Proof. Put \(M_0=2+\sum_{i=1}^dD(Y_i)\). The comparison (9) gives polynomial bounds in \(M_0\) for \(|g_i|\), \(|g_i|^{-1}\), and \((\Im g_i)^{-1}\). For \(A=V_o-\zeta-\sum_i g_i\), we have \(|A|\le CM_0\) and \(-\Im A\ge cM_0^{-1}\). Hence \(|m_o|\), \(|m_o|^{-1}\), and \((\Im m_o)^{-1}\) are also bounded by fixed powers of \(M_0\). In particular \(\Im m_o(e+i0)>0\) almost surely.

The inverse star Green matrix and its inverse have the explicit forms \[ S=\begin{pmatrix} V_o-\zeta&1&\cdots&1\\ 1&1/g_1&&0\\ \vdots&&\ddots&\\ 1&0&&1/g_d \end{pmatrix},\qquad S^{-1}=\begin{pmatrix} m_o&(-m_og_i)_i\\ (-m_og_i)_i^{\mathsf T}&(\delta_{ij}g_i+g_im_og_j)_{ij} \end{pmatrix}. \tag{17}\] Thus \(\|S\|+\|S^{-1}\|\le CM_0^C\). Also \[-\Im S=\mathop{\mathrm{diag}}\left(\eta, \frac{\Im g_1}{|g_1|^2},\ldots, \frac{\Im g_d}{|g_d|^2}\right), \qquad \Im S^{-1}=(S^{-1})^*(-\Im S)S^{-1}.\] Each neighbor diagonal in \(-\Im S\) is at least \(cM_0^{-C}\). For a vector \(x\) supported on a retained star, let the omitted neighbor be \(r\) and put \(y=S^{-1}x\). The unused row says \(0=y_o+y_r/g_r\), so \(|y_o|\le CM_0^C|y_r|\). Consequently \[|x|=|Sy|\le CM_0^C|(y_1,\ldots,y_d)|, \qquad x^*(\Im S^{-1})x\ge cM_0^{-C}|x|^2.\] Together with the upper bound for \(S^{-1}\), this bounds every eigenvalue of the retained covariance between \(cM_0^{-C}\) and \(CM_0^C\). The same bounds hold for an edge covariance: choose a retained star that contains the edge and restrict its quadratic form. This is possible since \(d\ge3\). The bounds hold equally at height zero by the displayed formulas and the almost sure positivity of all incoming imaginary parts.

All formulas are continuous when their inputs lie in \(\mathbb D\). Proposition 2 therefore gives convergence in probability. The polynomial bounds and sufficiently high, but finitely many, inverse-margin moments give the claimed \(L^{p_0}\) convergence by uniform integrability. Moreover \[|\log\det K|\le C(1+\log M_0).\] Any fixed positive moment of \(M_0\) bounds, for example, the second moment of this logarithm. The logarithms are consequently uniformly integrable and converge in \(L^1\). Tensoring \(K\) with the identity on \(m\) colors multiplies \(\log\det K\) by \(m\); it does not require any larger spatial moment order. ◻

Proof. By the definition of the density-of-states measure, \(\mathbb Em_o(\zeta)=\int(t-\zeta)^{-1}\,d\nu_{d,w}(t)\). Lemma 3 gives \(L^1\) continuity of \(m_o\) on \(J\times[0,h]\), uniformly on this compact parameter set. Thus \(\pi^{-1}\mathbb E\Im m_o(e+i\eta)\) converges uniformly on \(J\) to the continuous function in (18). For every continuous compactly supported \(f\) in the interior of \(J\), Poisson kernel approximation and Fubini’s theorem give \[\int f\,d\nu_{d,w} =\lim_{\eta\downarrow0}\frac1\pi \int f(e)\mathbb E\Im m_o(e+i\eta)\,de =\int f(e)\rho_{d,w}(e)\,de.\] This identifies the entire restricted measure, including the absence of a singular component. Since the root imaginary part is strictly positive almost surely, its finite expectation is strictly positive at every \(e\in J\). ◻

Proof. Condition on \(A\) and all potentials other than \(V_v\). Schur complementation gives \[ G_{vv}(\zeta)=(V_v-A_v(\zeta)-iB_v(\zeta))^{-1}, \qquad B_v(\zeta)>0, \tag{23}\] where \(A_v,B_v\) do not depend on \(V_v\). Integration of the Poisson kernel over the allowed interval gives \(\mathbb E_{V_v}\Im G_{vv}\le\pi/(2w)\). Integrating in the real spectral parameter, followed by Stieltjes inversion, gives the second inequality in (21). This is the usual rank-one spectral-averaging proof of the Wegner estimate [32]; the arbitrary finite-matrix scope is also recorded in [3].

For the last assertion, write the inverse as \((T+\mathop{\mathrm{diag}}(v_1,\ldots,v_k))^{-1}\) with \(\Im T<0\). Use \(v_1\) and the differences \(v_j-v_1\) as coordinates. The joint density is bounded, all the differences range over a fixed compact set, and the remaining variable adds a scalar multiple of the identity on a bounded interval. For any unit vector \(a\), the function \[f(z)=\langle a,(T_0-z)^{-1}a\rangle,\qquad z\in\mathbb H,\] where \(\Im T_0<0\), takes values in \(\mathbb H\) and satisfies \(|f(i)|\le1\). The function \(u(z)=\Re(e^{-i\pi/4}\sqrt{f(z)})\) is positive harmonic and satisfies \(u(z)\ge2^{-1/2}|f(z)|^{1/2}\). The Poisson inequality at \(i\), whose kernel is bounded below on each fixed bounded real interval, therefore bounds the integral of \(|f(t)|^{1/2}\) on that interval. This remains valid by boundary limits if necessary. Polarization, a fixed basis, and equivalence of norms in dimension \(k\) give a uniform bound on the integral of the half power of the inverse norm. Markov’s inequality proves (22). To apply this to \(G_{SS}\), condition also on the potentials outside \(S\) and take its Schur complement. Its effective matrix has strictly negative imaginary part. All conditioning and integration arguments remain valid for graph-selected sets and for graph modifications independent of the potentials. ◻

Proof. The identity \(G^*G=\eta^{-1}\Im G\), followed by averaging its diagonal, gives the equality. Lemma 6 gives the inequality. For the final assertion, begin each exploration at a uniform vertex or uniform oriented edge. At each occurrence of a vertex in the nonbacktracking lift, independently order the allowed outgoing stubs, excluding the incoming stub. In an edge-rooted exploration, exclude the marked base edge at its initial vertex as well. Every fixed position then has uniform vertex marginal, because the graph is regular. Different explorations are independent conditional on the graph. This remains true when their images intersect or fail to form trees; no claim of independence among positions within one exploration is used. ◻

Proof of Proposition 5. At \(\eta=h\), the cavity resolvent at a vertex with a sufficiently large fixed tree neighborhood is close to the clean-tree cavity value, first by polynomial approximation of the resolvent and then by the norm perturbation bound \(Cw/h^2\). Equivalently one may approximate the disordered resolvent locally and use its bounded potential perturbation. The polynomial degree is fixed after \(h\) and the desired accuracy are chosen; the Hamiltonians have uniformly bounded spectra. The fraction of vertices failing this fixed tree test has expectation \(O(n^{-1})\). Write \(\delta_*\) for the fixed \(L^2\) smallness threshold in Proposition 2. As permitted by its parameter order, take \(h\) small and then \(w_0/h^2\) small enough that the starting disk norm is below \(\delta_*/2\), and take the target tree norm below \(\delta_*/4\). Polynomial approximation with a sufficiently small fixed error then places \(P_H\) inside the threshold with failure probability \(O(n^{-1})\).

Fix \(r\in(0,1)\) sufficiently close to one. For any diagonal Green function of a fixed graph or cut graph, Schwarz–Pick applied to its map from \(\mathbb H\) to \(\mathbb D\) implies a uniform bound on its change between \(e+i\eta\) and \(e+ir\eta\) tending to zero as \(r\uparrow1\). Choose \(r\) so that this change is smaller than the gap between the improved smallness bound and the threshold. Starting at \(h\), descend through \(h,rh,r^2h,\ldots\), shortening the last step to reach the prescribed \(\eta\). At each height apply Proposition 8 for \(j=1,\ell\) and then Proposition 2. Its conclusion places \(P_H\) within \(C(n\eta)^{-c}\) of \(Q_\zeta\), hence back well inside the threshold when \(y_0\) is large. This closes the induction.

The failure probabilities sum geometrically from the smallest height: if \(\eta_*\) is the terminal height, then \[\sum_{\text{heights }\eta_k\ge\eta_*} C(n\eta_k)^{-c}\le C'(n\eta_*)^{-c}.\] The initial \(O(n^{-1})\) failure fits this bound after decreasing \(c\). We obtain (20) for \(P_H\). For the full-root law, use the full-root version of Proposition 8 with one internal level. Couple the cavity inputs to iid tree inputs optimally and apply the sibling estimate (11), with \(d\) children. Its coefficients involve inverse margins of the other tree inputs, whose required fixed moments are bounded. Independence of the coupled input pairs therefore bounds the output \(L^2\) error by a constant times the input \(W_2\) error. This proves the full statement. ◻

Proof. For the first item, take the augmented recursion of Proposition 8 with internal levels covering the prescribed neighborhood. By Proposition 5, its input law tends to the tree cavity law. Its internal iid potential coordinates are independent of these inputs by construction. The tree recursion with these data is exactly the joint tree law of the diagonal and the observed neighborhood. For any fixed additional depth this passage only needs weak convergence: all limiting input values lie in the open disk almost surely, where the finite recursion is continuous. Thus no higher inverse-margin order is required as the observed radius grows. Proposition 2 supplies the common-environment boundary continuity. Conversion back to the half-plane is valid since the limiting tree value is finite almost surely. The empirical-law conclusion gives product limits for independently sampled vertices, and their mutual Green entries vanish by (24). These arguments apply along every parameter sequence stated in the corollary. A diagonal choice of sequences gives the stated double-limit formulation. Letting the finite radius increase identifies the limit with the measurable function \(m_o(E+i0)\) of its complete tree environment.

