Proof. Write \(\xi=h_\varphi^{1/2}\) and introduce the symmetric map \[T(z)=h_\varphi^{1/4}z h_\varphi^{1/4}\colon M\longrightarrow L^2(M).\] It is bounded and takes weak-star convergence to weak convergence: testing against \(\zeta\in L^2(M)\) uses the \(L^1\) element \(h_\varphi^{1/4}\zeta^*h_\varphi^{1/4}\). For a finite partition \(P\) in \(A\), the pinching \(S_P=\sum_{e\in P}e(\,\cdot\,)e\) also acts on \(L^2(M)\). There it is an orthogonal projection. These projections decrease under refinement to the projection onto the \(A\)-central vectors. Such a vector must be zero: otherwise its left vector functional would be a nonzero \(A\)-central normal positive functional. To see centrality, move a unitary of \(A\) from the left of the vector to its right, where it commutes with the left action of \(M\). Consequently \[\|S_P(\xi)\|_2\longrightarrow0.\] For \(z\in M\), cyclicity and Hölder’s inequality give \[ \sum_{e,f\in P}\|T(ezf)\|_2^2 =\mathop{\mathrm{Tr}}\bigl(z^*S_P(\xi)zS_P(\xi)\bigr) \leq\|z\|^2\|S_P(\xi)\|_2^2. \tag{2}\] In particular \(\|T(eze)\|_2^2\leq\|e\xi e\|_2^2\) when \(\|z\|\leq1\).

Fix \(\kappa>0\) and a coarse partition \(P_0\) with \(\|S_{P_0}(\xi)\|_2<\kappa\). A weak-star convergent subnet of \(S_P(x)\), as \(P\) refines \(P_0\), has a self-adjoint contraction limit \(b\in A\): the limit commutes with every partition in \(A\), hence belongs to the masa. Since the weak and norm closures of a convex subset of a Hilbert space agree, there are refinements \(P^1,\ldots,P^d\) of \(P_0\) and probabilities \(t_1,\ldots,t_d\) such that \[\left\|\sum_{j=1}^d t_jT(S_{P^j}(x))-T(b)\right\|_2<\kappa.\] For each \(e\in P_0\), independently choose \(j_e\) with this distribution and use the partition \(P^{j_e}\) inside \(e\). The resulting partition \(R\) refines \(P_0\). The mean of \(T(S_R(x))\) is the displayed convex combination. Independence and Equation (2) bound its variance by \[\sum_{e\in P_0}\|e\xi e\|_2^2 =\|S_{P_0}(\xi)\|_2^2<\kappa^2.\] Thus one realization satisfies \(\|T(S_R(x)-b)\|_2<\sqrt2\kappa\).

Now color the pieces of this fixed \(R\) independently and uniformly with \(r_{\rm p}\) colors, and let \(S_{\rm col}\) be the resulting color pinching. Its mean is \[\mathbb E S_{\rm col}(x) =\frac{x}{r_{\rm p}}+\left(1-\frac1{r_{\rm p}}\right)S_R(x).\] The indicators of equal colors for distinct unordered pairs of pieces are uncorrelated, even when the pairs share a piece. Applying \(\|u+v\|_2^2\leq2\|u\|_2^2+2\|v\|_2^2\) and Equation (2), the variance after applying \(T\) is at most \(2\|S_R(\xi)\|_2^2<2\kappa^2\). Therefore a coloring can be chosen so that the self-adjoint element \[y=S_{\rm col}(x)-\left(1-\frac1{r_{\rm p}}\right)b -\frac{x}{r_{\rm p}}\] satisfies \(\|T(y)\|_2<2\sqrt2\kappa\). Also \(\|y\|\leq2\). Since \(\kappa\) is arbitrary, this gives arbitrarily small symmetric seminorm with the number of colors already fixed.

It remains to convert this seminorm into a large compression. Set \(t=\|T(y)\|_2\) and \(\psi(z)=\varphi(y^*zy)\). Small \(t\) need not control \(\|y\|_\varphi\), so we first seek \(q_0\) with both \(\varphi(1-q_0)\) and \(\psi(q_0)\) small; these bounds will control \(\|q_0yq_0\|_\varphi\) for the spectral cut. Polar decomposition gives \(h_\psi=yh_\varphi y^*\) and \(h_\psi^{1/2}=y\xi u^*\) for a partial isometry \(u\in M\). Hence \[0\leq\mathop{\mathrm{Tr}}(\xi h_\psi^{1/2}) \leq\|\xi y\xi\|_1 =\|h_\varphi^{1/4}T(y)h_\varphi^{1/4}\|_1 \leq t.\] Here the first inequality uses the positive cone, and the last uses \(\|h_\varphi^{1/4}\|_4=1\). Let \(q_0\) support the positive part in the Jordan decomposition of \(\varphi-\psi\). Equation (1) implies \[ \varphi(1-q_0)+\psi(q_0) =\frac{\varphi(1)+\psi(1)-\|\varphi-\psi\|}{2} \leq t. \tag{3}\] Moreover \[\|q_0yq_0\xi\|_2 \leq\|q_0y\xi\|_2+\|q_0y(1-q_0)\xi\|_2 \leq\psi(q_0)^{1/2}+2\varphi(1-q_0)^{1/2} \leq3\sqrt t.\] Inside \(q_0Mq_0\), let \(q\) be the spectral projection of \(q_0yq_0\) for \([-\varepsilon/2,\varepsilon/2]\). The spectral Markov inequality gives \[ \varphi(1-q) \leq\varphi(1-q_0)+\frac4{\varepsilon^2}\varphi((q_0yq_0)^2) \leq\left(1+\frac{36}{\varepsilon^2}\right)t. \tag{4}\] Taking \(\kappa\) sufficiently small makes this less than \(\rho\). With \(a=(1-1/r_{\rm p})b\), we have \(\|a\|\leq1\) and \[\|q(S_{\rm col}(x)-a)q\| \leq\frac\varepsilon 2+\frac1{r_{\rm p}} \leq\frac{3\varepsilon}{4}.\] This proves the lemma. ◻

Proof. Let \(p\) be the supremum of the supports of the \(A\)-central normal positive functionals on \(M\). Each support belongs to \(A\), since conjugation by every unitary of \(A\) preserves the functional and its support. These supports form a directed family: the support of a sum is the join of the supports. The corner \((1-p)M(1-p)\) has no nonzero \(A(1-p)\)-central normal positive functional, since compression would extend one to an \(A\)-central functional on \(M\).

Given a finite \(F\subset H\) and \(\delta>0\), set \[\chi(z)=\sum_{\xi\in F}\langle z\xi,\xi\rangle.\] Normality supplies a support \(e\leq p\) in the directed family with \(\chi(p-e)\) as small as desired. On the nonzero corner \(eMe\), the corresponding functional restricts, after normalization, to a faithful normal state central on \(Ae\).

For any faithful normal state \(\omega\), small \(\omega\)-mass on projections implies uniformly small mass for each fixed normal positive functional. Indeed, a contrary sequence in the positive unit ball would have a weak-star convergent subnet whose limit has zero \(\omega\)-mass but positive mass for that functional. Faithfulness forces the positive limit to be zero, a contradiction. Apply the assumed result on \(eMe\), using this observation to make the lost \(\chi\)-mass small. On \((1-p)M(1-p)\) apply Lemma 3 to the normalized restriction of \(\chi\) if it is nonzero; otherwise take \(q=0\) on this corner. Empty corners require no construction.

