Proof. Expand each determinant by permutations. Since \(W\) has zero diagonal, the off-diagonal terms are collections of disjoint nontrivial permutation cycles. Each such cycle \(C\) has sign minus and weight \[ w_C=\left(\prod_{(v,w)\in C}W_{vw}\right) \sum_{i=1}^r\prod_{v\in C}t_{vi} \tag{4}\] in the expectation; unused vertices contribute \(Y_v+a_v\). Indeed, all vertices on a cycle must receive one common color, and disjoint cycles involve disjoint independent choices.

In the product on the right of (3), suppose a vertex occurs in nontrivial cycles of \(k\) different determinant factors. Its degree in \(Y_v+a_v\) is \(r-k\). If \(k\geq2\), the indicated derivative kills the term. If \(k=0\), the normalized derivative sends \((Y_v+a_v)^r\) to \(Y_v+a_v\). If \(k=1\), it sends \((Y_v+a_v)^{r-1}\) to \(1/r\), cancelling the factor \(r\) from the matrix scaling at that vertex. The remaining terms are precisely (4) for disjoint cycles. The two square-root factors at each cycle vertex multiply to \(t_{vi}\), so this argument also proves the complex-parameter identity independently of the choices of square roots. This cycle calculation is algebraic in the entries of \(W\) and does not require them to be Hermitian; that hypothesis is used for the real-data stability conclusion, not for the identity.

For the half-plane assertion, proceed by induction on the subset \(D\) of vertices at which \(r-1\) derivatives have already been taken, simultaneously for all principal restrictions. Fix the other variables in \(\mathcal H\) and take a vertex \(v\notin D\). In each determinant, the coefficient of \(Y_v\) is its principal determinant with \(v\) removed, even for complex penalties. Extracting the coefficient of \(Y_v^r\) from the product commutes with all normalized derivatives at vertices in \(D\). Thus this coefficient is exactly the already-processed product for the principal set with \(v\) removed. It is nonzero by induction, so the polynomial in \(Y_v\) has degree exactly \(r\). Its roots lie in the closed convex complement of \(\mathcal H\). Gauss–Lucas implies that each of its successive derivatives has all roots in that complement and is nonzero. This proves the induction step.

For completeness, the determinant hypothesis holds whenever \(\mathcal H\) lies strictly beyond a numerical-range bound for \[ B_i=r t_i^{1/2}Wt_i^{1/2}-\mathop{\mathrm{diag}}(a). \tag{5}\] For example, if \(\operatorname{Im}\left\langle B_i u,u\right\rangle\leq\beta\left\lVert u\right\rVert^2\) and \(\operatorname{Im}Y_v>\beta\) for all \(v\), a null vector of \(\mathop{\mathrm{diag}}(Y)-B_i\) contradicts the imaginary part of its quadratic form. The same reasoning applies after a common rotation of the half-plane; principal compressions inherit numerical-range bounds. For Hermitian \(W\) and real probabilities and penalties, the matrices (5) are Hermitian, giving upper-half-plane stability. Real coefficients and complex conjugation then give real-rootedness of the univariate specialization. ◻

Proof. Fix every variable except \(Y_v\) at \(z\) in the upper half-plane. By (2), division by \(g(z)\ne0\) produces a monic affine polynomial \(Y_v-\alpha(z)\). Its root cannot be in the upper half-plane by Lemma 6. Hence \(h(z)=z-\alpha(z)\) satisfies (6).

The rational function \(h\) has real coefficients and only real poles. Its upper-half-plane property rules out poles of order two or higher: the imaginary part of a real leading polar term of such order has both signs along rays approaching the pole from above. At an actual simple pole the residue must be negative, as seen by approaching vertically. The polynomial part is \(z+a_v\), by comparing the two highest coefficients of the monic polynomials \(f\) and \(g\). This gives (7). For repeated roots of \(g\), assign the total residue to one copy and pad with zero masses at the others. The Schur complement proves (8) and (9), first away from the poles and then as identities of rational functions. The first two moments follow from the first diagonal entries of \(B\) and \(B^2\). Expanding \(f/g\) at infinity, the coefficient of \(1/z\) is minus the sum of the length-two cycle weights through \(v\), giving the final identity in (10).

For real \(x\) away from roots and poles, the Schur complement of \(xI-S\) in \(xI-B\) is \(h(x)\). Inertia is additive under this congruence, so the number of negative eigenvalues of \(xI-B\) exceeds that of \(xI-S\) by exactly \(\mathbf1_{\{h(x)<0\}}\). This proves the assertion about \(\xi_h\) and in particular \(0\leq\xi_h\leq1\) almost everywhere.

Factor \(f\) and \(g\) over their real roots. On the upper half-plane, take each factor \(z-\lambda\) with argument in \((0,\pi)\). The sum of the numerator arguments minus the denominator arguments equals \(\arg h\): their difference is a continuous multiple of \(2\pi\) and tends to zero at \(z=iR\) as \(R\to\infty\). The argument of \(x+i\eta-\lambda\), divided by \(\pi\), is the Poisson integral of \(\mathbf1_{(-\infty,\lambda)}\). Summing and subtracting gives (12). Finally the chain rule and (2) give the first identity in (13). The second follows by comparing the finite Stieltjes transforms on the two sides.

Pinning changes only the probability at \(v\), so \(g\) is unchanged and \(f\) is the weighted sum of its pinned versions. This gives the first identity in (14). Each pinned polynomial is stable by Lemma 6, and thus has the same type of quotient representation. Comparing residues at each distinct pole gives the second identity; use the same placement on repeated copies for every choice. Finally changing the \(v\)-penalty adds \((K-1)a_v g\) to the polynomial, proving (15). ◻

Proof. Set \(D_0=I+T\). Then \(0\leq D_0\leq2I\) and \((D_0)_{vv}=1\). Independently form the columns \[u_v=D_0^{1/2}e_v\otimes e_{\mathrm{color}(v)} \quad\text{in }\mathbb C^Q\otimes\mathbb C^r.\] They have squared norm one, their Gram matrix is \(I+T_{\mathrm{pav}}\), and, with \(d=2/r\), \[\sum_{v\in Q}\mathbb Eu_vu_v^*=(D_0/r)\otimes I_r\leq dI.\] Write the positive defect as a finite sum of deterministic outer products. Splitting its positive eigenvalues into finitely many summands at most one makes each added column have squared norm at most one. For all original and added columns together, their covariances \(A_j\) satisfy \(\sum_j A_j=dI\) and \(\mathop{\mathrm{Tr}}A_j\leq1\).

Theorem 4.1 of [12] identifies the expected characteristic polynomial of a sum of independent, finitely supported rank-one outer products with the mixed characteristic polynomial of the covariances. Theorem 5.1 of [12] gives largest root at most \((1+\sqrt{\epsilon})^2\) for positive semidefinite covariances summing to \(I\) with individual traces at most \(\epsilon\). Apply these results to \(A_j/d\), with \(\epsilon=1/d\), and rescale the spectral variable. The expected characteristic polynomial of the enlarged sum has largest root at most \[ d(1+\sqrt{1/d})^2=1+2\sqrt d+d. \tag{19}\]

We justify carefully the removal of the added columns. For independent, finitely supported columns \(u_1,\ldots,u_m\) in a finite-dimensional space, let \(A_j=\mathbb Eu_ju_j^*\) and let \(U\) be the matrix of columns. Then \[ F(Y):=\mathbb E\det\bigl(\mathop{\mathrm{diag}}(Y)-U^*U\bigr) =\left. \left(\prod_{j=1}^m(Y_j-\partial_{s_j})\right) \det\left(I+\sum_{j=1}^m s_j A_j\right) \right|_{s=0}. \tag{20}\] Indeed, the coefficient indexed by a subset \(J\) of columns on either side is \((-1)^{|J|}\) times the expected principal Gram determinant \(\mathbb E\det(U_J^*U_J)\). This follows first for deterministic rank-one matrices by determinant expansion, and then for the mean outer products by independence and multilinearity in the distinct matrices indexed by \(J\).

The determinant on the right of (20) is stable in the variables \(s_j\). If it had a null vector with all \(\operatorname{Im}s_j>0\), taking the imaginary part of its quadratic form would force the vector into \(\ker A_j\) for every \(j\); the identity term then forces the vector to be zero. If a polynomial \(p\) is stable, fixing all variables except \(s_j\) at upper-half-plane arguments gives \[\frac{\partial_{s_j}p}{p}= \sum_k\frac{1}{s_j-\lambda_k},\qquad \operatorname{Im}\lambda_k\leq0.\] The sum is zero when the degree is zero, and otherwise has negative imaginary part. Thus a new upper-half-plane variable \(Y_j\) cannot equal this logarithmic derivative, and \((Y_j-\partial_{s_j})p\) is stable. Repeating this observation, and then specializing the \(s_j\) to zero by the zero-free limit theorem, proves stability of \(F\). The limit is not the zero polynomial, since its leading monomial \(\prod_jY_j\) has coefficient one. The same proof works for every principal set of columns.

