Proof. Zero extension sends \(H_0^1\) of the allowed set into \(H^1\) of the compact configuration torus. The compact Sobolev embedding proves compactness of the form embedding, and hence compact resolvent. For a labelled form-domain function \(f\), its root-mean-square symmetrization \[F(X)=\left(\frac1{n!}\sum_{\pi\in S_n}|f(\pi X)|^2\right)^{1/2}\] has the same norm and no larger Dirichlet energy, by the gradient inequality for a vector norm. To justify the Dirichlet condition, first apply this construction to smooth compactly supported approximants. Their symmetrizations have compact support in the allowed set, admit interior smooth approximation, and are bounded in \(H^1\). Weak closedness of \(H_0^1\) gives the assertion for \(f\). The full and bosonic variational infima consequently agree. The modulus contraction gives the statement about \(|\Phi|\) by the same variational principle.

For slicing, take smooth compactly supported approximants converging in the form norm. Fubini’s theorem gives a subsequence converging in the Sobolev norm of the unfixed variables for almost every fixed tuple. Each approximant has compact support in its open allowed slice, proving the stated Dirichlet membership. Removing fixed obstacles enlarges that open set, so zero extension preserves membership. This argument does not require regularity or connectedness of its boundary. These are the ground-space and slicing arguments of [24]. ◻

Proof. We give separately the short-distance energy estimate and the recovery of the kinetic term. Choose equal subdivisions of every \(d\)-cube with side \(\ell\asymp D^{-0.31}\), and set \[s\asymp D^{-0.36},\qquad \varepsilon_D=D^{-0.002}.\] Rounding the number of subdivisions changes only fixed constants. For large \(D\), the support radius \(r\) is smaller than \(s/2\). Restrict configuration space to sectors specifying which small cube contains each labeled particle. Restriction of an \(H^1\) function to these sectors incurs no gradient cost; their faces carry Neumann form conditions. Discard interactions between different small cubes.

The radial replacement. Fix one particle coordinate and all the others in its small cube. Partition that cube into Voronoi regions of the other particles, clipping at the cube boundary. Each region is star-shaped about its center. On a ray from a center, the scattering variational principle gives, for \(R\ge r\), \[ \int_0^R\left((1-2\varepsilon_D)|f'(t)|^2 +\tfrac12v(t)|f(t)|^2\right)t^2\,\mathrm dt \ge (1-2\varepsilon_D)a'|f(R)|^2. \tag{29}\] For completeness, with kinetic coefficient one the minimizer at fixed endpoint is the regular scattering solution divided by its endpoint value. Its minimum is \(a_v(1-a_v/R)^{-1}|f(R)|^2\), where \(a_v\) is the scattering length of the potential in the integrand. This follows by integrating the radial scattering equation against the minimizer; the boundary term is \(R^2\overline{f(R)}f'(R)\). Completing the quadratic form around the minimizer proves the same lower bound for every ray function. Factoring out \(1-2\varepsilon_D\) replaces \(v\) by \(v/(1-2\varepsilon_D)\), whose scattering length is at least \(a'\), by the variational definition and monotonicity. Dropping the factor \((1-a_v/R)^{-1}\ge1\) proves (29). This argument applies first to smooth functions and then by form approximation.

Let \(U\) be constant on \(s/2\le |x|\le s\), zero elsewhere, and normalized by \(\int_{\mathbb R^3}U=4\pi\); thus \(\int_0^\infty U(t)t^2\,\mathrm dt=1\). On a ray ending at distance \(T\), average (29) over radii \(R\le T\) with weight \(U(R)R^2\,\mathrm dR\). The total weight is at most one, so only the energy at distances at most \(s\) is used. Integrating the rays proves the lower bound \((1-2\varepsilon_D)a'U(d_{\mathrm{nn}})\), where \(d_{\mathrm{nn}}\) is the nearest-neighbor distance within the small cube. If there is no other particle this expression is zero. Half of every pair interaction is assigned to each endpoint, so summing over the coordinates uses no interaction twice.

More explicitly, let \(S_i\) be the set of positions of coordinate \(x_i\) within distance \(s\) of another particle in its small cube. The preceding argument leaves the form \[ \mathcal T_{\mathrm{rem}}[\psi] =\sum_i\int\left[\varepsilon_D+(1-2\varepsilon_D)\mathbf 1_{S_i^c}\right] |\nabla_i\psi|^2. \tag{30}\] There is also a separate reserve \(\varepsilon_D\sum_i\|\nabla_i\psi\|^2\) available for the next step. Indeed, these two reserves and the coefficient \(1-2\varepsilon_D\) used inside \(S_i\) sum to the original kinetic coefficient at every point.

The energy in a small cube. The cases \(k=0,1\) have zero scalar lower bound and are immediate. For \(k\ge2\) particles in a cube of side \(\ell\), put \[W_k=(1-2\varepsilon_D)a'\sum_{i=1}^kU(d_{\mathrm{nn},i}).\] The first nonzero eigenvalue of the reserved kinetic operator is \(\Delta_\ell=\varepsilon_D\pi^2\ell^{-2}\). Its normalized constant vector will be denoted by \(\Omega_k\). For any fixed \(C_0\), uniformly for \(k\le C_0D\ell^3\), \[ \langle W_k\rangle_{\Omega_k} \ge\frac{4\pi(1-2\varepsilon_D)a'k(k-1)}{\ell^3} \left(1-C\frac{s}{\ell} -C\frac{ks^3}{\ell^3}\right). \tag{31}\] To see this, keep the first coordinate a distance at least \(s\) from the cube boundary. For each possible nearest neighbor integrate that neighbor over the shell of radii \(s/2,s\). The other \(k-2\) points lie outside the corresponding ball with probability at least \(1-Cks^3/\ell^3\), by the union bound. These events for different nearest neighbors are disjoint up to null sets. The removed boundary layer occupies at most \(Cs/\ell\) of the cube volume. Summing over the first coordinate proves (31).

The multiplication operator satisfies \[0\le W_k\le Cka's^{-3},\qquad \langle W_k^2\rangle_{\Omega_k} \le Cka's^{-3}\langle W_k\rangle_{\Omega_k}.\] The scalar mean in (31) is less than \(\Delta_\ell/2\) for large \(D\). Temple’s inequality therefore gives \[ \inf\operatorname{spec}\left( \varepsilon_D\sum_{i=1}^k(-\Delta_i^{\mathrm N})+W_k\right) \ge \langle W_k\rangle_{\Omega_k} -\frac{\langle W_k^2\rangle_{\Omega_k}} {\Delta_\ell-\langle W_k\rangle_{\Omega_k}}. \tag{32}\] This version of Temple’s bound follows by decomposing the constant trial vector into the ground eigenspace and its orthogonal complement; the latter spectrum is at least \(\Delta_\ell\), by the min–max principle and \(W_k\ge0\), and the variance bounds its spectral weight. The powers used here are \[D\ell^3\asymp D^{0.07},\quad s/\ell\asymp D^{-0.05},\quad Ds^3\asymp D^{-0.08},\quad ka's^{-3}\le CD^{0.15},\quad \Delta_\ell\asymp D^{0.618}.\] Consequently there is \(\delta_D\le CD^{-0.002}\) such that the energy available in each small cube is bounded below by \[ \frac{4\pi r}{\ell^3}(1-\delta_D)k(k-1), \qquad 0\le k\le C_0D\ell^3. \tag{33}\] The replacement of \(a'\) by \(r\) is included in \(\delta_D\).

For larger \(k\), divide its labels into groups of sizes between \(m/2\) and \(m\), where \(m=\lfloor C_0D\ell^3\rfloor\). Such a partition exists for \(k>m\), by taking \(\lceil k/m\rceil\) groups of nearly equal sizes; decreasing the lower endpoint by one accommodates integer rounding. Drop interactions between groups before applying the radial argument. A position close to a member of its group is also close to a particle of the original cube, so the reserve (30) remains available. After reducing the harmless lower endpoint to \(m/3\), the group bound is at least \[\frac{4\pi r}{\ell^3}(1-\delta_D)(m/3-1)k.\] Choose once and for all \(b>12\) and then \(C_0>12b\). Define the convex function on \([0,\infty)\) \[F_b(y)= \begin{cases} y^2,&0\le y\le b,\\ 2by-b^2,&y>b. \end{cases}\] Both the small-\(k\) estimate and the group estimate imply the common bound \[ \frac{4\pi r}{\ell^3}(1-\delta_D) \left[(D\ell^3)^2F_b\left(\frac{k}{D\ell^3}\right)-k\right]. \tag{34}\] For \(k\le m\) this follows from \(F_b(y)\le y^2\). For \(k>m\), use \(F_b(y)\le2by\) and \(m/3-1\ge2bD\ell^3-1\) for sufficiently large \(D\).

From small-cube counts to the desired count penalty. Set \(\vartheta=n/(Dv_b)\in[0,6]\). For all \(y\ge0\), \[ F_b(y)\ge\vartheta^2+2\vartheta(y-\vartheta) +c\mathfrak h(y-\vartheta). \tag{35}\] If \(y\le b\), the remainder after the tangent is \((y-\vartheta)^2\). If \(y>b\), write \(t=y-\vartheta\) and \(a=b-\vartheta\); the remainder is \(2at-a^2\ge at\), and \(a\ge b-6>0\). These two observations prove (35) with a uniform positive constant. Sum (34) over the small cubes. The linear tangent terms cancel because their counts sum to \(n\). The subtracted self-count terms cost at most \[C r\ell^{-3}n\le Cv_bD^{0.93}.\] The factor \(1-\delta_D\) changes the leading scalar by at most \(Cv_bD^{1-0.002}\). Since \(\mathfrak h\) is convex, Jensen’s inequality within each \(d\)-cube gives \[\sum_{\text{small cubes }C\subset A}|C|\, \mathfrak h\left(\frac{n_C}{D|C|}-\vartheta\right) \ge |A|\,\mathfrak h\left(\frac{n_A}{D|A|}-\vartheta\right).\] This proves the scalar and count terms in (28), while leaving (30) unused.

Recovery of the one-particle kinetic term. We now sum the sector restrictions back on the original \(H^1\) form domain of the \(E\)-cube; the following estimates concern its global one-coordinate slices, not an arbitrary function in the larger direct sum of the cut Neumann form domains. Let \(u_p\) be the normalized cosine basis of the Neumann cube, with \(p\in(\pi/E)\mathbb Z_{\ge0}^3\). The vector fields \(\nabla u_p/|p|\), for \(p\ne0\), are orthonormal. Let \(\mathcal P\) be their projection for \(0<|p|\le D^{0.003}\). The explicit cosine formulas give \[ \|\mathcal P\|_{L^2\to L^\infty}^2 \le\frac{C}{E^3}\#\{p:0<|p|\le D^{0.003}\} \le CD^{0.009}. \tag{36}\] For a fixed slice in the other particle coordinates, enlarge \(S_i\) to the union \(S\) of the balls of radius \(s\) about all the other particles. Its volume is at most \(Cns^3\). The integral kernel bound in (36), or duality, implies \[\|\mathcal P\mathbf 1_S F\|_2^2 \le CD^{0.009}ns^3\|F\|_2^2 \le CD^{-0.071+3\theta}\|F\|_2^2.\] Thus, for \(F=\nabla f\), \[\|\mathcal P\nabla f\|_2^2 \le2\|\mathbf 1_{S^c}\nabla f\|_2^2 +CD^{-0.071+3\theta}\|\nabla f\|_2^2.\] For example, if \(\theta<10^{-3}\), the last coefficient is \(o(\varepsilon_D)\). Integration by parts, using the Neumann condition on \(u_p\), gives \(\langle\nabla u_p/|p|,\nabla f\rangle =|p|\langle u_p,f\rangle\). A fixed fraction of (30) therefore controls \(\sum_{0<|p|\le D^{0.003}}p^2|\langle u_p,f\rangle|^2\). Another fraction of the everywhere reserve controls the remaining weights one, since \(\varepsilon_Dp^2\ge D^{0.004}\) for \(|p|>D^{0.003}\). Sum the slice estimates over the coordinates. Together with the count bound already obtained this proves (28) with \(c_{\mathrm r}=10^{-3}\). ◻

Proof. Let \(\Pi_k\) project onto \(n_L=k\), and write \(\psi_k=\Pi_k\psi\). The kinetic operator commutes with every cutoff. A two-particle multiplication operator changes \(n_L\) by at most two, so \(\Pi_kV_n\Pi_l=0\) for \(|k-l|>2\). The difference of the potential forms is exactly \[-\frac12\sum_{k,l}\sum_{j=1}^3 (F_j(k)-F_j(l))^2\langle\psi_k,V_n\psi_l\rangle.\] The cutoffs have Lipschitz constants at most \(C/M\). Positivity gives \[|\langle\psi_k,V_n\psi_l\rangle| \le\langle\psi_k,V_n\psi_k\rangle^{1/2} \langle\psi_l,V_n\psi_l\rangle^{1/2}.\] Since each row has at most five nonzero blocks, the absolute form difference is at most \[ \frac{C}{M^2}\langle\psi,\mathcal D(V_n)\psi\rangle, \qquad \mathcal D(V_n)=\sum_k\Pi_kV_n\Pi_k =\frac1{2\pi}\int_0^{2\pi} e^{-itn_L}V_ne^{itn_L}\,\mathrm dt. \tag{40}\]

The cosine evaluation bound on the low band is \(\sup_x P_L(x,x)\le CK_H^3/v_b\). For each pair it implies \[ P_{L,i}v(x_i-x_j)P_{L,i} \le C\|v\|_1\frac{K_H^3}{v_b}P_{L,i}. \tag{41}\] Indeed, condition on \(x_j\) and all other coordinates, apply \(|P_Lg(x_i)|^2\le(\sup_xP_L(x,x))\|P_Lg\|_2^2\), and integrate the potential in \(x_i\). For the pair \((i,j)\), the rotations on the other coordinates cancel. Writing \(z=e^{it}-1\), the remaining rotation is \[(1+zP_{L,i})(1+zP_{L,j}) =1+zP_{L,i}+zP_{L,j}+z^2P_{L,i}P_{L,j}.\] The squared norm after multiplication by \(v_{ij}^{1/2}\) is bounded by a fixed constant times the sum of the four squared norms. Use (41) on the last three terms, and \(P_{L,i}P_{L,j}\le P_{L,i}\), to obtain \[\langle\psi,\mathcal D(V_n)\psi\rangle \le C\langle\psi,V_n\psi\rangle +C\|v\|_1\frac{K_H^3}{v_b}\, n\langle\psi,n_L\psi\rangle.\] This and \(n_L\le n_+\) prove (39). A one-particle operator has block width one instead of two. Using its second-quantized norm at most \(n\), and \(2\|\psi_k\|\|\psi_l\|\le \|\psi_k\|^2+\|\psi_l\|^2\), proves the final assertion. ◻

Proof. We use the Bogoliubov identity (2.39), the completion of a square (7.18)–(7.20), and the scalar integral estimate in [13]. We give the intervening bounds, in particular an estimate of the creation term with every commutator retained. This obtains the required gap fraction directly; no quantifier about the coefficient in that paper’s free-energy Theorem 2.9 is needed. By homogeneity suppose \(\|\xi\|=1\), and abbreviate \(K=K_H\), \(N_+=\langle n_+\rangle_\xi\), \(\lambda=(Dv_b)^{-1/2}\), and \(h(k)=|\widehat g(k)|/(8\pi a')\le1\).

