The model and the result

We use units in which \(\hbar^2/(2m)=1\). Let \(v:\mathbb R^3\to[0,\infty)\) be bounded, measurable, radial, and compactly supported, and suppose that \(v\) is nonzero on a set of positive measure. Its scattering length \(a_v>0\) is determined by the radial solution of \[(-\Delta+\tfrac12v)f=0, \qquad f(x)=1-\frac{a_v}{|x|} \quad\text{for sufficiently large }|x|.\] On the torus \(\Lambda_L=(\mathbb R/L\mathbb Z)^3\), set \[v_L(x)=\sum_{z\in\mathbb Z^3}v(x+Lz), \qquad H^v_{N,L}=\sum_{j=1}^N(-\Delta_j) +\sum_{i<j}v_L(x_i-x_j),\] acting on \(L^2_{\mathrm{sym}}(\Lambda_L^N)\) with periodic boundary conditions. Write \(\mathcal G^v_{N,L}\) for the positive trace-one operators supported on its lowest eigenspace. For \(\Gamma\in\mathcal G^v_{N,L}\), define \[\gamma^{(1)}_\Gamma=N\mathop{\mathrm{Tr}}_{2,\ldots,N}\Gamma, \qquad u_0=L^{-3/2}, \qquad B_\Gamma=\frac{\langle u_0,\gamma^{(1)}_\Gamma u_0\rangle}{N}.\] Thus \(B_\Gamma\) is the occupation fraction of the constant orbital and \(1-B_\Gamma\) is its depletion.

The potential remains fixed throughout both limits. In particular, the statement applies to every thermodynamic accumulation value of the occupation; existence of an occupation limit at each positive density is not required. Neither monotonicity of the radial potential nor positivity of its Fourier transform is assumed. Boundedness makes the finite-volume heat semigroup positivity improving, so the ground state is in fact unique. The formulation with density matrices records the occupation observable without a choice of ground-state vector.

Historical context and antecedents

Bogoliubov’s quasiparticle theory describes the excitations and depletion of a weakly interacting condensate (Bogoliubov 1947). Lee, Huang, and Yang calculated the leading depletion and the second-order energy of a dilute hard-sphere gas (Lee et al. 1957). The two predictions concern different features of the many-particle ground state: its energy and its one-body density matrix. In the ordinary thermodynamic limit, long-wavelength particles can have arbitrarily small kinetic energy. Consequently, an accurate energy expansion does not by itself determine the occupation of the constant orbital.

The rigorous energy theory began with Dyson’s upper and lower bounds for hard spheres (Dyson 1957). Lieb and Yngvason proved the matching leading lower bound for repulsive interactions (Lieb and Yngvason 1998). Yau and Yin obtained the Lee–Huang–Yang upper bound for smooth repulsive potentials (Yau and Yin 2009); Basti, Cenatiempo, and Schlein extended the upper bound to nonnegative radial compactly supported \(L^3\) interactions, with a power-saving remainder (Basti et al. 2021). Fournais and Solovej proved the corresponding lower bound, first for integrable interactions and then in a class including hard cores (Fournais and Solovej 2020, 2023). The sharp hard-sphere upper bound was subsequently obtained by Basti, Brooks, Cenatiempo, Olgiati, and Schlein (Basti et al. 2026). For the present class of potentials, we use the upper bound of (Basti et al. 2021, Theorem 1.1) and retain the additional positive operators in a local lower bound to recover occupation observables.

Condensation and depletion are also understood in regimes where the spatial and dilute limits are linked. Lieb and Seiringer proved complete condensation for trapped gases in the Gross–Pitaevskii limit (Lieb and Seiringer 2002). Boccato, Brennecke, Cenatiempo, and Schlein determined the depletion in a periodic unit box with scattering length of order \(N^{-1}\) (Boccato et al. 2019, Appendix A). Their answer is a discrete Bogoliubov momentum sum. Results of Fournais and later developments extend condensation estimates to boxes larger than the healing length (Fournais 2021; Junge 2026). Such estimates do not directly control the ordinary thermodynamic limit, in which the one-particle gap closes at every fixed density. For fixed bounded interactions, the companion (OpenAI 2026a) establishes condensation at sufficiently small fixed densities and sufficiently low positive temperatures, and supplies the local density tails used here. The companion (OpenAI 2026b) establishes the depletion asymptotic for hard spheres. The present proof adapts its separation of local energy estimates from long-distance comparisons to bounded interactions.

Several established methods enter that adaptation. The energy analysis uses scattering-renormalized positive squares, Neumann localization, and Bogoliubov diagonalization, as developed in (Fournais and Solovej 2020, 2023) and in the low-temperature free-energy work (Haberberger et al. 2023; Fournais et al. 2026). The coherent-state substitution has the standard upper-symbol corrections of Lieb, Seiringer, and Yngvason (Lieb et al. 2005). The probabilistic comparison averages a change of law over routes before taking its moments. This has a direct antecedent in the deletion-tolerance argument of Peres and Sly (Peres and Sly 2014, Proposition 2.1). Its geometric ingredient is the unpredictable-path method of Benjamini, Pemantle, and Peres (Benjamini et al. 1998), which produces paths with exponentially decaying intersection tails. The precise finite-volume version, with rare encounters away from an endpoint, is supplied by (OpenAI 2026b, Proposition 7.1).

From local occupations to the constant mode

Here is the obstruction and the division of the proof. Let \(\Psi\) be the positive normalized ground function. For a partition of the torus into cubes of side \(\ell\), let \(P_\ell\) be the one-particle projection onto functions constant on each cube, and let \(P_0\) project onto the single global constant. Since \(P_0\le P_\ell\), the depletion splits exactly as \[ 1-B_{|\Psi\rangle\langle\Psi|} =\langle\Psi,(1-P_\ell)_1\Psi\rangle +\|(P_\ell-P_0)_1\Psi\|^2. \tag{1}\] The subscript indicates action on the first particle. The first term measures variations inside the cubes; the second measures variations between their mean values.

For a suitable mesoscopic choice of \(\ell\), Section 3 determines the first term at the depletion scale, together with second and fourth moments of particle counts. The argument follows the occupation estimates in (OpenAI 2026b, sec. 3), but deals directly with the bounded potential. In particular, the Neumann mirror interaction is compared with the original interaction in expectation using a two-particle bound. This avoids a pointwise radial-monotonicity comparison. The local lower bound retains both a kinetic gap and positive quasiparticle terms; these are needed to transfer the energy calculation to the one-body observable in (1).

The second term requires information that is uniform in the total volume. To obtain it, condition a distinguished particle to lie in a specified cube and integrate out its position. The resulting law of the other \(N-1\) particles has density proportional to the cube average of \(\Psi^2\). Comparing these deletion laws controls the spatial variation of the root mean squares of \(\Psi\). A local comparison of deletion laws, together with a fine-scale energy estimate, then connects those root mean squares to the arithmetic means appearing in \(P_\ell\Psi\).

Sections 4–6 perform these comparisons using ground-state Brownian paths. The construction pairs the midpoints of paths in two copies so that deleting different endpoint particles leaves exactly the same observed particles. For a bounded interaction the conditional path law has Boltzmann weights. We represent each weight by an auxiliary uniform-variable test and keep these tests in every conditional normalization. This gives the exact interacting law while allowing the collision estimates used in the hard-sphere construction. The density tails of (OpenAI 2026a, Lemmas 4.2 and 4.3) replace the deterministic count bounds available for hard spheres.

Two different accuracies are needed. A fourth-moment comparison between neighboring small cubes controls the passage from fine to coarse averages. A second-moment comparison between distant cubes controls the remaining variation across the entire torus. For the latter, routes may have arbitrarily many steps. The estimate therefore charges only encounters between two independently sampled routes, rather than one fixed loss at each step. In the copy used to measure the change of law, paths near the common endpoint remain unchanged. The route theorem makes encounters between the changed portions rare. These conditional-law and normalization estimates extend the local and distant comparisons of (OpenAI 2026b, secs. 5, 6, and 8) to bounded repulsion; the hard-sphere depletion theorem itself is not an input.

Section 2 fixes the scales and states the density-tail inputs. Section 3 proves the local occupation and count estimates, Section 4 constructs the conditional path law, and Sections 5 and 6 prove the two deletion comparisons. Section 7 combines them through (1) and restores the original units and order of limits.

Ground states and scaled variables

For finite \(N,L\), the potential in the Hamiltonian is a bounded multiplication operator. The heat kernel on the labelled configuration torus is strictly positive, and the Feynman–Kac formula preserves this positivity. Compact resolvent and positivity improvement therefore give a unique normalized positive ground function. Permutations of coordinates and simultaneous translations commute with the Hamiltonian. Uniqueness implies that this ground function is symmetric and translation invariant. In particular the bosonic ground space is one-dimensional. We shall use positivity when comparing spatial arithmetic means with root mean squares in Section 7.

Let \(R_v>0\) contain the support of the physical potential \(v\), and use a length unit \(\sigma>0\). In the resulting coordinates set \[ \begin{gathered} K=L/\sigma,\qquad V=K^3,\qquad D=N/V,\qquad a=a_v/\sigma,\\ \alpha=Da,\qquad r=R_v/\sigma,\qquad W(y)=\sigma^2v(\sigma y). \end{gathered} \tag{2}\] The scaled Hamiltonian has kinetic energy \(-\Delta\) and pair potential \(W\), periodized on the torus of side \(K\). Its normalized positive ground function is denoted by \(\Phi\), and \[\mu(\,\mathrm dX)=\Phi(X)^2\,\mathrm dX.\] Integrals in all particle coordinates use Lebesgue measure. We write \(\langle O\rangle_\psi=\langle\psi,O\psi\rangle\); this notation does not normalize a vector that is not already normalized.

Fix a sufficiently large constant \(B_0\), depending on \(v\), and put \[ \sigma_*=B_0\rho^{-1/2},\qquad D_*=\rho\sigma_*^3=B_0^3\rho^{-1/2},\qquad \alpha_0=a_vB_0^2. \tag{3}\] Along the thermodynamic sequence we choose \(\sigma\to\sigma_*\), as specified below. Then \(D\to D_*\) and \(\alpha\to\alpha_0\). All estimates are uniform for \(\alpha\) in a fixed compact positive neighborhood of \(\alpha_0\). Constants may depend on the fixed potential and this neighborhood. In these units, \[ a\asymp r\asymp D^{-1},\qquad \int_{\mathbb R^3}W\le C D^{-1},\qquad 0\le W\le C D^2. \tag{4}\] A bounded measurable radial potential can be replaced by a Borel version with its range and radial bounds holding everywhere. Such null changes affect neither the operator nor the Brownian path integrals.