Use disjoint palettes for the two corner partitions, and the additional projection \(p-e\) as the last piece. Take corner sums of the two diagonal terms and of the two large projections, with zero on \(p-e\). Every pinching piece is supported in one of the three corners, so all off-corner terms of \(x\) disappear. Thus the compressed operator norm is at most \(\varepsilon\), and the diagonal norm is at most one. The choices can ensure \(\chi(1-q)<\delta^2\), which gives \[\|(1-q)\xi\|^2\leq\chi(1-q)<\delta^2\qquad(\xi\in F).\] Zero projections pad either palette when necessary. The total number is the stated \(r\), independently of \(F\) and \(\delta\). Scaling handles an arbitrary nonzero self-adjoint \(x\); for \(x=0\) the assertion is immediate. ◻

Proof. Takesaki’s Conditional Expectation Theorem [17] applies because the modular group fixes \(A\) pointwise; both the state and its restriction are faithful, normal, and finite. Let \(\Omega\) be the GNS state vector. It is separating and therefore cyclic for the commutant. If a bounded net \(z_j\) satisfies \(z_j\Omega\to0\), then \(z_jc'\Omega=c'z_j\Omega\to0\) for every commutant element \(c'\). Density and boundedness give strong convergence. The converse follows by testing on \(\Omega\).

Since the partition projections lie in the centralizer, \(S_P\) induces an orthogonal projection on the GNS space; equivalently, it is the state-preserving conditional expectation onto the commutant in \(M\) of the projections of \(P\). These Hilbert-space projections decrease under refinement. On the vector \(z\Omega\), let their limit be \(\eta\). A bounded weak-star cluster point \(b\) of \(S_P(z)\) commutes with every partition of \(A\) and hence lies in \(A\). Its state vector is \(\eta\). Moreover \(E_A S_P(z)=E_A(z)\), so \(b=E_A(z)\). This proves Equation (5). If an increasing sequence generates \(A\), its limit commutes with the generating projections, and the same argument applies. ◻

Proof. Start with \(D_1=W^*(1,x)\). Inductively, choose a countable \(\|\cdot\|_\varphi\)-dense subset \(\mathcal D_j\) of the unit ball of \(D_j\). Such a set exists for every countably generated \(D_j\): the separable C*-algebra of its generators has dense state vectors by Kaplansky’s Density Theorem. Its GNS representation is faithful and normal on a separable Hilbert space, so \(D_j\) has separable predual. Let \(D_{j+1}\) be generated by \(D_j\), all \(E_A(z)\) for \(z\in\mathcal D_j\), and the projections of finite partitions \(P(z,k)\subset A\) satisfying \[\|S_{P(z,k)}(z)-E_A(z)\|_\varphi<1/k \qquad(z\in\mathcal D_j,\ k\geq1).\] Lemma 5 supplies these partitions. Only countably many generators have been added.

Put \(M_0=W^*(\bigcup_jD_j)\) and let \(A_0\) be generated unitally by all the adjoined elements of \(A\). Bounded Kaplansky approximation by the union algebra, followed by approximation from the sets \(\mathcal D_j\), shows that their union is state-norm dense in the unit ball of \(M_0\). Here one may first approximate in the norm closure of the union and then make a norm approximation and a harmless rescaling to preserve the bound. Since \(E_A\) is state-norm contractive, every \(E_A(z)\) for \(z\in M_0\) is a bounded state-norm limit of elements of \(A_0\). Lemma 5 and strong closedness imply \(E_A(z)\in A_0\).

If a unit-ball element \(z\in M_0\) commutes with \(A_0\), each selected partition satisfies \(S_P(z)=z\). For \(z'\in\mathcal D_j\) and its selected partition \(P=P(z',k)\), contractivity gives \[\|z-E_A(z)\|_\varphi \leq 2\|z-z'\|_\varphi +\|S_P(z')-E_A(z')\|_\varphi.\] The right side can be made arbitrarily small; faithfulness gives \(z=E_A(z)\in A_0\). Thus \(A_0\) is maximal abelian in \(M_0\). The algebra \(M_0\) is countably generated and has separable predual by the same GNS argument. All the assertions about the restricted state are immediate. Any paving produced in \(M_0\) has the same operator-norm bound and the same lost \(\varphi\)-mass when regarded in \(M\). This is the faithful-state estimate required by Proposition 4. ◻

Proof. Use the standard direct integral decomposition of a type I algebra over its center into full operator algebras; see [6]. Integrating ordinary fiber traces against a probability measure in the center measure class gives a faithful normal semifinite trace \(\tau\). The given state has a positive trace density \(h\) of full support, with \(\tau(h)=1\); this is the predual identification in the semifinite case [5]. Every unitary of \(C\) preserves the state and hence its density. Thus \(h\) commutes with \(C\) in the affiliated-operator sense, and its spectral projections belong to \(C\) by maximal abelianness. The increasing projections \[f_n=1_{[1/n,\infty)}(h),\qquad n\geq1,\] exhaust \(1\) and satisfy \(\tau(f_n)\leq n\). Their successive differences form a countable partition of \(1\) in \(C\) into trace-finite projections. Such a projection has finite rank almost everywhere in the type I fibers. Split it further by central projections according to its rank. Each resulting corner is a measurable field of fixed finite matrix size, with its compressed \(C\) a masa.

We justify explicitly the measurable diagonalization needed in these finite corners. Trivialize the finite-rank field and choose countably many commuting self-adjoint measurable matrix fields generating the compressed \(C\). In each fiber, the algebras generated by the first \(j\) fields eventually equal the algebra generated by the whole list, since their dimensions are nondecreasing bounded integers. The first index attaining the limiting dimension is measurable: each finite dimension is determined by ranks of finite lists of polynomial matrices, and the limiting dimension is their countable supremum. Split the center according to this first index. On each piece, successive eigenspace decompositions measurably diagonalize the finite commuting list. Ordered eigenvalues give measurable spectral projections, and Gram–Schmidt applied to fixed coordinate vectors gives measurable orthonormal bases of their ranges. The resulting rank-one projection fields commute with all generators, not just the chosen initial list. As bounded elements of the ambient field algebra, they therefore belong to the masa.

We have obtained countably many rank-one fields \(e_j\in C\), allowing zero outside their central domains, with \(\sum_j e_j=1\). On \(e_jDe_j\) the algebra \(Ce_j\) consists precisely of the center scalars over that domain, since the center of \(D\) is contained in \(C\). Measurable matrix units between \(e_j\) and \(e_k\) on their common central domain therefore partially normalize \(C\). These matrix units and center coefficients generate \(D\): compress a bounded field first by finite sums of the \(e_j\), express its matrix entries using these units, and pass to the strong limit.

