proof of \cite[Proposition~5.7]{OpenAI2026BEC}. We attach the factor $e^{\beta C_{\rm inc}\alpha}$ to each marked particle. With $W_0\le1$ and \eqref{eq:temp-gaps}, the one-particle operator of color $n$ then satisfies, in the sense of forms, \begin{equation}\label{eq:temp-one-payment} -3\Delta_{O_n}+\mathfrak g_n-W_0-C_{\rm inc}\alpha\ge-3\Delta_{O_n}+\frac{\mathfrak g_n}2. \end{equation} This single inequality pays for the removal of the marked particles and for the attractive field $W_0$, and it leaves the positive gap $\mathfrak g_n/2$. The insertion entropy of the removal is contained in $C_{\rm inc}$ by \eqref{eq:temp-log}, and no other factor is charged for it. The variational principle then bounds each marked canonical trace by that of $-3\Delta_{O_n}+\mathfrak g_n/2$. Since $J$ is determined by the occupations $k_n$, dropping the restriction $\sum_nk_n\le N$ introduces no sum over $J$. \emph{The marked traces.} For $s>0$, the periodic Gaussian image sum and killing give $\tr e^{3s\Delta_{O_n}}\le C\abs{O_n}(V^{-1}+s^{-3/2})$, at every $s>0$. Let $\Xi_n$ be the ideal Bose grand partition function of $-3\Delta_{O_n}+\mathfrak g_n/2$ at inverse temperature $\beta$. Since $\beta\mathfrak g_n/2\ge1$ by \eqref{eq:temp-gaps}, \[ \log\Xi_n=\sum_{a\ge1}\frac{e^{-a\beta\mathfrak g_n/2}}a\tr e^{3a\beta\Delta_{O_n}}\le C\abs{O_n}\bigl(V^{-1}+T^{3/2}\bigr)e^{-\beta\mathfrak g_n/2}. \] The term $V^{-1}$, from the zero mode of the torus, is kept. Combining the last three steps with $\abs{O_n}\le C(n+1)^3\abs A$, we obtain \[ \log\frac{\tr e^{-\beta\widetilde{\mathbb H}}}{Z_N}\le\frac{\beta C}\delta D\abs A\sum_{n\le n_*}(n+1)^3\varkappa_n+C\abs A\bigl(V^{-1}+T^{3/2}\bigr)\sum_{n\le n_*}(n+1)^3e^{-\beta\mathfrak g_n/2}. \] \emph{The factor $\delta/\beta$.} The comparison bounds the logarithm of the truncated moment by $\delta/\beta=\delta T$ times the last display. This factor cancels the factor $\beta/\delta$ of \eqref{eq:temp-field} exactly, and it turns the second term into $R_{D,K,\delta}\abs A$, since $\beta\mathfrak g_n/2=(m_0+n)/(2\delta T)$. The bounds do not depend on $n_*$, and monotone convergence removes the truncation, as in \cite{OpenAI2026BEC}. This proves \eqref{eq:temp-moment}. \emph{The remainder.} By \eqref{eq:temp-small}, $T\le1/4$, and $\delta\le2$; hence $x=1/(2\delta T)\ge1$ and \[ \sum_{n\ge0}(n+1)^3e^{-(m_0+n)x}\le e^{-m_0x}\sum_{n\ge0}(n+1)^3e^{-n}\le Ce^{-m_0x}. \] Since $\delta\le2$, this gives $R_{D,K,\delta}\le CT(V^{-1}+T^{3/2})e^{-m_0/(4T)}$. The functions $T\mapsto Te^{-m_0/(4T)}$ and $T\mapsto T^{5/2}e^{-m_0/(4T)}$ are increasing on $(0,\infty)$, and $e^{-m_0/(4T)}=D^{-m_0/(4\tau_0)}$ at $T=\tau_0/\log D$. This gives the first inequality in \eqref{eq:temp-remainder} for every $00$ there are $H_1$ and $\bar t\in(0,1]$, and for each $t\in(0,\bar t]$ there is $D_0$, such that, for every set $A$ of distinct cell centers, \[ \PP^{\mathrm{pur}}\{\text{no site of $A$ has a certificate}\}\le p_1^{\abs A} \] whenever $K\ge100$, $D\ge D_0$, $\alpha\in[\alpha_0/2,2\alpha_0]$, and $0\delta$, \cite[Proposition~5.1]{OpenAI2026BEC} shows that $X(0)$ and $X(2t)$ have law $\mu$ under $\PP^{\mathrm{pur}}$. By \cref{prop:temp-spatial}(3) and (4), the three endpoint tests fail at every site of $A'$ with probabilities at most $e^{-cD\abs{A'}}$ and $\exp(-c\theta_0b_0D^{2/3}\abs{A'})$, and both bases are at most $p'$ for $D$ large. \emph{Visitors.