Choose small exponents \(u>0\) and \(0<w<u/10000\), and let \[ R=\lceil D_*^w\rceil,\qquad B=R\lceil D_*^u/R\rceil,\qquad d=1/j\quad(j\in\mathbb N). \tag{5}\] The fixed microcell size \(d\) resolves variation within cells. The slowly growing scale \(R\) is used for the distant deletion comparison, while the still larger \(B\)-boxes give the energy calculation room to resolve variation on the \(R\)-scale. For each fixed \(j\), the dilute hierarchy is \(r\ll d\le1\ll R\ll B\). All partition sizes are fixed during the thermodynamic limit, when \(K\to\infty\); \(j\) is sent to infinity only after the estimates at each fixed \(j\).

The restrictions on \(u\) arise in the energy estimates; it may be taken smaller than \(10^{-9}\) and decreased further there. The path exponent \(x>0\) will be chosen after \(u,w\). At fixed \(\rho\), choose \(K\) to be the nearest positive multiple of \(B\) to \(L/\sigma_*\) and set \(\sigma=L/K\). This gives the asserted convergence of \(\sigma\) on a tail of every thermodynamic sequence.

We use nested cubic partitions \(\mathcal C_l\) of the scaled torus for \(l=B,R,1,d\). Unit cells \(U_z\) are centered at \(z\in\mathcal L=(\mathbb Z/K\mathbb Z)^3\), with compatible half-open conventions on boundaries. If \(A\) is a spatial set, \(n_A\) denotes its particle count. Let \(P_l\) be the one-body orthogonal projection onto functions constant on every cell of \(\mathcal C_l\), and set \[ W_l=1-P_l,\qquad \mathrm d\Gamma(O)=\sum_{i=1}^N O_i. \tag{6}\] Thus \(W_l\) is a projection, whereas \(W\) without a subscript is the pair potential. Averages over cells or cell pairs are uniform unless a different measure is specified.

Local density tails

We use two results from the bounded-interaction companion (OpenAI 2026a). We record both the physical form of the input and the consequence needed here, so that the dependence on temperature and volume is explicit. For a canonical Gibbs state let its position law be the diagonal of its density matrix.

The density-tail inputs are supplied by the localized density and occupation estimates of (OpenAI 2026a, Lemmas 4.2 and 4.3); for the upper tail we take exponential parameter one. Their proofs use local energy and entropy estimates and a localized replacement argument. They do not require a condensation or depletion conclusion. Their temperature uniformity lets us apply them to the ground state at each fixed finite volume: canonical Gibbs states converge in trace norm to its rank-one projection as temperature tends to zero.

For \(A\subset\mathcal L\) define \[F_A(X)=\sum_{i=1}^N e^{-d_K(x_i,A)},\qquad F_\varnothing=0,\] where \(d_K\) denotes torus distance and the sites of \(A\) are regarded as points of the torus. The tails imply the exponential moment bound \[ \log\mathbb E_\mu e^{hF_A}\le ChD|A|\qquad(0\le h\le h_*), \tag{8}\] for a fixed \(h_*>0\). Indeed the upper tail first gives this statement for the sharp count in a site set and one small positive value of \(h\); convexity extends it to the interval from zero. The sites at distance at most \(m+1\) from \(A\) number at most \(C(1+m)^3|A|\). Write the exponentially weighted count as a sum of counts in these enlargements, with exponentially decaying coefficients. Jensen’s inequality with normalized summable weights and the sharp-count bound prove (8). The same proof works if distance is multiplied by any fixed positive constant.

In particular, each fixed \(q\)th moment of a count in a fixed number of unit cells is \(O(D^q)\). Counts in polynomially many cells have polynomial fixed moments. If one of those cells exceeds a sufficiently large fixed multiple of \(D\), its probability is exponentially small in \(D\). Cauchy–Schwarz therefore makes every fixed product of these counts on that exceptional event smaller than every power of \(D^{-1}\). We will use this observation when removing count cutoffs.

Ground-state paths and local pair bounds

Let \(E_m\) be the ground energy of \(m\) particles on the scaled torus, with the same potential \(W\). Adjoining a constant last coordinate to a normalized \((m-1)\)-particle trial function gives \[ E_m-E_{m-1}\le\frac{m-1}{V}\int W\le C \quad(m\le N),\qquad E_N-E_{N-m}\le Cm. \tag{9}\] This trial need not be symmetric: the positive ground energy on the labelled space equals the bosonic ground energy by the uniqueness already proved.

For \(0<T\le1\), consider independent torus Brownian motions on \([-T,T]\) with generator \(\Delta\) and initial Lebesgue measure. Weight their joint law by \[ e^{2TE_N}\Phi(\omega(-T))\Phi(\omega(T)) \prod_{i<k}\exp\left[-\int_{-T}^T W(\omega_i(q)-\omega_k(q))\,\mathrm dq\right]. \tag{10}\] Here and below the pair potential in torus expressions is periodized. The Feynman–Kac formula makes (10) a probability measure with marginal \(\mu\) at every time. Restrictions to shorter slabs are consistent, by the ground-state eigenfunction identity.

Apply this lemma on the longer slab \([-1,1]\). At any time \(|q|\le1/2\), the probability for a free path started at \(Y_i\) to lie in a cube of side \(h\le1\) within a fixed distance of a site \(z\) is bounded by \(Ch^3e^{-2d_K(Y_i,z)}\), by the torus heat kernel. The bound also holds for the reversed observation. Summing (11) over distinct labels and using (8) shows that two specified such cubes have expected ordered pair count at most \(CD^2h^6\). Covering a fixed site neighborhood by \(h\)-cubes and summing over the bounded number of neighboring cubes for each one gives \[ \mathbb E\#\{(i,k):i\ne k,\ X_i,X_k\text{ in the neighborhood},\ d_K(X_i,X_k)\le h\}\le CD^2h^3. \tag{12}\] These bounds are uniform in the site and in the observation time on the central slab. They provide the required short-distance control using only the imported count tails and the elementary removal-energy estimate.

The upper bound and Neumann boxes

Put \(c_{\rm L}=128/(15\sqrt\pi)\) and define, holding \(a\) fixed when taking derivatives, \[e(y)=4\pi ay^2+4\pi c_{\rm L}a^{5/2}y^{5/2}.\] The Lee–Huang–Yang upper bound of (Basti et al. 2021, Theorem 1.1) applies because the fixed physical potential is radial, nonnegative, compactly supported and belongs to \(L^3\). Its error is \(O(\rho^{5/2+c})\) for some \(c>0\), with a constant depending on that potential. Under our rescaling energy density is multiplied by \(\sigma^5\), and \(\rho\asymp D^{-2}\). Consequently, for some \(c_*>0\), \[ E_N\le V\bigl(e(D)+O(D^{-c_*})\bigr). \tag{14}\] As elsewhere, this denotes a bound on the thermodynamic limsup; no finite-volume convergence rate is required. To pass from the Dirichlet upper bound to any periodic density-convergent sequence, place a Dirichlet trial box inside the periodic box with a range margin and side ratio tending to one. Use a slightly larger fixed density there, so its particle number eventually exceeds the desired one. Slicing out excess coordinates and averaging cannot increase the energy, since the omitted kinetic and interaction forms are nonnegative. Extending the remaining slices by zero respects the periodic form domain and the range margin makes the interactions agree. Let the larger density decrease to the prescribed density, using continuity of the explicit upper bound. This proves (14) along the sequences used here.

We use the same box estimates at two scales: \(E=R\) yields the coarse count variance, and \(E=B\) yields the microcell fourth moment and both occupation estimates. For a generic box side \(E\), abbreviate \(Q=E\) and \(v_b=E^3\). Thus \(Q\asymp D^\theta\) for \(\theta=u\) or \(w\). Restrict configuration space to sectors assigning each label to a box and discard interactions between boxes. These are Neumann form restrictions: we restrict gradient integrals, without differentiating the sector indicators. Each local vector is symmetric in the labels assigned to its box, and may take values in the Hilbert space of the spectator coordinates.

It is convenient to reflect the interaction at each wall. For a radial function \(g\) supported in \(\{|x|\le r\}\) define on \([0,E]^3\) \[g^s(x,y)=\sum_{z\in\mathbb Z^3}g(p_z(x)-y),\] where, coordinatewise, \(p_z\) maps \([0,E]^3\) onto the cube translated by \(Ez\), reversing a coordinate precisely when the corresponding \(z_i\) is odd. Write \(H_b\) for the Neumann Hamiltonian with pair potential \(W^s\). The added mirror terms are nonnegative, but their expectation on the original ground vector is small: \[ \sum_b\langle H_b\rangle\le E_N+CV/E. \tag{15}\] Indeed an added interaction requires two particles within \(Cr\) of each other and within \(Cr\) of a wall. Its multiplicity is bounded. At most \(C(V/E)r^{-2}\) cubes of side comparable with \(r\) cover all walls. The small-cube pair estimate proved before (12) gives \(CD^2r^6\) ordered pairs per such cube, and \(\|W\|_\infty\le CD^2\). Since \(r\asymp D^{-1}\), the resulting bound is \(CV/E\). All later excitation localizations use this same mirror Hamiltonian.

Let \(u_p\), \(p\in(\pi/E)\mathbb N_0^3\), be the real normalized Neumann cosines: \[u_p(x)=v_b^{-1/2}\prod_{i=1}^3c_{p_i}\cos(p_ix_i),\qquad c_0=1,\quad c_s=\sqrt2\ (s>0).\] Even extension shows that the integral operator with kernel \(g^s\) has eigenvalue \(\widehat g(p)\) on \(u_p\), with Fourier convention \(\widehat g(p)=\int g(x)e^{-ip\cdot x}\,\,\mathrm dx\). In particular its row integral is the constant \(\widehat g(0)\). Write \(a_p,a_p^*\) for the corresponding bosonic operators and set \[ J=Q^{100},\qquad n_+=\sum_{p\ne0}a_p^*a_p,\qquad n_L=\sum_{0<|p|\le J/E}a_p^*a_p,\qquad n_H=n_+-n_L,\qquad M\asymp Dv_bQ^{-400}. \tag{16}\] Take \(M\) integral. We refer to \(0<|p|\le J/E\) as the low band and the remaining nonzero modes as the high band. All power inequalities below hold if \(u\) is sufficiently small, for example \(u<10^{-9}\) and sufficiently small relative to \(c_*\). Further decreases are harmless.