We next obtain a useful quantitative bound on small intercepts. For a half-plane value \(g\) and \(x=X(g)\), the comparison used in (9) gives \[\frac{\Im g}{|g|^2}\ge c_J(1-|x|).\] Couple \(P_H^0\) with the tree law using (20). The bounded diameter of the disk also bounds the expectation of its squared transportation error by a negative power of \(y\). Markov’s inequality for this coupling and the tree inverse-margin moments show, for a sufficiently small \(\epsilon>0\), that \[ \mathbb P_{H,v}\{B_v(\zeta)<y^{-\epsilon}\}\le Cy^{-c}. \tag{33}\] For example, if the tree disk margin is at least \(2\delta\) and the coupling distance is below \(\delta\), the finite margin is at least \(\delta\); take \(\delta\) to be a sufficiently small power of \(y^{-1}\). This proves the bound directly from the two respective tail estimates.

The intercept formula gives \(\Im G_{vv}\le B_v^{-1}\). Since \(B_v\) is independent of \(V_v\), conditional spectral averaging implies, for every \(T>0\), \[ \mathbb E\bigl[\Im G_{vv}\mathbf1_{\{\Im G_{vv}>T\}}\bigr] \le \mathbb E\bigl[\Im G_{vv}\mathbf1_{\{B_v<T^{-1}\}}\bigr] \le C_w\mathbb P\{B_v<T^{-1}\}. \tag{34}\] This estimate applies after averaging over the uniform vertex as well. Thus truncating the imaginary part at \(T=y^\epsilon\) loses only a negative-power amount in expectation, by (33). Tree tails have the same property by their moment bounds.

To compare truncated imaginary parts through the disk local law, first introduce a disk cutoff with margin \(y^{-\epsilon'}\). On the remaining compact subdisk, the function \(x\mapsto\Im X^{-1}(x)\) has a Lipschitz constant bounded by a fixed power of \(y^{\epsilon'}\). Choose \(\epsilon'\) small enough for the transportation estimate to absorb this factor. The mass removed by the cutoff has a negative-power bound by the same coupling argument. Choose the clipping exponent \(\epsilon\) still smaller, so that multiplying this mass by \(y^\epsilon\) retains a negative-power bound. Combining the comparisons and then using Markov’s inequality gives a negative-power bound for the difference between the empirical imaginary trace and \(\mathbb E\Im m_o(e+i\eta)\). In the microscopic substitution with fixed \(s,y\), continuity of the tree expectation gives \(\mathbb E\Im m_o\to\pi\rho_{d,w}(E)\) by Corollary 4. This proves (31), after decreasing \(c\). All finite-graph constants were uniform in \(e\in J\), hence are independent of fixed \(s\).

Finally fix a possibly small microscopic height \(y>0\). The quantity \(B_v(\zeta)\) is the imaginary part of the Herglotz function \[\zeta+\langle a_v,(H^{(v)}-\zeta)^{-1}a_v\rangle,\] where \(H^{(v)}\) deletes the vertex and \(a_v\) is its adjacency column. For \(Y\ge y\), Harnack’s inequality at the same real part yields \[B_v\left(E+\frac{s+iy}{n}\right) \ge\frac yY B_v\left(E+\frac{s+iY}{n}\right).\] Choose a sufficiently large fixed \(Y\). Once \(\delta<(y/Y)Y^{-\epsilon}\), the probability that the left intercept is below \(\delta\) is at most the small-intercept probability at height \(Y\). Consequently \[\lim_{\delta\downarrow0}\limsup_n \mathbb P\left\{B_v\left(E+\frac{s+iy}{n}\right)<\delta\right\} \le CY^{-c}.\] Let \(Y\to\infty\) and apply (34) to obtain (32). This last conclusion will retain the normalization of eigenvector marks when taking microscopic limits. ◻

Proof. The Wegner bound of Lemma 6 gives tightness of the point measures on each compact interval. Choose interval endpoints uncharged almost surely by a subsequential limit; such an exhaustion exists, since the expected limiting counts are bounded by the same constant times interval length. Truncate each list length and then stratify by that length. At a uniformly sampled vertex the sum of squared coordinates of a list of length \(m\) has mean \(m\). This proves tightness of all finite-radius observations. A diagonal extraction retains all radii, list truncations, and interval restrictions, as well as their positions.

For each fixed radius the fraction of vertices with a non-tree neighborhood tends to zero. Uniform rooted orderings give invariance under root-preserving automorphisms. Sampling an oriented edge and reversing it gives invariance under re-rooting across an edge; together these give full tree-automorphism invariance. The empirical law of the potential labels alone converges in probability to their iid tree law: one and two sampled bounded neighborhoods have this law asymptotically, and two disjoint neighborhoods use independent potentials. Thus this marginal is deterministic inside each limiting \(\mathcal L\).

The second-moment assertion needs more than tightness. If the list comes from \([-L,L]\), spectral decomposition gives \[ \sum_{i\text{ in list}}u_i(v)^2 \le (1+L^2)\Im G_{vv}(E+i/n). \tag{37}\] The right side is uniformly integrable at a sampled vertex by Corollary 9. Consequently the limiting empirical second moments lose no mass. More explicitly, represent the joint convergence almost surely, first pass bounded second-moment truncations, and then remove the truncation using the tail bound in expectation. The empirical moments before taking the limit are exactly \(\delta_{ij}\), so their limits are the same almost surely; polarization handles the mixed moments. The eigenvalue equations pass to the limit on every finite neighborhood because \(\lambda_i=E+O_L(n^{-1})\). Countable consistency gives the assertions simultaneously at all vertices and for all retained lists. ◻

Proof. Let \(f\) be a bounded Lipschitz function of the potentials in a fixed rooted ball. Average over its branch orderings and put the corresponding finite-graph value \(f(v)\) equal to zero at tree failures. First require that \(f\) vanish unless the central potential belongs to a compact subinterval of \((-w,w)\). Choose \(b_0\in C_c^\infty((-w,w))\), nonnegative and bounded below by a positive constant on that subinterval. For \(0\le t\le a_n:=n^{-2/3}\), change every potential by \[V_v\longmapsto V_v+t b_0(V_v).\] For large \(n\) this is a diffeomorphism of \([-w,w]\), identical near its endpoints. The one-site density ratio is \(1+O(t)\), has integral one, and has second moment \(1+O(t^2)\). Independence therefore bounds the total variation distance of the product laws by \(O(\sqrt n\,t)=o(1)\), uniformly over this range. The normalized test vector \(f/\sqrt n\) changes by \(O_f(t)\) in Euclidean norm.

For the original configuration set \(B=\mathop{\mathrm{diag}}(b_0(V_v))\). The support condition writes \(f/\sqrt n=B^{1/2}q_f\), with \(\|q_f\|\le C_f\). Scalar spectral averaging gives, for every interval \(I\), \[ \int_0^{a_n} \left\langle f/\sqrt n,\mathbf1_I(H+tB)f/\sqrt n\right\rangle\,dt \le C_f|I|. \tag{39}\] For completeness, compress the resolvent by Schur complement to the indices where \(B>0\), and conjugate by \(B^{1/2}\). It becomes \((t\mathrm I-T)^{-1}\) with \(\Im T>0\) at positive spectral height. The integral of its imaginary part in a vector, over the real \(t\)-axis, is at most \(\pi\) times that vector’s squared norm. The Herglotz representation, followed by spectral inversion, proves (39); restricting the nonnegative integrand to \([0,a_n]\) preserves the bound.

Take \(I=E+[-L,L]/n\), divide by \(a_n\), and average over the original data. Total variation comparison and the change in the test vector show that \[\mathbb E\left\langle f/\sqrt n,\mathbf1_I(H)f/\sqrt n\right\rangle \le O_f(n^{-1/3})+o(1).\] The left side is \(\mathbb E\sum_{x_i\in[-L,L]}(n^{-1}\sum_v f(v)u_i(v))^2\). Hence every empirical root average against \(f\) vanishes in the limit. Passing these first moments is justified by the uniform second-moment bounds. A countable dense collection of tests on all finite balls, followed by an exhaustion of \((-w,w)\) in the central coordinate, determines the full conditional mean. Averaging tests over branch orderings loses no information, since the rooted joint law is invariant under those permutations. Re-rooting proves the claim at every vertex. ◻

Proof. For fixed environment radius \(R\) and number of bins \(Q\), let \(A_{R,Q}\) and \(B_{R,Q}\) be the tuples of environment labels seen by the ordered star and edge, respectively. The count first gives an inequality with these finite conditioning data: \[\begin{align*} &m\left[\frac12\log(2\pi\mathrm e\,\eta) -\int\log|t-E-i\eta|\,d\nu_{d,w}(t)\right]\\ &\quad\le \mathcal H(F^\eta_{\rm star}\mid A_{R,Q}) -\frac d2\mathcal H(F^\eta_{\rm edge}\mid B_{R,Q}) +H(A_{R,Q})-\frac d2H(B_{R,Q})-\log Q. \tag{43}\end{align*}\] We shall derive this inequality by removing the mark bins at fixed \(R,Q\). The remaining environment contribution is \[ H(A_{R,Q})-\frac d2H(B_{R,Q})-\log Q. \tag{44}\] Its limit is zero as \(Q\to\infty\) at fixed \(R\); after that cancellation, increasing \(R\) gives conditioning on \(\omega\).