The Bogoliubov identity (2.39) of [13] gives the exact separation \[ \mathcal H_0(z)=\mathcal E_{\rm B}(z) +\sum_{p\ne0}D_p(z)b_p(z)^*b_p(z) +\mathcal R_{\rm int}(z)+Q_3(z), \tag{52}\] where \[\begin{align*} \mathcal E_{\rm B}(z) &=\frac12\sum_{p\ne0} \bigl(D_p(z)-\tau(p)-\rho_z\widehat g(p)\bigr) +\frac{\widehat{g\omega}(0)|z|^4}{2v_b},\\ \mathcal R_{\rm int}(z) &=\rho_z\sum_{p\ne0} \{\widehat{g\omega}(0)+\widehat{g\omega}(p)\}a_p^*a_p. \end{align*}\] We shall bound \(Q_3\) using part of the positive quadratic energy, cancel its normal-ordering remainder against \(\mathcal R_{\rm int}\), and evaluate the scalar \(\mathcal E_{\rm B}\) at the end.

First remove the part of (49) with \(k\) low. Put \(l=|p-k|\). Bounded multiplicity of \(k\mapsto l\) for fixed \(p\), commutation of \(a_l\) past \(a_l^*\), and \(n_L\le M/4\) give \[\sum_{p,k}\|a_l^*a_p\xi\|^2\le C(M+K^3)N_+, \qquad \sum_{p,k}\|a_k\xi\|^2\le CK^3N_+.\] Here both \(p\) and \(k\) range over their low bands, with the signed multiplicity in \(p\) bounded by eight; the terms with \(l=0\) are omitted. Cauchy–Schwarz in the expectation \(\langle a_l^*a_p\xi,a_k\xi\rangle\) therefore bounds this part by \[C\lambda K^{3/2}(M+CK^3)^{1/2}N_+.\] Its coefficient relative to \(E^{-2}N_+\) is \(O(Q^{-48}+D^{-1/2}Q^{300.5})=o(1)\).

In the remaining cubic term \(k\) is high. Substitute \(a_k=U_kb_k+V_kb_k^*\) and write it as \(\mathcal T_1+\mathcal T_V\). For high \(k\), \(|U_k|\le C\) and \(|V_k|\le Ch(k)/k^2\). Thus the coefficients of \(\mathcal T_V\) are at most \(C\lambda h(k)^2/k^2\) in absolute value. Let \(l=|p-k|\). Since \(p\) is low and \(k\) high, the exact commutation is \[ a_p^*a_l b_k^*=b_k^*a_p^*a_l +U_k\mathbf 1_{\{l=k\}}a_p^* . \tag{53}\] The harmless common phase of \(z\) can be included in the coefficients. Set \[B_H=\sum_{k\ {\rm high}}\frac{k^2}{K}\|b_k\xi\|^2.\] For the first term on the right of (53), Cauchy–Schwarz in \(p\), followed by weighted Cauchy–Schwarz in \(k\), gives \[\begin{align*} &C\lambda\sum_{\substack{k\ {\rm high}\\p\ {\rm low,\ signed}}} \frac{h(k)^2}{k^2}\|b_k\xi\|\,\|a_p^*a_l\xi\|\\ &\qquad\le C\lambda K^2 B_H^{1/2} \left(\sum_{p,k}\frac{h(k)^4}{k^6} \|a_p^*a_l\xi\|^2\right)^{1/2}. \tag{54}\end{align*}\] The summation ranges on the last line are those on the first. The identity \(\|a_p^*a_l\xi\|^2=\|a_pa_l\xi\|^2+\|a_l\xi\|^2\) and bounded folding multiplicity imply \[\sum_{p,k}\|a_pa_l\xi\|^2\le CMN_+,\qquad \sum_{p,k}\|a_l\xi\|^2\le CK^3N_+.\] For the first bound use \(\sum_p a_p^*n_+a_p\le Cn_Ln_+\); it holds also when \(l=p\). Since \(k^2\ge K^2/E^2\), Young’s inequality in (54) yields, for every \(\varepsilon>0\), \[\begin{align*} |\langle\mathcal T_V^{\,\rm reordered}\rangle| &\le\varepsilon B_H+ C_\varepsilon\lambda^2K^4(E/K)^6(M+K^3)N_+\\ &\le\varepsilon B_H+ C_\varepsilon(Q^{-594}+D^{-1}Q^{103})N_+. \tag{55}\end{align*}\]

The second term in (53) is also summable. The condition \(|p-k|=k\) requires, in each coordinate, \(p_i=0\) or \(p_i=2k_i\). Some \(p_i\) is nonzero, so for fixed \(p\) the admissible \(k\) lie in at most three planes \(k_i=p_i/2\). Two-dimensional lattice summation, retaining \(h(k)^2\), gives \[\sum_{\substack{k\ {\rm high}\\|p-k|=k}}\frac{h(k)^2}{k^2} \le CE^2(1+\log D).\] Indeed the dyadic sums below a sufficiently large polynomial frequency are \(O(E^2)\) per annulus, and the radial transform bound \(h(k)\le D^C/|k|\) makes the remaining tail summable. Cauchy–Schwarz in the \(O(K^3)\) possible values of \(p\) now gives \[\begin{align*} |\langle\mathcal T_V^{\,\rm commutator}\rangle| &\le C\lambda E^2K^{3/2}(1+\log D)\sqrt{N_+}\\ &\le\delta\langle S\rangle+ C_\delta D^{-1}Q^{303}(1+\log D)^2. \tag{56}\end{align*}\] This ordering of the operators avoids an unweighted sum of vacuum commutators over all high modes. Choose \(\varepsilon\) small enough that \(\varepsilon B_H\le K^{-1}\sum_{k\ {\rm high}}D_k\|b_k\xi\|^2\). The other terms in (55) are \(o(\langle S\rangle)\); the scalar in (56) is \(o(v_bQ^{-1})\). The coefficient \(\delta\) here can be any prescribed fixed fraction of the one in the statement.

It remains to treat \(\mathcal T_1\). Complete a square with \((1-2K^{-1})\sum_{k\ {\rm high}}D_kb_k^*b_k\). The square is nonnegative. Its negative remainder, after normal ordering, is the sum of a commutator term \(\mathcal T_c\) and a quartic term \(\mathcal T_0\), as in (7.18)–(7.20) of [13]. The coefficient in that square is \[A_k=\frac{z\widehat g(k)} {v_b(1-2K^{-1})D_k\sqrt{1-\beta_k^2}} \sum_{\substack{p\in\mathcal P_L^{\mathbb Z}\\|p-k|\ne0}} c(p,k)a_{|p-k|}^*a_p.\] In \(A_k^*A_k\), the commutator is nonzero only when \(|p-k|=|s-k|\). Either \(p=s\), or at least one coordinate of \(k\) lies in the low-frequency slab \(|k_i|\le K/E\). The exact signed-cosine weight identity \(\sum_p c_p^{-2}a_p^*a_p=n_L\), followed by [13], gives \[ \langle\mathcal T_c\rangle \le2\rho_z\widehat{g\omega}(0)N_+ +C(Q^{-97.75}+D^{-1}Q^{200.25})E^{-2}N_+ . \tag{57}\] The first term cancels the remaining quadratic interaction term in (50), up to an error. Splitting that interaction into its low and high frequencies, Lemma 7.2(1) of the same paper bounds this error by \[Cn_H+CD^{-2}Q^{198}n_+ \le C(Q^{-98}+D^{-2}Q^{200})S.\] The quartic term has the explicit bound \[\mathcal T_0\le C\,\frac{a'K_\ell^3K}{v_b}n_Ln_+,\] obtained by Cauchy–Schwarz in its two low indices and summing the remaining high occupation, as in (7.28)–(7.29) of [13]. On the present subspace its ratio to \(E^{-2}n_+\) is at most \(CMa'K_\ell^3K/E=O(Q^{-295})\). All errors in this paragraph are therefore \(o(S)\). These estimates are uniform for \(0\le\rho_z\le2n/v_b\). Lemma 7.2 of [13] concerns the potential alone, and Lemma 7.3 is applied at \(\rho_{\rm aux}\) and multiplied by \(\rho_z/\rho_{\rm aux}\), which is uniformly bounded. At \(\rho_z=0\) the cubic and interaction terms vanish and \(b_p=a_p\).

In the notation of (52), the cancellation just proved reads \[\langle\mathcal R_{\rm int}-\mathcal T_c\rangle \ge-o(1)\langle S\rangle.\] The completed square spends \((1-2K^{-1})\) of the high quadratic energy; the reordered creation term spends at most another \(K^{-1}\). Together with the low-cubic and quartic bounds this leaves \[\langle\mathcal H_0(z)\rangle \ge\mathcal E_{\rm B}(z) +\sum_{p\ne0}\widetilde D_p(z)\|b_p(z)\xi\|^2 -\delta\langle S\rangle-C_\delta v_bQ^{-1}.\] We have retained every low Bogoliubov mode and at least a fraction \(K^{-1}\) of every high mode. Lemma A.1 of [13] evaluates \(\mathcal E_{\rm B}(z)\) as \(4\pi c_{\rm L}(a')^{5/2}v_b\rho_z^{5/2}\) with error \(Cv_bQ^{-1/4}\), uniformly on \(|z|^2\le2n\). Combining the two bounds, choosing the fixed gap fractions first and then increasing \(D\), proves (51). ◻

Proof. Extend a cosine subscript by componentwise absolute value. In a term \(a_p^*a_l a_k\), \(p\) is a signed low frequency and \(l=|p-k|\); terms with \(l=0\) are omitted. The product-to-sum identity in each coordinate shows that the printed coefficient agrees with the exact expansion unless \(k_i=0\) or \(k_i=|p_i|\) for some \(i\). All multiplicities and discrepancies are bounded by an absolute constant. For example, when \(k_i=0\) and \(p_i\ne0\), the two choices of the sign of \(p_i\) must share one coefficient; when \(k_i=p_i\ne0\) and \(l_i=0\), the zero cosine has normalization one. These are precisely the two exceptional cases.

Put \(\lambda=(Dv_b)^{-1/2}\) and \(h(k)=|\widehat g(k)|/(8\pi a')\le1\). Every correction has coefficient bounded by \(C\lambda h(k)\). We estimate its expectation in a normalized excitation vector \(\xi\) with \(n_+\le n\), \(n_L\le M/4\). For \(|k|\le3K_H/E\), Cauchy–Schwarz applied to \(\langle a_l^*a_p\xi,a_k\xi\rangle\) gives \[ C\lambda K_H^{3/2} \{M\langle n_+\rangle+CK_H^3\langle n_+\rangle\}^{1/2} \langle n_+\rangle^{1/2}. \tag{58}\] To see the first factor, sum the occupation products after commuting \(a_l\) past \(a_l^*\); for each \(p\), the multiplicities of \(l\) are bounded. The second Cauchy–Schwarz factor sums \(a_k^*a_k\) at most \(CK_H^3\) times. The two coefficients multiplying \(\langle n_+\rangle\) in (58) are \(O(Q^{-50})\) and \(O(D^{-1/2}Q^{298.5})\), respectively, and are \(o(E^{-2})\).

For \(|k|>3K_H/E\), both \(k\) and \(l\) are high frequencies. Use \(a_k=U_kb_k+V_kb_k^*\), where \(|U_k|\le C\) and \(|V_k|\le C/k^2\). The terms involving \(\|b_k\xi\|\), including the excitation part of \(\|b_k^*\xi\|\), are bounded by the product of \[\left(\sum_k k^2K_H^{-1}\|b_k\xi\|^2\right)^{1/2}\] and a factor whose square is at most \[ C\lambda^2K_H^4\left\{ (E/K_H)^2M\langle n_H\rangle +E^2(1+\log D)\langle n_+\rangle\right\}. \tag{59}\] Here is the complete weighted sum underlying this estimate. Restrict \((p,k)\) to the exceptional planes of the correction and put \(F_{pk}=\|a_l^*a_p\xi\|\), with \(l=|p-k|\). Cauchy–Schwarz in the at most \(CK_H^3\) values of \(p\), then in \(k\) with weight \(k^2/K_H\), gives \[C\lambda\sum_k h(k)\|b_k\xi\|\sum_pF_{pk} \le C\lambda K_H^2 \left(\sum_k\frac{k^2}{K_H}\|b_k\xi\|^2\right)^{1/2} \left(\sum_{p,k}\frac{h(k)^2}{k^2}F_{pk}^2\right)^{1/2}.\] Since \(l\) is high and \(p\) low, \(F_{pk}^2=\|a_la_p\xi\|^2+\|a_p\xi\|^2\) and bounded folding multiplicity gives \[\begin{align*} \sum_{p,k}\frac{h(k)^2}{k^2}\|a_la_p\xi\|^2 &\le C(E/K_H)^2\langle n_Ln_H\rangle \le C(E/K_H)^2M\langle n_H\rangle,\\ \sum_{p,k}\frac{h(k)^2}{k^2}\|a_p\xi\|^2 &\le CE^2(1+\log D)\langle n_+\rangle. \end{align*}\] The second line uses the two-dimensional plane sums \[ \sum_{\substack{k\text{ on an exceptional plane}\\|k|>3K_H/E}} \frac{h(k)^i}{k^2}\le CE^2(1+\log D),\qquad i=1,2. \tag{60}\] For (60), use dyadic annuli up to a sufficiently large power of \(D\). Beyond that scale the radial transform formula \(\widehat g(k)=4\pi|k|^{-1}\int_0^r t g(t)\sin(|k|t)\mathrm dt\) gives \(h(k)\le D^C/|k|\) and a summable tail. The same argument with \(k^{-4}\) gives a bound \(CE^2\).