A rough bound retaining count and kinetic penalties

We first obtain a lower bound with an error much larger than the Lee–Huang–Yang term. Its purpose is to control vectors with many excitations, leaving only a small-excitation sector for the sharper calculation. The construction is the radial replacement and small-box method of Dyson and Lieb–Yngvason (Dyson 1957; Lieb and Yngvason 1998), with an explicit reserve of kinetic energy.

Restricting the excitation number

The cutoff below is a finite-band instance of the matrix localization method of (Lieb and Solovej 2001, Appendix A). We give its form estimate directly so that the retained positive terms remain explicit.

Comparing Lemma 6 with the tangent to \(4\pi ay^2\) at \(D\), and using (14) and (15), gives the preliminary estimate \[ V^{-1}\sum_{b,\ n\le6Dv_b}\langle n_+\rangle\le CQ^2D^{1-0.001}. \tag{19}\] For completeness the higher number sectors do not spoil this comparison: partition their labels into groups of sizes between \(2.5Dv_b\) and \(6Dv_b\), apply the rough bound to each group and drop intergroup interactions. The energy per particle is then at least \(10\pi\alpha-o(1)\), exceeding the tangent slope \(8\pi\alpha\) by a fixed positive amount. Thus those sectors have a surplus of order \(n\). For \(n\le6Dv_b\), the scalar tangent remainder is nonnegative and \(\min(p^2,1)\ge cE^{-2}\) on the nonconstant modes, proving (19).

Localize only the middle number sectors \[Dv_b/2\le n\le6Dv_b.\] Choose real Lipschitz functions \(f,h\) with \(f^2+h^2=1\), \(f=1\) below \(M/8\), \(f=0\) above \(M/4\), and Lipschitz constants at most \(C/M\). The three localizing operators are \[ F_1=f(n_L)^2,\qquad F_2=h(n_L)^2,\qquad F_3=\sqrt2f(n_L)h(n_L). \tag{20}\] They have \(\sum F_i^2=1\). The first and third pieces are supported on \(n_L\le M/4\) and will be called low pieces; the second is supported on \(n_L\ge M/8\). Also \(\psi=F_1\psi+F_2\psi\), which later transfers count estimates even though counts need not commute with these cutoffs. Finite-rank smooth one-body projections preserve the Neumann form domain.

Here is the localization error, with \(V_n^s\) the sum of mirror pair potentials: \[ \left|\sum_{i=1}^3\langle H_b\rangle_{F_i\psi} -\langle H_b\rangle_\psi\right| \le\frac C{M^2}\left(\langle V_n^s\rangle_\psi +\|W\|_1\frac{J^3}{v_b}n\langle n_+\rangle_\psi\right). \tag{21}\] To prove it, let \(\Pi_k\) project onto \(n_L=k\). Kinetic energy commutes with the cutoffs, and the potential has block width two. The difference of its forms is \[-\frac12\sum_{k,l,i}(F_i(k)-F_i(l))^2 \langle\Pi_k\psi,V_n^s\Pi_l\psi\rangle.\] Positivity, form Cauchy–Schwarz and the finite block width bound this by \(CM^{-2}\) times the block-diagonal expectation. The latter is the phase average of \(e^{-isn_L}V_n^se^{isn_L}\). For a pair only the two coordinate rotations matter. Expanding each as \(1+(e^{is}-1)P_L\), every additional term has a low projection on at least one coordinate. The evaluation bound \(P_L(x,x)\le CJ^3/v_b\) and the constant row integral of \(W^s\) give \[P_{L,i}W^s(x_i,x_k)P_{L,i} \le C\|W\|_1J^3v_b^{-1}P_{L,i}.\] Expanding squared norms and summing pairs proves (21). The analogous error for \(\mathrm d\Gamma(A)\), \(\|A\|\le1\), is at most \(CnM^{-2}\|\psi\|^2\), by block width one and its norm bound \(n\).

The summed potential expectation is \(O(VD)\), including the mirror correction. Together with (19), (21) has total error \(O(VQ^{-1})\). Even replacing \(\|W\|_1\le C/D\) by the weaker \(CQ\) gives the per-volume bound \[O\bigl(D^{-1}Q^{794}+D^{-0.001}Q^{1097}\bigr)=O(Q^{-1}).\] All these inequalities include sector weights and spectator integration.

The sharp box estimate

We now analyze a low piece. The aim is to recover the Lee–Huang–Yang scalar while retaining two positive terms: a weak particle-excitation gap and a quasiparticle energy. The latter will identify the occupation observable.

Let \(\varphi\) be the scattering function of \(W\), and set \[\omega=1-\varphi,\qquad g=W\varphi,\qquad m=g\omega.\] The maximum principle and the radial scattering equation give \(0\le\varphi\le1\), \(\widehat g(0)=8\pi a\) and \(\|m\|_1\le C/D\). For \(p\ne0\) put \[ S=E^{-2}(n_++Jn_H),\qquad \tau_p=p^2-E^{-2}\bigl(1+J\mathbf 1_{\{p\ \text{high}\}}\bigr). \tag{22}\] Thus \(\tau_p\asymp p^2\). For the constant mode use coherent vectors \(|z\rangle=e^{-|z|^2/2}\sum_{q\ge0}z^q|q\rangle/\sqrt{q!}\), measure \(\,\mathrm d\lambda(z)=\pi^{-1}\,\mathrm d^2z\), and coherent projections \(\psi_z=\langle z|\psi\rangle\) onto the excitation Fock space. For \(|z|^2\le2n\) define \[\rho_z=|z|^2/v_b,\qquad t_p=\rho_z\widehat g(p),\qquad D_p=(\tau_p^2+2t_p\tau_p)^{1/2}.\] These roots are positive: \(|t_p|\le C\), while \(\widehat g(p)\ge0\) on every fixed momentum ball for large \(D\), by positivity and shrinking support. Large \(p\) are controlled by \(\tau_p\asymp p^2\). After rotating excitation modes by the phase of \(z\), the change of variables used below diagonalizes the one-mode quadratic form \[(\tau_p+t_p)a_p^*a_p+\tfrac12t_p(a_p^2+a_p^{*2}).\] Specifically, define \(b_p\) through \[ \begin{gathered} a_p=U_pb_p+V_pb_p^*,\qquad U_p>0,\quad U_p^2=1+V_p^2,\\ V_p^2=\tfrac12\left(\frac{\tau_p+t_p}{D_p}-1\right),\qquad 2U_pV_p=-t_p/D_p. \end{gathered} \tag{23}\] With this choice, \[(\tau_p+t_p)a_p^*a_p+\tfrac12t_p(a_p^2+a_p^{*2}) =D_pb_p^*b_p+\tfrac12(D_p-\tau_p-t_p).\] The formulas with \(z=0\) have \(U_p=1,V_p=0\).

The proof has three parts. Scattering-square splitting produces a quadratic Hamiltonian and a cubic term. Its quadratic vacuum energy is the Lee–Huang–Yang scalar. Completing the high-frequency cubic square creates a contraction that cancels against the retained \(m\) term. We supply each part, since the cancellation must leave the positive remainders in (24).

The energy surplus and count moments

The sharp estimate now applies to the low pieces; the rough estimate handles their complement and the outlying particle-number sectors. The bookkeeping is most transparent relative to the affine function \[t_b(n)=v_be(D)+e'(D)(n-Dv_b).\] Its sum over boxes is exactly \(Ve(D)\) in every assignment sector.

At \(E=R\), the first conclusion immediately gives a clipped second moment of \(q_b-1\) bounded by \(CD^{-1}Q^{-1/5}\). We next prove the microcell fourth moment at \(E=B\). This is where retaining the excitation gap, rather than only the scalar energy, is useful.

For a microcell \(A\subset b\), put \[X_A=n_A-n|A|/v_b =\mathrm d\Gamma\bigl(\mathbf 1_A-|A|/v_b\bigr).\] Its one-body summand has norm at most one and zero constant-mode diagonal. Hence for every \(n\)-particle vector \(\eta\), \[ \|X_A^2\eta\|^2 \le Cn^2\langle(n_++2)^2\rangle_\eta. \tag{38}\] Indeed its three blocks relative to \(P\oplus P^\perp\) change \(n_+\) by \(1,-1,0\). Between the \(k\) and \(k+1\) sectors their norms are bounded by \(\sqrt{(n-k)(k+1)}\), and the diagonal norm is at most \(k\). Thus \(X_A^2\) has bandwidth two and blocks of norm at most \(Cn(k+2)\). Squaring and using Cauchy–Schwarz over the bounded overlaps proves (38).

Lemma 8 gives \[ \frac1V\sum_{\rm low}\langle n_+\rangle\le CQ^{2-1/5},\qquad \frac1V\sum_{\rm low}\langle n_H\rangle\le CQ^{2-1/5}/J. \tag{39}\] Combine these with (28) and (38). Dividing by \((D|A|)^4\) and uniformly averaging all microcells gives the bound \[ C_j\left(D^{-1}Q^{-386-1/5}+D^{-1}Q^{-86-1/5} +D^{-2}Q^6+D^{-2}Q^{11}\right) \tag{40}\] for the low-piece fourth moment centered at the relative box mean. To check the volume factors, \(n^2\le CD^2Q^6\), and averaging the \(O_j(Q^3)\) microcells of each box gives prefactor at most \(C_jD^{-2}Q^9/V\) before the summed excitation moments. The terms \(Mn_+\) and \(nn_H\) then give the first two powers in (40); constants and the linear term in \((n_++2)^2\) give the last two.

Fix a finite clipping level, allowed to depend on \(j\). On the middle number range, a clipped fourth power of the box mean minus one is bounded by a constant times its square, so the box penalty recenters the preceding estimate at one. On high pieces a clipped fourth power of \(q_A-1\) is bounded by \(C_j\mathfrak h(q_A-1)\). On outlying number sectors it is bounded by a constant times the norm penalty of Lemma 8 after averaging cells. Since \(\psi=F_1\psi+F_2\psi\), the triangle inequality applied to the square root of the clipped nonnegative observable bounds its expectation in the original middle vector by twice the sum of those in \(F_1\psi\) and \(F_2\psi\). The extra low piece \(F_3\psi\) needed for energy localization has a nonnegative contribution and may be omitted here. This proves the clipped fourth-moment bound for the original ground state. The estimates beat \(D^{-1}Q^{-1/6}\) for sufficiently small \(u\).