Differentiating \(F\) in a diagonal variable deletes the corresponding column from the expected Gram polynomial. The monic affine argument of Lemma 7 therefore applies to the quotient of its univariate specialization by the deleted polynomial. It has imaginary part at least \(\operatorname{Im}z\), and its polynomial part is \(z-\mathbb E\left\lVert u_j\right\rVert^2\). The residue argument in that Lemma realizes the full polynomial as the characteristic polynomial of a Hermitian arrow matrix whose principal compression has the deleted polynomial as characteristic polynomial. The largest eigenvalue of a principal compression cannot exceed the largest eigenvalue of the whole matrix. Thus deleting columns cannot increase the largest root.

If the ambient column space has dimension \(q\), the exact identity \[z^q\mathbb E\det(zI_m-U^*U) =z^m\mathbb E\det(zI_q-UU^*)\] shows that the enlarged Gram polynomial and the enlarged sum polynomial differ only by zero roots. Since the bound (19) is positive, it also bounds the largest root of the enlarged Gram polynomial. Delete every added column using the preceding paragraph. The original Gram polynomial is \(\mathbb E\det(zI-(I+T_{\mathrm{pav}}))=f_Q(z-1)\). Shifting by minus one proves (18). A common penalty \(b\) replaces \(f_Q(z)\) by \(f_Q(z+b)\), giving the final assertion. ◻

Proof. The first spectral moment is \(-1\), so \(H=1+2m\geq1\). By Markov’s inequality, \[\gamma([-4H,4H])\geq3/4.\] Writing \(G=1/h\), we obtain \[-\operatorname{Im}G(i10H) =\int\frac{10H}{x^2+100H^2}\,d\gamma(x) \geq\frac{15}{232H}.\] Since \(\operatorname{Im}(1/G)\leq1/(-\operatorname{Im}G)\), \[10H+\sum_j\frac{10H\nu_j}{s_j^2+100H^2} =\operatorname{Im}h(i10H)\leq\frac{232}{15}H<16H.\] For \(|s_j|\leq60H\), the denominator is at most \(3700H^2\). It follows that the left side of (23) is at most \(2220H^2\).

Let \(e\) be the first coordinate vector of the arrow matrix \(B\), and let \(e_j\) be the other coordinate vectors, so that \((B-s_j)e_j\) is a scalar multiple of \(e\) of modulus \(\sqrt{\nu_j}\). Fix \(\lambda\geq4H\) and \(s_j\geq2\lambda\). Spectral calculus on \((-\infty,\lambda]\), which is separated from \(s_j\), and \(\left\langle e_j,e\right\rangle=0\) give \[\begin{align*} \left|\left\langle e_j,\mathbf 1_{(\lambda,\infty)}(B)e\right\rangle\right| &=\left|\left\langle e_j,\mathbf 1_{(-\infty,\lambda]}(B)e\right\rangle\right|\\ &=\sqrt{\nu_j}\int_{x\leq\lambda}\frac{d\gamma(x)}{s_j-x} \geq\frac{\sqrt{\nu_j}}{2s_j}. \end{align*}\] The last inequality uses \(\gamma([-\lambda,\lambda])\geq3/4\) and \(s_j+\lambda\leq3s_j/2\). Bessel’s inequality now yields \[\sum_{s_j\geq2\lambda}\frac{\nu_j}{s_j^2} \leq4\gamma((\lambda,\infty)).\] Integrating for \(\lambda\geq4H\) gives \[\sum_{s_j\geq8H}\frac{\nu_j}{s_j^2} \left(\frac{s_j}{2}-4H\right) \leq4\int_{4H}^\infty\gamma((\lambda,\infty))\,d\lambda \leq4m.\] For \(s_j\geq20H\) the summand on the left is at least \(3\nu_j/(10s_j)\). Thus (24) holds with constant \(40/3\). The same positive-tail calculation applied to \(-B\) has first absolute moment \(H\) and positive first moment \(\int(-x)_+\,d\gamma(x)=1+m\leq H\). It gives (25) with the same constant. These calculations include zero residues and repeated poles. ◻

Proof. Merge coincident poles and omit zero weights. If there are no remaining poles the assertion is immediate. Otherwise list them in increasing order. The rational function is strictly decreasing between poles. Immediately to the right of each pole it tends to \(+\infty\); before the next pole it tends to \(-\infty\), and after the last pole it tends to zero. Thus the positive-level exceedance consists of one interval from each pole to the ensuing root of the level equation. Clearing denominators in \(\sum_jw_j/(\lambda-s_j)=c\), comparison of the two highest coefficients shows that the sum of the level roots exceeds the sum of the poles by \((\sum_jw_j)/c\). This is precisely the total interval length. Reflection of the variable gives the negative-level identity. With positive poles the rational function is nonpositive on \((-\infty,0]\), proving the last assertion. Values at the finitely many poles and endpoints do not affect these lengths. ◻

Proof. Dividing the arrow matrix by \(a\) sends \(s_j\) to \(s_j/a\), the masses to \(\nu_j/a^2\), and both lengths and \(m\) to their values divided by \(a\). It therefore suffices to prove the assertion for \(a=1\). Write \(L=L(1,\nu)\) and \(H=1+2m\), and fix \(C_0\) from Lemma 9. Only positive-probability choices are used below.

Suppose first that \(m\geq1\). The means of the two nonnegative statistics \[A_i=\sum_{|s_j|\leq20H}\nu_{ij},\qquad B_i=\sum_{|s_j|>20H}\frac{\nu_{ij}}{|s_j|}\] are at most \(C_0H^2\) and \(C_0(m+H)\leq2C_0H\), respectively. Markov’s inequality and a union bound give a choice satisfying \[ A_i\leq C_2H^2,\qquad B_i\leq C_2H, \qquad C_2=8C_0. \tag{28}\] For \(\lambda>DH\) and \(D\geq40\), all denominators from poles \(|s_j|\leq20H\) are positive and their total contribution is at most \(C_2H^2/(\lambda-20H)\leq C_2H\). The contribution from poles \(s_j<-20H\) is at most \(C_2H\) as well. Put \(C_3=2C_2\). For the remaining positive poles the identity \[ \sum_{s_j>20H}\frac{\nu_{ij}}{\lambda-s_j} = (\lambda+H)\sum_{s_j>20H} \frac{\nu_{ij}}{(s_j+H)(\lambda-s_j)} -\sum_{s_j>20H}\frac{\nu_{ij}}{s_j+H} \tag{29}\] holds. Choose \(D\geq\max\{40,2C_3+1\}\). For any \(K\geq1\), if the entire rational sum exceeds \(\lambda+K\), then (29) forces \[\sum_{s_j>20H}\frac{\nu_{ij}}{(s_j+H)(\lambda-s_j)}>\frac12.\] Indeed the necessary lower bound for this sum is at least \((\lambda-C_3H)/(\lambda+H)\geq1/2\). By Lemma 10 and (28), its exceedance length is at most \[2\sum_{s_j>20H}\frac{\nu_{ij}}{s_j+H}\leq2C_2H.\] Including the initial interval \((0,DH)\), we obtain \[ L(K,\nu_i)\leq(D+2C_2)H\leq3(D+2C_2)m. \tag{30}\] This proves the required bound in this case, with a universal constant, for every \(K\geq1\).

It remains to treat \(0\leq m<1\), so \(1\leq H\leq3\). Set \[P=\sum_{s_j>0}\frac{\nu_j}{1+s_j},\qquad M_-(d)=\sum_{s_j\leq0}\frac{\nu_j}{d+|s_j|}\quad(d>0),\] using the right limit \(M_-(0+)\) when necessary. Splitting the positive poles at \(60\) and using Lemma 9 gives \[ P\leq9C_0+C_0m\leq C_4, \qquad C_4=10C_0. \tag{31}\] Here \(60\leq60H\) controls the near part and \(60\geq20H\) controls the tail. For \(\lambda>0\) away from the poles, the parent exceedance condition is equivalent to \[ R(\lambda):=\sum_{s_j>0} \frac{\nu_j}{(1+s_j)(\lambda-s_j)} >1+\frac{P-M_-(\lambda)}{1+\lambda}. \tag{32}\] The threshold on the right is at most \(1+P\). All poles of \(R\) with positive weight are positive. Lemma 10 therefore gives \[ L\geq\frac{P}{1+P},\qquad P\leq C_5L,\qquad C_5=1+C_4. \tag{33}\]

We next prove the uniform bound \[ M_-(L/2)\leq C_6, \tag{34}\] with the right-limit interpretation when \(L=0\). If \(L\geq1\), split the negative poles at absolute value \(60H\). The near part is at most \(2C_0H^2\leq18C_0\), while the far part is at most \(C_0H\leq3C_0\). Thus \(21C_0\) suffices in this case. If \(L=0\), then (33) implies \(P=0\). If \(M_-(0+)>1\), including an infinite right limit, then \(M_-(\lambda)>1+\lambda\) on a nonempty interval immediately to the right of zero. With no positive-pole contribution this would be an interval of parent exceedance, contradicting \(L=0\). Hence \(M_-(0+)\leq1\).