The extra unit in \(\|b_k^*\xi\|^2=\|b_k\xi\|^2+1\) comes with \(V_k\). Its total contribution is at most \[ C\lambda K_H^{3/2}E^2 \left\{\sqrt{M\langle n_H\rangle} +(1+\log D)\sqrt{\langle n_+\rangle}\right\}. \tag{61}\] To verify (61), bound the unit contribution by \(C\lambda\sum_{p,k}h(k)k^{-2}F_{pk}\). Its occupation-product part is at most \[C\lambda \left(\sum_{p,k}\frac{h(k)^2}{k^4}\right)^{1/2} \left(\sum_{p,k}\|a_la_p\xi\|^2\right)^{1/2} \le C\lambda E^2K_H^{3/2}\sqrt{M\langle n_H\rangle}.\] Here we used the loose bound \(CE^4\) for each \(k^{-4}\) plane sum and at most \(CK_H^3\) values of \(p\). Its commutator part is bounded by first summing \(h(k)/k^2\) on the plane and then using \(\sum_p\|a_p\xi\|\le CK_H^{3/2}\sqrt{\langle n_+\rangle}\). These are the two terms in (61).

Young’s inequality absorbs the first factor above into a \(\delta\) fraction of the high-frequency quasiparticle energy. In (59), the coefficients of \(\langle n_H\rangle\) and \(\langle n_+\rangle\) are \(O(Q^{-198})\) and \(O(D^{-1}Q^{399}(1+\log D))\). They are negligible relative to the gaps \(Q^{98}n_H\) and \(Q^{-2}n_+\). Applying Young’s inequality to (61) leaves scalar errors bounded by \(C_\delta[Q^{-194}+D^{-1}Q^{303}(1+\log D)^2]\). They fit within \(C_\delta v_bQ^{-1/4}\) for sufficiently small \(u\). This proves the stated bound. On the omitted large-\(z\) region the correction has a polynomial bound on the truncated particle sectors: the \(k\) coefficients have a polynomially bounded lattice square sum, and \(p\) lies in a finite band. Thus (47) also covers that region. ◻

Proof of Lemma 10. We detail both the gap allocation and the resolution of the zero mode. The calculations may first be made with finitely many Neumann modes. These truncations commute with the excitation cutoffs and approximate the local form domain. The bounded-potential terms then converge in form, and the retained positive quasiparticle expression passes by lower semicontinuity. This justifies the infinite sums below. Set \[S=E^{-2}(\pi n_+/2+K_Hn_H),\qquad H_b=H_b^{\rm mod}+S.\] Apply the symmetrization estimate of [13] with the fixed choice \(\varepsilon=1/10\). It gives \[ H_b^{\rm mod}\ge H_b^{\rm sym}-CnDr(r/E)-\varepsilon G, \qquad G=\tfrac12(1+n_L/M)S. \tag{62}\] On the present vector \(G\le5S/8\), so \(\varepsilon G\le S/16\). The hypotheses with the constant associated to this fixed choice hold for large \(D\), since \(K_H/K_\ell^4\asymp Q^{96}\) and \(K_\ell K_H^3\sqrt{\rho_{\rm aux}(a')^3}=O(D^{-1}Q^{301})\). The scalar error is \(O(E^2)\) and hence fits within \(Cv_bQ^{-1/4}\).

In the second-quantized identity of [13], the terms involving only the number operators are \[\frac{\widehat g(0)}{2v_b} \{n(n-1)-n_+(n_+-1)\}.\] The negative term can be paid from an arbitrarily small fraction of \(S\). Indeed \[ n_+^2\le2Mn_++2nn_H, \qquad \frac r{v_b}n_+^2 \le C\{Q^{-400}n_++n_H\}=o(1)S. \tag{63}\] The replacement of \(n(n-1)\) by \(n^2\) costs \(O(1)\).

The coherent resolution is an exact identity for quadratic forms; this use of upper symbols is the one in [21]. The upper symbols of the two zero-mode monomials are \[ a_0^*a_0:\ |z|^2-1,\qquad a_0^{*2}a_0^2:\ |z|^4-4|z|^2+2. \tag{64}\] Monomials containing only creations or only annihilations have their ordinary polynomial symbols. Let \(\mathcal C(z)\) denote the exact cosine cubic operator minus the printed cubic operator (49); Lemma 12 bounds this difference. If \(x=|z|^2\), the upper-symbol correction is \[ \mathcal R(z)=\frac{\widehat{g\omega}(0)}{2v_b}(-4x+2) -\frac1{v_b}\sum_{p\ne0} \{\widehat g(p)+\widehat{g\omega}(0)+\widehat{g\omega}(p)\}a_p^*a_p. \tag{65}\] The complete form of the second-quantized identity [13], with the exact cosine coefficients, is therefore \[\begin{align*} \langle\psi,H_b^{\rm sym}\psi\rangle &=\frac{\widehat g(0)}{2v_b} \left[n(n-1)\|\psi\|^2 -\langle\psi,n_+(n_+-1)\psi\rangle\right] \\ &\quad+\int_{\mathbb C} \langle\psi_z, (\mathcal H_0(z)+\mathcal R(z)+\mathcal C(z))\psi_z\rangle \frac{\mathrm d^2z}{\pi}. \tag{66}\end{align*}\] Here \(\mathcal H_0\) contains the modified kinetic symbol \(\tau\) and the low-band cubic cutoff from (50); the artificial scalar convexity term in (2.36) of that paper is absent. The identity has no additional remainder: the interaction pieces removed before symmetrization are already included in the error of (62). The integrated absolute quadratic form of \(\mathcal R\) is bounded by \[C\frac r{v_b}\int\left[ (1+|z|^2)\|\psi_z\|^2+\langle\psi_z,n_+\psi_z\rangle\right] \frac{\mathrm d^2z}{\pi} \le C\|\psi\|^2.\] The last inequality follows from \(\int|z|^2\|\psi_z\|^2\mathrm d^2z/\pi =\langle n-n_++1\rangle_\psi\) and \(rn/v_b=O(1)\).

Apply Lemma 11 to the operator \(\mathcal H_0(z)\) on \(|z|^2\le2n\), with a fixed sufficiently small \(\delta\). The coherent projection preserves both excitation cutoffs. The lemma retains \(\sum\widetilde D_pb_p^*b_p\) and the LHY scalar at \(\rho_z\), while using at most \(\delta S\). Together with \(G\le5S/8\) in (62) and the \(o(S)\) loss in (63), this leaves a fixed positive fraction of \(S\). In terms of the published formula (2.36), the artificial scalar convexity term has been subtracted identically, not estimated by its sign. Our argument uses no trace inequality from that paper’s Theorem 2.9.

By (47), the discarded coherent region has exponentially small weighted mass. All negative terms of the upper symbol have polynomial norm bounds in \(D,n,z\) on excitation sectors at most \(n\); this follows either from the bounded potential or from its projected second-quantized blocks. Dropping positive kinetic energy, the omitted region thus has exponentially small lower-bound error.

It remains to compare the LHY scalar at \(\rho_z\) with that at \(n/v_b\). Convexity and its derivative, bounded by \(C/D\) in the relevant density range, give an integrated loss at most \(CD^{-1}(1+\langle n_+\rangle_\psi)\), apart from the negligible tail. This too is absorbed by the retained gap. Replacing \(a'\) by \(r\) costs \(O(v_bQ^{-1})\) by (23).

The cosine convention in the printed cubic formula requires one final check. Lemma 12 shows that using exact cosine products costs only a prescribed small fraction of \(S\) and of the quasiparticle term, together with \(Cv_bQ^{-1/4}\). Combining these estimates leaves fixed positive fractions of both terms and proves (48). ◻

Proof. The preliminary rough bound and (21) give \[ \frac1V\sum_b\langle n_+\rangle\le CQ^2D^{1-c_{\mathrm r}} \tag{71}\] on the middle sectors, where \(c_{\mathrm r}>0\) is the exponent in Lemma [en:rough]. Lemma [en:localization] therefore bounds the sum of localization errors by \(CVQ^{-1}\). In particular this localization does not consume a fixed fraction of the leading energy.

On a high piece, the support condition \(n_L\ge M/8\) gives \[\left\langle\sum_p\min(p^2,1)a_p^*a_p\right\rangle \ge cME^{-2}\|\psi\|^2.\] Since \(ME^{-2}\asymp DQ^{-399}\), this pays both the rough error \(Cv_bD^{1-c_{\mathrm r}}\) and an \(O(v_b)\) LHY scalar, with a fraction tending to zero, provided \(D^{c_{\mathrm r}}Q^{-402}\to\infty\). The remaining rough bound then retains a fixed fraction of the kinetic and microcell penalties. Its leading quadratic scalar, compared with (22), also retains \(cDv_b(n_b/(Dv_b)-1)^2\). The inequality \(\mathfrak h(s+t)\le C(\mathfrak h(s)+\mathfrak h(t))\) allows a fraction of this scalar remainder to recenter the microcell penalty from \(n_b/v_b\) to \(D\).

On a low piece use Lemma 10. The convexity of \(e_r\) and the bound \(e_r''(x)\ge8\pi r\) imply \[ v_be_r(n_b/v_b)-t_b(n_b) \ge4\pi\alpha Dv_b\left(\frac{n_b}{Dv_b}-1\right)^2. \tag{72}\] For \(n_b<Dv_b/2\), the rough bound itself has a positive difference of order \(Dv_b\) from the tangent; its \(D^{1-c_{\mathrm r}}v_b\) error is negligible. For \(n_b>6Dv_b\), divide the labels into \(k=\lfloor n_b/(3Dv_b)\rfloor\ge2\) batches of nearly equal size. Their densities lie between \(3D-o(D)\) and \(4.5D+o(D)\); hence they lie in \([2.5D,5D]\) for large \(D\). Dropping interactions between batches and applying the rough bound to each batch gives a linear lower bound whose slope exceeds \(e_r'(D)\) by a fixed positive constant. Thus it retains \(c n_b\) beyond the tangent. Finitely many rounding adjustments are harmless for large \(Dv_b\). On these two classes, \(n_b+Dv_b\) also controls every bounded count deviation and the number of particles to which a bounded one-body observable can apply.

The squared localization factors sum to one, so the scalar tangent sums to \(Ve_r(D)\). The same identity preserves the box-count penalty, because box number commutes with all localizations. Summing the bounds just obtained and subtracting (21) proves the assertion, with \(Q^{-1/5}\) in place of the smaller sum of error powers. The conditions \(u\ll c_*\) and \(u\ll c_{\mathrm r}\) ensure that each power error is absorbed. ◻

Proof. Decompose the one-body operator into its \(P\)–\((1-P)\), \((1-P)\)–\(P\), and \((1-P)\)–\((1-P)\) blocks. Their second quantizations change the excitation number by \(1,-1,0\), respectively. Between the \(k\) and \(k+1\) sectors the first block has norm at most \(\sqrt{(n-k)(k+1)}\); the diagonal block has norm at most \(k\). Thus \(X\) is tridiagonal with block norms bounded by \(C[\sqrt{n(k+1)}+k]\). The blocks of \(X^2\) have bandwidth two and norm at most \(Cn(k+2)\), since \(k\le n\). Squaring, summing the five possible output blocks, and using Cauchy–Schwarz for their bounded overlaps proves (74). ◻

Proof. The inverse coefficients satisfy \[ |U_p|^2=1+|V_p|^2,\qquad |V_p|^2=\frac12\left\{ \frac{\tau(p)+\rho_z\widehat g(p)}{D_p(z)}-1\right\} \le C\min(|p|^{-1},|p|^{-4}). \tag{82}\] Rationalizing the expression proves the bound, using \(\rho_z|\widehat g(p)|\le C\), \(\tau(p)\asymp p^2\), and positivity of \(\widehat g\) near zero. On \(\mathcal P_L\), \(p^2|U_p|^2\le CD_p(z)\); on its complement \(U_p\) is bounded and \(\widetilde D_p(z)\ge cK_H/B^2\). The same inequalities hold with \(|V_p|\) in place of \(|U_p|\).

Write \(\mathsf U=\operatorname{diag}(U_p)\), \(\mathsf V=\operatorname{diag}(V_p)\), and \(\mathsf D=\operatorname{diag}(\widetilde D_p)\) on the nonconstant cosine modes. Split an input into its low and high frequency parts before estimating its image under \(A_b^{1/2}\). On the low part use \(A_b\le W\le CR^2\operatorname{diag}(p^2)\); on the high part use \(A_b\le1\) and \(K_H/B^2\gg1\). The squared norm of a sum is at most twice the sum of squared norms, so the coefficient bounds give \[ 0\le\mathsf U^*A_b\mathsf U\le C(1+R^2)\mathsf D, \qquad 0\le\mathsf V^*A_b\mathsf V\le C(1+R^2)\mathsf D. \tag{83}\] The split is made before applying \(A_b^{1/2}\) because \(A_b\) need not preserve either frequency band.

Take inner products linear in the second variable. For a finite set of modes the observable is the squared norm of \(A_b^{1/2}(a_p\xi)_p\). Decompose this column using \(a_p=U_pb_p+V_pb_p^*\). The annihilation column has squared norm at most \(q=C(1+R^2)\mathcal Q_z(\xi)\) by (83). For the creation column put \(\mathsf M=\mathsf V^*A_b\mathsf V\). The canonical commutation relations give the exact identity \[\begin{align*} \sum_{p,q}\mathsf M_{pq}\langle b_p^*\xi,b_q^*\xi\rangle &=\mathop{\mathrm{Tr}}(\mathsf M)\|\xi\|^2 +\sum_{p,q}(\mathsf M^{\mathsf T})_{pq} \langle b_p\xi,b_q\xi\rangle. \tag{84}\end{align*}\] The second matrix is the transpose, including when \(A_b\) is complex. Since \(\mathsf M\) is Hermitian, \(\mathsf M^{\mathsf T}=\overline{\mathsf M}\), and complex conjugation preserves its order bounds in (83): the dominating matrix \(\mathsf D\) is real diagonal. The creation column therefore has squared norm \(t+r\), where \(t=T_A(z)\|\xi\|^2\) and \(0\le r\le q\). Cauchy–Schwarz for the mixed columns gives \[\left|\langle\xi,\mathrm d\Gamma(A_b)\xi\rangle-t\right| \le2q+2\sqrt{q(t+q)}\le4q+2\sqrt{qt}.\]

To bound the trace uniformly, set \(G(p)=\min(|p|^{-1},|p|^{-4})\). A nonempty lattice shell of upper radius \(t\) contains at most \(CB^3t^3\) nonzero cosine frequencies; nonemptiness implies \(Bt\ge\pi\). Dyadic summation consequently gives \[ \frac1{v_b}\sum_{0<|p|\le\delta}G(p)\le C\delta^2, \qquad \frac1{v_b}\sum_{|p|\ge J}G(p)\le C/J \quad(0<\delta<1<J), \tag{85}\] and \(v_b^{-1}\sum_{p\ne0}G(p)\le C\). These estimates include coordinate-plane and axial frequencies. Since \(0\le(A_b)_{pp}\le1\), we obtain \(T_A(z)\le Cv_b\). The preceding mixed-column estimate proves (81).

