Proof. Zero extension maps \(H_0^1(\Omega_n)\) isometrically into \(H^1(\Lambda_K^n)\). On a bounded set in that Sobolev space, the squared \(L^2\) Fourier tail outside frequencies of length at most \(R\) is bounded by \(C R^{-2}\). The remaining Fourier space is finite dimensional. This proves compactness of the form embedding, hence compact resolvent and attainment of the bottom energy. Restricting to the closed symmetric subspace preserves compactness.

For \(f\in H_0^1(\Omega_n)\) define its root-mean-square symmetrization by \[F(X)=\left(\frac1{n!}\sum_{\pi\in S_n}|f(\pi X)|^2\right)^{1/2}.\] The gradient inequality for the Euclidean norm of a vector gives \(\lVert F\rVert_2=\lVert f\rVert_2\) and \(q_n[F]\le q_n[f]\). It also preserves zero Dirichlet data. More explicitly, apply the construction to smooth compactly supported approximants of \(f\). Their vector norms have compact support in \(\Omega_n\) and admit interior smooth approximation. They converge to \(F\) in \(L^2\) and remain bounded in \(H^1\). Weak closedness of \(H_0^1\) and lower semicontinuity prove the assertion and the energy inequality for \(f\). Since \(F\) is symmetric, the bosonic bottom is at most the full bottom. Inclusion of the domains gives the reverse inequality.

The bosonic minimizer \(\Psi\) therefore minimizes the full Rayleigh quotient. First variation against arbitrary full form-domain functions gives its full weak eigenfunction equation. The modulus contraction and the full variational principle yield \[E_N\le q_N[\Phi]\le q_N[\Psi]=E_N, \qquad \lVert\Phi\rVert_2=1.\] Thus \(\Phi\) satisfies the same full weak eigenfunction equation. Interior elliptic regularity makes both functions smooth in the open set. Their almost-everywhere identity \(\Phi=|\Psi|\) holds everywhere there by continuity. Since \(\Delta\Phi=-E_N\Phi\le0\), the strong minimum principle makes \(\Phi\) either strictly positive on a connected component or identically zero there.

On a positive component set \(v=\Psi/\Phi\). It is smooth and has \(|v|=1\), so \(\operatorname{Re}(\overline v\nabla v)=0\) and \[|\nabla\Psi|^2=|\nabla\Phi|^2+\Phi^2|\nabla v|^2.\] There are at most countably many open components. Summing their energy identities and using \(q_N[\Psi]=q_N[\Phi]\) shows that the last, nonnegative term has zero integral on each. Hence \(\nabla v=0\), and \(v\) is constant on that component. On a component where \(\Phi=0\), \(\Psi\) is also zero. This leaves unrelated phases on distinct positive components; no assertion of global connectedness or uniqueness is used.

Finally, multiplication by a bounded Lipschitz function preserves the form domain. Test the weak equation for \(\Phi\) with \(u^2\Phi\) and expand its gradient. Subtracting the resulting identity from the expansion of \(q_N[u\Phi]\) gives Equation (7). ◻

Proof. Choose \(f_j\in C_c^\infty(\Omega_N)\) converging to \(f\) in the form norm, and zero-extend. Fubini’s theorem gives a subsequence converging in \(H^1\) in the unfixed coordinates for almost every bath. For each \(j\), the restricted function is smooth and compactly supported in its allowed slice, because the full support of \(f_j\) is compact in \(\Omega_N\). Taking the limit proves membership in the slice’s \(H_0^1\) space. This proof works for every choice of unfixed coordinates. Dropping obstacles contributed by the fixed coordinates enlarges that domain and preserves membership by zero extension.

The continuous map \(x\mapsto(x,Y)\) sends a connected subset of \(S_Y\) into one connected component of \(\Omega_N\). The final assertion now follows from Lemma 3. In particular it holds whenever two points can be joined by motion of the first coordinate entirely inside \(S_Y\). ◻

Proof. By Lemma 4, the real part of \(\Psi(x,Y)\overline{\Psi(y,Y)}\) equals \(\Phi(x,Y)\Phi(y,Y)\) whenever \(x,y\in U_Y\). This includes the case in which the wave function vanishes there. At all other points their difference is between zero and \(2\Phi(x,Y)\Phi(y,Y)\). Thus the double-integral formula for the occupation gives \[B(\Phi)-B(\Psi) \le\frac2V\int\!\!\int\!\!\int \bigl(\mathbf 1_{A_Y}(x)+\mathbf 1_{A_Y}(y)\bigr) \Phi(x,Y)\Phi(y,Y)\,\,\mathrm dx\,\,\mathrm dy\,\,\mathrm dY.\] For the term containing \(\mathbf 1_{A_Y}(x)\), apply Cauchy–Schwarz to \[V^{-1/2}\Phi(x,Y),\qquad V^{-1/2}\mathbf 1_{A_Y}(x)\Phi(y,Y).\] Their squared norms, with respect to \(\,\mathrm dx\,\,\mathrm dy\,\,\mathrm dY\), are \(1\) and \(\delta_{\rm geom}\), respectively. The corresponding term is therefore at most \(\sqrt{\delta_{\rm geom}}\). Interchanging \(x\) and \(y\) proves the same bound for the other term, giving Equation (8). The final assertion follows by substitution. This argument chooses no phase as a function of the bath; all quantities integrated are already measurable. ◻

Proof. Fix a partner \(y\in Q\). A first-coordinate slice \(f\) vanishes where \(|x-y|\le r\). On a ray \(x=y+s\omega\) in the cube, let \(R(\omega)\) be its terminal radius. Convexity makes the ray intersection an interval, and \(R(\omega)\le\sqrt3l\). For \(r<s<R(\omega)\), \[|f(y+s\omega)|^2 \le\left(\int_r^s t^{-2}\,\mathrm dt\right) \left(\int_r^s|\partial_tf(y+t\omega)|^2t^2\,\mathrm dt\right) \le\frac1r\int_r^{R(\omega)}|\partial_tf|^2t^2\,\mathrm dt.\] Integrating in \(s\) and \(\omega\) proves \[ \int_Q|f|^2\le\frac{\sqrt3l^3}{r}\int_Q|\nabla f|^2. \tag{12}\] Rays ending before \(r\) contribute zero. The argument first applies to smooth restrictions and then to Sobolev slices by approximation. In a torus cube lift, Euclidean distance at most \(r\) also implies torus distance at most \(r\), so the required zero region is present.

For \(n\ge2\), subdivide \(Q\) into \(j^3\) equal cubes, where \(j=\lfloor(n/2)^{1/3}\rfloor\). Then \(n/16\le j^3\le n/2\). In every sector assigning the \(n\) coordinates to these cubes, at most \(j^3\) coordinates are in singleton cubes. For each of the at least \(n/2\) remaining coordinates choose a partner in its cube, using a fixed label ordering. Applying (12) to each such coordinate gives at least \[\frac{rj^3}{\sqrt3l^3}\frac n2 \ge\frac{rn^2}{32\sqrt3l^3}\] times the sector norm. We only restrict the coordinate integrals to sectors; no derivative of a sector indicator is taken. Sum these inequalities over sectors. The cases \(n=0,1\) have zero lower bound, and impossible sectors have zero norm. This proves (9), with the stated smaller constant.