Finally suppose \(0<L<1\). If \(M_-(2L)>7C_5+9\), then for \(0<\lambda<2L\) we have \(P\leq C_5\), \(M_-(\lambda)\geq M_-(2L)\), and \(1+\lambda\leq3\). Consequently the right side of (32) is strictly less than \(-2(C_5+1)\). By the negative-level identity in Lemma 10, \[\left|\{\lambda\in\mathbb R:R(\lambda)<-2(C_5+1)\}\right| =\frac{P}{2(C_5+1)}<\frac L2.\] Outside this exceptional set, the parent exceedance holds throughout \((0,2L)\), apart from finitely many poles. Its length is therefore greater than \(3L/2\), contradicting the definition of \(L\). It follows that \(M_-(2L)\leq7C_5+9\). Termwise comparison of denominators gives \(M_-(L/2)\leq4M_-(2L)\). Thus (34) holds in all cases with, for instance, \[ C_6=\max\{21C_0,1,28C_5+36\}. \tag{35}\]

For each choice define \(P_i\) and \(M_{-,i}\) by replacing \(\nu\) with \(\nu_i\). Averaging these nonnegative quantities, followed by Markov’s inequality and a union bound, gives a single choice \(i\) with \[ P_i\leq4P,\qquad M_{-,i}(L/2)\leq4C_6,\qquad \sum_{s_j>60}\frac{\nu_{ij}}{s_j}\leq4C_0m. \tag{36}\] There is no exception when one of the means is zero: the corresponding nonnegative statistic then vanishes at every positive-probability choice. When \(L=0\), finiteness of \(M_-(0+)\) forces \(\nu_j=0\) at every pole \(s_j=0\). The nonnegative averaging identities then force \(\nu_{ij}=0\) at those poles for every positive-probability choice. The right-limit child statistics are therefore finite sums over strictly negative poles, and their averages equal the parent statistic. Thus the same selection argument applies at \(L=0\) and at \(m=0\).

For this choice put \(P_i^{(K)}=\sum_{s_j>0}\nu_{ij}/(K+s_j)\). The child’s exceedance condition is \[ \sum_{s_j>0}\frac{\nu_{ij}}{(K+s_j)(\lambda-s_j)} >1+\frac{P_i^{(K)}-M_{-,i}(\lambda)}{K+\lambda}. \tag{37}\] For \(\lambda>L/2\) the negative statistic is at most \(4C_6\) by monotonicity. Therefore \(K\geq8C_6\) makes the right side of (37) at least \(1/2\). The length of the child exceedance beyond \(L/2\) is at most \(2P_i^{(K)}\) by Lemma 10. Split the positive poles at \(60\). For \(0<s_j\leq60\), \[\frac{1}{K+s_j}\leq\frac{61}{K}\frac{1}{1+s_j},\] whereas for \(s_j>60\) it is at most \(1/s_j\). Thus \[ 2P_i^{(K)} \leq\frac{122}{K}P_i +2\sum_{s_j>60}\frac{\nu_{ij}}{s_j} \leq\frac{488C_5}{K}L+8C_0m. \tag{38}\] Choose once and for all \[K\geq\max\{1,8C_6,976C_5\}.\] The first term in (38) is at most \(L/2\). Adding the possible length \(L/2\) of the initial interval gives \(L(K,\nu_i)\leq L+8C_0m\). Together with (30), this proves (27) with \(C\geq\max\{8C_0,3(D+2C_2)\}\). Scaling back to arbitrary \(a>0\) completes the proof. ◻

Proof. We first establish locality directly at the level of formal Laurent series, so that the assertion does not depend on any limiting argument. For a finite principal set \(Q\) and \(v\in Q\), put \(H_{v,Q}=h_{v,Q}/z\). Expanding the determinant expectation according to the singleton or nontrivial permutation cycle containing \(v\) gives \[ H_{v,Q}=1+\frac{a_v}{z} -\sum_{\substack{C\ni v\\ |C|\geq2}} w_Cz^{-|C|} \prod_{x\in C\setminus\{v\}}H_{x,Q(x)}^{-1}. \tag{46}\] Here \(w_C\) is the cycle weight in Equation (4). Delete \(v\) first and then the other vertices of \(C\) in a fixed order; \(Q(x)\) denotes the principal set remaining just before \(x\) is deleted. The inverse quotients in the product telescope to \(f_{Q\setminus C}/f_{Q\setminus\{v\}}\) after the displayed powers of \(z\) are accounted for. Each oriented permutation cycle is counted once, with the convention of the finite expansion.

All series \(H\) have constant term \(1\). A coefficient of order at most \(k\) in a term coming from a cycle of length \(l\) uses only coefficients through order \(k-l\) in its inverse factors. Induction on \(k\) now proves the radius-\(k\) locality assertion: the cycle has length at most \(k\), and the further radii of the needed factors stay within distance \(k\) of \(v\). Formal inversion, multiplication, and differentiation preserve this property. This proves locality for all the listed series and for their perturbation derivatives, including complex perturbations; only the algebraic finite polynomial identities were used. The expansion of \(zG_v\) is the moment expansion of \(\gamma_v\), proving the moment assertion.

For real data, the measures \(\gamma_v\) are probability measures with support in the common compact interval \([-B,B]\). The residue measures have support there and bounded mass for each fixed vertex, uniformly in the finite principal subset. Locality stabilizes all their moments: for the residue measure use the Laurent expansion of \(h_v(z)=z+a_v-\int(z-x)^{-1}\,d\nu_v(x)\). Compact-support moment determinacy proves unique weak convergence. It also gives locally uniform convergence of \(h_v,G_v\) and their pin alternatives on \(\mathbb C\setminus[-B,B]\), in particular on the upper half-plane. Their limits satisfy \(\operatorname{Im}h_v(z)\geq\operatorname{Im}z\).

For the finite spectral-shift functions \(\xi_h\), we have \(0\leq\xi_h\leq1\). Every weak-* subsequential limit therefore has these bounds. The identity \[\pi^{-1}\arg h(x+i\eta)=(P_\eta*\xi_h)(x)\] passes to the limit, since the Poisson kernel belongs to \(L^1(\mathbb R)\) and the analytic functions converge. Poisson approximation uniquely determines the limit function from these identities. Thus the entire sequence converges weak-*; all functions vanish above \(B\), so their integrals over the positive axis converge. The integrals defining \(m\) also converge by the compact support of \(\gamma\). Apply this reasoning to every pin alternative. The inequalities in Equation (45) and Lemma 11 pass to the limit, the latter because there are only finitely many choices of color.

Finite graph balls and the corresponding finite polynomial computations are measurable in the countable Borel relation. Their limits are therefore measurable. The same construction applies to single-vertex response functions indexed by pairs in an orbit. Components disconnected in the distance graph do not affect these local functions, so they can be treated separately. Throughout this construction the heights and weights are absolute on the lifted orbit; they are not reset when the root changes.

Finally suppose all vertices are unassigned. On a finite set \(Q\), the weighted polynomial at common argument \(z\) is \(\prod_{v\in Q}d_v\) times the unweighted mixed polynomial for \(T\), with zero penalties, evaluated at the tuple \((b+z/d_v)_{v\in Q}\). Its multiaffine Taylor coefficients at the constant tuple \(b\) are the corresponding smaller principal polynomials evaluated at \(b\). All are strictly positive by Equation (41) and Lemma 8, including the empty polynomial \(1\). Hence the weighted polynomial is strictly positive for every real \(z\geq0\). All its roots are real, so its local spectral measures have no positive support. Taking the local limits gives \(m_u=0\). ◻

Proof. We prove the analytic comparisons first and then recover the integral data. The transform \(G\) determines the local probability measure; \(h=1/G\) and \(R_i=h_i/h\) then determine the parent and pinned spectral shifts. For high deletions we test their integrated differences against bounded complex directions and express a single deletion response as a logarithmic derivative. Stability bounds that derivative independently of height. We transport its local Laurent coefficients and use analytic uniqueness to recover the whole-deletion response. Low deletions instead cut a matrix of small norm. All pre- and post-deletion systems use the same absolute heights, penalties, and retained probabilities.

Proof. The normalized tuple obeys the exact translation rule \[ \mathcal D^I(v,s+\alpha)=\mathcal D^{I-\alpha}(v,s). \tag{55}\] Indeed translating the heights by \(\alpha\) multiplies all weights \(d\), the matrix \(W\), and the penalties in the orbit by \(e^\alpha\). The spectral quantities \(m,\ell,\ell_i'\) scale by this same factor, which is canceled by the normalization at the root. The lifted probabilities are unchanged because the partial assignment is on the base. This proves Equation (55) first in finite balls and then in the local limits.

Take a large integer \(R\), use the reference interval \([-R,R]\), and write its tuple as \(\mathcal D^0\) in this proof. For \(\alpha\in[-1,1]\), compare this interval and its translate \([-R,R]-\alpha\) to their intersection. The differences involve only far high and low deletions. Equations (55) and Lemma 14 imply \[ \int_{v\ \mathrm{unassigned}} \int_{\{s\in[0,1]:\ s+\alpha\in[0,1]\}} \left|\mathcal D^0(v,s+\alpha)-\mathcal D^0(v,s)\right| \,ds\,d\mu(v)\leq e(R), \tag{56}\] where \(e(R)\to0\) uniformly in the assignment. We enlarge \(e(R)\) if necessary so that it is positive. Integrating in \(\alpha\) turns the left side into the corresponding integral over all pairs \((s,t)\in[0,1]^2\). Fubini therefore gives a single \(s_0\in[0,1]\) such that \[ \int_{v\ \mathrm{unassigned}}\int_0^1 \left|\mathcal D^0(v,s)-\mathcal D^0(v,s_0)\right| \,ds\,d\mu(v)\leq3e(R). \tag{57}\] The slice can also be chosen so that the local data and the buffered pin inequality hold almost everywhere on it.