The finite-mode computation passes to the full excitation space as follows. At fixed parameters \(\mathsf U,\mathsf V\) are bounded and \(\sum_p|V_p|^2<\infty\). The inverse relations and finite \(\langle\xi,n_+\xi\rangle\) imply \(\sum_p\|b_p\xi\|^2<\infty\). The creation column is square summable as well, since \[\sum_p|V_p|^2\|b_p^*\xi\|^2 =\sum_p|V_p|^2\|b_p\xi\|^2 +\left(\sum_p|V_p|^2\right)\|\xi\|^2<\infty.\] Finite coordinate projections commute with \(\mathsf U,\mathsf V\) and \(\mathsf D\), so the compressed matrices inherit (83). Norm convergence of the columns and convergence of the scalar traces justify the limit. If \(\mathcal Q_z(\xi)=\infty\), the asserted bound is automatic. ◻

Proof of Proposition 16. Apply the sector restrictions and excitation cutoffs used in Lemma 14, now with \(E=B\) and \(Q=D^u\). A box-wise operator evaluates by summing over these restrictions. For each fixed box the original restriction norms, summed over all assignments and integrated over spectators, have total square one. Thus the squared cutoffs preserve both that normalization and the identity \(\sum_b\langle n_b\rangle=N\).

Reduction to coherent traces. On sectors outside (24), use \(\mathrm d\Gamma(A_b)\le n\) and the norm penalty in Lemma 14; their contribution is \(o(V)\). The total one-body localization error on the middle sectors, divided by \(V\), is at most \[CDM^{-2}\le CD^{-1}Q^{794}=o(1).\] By splitting the input at \(|p|=1\) before applying \(W\), (80) also gives \[A_b\le W\le C(1+R^2)\min(-\Delta_{\rm Neu},1).\] The high pieces therefore contribute at most \(CV(1+R^2)Q^{-1/5}=o(V)\).

Let \(\phi\) be any low piece and \(\phi_z\) its zero-mode coherent projection. Since \(A_b\) annihilates the box constant, coherent resolution leaves its expectation unchanged. The part with \(|z|^2>2n\) is negligible by (47) and \(\mathrm d\Gamma(A_b)\le n\) on its excitation sectors. Apply Lemma 17 on the remaining region. The normalization above gives \(\sum_{\rm low}v_b\|\phi\|^2\le V\). Cauchy–Schwarz in the piece, spectator, and \(z\) variables then bounds the total error per volume by \[ C(1+R^2)Q^{-1/5} +C\{(1+R^2)Q^{-1/5}\}^{1/2}=o(1). \tag{86}\] Here \(2w<u/5\), as follows from our choice of exponents.

Define a measure \(\nu\) on the low-piece and coherent variables by \[ \int f\,\mathrm d\nu =\frac1V\sum_{\rm low}v_b \int_{|z|^2\le2n}f(\phi,z)\|\phi_z\|^2 \frac{\mathrm d^2z}{\pi}. \tag{87}\] As throughout the piece sums, spectator integration is included in this notation. The measure is positive and has mass at most one. Our reduction reads \[ \frac1V\langle\mathrm d\Gamma(A)\rangle =\int\frac{T_A(z)}{v_b}\,\mathrm d\nu+o(1). \tag{88}\]

Mass and density of the coherent measure. The omitted outlying sectors have volume weight at most \(CD^{-1}Q^{-1/5}\). On a middle-sector high piece, \(n_L\ge M/8\) implies \(\mathrm d\Gamma(\min(p^2,1))\ge cM/B^2\); hence its total omitted volume weight is bounded by \[ \frac1V\sum_{\rm high}v_b\|\phi\|^2 \le C\frac{v_bB^2}{M}Q^{-1/5} \le CD^{-1}Q^{401.8}=o(1). \tag{89}\] The coherent tail removes exponentially small weight. Therefore \(\nu\) has mass \(1-o(1)\).

Here is an explicit form of its density concentration. Decompose a fixed \(n\)-particle piece into excitation-number components \(\phi^{(m)}\), \(0\le m\le n\). Its zero-mode number on that component is \(n-m\). Before restricting to \(|z|^2\le2n\), orthogonality of the excitation sectors shows that the radial coherent law of \(s=|z|^2\) is the mixture of gamma densities \[e^{-s}\frac{s^{n-m}}{(n-m)!}\,\mathrm ds\] weighted by \(\|\phi^{(m)}\|^2\). Conditionally on \(m\), the mean is \(n-m+1\), the variance is \(n-m+1\), and \(\mathbb E|s-(n-m)|\le1+\sqrt{n+1}\). On the middle range the box penalty controls the squared relative number deviation, so \[\int\left|\frac n{Dv_b}-1\right|^2\mathrm d\nu \le CD^{-1}Q^{-1/5}.\] Before the \(z\) cut, the box-volume weighted integral of \(m/(Dv_b)\) is \[\frac1{DV}\sum_{\rm low}\langle\phi,n_+\phi\rangle \le CD^{-1}B^2Q^{-1/5}.\] These nonnegative bounds remain valid after the cut. Combining them with the conditional gamma estimate yields \[ \int\left|\frac{\rho_z}{D}-1\right|\mathrm d\nu \le C\left(D^{-1/2}Q^{-1/10} +D^{-1}B^2Q^{-1/5}+(Dv_b)^{-1/2}\right)=o(1). \tag{90}\] In particular every fixed deviation of \(\rho_z/D\) from one has vanishing \(\nu\)-weight.

Identification of the covariance. On a fixed annulus \(0<\delta\le|p|\le J\), all modes eventually belong to \(\mathcal P_L\), so \(\tau(p)-p^2=-\pi/(2B^2)\) there. The support radius \(r\) and (23) also imply \(D\widehat g(p)=8\pi\alpha+o(1)\) uniformly on bounded momentum sets. The expression in (82) is continuous in these parameters on the annulus when \(\rho_z/D\) is close to one, with a denominator bounded away from zero. By (90), \[\int\sup_{\delta\le|p|\le J} \bigl||V_p(z)|^2-F_\alpha(p)\bigr|\,\mathrm d\nu=o(1).\] On the exceptional density set use the uniform majorant in (82). The same majorant bounds \(F_\alpha\), so (85) removes the small and large momentum regions, uniformly in \(A_b\). Since \(|(A_b)_{pp}|\le1\), the annular lattice count divided by \(v_b\) is bounded, and the matrices are identical in local coordinates, (88) becomes (79). All bounds used only \(0\le A_b\le W\) and are therefore uniform in this class.

For a complex contraction \(C_b\), write \(C_b=S_b+iT_b\), with \(S_b=(C_b+C_b^*)/2\) and \(T_b=(C_b-C_b^*)/(2i)\) Hermitian contractions. For either \(J_b=S_b,T_b\), \[WJ_bW=2W\frac{1+J_b}{2}W-W, \qquad 0\le W\frac{1+J_b}{2}W\le W.\] A fixed linear combination of the positive-operator identities proves the assertion for \(WC_bW\), preserving uniformity. Finally, simultaneous translation of the particles is unitary and preserves the ground space. Applying the uniform estimate to the translated vector proves the assertion for every translated nested grid. ◻

Proof. We give the separate estimates for supply, exceptional motion, and the fine ball counts. Their combination will also explain why the one-site power and the joint parameter in (101) have different forms.

First consider the cutoffs under free motion. Gaussian maximal estimates give probability at most \(CD^{-40}\) for maximal displacement from the initial point larger than \(b\) or failure of (96), after the choices of \(C_1,C_2\). For the increment estimate, first use a union bound over the \(O(r^{-4})\) pairs of mesh points and then control oscillations on individual intervals. For a trial bridge with endpoint displacement at most \(b\), or two legs pinned at a point within \(8b\) of \(m_i\), the increment cutoff and confinement to the radius-\(20b\) ball fail with probability at most \(CD^{-40}\). The maximal-displacement test at radius \(b\) is used only for exceptional visitors in the original sample. To check the trial estimates uniformly as \(t\) decreases, represent the centered part of a leg of length \(h\) by \(W(q)-(q/h)W(h)\). Its prescribed line has increment divided by \(\sqrt{|q-q'|+r^2}\) at most \(C b/\sqrt t=C C_1\sqrt{\log D}\), so it uses only a fixed fraction of \(L_0\). The Gaussian remainder and the distance from that line to the confinement boundary give the asserted tails. Reversal of any of these exceptional events preserves the bound.

Here is the transfer of these free tails to the interacting sample. For a set \(A\) of sites let \(J_A(\omega)\) count the neighborhoods in that set visited by a path. If a reversal-invariant path event \(\mathcal E\) has free probability at most \(\varepsilon\), the free visit estimate in [24], with a unit-cell covering for our fixed neighborhood radius, gives \[\mathbb E_{\mathrm{free}}^y [e^{\vartheta J_A\mathbf 1_{\mathcal E}}-1] \leq C_\vartheta\varepsilon^{1/3}e^{-d_K(y,A)}.\] Its constants are uniform for \(t\leq1\). Expanding the exponential as a product over labels, applying (94), and then using the count moment-generating estimate yields \[ \log\mathbb E\exp\!\left(\vartheta\sum_i J_A(\omega_i)\mathbf 1_{\mathcal E}(\omega_i)\right) \leq C_\vartheta D\varepsilon^{1/3}|A|, \tag{102}\] provided the coefficient in the last count estimate is at most one. For a single neighborhood the probability of even one exceptional visitor is bounded by the exponential moment minus one; it is \(O(D\varepsilon^{1/3})\). With \(\varepsilon=CD^{-40}\) this is smaller than \(D^{-6}\). For a joint bound, choose a fixed large \(\vartheta\) in (102) first, and then take \(D\) large. The probability of an exceptional visitor at every site in \(A\) is at most \(e^{-(\vartheta-o(1))|A|}\).

To obtain the interior supply, start with the lower endpoint count \(\ell_*D\) supplied by Proposition 5. Its failure probability is \(O(D^{-2})\), jointly \((C/D^2)^{|A|}\). Remove labels with a close endpoint at either time, and labels whose displacement exceeds one. Proposition 7 gives a joint small-parameter bound for a fixed positive fraction of close labels, since \(s\ll D^{-1/3}\). Its one-site probability here is sharper: by (95) and Markov’s inequality, removing a fixed fraction of \(D\) labels has probability \[ CD s^3=CD^{-7/5}. \tag{103}\] Labels initially at a site but with a nonseparated terminal endpoint are covered by the same estimate in a fixed neighborhood, except for paths of displacement greater than one. The latter have free probability \(Ce^{-c/t}\); apply (102) to the event that a fixed positive fraction of the local labels has this property. Its probability is \(e^{-c'D}\) for large \(D\). The stronger cutoff failures have already been treated. A fixed positive fraction of \(D\) eligible labels consequently remains. Conditional on all paths, independent flags leave at least \(cf_0D\) labels with probability \(1-e^{-c'f_0D}\). The lists based at distinct sites are disjoint, so the same binomial estimate is joint over any prescribed sites.

We have controlled the supply of eligible labels and the exceptional visitors. The remaining certificate tests prevent too many obstacles from concentrating in small balls. The upper counts in unit cells follow from the count exponential moment; the weighted bounds require a uniform argument down to the shrinking hard-core scale. Test separated sites, so far apart that a path obeying the motion cutoff can contribute to only one of their neighborhoods. Select one mesh time and one ball of radius \(h_v\in[r,1]\) at each tested site \(v\). Truncate the oscillation weight at the cutoff. Multiply each contribution by the indicator that its whole central-slab path satisfies the increment and diameter cutoffs; outside the exceptional-visitor event already estimated, these restricted counts equal the actual local counts. This is a restriction of an integrand, not a conditioning of the path law. Put \(a_D=c'(\log D)^{-2}\). Since \(A_J\leq C\sqrt{\log D}\) on that cutoff, \(a_D(1+A_J)^3=o(1)\). For the single-path exponential increment \(F_v\) we therefore have \[0\leq F_v\leq C a_D(1+A_J)^3 \mathbf 1_{\{\omega(q_v)\in B_v\}},\qquad \mathbb E_{\mathrm{free},[-1,1]}^y F_v \leq C a_D h_v^3e^{-3d_K(y,v)}.\] For a following interval the second inequality uses the bounded Gaussian density at \(q_v\) and the bounded moments of the normalized increment. For a preceding interval, integrate first over the increment and translate the ball by that increment; its Gaussian tail gives the identical bound. Central-slab times are uniformly away from the longer slab’s endpoints, so the constants are uniform. For flagged counts, first average the flag of this label; the bound acquires a factor \(f_0\).

Because a path passing the diameter cutoff visits at most one tested neighborhood, its exponential increment is the sum of the corresponding \(F_v\). This property survives time reversal: the increment and diameter gates are reversal invariant, and \(F_v\circ\mathrm{rev}\) still requires a visit to the same spatial neighborhood \(v\), merely at the reflected time and interval. Thus replacing the single-path increment by \(\sum_v(F_v+F_v\circ\mathrm{rev})\) makes it reversal invariant while preserving the same separated support and free expectation bound, up to a factor two. Expand the product over distinct labels, apply (94) to these weights, and bound the resulting product by an exponential. This gives \[ \log\mathbb Ee^{a_D\sum_v W_v} \leq C D a_D\sum_v h_v^3, \tag{104}\] where \(W_v\) is the truncated weighted count restricted to paths satisfying the motion cutoffs. In the flagged version the right side has an extra \(f_0\). To justify the use of the spatial count estimate with different \(h_v\), write \(h_v^3=\int_0^1\mathbf 1_{\{h_v^3\geq u\}}\mathrm du\). For \(A_u=\{v:h_v^3\geq u\}\), the lattice exponential sum satisfies \(\sum_{v\in A_u}e^{-3d_K(y,v)}\leq C e^{-d_K(y,A_u)}\). Convexity of the logarithm of the exponential moment, followed by the set-count bound and \(\int_0^1|A_u|\mathrm du=\sum_vh_v^3\), proves (104).