We remove clipping only at this point, in the original state where (7) applies. Choose a large fixed unit-count threshold \(C_0D\). Except on the event that some constituent unit cell exceeds that threshold, an \(R\)-cell has bounded relative count and a \(d\)-cell has relative count at most \(C_0d^{-3}\). A fixed count moment is polynomial in \(D\), also for regions with polynomially many unit cells, by (8). Cauchy–Schwarz and the exponential upper tail therefore bound every discarded second or fourth moment by a quantity smaller than every inverse power of \(D\). This proves both count assertions in (13), with their stated slightly weaker exponents.

Recovering the occupation from the quasiparticle vacuum

It remains to extract the two occupation statements. Use \(E=B\) in this subsection. For \(l=R\) or \(d\), the one-body operator \(W_l\) acts within each \(B\)-box, annihilates its constant, and satisfies \[ 0\le W_l\le1,\qquad W_l\le Cl^2(-\Delta_{\rm Neu}). \tag{41}\] The second inequality is the Poincaré inequality on its component \(l\)-cells. On high pieces these bounds control \(\mathrm d\Gamma(W_l)\) by \(C(1+R^2)\mathrm d\Gamma(\min\{p^2,1\})\): split at \(p^2=1\) and use form Cauchy–Schwarz for the cross term. Lemma 8 then makes the contribution \(o(V)\) because \(R^2B^{-1/5}=o(1)\). Outlying sectors contribute \(o(V)\) by their norm penalty.

For a low piece at a coherent parameter, the Bogoliubov coefficients satisfy \[ |V_p|^2\le C\min\{|p|^{-1},|p|^{-4}\},\qquad p^2|U_p|^2\le CD_p\quad(p\text{ low}). \tag{42}\] For high \(p\), \(U_p\) is bounded and \(\widetilde D_p\ge cJ/E^2\). Define the vacuum trace and positive form \[T_l(z)=\sum_{p\ne0}(W_l)_{pp}|V_p|^2,\qquad \mathcal Q_z(\eta)=\sum_{p\ne0}\widetilde D_p\|b_p\eta\|^2.\] Shell summation gives \(T_l(z)\le Cv_b\), uniformly in the allowed \(z\). Substituting (23) into the observable yields \[ \begin{split} \left|\langle\mathrm d\Gamma(W_l)\rangle_\eta-T_l(z)\|\eta\|^2\right| \le{}&C(1+R^2)\mathcal Q_z(\eta)\\ &+C\bigl[(1+R^2)\mathcal Q_z(\eta)T_l(z)\|\eta\|^2\bigr]^{1/2}. \end{split} \tag{43}\] Here is a useful way to see all terms. Apply \(W_l^{1/2}\) to the column \((U_pb_p\eta)_p\). Its squared norm is bounded by the first term on the right, using (41) and (42) on the low band, and \(W_l\le1\) on the high band. For the column \((V_pb_p^*\eta)_p\), commute \(b\) past \(b^*\). The commutator is exactly \(T_l(z)\|\eta\|^2\), and the remaining excitation term has the same bound, also for the transposed matrix. Cauchy–Schwarz bounds the two cross terms and gives (43).

Integrate (43) over coherent parameters and sum low pieces. By Lemma 8 its right side is \(o(V)\); for the square-root term use Cauchy–Schwarz and \(\sum v_b\|\psi\|^2\le V\). Coherent resolution is exact for this observable, and its large-\(z\) tail is negligible by the particle-number bound. The one-body localization error already controlled after (21) is also negligible. We have therefore reduced both occupations to the averaged traces \(T_l(z)\).

To evaluate them, weight each low coherent piece on \(|z|^2\le2n\) by \(v_b\|\psi_z\|^2/V\) and include its spectator and sector integrals. The resulting finite measure has mass tending to one, and \[ \rho_z/D\longrightarrow1 \quad\text{in its measure}. \tag{44}\] Indeed the retained gain \(cM/E^2\) bounds the high-piece mass by \(CD^{-1}Q^{402-1/5}=o(1)\); the number penalty controls outlier mass, and the omitted coherent tail is exponentially small. The box penalty makes \(n/(Dv_b)\) concentrate at one; (39) makes \(n_+/(Dv_b)\) negligible. Finally, after angular integration the coherent radial variables have gamma shapes \(n-n_++1\) and variances of that same order. Their fluctuations divided by \(Dv_b\) vanish. This proves (44) with the prescribed order of limits.

On every compact momentum annulus, shrinking support of \(g\), \(\tau_p\to p^2\), and (44) show that \(|V_p|^2\) converges in this average to \[F_\alpha(p)=\frac12\left( \frac{p^2+8\pi\alpha}{\sqrt{p^4+16\pi\alpha p^2}}-1\right).\] The summable majorant in (42) removes the small and large momentum tails, uniformly in the coherent parameters. Neumann Riemann sums then converge with full-space measure \(\,\mathrm d\bar p=(2\pi)^{-3}\,\mathrm dp\).

For \(W_R\), its diagonal entries tend uniformly to one on compact annuli. In fact the average of a cosine over a cell of side \(R\) tends to zero when one of its frequencies is bounded away from zero; on an annulus at least one component has this property, and the product cosine formula gives the uniform assertion. For \(W_d\), (41) gives \((W_d)_{pp}\le\min\{1,Cd^2p^2\}\). Thus its limiting trace density is bounded by \[\int_{\mathbb R^3}\min\{1,Cd^2p^2\} C\min\{|p|^{-1},|p|^{-4}\}\,\,\mathrm d\bar p,\] which tends to zero as \(d=1/j\to0\) by dominated convergence. For \(W_R\) the full limiting integral is \[\begin{align*} \int_{\mathbb R^3}F_\alpha(p)\,\,\mathrm d\bar p &=\frac{(8\pi\alpha)^{3/2}}{4\pi^2} \int_0^\infty\left( \frac{x(x^2+1)}{\sqrt{x^2+2}}-x^2\right)\,\mathrm dx\\ &=\frac8{3\sqrt\pi}\alpha^{3/2}. \end{align*}\] The radial integral is \(\sqrt2/3\), again by elementary integration. This completes the proof of Proposition 5.

Exposed data and the exact conditional law

Restrict the stationary ground-path law of Section 2 to \([-t,t]\subset[-1/2,1/2]\). Write \[Y_i=\omega_i(-t),\qquad X_i=\omega_i(0),\qquad Z_i=\omega_i(t).\] A lift \(\widetilde\omega_i\) to \(\mathbb R^3\) satisfies the motion cutoff if \[ |\widetilde\omega_i(q)-\widetilde\omega_i(q')| \le L_0\sqrt{|q-q'|+r^2},\qquad -t\le q,q'\le t. \tag{49}\] Call a label eligible when the following conditions hold:

1. At each endpoint time its position is more than \(s_1\) from every other label’s position. The two endpoints have representatives at Euclidean distance at most \(b_0\).

2. The path has the winding determined by these nearby representatives, satisfies (49), and remains within \(20b_0\) of their mean \(m_i=(Y_i+Z_i)/2\) in this lift.

For sufficiently large \(K\), these nearby displacements and all subsequent local comparisons have unique compatible lifts, up to a common torus translation.

Independently flag each label with probability \(f_0\), and let \(I\) be the set of flagged eligible labels. We expose the two endpoint configurations, all flags, membership in \(I\), and the complete paths of labels outside \(I\). We call those paths frozen. A further augmentation turns the soft interaction into an exact indicator constraint. Set \[a_{ik}(\omega)=\exp\left\{-\int_{-t}^t W(\omega_i(q)-\omega_k(q))\,\,\mathrm dq\right\}.\] In the Feynman–Kac density, replace each factor \(a_{ik}(\omega)\) by integration over \(u_{ik}\in[0,1]\) of the indicator \[ \mathbf 1_{\{u_{ik}\le a_{ik}(\omega)\}}. \tag{50}\] Thus the joint density is the original endpoint-weighted free-path density times these indicators and Lebesgue measure in all \(u_{ik}\). Its path marginal is unchanged. Equivalently, given the paths, one may sample \(u_{ik}\) uniformly on \([0,a_{ik}(\omega)]\). These variables are not independent uniforms on \([0,1]\) after conditioning on the paths. Expose all of them, and denote the entire exposed datum by \(\mathcal D\).

Let \(B_i\) be the probability law of the Euclidean Brownian bridge of generator \(\Delta\) on \([-t,t]\) between the nearby representatives of \(Y_i,Z_i\). Its path space uses the chosen lift. For \(i\in I\), let \(J_i\) be the indicator of the motion cutoff and the \(20b_0\) tube.

All subsequent conditional comparisons use this indicator law. The values \(u_{ik}\) remain fixed throughout a comparison; collision bounds will hold uniformly in them.

Good sites and their probability

For an interior choice at a lattice site \(z\in\mathcal L\), we use labels in \(I\) whose initial endpoint belongs to \(U_z\) and whose endpoint displacement is at most one. Later, a label at the end of a comparison may have displacement up to \(b_0\); its site will be the unit cell containing \(m_i\).

Choose a uniform mesh on each half of \([-t,t]\), with an even number of intervals and common interval length \(\Delta\in[r^2/2,r^2]\). The midpoints of the two halves are therefore mesh points as well. For a mesh interval \(e\), define \[A_e(\omega)=r^{-1}\operatorname{diam} \widetilde\omega(e).\] Fix neighborhoods large enough to contain every collision test for a label associated with \(z\). Balls of radius \(1000b_0\) about \(z\), with a fixed enlargement to include intersecting unit cells, suffice. The actual collision tests below use distances less than \(200b_0\). Fixed further enlargements do not affect any conclusion. The flagged paths are sparse, so endpoint counts will suffice to sum their collision costs. For the denser frozen collection we also record positions and short-interval oscillations, allowing a sharper sum.

A site \(z\) is good if the following conditions hold:

1. It has at least \(cf_0D\) interior labels.

2. In its neighborhood, every unit cell has at most \(CD\) endpoints at each endpoint time and at most \(Cf_0D\) flagged endpoints. At each endpoint time, every ball of radius \(h\in[r,1]\) in the neighborhood has flag count at most \[ C(f_0Dh^3+D^{.01}). \tag{52}\]

3. Each frozen path visiting the neighborhood satisfies (49) and has lifted diameter at most \(10b_0\). For every mesh point \(q\), either adjacent interval \(e\), and every ball of radius \(h\in[r,1]\) there, the sum over frozen paths whose positions at \(q\) are in the ball, with weights \((1+A_e)^3\), is at most \[ C(Dh^3+D^{.01}). \tag{53}\]

Here \(c\) is sufficiently small and \(C\) sufficiently large. These are conditions on \(\mathcal D\). To make their measurability explicit, one may use countably many ball centers and radii. Equivalently, strengthened finite tests on enlarged ball nets at dyadic scales imply the stated bounds with a fixed change in \(C\).