At this slice choose the least minimizing color for \(\ell_{(v,s_0),i}'\). This is a measurable function \(i(v)\) because there are finitely many colors. Lemma 11 gives \[e^{-s_0}\bigl(\ell_{(v,s_0),i(v)}'-\ell_{(v,s_0)} -Cm_{(v,s_0)}\bigr)\leq0.\] The tuple norm contains all color alternatives. Consequently Equation (57) implies that the integral of this same normalized difference over \(v\) and \(s\in[0,1]\), now using the data at \(s\), is at most a fixed constant times \(e(R)\).

Consider a unit band \([k,k+1]\) in \([0,L]\), where \(k\) is an integer with \(k\geq R\) and \(L-k\geq R\). Translate it to \([0,1]\); its ambient interval becomes \([-k,L-k]\). Trimming this interval to \([-R,R]\) removes only far high and low vertices. Lemma 14 and Equation (55) show that its tuple differs from \(\mathcal D^0(v,s)\) in integrated norm by at most \(e(R)\), after enlarging the same uniformly vanishing error function. Thus the \(\mu\,ds\) integral over this unit band of \[e^{-s}\bigl(\ell_{(v,s),i(v)}'-\ell_{(v,s)}-Cm_{(v,s)}\bigr)\] on unassigned vertices is at most a constant times \(e(R)\). All remaining boundary bands have total length at most \(2R+2\). Their normalized data are uniformly bounded by Equation (45).

Absolute costs against \(d\widetilde\mu\) are exactly normalized costs against \(d\mu\,ds\). Summing the unit-band estimates and dividing by \(L\) bounds the total error by \(C_1e(R)+C_2(R+1)/L\), with constants independent of the assignment. Choose \(R\) sufficiently large, then choose \(L\) sufficiently large and satisfying any prescribed lower bound. This makes the error at most \(\tau_0\). Finally \(m\geq0\) permits replacing its integral over unassigned vertices by the integral over the whole slab, yielding Equation (54). All choices of the numerical parameters used only uniform estimates, proving the assertion about using the same \(L\) at different stages. Null modifications of base data can be absorbed into an invariant null set by nonsingularity. ◻

Proof. We first construct a partial assignment with small mixed-polynomial cost. Polynomial moments depend on finite graph balls, so sufficiently sparse simultaneous pins have an additive first-order response except on an event of quadratic order in the update probability. Approximating the derivative of the positive-part function controls the error in that response independently of the number of roots. Parameters used to select the slab will be specified below. On a fixed slab \([0,L]\), write \[\Phi=\frac1L\int_{V\times[0,L]}m_u\,d\widetilde\mu(u).\] Choose a common spectral support bound \([-B,B]\) for all the finite principal systems, pin alternatives, and deletion configurations on this slab, as in Lemma 13. For \(\tau>0\), choose a real polynomial \(P\) such that \[ P(-B)=0,\qquad \int_{-B}^B\left|P'(x)-\mathbf 1_{(0,\infty)}(x)\right|\,dx\leq\tau. \tag{60}\] For example, approximate the indicated step function first by a continuous function in \(L^1([-B,B])\), then uniformly by a polynomial, and integrate that polynomial from \(-B\). It follows that \[\sup_{[-B,B]}\left|P(x)-x_+\right|\leq\tau.\] Define \[m_{P,u}=\int P(x)\,d\gamma_u(x),\qquad \Phi_P=\frac1L\int_{V\times[0,L]}m_{P,u}\,d\widetilde\mu(u).\] Since \(\gamma_u\) is a probability measure and \(\widetilde\mu(V\times[0,L])=1-e^{-L}\leq1\), we have \[ \left|\Phi-\Phi_P\right|\leq\tau \qquad(L\geq1). \tag{61}\]

Proof. Put \(g(x)=(x-(Kb+\sigma))_+^2\). Since the paved operator is a compression by mutually orthogonal color projections, it remains a self-adjoint contraction; in particular \(0\leq g(T_{\mathrm{pav}}) \leq\mathbf 1\).

For an assigned base vertex \(v\), let \(B_k(v)\) be its radius-\(k\) ball in the base graph, and let \(T_v\) be the finite compression of the paved matrix to the assigned vertices of this ball. Polynomial diagonal moments through degree \(k\) at \(v\) are unchanged by this compression. Indeed every walk contributing to such a moment stays within \(B_k(v)\), and edges from assigned to unassigned vertices are zero after paving. Consequently \[(p(T_{\mathrm{pav}}))_{vv}=(p(T_v))_{vv}.\] Let \(\mathcal F\) be the measurable set of assigned vertices for which \(\lambda_{\max}(T_v)>Kb+\sigma\). Measurability follows by finite matrix evaluation on the Borel balls. If \(v\notin\mathcal F\) is assigned, then \(g(T_v)=0\) and the two polynomial approximation errors give \[(g(T_{\mathrm{pav}}))_{vv}\leq2\beta.\] On \(\mathcal F\cup\mathcal U\) its diagonal is at most \(1\). Therefore \[ \int_V(g(T_{\mathrm{pav}}))_{vv}\,d\mu(v) \leq2\beta+\mu(\mathcal U)+\mu(\mathcal F). \tag{71}\]

It remains to bound \(\mu(\mathcal F)\). Set \(I_{\mathrm{int}}=[k\log J,L-k\log J]\). For \(u=(v,s)\) with \(s\in I_{\mathrm{int}}\), the entire ball \(B_k(v)\) lifts into the slab: the height change along each graph edge has absolute value at most \(\log J\). Let \(Q_u\) project onto the assigned vertices of this lifted ball. With \(A_L\) as in (58), the order inequality \(A_L\leq(A_L)_+\) gives, after compression and diagonal congruence, \[ T_v-Kb\mathbf 1 \leq d_{\mathrm{ball}}^{-1/2}Q_u(A_L)_+Q_u d_{\mathrm{ball}}^{-1/2}. \tag{72}\] The base and lifted finite matrices here are identified up to the allowed diagonal unitary gauge, which does not affect the inequality or any diagonal spectral quantities. The right side is positive. If \(v\in\mathcal F\), its norm is greater than \(\sigma\) by (72), and hence its trace is greater than \(\sigma\). Equivalently, \[ \sum_{y\in B_k(u)\ \mathrm{assigned}} \frac{m_y^{\mathrm{surv}}}{d_y}>\sigma. \tag{73}\] Here \(B_k(u)\) denotes the lift of the base ball, which agrees with the radius-\(k\) enclosing lifted ball at these interior heights. The comparison uses only \(A_L\leq(A_L)_+\); it does not require positive part to commute with compression or diagonal congruence.

Integrate (73) against \(d\mu(v)\,ds/L\) over \(\mathcal F\times I_{\mathrm{int}}\), and enlarge the resulting nonnegative sum to all interior roots. Since \(d\mu(v)\,ds=d_u\,d\widetilde\mu(u)\), the resulting integrand is a sum of \((d_u/d_y)m_y^{\mathrm{surv}}\). On a radius-\(k\) ball, \(d_u/d_y\leq J^k\). Exchanging roots and targets by (43), each target \(y\) has at most \(N_k\) possible roots. Deleted vertices may be retained as locations for the enclosing-ball relation, with \(m_y^{\mathrm{surv}}=0\) there. Thus \[\sigma\frac{\left|I_{\mathrm{int}}\right|}L\mu(\mathcal F) \leq J^kN_k\Psi.\] Our lower bound on \(L\) ensures \(\left|I_{\mathrm{int}}\right|/L\geq1/2\), including the case \(J=1\). We conclude that \[\mu(\mathcal F)\leq\frac{2J^kN_k}{\sigma}\Psi.\] Combining this with (71) proves (70). ◻

Completion of the proof of Theorem 12. Fix the prescribed \(\sigma,\rho>0\). Choose \(\beta>0\) with \(2\beta<\rho/4\), and then choose the polynomial and its degree \(k\geq1\) in Lemma 17. Thus \(N_k\) is fixed before selecting the slab. Take \[\eta=\rho/4,\qquad \zeta=\frac{\rho\sigma}{8J^kN_k},\qquad L_0=\max\{1,4k\log J\}.\] Proposition 16 supplies a partial assignment and a slab of length at least \(L_0\) with \(\mu(\mathcal U)<\eta\) and \(\Psi<\zeta\). Give \(\mathcal U\) the additional color and apply Lemma 17. Its right side is strictly less than \(\rho\), as required.