Apply the cube bound to all unit-cell assignment sectors of the labels in \(I\), fixing the other coordinates, and integrate them. This gives (10). Finally, \(\sum_v n_v(n_v-1)\ge N^2/V-N\), so \(E_N/N\ge c_{\rm cube}r(D-1)\ge c_E\alpha\). ◻

Proof. Partition each coordinate circle into \(\lfloor K/(8s)\rfloor\) equal intervals and translate the resulting grid uniformly. Its cube side \(l\) belongs to \([8s,16s]\). Two points at distance at most \(2s\) lie in one cube with probability at least \((1-2s/l)^3\ge27/64\). For each coordinate having a partner in its cube, (12) gives kinetic energy at least \(r/(\sqrt3l^3)\) times its squared norm. Choose a nearest partner for each \(i\) with \(d_i^I\le2s\), and average the restricted inequalities over the grid translation. Each such coordinate is counted with probability at least \(27/64\). Since \(l^3\le4096s^3\), this proves the displayed constant. ◻

Proof. Choose a nondecreasing Lipschitz function \(h_r\) between zero and one, zero up to \(1.1r\), one beyond \(2r\), and with Lipschitz constant at most \(C/r\). For old positions \(X\) put \[f_X(y)=w(y)\min_{1\le i\le n}h_r(d(y,x_i)),\qquad Z(X)=\int f_X(y)^2\,\mathrm dy.\] The empty minimum is one. The union of the relevant radius-\(2r\) balls has volume at most \(Ckr^3\), so \(Z\ge3/4\) under (14). Where a nearest minimizer is unique, only that old coordinate and \(y\) contribute to its cutoff derivative. The same bound holds almost everywhere for the Lipschitz minimum. Therefore \[ \int\left(|\nabla_yf_X|^2+ \sum_{i=1}^n|\nabla_if_X|^2\right)\,\mathrm dy \le C\left(\lVert\nabla w\rVert_2^2+\lVert w\rVert_\infty^2kr\right). \tag{16}\] Indeed cutoff derivatives have squared size at most \(C/r^2\) and are supported in the relevant balls.

Set \(a_X=f_X/\sqrt Z\) wherever \(Z>1/4\). For an old-coordinate derivative, normalization gives the orthogonal-projection identity \[\partial a_X=Z^{-1/2} \bigl(\partial f_X-a_X\langle a_X,\partial f_X\rangle_{L^2(\,\mathrm dy)}\bigr).\] It follows that its squared \(L^2(\,\mathrm dy)\) norm is at most \(Z^{-1}\lVert\partial f_X\rVert_2^2\). The new-coordinate derivative has the same upper bound because \(Z\) does not depend on \(y\). Also \[\int a_X^2\,\mathrm dy=1, \qquad\int a_X\nabla_i a_X\,\mathrm dy=0.\] Thus the cross term vanishes on differentiating \(F(X)a_X(y)\) and integrating \(y\). Equation (16) proves (15).

For a globally defined multiplier replace \(\sqrt Z\) by \(\max(\sqrt Z,1/2)\). The resulting \(y\)-norm is at most one and equals one on a neighborhood of the support where the hypothesis is used. A Sobolev function and its weak gradient vanish almost everywhere on its zero set, so this extension changes none of the preceding integrals. The extra clearance \(1.1r\) and compactly supported approximation of the old function show that \(\widetilde F=Fa_X\) belongs to the enlarged Dirichlet form domain. This also justifies the calculation for arbitrary old form functions. ◻

Proof. Use \(w=V^{-1/2}\) in Lemma 8. For at most \(N\) old particles its norm loss is at most \(CDr^3=C\alpha^3/D^2\), and its energy cost is at most \(CDr=C\alpha\). Starting from the constant one-particle function constructs nonzero trials at every particle number up to \(N\). Apply the variational principle at each addition and sum the costs to obtain (17).

For the last assertion, fix \(k<n\) and average the kinetic energy of all \(k\)-label subsets of a normalized \(n\)-particle form function. Each coordinate is counted with probability \(k/n\), whereas slicing in the other coordinates bounds each subset energy below by \(E_k\). Consequently \((k/n)q_n[F]\ge E_k\); taking the infimum gives \(E_n/n\ge E_k/k\). ◻

Proof. Fix translates \(w_v\) of a smooth normalized real orbital supported strictly inside \(B_v\). Choose translates \(\chi_v\in[0,1]\) supported inside \(B_v\), equal to one on a fixed neighborhood of \(\mathop{\mathrm{supp}}w_v\), and satisfying \(|\nabla\chi_v|^2\le C\chi_v\). Put \(m_v=\sum_i\chi_v(x_i)\). The neighborhood is chosen before \(r\), so for sufficiently small \(r\) any old center whose cutoff ball reaches \(\mathop{\mathrm{supp}}w_v\) has \(\chi_v=1\).

Suppose \(F\) vanishes unless \(m_v\le2\ell D\) for all \(v\in A\). The case \(F=0\) is trivial, so assume \(\lVert F\rVert_2>0\). Insert \(\lfloor\ell D\rfloor\) particles at each of these sites, successively using \(w_v\). At each insertion at most \(3\ell D\) old centers can affect its orbital. Supports at distinct sites have a fixed positive separation, so insertions at other sites cause no additional cutoff. Lemma 8 applies for large \(D\) and gives cost at most \(C(1+3\ell\alpha)\) per inserted particle.

First choose \(\alpha_0\) large, and then \(\ell\) small, so that this cost is at most \(c_E\alpha/2\) throughout the \(\alpha\) window. Writing \(k=\lfloor\ell D\rfloor|A|\), the enlarged trial has norm \(\lVert F\rVert_2\) and energy at most \(q_N[F]+(c_E\alpha/2)k\lVert F\rVert_2^2\). In particular it is nonzero, so \(E_{N+k}<\infty\). The monotonicity in Corollary 9 and (11) give \(E_{N+k}\ge E_N+kE_N/N\ge E_N+c_E\alpha k\). The variational principle therefore proves \[ q_N[F]-E_N\lVert F\rVert_2^2 \ge c\alpha D|A|\lVert F\rVert_2^2, \tag{19}\] where \(c>0\) includes the fixed \(\ell\) and \(D\) is large enough that \(\lfloor\ell D\rfloor\ge\ell D/2\).

Let \(\zeta:[0,\infty)\to[0,1]\) be a Lipschitz function equal to one up to \(1\) and zero from \(2\) onwards, and set \(u_A=\prod_{v\in A}\zeta(m_v/(\ell D))\). Since distinct \(\chi_v\) have disjoint supports, \[|\nabla m_v|^2=\sum_i|\nabla\chi_v(x_i)|^2\le Cm_v, \qquad |\nabla u_A|^2\le\frac C D \sum_{v\in A}u_{A\setminus\{v\}}^2.\] The second estimate uses \(m_v\le2\ell D\) wherever the corresponding cutoff derivative is nonzero. Apply (19) and (7) to \(u_A\Phi\). With \(a(A)=\int u_A^2\,\mathrm d\mu\), this gives \[a(A)\le\frac{C}{D^2|A|}\sum_{v\in A}a(A\setminus\{v\}) \quad(A\ne\varnothing).\] Starting from \(a(\varnothing)=1\), induction on \(|A|\) yields \(a(A)\le(C/D^2)^{|A|}\). Finally \(m_v\le n_v\), so every configuration in the event of (18) has \(u_A=1\). ◻

Proof. Partitioning the particles. Let \(H\ge0\) be an auxiliary random radius with \(\mathbb P(H\ge h)=e^{-h}\). Choose a fixed Lipschitz profile \(\vartheta\) equal to \(\pi/2\) on \((-\infty,0]\), zero on \([1,\infty)\), and with bounded derivative. Set \[a_H(x)=\sin\vartheta(d(x,A)-H),\qquad c_H(x)=\cos\vartheta(d(x,A)-H).\] Thus \(a_H^2+c_H^2=1\), \(a_H=1\) at distance at most \(H\), and its support is within distance \(H+1\). Both squared gradients sum to at most \(C\mathbf 1_{\{H<d(x,A)<H+1\}}\). For \(I\subset\{1,\ldots,N\}\) define \[F_{I,H}(X)=F(X)\prod_{i\in I}a_H(x_i) \prod_{j\notin I}c_H(x_j).\] Expanding the gradients, using \(a_H\nabla a_H+c_H\nabla c_H=0\), gives the exact localization identity \[ \sum_I q_N[F_{I,H}] =q_N[F]+\int\sum_i\bigl(|\nabla a_H(x_i)|^2+ |\nabla c_H(x_i)|^2\bigr)|F|^2. \tag{24}\] The squared norms sum to \(\lVert F\rVert_2^2\).