We also define an event concerning the original sampled paths in a fixed neighborhood: require (53) for all paths, require (49) for every visiting path, and require its supremum distance from each of its endpoints to be at most \(b_0\) in the lift. We call these the full-path bounds. They will estimate the mass of successful comparisons. We do not add this event to \(\mathcal D\) or condition the bridge law on it.

The cost of one pair check

We now work at fixed good data. Paths of other labels in \(I\) may take arbitrary values satisfying their individual cutoffs; they need not be sampled from the conditional law. This uniformity will allow us to change subsets of pair checks in later comparisons.

For distinct labels \(i,k\), with trial label \(i\) eligible, define \[ \begin{split} \beta_{ik}&=b_D\left(1+ \frac1{s_1\vee d_K(Y_i,Y_k)}+ \frac1{s_1\vee d_K(Z_i,Z_k)}\right),\\ b_D&=C_xD^{-1}(\log D)^C. \end{split} \tag{57}\] The fixed polylogarithmic factor may be enlarged throughout this section. In particular, \(\beta_{ik}\le C_xD^{-.19}\) for large \(D\).

Uniform insertion probabilities

The one-check bound is already enough for the sparse set of flagged opponents. The frozen paths are much more numerous. Their weighted mesh counts give a sharper sum, of order \(rDt\), that still tends to zero on the chosen short slab.

We finish by recording the density cost of prescribing a midpoint. Under \(B_i\) the midpoint density in its local lift is \[ g_i(y)=(2\pi t)^{-3/2} e^{-|y-m_i|^2/(2t)}. \tag{61}\] Replacing it by a uniform midpoint in a unit cell at an absolute bounded distance from \(m_i\) costs at most \(CD^{Cx}\) in density. For two labels with \(|m_i-m_k|\le C\) for an absolute constant, the Gaussian formula gives, throughout either label’s pin range, \[ \frac{g_i(y)}{g_k(y)} \le C_xD^{C\sqrt x}. \tag{62}\] Indeed the difference of the two squared distances is at most \(Cb_0\), and \(b_0/t=O(\sqrt x\log D)\) with the fixed constants in (48). Conditioning a pair trial from Assertion 2 on success costs at most one additional factor two. These density statements concern a primary path, with the auxiliary bridge kept conditional on its midpoint if needed. A pair constrained to share its midpoint has no density relative to two independent full bridge laws; no such density will be used.

Augmented deletion measures and count errors

For a cell \(v\) of \(\mathcal C_d\) or \(\mathcal C_R\), define \[T_v=f_0D|v|,\qquad h_{vi}=\mathbf 1_{\{X_i\in v\}},\qquad A_v=\frac1{T_v}\sum_{i\text{ flagged}}h_{vi}.\] Here \(X_i=\omega_i(0)\), and the expectation \(\mathbb E\) includes the ground path sample, its independent flags, and the auxiliary variables in (50). Define a finite measure on this augmented sample together with a distinguished flagged label by \[\int H\,\,\mathrm dP_{\mathrm{aug},v} =\mathbb E\sum_{i\text{ flagged}}\frac{h_{vi}}{T_v}H(\omega,\mathcal D,i).\] Delete the distinguished midpoint and order the other \(N-1\) midpoints uniformly at random, independently of all existing variables. The resulting projected measure is \(P_v\) from (45). Indeed the midpoint law is \(\Phi^2\,\mathrm dX\), a specified flag has probability \(f_0\) independently of that law, and symmetry makes all \(N\) tag terms identical after projection. Their total coefficient is \(Nf_0/T_v=V/|v|\). The mass of both the augmented measure and its projection is \(\mathbb EA_v=1\). We keep these reference masses explicit when restricting tags and comparing the finite measures below.

We use normalized averages over cells and over adjacent ordered cell pairs. The periodic cubic adjacency graph is regular, so either cell marginal of its full edge average is a uniform cell average.

The comparisons will also discard failures of good data near their endpoint cells. For adjacent \(d\)-cells \(v,w'\), let \(\mathcal G_{v,w'}\) mean that every site in a fixed neighborhood of radius \(C b_0\) of the two cells is good. Choose \(C\) large enough to contain every possible tag mean and the coordinate routes used below. This event depends only on \(\mathcal D\). For two \(R\)-cells the corresponding neighborhoods have radius \(C_xR\). By (54), their failure costs, in either augmented reference, at most \[ C_{x,j}D^{-1-1/8+x}\quad\hbox{at scale }d, \qquad C_xR^3D^{-1-1/8+x}\quad\hbox{at scale }R. \tag{65}\] To see this, use the union bound for the number of tested sites, and bound the reference weight by \(A_v\le C_jD^x\) off the negligible count tail. Multiplication of any omission by a fixed product of relative counts is handled by the last assertion of Lemma 14. Also \(M_v'\le A_v\), so on the same count event \((M_v')^q\le(C_jD^x)^{q-1}M_v'\).

Two copies with the same remaining midpoints

Fix adjacent \(d\)-cells \(v,w'\) and retain \(\mathcal G_{v,w'}\). Choose distinct tags \(i,i_m\in I\) with weight \[ \frac{h_{vi}h_{w'i_m}}{T_vT_{w'}}. \tag{66}\] The site of a tag is the unit cell containing its near endpoint mean \(m_i\). The tube condition places this site in a fixed enlargement of the indicated cell whenever its tag indicator is nonzero. Join the two tag sites by a simple coordinate route of nearest-neighbor sites. At each interior vertex choose an interior label uniformly from the available supply, excluding the two tags. Choices at different sites are independent given \(\mathcal D\) and the tags. This gives a list \[ i_0=i,i_1,\ldots,i_m, \qquad m\le C_x=O(x^{-1/2}). \tag{67}\] If the two tag sites coincide, take just the two tags. The labels are distinct, consecutive means are at an absolute bounded distance, and each interior choice has point probabilities at most \(C_x/(Df_0)\). The route and these probability kernels depend only on the data and the tag identities; the weight (66) does not alter their definition.

Construct a left and a right path configuration. Keep all unlisted paths in both copies, the original path of \(i_0\) on the left, and the original path of \(i_m\) on the right. For \(k=1,\ldots,m\), generate the left path of \(i_k\) together with the right path of \(i_{k-1}\). Choose a common midpoint uniformly in a unit cell \(C_k\) at an absolute bounded distance from both means. Fix \(C_k\) as a measurable function of the endpoints before drawing any paths. Given the midpoint, propose the two paths independently as pinned free bridges. Condition this entire joint proposal, including its random midpoint, on the following tests: each path satisfies its individual cutoffs and every check (50) against retained paths and paths already inserted in its own copy. Proposition 12(2) gives success probability at least \(1/2\), so every stage is a probability kernel. The accepted midpoint need not be uniformly distributed.

Every interaction in a finished copy is checked when the later of its two paths is inserted, or was already checked between retained paths. Both copies therefore belong to the same conditional data fiber. Their midpoint identities are \[X_{i_k}^{\rm left}=X_{i_{k-1}}^{\rm right}, \qquad 1\le k\le m.\] Deleting \(i_0\) on the left and \(i_m\) on the right leaves the same midpoint multiset. Give the left remaining labels a uniform random order and give the right labels the order induced by these pairings and the unchanged labels. Each order is marginally uniform. Weighting the original sample by (66), and averaging tags, choices, and these kernels, defines a single finite measure \(\nu_{v,w'}\) on the remaining coordinates. It has a lift to either augmented reference: observe the left copy with tag \(i_0\), or the right copy with tag \(i_m\). Figure 1 displays the midpoint pairing.

We calculate a lift density before estimating it. Treat the left as the observed copy, fix \(\mathcal D\), the tags and list, and condition on the paths \(\eta\) outside \(S=\{i_1,\ldots,i_m\}\). Put \(B_S=\prod_{l\in S}B_l\). Let \(c_S(o;\eta)\) be the product of all individual cutoff indicators on \(S\) and all checks involving at least one label in \(S\). By Lemma 9, the original conditional law is \[ \lambda_S(\,\mathrm do\mid\eta) =\frac{c_S(o;\eta)}{Z_S(\eta)}B_S(\,\mathrm do), \qquad Z_S(\eta)=\int c_S(o;\eta)B_S(\,\mathrm do)>0. \tag{68}\] Checks wholly outside \(S\) are fixed and do not enter this formula.

The generation kernel on the changed left paths has a density \(k_S(o,\eta,z)\) with respect to \(B_S(\,\mathrm dz)\). To verify this precisely, write \(B_l^y\) for the free path law of label \(l\) conditional on midpoint \(y\), and \(g_l\) for its Gaussian midpoint density. A pair with observed label \(l\) and auxiliary label \(l'\) has proposal measure \[ \frac{\mathbf 1_C(z_l(0))}{g_l(z_l(0))} B_l(\,\mathrm dz_l)B_{l'}^{z_l(0)}(\,\mathrm dz'_{l'}). \tag{69}\] Multiplication by its success indicators and division by its success probability describes the actual insertion kernel. Sequential integration of the auxiliary paths gives \(k_S\). This is a density for the observed paths only: the paired paths have a common midpoint, and their joint law has no density against two independent bridges.

Cancellation in a fourth moment

The likelihood for one chosen list need not be close to one. Averaging many possible tags and interior labels produces the needed small error. Conditional centering cancels terms involving an isolated list; the terms that survive have either repeated labels or at least two rare interactions between lists. We make both statements precise.

For fixed tags and data, put a bar over a quantity to denote averaging over the route and interior label choices. We claim that, for sufficiently small fixed \(x\), \[ \mathbb E\left[\mathbf 1_{\mathcal G_{v,w'}} \sum_{i\in I}\frac{h_{vi}}{T_v} \left|\sum_{i_m\ne i}\frac{\overline{F_S}}{T_{w'}}\right|^4 \right]\le C_{x,j}D^{-1-1/20}. \tag{73}\] Each tag sum is restricted to the fixed enlargement of its cell and to labels in \(I\). We first prove the conditional interaction estimate needed to expand this fourth moment.