We first keep the environment radius, its bins, and the noisy-mark bins fixed. Conditional on the finite graph, potentials, and list, only the Gaussian noise is random. The entropy of its binned labeling is at most the entropy of its histogram type plus the logarithm of the number of labelings of that type. The first term is \(O(\log n)\). Using (40) for the second term therefore bounds the entropy per vertex by \[\mathbb E_{\rm noise}H(T)-\frac d2\mathbb E_{\rm noise}H(S)-\log Q+o(1).\] We may replace the random histograms by their conditional means in this expression. Indeed their individual cell frequencies have variance tending to zero. To see this, \(\mathop{\mathrm{Tr}}\Gamma_\eta^2=O_\eta(n)\) implies that the average squared cross-covariance between two independently sampled bounded local blocks is \(o(1)\). Their marginal covariances have deterministic upper and positive lower bounds at fixed \(\eta\). Gaussian continuity consequently makes the average total variation distance to the product of the two marginals tend to zero. This remains true after arbitrary shifts by the fixed eigenvectors and after binning. Overlapping blocks have vanishing probability. Thus all histogram entropies converge to those of the corresponding limiting observations. Gaussian smoothing removes mark-bin boundary atoms, and the potential bins have null boundaries as well.

We next remove the mark binning, while retaining the finite environment labels. Use cubes of side \(\Delta\) in \(\mathbb R^m\), clipped outside a large box. Before clipping, the entropy of the global quantization is at least the global Gaussian differential entropy minus \(nm\log\Delta\): the entropy within any cube is at most its log volume. Clipping changes the entropy per vertex by an amount tending to zero with the box size, at fixed \(\Delta\). For this last assertion it suffices to bound the average second moment, which is bounded by normalization of the marks and the fixed-height Gaussian covariance bound. For example, compare the tail-bin distribution with summable lattice weights proportional to \((1+|j|)^{-C}\); its entropy is bounded by its tail probability times a logarithmic moment bound. The same argument applies to each finite-dimensional limiting observation.

After removing the clipping, send \(\Delta\downarrow0\). The smoothed densities are bounded and continuous, and have finite second moments. Replacing a density by its uniform filling in each bin gives \(L^1\) convergence; the density bound and the preceding tail estimate give entropy convergence. The log-volume terms cancel because \(m(d+1)-(d/2)(2m)=m\). The lower bound furnished by the full Gaussian array is \[\begin{align*} \frac1n\mathcal H(\text{global Gaussian noise}) &=\frac m2\log(2\pi\mathrm e) +\frac m{2n}\log\det\Gamma_\eta\\ &=m\left[\frac12\log(2\pi\mathrm e\,\eta) -\frac1n\sum_{i=1}^n\log|\lambda_i-E-i\eta|\right]. \end{align*}\] The empirical spectral measure converges to \(\nu_{d,w}\), by polynomial approximation and convergence of bounded-neighborhood observations. Splitting each limiting joint entropy into the entropy of its environment labels and its conditional mark entropy gives (43).

We now compute the environment contribution (44) before increasing the radius. Set \[s_R=\sum_{k=0}^{R}b^k,\qquad p_R=\sum_{k=0}^{R-1}b^k, \qquad p_0=0.\] The star’s collection of labels reads \(1+ds_R\) potential bins: the ball of radius \(R+1\), with its first neighbor positions ordered and its remaining branches unordered. The edge reads \(2s_R\) bins in its two outward half-balls. When all these bins are distinct the overlaps identify the vertices uniquely. There are \(dp_R\) independent outward permutations of \(b\) branches in the star reading and \(2p_R\) in the edge reading. Thus \[\begin{align*} H(A_{R,Q})&=(1+ds_R)\log Q-dp_R\log(b!)+o_Q(1),\\ H(B_{R,Q})&=2s_R\log Q-2p_R\log(b!)+o_Q(1). \end{align*}\] For fixed \(R\), bin collisions have probability \(O_R(Q^{-1})\); their entropy cost is \(O_R(Q^{-1}\log Q)\), which justifies the errors. Both the bin counts and the permutation losses cancel in (44), leaving \(o_Q(1)\).

Take nested potential partitions as \(Q\to\infty\). Almost surely all finitely many potentials in a reading are distinct, so its limiting information is exactly the corresponding continuous potential configuration modulo the indicated permutations. Conditioning instead on fully ordered potentials in that union does not change the observation entropy: each omitted permutation fixes the observed star or edge, and invariance of \(\mathcal L\) makes its conditional observation law identical under that permutation. Finally let \(R\to\infty\); these ordered configurations exhaust \(\omega\).

Here the conditioning limits can be justified without subtracting infinite entropies of continuous environments. Use relative entropy with respect to a fixed product Gaussian reference on the observation coordinates and apply monotone convergence on the increasing conditioning sigma-fields. The conditional relative entropy is finite: fixed-height smoothing bounds conditional entropy below by noise entropy, while second moments bound it above. This proves convergence of the conditional entropies and hence (42).

All passages have been made for fixed finite choices before refinement. To make them hold inside almost every empirical law, include the countably many bin choices and heights in the joint subsequence construction, and take a further subsequence on which the high-probability count bounds hold eventually at every fixed precision. An almost sure representation of the empirical data can retain the original finite data by sampling their conditional law. Their deterministic relations, including the entropy counts and eigenvector normalization, are then preserved. The preceding limits apply to each represented realization and give the stated almost sure assertion. ◻

Proof. The inverse star Green matrix has central diagonal \(V_o-E\), root-neighbor entries \(1\), and neighbor diagonals \(1/g_i\). Its negative imaginary part is diagonal on the neighbors, with entries \(b_i/|g_i|^2\). The real central row implies that the imaginary covariance annihilates the constraint functional in (45).

To verify its law and its volume normalization directly, start with independent real Gaussians \[Y_i\sim N(0,b_i),\qquad Z\sim N(0,B_*^{-1}),\] and condition on \(t_0Z+\sum_iY_i=0\). Put \(X_i=Y_i-c_iZ\). The quadratic form of the resulting density is exactly (46). Gaussian conditioning gives the imaginary parts of the Green entries \[m_o,\qquad -m_o g_i,\qquad \delta_{ij}g_i+g_i m_o g_j,\] in the root, root-neighbor, and neighbor-neighbor positions, respectively. Hence this conditioned Gaussian has covariance \(K_o\). Integrating its density against \(\delta(C)\) gives the normalization \[(2\pi)^{d/2}\Bigl(\prod_i b_i\Bigr)^{1/2}|m_o|,\] because \(\mathop{\mathrm{Var}}(t_0Z+\sum_iY_i)=(t_0^2+B_*^2)/B_*\). The coefficient of the forgotten neighbor in \(C\) is one, so this is precisely the normalization in retained-coordinate volume, not Euclidean surface volume. It proves (47).

On an edge the inverse Green matrix has diagonal entries \(1/g'\) and \(1/g_i\), and off-diagonal entries \(1\). Using its diagonal negative imaginary part, or multiplying the two-by-two matrices, gives (48) and (49). Each of the two precision summands is positive definite: for \(g=c+i\beta\), its matrix is \[ Q(g)=\frac1\beta \begin{pmatrix}|g|^2&c\\c&1\end{pmatrix}, \qquad \det Q(g)=1. \tag{50}\] This also proves edge nondegeneracy. ◻

Proof. Use real parts of analytic logarithms in the upper half-plane. The edge factor is nonzero, as follows also from its invertible Schur matrix. Differentiating the full-root recursion yields \[\partial_\zeta\log m_o =m_o\left(1+\sum_i\partial_\zeta g_i\right).\] In the expected derivative of the edge term, edge reversal pairs the two differentiated cavity messages. Its contribution is \[-d\,\mathbb E_\omega\frac{(\partial_\zeta g_i)g'}{1-g_i g'} =-\mathbb E_\omega m_o\sum_i\partial_\zeta g_i,\] where \(m_o=g'/(1-g_i g')\). The indirect derivatives therefore cancel, leaving \(\mathbb E_\omega m_o\), the analytic derivative whose real part differentiates \(-U\). At fixed positive height the resolvent bounds justify differentiation and expectation. Both sides of (53) are asymptotic to \(-\log|\zeta|\) at imaginary infinity, which fixes the constant.

The boundary continuity and logarithmic integrability in Lemma 3, together with the Schur identities, pass the right side to the boundary in expectation. They apply also to the two incoming messages in an edge factor, and prove finiteness of the limit. Finally insert (47) and (49) into the Gaussian entropy formula. The \(\log(2\pi\mathrm e)\) terms cancel by dimension. The \(\log b_i\) terms cancel in expectation because the two cavity messages on an edge have the same marginal law. The remaining expression is precisely the boundary value of (53), proving (54). ◻

Proof. Fix \(\mathcal L\), and replace the noise in (56) by \(\sqrt s\,\gamma\), defining \(\mathcal D(s)\) for \(s>0\). We first prove \(\mathcal D'(s)\ge0\), keeping the environment dependence explicit.