Only the \(r\) assigned colors and one extra color occur in the final partition. The auxiliary graph colors, the slab length, the approximation degrees and the number of updates affect its construction but not this number of final colors. In particular the graph and cocycle bounds do not enter the condition on \(r\) in the theorem. ◻

Proof. Bounded Kaplansky approximation, followed by norm approximation inside the generating algebra, gives self-adjoint finite graph expressions converging strongly-star to \(Y\), with norms at most \(3\). Subtracting their \(A\)-expectations makes their diagonals zero and increases the norm bound to \(6\); state-norm contractivity of \(E_A\) preserves convergence. For each such expression, cut down by the base set on which every incident edge in its finite-degree enclosing graph has cocycle ratio between \(J^{-1}\) and \(J\). These sets increase to full measure as \(J\to\infty\), and the cuts converge strongly-star to the expression. A diagonal choice therefore gives self-adjoint \(Y_j\) with \[E_AY_j=0,\qquad \left\lVert Y_j\right\rVert\leq6,\qquad \left\lVert Y_j-Y\right\rVert_\varphi^\#\longrightarrow0,\] whose orbit matrices have bounded-degree support and bounded cocycle ratios, with the bounds allowed to depend on \(j\).

Apply Theorem 12 to \(Y_j/6\) and separately to \(-Y_j/6\), using state-error tolerances tending to zero. The rooted diagonal formula for the state says that the positive part above \(Kb+\sigma\) has \(\varphi\)-norm tending to zero in each case. These positive parts are self-adjoint and uniformly bounded, so they belong to the null ideal of Lemma 5. The sequence \(Y_j-Y\) belongs to that ideal as well. The resulting two quotient partitions, each of size \(r+1\), therefore give respectively \[C_{\mathcal P^+}[Y/6]\leq(Kb+\sigma)1, \qquad C_{\mathcal P^-}[Y/6]\geq-(Kb+\sigma)1.\] Take their common refinement. All partition projections commute, and compression is positive and unital, so both scalar bounds hold after refinement. This proves the lemma. Further finite \(A\)-refinement preserves the bound. ◻

Proof. Choose Borel representatives for the kernel. The set \[\mathcal T=\{(v,w):k_X(v,\{w\})>0\}\] is Borel and has countable sections in the second coordinate. For example, joint measurability of the singleton mass follows by using shrinking neighborhoods in a compatible separable metric on the standard Borel space. The countable-section uniformization theorem expresses \(\mathcal T\) as a countable union of graphs of partial Borel maps. If the conclusion fails, one such map \(F:S\to V\) has a domain of positive \(\mu\)-measure, with \[g(v):=k_X(v,\{F(v)\})>0\quad(v\in S).\] There is no assumption that \(F\) is injective. For a \(\mu\)-null Borel set \(B\subset V\), absolute continuity of the left marginal gives \[\int_{S\cap F^{-1}(B)}g(v)\,d\mu(v) \leq\nu_X(V\times B)=0.\] Thus \(F\) pulls back null sets to null sets. With \(e=1_S\in A\), composition with \(F\) consequently defines a normal \(*\)-homomorphism \[\theta:A\longrightarrow Ae,\qquad \theta(a)(v)=a(F(v))\quad(v\in S),\qquad \theta(1)=e.\] The graph will intertwine \(A\) with the possibly proper subalgebra \(\theta(A)\) of \(Ae\); its polar partial isometry need not yet normalize \(A\). We will use an expected type I intermediate algebra to split the right support into corners where these two abelian algebras coincide.

Apply the joint spectral projection of the graph of \(F\) to \(X\varphi^{1/2}\) and call the resulting vector \(\zeta\). It is nonzero and satisfies \[ \zeta e=\zeta,\qquad a\zeta=\zeta\theta(a)\quad(a\in A). \tag{81}\] Its right spectral density on \(S\) is \(g\). In particular, \(e\) is its smallest right supporting projection among the projections of \(A\): if \(f\in A\) is a projection and \(\zeta f=\zeta\), then \(e\leq f\).

Take the \(L^2\) polar decomposition \(\zeta=u h\), where \(h=|\zeta|\in L^2(M)_+\) and \(u\in M\) is a partial isometry. Put \(p=u^*u\leq e\) and \(q=uu^*\). We use here the usual polar decomposition and support properties of standard-form noncommutative \(L^2\); see, for example, [7]. For a unitary \(a\in A\), the operator \(b=\theta(a)+1-e\) is unitary and (81) reads \(a\zeta=\zeta b\). Taking absolute values gives \(h=b^*hb\). Uniqueness of the polar decomposition then gives \(au=ub=u\theta(a)\). Consequently, \[ h\theta(a)=\theta(a)h,\qquad p\theta(a)=\theta(a)p,\qquad au=u\theta(a)\quad(a\in A),\qquad q\in A. \tag{82}\] The identities for all \(a\) follow by linearity from unitaries. For the last assertion, the identities imply that \(q\) commutes with \(A\), and \(A\) is maximal abelian.

Consider the von Neumann algebra \[D=\theta(A)'\cap eMe.\] It contains \(p\) and \(Ae\), and \(Ae\) is a MASA in \(D\). Moreover \(\theta(A)\subset Z(D)\). If \(d\in pDp\), then \(udu^*\) commutes with \(Aq\), by (82), and hence belongs to \(Aq\). On the other hand, \[u\theta(a)p u^*=aq\quad(a\in A).\] Conjugation by \(u\) therefore identifies \(pDp\) with \(Aq\), and \[ pDp=\theta(A)p. \tag{83}\] Let \(c\) be the central support of \(p\) in \(D\). Since \(Z(D)\subset Ae\), the identity \(pc=p\) gives \(\zeta c=\zeta\); the minimal right \(A\)-support property gives \(c=e\). Thus \(D\) has an abelian projection with full central support and is type I. Finally, if \(d\in Z(D)\), then (83) gives \(dp=\theta(a)p\) for some \(a\in A\). The central element \(d-\theta(a)\) annihilates a projection of full central support and is zero. We have proved \[ Z(D)=\theta(A). \tag{84}\]

We next need a partition of \(e\) by projections \(e_j\in Ae\) such that \[ (Ae)e_j=\theta(A)e_j. \tag{85}\] The restriction of \(E_A\) to \(D\) is a normal conditional expectation onto \(Ae\). The type I MASA theorem of Akemann and Sherman [2] supplies a partition of \(e\) by projections in \(Ae\) that are abelian in \(D\). There are only countably many nonzero members, because \(D\) has the faithful normal state \(\varphi|_D/\varphi(e)\). For an abelian projection in a type I algebra, \[e_jDe_j=Z(D)e_j.\] This follows, for instance, by writing the type I algebra as a direct integral: an abelian projection has fiber rank at most one. Together with (84) and \(\theta(A)\subset Ae\subset D\), this proves (85).

Set \(\eta_j=\zeta e_j\) and take its polar decomposition \(\eta_j=u_j h_j\). Its right support is at most \(e_j\). Repeating (82) shows that \(h_j\) commutes with \(\theta(A)\) and that \[au_j=u_j\theta(a)\quad(a\in A).\] By (85) and the support condition, \(h_j\) commutes with all of \(A\). Therefore the right support \(p_j=u_j^*u_j\) belongs to \(A\); the left support \(q_j=u_ju_j^*\) belongs to \(A\) by the same polar argument as before. In addition, \[\theta(A)p_j=Ap_j,\qquad u_j(Ap_j)u_j^*=Aq_j.\] Thus each nonzero \(u_j\) is a groupoid normalizer and belongs to \(N\).

Let \(\psi_j\) be the normal positive functional whose positive implementing vector is \(h_j\). Since \(h_j\) commutes with \(A\), this functional centralizes \(A\). In particular \(\psi_j\circ C_{\mathcal Q}=\psi_j\) for every finite partition \(\mathcal Q\) in \(A\). Lemma 4, normality, and boundedness of the compression net imply \[\psi_j=\psi_j\circ E_A.\] Write \(f_j=d(\psi_j|_A)/d\mu\in L^1(V,\mu)_+\) and \(f_{j,m}=\min(f_j,m)\). The positive vectors \(f_{j,m}^{1/2}\varphi^{1/2}\) belong to \(A\varphi^{1/2}\) and converge in \(L^2\); indeed their squared pairwise distances are \[\int_V\big|f_{j,m}^{1/2}-f_{j,n}^{1/2}\big|^2\,d\mu.\] Their normal positive functionals are \(x\mapsto\int f_{j,m}E_A(x)\,d\mu\) and converge in norm to \(\psi_j\). The limit is therefore the unique positive implementing vector \(h_j\). It follows that \[h_j\in\overline{A\varphi^{1/2}},\qquad \eta_j\in\mathcal H_N:=\overline{N\varphi^{1/2}}.\] The right projections \(e_j\) are orthogonal and sum to \(e\), so \(\zeta=\sum_j\zeta e_j\in\mathcal H_N\), with convergence in \(L^2\).

The closed space \(\mathcal H_N\) reduces both the left and right actions of \(A\): for the right action, use \(n\varphi^{1/2}a=na\varphi^{1/2}\). It therefore reduces their joint spectral projections, including the graph projection defining \(\zeta\). But \(E_N(X)=0\) gives, for \(n\in N\), \[\varphi(n^*X)=\varphi(n^*E_N(X))=0,\] so \(X\varphi^{1/2}\perp\mathcal H_N\). Thus \(\zeta\perp\mathcal H_N\), contradicting \(\zeta\ne0\) and the membership just proved. ◻

Proof. Choose an increasing sequence of finite partitions \(\mathcal Q\) in \(D\) which generates \(D\), separates points of a conull standard model, and satisfies \[ \max_{q\in\mathcal Q}\phi_0(q)\longrightarrow0. \tag{86}\] Such a sequence is obtained by adjoining a countable separating family and, at each stage, a sufficiently small-mass finite refinement; the latter exists by diffuseness. Give its cells independent labels \(Z_q\) uniform on the \(n\)th roots of unity, and put \(w_{\mathcal Q}=\sum_q Z_q q\).