The energy retained outside the localized set. The index set \(I\) consists of the particles localized near \(A\). For each \(I\), slicing in the coordinates of \(I\) and dropping their exclusions from the others gives \[q_{I^c}[F_{I,H}]\ge E_{N-|I|}\lVert F_{I,H}\rVert_2^2 \ge(E_N-C\alpha|I|)\lVert F_{I,H}\rVert_2^2.\] To sum these removal costs, regard the squared product factors at fixed \(X,H\) as probabilities of independent choices of membership in \(I\), with probabilities \(a_H(x_i)^2\). In particular, \[\sum_I |I|\prod_{i\in I}a_H(x_i)^2 \prod_{j\notin I}c_H(x_j)^2 =\sum_i a_H(x_i)^2.\] Since \[\mathbb E_H a_H(x)^2\le e\,e^{-d(x,A)},\qquad \mathbb E_H\mathbf 1_{\{H<d(x,A)<H+1\}}\le e\,e^{-d(x,A)},\] the mean removal cost and the mean localization error together are at most \(C\alpha\int f_A|F|^2\); we used the fixed positive lower bound for \(\alpha\) to absorb the error without \(\alpha\).

The local energy that pays for removal. The kinetic energy in the coordinates of \(I\) is still available. Apply half of (10) and half of (13) to \(q_I[F_{I,H}]\). Put \(p_v=e^{-d(v,A)}\). If \(H\ge d(v,A)+1\), the whole centered unit cell \(B_v\) is on the \(a_H=1\) side. Summing over \(I\) and averaging \(H\) therefore makes the first term at least \[cr\int\sum_v p_v n_v(n_v-1)|F|^2.\] For a coordinate counted by \(k_A\), some partner lies within \(2s_0<1\), and hence both are at distance less than \(7\) from \(A\). On \(H\ge8\) both labels are certainly in \(I\). This event has fixed probability \(e^{-8}\), so the second kinetic term is at least \[crs_0^{-3}\int k_A|F|^2 =c\alpha b^{-3}\int k_A|F|^2.\] In particular the partner used here belongs to the removed set before the subset-label inequality is applied.

Combining these estimates with (24) gives \[ q_N[F]-E_N\lVert F\rVert_2^2 \ge\int\left(cr\sum_vp_v n_v(n_v-1) -C\alpha f_A+c\alpha b^{-3}k_A\right)|F|^2. \tag{25}\] Leaving a positive count penalty. Within each unit cell its distance to \(A\) differs from its center’s distance by at most \(1\). Consequently \[f_A\le e\sum_vp_v n_v, \qquad \sum_vp_v\le\sum_{w\in A}\sum_v e^{-d(v,w)}\le C|A|.\] The last lattice exponential sum is uniformly bounded on every periodic unit grid. Since \(r=\alpha/D\), completing a scalar square gives, for all integers \(n\ge0\) and \(D\ge1\), \[\frac cD n(n-1)-(C+1)e n\ge-C'D.\] Multiply by \(\alpha p_v\) and sum. In (25) this leaves the positive term \(\alpha f_A\) and proves (23). None of the constants in this calculation depends on \(b\). ◻

Proof. Distance is \(1\)-Lipschitz and \(e^{-d}\le1\), so \(|\nabla f_A|^2\le f_A\) almost everywhere. Let \(M(z)=\int e^{zf_A}\,\mathrm d\mu\). Apply (7) and (23) with \(u=e^{zf_A/2}\), dropping the nonnegative close-count term. This gives \[\alpha\bigl(M'(z)-CD|A|M(z)\bigr) \le\frac{z^2}{4}M'(z).\] Our choice of \(\alpha_0\) ensures \(\alpha\ge1\). For \(0\le z\le1\) we therefore obtain \(M'(z)/M(z)\le CD|A|\). Integrating from \(M(0)=1\) proves (26). These differentiations are ordinary finite-volume ones, since \(0\le f_A\le N\).

For the second statement first assume that distinct sites of \(A\) have distance greater than \(20\). Choose translates \(a_v\in[0,1]\), equal to one on the radius-\(5\) ball and supported on the radius-\(6\) ball, with \(|\nabla a_v|^2\le Ca_v\). Let \(\chi\in[0,1]\) equal one on \([0,1]\), zero on \([2,\infty)\), and satisfy \(|\chi'|^2\le C\chi\). Such cutoffs are squares of fixed Lipschitz profiles. Define \[\widetilde m_v(X)=\sum_i a_v(x_i)\chi(d_i/s_0).\] Then \(\sum_{v\in A}\widetilde m_v\le k_A\), and each close count in (27) is at most \(\widetilde m_v\).

We verify the gradient estimate \[ |\nabla\widetilde m_v|^2\le Cs_0^{-2}\widetilde m_v. \tag{29}\] Away from a null set, every nearest neighbor is unique. If \(x_i,x_j\) have the same nearest neighbor \(x_k\), with the first incoming distance at least the second, then \(|x_i-x_j|\ge|x_i-x_k|\). The cosine law shows that the two directions at \(x_k\) have angle at least \(\pi/3\). For the edges of length at most \(2s_0\) used by the derivatives, consistent Euclidean lifts exist because \(s_0<1/2\) and \(K\ge100\). Packing directions on the unit sphere bounds the incoming degree by a numerical constant. Each coordinate derivative therefore meets only a bounded number of summands of \(\widetilde m_v\). The inequalities for \(a_v,\chi\) bound the squared full gradient of each summand by \(Cs_0^{-2}a_v(x_i)\chi(d_i/s_0)\). Summing with bounded overlap proves (29). All coordinates in those derivatives are within \(6+2s_0<7\) of \(v\).

Choose a Lipschitz increasing function \(\eta\) equal to zero on \([0,1/2]\) and one on \([1,\infty)\), and put \[u_A=\prod_{v\in A}\eta(\widetilde m_v/(\theta D)), \qquad a(A)=\int u_A^2\,\mathrm d\mu.\] On the nonzero set of \(u_A\) we have \(k_A\ge(\theta D/2)|A|\). Choose \(b\) so small that the constants in (23) satisfy \(cb^{-3}\theta/2-C\ge1\). That form inequality gives \[q_N[u_A\Phi]-E_N\lVert u_A\Phi\rVert_2^2 \ge\alpha D|A|a(A).\] At almost every configuration, gradients belonging to distinct sites involve disjoint particle coordinates, by the separation of the sites. Using (29) on the transition region \(\widetilde m_v\le\theta D\) therefore gives \[|\nabla u_A|^2\le\frac{C}{\theta Ds_0^2} \sum_{v\in A}u_{A\setminus\{v\}}^2.\] Equation (7) and induction from \(a(\varnothing)=1\) now imply \[a(A)\le\left(\frac{C}{\alpha\theta D^2s_0^2}\right)^{|A|} \le\left(C_\theta b^{-2}D^{-4/3}\right)^{|A|}.\] On \(A\subset\mathcal Q\), every cutoff equals one, proving this bound for separated \(A\).

Finally the graph joining distinct lattice sites at distance at most \(20\) has uniformly bounded degree, also on the torus. It admits a coloring with an absolute number \(Q\) of colors. Every prescribed \(A\) contains a separated color class \(A'\) of size at least \(|A|/Q\). The event \(A\subset\mathcal Q\) implies \(A'\subset\mathcal Q\). Applying the separated estimate, with its base at most one, proves (28) for all \(A\). ◻

Proof. For the Dirichlet Laplacian on the open allowed configuration set \(\Omega\), the killed heat kernel represents \(e^{-sH_N}\). This representation applies to the exact form domain \(H^1_0(\Omega)\). One way to see that no regularity assumption on the collision boundary is needed is to exhaust \(\Omega\) by increasing smooth relatively compact open sets. On each set the stopped heat equation gives the killed representation. The union of their form domains is dense in \(H^1_0(\Omega)\), since every smooth compactly supported function is supported in one exhaustion member. The variational characterization of resolvents therefore gives strong resolvent convergence, and then strong heat-semigroup convergence. A continuous path remaining in \(\Omega\) throughout a compact time interval has compact image in \(\Omega\), hence lies in one exhaustion member. Monotone convergence on paths identifies the limiting kernel.