Exponential Markov inequality at thresholds \(C(Dh_v^3+D^{1/100})\) now bounds any specified collection of ball violations by \[\exp\{-c a_D D^{1/100}|A|\}.\] The same proof applies to flagged counts at their stated thresholds. There are only polynomially many net balls and mesh times per site; their union therefore still has an arbitrarily large power saving, jointly as well as marginally. A fixed finite coloring separates arbitrary sites. Finally assign to every bad site one of the finitely many failed tests and select a largest class. The coloring and this assignment cost only fixed roots and fixed exponential factors in the joint parameter. Each underlying parameter can be made as small as needed before choosing \(D\), proving the joint assertion for any fixed \(p\). Marginally the largest remaining term is \(CD^{-7/5}\) from (103), which is bounded by \(CD^{-1-1/8}\). Applying the weighted-ball and motion arguments to all labels instead of only frozen labels proves the weighted part of the full-path bound. Either failure in (100) has free probability at most \(CD^{-40}\) by the first Gaussian estimate and its reversal. The visitor estimate consequently bounds their local union by \(O(D\,D^{-40/3})\), which is smaller than \(D^{-6}\). This proves the asserted full-path failure bound including its ordinary-motion requirements. ◻

Proof. Orient each half of an ordinary bridge from its nearer endpoint; for a pinned trial do this separately on each of its two legs. Let \(h\geq s\) be the separation between the trial and opponent at that endpoint, and let \(q\) be elapsed time from it. On the two increment cutoffs, a collision in a mesh interval is impossible unless \[q+C\Delta\geq c h^2/(\log D).\] The lower bound exceeds \(\Delta\) by a power of \(D\), since \(s^2/(r^2\log D)\to\infty\). A possible collision requires positions within \(Cr\sqrt{\log D}\) at a neighboring mesh point. On the oriented half-leg, the bridge density there is bounded by \(Cq^{-3/2}\). Thus the sum over these mesh points is at most \[C r^3(\log D)^{3/2} \sum_{q\gtrsim h^2/\log D}q^{-3/2} \leq \frac{Cr(\log D)^2}{h}.\] Summing the finitely many oriented pieces proves the stated endpoint and midpoint terms, after enlarging \(b_D\). A fixed additive term covers the harmless bounded-distance conventions. Confinement eliminates opponents outside the relevant neighborhood, and local lifts apply for all sufficiently large \(K\). Finally \(\int_Q(s\vee|y-z|)^{-1}\mathrm dy\leq C\) for every unit cube \(Q\) and every \(z\), whereas the volume where \(|y-z|\leq s\) is at most \(Cs^3\). Since \(s^3=o(b_D)\), integration proves the last assertion. ◻

Proof. For unexposed opponents, sum Lemma 20 over endpoint shells using (98). Both endpoint lists satisfy \[\sum_k\beta_{ik} \leq C b_D(f_0D+D^{1/100}/s) \leq C(\log D)^C(D^{-x}+D^{-19/100})=o(1).\] This also bounds the integrated cost for a uniform midpoint. The extra midpoint separation costs at most \(Cs^3\) per opponent, and there are \(O(f_0D)\) such opponents in a relevant neighborhood.

For frozen opponents the endpoint-shell bound alone would lose a logarithm. We instead use their weighted space-time count. Consider an oriented half-leg and put \[q_0=cs^2/(\log D).\] Endpoint separation and (96) imply that only intervals with elapsed time \(q\geq q_0\) can contribute. At their initial mesh point the trial position has Gaussian density at most \(Cq^{-3/2}e^{-c|z-a_q|^2/q}\), where \(a_q\) is its bridge mean. Given that position, the following increment has the representation \[X_{q+v}=X_q+\frac{v}{\ell-q}(a'-X_q)+R_v, \qquad 0\leq v\leq\Delta,\] where \(\ell\in\{t,2t\}\) is the bridge or leg duration, \(a'\) is its opposite prescribed endpoint, and \(\ell-q\geq\ell/2\) on the oriented half. The centered bridge \(R\) is independent of \(X_q\). Its supremum divided by \(r\), denoted by \(U\), has a uniform Gaussian tail. On confinement the drift over the mesh interval is at most \(C_b\Delta/t\leq r\) for large \(D\). A collision with frozen path \(k\) therefore requires \[ |\omega_i(q)-\omega_k(q)| \leq Cr(1+A_J(\omega_k)+U). \tag{106}\] After deriving this necessary event we discard the cutoff before integrating the unrestricted bridge, under which the stated independence holds. This conditional-increment argument is the one used in [24].

Truncate \(U\) at \(C\sqrt{\log D}\) with an arbitrarily small polynomial error, summed over the polynomial number of local paths and mesh intervals. The radii in (106) are then \(o(\sqrt q)\) uniformly for \(q\geq q_0\). Integrate the Gaussian density over these balls, then integrate \(U\). Its fixed moments are bounded, so the result is bounded by \[Cr^3q^{-3/2}\sum_k(1+A_J(\omega_k))^3 e^{-c|\omega_k(q)-a_q|^2/q}.\] Gaussian shells of radius \(\sqrt q\), and (99), bound this expression by \(Cr^3(D+D^{1/100}q^{-3/2})\). Shells of radius greater than one use a unit-ball covering; their Gaussian decay makes the sum finite. Summing the mesh intervals on all oriented pieces gives \[ C r\left(Dt+\frac{D^{1/100}}{\sqrt{q_0}}\right)+o(1) \leq \frac{C\alpha}{x\log D} +C\alpha D^{-19/100}\sqrt{\log D}+o(1)=o(1). \tag{107}\] For an ordinary bridge no midpoint is pinned, hence no midpoint separation is needed. For a uniform midpoint the excluded volume around frozen midpoint positions is at most \(CDs^3=o(1)\), by the unweighted count bound. For each chosen midpoint satisfying the separation condition, the preceding frozen estimate is uniform. The individual trial cutoffs fail with probability \(O(D^{-40})\). These estimates prove the first two assertions, and a union bound in the common-midpoint proposal proves the third.

On the full-path event the entire obstacle collection obeys (99); the same calculation proves the fourth assertion for any separated prescribed midpoint. Each added or substituted allowed obstacle contributes at most \(Cb_D/s=o(1)\) by Lemma 20. Deleting checks only increases each success probability. ◻

Proof. For a single label, its conditional density is an admissibility indicator divided by its free insertion probability. Proposition 21 bounds that denominator below by \(3/4\), uniformly in the other allowed paths. Successively integrate labels, or equivalently compare the partition integrals before and after each deletion. Each deletion costs at most \(4/3\), proving \(C^q\) for any specified collection. The midpoint statements were computed above. All arguments continue to apply with checks removed because the same lower insertion bound remains valid. ◻

Proof. The midpoint law is \(\Phi^2\,\mathrm dX\). Flagging a particular label has probability \(f_0\), independently of this law. Symmetry of \(\Phi^2\) and the uniform ordering make all \(N\) summands in (112) identical after projection. Their coefficient is \(Nf_0/T_v=V/|v|\). This identifies the projection with \(P_v\) in (109). The last assertion follows by summing the cell indicators, since \(N=DV\). ◻

Proof. Conditional on the configuration, the flagged count \(M_v\) in a cell is binomial with parameters \(n_v\) and \(f_0\). In particular, \[\mathbb E[(M_v-f_0n_v)^2\mid X]\le f_0n_v, \qquad \mathbb E[(M_v-f_0n_v)^4\mid X] \le C\bigl(f_0n_v+(f_0n_v)^2\bigr).\] Write \[T_v^{\mathrm{rel}}-1 =\frac{M_v-f_0n_v}{f_0D|v|} +\left(\frac{n_v}{D|v|}-1\right).\] The quadratic and quartic inequalities for this sum give the displayed bounds from the count moments of Proposition 13. The binomial contribution to the fourth moment is at most \(C_j(D^{-2+2x}+D^{-3+3x})\); the second term is smaller for \(x<1\). The required first and second moments of \(n_v/(D|v|)\) follow either from these count estimates or from Proposition 6.

For the last assertion, use the event that every unit cell meeting the indicated cells has count at most \(CD\), where \(C\) is a sufficiently large fixed constant. On that event \(T_v^{\mathrm{rel}}\le C_j/f_0\). Proposition 6 bounds the complement by a polynomial in \(D\) times \(e^{-cD}\), also for an \(R\)-cell since it meets \(O(R^3)\) unit cells. Hard packing bounds all counts in these bounded regions by a fixed power of \(D\). Consequently every fixed product on the exceptional event has expectation smaller than any prescribed power of \(D^{-1}\). ◻

Proof. Apart from flagging, failure of membership in \(\mathcal I\) means failure of an endpoint separation or an individual motion test. Flag independence cancels the factor \(f_0\) in \(T_v\). A tag in \(v\) satisfying the motion cutoffs has both endpoints in a bounded enlargement of \(v\). Equation (95) bounds the expected number of labels there having a second endpoint within distance \(s\) by \(CD^2s^3\) per unit volume. Division by \(D\) gives \(CDs^3=CD^{-7/5}\). The visitor and free-cutoff estimates in the proof of Proposition 19 give errors of arbitrarily high fixed power for the remaining motion failures. Summing over the unit cells meeting an \(R\)-cell and dividing by its volume leaves the same bound. Both estimates are stronger than the stated \(D^{-1-1/8}\) bound.

For a good-neighborhood failure use Proposition 19 and a union bound. There are \(O_{x,j}(1)\) tests for a local comparison and \(O_x(R^3)\) tests at scale \(R\). On the count event in Lemma 25, the total reference weight is bounded by \(C_j/f_0\). Its complement is negligible. This proves the second claim and the assertion about products. If \(M_v\) denotes the omitted relative count, then \(0\le M_v\le C_j/f_0\) on the same count event and \(M_v^q\le(C_j/f_0)^{q-1}M_v\). This proves the final convention. ◻

Proof. Join the tag mean-sites by a coordinate route of nearest-neighbor sites, erase loops, and choose one interior label at each interior site. The tag means lie within distance \(Cb\) of adjacent cells, so there are \(O(b)=O(x^{-1/2})\) entries. If the sites coincide the list consists just of the two tags. The good-neighborhood event supplies all labels. The interior supplies have size at least \(cf_0D\), so excluding the bounded number of tags preserves the claimed point bound.

For \(k=1,\ldots,m\), sample a common midpoint uniformly in a unit cell whose distance from both corresponding means is bounded. Fix that cell as a measurable function of the exposed endpoints alone, before sampling any latent original or proposed paths. Conditional on that point, propose independently the left path of \(i_k\) and the right path of \(i_{k-1}\) as two-leg free bridges. Condition this pair on its individual cutoffs and on avoidance, in each copy separately, of paths retained in that copy and paths already inserted in that copy. Paths to be inserted at later stages are not checked at this stage. Proposition 21 gives success probability at least \(1/2\) for every such proposal, including every allowed preceding history. Thus each stage is a probability kernel. Every pair of paths in a finished copy is checked when the later of them is inserted. The midpoint identities and equality of the two remaining lists follow directly from the cyclic shift of labels in (116). ◻

Proof. Before dividing by the reference density, the output density is the right side of (118) divided by \(Z_S\). The kernel produces only admissible left paths; on that support the reference density is \(1/Z_S\). These factors cancel. Integrating the resulting likelihood against (117) and using that \(K_S\) has total mass one gives \[\int L_S\,\mathrm d\lambda_S =Z_S^{-1}\int a_S(o;\xi)h_{w,j}(o)\prod_{l\in S}B_l(\mathrm do_l) =\int h_{w,j}\,\mathrm d\lambda_S.\] This proves the conditional centering, also with deleted checks.

For clarity, the density used here can be written directly from the pair proposals. If \(C_k\) is the chosen unit cell and \(g_l\) the free midpoint density for label \(l\), then the proposal for the \(k\)th pair, viewed from its left coordinate \(z_{i_k}\), is \[ \frac{\mathbf 1_{C_k}(z_{i_k}(0))}{g_{i_k}(z_{i_k}(0))} B_{i_k}(\mathrm dz_{i_k})\, B_{i_{k-1}}^{\,z_{i_k}(0)}(\mathrm dz'_{i_{k-1}}). \tag{121}\] Here \(B_l^y\) is the two-leg bridge pinned at midpoint \(y\). Its restriction to the individual cutoffs, together with the compatibility checks, is divided by a success probability at least \(1/2\). The first density factor in (121) is bounded by \(CD^{Cx}\) on \(C_k\), by Corollary 22. Integrating the auxiliary right paths successively, bounding each normalizer by \(2\), and using \(m=O(x^{-1/2})\) gives (120). The original integral has mass at most one. The same argument works with checks deleted because their success probabilities only improve. ◻

Proof. Expand each centered factor as \(L_S-h_{w,j}\) and use (118) and (121) for the likelihood factors. This represents the conditional integral using the following variables: reference output bridges on \(U\); a separate collection of original bridges for each likelihood; and the auxiliary right bridges in its pair proposals. Without avoidance checks these variables factor between blocks. Within one block the two members of a pair proposal have a common midpoint, as they must. Every indicator imposing an individual cutoff is retained in the estimates below.

There are three kinds of factors: indicators, success-probability reciprocals, and the densities in (121). The last factors do not depend on switches. When the changed sets are disjoint, combine each such density with its free left-output bridge and its pinned auxiliary bridge: this is exactly the probability measure of the common-uniform- midpoint pair proposal. We retain that measure when integrating a collision. In particular we do not replace it by the product of two independent bridges; that product would not even dominate the pair law. Thus the base measure within each singleton block is a product of plain original bridges and independent pair proposals, or of plain output bridges if its centered factor contributes the subtracted indicator. The different blocks have independent base measures, conditional on the fixed outside paths. Every proposal cell was fixed using the exposed endpoints, so its law has no remaining dependence on other blocks.

There are \(O(x^{-1/2})\) pairs, including all four lists. All success reciprocals are at most two. The inverse reference normalizer costs a fixed factor \(C_x\), by successive single-path insertions on the bounded set \(U\); keep that normalizer fixed when taking differences. With one identification, the nonsingleton block has repeated output coordinates. We leave those coordinates and their pinned auxiliary variables together until the singleton collision integrations have been carried out. Afterwards the remaining integral is bounded by the marginal-density argument proving (120), paying at most \(C_xD^{C\sqrt x}\). This estimate never requires a joint density for two paths with a common midpoint. These observations also bound all unused factors in a difference by the same displayed expense.

An indicator difference caused by a block-pair switch is supported on the event that at least one of the removed avoidance checks fails. Splitting this event by the union bound gives a collision between one path from each of the two blocks. The paths have distinct labels and meet their individual cutoffs. Integrating a free bridge or a marginal of a common-uniform-midpoint proposal against a deterministic allowed opponent costs at most \(C_x\beta_{lk}\), by Lemma 20. The same assertion holds with the roles of the two paths exchanged.

We give the analogous verification for denominators. At a fixed insertion stage and preceding history its success probability is \[p(\sigma)=\int Q(\mathrm dz)\,c(z)\prod_e A_e(z)^{\sigma_e}, \qquad p(\sigma)\ge\tfrac12.\] Here \(Q\) is a fresh pair proposal, \(c\) contains the individual cutoffs and unswitched checks, and \(A_e\) is the product of checks controlled by switch \(e\) at that stage. The convention \(A^0=1\) applies also when \(A=0\). A first difference of \(p\) is an integral with the factor \(A_e-1\); a mixed difference in \(e,f\) has the factor \((A_e-1)(A_f-1)\). They therefore force one or both corresponding collisions. The reciprocal identity \[p^{-1}-q^{-1}=\frac{q-p}{pq}\] and its difference in a second variable bound a first reciprocal difference by a constant times \(|\Delta_ep|\), and a second difference by a constant times \[|\Delta_e\Delta_fp| +\max_{\sigma}|\Delta_ep(\sigma)|\, \max_{\sigma}|\Delta_fp(\sigma)|.\] The product is represented using independent fresh proposals. These proposals belong to the block whose denominator is being tested. After dropping unused compatibility indicators, they are ordinary pair proposals with their own cutoff indicators. They introduce only a fixed number of additional density factors.