Finally, an abelian projection \(e\in D\) has rank one almost everywhere on its central support \(z\). Choose, fiber by fiber on \(z\), the first nonzero \(e_j\) from the countable decomposition. This gives a projection \(e'\in C\) with the same central support and rank. Measurable unit vectors spanning the ranges of \(e\) and \(e'\) give a partial isometry \(w\in D\) with \(ww^*=e\) and \(w^*w=e'\). The assertion also holds for \(e=0\). ◻

Proof. Let \(\Omega\) be the standard state vector for \(\varphi\). The commuting left and right actions of \(A\) have a joint spectral measure on \(X\times X\) on the vector \(y\Omega\). Since \(A\subseteq M_\varphi\), the right action of \(f\in A\) on this vector is \(yf\Omega\). Thus the joint vector measure \(\lambda\) satisfies \[ \lambda(S\times T) =\varphi(y^*1_Sy1_T) =\int_T E_A(y^*1_Sy)\,d\mu \tag{8}\] for Borel \(S,T\subseteq X\). Disintegrating relative to the right coordinate gives a kernel for \(K_y\) with mass \(E_A(y^*y)(v)\leq\|y\|^2\). Choose a Borel version and set its rows to zero on a Borel null exceptional set so that this mass bound holds for every row. For every fixed \(\mu\)-null \(S\), Equation (8) gives \(K_y(v,S)=0\) almost everywhere. This construction uses the joint spectral measure and does not require product-measure absolute continuity.

Suppose this kernel is not almost everywhere atomless. The function \((v,u)\mapsto K_y(v,\{u\})\) is measurable: express it as the decreasing limit of the masses of the pieces containing \(u\) in an increasing sequence of finite Borel partitions that separates points. For some integer \(k\geq1\), the Borel set where this function is at least \(1/k\) has finite vertical sections and a projection of positive measure. The Lusin–Novikov Theorem [7] supplies a measurable selection \(s\colon T\to X\) on a positive-measure Borel set \(T\) such that \[ K_y(v,\{s(v)\})\geq1/k\qquad(v\in T). \tag{9}\] For a fixed \(\mu\)-null Borel set \(S\), this implies \[\mu(\{v\in T:s(v)\in S\}) \leq k\int_T K_y(v,S)\,d\mu(v)=0.\] Consequently \[\theta(f)=1_T(f\circ s)\colon A\longrightarrow A1_T\] is a well-defined unital *-homomorphism into the corner. It is normal: the pushforward under \(s\) of any finite measure with an \(L^1\) density on \(T\) is absolutely continuous with respect to \(\mu\), and its Radon–Nikodym density gives the preadjoint of \(\theta\).

Choose increasing finite Borel partitions \(P_j\) of \(X\) which generate \(A\) and separate points, and put \[ b_j=\sum_{e\in P_j} e y\theta(e). \tag{10}\] The left support projections and the right support projections are orthogonal families, so \(\|b_j\|\leq\|y\|\). Also \(E_N(b_j)=0\) by \(N\)-bimodularity of \(E_N\). Their state vectors are the decreasing joint projection cuts of \(y\Omega\) to the sets where the left coordinate and \(s\) of the right coordinate lie in the same piece of \(P_j\), with the right coordinate in \(T\). Their limit \(\eta\) therefore satisfies \[\|\eta\|^2 =\int_T K_y(v,\{s(v)\})\,d\mu(v)>0.\] A weak-star convergent subnet of the bounded \(b_j\) has a limit \(b\in M^\circ\) with \(b\Omega=\eta\). Indeed testing the vectors against \(z\Omega\), \(z\in M\), gives the normal functionals \(\varphi(z^*b_j)\), so their vector and weak-star limits agree. In particular \(b\ne0\). We have \(b=b1_T\) and \[ fb=b\theta(f)\qquad(f\in A). \tag{11}\] For partition step functions this follows from refinement in Equation (10); bounded strong approximation and normality of \(\theta\) extend it to all \(f\in A\).

We show that Equation (11) forces \(b\in N\). Write its polar decomposition as \(b=u|b|\). The intertwining relations imply \[|b^*|\in A,\qquad fu=u\theta(f)\quad(f\in A), \qquad e:=u^*u\in D:=\theta(A)'\cap1_T M1_T.\] Indeed \(bb^*\) commutes with \(A\), while \(b^*b\) commutes with \(\theta(A)\), and polar decomposition preserves the resulting intertwining relation. For \(d\in eDe\), the element \(udu^*\) commutes with \(A\) and belongs to \(M\), so lies in \(A\). It follows that \(e\) is an abelian projection in \(D\). Let \(z\) be its central support in \(D\). The abelian-projection characterization of type I algebras [6] shows that \(Dz\) is type I: every nonzero central part of \(Dz\) meets \(e\) and hence contains a nonzero abelian projection. Moreover \(A1_T\) is a masa in \(D\), so its center, and in particular \(z\), belongs to \(A1_T\).

The restricted state on \(Dz\), after normalization, is faithful, normal, and central on the masa \(Az\). By Lemma 7, the algebra \(Dz\) is generated by groupoid normalizers of \(Az\). These also normalize \(A\) partially: their supports lie in \(Az\), and every element of \(A\) restricts on \(z\) to an element of \(Az\). Therefore \(Dz\subseteq N\). The same lemma supplies \(w\in Dz\) and \(e'\in Az\) with \[ww^*=e,\qquad w^*w=e'.\] Set \(v=uw\). Its range projection \(uu^*=\mathop{\mathrm{supp}}|b^*|\) and its initial projection \(e'\) belong to \(A\). Since \(w\) commutes with \(\theta(A)\), \[v^*Av=w^*u^*Auw=\theta(A)e'\subseteq Ae'.\] The left side is a masa in \(e'Me'\), by conjugating the masa on the range corner with \(v\). Hence the inclusion is equality, and \(v\) is a groupoid normalizer of \(A\). We conclude \[u=vw^*\in N,\qquad b=|b^*|u\in N,\] contrary to \(0\ne b\in\ker E_N\). The kernel is therefore atomless almost everywhere.

For the composition assertion, let \(K_1,K_2\) be two of the kernels. Choose one fixed \(\mu\)-null Borel set \(S_0\) outside which every row of \(K_2\) is atomless. For almost every \(v\), \(K_1(v,S_0)=0\). For each such \(v\) the measure \[(K_1\circ K_2)(v,E)=\int_X K_2(u,E)\,K_1(v,du)\] vanishes on every singleton \(E\); the conclusion holds for all singletons simultaneously because the integrand row measures are atomless off \(S_0\). The mass bound and the fixed-null-set property also pass to this composition. Iteration proves the assertion for any finite composition.

Finally the atomic part of the standard probability space can be represented by a countable set of positive-mass points. Atomlessness gives \(E_A(y^*p_{\rm at}y)=0\); faithfulness then gives \(p_{\rm at}y=0\). Applying the same argument to \(y^*\in M^\circ\) gives \(yp_{\rm at}=0\). ◻

Proof. The normalization agrees with Definition 5 of [15]. Real-rootedness is part of the mixed determinantal theory and also follows from Lemma 10 below. Equation (13) is [15], whose hypotheses are zero-diagonal Hermitian contractions and \(k\geq2\).