For a smooth positive density \(p\), its score is \(\nabla\log p\), and its Fisher information is \(\int|\nabla\log p|^2p\). At fixed \(\omega\), let \(X\) have the conditional law of \(L_o+\sqrt s\,\gamma\), let \(\rho_o\) be its score, and let \(\rho_i\) be the score of \(R_iX\). Differentiating a marginal density gives \[\mathbb E[ R_i\rho_o(X)\mid R_iX]=\rho_i(R_iX).\] Thus conditional Jensen and (52) give \[\begin{align*} I_o(s)&:=\mathbb E|\rho_o(X)|^2 =\sum_i\mathbb E\bigl|A_{oi}^{1/2}R_i\rho_o(X)\bigr|^2,\\ I_o(s)-\sum_i\mathbb E\bigl|A_{oi}^{1/2}\rho_i(R_iX)\bigr|^2 &=\sum_i\mathbb E\bigl|A_{oi}^{1/2} (R_i\rho_o(X)-\rho_i(R_iX))\bigr|^2 =:\Delta_o(s)\ge0. \tag{58}\end{align*}\] All expectations on these lines are conditional on \(\omega\).

The conditional edge law, read in common whitened coordinates, is the same from its two endpoints. Invariance under re-rooting therefore pairs the \(A_{oi}\) term with its \(A_{io}\) term after \(\mathbb E_\omega\). Their sum is the identity. Consequently \[ \mathbb E_\omega\sum_i\mathbb E |A_{oi}^{1/2}\rho_i(R_iX)|^2 =\frac12\mathbb E_\omega\sum_i I_{oi}(s), \tag{59}\] where \(I_{oi}(s)\) is the ordinary Fisher information of the edge marginal. De Bruijn’s heat identity now yields \[ \mathcal D'(s)=\frac12\mathbb E_\omega\Delta_o(s)\ge0 \quad\text{for almost every }s>0. \tag{60}\] The comparison of joint and marginal scores belongs to the Fisher-information approach to entropy subadditivity [13]; for fixed projection families, see [14]. Combining this comparison with the heat identity [29] follows the tree entropy argument of [7].

Here are the analytic justifications for this use of the heat identity. Gaussian smoothing of a finite-second-moment law has finite entropy, and its Fisher information in dimension \(q\) is at most \(q/s\), by conditioning the Gaussian score. On a compact positive-time interval, a fixed positive mass of the input in a bounded ball bounds the absolute logarithm of the convolved density by \(C(1+|x|^2)\). Gaussian-kernel derivative bounds and the input second moment then justify integration of the heat equation against the log density, first with spatial cutoffs and then without them. This gives \(d\mathcal H(X)/ds=I(X)/2\) in integrated form. Since \(0<A_{oi}<\mathrm I\), no inverse-weight moments enter (58); Fisher information is bounded by dimension over \(s\), also after environment integration. The entropy bounds used in the preceding subsection justify that integration on compact \(s\)-intervals. These are also the finite-second-moment hypotheses in [7].

As \(s\to\infty\), \(\mathcal D(s)\to0\). Indeed, divide each observation by \(\sqrt s\); the dimensional logarithms cancel, and at each environment its entropy tends to standard Gaussian entropy. The noise entropy is a lower bound, and the Gaussian entropy upper bound gives domination by a constant times a logarithmic second moment. Such a bound is integrable even if the whitened second moments themselves are not: with \(M(\omega)=\mathbb E_{\mathcal L} [\|U_o^{\rm ret}\|^2\mid\omega]\), \[\log\bigl(1+\mathbb E[\|L_o\|^2\mid\omega]\bigr) \le\log(1+\|(K_o^{\rm ret})^{-1}\|)+\log(1+M(\omega)).\] The first term is integrable by the spatial tree estimates, and the second by (36) and invariance. Edge moments obey the same bound. Dominated convergence proves the claim.

Together with \(\mathcal D(1)\ge0\), monotonicity and the zero limit imply \(\mathcal D(s)=0\) for all \(s\ge1\). Equation (60) makes every summand of (58) vanish for almost every such \(s\), almost surely in \(\omega\). Choose one \(s\) for which this holds simultaneously at all the countably many re-rootings. Positive definiteness of every \(A_{oi}\) gives \[R_i\rho_o(X)=\rho_i(R_iX).\] The densities are positive and smooth, so this identity holds everywhere. Differentiating shows that the symmetric log-density Hessian commutes with every \(P_i\).

We next determine what these commuting projections permit. At one star, they leave the central law free but force the complementary spatial directions to be Gaussian with a common color covariance. We will then read one edge from both endpoints to determine the central law as well.

For one spatial color, let \(w_c\) be the unit direction reading the central coordinate, standardized with its noise variance. The edge-reading space is \(\operatorname{span}\{w_c,a_i\}\), where \(a_i\) is the unit neighbor reading after its central component is removed. In the Gaussian reference field, conditional on its central value, the residual neighbor covariance is \[ \mathop{\mathrm{diag}}(b_i)-\frac{(b_i)_i(b_i)_i^{\mathsf T}}{B_*}. \tag{61}\] It follows that the \(a_i\) span the \((d-1)\)-dimensional orthogonal complement of \(w_c\), and, for \(i\ne j\), \[ \langle a_i,a_j\rangle =-\sqrt{\frac{b_i b_j}{(B_*-b_i)(B_*-b_j)}}\in(-1,0). \tag{62}\] Here strict inequality in magnitude uses \(d\ge3\). In particular these lines are pairwise distinct and nonorthogonal. For \(m\) colors all these spaces are tensored with \(\mathbb R^m\).

The intersection of the edge-reading spaces is exactly the central color space \(\operatorname{span}(w_c)\otimes\mathbb R^m\). A Hessian commuting with all their projections preserves this intersection and its orthogonal complement. On the complement it preserves each \(\operatorname{span}(a_i)\otimes\mathbb R^m\), so its restriction there has the form \(a_i\otimes v\mapsto a_i\otimes D_i v\). Self-adjointness and \(\langle a_i,a_j\rangle\ne0\) imply \(D_i=D_j\). Since the \(a_i\) span, the complementary Hessian is spatial identity tensored with one color matrix.

Vanishing mixed derivatives first split the log density into a central function and a complementary function. In any orthonormal spatial basis of the complement, the latter has zero mixed blocks and identical diagonal color Hessians. It therefore splits into a sum of \(d-1\) functions, each of its own color variable. Since \(d-1\ge2\), equality of their Hessians for all independent choices of these variables forces the common matrix to be constant. Integrability of the density makes it negative definite. The complementary law is consequently Gaussian, with covariance equal to spatial identity tensored with a positive color matrix, and is independent of the central law.

It remains to constrain that central law. Read a common edge from its two incident stars. In the whitened two-dimensional spatial edge, their standardized endpoint directions are distinct by (49). They are also nonorthogonal almost surely. Indeed the unstandardized endpoint covariance is \[\Im(-m_o g_i) =-\frac{t_0b_i+B_*c_i}{t_0^2+B_*^2}.\] Condition on the incoming branches and vary the independent continuously distributed \(V_o\). Its numerator is affine in \(V_o\), with nonzero slope \(b_i\), so it vanishes with probability zero. A countable union handles all edges.

In coordinates given by the first endpoint direction and its perpendicular, the edge log-density Hessian has block form \(\mathop{\mathrm{diag}}(D_c(z),D)\), with \(D\) the constant Gaussian color block already obtained. The other endpoint direction has coordinates \((\cos\theta,\sin\theta)\), with both entries nonzero. Commutation with its projection forces \[\cos\theta\sin\theta\,(D_c(z)-D)=0.\] Thus the central block too is the same constant \(D\). The entire smoothed star is Gaussian with covariance spatial identity tensored with a color covariance, and the color covariance agrees across every edge.

Dividing characteristic functions by that of the smoothing Gaussian now proves the same assertion for the unsmoothed marks, with a possibly singular positive semidefinite color covariance \(C_{\mathcal L}(\omega)\). Their means vanish by Lemma 11. Edge agreement makes \(C_{\mathcal L}(\omega)\) invariant under re-rooting, and the covariance is a measurable function of \(\omega\) by disintegration of the fixed law \(\mathcal L\).

The iid environment is ergodic under tree automorphisms. One direct proof approximates an invariant event by an event depending on a finite ball, moves that ball disjointly far away by an automorphism, and uses independence; approximation then makes the invariant event independent of itself. Root-preserving invariance and re-rooting invariance therefore make \(C_{\mathcal L}\) a deterministic matrix within this fixed empirical law. In original coordinates the root covariance is \((\Im m_o)C_{\mathcal L}\). Its expectation is \(\mathrm I_m\) by Proposition 10, so \[(\mathbb E_\omega\Im m_o)C_{\mathcal L}=\mathrm I_m, \qquad C_{\mathcal L}=\frac1{\pi\rho}\mathrm I_m.\] This proves (57) and fixes the same answer for every admissible empirical law.

Finally, finite vertex samples were conditionally independent samples from their empirical law before taking limits. Their limiting joint law is therefore the corresponding product conditional on \(\mathcal L\) and the retained positions. The one-root marked-environment marginal has just been shown to be deterministic, with iid environment and the Gaussian conditional kernel (57). Its product has no remaining dependence on \(\mathcal L\) or on \(\nu\). This proves the sampled-root assertion, including the conditional formulation and multiplicities. ◻

Proof. Choose \(\varepsilon>0\) sufficiently small that \(2\varepsilon<c(1-\varepsilon)\), and consider dyadic \(R\). Set \(Y=R^{1-\varepsilon}\) and use a real grid of mesh \(R^{1-2\varepsilon}\) in \([-3R,3R]\). There are \(O(R^{2\varepsilon})\) grid points. By (68), the sum over dyadic \(R\) of their failure probabilities is finite. Moreover, changing the center of a Poisson kernel by at most \(R^{1-2\varepsilon}\) changes its ratio by \(1+O(R^{-\varepsilon})\), uniformly in the position variable. Borel–Cantelli and positivity therefore give, for some \(c_1>0\), \[ I(s,Y)=\pi\rho+O(R^{-c_1}),\qquad |s|\le3R, \tag{71}\] almost surely for all sufficiently large dyadic \(R\).