Apply the finite-product difference rule. A first difference is thus a finite sum of integrals with one forced cross-block collision. To estimate it, integrate a singleton block against its fixed opponents whenever there is an overlapping block. In the application with one identification every block pair has a singleton side unless both blocks are singletons, which is easier. The remaining factors are bounded as above. This proves the first assertion in precisely the overlapping case needed below; with four disjoint lists either side can be integrated.

For the second assertion the two different block-pair edges form either a two-edge tree or two disjoint edges. In the tree case fix all variables in the central block and integrate the two leaf blocks in succession. Each leaf participates in just one of the two forced collisions, so its integration costs the appropriate \(\beta\) bound uniformly in the central block. No independence of collisions inside the central block is used. If a forced collision comes from a denominator test, its fresh proposal belongs to its generating block; the same leaf integration applies. The two disjoint edges are estimated separately. This proves the product bound.

Extra differences in other switches, when they occur, are bounded by a finite sum of evaluations of the same two-difference expression. This does not increase the number of forced collisions that must be estimated. Finally \(\beta_{lk}\le b_D(1+2/s)\), which is bounded by \(C_xD^{-19/100}\) for sufficiently large \(D\). This proves the stated worst bound. ◻

Proof. Expand the fourth power into four independently randomized routes and label lists, conditionally on the datum and the common tag \(i\). The total tag counting weights are bounded by \(C_{x,j}\) on good data: there are at most \(C_xDf_0\) candidates in the bounded enlargement, and \(T_v,T_w\) are fixed multiples of \(Df_0\). Each normalized tag weight, and each interior label probability, is at most \(C_{x,j}/(Df_0)\).

First split the tag summations by the mean-sites of the second tags and the resulting route site patterns, which have only \(O_x(1)\) possibilities. Retain the original summation weights in each subsum; there is no renormalization by the mass of a site pattern. For every fixed tag choice in such a subsum, each tag counting weight and each interior-choice probability is bounded by \(C_{x,j}/(Df_0)\). The set of candidate labels for any slot has at most \(C_xDf_0\) elements. In bounding a positive sum we may discard within-list exclusions and dominate its joint weight by the product of these point bounds on the respective candidate sets. This domination requires no claim that conditioning on an arbitrary previously revealed tag preserves the stated point-mass bound.

Two or more independent identifications between the four changed lists therefore cost at most \(C_{x,j}(Df_0)^{-2}\). An identification means one equality needed to merge previously separate label slots; a triple equality counts as two. Indeed an equality pattern with \(q\) independent identifications leaves \(q\) fewer freely chosen labels. Summing the product point bounds over that pattern gives at most \(C_{x,j}(Df_0)^{-q}\), since every free label has at most \(C_xDf_0\) possible values. There are only \(O_x(1)\) slot patterns. A label cannot occur twice in one list, and none of the changed lists contains \(i\). The supremum bound (120) therefore bounds all contributions with at least two identifications by \[ C_{x,j}D^{-2+2x+C\sqrt x}. \tag{123}\]

There remain the cases of no identification and exactly one. Condition on paths outside \(U\) and write the conditional expectation as \(Z_U^{-1}\) times an integral against the free product bridges on \(U\) with its admissibility numerator. Keep \(Z_U\) fixed. Regard this numerator, including the four \(F_a^\sigma\), as a function \(G\) of the block-pair switches. The elementary expansion at the all-off vertex of the Boolean cube is \[ G(\boldsymbol1)=\sum_{E}\Delta_EG(\boldsymbol0). \tag{124}\] In a term with an isolated singleton block, the integrand factors with respect to that block and the remaining blocks. Integrating the isolated block gives zero by (119), multiplied by its unnormalized reference density. This is a statement about each term of the mixed difference: its absent edges are off in every evaluation.

With exactly one identification there are three blocks, two of them singletons. A surviving term in (124) must therefore contain at least one edge. Lemma 29 and the identification cost bound its total by \[ C_{x,j}D^{-1+x-19/100+C\sqrt x}. \tag{125}\] With no identification all four blocks are singletons. A surviving term must contain at least two different edges. Choose two of them, apply the two-difference bound, and estimate the remaining differences by the finite sum described in that lemma.

One of the two collision factors can be averaged over a free label choice. There is always such a label: every cross-block check has a label of a changed list on at least one side, whereas \(i\) belongs to none of those lists. The flagged-endpoint bound in Definition 18, summed over dyadic distance annuli, gives, uniformly in the other endpoint, \[ \frac{C_{x,j}}{Df_0}\sum_{k\text{ in a candidate list}} \frac1{s\vee d(Y_k,y)} \le C_{x,j}\left(1+\frac{D^{1/100}}{Df_0s}\right)\le C_{x,j}. \tag{126}\] The same holds at the other endpoint time. The candidate list is contained in a fixed bounded neighborhood; the annulus estimate uses \(\#B_h\le C(Df_0h^3+D^{1/100})\). The last inequality follows from \(x<19/100\). Thus the averaged collision factor is at most \(C_{x,j}b_D\). We bound the other factor by \(\beta_*\). Excluding additional label coincidences can only decrease these positive upper bounds, so their restriction need not be retained during this summation. The no-match contribution is consequently \[ C_{x,j}D^{-1-19/100+C\sqrt x}(\log D)^C. \tag{127}\] The factor \(h_{v,i}\) depends only on the fixed outside path \(i\) and can be retained throughout the cancellations. It is bounded by one when estimating the errors. Summing the first tags has total weight at most \(C_{x,j}\) on good data. Combining (123), (125), and (127) proves (122) when \(x\) is small enough that all power losses leave a saving of \(1/20\). ◻

Proof. Let \(M_w\) be the omitted relative count in \(w\). On the retained left reference, (128) gives \[\ell_v-1=\sum_{j\ne i}T_w^{-1}\overline{F_S} +(T_w^{\mathrm{rel}}-1)-M_w.\] The inequality \(|a+b+c|^4\le27(|a|^4+|b|^4+|c|^4)\) separates the three terms. Lemma 30 treats the first. For the second, the count-tail bound gives \[\mathbb E\bigl[T_v^{\mathrm{rel}}|T_w^{\mathrm{rel}}-1|^4\bigr] \le C_jD^x\mathbb E|T_w^{\mathrm{rel}}-1|^4+O(D^{-A})\] for any fixed \(A\), after averaging cells. Neighboring pairs have uniformly bounded degrees, so Lemma 25 applies. For the third term use \(\mathbb E[T_v^{\mathrm{rel}}M_w^4] \le C_jD^{4x}\mathbb EM_w+O(D^{-A})\) and Lemma 26. On omitted first tags or data the density is zero, so the fourth error is exactly the omitted reference mass. The same lemma bounds it. Choose \(x\) so that, in addition to the requirements of Lemma 30, \[x<u/24,\qquad 4x+u/8<1/8.\] The resulting powers are bounded by \(D^{-1-u/8}\); the binomial \(D^{-2+2x}\) contribution is smaller. This proves the left half of (129). For the right half, analyze the same generated kernel with the observed and auxiliary copies exchanged, retaining its original insertion order. Every preceding density and switch estimate holds for any fixed insertion order: only the specified previous-path checks enter its success probabilities. Thus this argument uses no symmetry of the success normalizers. Interchanging the two tag roles gives the other half. ◻

Proof of Proposition 23. Let \(q\) be the Lebesgue density of the common projection \(\nu_{v,w}\). Conditional Jensen under the projection gives \[\int\left|\frac q{s_v^2}-1\right|^4s_v^2\,\mathrm dY' \le\int|\ell_v-1|^4\,\mathrm dP_{\mathrm{aug},v},\] with the analogous estimate for \(w\). The measure \(\nu_{v,w}\) is absolutely continuous with respect to both references, so \(q=0\) where either corresponding density vanishes.

For completeness, for \(a,b\ge0\) and \(q\ge0\) with this zero-set convention there is an absolute constant \(C\) such that \[ \frac{(a-b)^4}{a^2+b^2} \le C\left\{a^2\left|\frac q{a^2}-1\right|^4 +b^2\left|\frac q{b^2}-1\right|^4\right\}. \tag{130}\] For \(a,b>0\), put \(c=\sqrt q\). The inequalities \(|a-b|^4\le8(|a-c|^4+|b-c|^4)\) and \(|1-\sqrt z|\le|1-z|\) for \(z\ge0\) give \[\frac{|a-c|^4}{a^2+b^2} \le a^2\left|1-\frac c a\right|^4 \le a^2\left|1-\frac q{a^2}\right|^4,\] with the same estimate for \(b\). This proves (130) with \(C=8\). If one of \(a,b\) vanishes, absolute continuity forces \(q=0\), and the inequality follows directly; if both vanish, both sides are defined to be zero.

Apply (130) to \(a=s_v\), \(b=s_w\) and sum the Jensen bounds. This proves (110). ◻

Proof. Choose an ordered ternary tree of depth \(m\) with \(3^m\ge T\). Give the root spin \(+1\). At every nonleaf vertex the first two children inherit its spin and the third receives an independent fair spin. If the leaf spins in order are \(X_1,\ldots,X_{3^m}\), let \(S_j=\sum_{i=1}^jX_i\).

The leaf sum of a depth-\(d\) subtree with root spin \(s\) equals \[2^ds+\sum_{a=1}^d2^{d-a} \sum_{i=1}^{3^{a-1}}\varepsilon_{a,i},\] where the displayed fresh spins are independent. Select one fresh spin at each depth. Their weighted sum is uniform on \(2^d\) distinct integers: writing \(\varepsilon=2\eta-1\) reduces this assertion to binary expansion. Independently adding the remaining terms shows that every atom of the subtree sum is at most \(2^{-d}\).

For \(k\ge2\), choose \(d\ge0\) with \(2\cdot3^d\le k<2\cdot3^{d+1}\). The interval of leaves \(\{u+1,\ldots,u+k\}\) contains a complete aligned depth-\(d\) subtree. Condition on every fresh spin except those strictly below its root. This fixes the history through time \(u\), its root spin, and all leaves outside the subtree. The conditional atom bound just proved is \(2^{-d}\le6^\sigma k^{-\sigma}\). Conditioning down to the history gives (135). The bound for \(k=1\) is immediate. ◻

Proof. Set \(k(0)=1\) and \(k(t)=t^{-2\sigma}\) for \(t\ge1\). Successive conditioning in (136) bounds a summand on the left of (137) by \[C^j\prod_{i=1}^j(1+l_i)^2\, k(s_1)\prod_{i=2}^j k(s_i-s_{i-1}).\] Since \(2\sigma>1\), the sum of \(k\) is finite. Summing over the initial time and positive gaps proves the first assertion. Expanding the product into ordered subsets of times and using nonnegative summation proves the second. ◻

Proof. The omitted tags, failed endpoint neighborhoods and route-existence rejections have total weighted mass \(O(D^{-1-c})+o_K(1)\) by Lemma 26 and Proposition 32. For the latter use the marginal bad-site bound in an \(O(R^3)\) neighborhood, the exponentially small probability of a filled component reaching across the margin, and the thermodynamically vanishing probability of a large component. Multiplying by the bounded number of relative-count factors costs at most \(D^{Cx}\), outside events smaller than any power of \(D\).

It remains to check the subprobability steps near the two ends. Use the full-path count and motion event of Proposition 19 in the enlarged endpoint neighborhoods solely to estimate the discarded mass; it is not added to the conditional bridge law. Its ordinary-original motion tests give \(\sup_q|\omega(q)-Y|\le b\) and \(\sup_q|\omega(q)-Z|\le b\) in the chosen lift. In particular its midpoint \(y\) has distance at most \(b\) from the source mean. Consecutive means have distance bounded by a fixed constant, which is at most \(b\) after increasing the fixed cutoff constant. Thus every required pin has \(|y-m_{\rm insert}|\le2b<8b\). The full-path event fails with weighted mass smaller than \(D^{-1-c}\) after the \(O(R^3)\) union bound. On this event, a pinned trial with a separated midpoint has probability at least \(3/4\). Indeed, start with the full original obstacle collection in Proposition 21, delete any replaced originals, and add the already generated paths. Only boundedly many added paths can meet the trial neighborhood, by simplicity of the route. The last assertion of that lemma applies to these additions.

For a middle path the necessary separation from a later pinned midpoint was already checked in the copy containing the retained original that provides that midpoint. The paired middle paths have the same midpoint. The two silent ends are separated by a distance of order \(K\), so no check joins them for sufficiently large \(K\). Thus the only remaining separation failure is that a selected silent original has midpoint within \(s\) of another original midpoint. In each bounded region the expected number of such ordered pairs is at most \(CD^2s^3\). Summing the selected interior labels uses (156); summing an end tag uses its factor \(T_v^{-1}\) or \(T_w^{-1}\). Enlarging all site sums to the endpoint neighborhoods and using the other relative-count bounds gives at most \[C R^3D^{Cx}\frac{D^2s^3}{Df_0} \le C D^{-7/5+3w+C'x}.\] The midpoint range failure is already covered by the displacement cutoffs chosen above. Outside these discarded events every pin has \(p\ge3/4\), so (157) equals \(p\) and the pin kernel has mass one. Since \(w\) and then \(x\) were chosen small, all displayed losses have a strict power saving beyond \(D^{-1}\). This proves the lemma. ◻

Proof. The conditional distribution of any one original path, given all the others, is bounded by \(c_0^{-1}B_l\) with \(c_0=3/4\), by Proposition 21. This remains true after arbitrary checks have been removed. Every reduced normalizer is positive by inserting its coordinates successively; the resulting lower bound may depend on their number and is used only for positivity.

Suppose first that \(S,T\) are disjoint. With their cross checks removed the integral factorizes as \(Z_S^-Z_T^-\). Add the cross checks in any fixed order. For a check between \(l\) and \(k\), condition on all paths except \(l\). Its failure probability is at most \(c_0^{-1}\beta_{lk}\). Consequently this insertion multiplies the inverse normalizer by at most \[(1-c_0^{-1}\beta_{lk})^{-1} \le \exp(2c_0^{-1}\beta_{lk})\] for large \(D\). Multiplication gives (164). Under the normalized product with all cross checks deleted, their union fails with probability at most \(c_0^{-1}\sum\beta_{lk}\). Dropping \(H\le1\) and using the normalizer ratio proves (166).