} We take $E=E_{\rm all}$, for which $\epsilon_E=1$, and $\lambda=c't$ with a constant $c'\in(0,1]$. Then $\sum_n\varkappa_n\le Cc't$ with $C$ independent of $t$, so the smallness condition of \cref{prop:temp-moment} holds for every $t\in(0,1]$ once $c'$ is small, and the first term in \eqref{eq:temp-moment} is at most $C_4\lambda D\abs{A'}$ with $C_4$ independent of $t$. If every site of $A'$ has more than $H_1D$ visitors, then $\sum_iJ_{A'}(\omega_i)>H_1D\abs{A'}$, and the exponential Markov inequality and \cref{prop:temp-moment} bound the probability by \[ \exp\bigl\{[-\lambda(H_1-C_4)D+r_D]\abs{A'}\bigr\}. \] We choose $H_1=C_4+1$. For each $t\in(0,1]$, the base $\exp(-c'tD+r_D)$ is at most $p'$ once $D$ is large depending on $t$. Decreasing $t$ does not affect $H_1$. \emph{Visitors of large diameter.} We take $E=E_{\rm diam}$ and $\lambda=1$. Then $\epsilon_E^{1/3}\le Ce^{-c/t}$, and both $\sum_n\varkappa_n$ and $\sum_n(n+1)^3\varkappa_n$ are at most $Ce^{-c/t}$. We choose $\bar t\in(0,1]$ such that, for every $t\in(0,\bar t]$, the smallness condition holds and the first term in \eqref{eq:temp-moment} is at most $(\theta_0D/2)\abs{A'}$. If every site of $A'$ is visited by at least $\theta_0D$ paths of diameter greater than one, then $\sum_iJ_{A'}(\omega_i)\ind_E(\omega_i)\ge\theta_0D\abs{A'}$, and the probability is at most $\exp[-(\theta_0D/2-r_D)\abs{A'}]\le(p')^{\abs{A'}}$ for $D$ large. We now fix $t\in(0,\bar t]$. \emph{Irregular visitors.} We take $E=E_{\rm irr}$ and $\lambda=\log(2/p')$. Then $\epsilon_E^{1/3}=C_tD^{-4}$, so that $\sum_n(n+1)^3\varkappa_n\le C_{p',t}D^{-4}$; the smallness condition holds for $D$ large, and the first term in \eqref{eq:temp-moment} is at most $C_{p',t}D^{-3}\abs{A'}$. If every site of $A'$ is visited by an irregular path, then $\sum_iJ_{A'}(\omega_i)\ind_E(\omega_i)\ge\abs{A'}$, even if one path visits several of these sites. The probability is at most \[ \exp\bigl[-\bigl(\log(2/p')-C_{p',t}D^{-3}-r_D\bigr)\abs{A'}\bigr]\le(p')^{\abs{A'}} \] once $C_{p',t}D^{-3}+r_D\le\log2$. Here the remainder enters once per site, without a factor $D$, so $r_D\to0$ is needed. \emph{The threshold.} Let $D_0=D_0(t)$ be at least $D_2$, at least the thresholds of the six tests above, and large enough that the scale conditions after \eqref{eq:thermal-mesh} hold and \cite[Lemma~5.5]{OpenAI2026BEC} applies. None of these thresholds depends on $K$ or $T$. \end{proof} \section{The volume threshold} \begin{lemma}\label{lem:temp-volume} The bounds of \cref{prop:temp-free-energy,prop:temp-spatial,prop:temp-moment,prop:temp-certificates} hold for every integer $K\ge100$ and every $00$, with coefficients that do not depend on $T$. The bounds therefore hold for every $00$, \cref{prop:temp-certificates} provides $H_1$ and $\bar t\in(0,1]$, and, for each $t\in(0,\bar t]$, a threshold $D_0$ such that \eqref{eq:thermal-certificates} holds whenever $K\ge K_1$, $D\ge D_0$, $\alpha\in[\alpha_0/2,2\alpha_0]$, and $0