Every interval of length at most \(Y\) in this range has \(O(Y)\) points: evaluate \(I\) at its midpoint and note that each point of the interval contributes at least a constant times \(Y^{-1}\). Covering by such intervals gives \(\nu([-R,R])=O(R)\), first dyadically and then at all large radii. It also bounds the number of points in a strip of width \(W\ge Y\), lying in \([-2R,2R]\), by \(O(W)\).

For \(-R\le v\le u\le R\), integrate \(\pi^{-1}I(s,Y)\) over \([v,u]\). By (71), this gives \[\int P_{v,u,Y}(x)\,d\nu(x) =\rho(u-v)+O(R^{1-c_1}),\qquad P_{v,u,Y}(x)=\frac1\pi\int_v^u \frac{Y\,ds}{(x-s)^2+Y^2}.\] Take \(W=R^{1-\varepsilon/2}\). The strips of width \(W\) around the two endpoints contain \(O(W)\) points. Outside these strips, within \([-2R,2R]\), the difference between \(P_{v,u,Y}\) and the interval indicator is \(O(Y/W)\); the linear count bound makes its total contribution \(O(RY/W)\). For \(|x|>2R\), \(P_{v,u,Y}(x)\le CRY/x^2\), so dyadic summation using linear counts bounds the remaining contribution by \(O(Y)\). Hence the sharp and smoothed counts differ by \(O(R^{1-\varepsilon/2})\), uniformly in the endpoints. This also handles atoms at the endpoints. Taking \(0<\chi\le\min(c_1,\varepsilon/2)\) proves (69), with a neighboring dyadic radius covering an arbitrary \(R\).

Apply (69) separately to the positive and negative half-lines. Their counting errors are \(O(1+t^{1-\chi})\) up to distance \(t\). In the symmetric sum of \(1/x_j\), the two constant-density contributions cancel. Partial summation bounds the remaining tail by \(O(R^{-\chi})\), since the integrated error is bounded by a constant times \(\int_R^\infty t^{-1-\chi}\,dt\). The inverse-square bound follows directly from linear counts. Possible points at zero play no role in these tail statements. ◻

Proof. Condition on \(\nu\) and the sampled environments, and put \(Z_j=b_jb_j^{\mathsf T}-C_k\). Its entries are independent across \(j\), centered, and have bounded second moments, with bounds depending on the fixed conditioning data. By (70), \(\sum_j Z_j/(x_j-i)\) converges almost surely entrywise when ordered by increasing \(|x_j|\). Also \[\sum_{|x_j|>1}\frac{\|Z_j\|}{x_j^2}<\infty\] conditionally almost surely, because its conditional expectation is finite. Thus the difference series between \(z\) and \(i\) converges absolutely and locally uniformly. Combining it with \(C_km(z)\) proves local uniform convergence of the marked principal value, say \(S_k(z)\). Equations (66) and (67) show that \(M_k-S_k\) is a real symmetric matrix constant in \(z\).

Linear counts imply \(\sum_j|x_j-iy|^{-2}=O(y^{-1})\). Each entry of the centered marked sum at \(iy\) consequently has conditional second moment \(O(y^{-1})\). Together with (73), this yields \[ S_k(iy)\longrightarrow i\pi\rho C_k \quad\text{in probability}. \tag{77}\] The convergence remains true without conditioning, by bounded convergence applied to the conditional probabilities.

We must compare this limit with \(M_k(iy)\) on the same space as its sampled environments. Corollary 9 gives the large-microscopic-height limit jointly with every fixed-radius environment. Its limiting tree diagonal is the measurable function \(\mathcal E_\alpha\mapsto m_\alpha^*\) of the entire environment. It follows that \[ M_k(iy)\longrightarrow\mathop{\mathrm{diag}}(m_1^*,\ldots,m_k^*) \quad\text{in probability}. \tag{78}\] Here is the measurable-function point in this inference. Approximate \(m_\alpha^*\) in probability, under the fixed iid environment law, by a bounded continuous function of finitely many potential coordinates. Joint convergence with those coordinates controls the distance to this approximation. The two approximation errors have the same environment law before and after the limit; letting their probabilities tend to zero proves convergence to \(m_\alpha^*\) itself. The off-diagonal entries in (78) tend to zero by the sampled Ward estimate in Lemma 7. Thus this is a statement about the existing environments, rather than only about the marginal distributions of the diagonals.

Subtract (77) from (78). The constant \(M_k-S_k\) equals \(\mathop{\mathrm{diag}}(m_\alpha^*)-i\pi\rho C_k=A_k\) almost surely, proving (76). Finally, (69) gives infinitely many positions. Conditional on positions and environments, the first \(k\) Gaussian columns are linearly independent almost surely. Since all \(c_\alpha\) are positive, their positive weighted outer products already make \(\Im M_k(z)\) positive definite for every \(z\in\mathbb H\). ◻

Proof. We first prove a pointwise implication under a fixed bound on \(m\). For one sampled site, write \[M_1(z)=a_1+c_1\bigl(m(z)+Q(z)\bigr),\qquad Q(z)=\mathop{\mathrm{PV}}\sum_j\frac{g_j^2-1}{x_j-z}.\] For every integer \(p\ge1\), conditionally on \(\nu\) and the environment, \[ \mathbb E\bigl(|Q(s+iy)|^{2p}\mid\nu,\mathcal E_1\bigr) \le C_p\left(\sum_j|x_j-s-iy|^{-2}\right)^p =C_p\left(\frac{\Im m(s+iy)}y\right)^p. \tag{82}\] To verify the inequality, first use a finite sum and expand the \(2p\)th absolute moment. Every surviving spectral index occurs at least twice because \(g_j^2-1\) is centered. Each partition of these indices is bounded by the appropriate Gaussian moments and by the \(p\)th power of the sum of squared weights; for a block of size \(q\ge2\), use \(\sum |w_j|^q\le(\sum|w_j|^2)^{q/2}\). There are only finitely many partitions for fixed \(p\). Convergence in these moments then proves the assertion for the series.

In particular, for every fixed \(C_0,A<\infty\), at any fixed \(s,y\), \[ \mathbb P\bigl(|m(s+iy)|\le C_0,\ |Q(s+iy)|>y^{-1/4}\bigr) \le C_{A,C_0}y^{-A}. \tag{83}\] Choose \(p\ge2A\) in (82). The identical assertion holds for the hatted ensemble, with any fixed enlarged bound on \(|\widehat m|\).

Choose an event \(\mathcal F\), of probability \(p_0>0\), on the old environment and the independent new potential such that \[ |m_1^*|+|\widehat m_1^*|\le C_1,\qquad \Im m_1^*,\Im\widehat m_1^*\ge c_*>0,\qquad |D_1|\ge c_*. \tag{84}\] Here all constants are deterministic for the fixed positive \(w\). Such an event exists directly from the cavity formula. The branch sum has finite entries and strictly positive imaginary part almost surely, so with positive probability its modulus is bounded and its imaginary part is bounded below. On that event the root Green function is bounded, with imaginary part bounded below, uniformly for every root potential in \([-w,w]\). Impose additionally \(|\widehat V_1-V_1|\ge w/2\), an event of positive probability. The old environment is independent of \(\nu\), and the new potential is independent of all old data; hence \(\mathcal F\) is independent of the old point process.

Suppose \(|m(z)|\le C_0\). By (80), \(|\widehat m(z)|\) lies in a fixed slightly larger range for \(y\ge1\). On \(\mathcal F\), outside the two exceptions in (83), insert the two marked approximations into (79). Replacing \(\widehat m\) by \(m\) costs \(O(y^{-1})\) and gives \[ \begin{split} P(m)&=O(y^{-1/4}),\\ P(t)&=(1+D_1(a_1+c_1t)) (\widehat a_1+\widehat c_1t)-(a_1+c_1t). \end{split} \tag{85}\] The tree branches are unchanged by resampling, so \((1+D_1m_1^*)\widehat m_1^*=m_1^*\). Thus the real-coefficient polynomial \(P\) vanishes at \(i\pi\rho\) and at its conjugate. Its leading coefficient is \(D_1c_1\widehat c_1\), whose modulus is bounded below on \(\mathcal F\). Consequently \[P(t)=D_1c_1\widehat c_1(t-i\pi\rho)(t+i\pi\rho).\] Since \(\Im m(z)>0\), we have \(|m(z)+i\pi\rho|\ge\pi\rho\). Equation (85) therefore implies \(|m(z)-i\pi\rho|\le C_2 y^{-1/4}\).

This conclusion gives an arbitrary-power failure estimate for the old point process alone. More precisely, if \[\mathcal B_z=\{|m(z)|\le C_0, \ |m(z)-i\pi\rho|>C_2y^{-1/4}\},\] then independence gives \(\mathbb P(\mathcal B_z\cap\mathcal F)=p_0\mathbb P(\mathcal B_z)\). But this intersection is contained in the union of the old and new marked-sum exceptions, each of which has unconditional probability at most \(C_Ay^{-A}\). We have proved \[ \mathbb P(\mathcal B_{s+iy})\le C_{A,C_0}y^{-A} \quad\text{for every }A>0, \tag{86}\] uniformly in the fixed real part \(s\). In particular, this reasoning does not require the switched process to have its original law conditionally on the old positions or environments.