We first make the averaged reduced moments vanish by estimating return probabilities for diffuse kernels. We then control their fluctuations using the small masses of the partition cells, so that one deterministic labeling works for each prescribed finite list of moments.

Proof. Let \(E_0\) be the Hilbert-space projection onto the vacuum and all reduced words not starting in the centered part of the \(w\)-algebra. For a spectral projection \(p\) of \(w\), put \(t=1/n\). On this subspace, left multiplication by \(p\) is the sum of the scalar action \(t\) and creation by \(p-t1\). The two resulting subspaces are orthogonal, and \(\left\lVert p-t1\right\rVert_2^2=t(1-t)\); hence \[\left\lVert pE_0\right\rVert=\left\lVert E_0p\right\rVert=\sqrt{t}.\] Since \(U_{\mathrm f}\) is centered, it sends a word starting in the \(w\)-centered space to a word starting in the other factor. Therefore \(U_{\mathrm f}(1-E_0)=E_0U_{\mathrm f}(1-E_0)\), and \[pU_{\mathrm f}p =pU_{\mathrm f}E_0p+pE_0U_{\mathrm f}(1-E_0)p.\] Unitarity gives norm at most \(2\sqrt t\). The sum of the compressions is block diagonal for the mutually orthogonal \(p_j\), so its norm is the largest block norm. ◻

Proof of Proposition 20. By the atomic reduction following Lemma 21, it is enough to work on the central summand where \(A\) is diffuse. Normalize the state on that summand and retain the same notation. Put \(z=Z/2\), so \(z\) is a self-adjoint contraction and \(E_Nz=0\).

The ordinary two-by-two unitary dilation need not have a diffuse kernel over \(A\oplus A\): already at \(z=0\), its identity off-diagonal entry gives an atomic cross-kernel. We replace the second diagonal algebra by a free diffuse abelian algebra, turning the cross terms into normal functionals on diffuse spaces. This is the symmetry form of the free-product projection dilation in [16]: with \(y=(1+z)/2\), the symmetry below is twice that projection minus the identity. The kernel and moment lemmas above supply the required freeness in the present nontracial setting.

Let \((B_0,\tau_B)\) be a diffuse standard separable abelian probability algebra and form the reduced von Neumann free product \[(P,\psi)=(M,\varphi)*(B_0,\tau_B).\] The free-product state is faithful and normal, the two factor expectations preserve it, and its Hilbert-space representation is the reduced-word representation; see, for example, [8]. Both \(A\) and \(B_0\) remain MASAs in \(P\). We give the argument to make its nontracial applicability explicit. There are unitaries \(v_k\in A\) tending weakly to zero, since \(A\) is diffuse. They centralize \(\psi\), by the expectation onto \(M\). On \(L^2(P,\psi)\ominus L^2(M,\varphi)\), a dense family consists of vectors of words \[x_0 b_1 x_1\cdots b_hx_h,\qquad h\geq1,\] where \(b_i\in B_0\) are centered, the internal \(x_i\in M\) are centered, and the endpoints are arbitrary. To compute a matrix coefficient of conjugation by \(v_k\), take another such word with initial endpoint \(y_0\), and split the middle product \(y_0^*v_kx_0\) into its centered part and its scalar mean. The centered part gives a reduced word with zero expectation onto \(M\). The scalar part is \(\varphi(y_0^*v_kx_0)\) times an expression with a fixed norm bound and tends to zero. The additional right factor \(v_k^*\) in the conjugation does not change that norm bound. Conjugation is unitary on the state Hilbert space; density proves that it tends weakly to zero on the indicated complement. If an element of \(P\) commutes with \(A\), its vector after subtracting its \(M\)-expectation lies in this complement and is fixed by all these conjugations. It must be zero. The element belongs to \(M\), and then to \(A\). Interchanging the factors proves the assertion for \(B_0\).

In \(\widetilde M=\operatorname{Mat}_2(P)\) use the state \(\widetilde\varphi=\mathop{\mathrm{tr}}_2\otimes\psi\), with normalized matrix trace. The diagonal algebra \(D=A\oplus B_0\) is a diffuse MASA in its centralizer. Define \[ U=\begin{pmatrix}z&s\\s&-z\end{pmatrix},\qquad s=(1-z^2)^{1/2}. \tag{98}\] Since \(z\) commutes with \(s\), this is a self-adjoint unitary. Furthermore \(E_D(U)=0\). For \(a\in A\) and \(b\in B_0\), direct free-product expectation gives \[ E_D\bigl(U(a\oplus b)U\bigr) =\bigl(E_A(zaz)+\tau_B(b)E_A(s^2)\bigr) \oplus \bigl(\varphi(sas)1+\tau_B(b)\varphi(z^2)1\bigr). \tag{99}\] In particular, \(E_{B_0}(zbz)=\tau_B(b)\varphi(z^2)1\) because \(\varphi(z)=0\). Lemma 21 makes the first kernel \(a\mapsto E_A(zaz)\) diffuse. The other kernels in (99) are diffuse because the base algebras are diffuse and the displayed functionals are normal. Thus the kernel of \(a\mapsto E_D(UaU)\) is almost everywhere diffuse.

Apply Lemma 22 with \(\phi_0= \widetilde\varphi\). Let \(P_{j,l}\in D\), \(1\leq j\leq n\), be the spectral projections of the resulting finite-order \(w_l\), padded with zero projections when necessary, and set \[S_l=\sum_{j=1}^n P_{j,l}UP_{j,l}.\] Each projection is a polynomial in \(w_l\) and \(w_l^*\). Consequently the scalar moments of the self-adjoint contractions \(S_l\) tend to the moments of \(S_{\mathrm f}=\sum_jp_jU_{\mathrm f}p_j\) in the reduced free law. Let \(c=2/\sqrt n\). Lemma 23 gives \(\left\lVert S_{\mathrm f}\right\rVert\leq c\). Uniform polynomial approximation on \([-1,1]\), applied to the continuous function \(t\mapsto(\left|t\right|-c)_+^2\), gives \[ \left\lVert(\left|S_l\right|-c)_+\right\rVert_{\widetilde\varphi}\longrightarrow0. \tag{100}\] This passage uses moments to obtain a state-norm estimate, not operator-norm convergence in \(\widetilde M\).

The sequence \((S_l)\) belongs to the multiplier algebra of Lemma 5: \(U\) is fixed and each \((P_{j,l})_l\) is a bounded centralizer sequence. Its excess in (100) is self-adjoint, so it belongs to the strong-star null ideal. Continuous functional calculus in the quotient gives \(\left\lVert[S_l]\right\rVert\leq c\).

Compress by the fixed first matrix projection \(e_{11}\). Write \(P_{j,l}=p_{j,l}\oplus b_{j,l}\), so the \((p_{j,l})_{j=1}^n\) form a partition in \(A\). The compressed sequence is \[e_{11}S_le_{11}=\sum_{j=1}^n p_{j,l}z p_{j,l}\] in the first corner. Its quotient norm upstairs is at most \(c\). Functional calculus there makes its absolute-value excess above \(c\) strong-star null. These compressed elements belong to \(M\), and the state on their corner restricts to \(\varphi/2\); hence the same excess is strong-star null for \((M,\varphi)\). The compressed sequence is also a multiplier sequence for \(M\), since its partition projections centralize \(\varphi\). Its original quotient norm is therefore at most \(c\). Multiplying by \(2\) proves (78) on the diffuse summand. Extending the same colors over the central summand where \(Z=0\) proves the full assertion. ◻

Proof of Theorem 18. Choose \[b=\sigma=\frac{\gamma}{24(K+1)},\] choose \(r\) with \(2\sqrt{2/r}+2/r<b\), and choose \(n\geq2\) with \(4/\sqrt n\leq\gamma/2\). Use Lemma 19 and Proposition 20, and take the common refinement of their partition sequences. It has \(R_{\mathrm{st}}(\gamma)=(r+1)^2n\) members after padding. Finite refinement preserves each quotient bound, so for \[T_j^0=C_{\mathcal P_j}(x-E_Ax)\] we have \(\left\lVert[T_j^0]\right\rVert\leq\gamma\), since the two contributions are at most \(\gamma/4\) and \(\gamma/2\).