The integral in Equation (30) is \(e^{2tE_N}\langle\Phi,e^{-2tH_N}\Phi\rangle=1\). Splitting the killed kernel at time \(u\) gives the marginal density \[e^{2tE_N}(e^{-uH_N}\Phi)(x) (e^{-(2t-u)H_N}\Phi)(x)=\Phi(x)^2.\] Integrating an added initial or terminal interval proves the restriction statement. Symmetry of the killed kernel proves reversal. At fixed endpoints the two factors \(\Phi\) are constant, so disintegration of the free measure leaves exactly its independent bridges and the avoidance indicator. Resolving the endpoint heat kernel into its Euclidean translates gives the last assertion. All conditional statements are understood for almost every datum under its stated law. ◻

Proof. First take bounded \(h\). Drop every collision restriction involving a label in \(I\). For fixed selected endpoint vectors \(y,z\), integrate the other paths with their killed semigroup. The resulting matrix element between the corresponding slices of \(\Phi\) is at most \[e^{-2tE_{N-m}}a(y)a(z),\qquad a(y)^2=\int\Phi(y,x')^2\,\mathrm dx'.\] The free selected paths, weighted by \(\prod h\), give a symmetric endpoint kernel, by reversal invariance. More explicitly, let \(\nu_I\) be their product free-path measure with Lebesgue initial measure, and write \(y=\omega_I(0)\) and \(z=\omega_I(2t)\). Then \[\begin{align*} \int a(y)a(z)\prod_{i\in I}h(\omega_i)\,\mathrm d\nu_I &\le\frac12\int\bigl(a(y)^2+a(z)^2\bigr) \prod_{i\in I}h(\omega_i)\,\mathrm d\nu_I\\ &=\int a(y)^2\prod_{i\in I}\mathbb E_{\rm free}^{y_i}h\,\,\mathrm dy. \end{align*}\] Here the equality reverses the paths in the term containing \(a(z)^2\); independence is used only under the free measure. By the definition of \(a\), the last integral is \(\int\mu(\,\mathrm dY)\prod_{i\in I}\mathbb E_{\rm free}^{Y_i}h\). Reinstating the factor \(e^{2t(E_N-E_{N-m})}\) and applying Corollary 9 proves Equation (31). Applying this to \(h\wedge M\) and letting \(M\) increase proves the general case. ◻

Proof. For Brownian motion, the variables \(A_j\) use independent increments. Their tails are uniformly Gaussian, since \(\tau/r^2\in[1/2,1]\). Thus the maximum condition fails with probability at most \(C m_t e^{-cD^{2\gamma}}\). Put \(U_j=(1+A_j)^3\). Their moments of every fixed order are uniformly bounded. In the expansion of the \(2k\)th moment of \(\sum_j(U_j-\mathbb EU_j)\), a nonzero term uses each of its indices at least twice; there are at most \(C_k m_t^k\) such terms. Consequently \[\mathbb E\left|\frac1{2m_t}\sum_j(U_j-\mathbb EU_j)\right|^{2k} \le C_k m_t^{-k}.\] Choose \(C_{\rm reg}\) greater than a fixed multiple of \(1+\sup\mathbb EU_j\). Since \(m_t\) is comparable to \(D^2\) at fixed \(t\), Markov’s inequality proves every desired polynomial bound for the average condition. On an endpoint window the maximal Gaussian bound has exponent a constant times \[s_0^2/h_0=b^2D^{1/3},\] which proves the remaining free estimate.

For a bridge leg of duration \(t\), write its centered part as \(W(u)-(u/t)W(t)\), using a Brownian motion \(W\), and add its fixed endpoint line. The extra increment on a mesh interval, divided by \(r\), is at most \[\frac{r}{t}\bigl(|W(t)|+C\bigr).\] Outside an event of arbitrarily small polynomial probability this tends to zero, uniformly for bounded endpoint displacement. The inequality \((x+y)^3\le4x^3+4y^3\) transfers the averaged cutoff bound after the numerical choice of \(C_{\rm reg}\) is enlarged once. On an endpoint window the extra displacement is \(h_0(|W(t)|+C)/t=o(s_0)\) with the same probability control. These estimates prove the assertion for two legs, using their two Brownian representations and a union bound. A duration-\(2t\) bridge uses the identical representation with \(2t\) in place of \(t\).

Finally the endpoint lines of the two legs remain in the radius-\(3\) ball. Leaving the radius-\(10\) ball requires a centered fluctuation of size at least seven on one leg. The Brownian representation and the maximal Gaussian bound give the stated \(Ce^{-c/t}\) estimate. ◻

Proof. Mark the lifted Brownian path at its initial point and at each first exit from the unit ball about its last mark. If \(Q\) is the number of completed exits by time \(2t\), then \(J_A\le C(Q+1)\): between successive marks the path remains in a unit ball, so it can visit radius-\(12\) balls about only a bounded number of lattice sites. Projection to the torus cannot increase this bound.

The exit durations are iid copies of a positive random variable \(\sigma\). For \(s>0\), \[\mathbb P(Q\ge k)\le e^{2s}(\mathbb Ee^{-s\sigma})^k.\] Since \(\mathbb Ee^{-s\sigma}\to0\) as \(s\to\infty\), \(Q\) has every fixed exponential moment, uniformly for \(t\le1\). Moreover \(J_A>0\) requires a lifted displacement at least \((d(y,A)-12)_+\), and hence has a Gaussian tail in that distance. Using \(e^{\lambda J_A}-1\le\lambda J_A e^{\lambda J_A}\) and Hölder with three equal exponents gives \[\mathbb E[(e^{\lambda J_A}-1)\mathbf 1_{\mathcal D}\mathbf 1_{\{J_A>0\}}] \le \lambda\bigl(\mathbb E[J_A^3e^{3\lambda J_A}]\bigr)^{1/3} \varepsilon_{\mathcal D}^{1/3} \mathbb P(J_A>0)^{1/3}.\] The exponential moment and Gaussian travel bound imply Equation (34), after changing its constant. ◻

Proof. Set \(h(\omega)=e^{\lambda J_A(\omega)\mathbf 1_{\mathcal D}(\omega)}-1\). Expand \(\prod_i(1+h(\omega_i))\) over subsets of labels and apply Lemma 14 to each term. Lemma 17 then bounds the result by \[\int\mu(\,\mathrm dY)\prod_i [1+e^{Ct\alpha}\zeta e^{-d(Y_i,A)}] \le \int e^{e^{Ct\alpha}\zeta f_A}\,\mathrm d\mu, \qquad f_A=\sum_i e^{-d(Y_i,A)}.\] Equation (26) applies with the indicated exponent at most one, and proves Equation (35). ◻

Proof. We bound simultaneous failures of each individual test first. The low-count estimate in Proposition 10 and the close-count estimate in Equation (28) apply at both endpoints by Lemma 13. Their per-site parameters tend to zero as \(D\) increases.

For excessive visiting paths use \(\mathcal D\) equal to all paths. Choose a small fixed \(\lambda>0\) so that \(e^{Ct\alpha}C_\lambda\lambda\le1\) for every \(t\le1\) and the allowed \(\alpha\). If every site in \(A\) has more than \(H_1D\) visitors, then \(\sum_iJ_A(\omega_i)>H_1D|A|\). Equation (35) and exponential Markov inequality give \[\mathcal R(\text{this event}) \le\exp\{-[\lambda H_1-Ce^{Ct\alpha}C_\lambda\lambda]D|A|\}.\] Take \(H_1\) large enough to make the bracket positive uniformly.

For visitors of diameter greater than one take \(\lambda=1\) and \(\mathcal D=\{\mathop{\mathrm{diam}}\widetilde\omega>1\}\). Its free probability is at most \(Ce^{-c/t}\). Choose \(t\) small enough that the right-hand coefficient in Equation (35) is at most \(\theta/2\). If every site in \(A\) has at least \(\theta D\) such visitors, their sum of visits is at least \(\theta D|A|\), so the probability is at most \(e^{-\theta D|A|/2}\).

Finally let \(\mathcal D\) be failure of regularity. For any fixed large \(\lambda\), Lemma 16 gives \(D\zeta\to0\). If every site in \(A\) has a nonregular visitor, the sum of visits by such paths is at least \(|A|\). Thus Equation (35) bounds this probability by \(\exp[-(\lambda-o(1))|A|]\). Choosing \(\lambda\) first and then \(D\) makes its per-site parameter as small as desired. The same increase of \(D\) makes the preceding five individual failure bounds arbitrarily small.