By [15], for a tuple of zero-diagonal Hermitian matrices, replacing one member by zero cannot increase its largest root. Expansion in principal minors gives \[\chi[L,0](z)=\det(zI-L/2):\] a fixed principal minor on \(S\) occurs with probability \(2^{-|S|}\). Applying monotonicity while retaining either \(L\) or \(-L\) proves Equation (14). The all-unassigned case uses the tuple containing \(r\) copies each of \(T\) and \(-T\), so \(k=2r\). For a fully assigned coloring, independence between the color classes gives the stated factorization. ◻

Proof. Each determinant factor is affine in any of its diagonal variables. When differentiating \(k_v-1\) times in \(z_v\), every surviving Leibniz term leaves \(v\) in exactly one of its \(k_v\) available factors and deletes it from the others. The multiplicity \((k_v-1)!\), divided by \(k_v!\), gives probability \(1/k_v\). Processing all vertices gives the outcome average, and differentiation once more in \(z_v\) deletes that site. This proves Equation (15), including for the more general matrices.

If \(\operatorname{Im}z_v>h\), each matrix \(Z-B_j\) on \(U_j\) has strictly positive imaginary part and is invertible. Its determinant is therefore nonzero. Differentiation preserves nonvanishing in the specified half-plane by the Gauss–Lucas Theorem. Here the degree cannot drop at a specialization of the other variables: after any subset of the vertex operations has been performed, the leading coefficient of degree \(k_v\) in an unprocessed variable \(z_v\) is the corresponding partially processed expression with \(v\) deleted. Induct simultaneously over the processed vertices and all principal restrictions. That leading coefficient is nonzero by the induction hypothesis, so every required derivative has its indicated degree and is nonzero in the half-plane. This also proves the assertion for all deletion polynomials.

For fixed other coordinates in that region, \(P_I/P_{-v}=z_v+\zeta_0\), by the derivative identity and multiaffineness. Its zero has imaginary part at most \(h\). This gives Equation (16).

For Hermitian data, the uniform-variable polynomials have real coefficients; half-plane nonvanishing and conjugation therefore show that all their roots are real. Equation (16) with \(h=0\) implies that \(F_v(z)-z\) has nonnegative imaginary part. Its constant at infinity is \(a_v\), since every outcome has \(v\) diagonal \(-a_v\). A nonremovable real pole of a real rational function with nonnegative imaginary part must be simple with nonpositive residue: approach the pole from different angles in the upper half-plane to exclude higher orders, and then approach vertically to determine the sign. This proves Equation (17).

The Schur complement of \(z-\Lambda_v\) in \(z-H_v\) is \(F_v(z)\). Thus \(\det(z-H_v)=Q_v(z)F_v(z)=P_I(z)\), and its first resolvent entry is \(1/F_v(z)=G_v(z)\). The spectral theorem proves the first identity in Equation (19), and principal-matrix interlacing proves the interlacing assertion. Finally, uniform specialization of Equation (15) gives \[\sum_{v\in I}G_v(z)=\frac{P_I'(z)}{P_I(z)}.\] The right side is the Cauchy transform of the root-counting measure. Uniqueness of partial fractions proves the second identity in Equation (19). ◻

Proof. For \(J\subseteq I\), let \(P_J^{\mathrm{un}}(Z)\) be the expected polynomial with the same pair laws, all weights equal to one, and all diagonal shifts removed. Factoring the weight matrix from every outcome determinant gives \[ P_J(x)=\left(\prod_{v\in J}W_v\right) P_J^{\mathrm{un}}\bigl((\tau_v+x/W_v)_{v\in J}\bigr). \tag{22}\] Both sides equal one for \(J=\varnothing\). We also use the following consequence of multiaffineness and the deletion identity in Equation (15). For real coordinates \(b=(b_v)_{v\in I}\) and increments \(h_v\geq0\), \[P_J^{\mathrm{un}}((b+h)|_J) =\sum_{S\subseteq J} P_{J\setminus S}^{\mathrm{un}}(b|_{J\setminus S}) \prod_{v\in S}h_v.\] Thus positivity of all principal evaluations at \(b\) is preserved when any coordinates are increased.

If every site is unassigned and \(t>3/\sqrt r\), Lemma 9 puts \(t\) strictly above the largest root of every nonempty \(P_J^{\mathrm{un}}(z)\). All principal evaluations at the common coordinate \(t\) are therefore positive. For \(x\geq0\), the coordinates \(t+x/W_v\) in Equation (22) are at least \(t\), so \(P_I(x)>0\). Real-rootedness then gives \(s(P_I)=0\).

Now suppose every site is assigned and \(\|T_c\|>2\beta t+\alpha\). The fully assigned factorization and the lower bound in Lemma 9 show that \(P_I^{\mathrm{un}}(z)\) has a root greater than \(\beta t+\alpha/2\). Set \(x_0=(\alpha/2)\min_{v\in I}W_v>0\). If the largest root of \(P_I\) were smaller than \(x_0\), repeated interlacing would put \(x_0\) strictly above every root of every nonempty principal restriction. Thus \(P_J(x_0)>0\) for all \(J\subseteq I\). By Equation (22), all the corresponding unweighted principal evaluations are positive at \[b_v=\beta t+x_0/W_v\leq\beta t+\alpha/2.\] Increasing these coordinates to any common value \(q\geq\beta t+\alpha/2\) preserves that positivity. In particular \(P_I^{\mathrm{un}}(q)>0\) for every such \(q\), contradicting the root above \(\beta t+\alpha/2\). Hence the largest root of \(P_I\) is at least \(x_0\), which also gives the stated lower bound for \(s(P_I)\). ◻

Proof. Equation (23) follows from the principal interlacing in Lemma 10; integration gives Equation (24). To prove the tail estimate, write \(H=H_v\), \(\Lambda=\Lambda_v\), \(a=a_v\), and \(L=H-BI\). Let \(E\) be the first-coordinate projection, \(D=1-E\), and \(R=1_{(0,\infty)}(L)\). The compression \(DRD\) is a positive contraction on the deletion coordinates. The variational formula for the positive trace therefore gives \[\mathop{\mathrm{Tr}}(\Lambda-BI)_+\geq\mathop{\mathrm{Tr}}\bigl((\Lambda-BI)DRD\bigr).\] Since \(L-DLD=EL+LE-ELE\), it follows that \[\begin{align*} \mathop{\mathrm{Tr}}(H-BI)_+-\mathop{\mathrm{Tr}}(\Lambda-BI)_+ &\leq\mathop{\mathrm{Tr}}\bigl((EL+LE-ELE)R\bigr)\\ &=2\langle e_0,LR e_0\rangle+(a+B)\langle e_0,R e_0\rangle\\ &=2\int(x-B)_+\,\rho_v(dx)+(a+B)\rho_v((B,\infty)). \end{align*}\] Here \(e_0\) is the first unit vector of the arrow matrix. The left side is the tail integral in Equation (25). Taking \(B=0\) and using that \(\rho_v\) is a probability measure gives the last assertion. ◻

Proof. Put \(u_v=z_v^{-1}\) and consider the formal polynomial \[\mathcal H_I(u)=P_I(Z)\prod_{v\in I}u_v =\sum_{S\subset I}c_S\prod_{v\in S}u_v.\] The coefficient \(c_S\) is the expected determinant of the negative outcome matrix restricted to \(S\). If \(S\) splits into graph components, the restricted outcome matrix splits into the same blocks, and their choices are independent. Hence \(c_S\) factors over the components. Since \(\mathcal H_I(0)=1\), its formal logarithm is defined. A coefficient in this logarithm depends only on its supported subset. On a disconnected subset \(\mathcal H_I\) factors, so its logarithm is a sum over components. Thus every nonconstant monomial in \(\log\mathcal H_I\) has connected support.