Let \(\operatorname{clip}_\gamma(t)=\max(-\gamma, \min(t,\gamma))\) for real \(t\), and define \[w_j=T_j^0-\operatorname{clip}_\gamma(T_j^0).\] The quotient norm bound and continuous functional calculus show that \((w_j)\) has zero quotient image. It is self-adjoint, hence \(\left\lVert w_j\right\rVert_\varphi\to0\). Also \(\left\lVert T_j^0\right\rVert\leq2\), so \(\left\lVert w_j\right\rVert\leq2\). The operator \(T_j^0\), and therefore \(w_j\), is block diagonal for \(\mathcal P_j\). Since \(E_Ax\) belongs to \(A\), this gives \[C_{\mathcal P_j}(x-w_j)-E_Ax =T_j^0-w_j=\operatorname{clip}_\gamma(T_j^0),\] of norm at most \(\gamma\). These are all the assertions of (74). ◻

Proof. Let \(\varphi\) be a faithful normal state on \(M\), let \(A\subset M\) be a MASA in its centralizer, and fix a self-adjoint contraction \(x\in M\). Construct countable sets of generators as follows. Start with \(\mathbf 1,x\). At each stage form the unital \(\mathbb Q(i)\)-algebra generated by the current set and its adjoints. For every element \(t\) of this countable algebra, add \(E_A(t)\) and the projections of finite \(A\)-partitions \(\mathcal Q_{t,k}\) satisfying \[\left\lVert C_{\mathcal Q_{t,k}}(t)-E_A(t)\right\rVert_\varphi^\#<1/k, \qquad k\geq1.\] These partitions exist by Lemma 4. Repeat this procedure countably many times, let \(B\) be the union of the resulting countable algebras, and set \[M_1=W^*(B),\qquad A_1=A\cap M_1.\]

Every element of \(M_1\) is the limit of an operator-norm bounded sequence from \(B\) in \(\left\lVert\cdot\right\rVert_\varphi^\#\). Indeed, Kaplansky density gives bounded strong-star approximants from the norm closure of \(B\), and one can then approximate in norm by elements of \(B\). On bounded sets the state sharp seminorm determines the strong-star topology, so sequences suffice here. Taking real and imaginary parts separately gives this assertion for arbitrary elements. Since \(E_A\) is contractive for the state sharp seminorm and in operator norm, it follows that \(E_A(M_1)\subset A_1\).

Suppose \(z\in M_1\) commutes with \(A_1\). Choose bounded \(t_j\in B\) converging to \(z\) in the state sharp seminorm, and choose one of the stored partitions \(\mathcal Q_j\subset A_1\) for each \(t_j\) so that its compression error is less than \(1/j\). Since \(C_{\mathcal Q_j}(z)=z\), contractivity gives \[\left\lVert z-E_A(z)\right\rVert_\varphi^\# \leq 2\left\lVert z-t_j\right\rVert_\varphi^\# +\left\lVert C_{\mathcal Q_j}(t_j)-E_A(t_j)\right\rVert_\varphi^\# \longrightarrow0.\] Thus \(z=E_A(z)\in A_1\), proving that \(A_1\) is a MASA in \(M_1\). It lies in the centralizer of \(\varphi|_{M_1}\). The GNS representation of this restricted faithful normal state is faithful and normal, and its Hilbert space is separable because the countable set \(B\) applied to the cyclic vector is dense. Hence \(M_1\) has separable predual. The restriction of \(E_A\) is its state-preserving expectation onto \(A_1\). Applying Theorem 18 in \(M_1\) proves the assertion, including the unchanged bound on the partition size. ◻

Proof. In standard form, each finite partition \(\mathcal Q\) in \(D\) defines an orthogonal projection \[P_{\mathcal Q}\xi=\sum_{q\in\mathcal Q}q\xi q.\] These projections decrease under refinement to the projection onto the \(D\)-central vectors. A nonzero central vector \(\zeta\) would give a nonzero \(D\)-central normal positive functional \(a\mapsto\left\langle a\zeta,\zeta\right\rangle\): conjugation by a unitary of \(D\) can be transferred to the commuting right action. The hypothesis therefore implies \(P_{\mathcal Q}\to0\) strongly along the net of finite partitions.

Choose independent random labels \(Z_q\), uniform on the \(m\)th roots of unity, and put \(u=\sum_{q\in\mathcal Q}Z_q q\). For \(1\leq k<m\), independence and \(\mathbb EZ_q^k=0\) give \[\mathbb E\left\langle u^k\xi_\theta u^{-k},\xi_\theta\right\rangle =\left\langle P_{\mathcal Q}\xi_\theta,\xi_\theta\right\rangle \longrightarrow0.\] Each overlap is real and nonnegative, by the natural-cone properties. Consequently, for a sufficiently fine partition there is one choice of labels making all these overlaps arbitrarily small. The states \[\theta_j=\theta\circ\mathop{\mathrm{Ad}}(u^{-j}),\qquad 0\leq j<m,\] have natural-cone vectors \(\xi_j=u^j\xi_\theta u^{-j}\). Their pairwise overlaps are among those just considered. Equation (101) shows that \(\left\lVert\theta_j-\theta_k\right\rVert_{P_*}\) is arbitrarily close to \(2\) for every \(j\ne k\).

Fix a small \(\eta>0\). The positive support projection of \(\theta_j-\theta_k\) attains half the norm of that Hermitian functional, since its value at \(\mathbf 1\) is zero. Thus the labels can be chosen so that there are projections \(q_{jk}\in P\) with \[ \theta_j(q_{jk})\geq1-\eta,\qquad \theta_k(q_{jk})\leq\eta\qquad(j\ne k). \tag{103}\] For \(0<t<1\) put \[a_j=\frac1{m-1}\sum_{k\ne j}q_{jk},\qquad b_j=1_{[1-t,1]}(a_j).\] Spectral calculus and Equation (103) yield \[ \theta_j(b_j)\geq1-\frac\eta t,\qquad \theta_k(b_j)^{1/2}\leq\sqrt\eta+\sqrt{(m-1)t}\quad(k\ne j). \tag{104}\] For the first estimate, use \(\theta_j(1-a_j)\leq\eta\) and \(1-b_j\leq t^{-1}(1-a_j)\). For the second, the positive-summand inequality \(1-q_{jk}\leq(m-1)(1-a_j)\) gives \[\left\lVert(1-q_{jk})b_j\right\rVert^2 =\left\lVert b_j(1-q_{jk})b_j\right\rVert\leq(m-1)t.\] Splitting \(b_j\xi_k=b_jq_{jk}\xi_k+b_j(1-q_{jk})\xi_k\) proves the claim, using the adjoint of this operator-norm estimate.

The pairwise separating projections need not be mutually orthogonal, so they cannot simply be added to form a partition. Sequential products give positive effects with sum at most the identity while retaining almost full mass for their respective states. Define \[V_0=\mathbf 1,\qquad V_j=(1-b_{j-1})\cdots(1-b_0),\qquad s_j=V_j^*b_jV_j\quad(0\leq j<m).\] These positive elements need not be projections. They satisfy \[ s_j=V_j^*V_j-V_{j+1}^*V_{j+1},\qquad \sum_{j=0}^{m-1}s_j=\mathbf 1-V_m^*V_m\leq\mathbf 1. \tag{105}\] Telescoping the products of contractions also gives \[\left\lVert V_j\xi_j-\xi_j\right\rVert \leq\sum_{k<j}\left\lVert b_k\xi_j\right\rVert.\] By Equation (104), whenever \(\eta<t\), \[\left\lVert b_jV_j\xi_j\right\rVert \geq\sqrt{1-\eta/t} -j\bigl(\sqrt\eta+\sqrt{(m-1)t}\bigr).\] For fixed \(m\), choosing \(t\) small and then \(\eta\) sufficiently small makes every \(\theta_j(s_j)=\left\lVert b_jV_j\xi_j\right\rVert^2\) arbitrarily close to \(1\).

Set \[s=\frac1m\sum_{j=0}^{m-1}u^{-j}s_ju^j.\] Then \(0\leq s\leq\mathbf 1\) and \(\theta(s)\) is the average of the \(\theta_j(s_j)\), so it exceeds \(1-\alpha\) with the choices above. Since \(u^m=\mathbf 1\), Equation (105) gives \[ \sum_{\ell=0}^{m-1}u^\ell s u^{-\ell} =\frac1m\sum_{k=0}^{m-1}u^k \left(\sum_{j=0}^{m-1}s_j\right)u^{-k} \leq\mathbf 1. \tag{106}\] Let \(p_1,\ldots,p_m\) be the spectral projections of \(u\) for the \(m\)th roots, allowing zero projections. Compressing Equation (106) by \(p_i\) gives \(mp_i s p_i\leq p_i\), as required. ◻

Proof. For test vectors \(\xi_1,\ldots,\xi_d\), normalize the sum of their vector functionals and those of \(x\xi_1,\ldots,x\xi_d\) to obtain a normal state \(\theta\). If this sum vanishes, all tests are zero and one may take \(s=0\). Otherwise Lemma 25 makes \[\theta((1-s)^2)\leq1-\theta(s)\] arbitrarily small. Thus \(1-s\) is arbitrarily small on every test vector and its image under \(x\). The identity \[sxs-x=sx(s-1)+(s-1)x\] proves the strong approximation. Finally, \(-s^2\leq sxs\leq s^2\) and \(s^2\leq s\) imply \(-p_i/m\leq p_i sxs p_i\leq p_i/m\). The norm of the block diagonal sum is the maximum of its block norms. ◻

Proof. The support of an \(A\)-central normal positive functional is invariant under every unitary in \(A\), and hence belongs to \(A\) by maximal abelianness. Such a functional is faithful on its support corner, and remains faithful and centralizing upon restriction to any nonzero \(A\)-subprojection of its support.