There are six failure types. If every site of \(A\) fails some test, assign one type to each failing site. In each of the at most \(6^{|A|}\) assignments, one type occurs on a subset of size at least \(|A|/6\). If each single-type bound has parameter at most \(\varepsilon\), the union has probability at most \((6\varepsilon^{1/6})^{|A|}\). Choose \(\varepsilon\le(p_1/6)^6\). This proves Equation (36). The free estimates remain valid when the already chosen \(t\) is reduced and \(D_0\) is then increased, which proves the final assertion. ◻

Proof. The means are \(a y_i+(1-a)u\), where \(a=1-s/t\ge1/2\), and the covariance matrix is \(2s(1-s/t)\) times the identity. The means are therefore \(s_0/2\)-separated. For points \(q_i\) separated by \(s_0/2\), disjoint balls of radius \(s_0/4\) give \[\#\{i:|q_i-x|\le R\}\le C(1+R/s_0)^3.\] Split the Gaussian sum into the ball of radius \(\sqrt{s}\) about \(x\) and the annuli with radii \(j\sqrt{s},(j+1)\sqrt{s}\), \(j\ge1\). The last bound, multiplied by the Gaussian decay on each annulus, yields \[\sum_i \exp\!\left(-\frac{|x-q_i|^2}{4s}\right) \le C\left(1+\frac{s^{3/2}}{s_0^3}\right).\] The covariance lies between \(s\) and \(2s\) times the identity, so the averaged density is at most \[\frac{C}{m s^{3/2}}+\frac{C}{m s_0^3} \le C\left(\frac{1}{D s^{3/2}}+1\right).\] ◻

Proof. We estimate the first leg from an outer endpoint \(y_i\) to the random midpoint \(u\); reversing time gives the identical estimate for the leg from \(z_i\) to \(u\). On \(\mathcal C_v\), the endpoint windows of both paths have diameter at most \(s_0/8\). Endpoint separation consequently leaves distance at least \(3s_0/4>r\) throughout \([0,h_0]\). Every mesh interval on which a collision remains possible starts at \(s\ge h_0-\tau\ge h_0/2\).

We first explain the use of lifts. On a mesh interval, the obstacle has diameter at most \(rD^\gamma<1\). A collision with a confined replacement therefore requires \(d(c(s),v)<12\) at the interval’s starting time. For \(K>50\) this determines a unique lift \(\widetilde c(s)\) within distance \(12\) of the chosen lift of \(v\); continue that lift over the interval. Its displacement is at most \(rA_k(c)\). Intervals for which no such lift exists make no contribution. We may thus use Euclidean Gaussian densities without summing over windings.

Proof. For \(Q_{y,z}^{B}\), repeat the preceding proof with one endpoint. At \(s\le t/2\) its Gaussian density is at most \(Cs^{-3/2}\); the bound (44) becomes \(Cr^3(1+A_k(c))^3s^{-3/2}\). The second half of each leg retains (45). The same two sums give the first inequality.

For \(Q_{y,z}^{0}\), use the forward orientation up to time \(t\) and the reversed orientation for the rest. In either orientation the remaining bridge duration is at least \(t\). Formula (41) holds with \(2t\) in place of \(t\), its drift on the cutoff event is at most \(C\tau/t\), and its time-\(s\) density is bounded by \(Cs^{-3/2}\) for \(0<s\le t\). Endpoint separation again excludes the window \([0,h_0]\). Consequently the first-half argument and its mesh sum apply over both orientations, giving \(CrD^{3\gamma}/\sqrt{h_0}\). Adding the nonnegative term \(Ctr\) yields the stated common bound. Finally \(r=\alpha/D\), \(h_0=D^{-1}\) and \(3\gamma=1/4\) give the last inequality. ◻

Proof. The linear means of both bridge legs stay within distance \(3\) of \(v\). The Gaussian maximal bound therefore makes confinement failure at most \(Ce^{-c/t}\). Lemma 16 gives a regularity failure probability \(\varepsilon_D(t)\to0\), uniformly over the listed endpoints and the target points. A confined trial can collide only with an obstacle visiting the ball of radius \(12\), since \(r<1\). Union-bounding over those obstacles and using Proposition 21, with a self-label summand omitted when necessary, gives for each target cell \[\begin{align*} \frac1m\sum_i\int_B Q_{y_i,z_i}^{u}(\text{failure})\,\mathrm du &\le Ce^{-c/t}+\varepsilon_D(t) +CH_1D\left(tr+\frac{rD^{3\gamma}}{D\sqrt{h_0}}\right) \\ &\le Ce^{-c/t}+\varepsilon_D(t) +CH_1\alpha\bigl(t+D^{-1/4}\bigr). \tag{48}\end{align*}\] The constant multiplying \(t\) is independent of small \(t\), which is the reason for the two parts of the bridge estimate. First choose \(t\) small and then \(D\) large so that the last bound is less than \(1/70000\) for every \(\alpha\) in the fixed window.

If \(|T_{i,B}|<99/100\), the integral of the failure probability over \(B\) is greater than \((1/100)(1/10)=1/1000\). Thus fewer than \(m/70\) labels fail the conclusion for a specified target cell. There are seven target cells, so fewer than \(m/10\) labels fail for at least one of them. All the remaining labels have the asserted property. ◻

Proof. Start with the at least \(\ell D\) labels having \(Y_i\in B_v\). Every such path visits the radius-\(12\) ball and is therefore regular. Discard the fewer than \(\theta D\) visiting paths of diameter greater than one. The remaining paths stay within distance \(2\) of \(v\) and end within distance \(2\) of \(v\). Discard also those whose initial or terminal endpoint has another endpoint within distance \(s_0\). The certificate bounds these two losses by \(\theta D\) each, since the endpoints in question lie in the radius-\(5\) ball about \(v\). The remaining labels are eligible, and their number is at least \((\ell-3\theta)D\ge\ell D/2\).

We now apply Corollary 24 to the entire eligible list at this site, not merely the subset just used to prove its lower cardinality bound. Every eligible label has its initial endpoint in \(B_v\), terminal endpoint within distance three of \(v\), and both endpoints separated from all corresponding endpoints of other labels by more than \(s_0\). The certificate supplies at most \(H_1D\) visiting obstacles, all regular. These are precisely the hypotheses of that corollary. It gives Equation (50) for at least nine tenths of the entire eligible list, with choices uniform in the certificate. ◻

Proof. Condition first on the endpoints. By Lemma 13, the free reference is a product of torus bridges and the remaining interaction is the hard-avoidance indicator. At fixed endpoints, eligibility of an individual path consists of its own regularity and confinement conditions: its starting cell, endpoint displacement and endpoint separation are already fixed. The probabilities of all allocations depend only on the recorded eligibility indicators, and are consequently constant on the conditional fiber.

Now expose all paths outside \(\mathcal I\). Every retained label was eligible, so the only remaining individual restrictions are exactly its two cutoffs and avoidance of the fixed frozen paths. Confinement to its radius-\(10\) ball selects a single Euclidean lift when \(K\) is large. The winding weights of the original torus bridge then depend only on the fixed endpoints and cancel in normalization. The remaining nonfactorizing restriction is mutual avoidance inside \(\mathcal I\). This proves Equation (53) by disintegration on the continuous-path spaces. Equivalently, integrate the asserted formula against bounded functions of the exposed and unexposed variables and use Fubini; all factors just identified reproduce Equation (30) and the allocation probabilities.

The integral of the conditional density is positive for almost every sampled datum. Its positivity implies positivity of each \(a_i\) and of \(Z_{\mathcal I}\); otherwise that fiber would have zero weight. Finally a retained trajectory stays within distance ten of its home site. Two such tubes can meet at distance \(r<1\) only when their home sites are within a fixed lattice distance. There are at most \(M\) labels per site and a fixed number of such sites, proving the last assertion. In particular Equation (53) is a conditioned product, not a claim that its trajectories are independent. ◻

Proof. Choose \(p_1<p/2\) in Proposition 19 and meet also the requirements of Lemma 25. Fix the full path sample before the allocation randomizations. At a certified cell there are at least \(\ell D/2\ge MG_D\) eligible labels, and at least nine tenths satisfy Equation (50). For a fixed group, its \(M\) labels form a uniform sample without replacement from this list. For any \(k\) specified positions in that sample, the probability that all \(k\) lack the stronger property is at most \(10^{-k}\). A union bound therefore gives \[ \mathbb P\{\text{fewer than $M/2$ stronger labels in the group} \mid\text{full sample}\} \le q_M:=2^M10^{-M/2}. \tag{56}\] This tends to zero exponentially in \(M\).