It remains to establish the bound on \(m\) needed to use this implication across an expanding region. Fix \(0<\delta<1\) and use, for dyadic \(R\), a unit grid in the padded region \[|e|\le2R,\qquad \tfrac12R^\delta\le y\le2R^2.\] There are \(O(R^3)\) grid points. At its top height, the uniform variant of (73) gives \(m(e+iy)=i\pi\rho+o(1)\) uniformly for \(|e|\le2R\). For two heights with \(y/2\le y'\le y\), the absolutely convergent difference series gives \[ |m(e+iy')-m(e+iy)| \le\frac{y-y'}{y'}\Im m(e+iy) \le\Im m(e+iy). \tag{87}\] Choose \(C_0\) large enough that this inequality places the next height inside \(|m|\le C_0\) whenever the preceding value is within \(1\) of \(i\pi\rho\). Decrease heights by one unit, shortening the last step if necessary; for large \(R\) all successive ratios are within the range in (87). At each step (86) then returns the value to within \(C_2 y^{-1/4}\) of \(i\pi\rho\).

Choose \(A\) so large that \(R^{3-\delta A}\) is summable over dyadic \(R\). The union bound and Borel–Cantelli exclude all implication failures on these grids for sufficiently large \(R\), almost surely. The preceding descent proves the estimate on the grids without any additional trace-bound assumption. Finally, \[ |m'(z)|\le\sum_j|x_j-z|^{-2}=\frac{\Im m(z)}{\Im z}. \tag{88}\] Nearby positive Poisson kernels are comparable. Thus the grid bounds on \(\Im m\) and (88) fill unit grid gaps at cost \(O(y^{-1})\). Padding permits arbitrary large radii. Taking, for example, any fixed \(0<\xi<1/4\) absorbs constants. Apply the argument to a countable collection of positive \(\delta\) tending to zero; for any prescribed \(\delta\), a smaller one in that collection implies the asserted estimate. ◻

Proof. First suppose the selections depend on fixed-radius potential balls through continuity sets, and the maps are piecewise translations with only null sets of discontinuities. At finite \(n\), use these same selections and maps when the \(k\) sampled balls are disjoint trees; otherwise make no change. Conditional on the graph, the sampling data, and all potentials except the \(k\) central ones, every selection is fixed and every central potential remains independent and uniform. Applying the selected maps therefore preserves the joint finite-graph law. The probability of ball or separation failure is \(O_k(n^{-1})\) for the fixed radii.

Write \(H'=H+W_{n,k}D W_{n,k}^{\mathsf T}\) on this event, and let \(x_i'\) be its microscopic eigenvalues. The determinant identity and differentiation in the microscopic variable give exactly \[ \sum_i\frac1{x_i'-z} =\sum_i\frac1{x_i-z} -\partial_z\log\det(\mathrm I+D M_{n,k}(z)). \tag{90}\] Indeed the determinant ratio is \(\det(H'-E-z/n)/\det(H-E-z/n)\); its logarithmic derivative is \(-n^{-1}\mathop{\mathrm{Tr}}G'(E+z/n)+n^{-1}\mathop{\mathrm{Tr}}G(E+z/n)\). This also verifies the sign and the absence of any further factor of \(n\) in (90).

Take imaginary parts. The scalar sums converge by the tail argument preceding (65); the compression functions and their derivatives converge locally uniformly. On the coordinates where \(D_\alpha\ne0\), write the logarithmic derivative as \[ \mathop{\mathrm{Tr}}\bigl((D^{-1}+M_k(z))^{-1}M_k'(z)\bigr), \tag{91}\] using the corresponding compression. Positive definiteness of \(\Im M_k(z)\) makes \(D^{-1}+M_k(z)\) invertible. Zero coordinates can simply be removed. Equivalently, the determinant of \(\mathrm I+D M_k\) is nonzero, and its logarithmic derivative has no choice-of-branch ambiguity. The continuous mapping theorem now passes (90) to the limit, jointly in any finite list of arguments, and law preservation gives the claimed identity for these selections and maps.

For completeness, no continuity assumption on a fixed measure-preserving map is needed in the stated result. A bounded measurable \(T_\alpha\) can, for convergence purposes, be approximated in probability under the uniform root law by continuous functions (or by functions with finitely many interval discontinuities). This follows from approximation of measurable sets in Lebesgue measure by finite unions of intervals, applied to a finite binning of its range. The approximants need not preserve measure: the finite-graph law-preservation identity always uses \(T_\alpha\) itself. They only establish joint convergence of \(T_\alpha(V_\alpha)\) with the resolvents and environment data. The approximation probabilities are the same at every \(n\) and in the limit because the root marginals are uniform. This proves the fixed-radius assertion for every measurable measure-preserving map.

Finally, approximate each general \(B_\alpha\) in probability by events in finitely many root-excluded coordinates. Such approximation follows from conditional expectations on increasing finite-radius sigma-fields. Each finite-coordinate event can in turn be approximated by a continuity set for its finite product uniform law. All these approximations exclude the root potential. For fixed \(k\), the probability that any of the selected maps differs from its approximation tends to zero. On their agreement event, (89) is exactly unchanged; therefore its finite-dimensional law converges in the approximation. The preceding identity passes to the limit. Independent branch orderings can be included among the sampling data when an event uses ordered coordinates. ◻

Proof. The centered matrix in (101) is a sum of independent self-adjoint matrices \[\sum_{\alpha=1}^{r}\beta_{\alpha\alpha} (v_\alpha v_\alpha^\top-P),\qquad P=U_{\mathcal A}^2,\] where \(v_\alpha\) is Gaussian of covariance \(P\). Its coefficients may have either sign. Gaussian fourth moments give \[ \mathbb E[(v_\alpha v_\alpha^\top)^2] =(\mathop{\mathrm{Tr}}P)P+2P^2. \tag{103}\] Truncate each original Gaussian column on the event that all its entries have absolute value at most \(\log r\), and center the truncated summand. There are only polynomially many entries, so this alters the matrices with superpolynomially small probability. Gaussian tails also make the resulting centering correction superpolynomially small.

Each truncated summand has norm at most \[C\|\beta\|(\log r)^2\mathop{\mathrm{Tr}}P \le \theta\operatorname{polylog}(r).\] If \(X\) is a self-adjoint random matrix, then \(\mathbb E[(X-\mathbb EX)^2]=\mathbb E[X^2]-(\mathbb EX)^2\le\mathbb E[X^2]\). Also truncating \(v_\alpha v_\alpha^\top\) by an event can only decrease the expectation of its square in positive semidefinite order. Thus (103), including for negative \(\beta_{\alpha\alpha}\), bounds the matrix variance parameter by \[C s_2(\mathop{\mathrm{Tr}}P)\|P\| \le \frac{T}{a_{\mathcal A}}\operatorname{polylog}(r).\] The bounded self-adjoint matrix Bernstein inequality [30], applied to both signs, gives, for a sum of independent centered self-adjoint matrices of norm at most \(b\) and variance parameter \(\sigma_{\rm mat}^2\), a two-sided tail bounded by \[2|\mathcal A|\exp\left( -\frac{c t^2}{\sigma_{\rm mat}^2+bt}\right).\] Its dimension factor is polynomial in \(r\). Taking a sufficiently large power of \(\log r\) in \(t\) proves (101).

For the cross block, independence of distinct Gaussian rows and direct second-moment calculation give \[\mathbb E\|P_0\|_{\mathrm{HS}}^2 =s_2(\mathop{\mathrm{Tr}}U_I^2)(\mathop{\mathrm{Tr}}U_F^2).\] Equation (102) follows from (100) and Markov’s inequality, with one additional logarithmic factor. ◻

Proof. Conjugating the matrix to be inverted by \(U_F\) gives \[U_F(D_F+A'_{FF}-z)U_F =\operatorname{sign}(D_F)+U_F(A'_{FF}-z)U_F.\] The second term has norm at most \(\delta_F\), by Lemma 21, the bound \(|s_1|=O(T)\), and (100). Since \(\delta_F=o(1)\), a Neumann expansion proves (104). In particular, \(Q_F=U_F^{-1}G_FU_F^{-1}\) has bounded norm. By (96), \[\left|\sum_{j\in F}\frac1{x_j}\right|=O(K^{-\chi}), \qquad |\mathop{\mathrm{Tr}}(G_F-D_F^{-1})| \le\delta_F\mathop{\mathrm{Tr}}U_F^2=o(1).\] This proves the assertion about the far trace.

For the Schur correction, replacing \(G_F\) by \(D_F^{-1}\) costs at most \[ \|P_0\|_{\mathrm{HS}}^2\delta_F \le T\operatorname{polylog}(r)\delta_F \tag{106}\] in the norm weighted on both sides by \(U_I\). It remains to estimate \(A'_{IF}D_F^{-1}A'_{FI}\). Condition further on the inner Gaussian row block \(\mathcal G_I\), and set \[Q=\mathcal G_I\beta^2\mathcal G_I^\top.\] The columns of \(A'_{IF}\), indexed by \(j\in F\), are then independent centered Gaussian vectors \(a_j\) with covariance \(Q\). Consequently \[\begin{align*} \mathbb E\left[\sum_{j\in F}\frac{a_ja_j^\top}{x_j} \,\middle|\,\mathcal G_I,\mathcal E\right] &=Q\sum_{j\in F}\frac1{x_j},\\ \mathop{\mathrm{Var}}\left(\sum_{j\in F}\frac{(a_j)_a(a_j)_b}{x_j} \,\middle|\,\mathcal G_I,\mathcal E\right) &=(Q_{aa}Q_{bb}+Q_{ab}^2) \sum_{j\in F}\frac1{x_j^2}. \end{align*}\] Also \(\mathbb E[Q_{aa}^2\mid\mathcal E] =s_2^2+2\mathop{\mathrm{Tr}}\beta^4=O(T^2)\). Since \(Q\) is positive semidefinite, \(|Q_{ab}|^2\le Q_{aa}Q_{bb}\); Cauchy–Schwarz bounds the remaining second moments by \(O(T^2)\). Multiplying entry \(a,b\) by \((U_I)_{aa}(U_I)_{bb}\), summing its second moment over \(a,b\), and using (96) and (100), we obtain \[\mathbb E\left\| U_IA'_{IF}D_F^{-1}A'_{FI}U_I \right\|_{\mathrm{HS}}^2 \le C T^2(\log r)^2(K^{-2\chi}+K^{-1}).\] Markov’s inequality therefore bounds this norm by \(T\operatorname{polylog}(r)(K^{-\chi}+K^{-1/2})\) with probability tending to one.