Fix four lists with the same first tag \(i\), and let \(S_a\) be their changed sets, \(1\le a\le4\). Condition on \(\mathcal D\) and on all paths outside \(U=\bigcup_aS_a\). Join two lists if their changed sets intersect; the connected components are called blocks. A singleton block contains one list, not necessarily one label. Distinct blocks have disjoint label sets. Introduce a switch \(\sigma_e\in\{0,1\}\) for each unordered pair \(e\) of blocks. Turning it off deletes all checks between those blocks from the following three places:

1. the union reference on \(U\);

2. the original-path indicators in each likelihood integral;

3. the insertion indicators and their success probabilities.

In the last two places, paths outside a given changed set are the reference paths. All individual cutoffs are retained. The unchanged left path of the common tag \(i\) is outside every block, so checks against it are never switched. A newly generated right path with label \(i\), however, belongs to its generating block; its checks against other blocks are switched. It is never checked against the left copy of the same label. These rules also specify the switches when two lists overlap.

Let \(c_U^\sigma\) denote the switched union indicator and \(F_a^\sigma\) the switched centered likelihood. Define the unnormalized integral \[ G(\sigma)=\int c_U^\sigma(z) \prod_{a=1}^4F_a^\sigma(z)B_U(\,\mathrm dz). \tag{74}\] The conditional fourth-moment summand before switching is \(G(\boldsymbol1)/Z_U\), where \(Z_U=\int c_U^{\boldsymbol1}\,\mathrm dB_U\). We hold this denominator fixed throughout the switch expansion. Successive single-path insertions give \(Z_U^{-1}\le C_x\), because \(U\) has at most \(4C_x\) labels.

For a block-pair edge \(e\), let \(\mathcal Q_e\) be the finite list of distinct-label pairs whose checks are controlled by that switch. It includes checks using generated auxiliary paths and original paths, with bounded multiplicity. All its labels and endpoints are fixed by the data and the four lists.

We can now prove (73). Expanding the fourth power gives four independently chosen lists, conditional on the common first tag and the data. There are at most \(C_xDf_0\) candidates for each tag or interior slot, by the good endpoint counts and the tube restriction. A normalized tag counting weight and every interior-choice probability are at most \(C_{x,j}/(Df_0)\). Split the sums by the tag mean-sites and the route site patterns; there are only \(C_x\) patterns. We retain the original weights in each subsum, without conditioning on a pattern and renormalizing.

An equality pattern with \(q\) independent identifications has \(q\) fewer free label choices. Dominating all slot weights by their point bounds and counting the remaining choices gives total weight at most \(C_{x,j}(Df_0)^{-q}\). A triple equality counts as two identifications. These are joint counting estimates; they assert no point-mass bound conditional on the occurrence of a coincidence. There are only \(C_x\) equality patterns. Consequently all patterns with \(q\ge2\) contribute, by (72), at most \[ C_{x,j}D^{-2+2x+C\sqrt x}. \tag{77}\]

For \(q=0\) or \(1\), use the switches above. For an edge set \(E\), let \(\Delta_E=\prod_{e\in E}\Delta_e\). The elementary Boolean identity is \[ G(\boldsymbol1)=\sum_E\Delta_EG(\boldsymbol0). \tag{78}\] If the graph with edge set \(E\) has an isolated singleton block, its term vanishes. Indeed all edges from that block are off in every evaluation comprising the difference. Its variables separate from the others, and their integral is \[\int c_S F_S\,\,\mathrm dB_S=0\] by (71). This is why the common denominator \(Z_U\) was held fixed rather than switched.

When \(q=1\) there are three blocks, two of them singletons, so a nonzero term must contain an edge. Its identification cost and (75) give total contribution at most \[ C_{x,j}D^{-1+x-19/100+C\sqrt x}. \tag{79}\] When \(q=0\), all four blocks are singletons and a nonzero term has at least two distinct edges. Apply (76) to any two of them, and bound any remaining differences by a finite sum of evaluations with the other switches fixed.

One of these two collision factors can be averaged over a label slot. Every cross-block check has at least one label in a changed set, whereas the common tag \(i\) belongs to none of those sets. The endpoint bound (52), summed over dyadic annuli in a fixed neighborhood, gives uniformly in the other endpoint \(y\) \[ \frac{C_{x,j}}{Df_0} \sum_{k\text{ in a slot range}} \frac1{s_1\vee d_K(y,Y_k)} \le C_{x,j}\left(1+\frac{D^{1/100}}{Df_0s_1}\right) \le C_{x,j}. \tag{80}\] For example, a dyadic annulus of radius \(h\le1\) contributes at most \(C_{x,j}(h^2+D^{1/100}/(Df_0h))\); the innermost ball is bounded at radius \(s_1\), and the unit-scale and larger annuli use the unit-count bound. The geometric sums give the displayed expression. The same estimate holds with \(Z_k\) in place of \(Y_k\). Thus averaging this factor costs at most \(C_{x,j}b_D\). Bound the other factor by \(C_xD^{-19/100}\). In these positive bounds we may remove all further noncoincidence restrictions and enlarge the label sums. The total contribution with no identification is therefore at most \[ C_{x,j}D^{-19/100+C\sqrt x}b_D. \tag{81}\] The first-tag factor \(h_{vi}\) is fixed in every conditional integral, so it does not disturb centering. Its total normalized tag sum on good data is bounded by \(C_{x,j}\). Equations (77)–(81) prove (73) once \(x\) is chosen sufficiently small. Indeed \(b_D=C_xD^{-1}(\log D)^C\), and all three powers then save more than \(1/20\) beyond \(D^{-1}\).

The local comparison after projection

Let \(\ell_v\) and \(\ell_{w'}\) be the two lifted densities of \(\nu_{v,w'}\) relative to their augmented references. On retained data with first tag in \(I\), the left density is \[\ell_v=\sum_{i_m\ne i}\frac{\overline{L_S}}{T_{w'}};\] set it to zero on the omitted part of the reference. The centered subtraction sums to \(A_{w'}-M_{w'}'\). In fact the candidate mean range contains every tag with nonzero \(h_{w'i_m}\), by confinement, and the excluded first tag has \(h_{w'i}=0\) because the cells are disjoint. Thus on the retained part \[ \ell_v-1=\sum_{i_m\ne i}\frac{\overline{F_S}}{T_{w'}} +(A_{w'}-1)-M_{w'}'. \tag{82}\]

The geometric input and its application

We use the following complete statement of (OpenAI 2026b, Proposition 7.1). It concerns a random set of lattice sites only. The proof in (OpenAI 2026b, sec. 7) uses folded paths and the unpredictable-path method of (Benjamini et al. 1998).

We check the hypotheses and fix how this result is used. The bad sites are those of Section 4. Equation (54) gives the required joint bound with any sufficiently small fixed \(p\), by increasing the threshold for \(D\). All possible mean-sites of tags whose midpoint lies in \(v\) belong to a fixed enlargement \(Q_v\) of that cell, of side \(O_x(R)\); this follows from the individual tubes. The same holds for \(w'\). Since \(K/R\to\infty\) in the thermodynamic limit, the displacements from any site of \(Q_v\) to any site of \(Q_{w'}\) belong to the wider interval in Proposition 18.

Choose \(r_c\) large enough, in terms of \(b_0\), to contain every possible cross-route interaction between paths in their individual tubes, as well as the bounded offsets between sites and means. Take \(q=2\) to allow the adjacent labels used by a paired insertion. These choices are fixed once \(x\) is fixed. We retain a bad-free margin of radius \(C_xR\) around both cells and impose the route-availability events for every endpoint site in their enlarged cubes, for both orientations. There are only polynomially many such sites in \(R\). Thus (86) bounds the additional rejection by \(o_K(1)+C_xR^{C_x}e^{-c_xR}\). This extra retention can only decrease the nonnegative quantities in (84)–(85).

We always sample the routes by the construction of Proposition 18, conditional on the datum and the endpoint sites. In a likelihood product, the datum is shared and the route randomizations are independent. In particular, we do not condition their law on a subsequent successful path insertion. The exponent \(c_g\) is independent of \(x\), although the constants and \(\kappa\) may depend on \(x\).

A common measure with retained end portions

Use the tag weight (66) for \((v,w')\in\mathcal P_R\), retained data, and distinct tags \(i_0,i_m\in I\). On the sampled route from their mean-sites choose one interior label at each intermediate site, uniformly and independently given the tags and sites, excluding the two tags. The resulting list \(i_0,\ldots,i_m\) has distinct labels; each interior choice has atom at most \(C_x/(Df_0)\). Successive means are at bounded absolute distance.

Write the initial silent portion of the route as \(k<a_0\), and its terminal silent portion as \(k>a_1\). Thus \(a_0\) is one plus the last visit within distance \(H\) of the initial site, while \(a_1\) is one less than the first index in the terminal portion. Both tags lie in these portions, and the portions are disjoint. Keep all unlisted paths in both copies. Keep the prefix originals \(k<a_0\) on the left, and the suffix originals \(k>a_1\) on the right. The remaining paths are generated as follows, pairing left label \(i_k\) with right label \(i_{k-1}\) for \(1\le k\le m\).

1. For \(a_0\le k\le a_1+1\), propose a common uniform midpoint in a unit cell within bounded distance of both means and propose the two paths independently conditional on that midpoint. Condition jointly on their individual cutoffs, the checks (50) against retained and previously generated paths in their respective copies, and midpoint separation by \(s_1\) from those opponents. These middle pairs are probability kernels: Proposition 12, Assertion 2, bounds each success probability below by \(1/2\). Use any fixed order for them.

2. In every other pair, one path is a retained original. Pin the other path to its midpoint \(y\). Give this step mass zero unless \(y\) is within \(8b_0\) of the new label’s mean and is \(s_1\)-separated from all opponent midpoints already present in the copy receiving the new path. If these conditions hold, propose the bridge pinned at \(y\), impose its individual cutoffs and checks (50), and divide by \[ \max\{1/2,p_{\rm pin}\}, \tag{87}\] where \(p_{\rm pin}\) is their success probability under the pinned proposal. This is a subprobability kernel. Perform these steps after all middle pairs, in fixed orders. Prefix pins are inserted on the right and suffix pins on the left; neither family uses paths generated by the other family.

The final copies satisfy all conditional reference checks. Every left midpoint except that of \(i_0\) agrees with its paired right midpoint, and every right midpoint except that of \(i_m\) is paired. Deleting these respective tags and using the common ordering of the bath coordinates therefore gives one measure on the bath with lifts to both augmented references. Figure 2 shows the pairing and the retained end portions.