By Equation (15), \[G_v(Z)=\partial_{z_v}\log P_I(Z).\] Apart from the term \(1/z_v\), its monomials arise from connected supports containing \(v\). After uniform specialization, the coefficient of order \(z^{-1-k}\) therefore uses only the radius-\(k\) ball at \(v\). The Cauchy-transform expansion identifies that coefficient with the \(k\)-th rooted moment. Formal inversion of the series with leading term \(1/z\) gives the same locality statement for \(F_v\). Conditioning or changing the shift at \(v\) preserves this argument.

For singleton and two-point sets the coefficients are \[c_{\{v\}}=a_v,\qquad c_{\{u,v\}}=a_ua_v-W_uW_v|T_{uv}|^2 \mathbb P(\omega_u=\omega_v).\] Expanding the logarithmic derivative to order \(z^{-3}\) gives the first identity in Equation (26). Alternatively, the arrow matrix has first diagonal \(-a_v\) and off-diagonal squared norm \(\sum_\lambda b_{v,\lambda}\); comparison with its second rooted moment gives the second identity.

Under the neighboring-ratio hypothesis, the sum in Equation (26) is at most \[K W_v^2\sum_{u\ne v}|T_{vu}|^2\leq K W_v^2.\] This holds for each conditioned law as well, since every collision probability is at most one. Also \(a_v\leq\beta tW_v\). Cauchy–Schwarz for the probability measure \(\rho_v\), followed by Lemma 12, proves Equation (27).

Finally every outcome has norm at most \((1+\beta t)H_0\), because principal compression and pinching preserve the contraction bound of \(T\). For real \(z>L\), all its characteristic determinants are positive; for \(z<-L\), all have sign \((-1)^{|I|}\). Their average therefore has no root outside \([-L,L]\). Real-rootedness proves Equation (28). The same reasoning applies after restriction or a site option, and the spectral representation gives the assertion for rooted measures. ◻

Proof. Write \(a=a_v=tW_v>0\), \(m=m_v\), \(d=d_v\), and \[H=\begin{pmatrix}-a&g^*\\g&\Lambda\end{pmatrix}\] for the old arrow matrix. By Equation (20), the color-conditioned residues \(b_{i,\lambda}\) are nonnegative and have average \(b_\lambda\). A permanent assignment replaces these residues by \(b_{i,\lambda}\) and the first diagonal by \(-\beta a\). All deletion eigenvalues \(\Lambda\) stay fixed.

Separate the deletion coordinates into those with \(\lambda>0\) and those with \(\lambda\leq0\), and use subscripts \(+\) and \(-\) for these two blocks. Write \(H=H_+-H_-\) for its positive and negative parts; subscripts on \(H_\pm\) below specify compressions to the deletion blocks, not additional positive or negative parts. Set \[e_+=\mathop{\mathrm{Tr}}(H_-)_{++},\qquad e_-=\mathop{\mathrm{Tr}}(H_+)_{--}.\] Since the first diagonal of \(H_+\) is \(m\), comparison of the diagonal blocks in \(H=H_+-H_-\) gives \[ d=m+e_++e_-, \qquad (H_-)_{00}=m+a. \tag{30}\] We treat separately the cases in which the rooted cost is at least the shift and in which it is smaller.

Case \(m\geq a\). Consider the two side arrows obtained by retaining the first coordinate and just one deletion block, initially with first diagonal \(-a\). For the positive block there is exactly one negative eigenvalue, say \(-L_0\), and \[L_0=a+\sum_{\lambda>0}\frac{b_\lambda}{L_0+\lambda}, \qquad d^+=L_0-a.\] Here \(d^+\) is its positive-trace increment over its deletion block. The formulas remain valid for an empty positive block, with \(L_0=a\). For the nonpositive block there is at most one positive eigenvalue. Let \(s_0\) denote that eigenvalue if present, and put \(s_0=0\) otherwise. Its increment is \(d^-=s_0\), and \[ \sum_{\lambda\leq0}\frac{b_\lambda}{s_0+|\lambda|} \leq s_0+a. \tag{31}\] If \(s_0=0\), all residues at \(\lambda=0\) must vanish; otherwise the side quotient would have a positive zero. Omit those terms in Equation (31). If the remaining sum at zero were greater than \(a\), continuity and the behavior at infinity would again produce a positive zero, which proves the displayed inequality also in this case.

Pinching off the other coupling in \(H\) cannot increase its positive trace. The uncoupled block contributes exactly its own positive trace, so \(d^+\leq d\) and \(d^-\leq d\). By averaging the residues, at least three quarters of the colors satisfy \[\sum_{\lambda>0}\frac{b_{i,\lambda}}{L_0+\lambda} \leq4d^+,\] and at least three quarters satisfy \[\sum_{\lambda\leq0}\frac{b_{i,\lambda}}{s_0+|\lambda|} \leq4(s_0+a).\] Thus some color satisfies both bounds. A zero old residue is zero for every color by nonnegativity, so the omitted zero-denominator terms remain absent.

For this color, the positive-block arrow with first diagonal \(-a\) has increment at most \(4d^+\). Indeed, if its negative eigenvalue magnitude is at most \(L_0\), the assertion is immediate; otherwise its root equation bounds the increment by the preceding test sum. The nonpositive-block arrow with first diagonal \(-4a\) has increment at most \(4s_0\). If it had a positive root \(s>4s_0\), its decreasing residue sum would be at most its value at \(s_0\), hence at most \(4(s_0+a)\), whereas its root equation would require that sum to be \(s+4a>4(s_0+a)\).

Extend the two side arrows by zero and add them. The resulting full arrow has first diagonal \(-5a\). Subadditivity of positive trace, which follows from its variational formula, shows that its increment is at most \(4d^++4d^-\). Decreasing the first diagonal further to \(-\beta a\) can only decrease positive trace. Therefore \[d_v^{\,i}\leq4(d^++d^-)\leq8d,\qquad d_v^{\,i}-d\leq7d\leq7(2m+a)\leq21m,\] where Lemma 12 was used in the last step.