Combining this estimate with (106) gives \[\|U_I\Sigma U_I\| \le \operatorname{polylog}(r)\,O\left( TK^{-\chi}+TK^{-1/2}+\frac{T(1+T)}K +T\sqrt{T/K}+T\theta\right)\] with probability tending to one. After multiplication by \(T^5\), all terms tend to zero by a fixed power of \(r\). For example, the three potentially slow terms have exponents \[6\tau-\frac{\chi}{100},\qquad \frac{13}{2}\tau-\frac1{200},\qquad \frac{13}{2}\tau-\frac12,\] respectively. They are negative under (92); the terms \(T^6K^{-1/2}\) and \(T^6(1+T)/K\) have still ample margins. This proves (105). ◻

Proof. Every diagonal entry of \(U_I(D_I-ih_1)U_I\) has modulus at least \(1/\sqrt2\). The added term has norm at most \[\|U_IA'_{II}U_I\| \le C T/h_1+\operatorname{polylog}(r) \left(\sqrt{T/h_1}+\theta\right)=o(1)\] with probability tending to one. Hence \[\|U_I^{-1}G_I(ih_1)U_I^{-1}\|\le C.\] The identity \(G_I(ih_1)^*G_I(ih_1)=h_1^{-1}\Im G_I(ih_1)\) now gives \[\|G_I(ih_1)U_I^{-1}\|+\|U_I^{-1}G_I(ih_1)\| \le C h_1^{-1/2}.\] Alternatively the same mixed-weight estimates follow from the preceding weighted inverse and \(\|U_I\|\le h_1^{-1/2}\). Use the exact twice-iterated resolvent identity \[\begin{align*} G_I(z) ={}&G_I(ih_1)+(z-ih_1)G_I(ih_1)^2\\ &+(z-ih_1)^2G_I(ih_1)G_I(z)G_I(ih_1). \end{align*}\] The unweighted middle resolvent in the last term has norm at most \((\Im z)^{-1}\). Conjugating the identity by \(U_I^{-1}\), the three terms are bounded by \[C,\qquad C|z-ih_1|/h_1,\qquad C|z-ih_1|^2/(h_1\Im z),\] respectively. This proves (107).

The Schur complement formula for the full inverse has upper-left block \(\widetilde G_I=(G_I^{-1}-\Sigma)^{-1}\). Set \(R_I=U_I^{-1}G_IU_I^{-1}\) and \(S_I=U_I\Sigma U_I\). Then \[U_I^{-1}\widetilde G_IU_I^{-1} =(R_I^{-1}-S_I)^{-1}.\] By (107) and (105), a Neumann expansion gives \[\|U_I^{-1}\widetilde G_IU_I^{-1}\|=O_{\mathbb P}(h_1),\qquad \|U_I^{-1}(\widetilde G_I-G_I)U_I^{-1}\| \le O_{\mathbb P}(h_1^2)o_{\mathbb P}(T^{-5})=o_{\mathbb P}(T^{-1}).\] Multiplying the last bound by \(\mathop{\mathrm{Tr}}U_I^2=O(\log r)\) shows that their traces differ by \(o_{\mathbb P}(1)\).

The full block-inverse trace is \[\mathop{\mathrm{Tr}}\widetilde G_I+\mathop{\mathrm{Tr}}G_F+ \mathop{\mathrm{Tr}}\bigl(\widetilde G_IA'_{IF}G_F^2A'_{FI}\bigr).\] The far trace is \(o_{\mathbb P}(1)\) by Lemma 22. For the last term recall \(Q_F=U_F^{-1}G_FU_F^{-1}\). It has bounded norm, and \[U_F^{-1}G_F^2U_F^{-1}=Q_FU_F^2Q_F,\qquad \|Q_FU_F^2Q_F\|=O_{\mathbb P}(K^{-1}).\] No positivity is asserted for this complex square. The ordinary operator-norm and Hilbert–Schmidt trace inequality instead gives \[\begin{align*} \left|\mathop{\mathrm{Tr}}\bigl(\widetilde G_IA'_{IF}G_F^2A'_{FI}\bigr)\right| &\le \|U_I^{-1}\widetilde G_IU_I^{-1}\|\, \|P_0\|_{\mathrm{HS}}^2\, \|Q_FU_F^2Q_F\|\\ &\le \operatorname{polylog}(r)\,O_{\mathbb P}(T^3/K)=o_{\mathbb P}(1). \end{align*}\] This proves (108). All estimates used the same Gaussian matrices, so taking a finite union gives the joint assertion. ◻

Proof. Write \[A'_{II}-s_1\mathrm I_{m_I} =\sum_{\alpha=1}^{r} \beta_{\alpha\alpha}(g_\alpha g_\alpha^\top-\mathrm I_{m_I}),\] where the \(g_\alpha\) are independent standard \(m_I\)-dimensional Gaussian vectors. The covariance identity \[\mathop{\mathrm{Cov}}(g_ag_b,g_cg_d) =\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}\] shows that \(gg^\top-\mathrm I\) and a raw GOE matrix have the same entry covariances. Their means are both zero. Thus replace the summands one at a time by \(\beta_{\alpha\alpha}\mathcal W_\alpha\), with independent raw GOE matrices \(\mathcal W_\alpha\). Negative coefficients are allowed: both covariances are multiplied by \(\beta_{\alpha\alpha}^2\).

For either kind of summand \(X\), \[\mathbb E\|X\|^3\le C\theta^3m_I^3.\] For the rank-one term this follows from Gaussian moments of \(\|g\|^2\); for the GOE term the Hilbert–Schmidt norm gives a sufficient bound. Fix spectral parameters \(z_1,\ldots,z_b\in\mathbb H\), and a bounded smooth test \(\Phi\) on the corresponding real and imaginary trace coordinates, with bounded derivatives. For any real symmetric matrix \(A\), its resolvents satisfy \(\|(A-z_\ell)^{-1}\|\le(\Im z_\ell)^{-1}\). Differentiating the resolvent in a real symmetric direction \(\Delta\) gives, for \(j=1,2,3\), a trace derivative bounded by \(C_{z,j}m_I\|\Delta\|^j\). The chain rule therefore bounds the third directional derivative of the composed test by \[C_{z,\Phi}m_I^3\|\Delta\|^3.\] This estimate holds at every intermediate real symmetric matrix, independently of eigenvalue gaps.

The Lindeberg replacement argument (see [15]) uses Taylor expansion to second order, which cancels the means and covariances in each replacement. Summing the third-order remainders gives total error \[C_{z,\Phi}m_I^6 r\theta^3 =O\bigl(r^{-0.44+(3/2)\tau}\bigr)=o(1).\] Finally \(\sum_\alpha\beta_{\alpha\alpha}\mathcal W_\alpha\) has exactly the law \(\sqrt{s_2}\,\mathcal W\), proving the lemma. ◻

Completion of Proposition 20. The coefficient bounds were proved in (94). Combine (98) with the trace cut in Lemma 23 and the Gaussian replacement in Lemma 24. They give a conditional comparison of bounded smooth tests of finitely many trace values with \(\mathcal S_r\). Its error tends to zero for almost every \(\mathcal E\), and is bounded, so bounded convergence removes the conditioning. The unconditional potential-switch identity from Proposition 19 therefore gives \[ \bigl(\Im\mathop{\mathrm{Tr}}(H_r-z_\ell)^{-1}\bigr)_{\ell=1}^b \ \Longrightarrow\ \bigl(\Im m(z_\ell)\bigr)_{\ell=1}^b \qquad(z_1,\ldots,z_b\in\mathbb H). \tag{109}\] Bounded smooth tests determine finite-dimensional weak convergence. In particular, switching invariance has only been used after averaging out the conditioning data.

We finish by explaining why these trace limits imply convergence of point measures, without assuming a uniform bound on their total number of points. Let \(\nu^{(r)}\) be the eigenvalue counting measure of \(H_r\), and define \[\sigma_r(\,d x)=\frac{\nu^{(r)}(\,d x)}{1+x^2}.\] Its total mass is \(\Im\mathop{\mathrm{Tr}}(H_r-i)^{-1}\), whose laws are tight by (109). Regarding \(\sigma_r\) as a finite positive measure on the one-point compactification \(\overline{\mathbb R}=\mathbb R\cup\{\infty\}\), its laws are therefore tight. For \(z=e+i\eta\in\mathbb H\), the kernel \[k_z(x)=(1+x^2)\frac{\eta}{(x-e)^2+\eta^2}\] extends continuously to \(\overline{\mathbb R}\), with \(k_z(\infty)=\eta\). Its integral against \(\sigma_r\) is \(\Im\mathop{\mathrm{Tr}}(H_r-z)^{-1}\). It follows that in any subsequential limit \(\sigma\), the joint law of these integrals at a countable dense set of \(z\)’s is that of \(\Im m(z)\). Continuity extends this conclusion to all \(z\).