For general \(S,T\), put \(A=S\cap T\). Delete the labels of \(A\) and their incident checks from both separate reference integrals. This increases \(Z_S^-\) and \(Z_T^-\). The remaining sets \(S\setminus A,T\setminus A\) are disjoint, so the preceding comparison applies to them. Their joint normalizer is at most \(c_0^{-q}Z_U\), since one can add the \(q\) labels in \(A\) using their uniform insertion lower bounds. The cross-check costs used for the disjoint remainder are among those in \(B_{ST}\). These two inequalities prove (165). ◻

Proof. Fix the history before one generated pair or pin. Its proposal law before checks is unchanged by deletion of outside obstacles. Let \(p\) and \(p^-\) be the acceptance probabilities with and without the removed checks; then \(p\le p^-\). For a middle pair \(p\ge1/2\), and hence \[\frac{p^-}{p}\le 1+2(p^- -p).\] For a pin, use \(w=\max(1/2,p)\) and \(w^- =\max(1/2,p^-)\). The map \(p\mapsto\max(1/2,p)\) is increasing and \(1\)-Lipschitz, so \[\frac{w^-}{w}\le 1+2(p^- -p).\] The ratio of admissibility indicators is bounded by one in this upper comparison. A midpoint separation indicator outside the pin proposal can also be increased. On the support of the old nonzero integrand, its midpoint is separated from every old opponent, so the pinned collision estimate applies. Off that support the inequality is automatic. The capped expression (162) thus gives a valid bound everywhere.

For a middle proposal the union bound gives \(p^- -p\le C\sum\beta_{lk}\) over its removed checks. For a pin the bound is \(C\sum(\beta_{lk}+m(y,X_k))\). The proposal may be a coupled pair, but the removed checks involve one of its two marginals, and the uniform-midpoint insertion bounds apply to those marginals. Conditioning the original proposal does not enter this estimate of \(p^- -p\).

Multiplying the displayed ratio bounds over the affected transitions of both protocols gives \(\mathcal R\le\exp(C(B_{ST}+M_{ST}))\). The inequalities \(e^a-1\le ae^a\) and (163) give both conclusions. Unaffected transitions have identical numerators and divisors at the same history. Thus the number of factors actually paid is bounded by the number of graph edges, not by the number of generated transitions. ◻

Proof. Integrate an arbitrary nonnegative measurable test function of the selected outputs. Starting with the last stage, integrate every unselected stage using its mass bound one. At a selected stage integrate its auxiliary variables and use the primary marginal bound. The new test function depends on the latent input and the earlier selected outputs, but no longer on the history variables generated at this stage. Induction backwards through the finite list proves (169). The argument first applies to bounded tests and then to all nonnegative tests by monotone convergence. ◻

Proof. The normalized separate latent-original law is its admissible product bridge law. A middle stage, conditional on all these originals and on the preceding history, has primary marginal at most \(C D^{Cx}B_l\), by Corollary 22 and its acceptance probability at least \(1/2\). A pin has primary conditional bridge law at most \(2\mathbf 1_{\rm range}(y) B_l(\mathrm dz\mid z(0)=y)\), where \(y\) is the midpoint of its retained original partner and \(\mathbf 1_{\rm range}(y)=\mathbf 1_{\{|y-m_l|\le8b\}}\); all separation and avoidance indicators are discarded in this upper bound, so the retained cutoff is independent of the preceding history. This cutoff conditional bridge is the kernel \(\nu_r\) in Lemma 40. Both estimates hold uniformly in the history, including histories obtained after deleting some outside checks.

Apply Lemma 40 to the specified stages. For each specified pin its source is a latent original in \(S\): these sources lie in the right-silent suffix. The sources for distinct pins are distinct labels. If there are \(p\) specified pins, their joint latent-source marginal is bounded by \(C^p\) times the corresponding product bridge law, by Corollary 22. Indeed the uniform conditional one-path density bound survives integration over any other latent coordinates and can then be iterated over these \(p\) coordinates.

Let \(a(l)\) be the source label of the pin with output label \(l\). Integrating out all parts of its source other than the midpoint gives \(g_{a(l)}(y)\,\mathrm dy\). On the retained range, neighboring means and the Gaussian ratio bound (108) give \(g_{a(l)}(y)\le C D^{C\sqrt x}g_l(y)\). Therefore \[\int \mathbf 1_{\rm range}(y)g_{a(l)}(y) B_l(\mathrm dz\mid z(0)=y)\,\mathrm dy \le C D^{C\sqrt x}B_l(\mathrm dz).\] Use this for each distinct source. Together with the middle-output bounds it proves (170). No assertion of independence between a pin and its source midpoint has been used: the midpoint is integrated against the conditional bridge before the resulting free bridge bound is obtained. ◻

Proof. By (160) and the enlargement of the kernels, the conditional product integral is bounded by the integral using the free output measure \(B_U\), the union admissibility indicator, and the single divisor \(Z_U\). In each likelihood keep a separate set of latent original variables. These copies are not identified even if \(S,T\) have common labels. Only shared output coordinates are identified by the integration against \(B_U\).

Suppose first that the sets are disjoint. Apply Lemma 39, delete the union output-admissibility indicator in an upper bound, and integrate with the separate protocols. The term one in (168) has integral at most \[\frac1{Z_U} \int A_S^-(x)A_T^-(x')h_{w,j}(x)h_{w,j'}(x') \,\mathrm dB_S(x)\mathrm dB_T(x').\] Here the generated kernels were integrated using their mass bounds one. By (166) this is at most the reference expectation of \(h_{w,j}h_{w,j'}\), plus \(Ce^{CB_{ST}}B_{ST}\). The terms from \(B_{ST}\) in (168) are bounded by the same normalizer ratio and the masses of the separate normalized protocols, each at most one.

Consider one term \(m(y,X_k)\) of \(M_{ST}\). Its midpoint \(y\) belongs to a latent original of one separate protocol, whereas \(X_k\) is an output path of the other. Conditional on the fixed outside the two extended protocol measures are independent. On the side providing \(y\), drop its tag indicator and integrate all generated transitions. The marginal of the source original is bounded by a fixed constant times its free bridge. For every fixed point \(z\), the Gaussian midpoint density then gives \[\begin{align*} \int_{d(y,z)\le C_b}\frac{g_l(y)}{s\vee d(y,z)}\,\mathrm dy &\le \|g_l\|_\infty\int_{|u|\le1}|u|^{-1}\,\mathrm du +\int_{d(y,z)>1}g_l(y)\,\mathrm dy\\ &\le C(1+t^{-3/2}). \end{align*}\] The local range \(C_b\) suffices because all other checks vanish; nearby lifts or the periodized Gaussian give the same estimate. This calculation does not require a density bound for the opposing output midpoint. There are at most \(CJ_*\) such terms. Use (164) for the normalizer ratio and absorb its exponent using (163). This proves (171).

For intersecting sets use (167). After dropping union admissibility, write the separate output measures as \(Z_S^-f_S B_S\) and \(Z_T^-f_T B_T\), with the original tag indicators included. Integrating unshared output coordinates leaves the product of their marginals on \(A=S\cap T\). By Lemma 41, one of these marginal densities is at most \(G_D^q\); the other has integral at most one. Thus the free output integral is at most \(Z_S^-Z_T^-G_D^q\). Division by \(Z_U\), (165), and absorption of \(C^q\) in \(G_D^q\) prove (172). ◻

Proof. Every possible mean-site of an end tag lies in the corresponding \(R\)-cell enlarged by a fixed radius. At each site the number of eligible tags is at most \(Cf_0D\), by the endpoint count estimate. Since \(T_v=T_w=f_0DR^3\), their counting sums with \(h\le1\) are dominated by a site weight \(CR^{-3}\) and a label subprobability measure with maximum atom \(C/(Df_0)\). These are deterministic upper bounds at the fixed data; the site weights will be removed before any environmental expectation. Zero or very small tag families cause no problem: their extracted label measure simply has mass less than one. Interior selections are probability measures and have the same maximum-atom bound (156).

For any fixed endpoint \(z\), a dyadic decomposition of the endpoint count bound in Definition 18 gives, for a candidate label family in a bounded neighborhood, \[ \frac{C}{Df_0}\sum_l\frac1{s\vee d(z,Y_l)} \le C\left(1+\frac{D^{1/100}}{Df_0s}\right)\le C, \tag{174}\] and the same holds for the \(Z\) endpoints. To see the first inequality, sum separately the innermost ball of radius \(s\) and the shells \(2^as<d\le2^{a+1}s\le C_b\). The volume term sums as a geometric series of order \(Df_0\), whereas the additive \(D^{1/100}\) term sums to \(CD^{1/100}/s\). Here \(D^{1/100}/(Df_0s)=D^{-19/100+x}\) is bounded for small \(x\).

Every edge contributing a \(\beta\) cost has a label choice on at least one side which is free to average: the common first tag is outside both changed sets. For an interior label use (174) conditional on the tags; for an end-tag label use the same bound in its candidate family at its fixed mean-site. Coincidence exclusions may be removed in this positive upper bound, because the reciprocal distance is capped at \(s\). Thus each averaged \(\beta\) costs at most \(Cb_D\), and there are at most \(CJ_*\) edges. The midpoint cost in (171) has already been integrated. This bounds the disjoint error by \[C e^{C D^{-19/100}J}J_*b_D(1+t^{-3/2}).\]

For the intersecting choices let \(\mathcal E\) be all pairs of slots in the two changed lists whose label choices could agree. Its cardinality is at most \(CJ_*\), with all such pairs among the \(CJ\) full contacts. Because each list has distinct labels, the actual matches form a matching: no slot belongs to two of them. The product label measure assigns each prescribed set of \(h\) disjoint matches mass at most \((C/(Df_0))^h\). For interior matches, this follows by conditioning on the tags and then on one list, and using independence between sites in the other list. A tag can match an interior slot only at its one possible base site. For a match of the two second tags, sum one tag with the same maximum-atom bound. There are only two noncommon tag slots, so their exclusions change these bounds by a fixed factor per prescribed match, independent of the route length.

The tag measures have mass at most one, so summing the remaining tag variables cannot increase this bound.

Expand \[G_D^q=\prod_{e\in\mathcal E} \bigl(1+(G_D-1)\mathbf 1_{\{e\ {\rm matched}\}}\bigr).\] To restrict to at least one match, designate one matched edge and bound the remaining product in the same way. The preceding probability bound and the elementary inequality \((1+a)^M\le e^{aM}\) give \[ \mathbb E_{\rm labels}[\mathbf 1_{\{q>0\}}G_D^q] \le \frac{C G_D}{Df_0}J_* \exp\left(\frac{C G_D}{Df_0}J\right). \tag{175}\] Use this in (172). Since \(G_D/(Df_0)=C D^{-1+x+C\sqrt x}\), the coefficient in this last exponent tends to zero. Absorb the fixed powers of \(\log D\) in an arbitrarily small power depending on \(x\), and combine the two estimates. They have the form (173), with for example \(\delta_D=C D^{-19/100}+C G_D/(Df_0)\). This proves the lemma. ◻

Proof. In the expansion of the squared likelihood keep the common first-tag factor \(h_{v,i}/T_v\). Conditional on the outside of \(S\cup T\) it is fixed, because neither changed set contains \(i\). Apply Lemma 42 to the two choices. The conditional expectation in its disjoint leading term is exactly the original expectation of the two end-tag indicators. All tag choices, route choices, and interior label choices are fixed independently of the latent paths, conditional on the data. We may therefore sum this leading term on the original path space. Removing all good-event and noncoincidence restrictions in this nonnegative upper bound yields \[\mathbb E\left[ \left(\sum_i\frac{h_{v,i}}{T_v}\right) \left(\sum_j\frac{h_{w,j}}{T_w}\right)^2\right] =\mathbb E[T_v^{\rm rel}(T_w^{\rm rel})^2].\] The errors and intersecting choices are controlled by Lemma 43 and (176). This proves (177). The mass bound is Lemma 37. For the right augmented estimate keep the same two-copy construction, and hence the same common measure. Reverse only the list indexing and designate the right copy primary; retain the actual chronological order of the generated middle kernels. Its common endpoint is then the last tag and its silent prefix is the previous silent suffix. The kernels pinned into the left copy can now be integrated out. The remaining right pins do not use them: the two end families lie in opposite copies, and no check joins the separated endpoint neighborhoods. Every kernel estimate and the reverse-integration lemma hold for an arbitrary fixed middle order, so this change of designation requires no resampling or reversal of the sampling order.

The original retention also gives the event required for the reversed route comparison. The fixed endpoint cube \(Q_v\) and its \(10R\) margin are bad-free. For every filled bad component, the actual first endpoint lies outside its fill by the original route event. Since \(Q_v\) is nearest-neighbor connected and contains that endpoint, a path in \(Q_v\) cannot enter the fill without meeting the bad component. Thus every site of \(Q_v\) lies outside every such fill. The component-size cutoff is unchanged, so Proposition 32 applies with the common terminal endpoint and the noncommon endpoints in \(Q_v\). The preceding calculation therefore gives the right augmented estimates for the original common measure. ◻

Proof of Proposition 36. The augmented reference has mass \(\mathbb ET_v^{\rm rel}\), while its common measure has mass \(\int\ell_{v,w}\,\mathrm dP_{{\rm aug},v}\). Subtract twice the mass lower bound from the second-moment upper bound and add the reference mass. Proposition 44 gives \[ \int|\ell_{v,w}-1|^2\,\mathrm dP_{{\rm aug},v} \le \mathbb E[T_v^{\rm rel}(T_w^{\rm rel}-1)^2] +o(D^{-1})+o_K(1) \tag{178}\] in the normalized pair average. The likelihood remains zero on the omitted reference events; their loss was included in the mass estimate.

Outside the negligible count tails, \(T_v^{\rm rel}\le C/f_0\). Lemma 25 and the uniform marginal weights of the ordered cell pairs therefore bound the main term of (178) by \[C D^x\bigl(D^{-1-w/6}+D^{-1+x}R^{-3}\bigr) \le C\bigl(D^{-1-w/6+x}+D^{-1+2x-3w}\bigr) =o(D^{-1}).\] The removed count tails remain negligible after multiplication by the relative counts, by the bookkeeping in Lemma 25. Choose \(x<w/12\), in addition to its preceding restrictions. The same centered bound holds in the right augmented reference.