Choose a maximal orthogonal family \((q_\beta)\) of nonzero projections in \(A\), each subordinate to the support of one such functional. Their sum is \(e\): a nonzero remainder under \(e\) would have nonzero intersection with one of the defining supports, contradicting maximality. On each \(q_\beta Mq_\beta\), normalize the restricted functional to a faithful normal state. The corner \(Aq_\beta\) is a MASA in its centralizer. Takesaki’s conditional expectation theorem [21], in the state formulation [7], gives a faithful normal expectation \(E_\beta:q_\beta Mq_\beta\to Aq_\beta\). Its hypotheses hold because the centralizer is fixed by the modular group and a state is finite on the subalgebra.

Define \[ E(z)=\sum_\beta E_\beta(q_\beta zq_\beta),\qquad z\in eMe. \tag{107}\] The orthogonal block sum is well defined, bounded in norm by \(\left\lVert z\right\rVert\), and belongs to \(Ae\). It is positive, unital, \(Ae\)-bimodular, and fixes \(Ae\). For a bounded increasing net of positive elements, normality of every \(E_\beta\) identifies the supremum of every \(q_\beta\)-block; this proves normality of \(E\), even for an uncountable family. If \(z\geq0\) and \(E(z)=0\), faithfulness on each corner gives \(q_\beta zq_\beta=0\), hence \(z^{1/2}q_\beta=0\) for all \(\beta\). Since \(\sum_\beta q_\beta=e\), this forces \(z=0\). Thus \(E\) is faithful.

An \(Ah\)-central normal positive functional on \(hMh\) would extend by \(z\mapsto hzh\) to an \(A\)-central one on \(M\) supported under \(h\). The definition of \(e\) excludes a nonzero such functional. All corner MASAs used here remain maximal abelian: an element in a corner commuting with the corner of \(A\), extended by zero, commutes with \(A\). ◻

Proof of Theorem 1. If \(x=0\), take \(y=a=0\) and the one-part partition, padded with zero projections to the size chosen below. Otherwise, by scaling it suffices to prove the assertion for \(\left\lVert x\right\rVert=1\), with an arbitrary strong-approximation tolerance \(\delta>0\). Fix \(0<\varepsilon<1\) and put \(\gamma=\varepsilon/4\). Let \(r_f\) be the partition-size bound \(R_{\mathrm{st}}(\gamma)\) in Theorem 18, and fix \(m\geq2\) with \(1/m\leq\gamma\). We shall use \[ R(\varepsilon)=r_f+m+1. \tag{108}\] More explicitly, the choices in the state-paving proof may be taken as \[b=\sigma=\frac{\varepsilon}{96(K+1)},\qquad r=\left\lceil\frac{32}{b^2}\right\rceil,\qquad n=\left\lceil\frac{1024}{\varepsilon^2}\right\rceil,\qquad m=\left\lceil\frac4\varepsilon\right\rceil.\] Here \(2\sqrt{2/r}+2/r\leq b/2+b^2/16<b\), \(4/\sqrt n\leq\gamma/2\), and \(1/m\leq\gamma\). Consequently \(r_f=(r+1)^2n\) and \[ R(\varepsilon)=(r+1)^2n+m+1 \leq C_{\mathrm{ap}}\varepsilon^{-6} \qquad(0<\varepsilon<1) \tag{109}\] for a universal constant \(C_{\mathrm{ap}}\). Only the universal buffer \(K\) enters this constant; graph, slab, and approximation parameters do not enter the color count. These choices precede the finite test set and its tolerance.

Let \(e,h\) and \(E\) be as in Lemma 27. Given a finite set \(F\) of test vectors, choose a unit vector \(\xi_0\) such that \(\left\lVert x\xi_0\right\rVert>3/4\), and put \(F'=F\cup\{\xi_0\}\). On \(Ae\) consider the normal positive functional \[\psi(a)=\sum_{\xi\in F'}\left\langle a e\xi,e\xi\right\rangle,\qquad a\in Ae,\] and let \(f=\mathop{\mathrm{supp}}(\psi)\in Ae\) and \(g=e-f\). The equality \(\psi(g)=0\) implies \(g\xi=0\) for every \(\xi\in F'\). Empty corners below are interpreted as zero.

If \(f\ne0\), then \(\psi|_{Af}\) is faithful. After normalizing it, compose it with the restriction \(E:fMf\to Af\) to obtain a faithful normal state \(\varphi_f\) on \(fMf\). Bimodularity of \(E\) and commutativity of \(Af\) show that \(Af\) is in the \(\varphi_f\)-centralizer. Apply Theorem 18 and Lemma 24 to the self-adjoint contraction \(x_f=fxf\). They give \[y_f=x_f-w,\qquad w=w^*,\quad\left\lVert w\right\rVert\leq2, \qquad \left\lVert C_{\mathcal P_f}(y_f)-d_f\right\rVert\leq\gamma, \quad d_f=E(x_f),\] where \(\mathcal P_f\) is an \(r_f\)-partition of \(f\), and \(w\) can be chosen arbitrarily small on the vectors \(f\xi\) for \(\xi\in F'\). Indeed, the bounded self-adjoint corrections in that theorem have state norm tending to zero, hence converge strongly in the given normal representation.

On \(g\) put \(y_g=0\). On \(h\) put \(x_h=hxh\) and use Corollary 26 to choose \(y_h=sx_hs\) arbitrarily close to \(x_h\) on the vectors \(h\xi\), \(\xi\in F'\), together with an \(m\)-partition \(\mathcal P_h\) of \(h\) such that \[\left\lVert y_h\right\rVert\leq1,\qquad\left\lVert C_{\mathcal P_h}(y_h)\right\rVert\leq\gamma.\]

Keep all off-diagonal blocks of \(x\) between \(f,g,h\), and replace its diagonal blocks by the operators just constructed. Explicitly, set \[y=y_f+y_g+y_h+ \sum_{\substack{b,c\in\{f,g,h\}\\b\ne c}}bxc.\] This operator is self-adjoint. The difference \(y-x\) is block diagonal: its \(f\)-block has norm at most \(2\), its \(g\)-block has norm at most \(1\), and its \(h\)-block has norm at most \(2\). Hence \[ \left\lVert y-x\right\rVert\leq2,\qquad\left\lVert y\right\rVert\leq3. \tag{110}\] The \(g\)-block of the difference annihilates \(F'\). Choosing the other two approximation errors sufficiently small therefore ensures \[ \left\lVert(y-x)\xi\right\rVert<\min\{\delta,1/4\}\qquad(\xi\in F'). \tag{111}\] In particular \(\left\lVert y\right\rVert>1/2\), by applying this inequality to \(\xi_0\).

Use the projections of \(\mathcal P_f\), the projections of \(\mathcal P_h\), and the single projection \(g\) as distinct members of one \(A\)-partition \(\mathcal P\), padding with zeros in absent corners. It has the size in Equation (108) and kills every off-diagonal block between \(f,g,h\). With \(d=d_f\) on \(f\) and zero on \(g+h\), we obtain \[d=d^*\in A,\qquad\left\lVert C_{\mathcal P}(y)-d\right\rVert\leq\gamma.\] Since compression is norm-contractive, \(\left\lVert d\right\rVert\leq\left\lVert y\right\rVert+\gamma\). Clip \(d\) by functional calculus to the interval \([-\left\lVert y\right\rVert,\left\lVert y\right\rVert]\), obtaining \(a\in A\) with \(\left\lVert a\right\rVert\leq\left\lVert y\right\rVert\) and \(\left\lVert a-d\right\rVert\leq\gamma\). It follows that \[\left\lVert C_{\mathcal P}(y)-a\right\rVert \leq2\gamma=\varepsilon/2\leq\varepsilon\left\lVert y\right\rVert.\] Equations (110) and (111) give the remaining requirements. The partition size depends only on \(\varepsilon\), since the bound in Theorem 18 is uniform over the state-preserving inclusions used here. Finally, scaling \(y\) and \(a\) by the original \(\left\lVert x\right\rVert\), and first dividing the requested strong tolerance by that norm, proves the theorem with the universal constant \(\kappa=3\). ◻

Proof of Corollary 2. By Theorem 1, every such \(x\) is \((\varepsilon/12,R(\varepsilon/12);3)\)-approximately pavable in the terminology of [16]. Apply [16] with \(\varepsilon'=\varepsilon/6\). Its error is \(2\cdot3\cdot(\varepsilon/6)=\varepsilon\), so \(R_s(\varepsilon)=R(\varepsilon/12)\) suffices. ◻

Proof of Corollary 3. By [13], every self-adjoint \(x\in M\) is \((\varepsilon/2,r_{\varepsilon/2})\) so-pavable, where \(r_{\varepsilon/2}\leq C\varepsilon^{-2}\) for a universal \(C\). Since \(\varepsilon/2<\varepsilon\), [16] makes \(x\) \((\varepsilon,r_{\varepsilon/2};3)\)-approximately pavable. Its proof supplies self-adjoint approximants and diagonal witnesses, with paving error relative to the approximant’s norm. The strong-limit net in [16] supplies the finite-vector conclusion with the fixed count \(N(\varepsilon)=r_{\varepsilon/2}\). ◻