These integrals determine the de-weighted restriction of \(\sigma\) to \(\mathbb R\). Indeed, for \(f\in C_c(\mathbb R)\), integrate the kernel against \(\pi^{-1}f(e)\,d e\) and let \(\eta\downarrow0\). On \(\mathbb R\) the resulting kernel tends to \((1+x^2)f(x)\), by the Poisson approximate identity. The integrated kernels are bounded uniformly over \(x\in\overline{\mathbb R}\) and \(0<\eta\le1\): near the compact support this is the Poisson mass bound, and away from it the factor \(1+x^2\) cancels the quadratic denominator. At infinity their value is \(\eta\pi^{-1}\int f(e)\,d e\), which tends to zero. Dominated convergence is thus applicable to each finite measure \(\sigma\), including any mass at infinity.

By Proposition 17, applying the same inversion to \(\Im m(z)\) gives \(\int f\,\,d\nu\). Taking a countable determining family of compactly supported tests, the joint distribution of these integrals is therefore exactly the law of \(\nu\). The map from \(\sigma\) to its de-weighted restriction is continuous for local vague convergence: for a compactly supported \(f\), the function \((1+x^2)f(x)\), extended by zero at infinity, is continuous on \(\overline{\mathbb R}\). Every subsequential local vague limit of \(\nu^{(r)}\) consequently has the law of \(\nu\). This proves the point-process assertion. ◻

Proof. The marginal statement is [25], with Definition 2.1 and Equations (2.5)–(2.9) there, specialized to energy zero and equal densities \(\rho_{\mathrm{fc},t}(0)=\rho_{\mathrm{sc}}(0)=1/\pi\). The energy lies in the required interior window for any fixed interior fraction. That theorem gives a power-decaying bound for [eq:dbm-marginal-test]. To pass to the last assertion, multiply each integral by \[\frac{(m)_a}{(m/\pi)^a}, \qquad (m)_a=m(m-1)\cdots(m-a+1).\] This is the Jacobian and ordered-tuple multiplicity for the unit-mass marginal convention, and it tends to \(\pi^a\) for fixed \(a\). ◻

Proof. First, \(\|V_*\|\) is bounded by (116), since \(T=o(K)\). We verify a slightly enlarged regularity domain, \[|\Re\zeta|\le2G,\qquad g/2\le\Im\zeta\le20,\] so that subsequent dilation by \(c_r=1+o(1)\) will be harmless. The normalized resolvent trace is \[m_{V_*}(\zeta)= \alpha_*^{-1}\sum_{j\in I} \frac1{x_j-(m_I\zeta/\alpha_*-s_1)}.\] Writing \(e=m_I\Re\zeta/\alpha_*-s_1\) and \(u=m_I\Im\zeta/\alpha_*\), the enlarged domain has \[ |e|=O(K^{7/8}+T)=o(K),\qquad cT^{1/2}\le u\le CK. \tag{118}\] The lower height exceeds a fixed positive power of \(K\). Proposition 18, with a radius comparable to \(K\) and any fixed exponent smaller than \(50\tau\), therefore gives \[\sum_j\frac{u}{(x_j-e)^2+u^2} =\pi\rho+o(1)\] uniformly in (118). For \(u/K\) sufficiently small, the omitted terms with \(|x_j|>K\) sum to \(O(u/K)\), uniformly for \(|e|=o(K)\), by the linear count bound from Lemma 16. Hence the retained sum has a positive lower bound. For \(u/K\) bounded below, every retained point has \(|x_j-e|\le2K\) eventually, so \[\sum_{j\in I}\frac{u}{(x_j-e)^2+u^2} \ge \frac{m_Iu}{4K^2+u^2}\ge c>0.\] Its upper bound follows by positivity from the full sum. This proves regularity on the enlarged domain.

With \(m=m_I\), the relevant powers are \[g=m^{-1+50\tau+o(1)},\qquad t_*=m^{-1+100\tau+o(1)},\qquad G=m^{-1/8}.\] In particular, \[\frac{t_*}{g m^\tau}=m^{49\tau+o(1)}\longrightarrow\infty, \qquad \frac{t_*}{m^{-\tau}G^2} =m^{-3/4+101\tau+o(1)}\longrightarrow0.\] The inequalities \(m^{\tau-1}\le g\le m^{-\tau}\) and \(G\le m^{-\tau}\) follow as well. Our bound on \(\tau\) makes all these exponent inequalities strict.

It remains to determine the free-convolution density. At \(\zeta=0\), Equation (112) becomes \[ \mathfrak m=F_r(\mathfrak m),\qquad F_r(\mathfrak m)=\alpha_*^{-1}\sum_{j\in I} \frac1{x_j-(\alpha_*s_2\mathfrak m-s_1)}. \tag{119}\] For \(|\mathfrak m-i|\le1/4\), the argument \(z=\alpha_*s_2\mathfrak m-s_1\) has \(\Im z\asymp T\) and \(|z|=O(T)\). Propositions 18 and 17 imply \[\mathop{\mathrm{PV}}\sum_j\frac1{x_j-z}=i\pi\rho+o(1)\] uniformly in this disk, with the symmetric principal-value convention of Section 6. Furthermore, \[\mathop{\mathrm{PV}}\sum_{|x_j|>K}\frac1{x_j-z} =\mathop{\mathrm{PV}}\sum_{|x_j|>K}\frac1{x_j} +\sum_{|x_j|>K}\frac{z}{x_j(x_j-z)} =O(K^{-\chi}+T/K).\] The first estimate follows by partial summation in Lemma 16; the second uses its linear count bound and \(|z|=o(K)\). Thus \(F_r(\mathfrak m)=i+o(1)\) uniformly on \(|\mathfrak m-i|\le1/4\).

Rouché’s theorem applied to \(\mathfrak m-F_r(\mathfrak m)\) and \(\mathfrak m-i\) shows that (119) has exactly one zero, counted with multiplicity, in this disk for all large \(r\). The zero \(\mathfrak m_r\) is therefore simple, and \(\mathfrak m_r\to i\). The implicit function theorem continues it to a solution near \(\zeta=0\) of the full fixed-point equation, still with positive imaginary part. This solution is the physical branch. Indeed, every upper-half-plane solution \(\mathfrak m\), at \(\Im\zeta>0\), satisfies \[ \frac1{m_I}\sum_{j\in I} |(V_*)_{jj}-\zeta-t_*\mathfrak m|^{-2} =\frac{\Im \mathfrak m}{\Im\zeta+t_*\Im \mathfrak m}<\frac1{t_*}. \tag{120}\] If two distinct such solutions existed, subtracting their equations and dividing by their difference would give \[1=\frac{t_*}{m_I}\sum_{j\in I} \frac1{((V_*)_{jj}-\zeta-t_*\mathfrak m_1) ((V_*)_{jj}-\zeta-t_*\mathfrak m_2)}.\] Cauchy–Schwarz and (120) make the absolute value of the right-hand side strictly less than one. This contradiction proves uniqueness and identifies the analytic continuation. Its boundary value gives \(\rho_{\mathrm{fc},t_*}(0)=\Im \mathfrak m_r/\pi\to1/\pi\).

Finally, simultaneous replacement of \(V_*,t_*\) by \(c_rV_*,c_r^2t_*\) dilates the free-convolution law by \(c_r\). For example its transform is \(c_r^{-1}m_{\mathrm{fc},t_*}(\zeta/c_r)\), as follows directly from (112). Its density at zero is exactly \[c_r^{-1}\rho_{\mathrm{fc},t_*}(0)=\frac1\pi.\] Since \(c_r\to1\), the enlarged regularity domain proves (110) for \(c_rV_*\) with the original \(g,G\). The strict power margins preserve (113) for \(c_r^2t_*\), completing the proof. ◻

Proof of Theorem 1. Choose the fixed compact bulk interval \(J\) with \(I_{d,\kappa}\subset\operatorname{int}J\). The finitely many smallness choices in Proposition 2 and the subsequent arguments give one positive number \(w_0(d,\kappa)\). Fix any \(0<w<w_0(d,\kappa)\) and \(E\in I_{d,\kappa}\). Corollary 4 gives a continuous density on an open neighborhood of \(I_{d,\kappa}\), strictly positive there; write \(\rho=\rho_{d,w}(E)>0\).

Start with any sequence of admissible graph sizes tending to infinity. Proposition 10 supplies a further subsequence on which the unscaled microscopic measures \(\nu_n\) converge in law to a process \(\nu\) of the form used above. The graph point measure in the theorem is \(\mathcal D_\rho\nu_n\). Bounded continuity of its Laplace functional and (126) show that its Laplace expectation, along this further subsequence, tends to the Laplace expectation of \(\mathcal S_{\mathrm{GOE}}\). The reference GOE expectations at dimension \(n\) tend to the same value, along all sizes. Every original subsequence therefore has a further subsequence along which their difference tends to zero, which proves the required full-sequence limit. All spectral estimates and negligible exclusions used to construct the subsequential law were events in the original simple-graph model; no conditioning on connectedness, bipartiteness, or any additional graph property enters the conclusion. The constants in estimates that use spectral averaging may depend on this fixed positive \(w\). The argument therefore proves the assertion for every fixed \(w\in(0,w_0(d,\kappa))\), with the limit in graph size taken afterward. ◻