A label with the stronger property is accessible after allocation: its frozen obstacle family is a subset of all other sampled paths, so removing obstacles can only increase its success probabilities. Moreover the auxiliary sampling tests at distinct certified cells are independent conditional on the full sample. For each subset \(A_0\subset A\), designate the cells of \(A_0\) as lacking certificates and the other cells as certified but failing the sampling test. The probability of this event is at most \(p_1^{|A_0|}q_M^{|A\setminus A_0|}\), by conditioning first on the full sample and then using Equation (36). Summing over \(A_0\) bounds the probability in Equation (55) by \((p_1+q_M)^{|A|}\). Take \(M\) large enough that \(q_M<p/2\). The argument applies to each group without conditioning on its tag or on its unexposed trajectories. ◻

Proof. The midpoint density \(g_i\) of \(B_i\) is the Euclidean Gaussian with mean \((Y_i+Z_i)/2\) and covariance \(tI\). On every target cell it has a lower bound \(c_t>0\), uniform in the eligible endpoint data. Indeed those endpoints are within distance three of \(v\), and every target cell lies in a fixed bounded neighborhood of \(v\); the explicit Gaussian density gives such a bound. Disintegrating \(B_i\) at its midpoint gives the conditional laws \(Q_i^u\). Hence \[a_i=\int g_i(u)Q_i^u(C_i(F))\,\mathrm du \ge 0.99\cdot0.9\,c_t=:a_t>0.\] At a successful trajectory with midpoint \(u\), the proposal density relative to \(P_i\) is \[\frac{a_i\mathbf 1_U(u)}{|U|g_i(u)Q_i^u(C_i(F))} \le \frac1{0.98\cdot0.9\,c_t},\] since \(a_i\le1\). Taking \(G\) to be the maximum of two and this constant proves Equation (57); its final identity holds because both measures are probabilities.

For collision with the fixed opponent, first use the unconditioned bridge \(B_i\). The event contributing under \(P_i\) satisfies the regularity and confinement cutoffs. Proposition 22 bounds its probability by \(C(tr+rD^{3\gamma}h_0^{-1/2})\). Dividing by \(a_i\ge a_t\) proves the required estimate for \(P_i\). For the proposal, integrate that proposition’s two-leg estimate over the whole target cell; restricting the integral to \(U\) can only decrease it. Division by \(|U|\ge0.98\) and by the through-\(u\) success probability, which is at least \(0.9\), proves the proposal estimate. The endpoint separation required in that proposition holds because \(i\) was eligible and the opponent has a distinct recorded label.

Finally \(\gamma=1/12\), \(h_0=D^{-1}\) and \(r=\alpha/D\) give \(rD^{3\gamma}h_0^{-1/2}=\alpha D^{-1/4}\) and \(tr=O_t(D^{-1})\). The \(\alpha\)-window is fixed. This proves Equation (58), with constants depending on the already fixed \(t\) but not on the later choice of \(M\) or the dilution threshold. ◻

Proof. A vertex outside the coordinate box has a coordinate ray to infinity that misses \(C\), so it is not in a finite complementary component. For the shell’s connectivity we apply Kesten’s boundary-connectivity theorem in the formulation of Timár (Timár 2013, arXiv version 2, Theorem 4): for a finite \(*\)-connected subset of \(\mathbb Z^3\), its exterior \(*\)-neighbors, accessible from infinity through its nearest-neighbor complement, are nearest-neighbor connected. These are exactly the vertices in the shell just defined. There are at most \(26\) possible \(*\)-neighbors per vertex of \(C\). A bad shell vertex would be \(*\)-adjacent to \(C\) and hence belong to the same bad component, which is impossible for a vertex outside \(C\). ◻

Proof. Choose a spanning tree, lift its root, and lift each tree edge by its unique displacement in \([-a,a]^3\). Distinct torus vertices have distinct lifts. The difference between any two lifted vertices has maximum norm at most \(a(m-1)\), by their tree path. For a remaining edge, its assigned endpoint difference minus its short displacement belongs to \(K\mathbb Z^3\) and has maximum norm at most \(am<K\). It must therefore vanish. All edges are consistent with these lifts. The full inverse image is their disjoint translates by \(K\mathbb Z^3\); no edge connects two translates. ◻

Proof. With densities \(p,q,s\) relative to a common dominating measure, Hölder’s inequality with exponents \(4,4,2\) gives \[m=\int \left(\frac{s^2}{p}\right)^{1/4} \left(\frac{s^2}{q}\right)^{1/4}(pq)^{1/4} \le C_P^{1/4}C_Q^{1/4}\mathsf H(P,Q)^{1/2}.\] The ratios can be set to zero where their denominators vanish, since \(s\) is zero there. Squaring and rearranging proves the claim. ◻

Proof. Put all groups on the original stationary sample, with all allocation randomizations and a fresh uniform permutation of the full label set. After deleting a tag, the restricted permutation uniformly orders its bath. Each group’s conditional construction is a disintegration of this common sample with respect to its own exposed data. Thus the differing data spaces are projected away before the following comparison; no conditional symmetry of the unexposed-path law is asserted. For every sampled path configuration, group label sets are disjoint. Their sum of tag contributions is bounded by the sum over all \(N\) trajectory labels. The unrestricted midpoint vector has the symmetric density \(\Phi^2\). For each fixed tag label, uniform ordering of the bath therefore gives the same density \(\Phi(x,Y)^2\). Summing all tags gives \(N\) times that density. Since the extra target cell has volume one and \(V/N=1/D\), projecting away the data yields \[\frac1D\sum_g\operatorname{proj}\widehat P_g\le P_{vw}, \qquad \frac1D\sum_g\operatorname{proj}\widehat Q_g\le Q_{vw}.\] Use monotonicity, (75), and the increase of affinity under projection. Its homogeneity supplies the factor \(1/D\). ◻

Proof. Put \(a_\chi=1-\chi\) and \(c=|C_S|+|C_T|\). Fix the primary coordinates outside \(U\). All normalizers below are conditional on these fixed values; we first work on a configuration for which this conditional law is defined.

For any selected set, its reference normalizer is the integral of its reference-check indicator against its product references. In particular, write \(Z_U\) for this integral over \(U\). When the coordinates outside \(S\) are fixed, write \(Z_S\) for the corresponding integral over \(S\). Thus \(Z_S\) can depend on coordinates in \(T\setminus S\). Write \(Y_S\) for the integral of the proposal-check indicator \(J_S\) against \(\bigotimes_{i\in S}\Pi_{S,i}\), with the same outside values fixed.

Disintegrate each factor in the form \[\Pi_{S,i}(\,\mathrm dx_i\,\,\mathrm d\zeta_i) =P_i(\,\mathrm dx_i)K_{S,i}(x_i,\,\mathrm d\zeta_i), \qquad K_{S,i}(x_i,\text{all})=g_{S,i}(x_i).\] Here \(\zeta_i\) denotes all its auxiliary coordinates. Define \[H_S(x_S;x_{S^c}) =\int J_S\prod_{i\in S}K_{S,i}(x_i,\,\mathrm d\zeta_i).\] On the reference-admissible configurations, the exact likelihood is \[ L_S=\frac{Z_S}{Y_S}H_S. \tag{83}\] Indeed, the conditional reference density is its check indicator divided by \(Z_S\), whereas the new primary density is \(H_S/Y_S\), both relative to \(\bigotimes_{i\in S}P_i\). The outside marginal is identical. The inclusion of primary reference checks ensures that the new density is zero on reference-forbidden configurations.

Now retain only the factors indexed by \(F_S\), together with coordinates fixed outside \(U\), using the stated deletion convention. Denote their reference normalizer, proposal normalizer and primary proposal density by \(Z_{F_S},Y_{F_S},H_{F_S}\). Define the three corresponding \(F_T\) quantities using the factors from the \(T\) change. These quantities depend only on the outside of \(U\) and, for \(H_{F_S},H_{F_T}\), on their own primary coordinates. They do not depend on the omitted near coordinates.