Case \(m<a\). The positive and negative parts of \(H\) give more useful residue bounds in this case. For a positive block matrix, the outer product of its first coupling column is bounded by its first diagonal entry times its remaining block. Apply this to \(H_+\) and \(H_-\), and use \((p-q)(p-q)^*\leq2pp^*+2qq^*\) for the difference of their coupling columns. As \((H_+)_{++}=\Lambda_++(H_-)_{++}\) and \((H_-)_{--}=|\Lambda_-|+(H_+)_{--}\), this gives \[\begin{align*} g_+g_+^*&\leq 2m\Lambda_++(4m+2a)(H_-)_{++}, \tag{32}\\ g_-g_-^*&\leq 2(m+a)|\Lambda_-|+(4m+2a)(H_+)_{--}. \tag{33}\end{align*}\] Conjugate Equation (32) by \((32a+\Lambda_+)^{-1/2}\) and take the operator norm. The left side has rank at most one, so its norm is the corresponding residue sum. The first term on the right has norm at most \(2m\), and the error term has norm at most its trace. Since \(4m+2a\leq6a\), \[ \sum_{\lambda>0}\frac{b_\lambda}{32a+\lambda} \leq2m+\frac6{32}e_+. \tag{34}\]

Put \(L_1=e_-\). If \(L_1>0\), conjugating Equation (33) by \((L_1+|\Lambda_-|)^{-1/2}\) in the same way gives \[ \sum_{\lambda\leq0}\frac{b_\lambda}{L_1+|\lambda|} \leq2(m+a)+(4m+2a)\leq10a. \tag{35}\] Indeed the trace of \((H_+)_{--}\) is \(L_1\), so its conjugated norm is at most \(1\). If \(L_1=0\), positivity implies \((H_+)_{--}=0\). Equation (33) then forces zero coupling on the kernel of \(|\Lambda_-|\). Use its inverse only on its nonzero support and omit the zero residues at \(\lambda=0\); Equation (35) remains valid.

Choose a color for which both residue sums in Equations (34) and (35) are at most four times their indicated upper bounds. Such a color exists by the same averaging argument. For that color, the positive-block arrow with diagonal \(-32a\) has negative eigenvalue magnitude at least \(32a\); its root equation therefore bounds its positive-trace increment by \[8m+\frac{24}{32}e_+.\] The nonpositive-block arrow with diagonal \(-40a\) has increment at most \(L_1\). Indeed a positive root \(s>L_1\) would have residue sum at most its test value \(40a\), whereas its root equation requires \(s+40a>40a\). This also covers \(L_1=0\), with zero residues omitted as above.

Adding the side arrows gives the full arrow with diagonal \(-72a\). Subadditivity of positive trace and then decreasing the diagonal to \(-80a\) yield \[d_v^{\,i}\leq8m+\frac34e_++e_-.\] Together with Equation (30), this gives \[d_v^{\,i}-d\leq7m-\frac14e_+\leq7m.\] Both cases imply Equation (29). All arguments allow empty deletion blocks and empty sums, so they include a one-site matrix and every zero-residue case. ◻

Proof. By Lemma 13, each moment \(\int x^k\,\rho_{I,v}(dx)\) is determined by a bounded neighborhood of \(v\), and is unchanged once \(I\) contains that neighborhood. Compactness of the probability measures on \([-L,L]\) and uniqueness from their moments therefore give an exhaustion-independent weak limit \(\rho_v\). Its Cauchy transform is the limit of \(G_{I,v}\), locally uniformly on the upper half-plane. Moreover, \[-\operatorname{Im}G_v(z) =(\operatorname{Im}z)\int |z-x|^{-2}\,\rho_v(dx)>0,\] so this limit can be inverted. Hence \(F_{I,v}\) converges to \(F_v\) there. The same argument applies to the altered color law or shift at the root. The finite quotients also have the form \[F_{I,v}(z)-z-a_v=-\int\frac{1}{z-x}\,\kappa_{I,v}(dx),\] where the positive residue measure \(\kappa_{I,v}\) is supported on \([-L,L]\). Its mass is at most \(H_0^2\) by the second-moment formula in Lemma 13; the same bound holds for the conditional color laws. Compactness and uniqueness of Cauchy transforms therefore give a limiting residue measure as well. This also justifies taking the local Laurent-coefficient limits of the quotients below. Since \(x_+\) is continuous on \([-L,L]\), the costs \(m_{I,v}\) converge as well. Polynomial approximation on this interval, together with moment locality, makes this convergence uniformly approximable by finite neighborhood calculations.

Deletion increments require a different approximation, because their definition involves the roots at every vertex of a finite set. For a real polynomial \(f\), put \[d_{I,v}[f]=\sum_{P_I(x)=0}f(x) -\sum_{Q_{I,v}(x)=0}f(x).\] The identity that the sum of the rooted measures counts all polynomial roots gives \[ d_{I,v}[f] =\sum_{u\in I}\left( \int f\,d\rho_{I,u} -1_{\{u\ne v\}}\int f\,d\rho_{I\setminus\{v\},u} \right). \tag{38}\] Each summand is local. It vanishes unless \(u\) lies within a radius depending only on \(\deg f\) of \(v\). Thus the entire increment can be computed in a sufficiently large finite neighborhood of \(v\), independently of the ambient finite set. This defines its value on the infinite graph.

To pass from polynomials to \(x_+\), for an absolutely continuous real function \(f\) on \([-L,L]\) use interlacing in the form \[ d_{I,v}[f] =f(-L)+\int_{-L}^{L} f'(x) \bigl(n_{P_I}(x)-n_{Q_{I,v}}(x)\bigr)\,dx, \qquad 0\leq n_{P_I}(x)-n_{Q_{I,v}}(x)\leq1. \tag{39}\] Here \(n_P(x)\) counts roots strictly greater than \(x\). The formula follows by writing \(f(\lambda)=f(-L)+\int_{-L}^{\lambda}f'(x)\,dx\) for each root; the difference in degrees is one. Consequently \[ |d_{I,v}[f]-d_{I,v}[g]| \leq |f(-L)-g(-L)|+\|f'-g'\|_{L^1([-L,L])}. \tag{40}\] Approximate \(f'\) in \(L^1\) by polynomials and integrate them with prescribed initial value \(f(-L)\). This gives polynomial approximants with both uniform function error and the increment error in Equation (40) tending to zero. Apply this to \(f(x)=x_+\). The local polynomial increments are eventually constant, and their approximation error is uniform in every finite set. Hence \(d_{I,v}\) has the stated limit and uniform locality property. The same proof works for each of the finitely many virtual options at \(v\).

Finally, finite balls in a measurable bounded-degree graph, their finite matrix data, and their conjugacy-invariant polynomial calculations are measurable. The limiting constructions above are therefore measurable. Deleting vertices or restricting the relation does not change the argument: use the original graph for the locality radius and the retained vertices for each principal calculation. ◻

Proof. An injective measurable point code supplies such an ordering if none has already been specified. Begin with a real polynomial \(f\) in place of \(x_+\). Write \(m_v[f]=\int f\,d\rho_v\) at present vertices and zero at absent ones. For each root \(v\), this quantity depends only on a fixed finite-radius neighborhood. List the finitely many vertices of \(E_0\) in that neighborhood in their specified order and telescope its change over this list. Each term agrees with the change produced at that vertex in the global comparison environment: changes outside the neighborhood have no effect on this polynomial rooted datum. Thus no well-ordering or infinite telescoping at an individual root is being assumed.