For finite measures \(P\) and \(\nu\ll P\) with likelihood \(\ell\), their squared Hellinger distance is \(\int(1-\sqrt\ell)^2\,\mathrm dP\le\int(1-\ell)^2\,\mathrm dP\). Projection cannot increase this distance: conditional Cauchy–Schwarz increases the affinity and preserves both masses. Project both augmented comparisons to the common ordered bath. Their reference densities are \(s_v^2\) and \(s_w^2\), by Lemma 24, and their common projected density is that of \(\nu_{vw}\). The triangle inequality in \(L^2\) and the two bounds (178) now imply (154). Every volume-dependent rejection was recorded as \(o_K(1)\) at fixed dilution; taking that limit before \(D\to\infty\) completes the assertion. ◻

Proof. Use the unitary discrete Fourier transform on \((\mathbb Z/M\mathbb Z)^3\). The right-hand quadratic form before multiplication by \(C\) has Fourier multiplier \[a_M(k)=\frac1{|I_M|M^2} \sum_{h_1\in I_M}\sum_{h_2,h_3} |e^{2\pi i k\cdot h/M}-1|^2.\] The variance on the left has multiplier one for \(k\ne0\) and zero at \(k=0\), so it suffices to bound \(a_M(k)\) below away from zero for \(k\ne0\). If \(k_2\) or \(k_3\) is nonzero modulo \(M\), orthogonality gives \(a_M(k)=2\). Otherwise, write \(q=\min(k_1,M-k_1)\). For \(q\ge1\), the geometric-sum formula and \(\sin(\pi q/M)\ge2q/M\) give \[\left|\frac1{|I_M|}\sum_{h\in I_M}e^{2\pi i k_1h/M}\right| \le \frac{C_0}{q}.\] Thus \(a_M(k)\ge1\) when \(q\ge2C_0\). For each of the finitely many remaining positive \(q\), the common value \(a_M(q,0,0)=a_M(M-q,0,0)\) converges to \[\frac1{.15}\int_{.30}^{.45}|e^{2\pi i q t}-1|^2\,\mathrm dt>0.\] Their minimum is positive, uniformly for large \(M\). Parseval’s identity now proves (179), including Hilbert-valued arrays. ◻

Proof. Let \(Q\) denote the double average on the right of (182), before its factor \(1/2\). Weighted Cauchy–Schwarz and independence of the two uniform indices give \[[2(a-m^2)]^2 =\bigl[\mathbb E_{i,i'}(y_i-y_{i'})^2\bigr]^2 \le Q\mathbb E_{i,i'}(y_i^2+y_{i'}^2)=2aQ.\] If \(a>0\), divide by \(4(\sqrt a+m)^2\ge4a\); the case \(a=0\) is immediate. For the second statement factor \(y-z=(\sqrt y-\sqrt z)(\sqrt y+\sqrt z)\). The remaining ratio is \((\sqrt y+\sqrt z)^4/(y^2+z^2)\), which lies between \(1\) and \(8\). ◻

Proof. First let \(\Phi\ge0\). For Hilbert-valued arrays, the variance is the minimum of the mean squared distance to a constant array. Comparing the array \(m_A\) with the mean of the array \(s_A\) thus yields \[\frac1{|\mathcal C_R|}\sum_A\|m_A-\overline m\|_2^2 \le\frac2{|\mathcal C_R|}\sum_A\|m_A-s_A\|_2^2 +\frac2{|\mathcal C_R|}\sum_A\|s_A-\overline s\|_2^2.\] Here bars without cell subscripts denote averages over all \(R\)-cells. The left side is exactly \(\|(P_R-P_0)_1\Phi\|_2^2\), by (186) and the normalized spatial measure. Equations (181) and (189) prove (190) for nonnegative vectors, with coefficient \(4\) in place of \(16\). The multiplication of a volume remainder by \(D\) is harmless in the inner limit because \(D\to D_*\).

For a real bosonic ground vector \(f\), its positive and negative parts \(f_+,f_-\) belong to the Dirichlet form domain and remain symmetric. The Sobolev truncation identities give \[\|f\|_2^2=\|f_+\|_2^2+\|f_-\|_2^2, \qquad q[f]=q[f_+]+q[f_-].\] Each nonzero part has Rayleigh quotient at least \(E_N\), and their sum attains \(E_N\). Both nonzero parts therefore attain \(E_N\) and are ground vectors. Complex conjugation commutes with the operator, so the real and imaginary parts of a complex ground vector are ground vectors or zero. A normalized \(\Phi\) can consequently be written as \[\Phi=\sum_{a=1}^m c_a\Phi_a, \qquad m\le4,\quad \Phi_a\ge0,\quad\|\Phi_a\|_2=1, \qquad \sum_{a=1}^m|c_a|^2=1,\] where each \(\Phi_a\) is a ground vector. Cauchy–Schwarz gives \[\|(Q_R)_1\Phi\|_2^2 \le\sum_{a=1}^m\|(Q_R)_1\Phi_a\|_2^2.\] Thus at most a factor four is lost in the vanishing error in (190). No comparison of component phases or connectedness assumption is used. ◻

Proof. For fixed \(y\), the difference is \(T_h\) followed by multiplication by the fraction of partitions that put \(y\) and \(y+h\) in different boxes. In one coordinate, separation requires a box face between the two lifted coordinates. As the shift varies, possible faces are spaced by \(R\), so at most \(H/R+2\) of the \(B/R\) shifts separate the pair. A union bound over the three coordinates proves the uniform multiplier bound (200). The count also applies across the periodic boundary; the stated restriction on \(K\) allows the interval to be lifted without ambiguity. Cell faces themselves are null sets. ◻

Proof. Fix a shifted partition \(\mathcal C\). In local coordinates \([0,B)^3\), the restriction of \(T_h^{\mathcal C}\) is the contraction \[(C_{b,h}g)(y)=\mathbf 1_{[0,B)^3}(y+h)g(y+h).\] It is identical in every box. Write \(W\) for the restriction of \(W_R\) to a box, and \(e_p\) for its normalized Neumann cosine mode with \(p\in(\pi/B)\mathbb N_0^3\). Proposition 16, including its complex-contraction assertion, gives \[ \frac1V\langle\mathrm d\Gamma(W_RT_h^{\mathcal C}W_R)\rangle_\Phi =\frac1{B^3}\sum_{p\ne0} \langle e_p,WC_{b,h}We_p\rangle F_\alpha(p)+o(1). \tag{203}\] The estimate applies uniformly to these shifted grids: simultaneous translation maps the ground space unitarily to itself, and the local bound is uniform over its normalized vectors.

We compute the diagonal on a fixed annulus \(0<\delta\le|p|\le J\). At least one coordinate has \(|p_i|\ge\delta/\sqrt3\). The mean of its cosine over an \(R\)-interval is bounded by \(2/(R|p_i|)\), whereas each other cosine mean has absolute value at most one. Including the Neumann normalization factors and summing the squared cell means gives \[\|(1-W)e_p\|\le\frac{C}{\delta R}.\] Since \(C_{b,h}\) is a contraction, \[ \langle e_p,WC_{b,h}We_p\rangle =\prod_{i=1}^3\cos(p_i h_i) +O\left(\frac1{\delta R}+\frac HB\right). \tag{204}\] To verify the translation term, extend the cosine formula for \(e_p\) to \(\mathbb R^3\). Removing the clipping changes its integral by \(O(H/B)\): the boundary layer has relative volume \(O(H/B)\) and \(|e_p(y)e_p(y+h)|\le C/B^3\). On the full box the integral factors. For \(p_i>0\), the sine–cosine integral over \([0,B]\) is zero because \(p_iB\in\pi\mathbb N\); the cosine-square integral gives \(\cos(p_i h_i)\). For \(p_i=0\) the factor is one.

Uniformly for \(\alpha\in I_\alpha\), the explicit formula gives \[ F_\alpha(p)\le C\min(|p|^{-1},|p|^{-4}). \tag{205}\] The normalized Neumann-lattice tails in (85) therefore bound the contributions of \(0<|p|\le\delta\) and \(|p|\ge J\) by \(C\delta^2\) and \(C/J\), respectively, uniformly in \(B\), for \(0<\delta<1<J\). The same bounds hold for the corresponding integrals.

We can now truncate (203) to a fixed annulus, use (204), and take Riemann sums. The Neumann mesh has density \(\pi^{-3}\) in the nonnegative octant. On the annulus the integrands are bounded and equicontinuous for \(|h|\le H\) and \(\alpha\in I_\alpha\); continuous cutoffs at its edges may be used. Letting \(\delta\downarrow0\) and \(J\uparrow\infty\) after these estimates yields, uniformly in \(h\) and \(\mathcal C\), \[\begin{align*} \frac1V\langle\mathrm d\Gamma(W_RT_h^{\mathcal C}W_R)\rangle_\Phi &=\frac1{\pi^3}\int_{[0,\infty)^3} F_\alpha(p)\prod_i\cos(p_i h_i)\,\mathrm dp+o(1)\\ &=\int_{\mathbb R^3}F_\alpha(p)e^{ih\cdot p} \frac{\mathrm dp}{(2\pi)^3}+o(1). \tag{206}\end{align*}\] The last equality uses reflection symmetry in each coordinate. Averaging over \(\mathcal C\) and applying (198) and (201) proves (202). ◻

Proof of Theorem 1. Fix \(a>0\). For a sufficiently small density \(\rho>0\), put \(\eta=\rho a^3\) and \[ s_* =\sqrt{\frac{\alpha_0}{\rho a}},\qquad D_* =\rho s_*^3=\frac{\alpha_0^{3/2}}{\sqrt\eta}. \tag{207}\] Choose \(R,B\) from this \(D_*\) by (13). For any thermodynamic sequence \(N_k,L_k\to\infty\) with \(N_k/L_k^3\to\rho\), let \(K_k\) be a positive multiple of \(B\) nearest to \(L_k/s_*\). Set \[s_k=L_k/K_k,\qquad r_k=a/s_k,\qquad D_k=N_k/K_k^3,\qquad \alpha_k=D_kr_k.\] Then \(K_k\to\infty\), \(s_k\to s_*\), \(D_k\to D_*\), and \(\alpha_k\to\alpha_0\). Thus all the nested partitions and parameter conditions hold on the tail of every such sequence.

The unitary change of variables is \[ (U_k\Psi)(y_1,\ldots,y_{N_k}) =s_k^{3N_k/2}\Psi(s_ky_1,\ldots,s_ky_{N_k}). \tag{208}\] It maps the allowed sets and form domains to their scaled versions, with \(q_{K_k,r_k}[U_k\Psi]=s_k^2q_{L_k,a}[\Psi]\), and hence maps ground eigenspaces unitarily. The corresponding one-particle unitary sends a physical plane wave of momentum \(\kappa\) to the scaled one of momentum \(p=s_k\kappa\), preserving its occupation. Also \[ D_kr_k^3=(N_k/L_k^3)a^3\longrightarrow\eta, \qquad D_k\sqrt\eta\longrightarrow\alpha_0^{3/2}. \tag{209}\]

Suppress \(k\) temporarily. For a normalized physical ground vector \(\Psi\), put \(\Phi=U\Psi\) and \[ \lambda=s\sqrt{8\pi\rho a},\qquad \sigma_\Psi=\frac1{D\sqrt\eta} (p\mapsto p/\lambda)_*\mathfrak m_\Phi. \tag{210}\] The normalized momentum sum in (2), evaluated in the pure state \(|\Psi\rangle\langle\Psi|\), is \(\int f\,\mathrm d\sigma_\Psi\). In the inner limit, \(\lambda\to\lambda_0:=\sqrt{8\pi\alpha_0}\). The target measure \(\sigma_0:=\nu_{\mathrm{Bog}}\) from (1) satisfies \[ \sigma_0=(p\mapsto p/\lambda_0)_* \left(\alpha_0^{-3/2}F_{\alpha_0}(p) \frac{\mathrm dp}{(2\pi)^3}\right). \tag{211}\] This follows by substituting \(p=\sqrt{8\pi\alpha_0}\,t\) in the explicit formula (78).

Apply Proposition 49 at \(h/\lambda\) and use (209). Since \(\lambda\) stays bounded away from zero, we obtain \[ \int e^{ih\cdot t}\,\mathrm d\sigma_\Psi(t) =\int e^{ih\cdot t}\,\mathrm d\sigma_0(t)+o(1), \tag{212}\] uniformly on each compact set of \(h\) and uniformly over the ground vectors. Replacing \(\alpha,\lambda\) in the continuous integral by their inner limits costs only an inner-limit error: the majorant in (205) is integrable, and truncation in \(p\) makes this replacement uniform on compact \(h\)-sets. The case \(h=0\) gives total mass convergence and, in particular, a uniform mass bound in the iterated limit.

For \(g\in C_c^\infty(\mathbb R^3)\), Fourier inversion and its integrable Fourier transform imply \[\int g\,\mathrm d(\sigma_\Psi-\sigma_0)=o(1).\] Indeed, on a fixed compact Fourier set use (212); on its complement use the mass bound times the \(L^1\) tail of the Fourier transform. Uniform approximation and the same mass bound give this conclusion for all \(g\in C_c(\mathbb R^3)\).

Now let \(f\) be any bounded continuous real function. Choose \(\chi\in C_c^\infty(\mathbb R^3)\) with \(0\le\chi\le1\) and \(\int(1-\chi)\,\mathrm d\sigma_0\) as small as desired. Positivity gives \[\begin{align*} \left|\int f\,\mathrm d(\sigma_\Psi-\sigma_0)\right| &\le\left|\int f\chi\,\mathrm d(\sigma_\Psi-\sigma_0)\right|\\ &\quad+\|f\|_\infty\left\{ 2\int(1-\chi)\,\mathrm d\sigma_0 +|\sigma_\Psi(\mathbb R^3)-\sigma_0(\mathbb R^3)| +\left|\int\chi\,\mathrm d(\sigma_\Psi-\sigma_0)\right|\right\}. \tag{213}\end{align*}\] The compact tests and the mass difference tend to zero in the stated iterated sense. For a prescribed \(\varepsilon>0\), first choose \(\chi\) and the compact-test approximations, then choose the auxiliary parameters and a lower threshold on \(D_*\). Equation (207) turns that threshold into \(\rho_0(a,f,\varepsilon)>0\). It works for every fixed \(0<\rho<\rho_0\) and every density-convergent thermodynamic sequence. The volume threshold may depend on that fixed density and sequence; no convergence rate for \(N_k/L_k^3\) is required.

Finally, Lemma 3 makes every finite-volume ground eigenspace finite dimensional. Each state in \(\mathcal G_{N,L,a}\) has a spectral decomposition \(\Gamma=\sum_b\omega_b|\Psi_b\rangle\langle\Psi_b|\) with normalized ground vectors, \(\omega_b\ge0\), and \(\sum_b\omega_b=1\). Its momentum measure is \(\sum_b\omega_b\sigma_{\Psi_b}\). The absolute error for \(f\) is therefore at most the largest pure-state error at that volume. All preceding bounds were uniform in the pure ground vector, so the conclusion holds with the supremum over \(\Gamma\in\mathcal G_{N,L,a}\) required by the theorem. ◻