The silent portions have two purposes. They keep the common tag outside the changed set in a likelihood product, and they ensure that the changed sets start far enough from their common end for (85) to apply. The opposite-copy pins preserve the matching of bath midpoints across these retained portions.

Conditional likelihoods and removal of cross checks

We next bound the second moment of a lift. Unlike a uniform bound on its density, this estimate will have no factor depending on route length. Treat the left copy as the observed, or primary, copy. Its changed labels form \[S=\{i_k:k\ge a_0\}.\] Fix the datum, tags, route and list, and condition on the paths \(o\) outside \(S\). With \(B_S\) the product plain bridge law and \(c_S\) the conditional reference indicator, write \[ M_S(o;\,\mathrm dz)=\int c_S(y;o)h_{w'i_m}(y) K_S(y,o;\,\mathrm dz)\,B_S(\,\mathrm dy),\qquad Z_S(o)=\int c_S(y;o)\,B_S(\,\mathrm dy). \tag{89}\] Here \(y\) denotes latent original paths, \(z\) the primary outputs, and \(K_S\) the subprobability kernel of all generation steps. The likelihood, before the divisor \(T_{w'}\), is \(L_S=\,\mathrm dM_S/\,\mathrm dB_S\) on admissible reference outputs: the input and output conditional normalizer \(Z_S\) cancels exactly. Absolute continuity is verified below, including for the pins.

For an upper bound we may integrate out the pins generated on the right at the silent prefix and replace their total mass by one. No primary middle or pin step uses those paths. We use the same notation \(M_S\) for this enlarged measure when taking upper bounds.

Take a second, independently sampled route and list with the same first tag \(i_0=i\). Denote its changed set by \(T\) and the two terminal tags by \(j_1,j_2\). Condition now on all paths outside \(U=S\cup T\), as well as the datum. In the \(S\) construction remove all checks against outside labels in \(T\setminus S\), both from the latent reference and from every proposal and success normalization. In the \(T\) construction remove checks against \(S\setminus T\). Write \(A_S^-,Z_S^-,M_S^-\) for the resulting reference indicator, normalizer, and unnormalized output measure, and similarly for \(T\). These two constructions now use only their own latent and generated variables and the fixed paths outside \(U\).

This independence also holds at the edge of a silent prefix. Its right pins have been integrated out. If an old prefix path is removed as an outside obstacle, no other part of the construction uses that trajectory. A right middle partner uses the fixed endpoints of its label, not its old trajectory. Routes and label choices use only data, sites, and tag identities, all fixed here.

Let \(Z_U=\int c_U\,\,\mathrm dB_U\) be the union reference normalizer. List all checks affected by the removals: reference checks and checks of latent or generated paths against removed old obstacles, including those in the success probabilities. Every such check requires two route vertices within \(r_c\), allowing the fixed step offsets in \(J_*\). There are at most \(C_xJ_*\) such checks, with bounded multiplicity. The simplicity of each route gives this bound regardless of its length. Only distinct-label checks are listed; in particular, a proposal never gains a check against its own old trajectory if that trajectory was not an obstacle originally.

Let \(B_{ST}\) be the sum, with these bounded multiplicities, of the corresponding costs \(\beta_{lk}\) from (57). For pin checks also sum their costs \(m(y,\omega)\) from (58), writing the result as \(M_{ST}'\). The first quantity depends on the fixed endpoints and labels; the second can depend on latent or generated midpoints. Uniformly, \[ B_{ST}+M_{ST}'\le C_xD^{-.19}J_c. \tag{90}\]

The following estimate is the aim of the conditional comparison. For disjoint changed sets it preserves the original tag product, with an error charged only to cross-route checks. For overlapping sets it charges a density bound only to the shared output labels. Put \(G_D=C_xD^{C\sqrt x}\), with the constants sufficiently large.

We prove the lemma through three comparisons: first of the reference normalizers, then of the generation kernels when cross checks are removed, and finally of the primary output densities on shared labels.

There is also a comparison before any generated variables are integrated. The original two constructions are dominated by the separate constructions times a factor bounded in both forms \[ e^{C_xD^{-.19}J_c},\qquad 1+Ce^{C_xD^{-.19}J_c}(B_{ST}+M_{ST}'). \tag{95}\] To prove this, fix a history at an affected transition. Removing checks increases its success probability from \(p\) to \(p^-\). For a middle step \(p\ge1/2\); for a pin the denominator is the function \(\max(1/2,p)\), which is Lipschitz and bounded below by \(1/2\). Thus the expense in the likelihood density is at most \[\frac{\max(1/2,p^-)}{\max(1/2,p)}\le1+2(p^--p),\] with the same estimate for the uncapped middle denominators. Removing the preliminary pin-separation indicator only enlarges the integrand. For a middle step, integrate the removed checks against its common uniform midpoint proposal; Proposition 12 bounds the difference by the sum of \(C\beta_{lk}\), including failures of midpoint separation. For a pin step the same bound includes its costs (58); the needed midpoint separation holds on the support of the old integrand. Products of these bounds, and (90), prove (95). Indicators of fewer reference checks only enlarge an integrand. Every unaffected transition has ratio exactly one.

Primary output densities and the conditional product bound

We need one density estimate when two changed sets share labels. For any \(q\) specified primary labels, the marginal of \(M_S^-/Z_S^-\) has density at most \[ G_D^q\quad\hbox{against their product plain bridges}. \tag{96}\] To prove it, omit the terminal tag indicator, so the latent originals have their normalized separate reference law. Fix those originals, denoted by \(y\), and let \(K_S^-\) denote the separate generation kernel. Divide the specified primary indices into middle indices \(A_{\rm mid}\) and pin indices \(A_{\rm pin}\). A primary pin with index \(k\) uses the midpoint of the retained original label \(i_{k-1}\) in the right suffix. Write \(\mathcal R_k=\{\xi:|\xi-m_{i_k}|\le8b_0\}\) for its pin range in the compatible lift.

Conditional on the generation history, a specified middle output is bounded by \(C_xD^{Cx}B_{i_k}\), by (69). A specified pin is bounded by \(2\mathbf 1_{\mathcal R_k}(y_{i_{k-1}}(0)) B_{i_k}^{y_{i_{k-1}}(0)}\), where the superscript denotes a prescribed midpoint. These majorants no longer depend on earlier generated paths. Integrating the generation steps in reverse order against a nonnegative test of the specified outputs therefore gives the conditional marginal bound \[ \begin{split} K_S^-(y,o;\,\mathrm dz_A)\le{}& (C_xD^{Cx})^{|A_{\rm mid}|}2^{|A_{\rm pin}|} \prod_{k\in A_{\rm mid}}B_{i_k}(\,\mathrm dz_k)\\ &\times\prod_{k\in A_{\rm pin}} \mathbf 1_{\mathcal R_k}(y_{i_{k-1}}(0)) B_{i_k}^{y_{i_{k-1}}(0)}(\,\mathrm dz_k). \end{split} \tag{97}\] Here \(A=A_{\rm mid}\cup A_{\rm pin}\), and \(z_k\) denotes the primary output path of label \(i_k\). Every unspecified step contributes at most its mass, which is at most one. Thus the displayed product bound does not require independence of the accepted outputs.

The sources \(i_{k-1}\) of the specified pins are distinct latent labels in \(S\). Corollary 13 bounds their joint latent marginal by \(C^{|A_{\rm pin}|}\) times their product plain bridges. Keep their confinement indicators when making this comparison, so all source midpoints use the chosen local lifts. Dropping the remaining source cutoffs in those lifts only enlarges the integral. For a source \(l=i_{k-1}\), evaluate the resulting midpoint mixture by the measure identity \[ \int_{\mathcal R_k}g_l(\xi)B_{i_k}^{\xi}(\,\mathrm dz_k)\,\,\mathrm d\xi =\mathbf 1_{\mathcal R_k}(z_k(0)) \frac{g_l(z_k(0))}{g_{i_k}(z_k(0))} B_{i_k}(\,\mathrm dz_k). \tag{98}\] The two means are at bounded distance, and the indicator restricts the ratio to the primary pin range. Thus (62) bounds this ratio by \(C_xD^{C\sqrt x}\). Applying this to each pin in (97) proves (96). The same argument with the original checks in place proves absolute continuity of all primary outputs in (89). Only the primary marginal acquires this density; a pair with a common midpoint is still singular relative to two independent full bridges.

Averaging the labels and route encounters

Let \(\ell_v\) be the left lift likelihood, with zero value on omitted data and tags. On the retained part it is the sum of \(\overline{L_S}/T_{w'}\) over second tags, where the bar averages the route and interior label kernels. Squaring it gives two independent choices with common first tag \(i\). The augmented reference supplies its factor \(h_{vi}/T_v\) only once; this path is fixed outside \(U\). The disjoint leading term in (91), summed over choices, is bounded by \[ \mathbb E[A_vA_{w'}^2]. \tag{100}\] Here the indicators are the original reference indicators. Removing good-data and distinctness restrictions only increases their nonnegative sum, and the route and label kernels have mass one.

We detail the remaining sums because an estimate uniform in route length is essential. At every possible tag mean-site, the number of candidate tags is at most \(C_xDf_0\), by the good-data flagged counts and the bounded endpoint offsets. Since \(T_v=T_{w'}=f_0DR^3\), its normalized counting measure is dominated by a deterministic site weight \(C_xR^{-3}\) times a label subprobability measure whose atoms are at most \(C_x/(Df_0)\). The same atom bound holds for every interior slot, using its actual conditional probability law. Route sampling depends on the endpoint sites and datum, not on tag identities.

For a disjoint error, enumerate all slot pairs across the two routes that are within the contact radius, including the fixed step offsets. This overcounts the affected checks by a bounded factor. Every such check has an averageable label, and (80) gives an average cost at most \(C_xb_D\) for its \(\beta\). In detail, interior choices are independent across routes given the tags; a label in the opposite slot can be held fixed. When a tag must instead be summed, its point weight gives the same bound. A prefix label used by the pair at the change edge is assigned to its adjacent prefix slot. The common first tag is outside both changed sets. Exclusions such as omitting a label’s own old trajectory remove terms; we may discard disjointness restrictions in these positive, capped-cost sums. Thus the entire disjoint error after label averaging is at most \[ C_xe^{C_xD^{-.19}J_c}J_*b_D(1+t^{-3/2}), \tag{101}\] with the deterministic site weights still to be summed.