Integrate the resulting finite local sum and apply Equation (37) to interchange the root and the changed vertex. For a fixed changed vertex \(u\), the sum over affected roots equals the change in \(d_u[f]\). Indeed, this can be checked on a sufficiently large finite principal neighborhood using Equation (38): the old and new polynomials have the same deletion polynomial at \(u\). If the vertex is deleted, the new rooted datum there and the new deletion increment are both zero by convention. Bounded degree, bounded spectral support, and finite total measure justify all integrals for this fixed polynomial. We have proved \[\int_X(m_v^{\mathrm{new}}[f]-m_v^{\mathrm{old}}[f])\,d\nu(v) =\int_{E_0}\delta d_u[f]\,d\nu(u).\]

Choose the polynomial approximants to \(x_+\) constructed in the proof of Lemma 15. Uniform function approximation controls the two rooted terms on the left; the \(L^1\) derivative bound in Equation (40) controls both comparison increments on the right. Finite measure then permits passage to the limit and proves Equation (41). For a deletion the comparison increment is \(-d_u\leq0\), by interlacing. This proves the last assertion. ◻

Proof. We may discard initially absent vertices from \(E_{\mathrm{hi}}\): this leaves both configurations unchanged and only decreases its measure. Thus every deletion vertex below is present before the deletions begin. We first prove bounds for one deletion in a finite principal set. We then pass these bounds to infinite bounded graphs and integrate them using an analytic counterpart of the ordered-change identity.

A single deletion and the field \(G\). Fix a finite principal set \(I\), \(u\in I\), and \(z\) with \(y=\operatorname{Im}z>0\). For arbitrary complex numbers \(\lambda_v\) satisfying \(|\lambda_v|\leq1\), perturb the diagonal variables to \(z+\zeta\lambda_v\), \(v\in I\). On the disk \(|\zeta|<y/2\) they remain in the upper half-plane. By Lemma 10, the quotient \(P_I/P_{I\setminus\{u\}}\) in these variables has positive imaginary part. Its logarithm has a holomorphic branch with imaginary part in \((0,\pi)\) throughout the disk.

We use the elementary derivative estimate that a holomorphic function on a disk of radius \(R\), with bounded imaginary part, has derivative at the center bounded by a numerical constant times the oscillation of that imaginary part divided by \(R\). To see this, restrict to a concentric circle of radius \(R/2\) and recover the first Taylor coefficient from the first Fourier coefficient of the imaginary part. Applied to the logarithm just described, it bounds its derivative at zero by \(C_0/y\). Logarithmic differentiation of \(P_I\) in a diagonal variable gives \(G_{I,v}\), so this derivative is \[\sum_{v\in I}\lambda_v \bigl(G_{I,v}(z)-G_{I\setminus\{u\},v}(z)\bigr),\] where the second transform at \(v=u\) is zero. Choosing the phases of \(\lambda_v\) to make all summands nonnegative real yields \[ \sum_{v\in I} |G_{I,v}(z)-G_{I\setminus\{u\},v}(z)|\leq C_0/y. \tag{44}\] This estimate contains no bound on the weights.

A single deletion and the conditional color field. Now assume \(u\notin\mathcal U\), and take \(\lambda_v=0\) except at unassigned vertices in \(\mathcal U\). Perturb the off-diagonal parts of both pair matrices of color \(i\) by left and right diagonal scaling with entries \(1+\zeta\lambda_v\). There is no conjugation of \(\zeta\) in the right scaling, and the fixed diagonal shift is left unchanged. This gives an analytic perturbation, though its matrices need not be Hermitian for complex \(\zeta\).

Let \(D_\lambda=\mathop{\mathrm{diag}}(\lambda_v)\) and \(D_W=\mathop{\mathrm{diag}}(\sqrt{W_v})\) on one of the two relevant allowed-site sets. With the sign suppressed, its off-diagonal matrix is \(C_i=D_W T D_W\), restricted to this set. The perturbation of this matrix equals \[\zeta(D_\lambda C_i+C_iD_\lambda) +\zeta^2D_\lambda C_iD_\lambda.\] In the first row term only band rows and their graph neighbors occur. The row weights are at most \(b_2\), and these neighboring column weights are at most \(Kb_2\). Factoring the diagonal weights on those row and column sets and using \(\|T\|\leq1\) gives \(\|D_\lambda C_i\|\leq\sqrt K b_2\). The same argument applies to \(C_iD_\lambda\). In the quadratic term both indices lie in the band, giving norm at most \(b_2\). The estimates apply to every principal restriction. Consequently the full matrix perturbation has norm at most \[ 2|\zeta|\sqrt K b_2+|\zeta|^2b_2. \tag{45}\]

Choose a disk of positive radius depending only on \(y,b_2,K\) on which Equation (45) is less than \(y/2\). The imaginary parts of all perturbed pair matrices are then bounded above by \((y/2)I\). With diagonal variable still equal to \(z\), the non-Hermitian version of Lemma 10 shows that the quotient \(P_I/P_{I\setminus\{u\}}\) has positive imaginary part. Once again its logarithm has imaginary part in \((0,\pi)\), and the derivative estimate is uniform in the upper weight bound \(H_0\).

The derivative of \(\log P_I\) in this perturbation is \[ \frac2r\sum_{v\in I}\lambda_vJ_i(v,z). \tag{46}\] For completeness, consider the permutation expansion of an outcome determinant. At a target vertex \(v\) whose selected color is \(i\), a nontrivial permutation cycle through \(v\) uses one row factor and one column factor, hence has derivative multiplier \(2\lambda_v\). A fixed point contributes \(z+a_v\) and is unchanged because \(T\) has zero diagonal. Conditional on color \(i\) at \(v\), the sum of the nontrivial terms through \(v\) is \[P_{I\setminus\{v\}}(z) \bigl(F_{v,i}^0(z)-z-a_v\bigr).\] The probability of that color at an unassigned vertex is \(1/r\). Dividing by \(P_I\) proves Equation (46), including the contribution of both signs. The same formula holds after deleting \(u\). Apply the derivative bound to the logarithm of the quotient and choose the directions \(\lambda_v\) as before. We obtain \[ \sum_{v\in I\cap\mathcal U\cap\{c=0\}} |J_i^{I}(v,z)-J_i^{I\setminus\{u\}}(v,z)| \leq C_1(y,b_2,K,r). \tag{47}\] The constants in Equations (44) and (47) can be chosen uniformly for \(z\) in any fixed upper-half-plane compact set.

Passage to bounded infinite graphs. Exhaust the graph component of \(u\) by finite principal sets. The transforms converge by Lemma 15, including the conditional transforms in Equation (42). Fatou’s Lemma therefore passes both single-deletion estimates to the infinite graph. Their sums can be taken over the entire relation class: there is no effect on other graph components. Moreover, the bounds hold in every comparison environment formed by deleting other vertices first.

The single-deletion estimates are now uniform in the largest weight. It remains to convert them into a comparison for a measurable set of deletions. Directly summing pointwise changes in an arbitrary infinite order would not justify that step. We instead establish the integrated identity first for the local Laurent coefficients and then for the analytic transforms.

The analytic integral change identity. Fix measurable complex directions \(\lambda_v\), \(|\lambda_v|\leq1\), supported on the band targets: on \(\mathcal U\) for \(G\), and on \(\mathcal U\cap\{c=0\}\) for \(J_i\). Let \(Z_v(z)\) denote the respective field, extended by zero off those targets. Order \(E_{\mathrm{hi}}\) measurably and injectively along each class. At a deletion vertex \(u\), let \(\delta_u Z_v\) mean presence-minus-absence at \(u\), with earlier vertices already deleted and later vertices still present. In the following formula orient \(\delta_{\mathrm{tot}}\) as before-minus-after. We claim that \[ \int_X\lambda_v\delta_{\mathrm{tot}}Z_v(z)\,d\nu(v) =\int_{E_{\mathrm{hi}}} \left(\sum_{v\in[u]_{\mathcal R}} \lambda_v\delta_u Z_v(z)\right)d\nu(u). \tag{48}\] The left side contains one rooted term, whereas the right side sums over targets in the class of the deleted vertex.