Successive insertion using assumption (i), and noninteraction in assumption (ii), give \[\begin{align*} Z_U&\ge a_\chi^c Z_{F_S}Z_{F_T}, \tag{84}\\ Z_S&\le Z_{F_S},\qquad Y_S\ge a_\chi^{|C_S|}Y_{F_S}, \tag{85}\\ Z_T&\le Z_{F_T},\qquad Y_T\ge a_\chi^{|C_T|}Y_{F_T}. \tag{86}\end{align*}\] For the first inequality, the reference integral over the two far sets factorizes; add every remaining distinct coordinate with its uniform compatibility bound. There are at most \(c\) such coordinates. For the upper bound on \(Z_S\), delete the checks involving \(C_S\) and integrate those reference coordinates freely. Coordinates in \(T\setminus S\) have no checks with the far set. For its proposal lower bound, start with the reduced \(F_S\) integral and add the \(C_S\) factors. The fixed original coordinates in \(T\setminus S\) still do not meet the far factors. Each addition retains at least the factor \(a_\chi\). Applying these two arguments to the \(T\) change proves the last line. These statements are pointwise in the opposing coordinates; they do not assert that \(Z_S\) or \(Y_S\) is constant in them.

The insertion bounds also prove positivity of the divisors. In particular, integrating coordinates one at a time gives a positive product lower bound for every reduced normalizer. The full reference conditional is defined on the outside configurations under consideration.

Deleting proposal checks involving a near factor gives \[H_S\le H_{F_S}\prod_{i\in C_S}g_{S,i}, \qquad H_T\le H_{F_T}\prod_{i\in C_T}g_{T,i}.\] After dropping the reference admissibility indicator in an upper bound, the two far primary integrals equal \(Y_{F_S}\) and \(Y_{F_T}\). The remaining coordinates carry only the displayed marginal densities. A coordinate occurring once integrates to one. A coordinate occurring in both lists costs at most \(G\), by (81). Therefore \[ \int H_SH_T\mathbf 1_{\mathrm{admissible}} \prod_{i\in U}P_i(\,\mathrm dx_i) \le Y_{F_S}Y_{F_T}G^{|S\cap T|}. \tag{87}\] The conditional reference law on \(U\) has normalizer \(Z_U\). Combine (83)–(87). The two pointwise ratio bounds contribute \(a_\chi^{-c}Z_{F_S}Z_{F_T}/(Y_{F_S}Y_{F_T})\); division by \(Z_U\) contributes at most \(a_\chi^{-c}/(Z_{F_S}Z_{F_T})\). All four far normalizers cancel, leaving \(G^{|S\cap T|}a_\chi^{-2c}\). Finally integrate the outside coordinates. This proves (82). ◻

Proof. For large \(K\), at least \(\upsilon_0V^2\) ordered cell pairs meet the displacement condition of Proposition 29. For each such pair, Lemma 33, (91) and (94) give \[\mathsf H(P_{vw},Q_{vw}) \ge\frac{G_D}{4D C_2} \ge\frac\ell{32M C_2}.\] Sum these nonnegative contributions in (77) to obtain \(B(\Phi)\ge\upsilon_0\ell/(32M C_2)\), which is (95). All ingredients are uniform in the ground vector. The threshold and local-uniformity assertions follow from the preceding path and group estimates and Proposition 29. No parameter entering \(c_*\) is chosen after \(D_0\). ◻

Proof. There is nothing to prove when \(G(Y)\) is empty. Otherwise the bath is admissible and all lifted bad components are finite. Choose lifted microcube sites containing the two points. Each lies outside the fill of every lifted bad component, and hence in its infinite nearest-neighbor complement component.

Start with any finite nearest-neighbor path between these sites. If it visits a bad component \(C\), let \(u\) precede the first visit and let \(w\) follow the last visit. The path prefix before \(u\) avoids \(C\) and connects \(u\) to an endpoint outside its fill. The reversed suffix does the same for \(w\). Thus both vertices belong to the exterior \(*\)-boundary of \(C\) visible from infinity by nearest-neighbor paths. Lemma 30 joins them by a nearest-neighbor path in this boundary. All its vertices are good: a bad \(*\)-neighbor of \(C\) would already be in \(C\).

This replacement removes bad vertices and introduces none. Repetition therefore terminates, since the original path had finitely many bad vertices. First and last visits are taken for one fixed lifted component; different periodic copies are treated separately. Projecting the resulting good-cube path gives a route on the torus.

Join successive cube centers through their common faces, and join the original points to their cube centers. A point on a grid face can be assigned any adjacent cube: membership outside \(\mathcal A(Y)\) ensures that each such cube is good and outside every fill. Every closed good cube is disjoint from every closed forbidden ball. The finite union of cubes used by the route is compact, so its distance from the union of the closed forbidden balls is positive. All the line segments stay in these cubes. This proves both open-slice connectivity and strict clearance. ◻

Proof. The fill of a finite component lies in its coordinate bounding box by Lemma 30. Moreover, every column parallel to the first coordinate axis that meets the fill also meets the component. Otherwise that entire column would avoid the component and connect each of its vertices to infinity, precluding a finite hole there.

For a small component \(C\), there are at most \(|C|\) such columns. Each contributes length at most \(1+2h\) in physical units and cross-sectional area \(h^2\). Hence its filled cubes have volume at most \[ (1+2h)h^2|C|\le3h^2|C|. \tag{97}\] A closed radius-\(r\) ball meets at most \(C_0=343\) microcubes, since \(r/h\le2\) and in each coordinate at most seven closed grid intervals can meet its projection.

Choose one representative of each small component modulo the torus period. Its projection is injective because each physical coordinate span is at most one and \(K>10\). The projected components are disjoint, and their total number of bad microcubes is at most \(C_0(N-1)\). Summing Equation (97), and allowing overlaps of their fills in this upper bound, gives \(3C_0h^2(N-1)\le C r^2N\). ◻

Proof. Fix a lifted microcube index \(z\) in that component and in a lift of \(B_v\). There is a component vertex at scaled maximum distance greater than \(1/2\) from \(z\). For a finite component this follows from its diameter being greater than one; for an infinite component it follows from local finiteness. Follow a simple \(*\)-path from \(z\), stopping when its scaled maximum distance first reaches \(1/4\). It contains at least \(1/(4h)\) distinct vertices and remains within scaled distance \(1/4+h\) of \(z\).

Choose one bath ball witnessing badness at each path vertex. A fixed ball witnesses at most \(C_0\) of these vertices. All the chosen centers are in a fixed small neighborhood of \(v\), contained in its radius-\(5\) ball for sufficiently small \(r\). This neighborhood and the cubes under consideration have diameter less than \(K\), so no two periodic copies of the same label occur. The witness sequence therefore contains at least \(1/(4C_0h)\ge1/(4C_0r)\) distinct labels. Shrink \(r_1\) so this number is at least two, and retain the stated smaller constant \(c_{\rm g}\).

Repeated labels do not destroy the close-partner conclusion. Form a graph whose vertices are the distinct witness labels, joining different labels whenever they occur consecutively in the sequence. This graph is connected and has at least two vertices, so none is isolated. Two adjacent closed microcubes meet. The centers of balls intersecting those cubes are therefore at distance at most \[2r+2\sqrt3\,h\le6r.\] Every distinct label thus has a different close partner, including labels that recur in separated portions of the witness sequence. ◻

Proof. We estimate the bath-dependent volume by integrating under the full configuration law: \[ \delta_{\rm geom} =\int_{\Lambda_K^N}\frac{|\mathcal A(Y)|}{V}\,\mu(\,\mathrm dx\,\,\mathrm dY). \tag{103}\] This is the marginal identity \(\int\Phi(x,Y)^2\,\mathrm dx=m(Y)\). The coarse bad events used next may therefore depend on \(x\) as well as on the bath.

Set \(m_K=\lfloor K/10\rfloor\) and let \(\mathcal E_K\) be the event that the coarse bad set in (100) has a \(*\)-component of size at least \(m_K\). In a graph of degree \(26\), the number of connected vertex sets of size \(m\) containing a prescribed root is at most \(C_{\rm an}^m\). To see this, choose for each set a deterministic spanning tree and its depth-first traversal: the resulting walk has \(2(m-1)\) steps, and its visited set recovers the original set. There are at most \(26^{2(m-1)}\) such walks. A connected set of size at least \(m_K\) contains a connected subset of size exactly \(m_K\), by taking the first \(m_K\) vertices in a tree exploration. Equation (101) and a union bound over roots give \[ \mu(\mathcal E_K) \le K^3(C_{\rm an}p_D)^{m_K}. \tag{104}\]

On \(\mathcal E_K^c\), every coarse bad component has fewer than \(K/10\) vertices. We make its periodic lifting explicit. Lift a spanning tree of a component with \(m\) vertices. Each additional edge and its tree path form a loop of length at most \(2m-1<K\). Such a loop has zero winding: every step changes each coordinate by at most one, whereas nonzero winding requires displacement of magnitude at least \(K\) in one coordinate. Every additional edge therefore closes in the chosen finite tree lift. This proves that the full lifted component is finite, projects bijectively, and has coordinate span at most \(m-1\). It is the nearest-range case of Lemma 31.