For overlapping sets, identical labels across the two lists form a matching of their slots, since each individual list has distinct labels. The common first tag cannot participate. A match requires a contact, so the number \(M_*\) of eligible slot pairs is at most \(C_xJ_*\) and at most \(C_xJ_c\). Put \(p_D=C_x/(Df_0)\). Prescribing \(m'\) disjoint matches has total label weight at most \(p_D^{m'}\). To see the joint bound, first fix the two terminal tags. For every prescribed edge involving an interior slot, integrate one of its interior choices using the atom bound \(p_D\). These choices are independent given the tags, and disjoint matching edges use distinct choices. If a terminal-tag pair is prescribed, sum that match last using the subprobability point bound. This proves the bound also when different edges match each tag to an interior slot of the other list.

Write \(q=|S\cap T|\), and let \(\mathbb E_{\rm labels}\) denote the sum with the interior probability kernels and terminal-tag subprobability weights, at the fixed sites and data. Expand \(G_D^q\) as the product of \(1+(G_D-1)\mathbf 1_{\rm match}\) over eligible pairs. To restrict to \(q\ge1\), designate one actual match first. The preceding joint bound, with incompatible prescriptions contributing zero, gives \[\mathbb E_{\rm labels}[\mathbf 1_{\{q\ge1\}}G_D^q] \le M_*p_DG_D\exp\bigl(M_*p_D(G_D-1)\bigr).\] Combining this with (92) bounds the overlap contribution by \[ C_x\frac{G_D}{Df_0}J_* \exp\left(C_xD^{-.19}J_c+C_x\frac{G_D}{Df_0}J_c\right). \tag{102}\] Only matched slots incur a cost; unused route vertices incur none.

The sum of (101) and (102) has the form \[ C_xD^{-1+\zeta(x)}J_*e^{\delta_DJ_c},\qquad \zeta(x)\longrightarrow0\ (x\downarrow0),\quad \delta_D\longrightarrow0\ (D\to\infty), \tag{103}\] where the second limit is for a sufficiently small fixed \(x\). The factors \(b_D(1+t^{-3/2})\) have only polylogarithmic losses, and \(G_D/(Df_0)=C_xD^{-1+x+C\sqrt x}\), so these choices are possible.

We can now use the geometry. For each fixed common site, the deterministic weights of the two noncommon sites are bounded by \(C_xR^{-6}\). Enlarge their sum to a fixed cube of side comparable with \(R\) and compare with its uniform average. Equations (84)–(85) and Cauchy–Schwarz give \[\begin{align*} &\mathbb E_{\rm sites}\mathbb E\bigl[ \mathbf 1_{\rm retained}J_*e^{\delta_DJ_c}\bigr]\\ &\quad\le \Bigl(\mathbb E_{\rm sites}\mathbb E[ \mathbf 1_{\rm retained}\mathbf 1_{\{J_*>0\}}]\Bigr)^{1/2} \Bigl(\mathbb E_{\rm sites}\mathbb E[ \mathbf 1_{\rm retained}J_c^2e^{2\delta_DJ_c}]\Bigr)^{1/2} \le C_xR^{-c_g/2}. \end{align*}\] For the last factor take \(D\) large enough that \(2\delta_D<\kappa/2\) and absorb \(J_c^2\) into \(e^{\kappa J_c/2}\). The common-site weights, each bounded by \(C_xR^{-3}\) on \(O_x(R^3)\) sites, have bounded total mass. All these bounds precede the data average; the site weights are not probabilities conditioned on availability.

Choose \(x\) so small that \(\zeta(x)<wc_g/4\). Since \(R\asymp D^w\), (103) has total expectation \(O(D^{-1-c''})\) for some \(c''>0\). The permissible bad-set base and \(\kappa\) can then be fixed for this \(x\), and finally \(D\) can be increased. The absolute positive exponent \(c_g\) makes this order of choices possible. We have proved \[ \int\ell_v^2\,\,\mathrm dP_{{\rm aug},v} \le \mathbb E[A_vA_{w'}^2]+O(D^{-1-c''})+o_K(1), \tag{104}\] in particular after averaging over \(\mathcal P_R\).

The right lift of the same common measure satisfies (104) with \(v,w'\) exchanged. To verify this, make the right copy primary and read the indices in reverse, while preserving the actual generation order. Integrate out the unobserved pins near its common end; none of the remaining kernels uses them. All conditional comparisons above allow that order. For route averaging, use the common-terminal conclusions of Proposition 18 for the same route law, rather than a different coupling.

Completion of the distant comparison

The reference mass is \(\mathbb EA_v\) and the lift mass is the common mass. Subtract twice (88) from (104) and add the reference mass. This gives, after the indicated cell-pair average, \[ \int(\ell_v-1)^2\,\,\mathrm dP_{{\rm aug},v} \le \mathbb E[A_v(A_{w'}-1)^2]+o(D^{-1})+o_K(1). \tag{105}\] Up to negligible count-tail terms, \(A_v\le CD^x\). Equation (63), together with the uniform marginal distribution of each cell in \(\mathcal P_R\), therefore bounds the average of the main term by \[C\bigl(D^{-1-w/6+x}+D^{-1+2x}R^{-3}\bigr)=o(D^{-1})\] for sufficiently small \(x\) relative to \(w\). The same holds for the right lift.

Let \(q\) be the density of the common projected measure on the bath. Jensen’s inequality under projection preserves both squared likelihood bounds. Since \(P_v\) has density \(s_v^2\), \[\int|s_v-\sqrt q|^2\,\,\mathrm dY' =\int s_v^2\left|1-\sqrt{q/s_v^2}\right|^2\,\,\mathrm dY' \le\int s_v^2|1-q/s_v^2|^2\,\,\mathrm dY'.\] Use absolute continuity for the zero-density convention. Applying the same inequality with \(w'\) and then the squared triangle inequality proves (47).

A variance bound from distant pairs

Let \(s_v\) be the nonnegative bath function defined by (45), for \(v\in\mathcal C_R\). We regard these functions as a finite array with values in the Hilbert space \(L^2\) of the \(N-1\) bath coordinates. Its mean is \(\bar s=\mathbb E_{v\in\mathcal C_R}s_v\).

Apply the lemma with \(M=K/R\) and use (47). The result is \[ \mathbb E_{v\in\mathcal C_R}\|s_v-\bar s\|^2=o(D^{-1})+o_K(1). \tag{106}\] The scale-independent spectral gap here is the reason to average macroscopically separated pairs. Nearest-neighbor squared differences alone would introduce a volume-dependent factor.

Arithmetic means and root mean squares

Put \(F(y,Y')=\sqrt V\,\Phi(y,Y')\), where \(Y'\) denotes the bath coordinates. The first-coordinate spatial measure is now normalized as \(\,\mathrm dy/V\). For every cell \(v\) define \[m_v(Y')=\frac1{|v|}\int_v F(y,Y')\,\mathrm dy,\qquad s_v(Y')=\left(\frac1{|v|}\int_v F(y,Y')^2\,\mathrm dy\right)^{1/2}.\] The second definition agrees with (45). The quantities \(m_v\) are precisely the cellwise values used by \(P_l\). Positivity of \(\Phi\) gives \(0\le m_v\le s_v\) and \((s_v-m_v)^2\le s_v^2-m_v^2\). For microcells, integrating this inequality and using Proposition 5 gives \[ \mathbb E_{v\in\mathcal C_d}\|s_v-m_v\|^2 \le\frac1N\langle\mathrm d\Gamma(W_d)\rangle_\Phi \le D^{-1}(\epsilon_j+o(1)),\qquad \epsilon_j\longrightarrow0. \tag{107}\] We must transfer this estimate to an \(R\)-cell. Averaging the \(m_v\) over its microcells gives its arithmetic mean exactly; averaging the \(s_v\) does not give its root mean square. The next inequality controls that discrepancy with the fourth-order information supplied by (46).

Within an \(R\)-cell \(b\), apply this lemma pointwise in \(Y'\) to the microcell values \(s_v\). Their root mean square is exactly \(s_b\). Join any two microcells of \(b\) by at most \(C_jR\) adjacent steps inside \(b\), telescope their square roots, and use \((\sum_{k=1}^m |a_k|)^4\le m^3\sum_{k=1}^m|a_k|^4\). Even bounding each path sum by the sum over every internal edge gives, after averaging \(b\) and using (46), \[ \mathbb E_{b\in\mathcal C_R} \left\|s_b-\mathbb E_{v\subset b,\ v\in\mathcal C_d}s_v\right\|^2 \le C_jR^6D^{-1-u/8}=o(D^{-1}). \tag{110}\] Here \(R^3\) comes from the fourth-power path inequality and another \(R^3\) from the number of internal edges. The prescribed \(w<u/10000\) ensures \(6w<u/8\). Jensen’s inequality and (107) now give \[ \mathbb E_{b\in\mathcal C_R}\|s_b-m_b\|^2 \le D^{-1}(2\epsilon_j+o(1)). \tag{111}\]

Completion of the depletion asymptotic

Let \(P_0\) be the projection onto the global constant function on the scaled torus, and let a subscript \(1\) indicate action in the first particle coordinate. Centering a Hilbert-valued array is an orthogonal projection and hence a contraction. Combining (106) and (111), we obtain \[ \begin{split} D\|(P_R-P_0)_1\Phi\|^2 &=D\mathbb E_{b\in\mathcal C_R}\|m_b-\bar m\|^2\\ &\le4\epsilon_j+o(1)+o_K(1). \end{split} \tag{112}\] Multiplication of an inner-limit remainder by \(D\) is harmless: at fixed \(\rho\), \(D\) converges to the fixed number \(D_*\). Since \(j\) can be chosen arbitrarily large before the dilution threshold is taken, (112) makes the very-low-frequency contribution negligible.

The projections \(W_R\) and \(P_R-P_0\) are orthogonal and sum to \(1-P_0\). Permutation symmetry and Proposition 5 therefore yield, in the stipulated order of limits, \[ \begin{split} D\bigl(1-\|(P_0)_1\Phi\|^2\bigr) &=\frac1V\langle\mathrm d\Gamma(W_R)\rangle_\Phi +D\|(P_R-P_0)_1\Phi\|^2\\ &=\frac8{3\sqrt\pi}\alpha^{3/2}+o(1). \end{split} \tag{113}\] The equality of the leading term includes both bounds: the second summand is nonnegative, tends to zero, and the first already has the asymptotic displayed in Proposition 5.