The parent-cell map sends adjacent microcubes to equal or \(*\)-adjacent unit cells. Every large microcomponent is therefore contained in the unit cubes of one lifted coarse bad component. In particular it is finite on \(\mathcal E_K^c\): each of those finitely many unit cubes contains only \(q_r^3\) microcubes. Its fill lies in its own coordinate bounding box and thus in the coordinate box of those unit cubes.

Consequently, if a prescribed unit cell is affected by a large fill, there is a coarse bad animal of some size \(m\ge1\) with a root within maximum distance \(C m\) of that cell. There are at most \(C m^3\) possible roots, even after projecting to the torus. The rooted-animal count and Equation (101) give \[ \mu\{\mathcal E_K^c\text{ and the cell meets a large fill}\} \le\sum_{m\ge1} C m^3(C_{\rm an}p_D)^m\le C'p_D. \tag{105}\] The last inequality follows by summing the convergent power series on \(C_{\rm an}p_D\le1/2\). No independence of coarse sites is used.

Sum Equation (105) over unit cells and divide by \(V\). On \(\mathcal E_K\), bound the discarded fraction by one. Lemma 37 bounds the small-fill contribution by \(Cr^2N/V=Cr^2D\). Combining these bounds with Equation (104) proves Equation (102), by Equation (103). Lemma 36 gives the connectivity assertion.

All sets used here are measurable. Bad-cube membership is a finite family of distance tests in one period. A finite periodic component has a finite lift precisely when its spanning-tree lift closes along every additional edge. In that case its fill is determined by connectivity in a finite bounding box, with the exterior attached. These finite procedures determine the projected discarded cubes and their volume from the bath positions. ◻

Proof. We first list the choices that yield Proposition 35. Fix the local cutoff profiles and the exponent \(\gamma=1/12\) in the path regularity conditions. Choose \(\alpha_0\) large enough for the insertion and spatial estimates, and then choose the small lower-count constant \(\ell\). Choose the close-count threshold \(\theta\) so that \[3\theta<\ell/2,\qquad \theta<c_{\rm g}/(2\alpha_0).\] The first inequality leaves the required eligible labels; the second is the witness condition in Equation (98). Next fix the separation constant \(b\) in \(s_0=bD^{-1/3}\).

At the fixed encounter radius \(R=30\), Proposition 29 supplies a bad-site tolerance \(p\), an exponential-moment parameter \(\kappa>0\), and a bound \(C_{\rm path}\). Fix the visitor threshold \(H_1\) and the regularity threshold in the certificate estimates, and then the small time \(t>0\) in the bridge estimates. All these constants are fixed before \(D\) is increased. The local comparison bound \(G\ge2\) of Proposition 28 is now fixed as well. Choose the integer group size \(M\) large enough for the grouping failure estimate and for \[\frac{2(G-1)}M\le\frac\kappa2.\] Finally choose \(D_0\) large enough for all the preceding estimates, for \(\ell D\ge8M\), and for \[-4\log(1-\chi_D)\le\frac\kappa2, \qquad \chi_D=C_{\rm col}M\eta_D.\] Here \(\eta_D\to0\) is the deterministic-opponent collision bound from Proposition 28. We may replace each vanishing error by its nonincreasing upper envelope, so these requirements hold for all \(D\ge D_0\).

Proposition 35 then gives \[ B(\Phi)\ge c_*:=\frac{\upsilon_0\ell}{32M G C_{\rm path}}>0, \qquad \upsilon_0=\frac1{2000}. \tag{106}\] None of \(\upsilon_0,\ell,M,G,C_{\rm path}\) changes when the lower threshold on \(D\) is subsequently increased.

By Proposition 39, the retained points lie in a common component of each allowed slice, and their discarded-volume expectation satisfies \[ \delta_{\rm geom} \le C\frac{\alpha^2}{D}+C p_D +C K^3(C_{\rm an}p_D)^{\lfloor K/10\rfloor}, \qquad p_D\longrightarrow0. \tag{107}\] The constants in this estimate use only choices already fixed. Increase \(D_0\) to \(D_*\) so that, for every \(D\ge D_*\) and every \(\alpha\) in its fixed window, \[C\frac{\alpha^2}{D}+Cp_D\le\frac{c_*^2}{256}, \qquad C_{\rm an}p_D\le\frac12.\] For sufficiently large \(K\) the last term in Equation (107) is at most \(c_*^2/256\). In particular \(\delta_{\rm geom}\le c_*^2/64\), and Lemma 5 gives \(B(\Psi)\ge c_*/2\).

Proposition 35 supplies volume thresholds locally uniform in \(D\), independently of the ground vector and of \(\alpha\) in its fixed window. The geometric restrictions above hold for all \(D\ge D_*\), and the remaining volume error is uniformly at most \(C K^3 2^{-\lfloor K/10\rfloor}\). Combining these two facts proves the asserted local uniformity in \(D\) for the complex ground vector. ◻

Proof of Theorem 1. Let \(\alpha_0,D_*,c_*\) be the constants of Proposition 40, and set \[\varepsilon_0=\frac{\alpha_0^3}{4D_*^2},\qquad c_0=c_*/2.\] Decrease \(\varepsilon_0\) if needed to meet the fixed dilute-regime prerequisites. Fix \(a,\rho>0\) with \(\rho a^3<\varepsilon_0\), and let \(N_k,L_k,\Psi_k\) be any sequence in the theorem. We choose the integer side to keep the product of scaled density and exclusion distance close to \(\alpha_0\). Equation (4) will then turn the small physical gas parameter into a large scaled density. For all sufficiently large \(k\) define \[q_k=\sqrt{\frac{aN_k}{\alpha_0 L_k}},\qquad K_k=\lfloor q_k\rfloor,\qquad r_k=\frac{aK_k}{L_k},\qquad D_k=\frac{N_k}{K_k^3}.\] Since \(q_k/L_k\to\sqrt{a\rho/\alpha_0}>0\), we have \(K_k\to\infty\) and \(K_k/q_k\to1\). Consequently \[\begin{align*} \alpha_k:=D_kr_k &=\frac{aN_k}{L_kK_k^2} =\alpha_0\left(\frac{q_k}{K_k}\right)^2 \longrightarrow\alpha_0,\\ D_k&\longrightarrow D_\infty:=\frac{\alpha_0^{3/2}}{\sqrt{\rho a^3}} >2D_*. \end{align*}\] Thus \(\alpha_k\) lies in the required window and \(D_k\) lies in a compact subinterval of \((D_*,\infty)\) for all sufficiently large \(k\). The local uniformity in Proposition 40 applies to this varying sequence of scaled densities.

Use the unitary change of length from Section 2, with \(s_k=L_k/K_k\). It carries \(\Psi_k\) to a normalized bosonic ground vector on \(\Lambda_{K_k}\) with exclusion distance \(r_k\), and preserves \(B\). Proposition 40 therefore gives \(B(\Psi_k)\ge c_0\) for all sufficiently large \(k\). The argument is uniform over the choice of ground vector, so it proves Equation (2) for every such sequence. ◻

Proof of Corollary 2. The bosonic ground eigenspace is finite dimensional by compactness of the form embedding. Every density operator supported on it is a convex combination of rank-one projections onto normalized ground vectors. For all sufficiently large members of the sequence, the proof of Theorem 1 gives the same lower bound \(c_0\) for each of those vectors. Constant-orbital occupation is linear in the density operator. The lower bound therefore holds for their convex combination, including \(\Pi_k/\mathop{\mathrm{Tr}}\Pi_k\). A nonnegative ground vector is already covered by the theorem. ◻