To prove the identity, initially work near infinity in the upper half-plane. Every target is present in both configurations, since the deleted set is disjoint from the band and initially absent band vertices are excluded. At such a target, the bounded spectral support gives \[G_v(z)=z^{-1}\sum_{k\geq0}h_k(v)z^{-k},\qquad |h_k(v)|\leq L^k.\] At a \(J_i\) target, the corresponding expansion is \[F_{v,i}^0(z)-z-a_v=-\sum_{k\geq0}b_k(v,i)z^{-k-1},\qquad |b_k(v,i)|\leq B_{v,i}L^k,\] where \(B_{v,i}\) is the total mass of the limiting residue measure. These bounds hold because the finite quotients are the negative Cauchy transforms of their residue measures, all supported in \([-L,L]\). The total residue mass is uniformly bounded in this bounded setting by Lemma 13. These expansions persist in the limit: each coefficient is eventually fixed by a finite neighborhood, the coefficient bounds are uniform, and the series converge to the limiting transforms for sufficiently large \(|z|\). The product defining \(J_i\) therefore also has exponentially bounded Laurent coefficients. The coefficient at each order in either field depends on a neighborhood of radius at most a constant times that order, by Lemma 13 and multiplication of the series.

For each individual coefficient the ordered local telescoping argument from Lemma 16, followed by Equation (37), proves the coefficient version of Equation (48). The bounded degree ensures that the number of affected roots at a fixed deletion vertex grows at most exponentially in the coefficient order. Combining this with the coefficient bounds just obtained gives absolute summability of the resulting series for sufficiently large \(|z|\). Finite measure then allows the sums and integrals to be interchanged. This proves Equation (48) on a nonempty open region of the upper half-plane. The region may depend on \(H_0\), which causes no problem for the analytic continuation that follows.

Both sides of Equation (48) are holomorphic on the full upper half-plane. On the right, the sum over \(v\) converges absolutely at each \(z\), by the single-deletion estimates. Its analytic partial sums, for example over increasing neighborhoods, are locally bounded uniformly by those same estimates. Normal-family convergence therefore makes the sum holomorphic. The outer integral is over the finite measure set \(E_{\mathrm{hi}}\), and the bounds are uniform on compact sets, so it too is holomorphic. On the left, \(|G_v(z)|\leq(\operatorname{Im}z)^{-1}\); the residue bound on the band similarly bounds \(J_i(v,z)\) on compact sets. These estimates and the finite band measure justify holomorphy of that integral. The identity theorem now proves Equation (48) everywhere in the upper half-plane.

Finally fix \(z\) and choose the measurable directions \(\lambda_v\) so that \(\lambda_v\delta_{\mathrm{tot}}Z_v(z)= |\delta_{\mathrm{tot}}Z_v(z)|\), with arbitrary value when the difference is zero. These are fixed directions for the just-proved identity; they need not depend analytically on \(z\). Its right side is bounded by the single-deletion constant times \(\nu(E_{\mathrm{hi}})\). Apply this argument separately to \(G\) and \(J_i\) and add the estimates. Their constants were independent of \(H_0\) and of the comparison environments, which proves Equation (43) with the stated uniformity. ◻

Proof. The two cutoffs use different controls. The set of weights above \(H\) has invariant measure \(H^{-1}\), so the rank comparison controls the upper end. With the upper cutoff fixed, the edge-ratio bound makes sufficiently small weights invisible in any chosen finite neighborhood of a fixed band. We first prove uniform convergence on the band \(1\leq W\leq e\). The same argument applies to any fixed band bounded away from zero. Keep the lower cutoff \(l\) fixed and compare upper cutoffs \(H_2\geq H_1\geq e\). Passing from \(H_2\) to \(H_1\) deletes a set of invariant measure at most \(1/H_1\). Lemma 17 gives, on the appropriate sites of the band, integrated absolute differences of order \(1/H_1\) for \[G_v(z) \quad\text{and}\quad J_i(v,z)=\bigl(F_{v,i}^0(z)-z-a_v\bigr)G_v(z), \qquad a_v=\tau_vW_v.\] For \(J_i\) the targets are the unassigned band sites. The constants are uniform in \(l,H_1,H_2\), the configuration, and \(z\) on each compact subset of the open upper half-plane.

The second-moment estimate of Lemma 13 is \[ \int x^2\,\rho_v(dx) \leq\bigl((\beta t)^2+K\bigr)W_v^2. \tag{57}\] It also holds for each of the altered site choices used here. In particular, these second moments are uniformly bounded on the band. They give a uniform positive lower bound for \(|G_v(z)|\) on each compact subset of the upper half-plane: a fixed bounded interval carries a uniformly positive amount of the probability measure, and \[-\operatorname{Im}G_v(z) =(\operatorname{Im}z)\int |z-x|^{-2}\,\rho_v(dx).\] The residue bound in Lemmas 10 and 13 also bounds \(J_i\) uniformly on these compact sets. Division is thus controlled in the identities \[F_v=1/G_v, \qquad F_{v,i}^0=z+a_v+J_i/G_v, \qquad F_{v,i}=F_{v,i}^0+(\beta-1)tW_v.\] The integrated differences of these quotients consequently tend to zero as \(H_1\to\infty\), uniformly in the lower cutoff.

We next recover the costs from these transform comparisons. Probability measures with the uniform second-moment bound form a weakly compact family, and integration of \(x_+\) is continuous on this family by uniform integrability. Cauchy transforms on a countable dense subset of the upper half-plane separate probability measures. Compactness therefore implies that, for any prescribed accuracy of \(m_v\), comparisons of the transforms at finitely many arguments to sufficiently small accuracy suffice. The integrated transform estimates, together with the uniform bound on \(m_v\), give the desired integrated comparison for \(m\).

For deletion costs we use the argument of the quotient. In a finite restriction, let \(n_P(x)\) and \(n_Q(x)\) count roots strictly greater than \(x\), with multiplicity. Interlacing and the quotient properties give, for \(y>0\), \[ \arg F(x+iy) =\int_{\mathbb R}\frac{y}{(x-u)^2+y^2} (n_P-n_Q)(u)\,du, \qquad 0\leq\arg F\leq\pi. \tag{58}\] This follows by writing the arguments of the individual root factors; the branch is fixed by \(\operatorname{Im}F>0\). The spectral-shift function \(n_P-n_Q\) takes values between zero and one. Lemma 12 and Equation (57) imply that \[ \int_B^\infty(n_P-n_Q)(x)\,dx=O(B^{-1}) \quad(B\to\infty), \tag{59}\] uniformly on the band, including for the shifted virtual options. Indeed, the bound in that lemma is \[2\int(x-B)_+\,\rho_v(dx)+(a_v+B)\rho_v\{x>B\},\] whose two terms are controlled by the second moment and the band bound on \(a_v\).

On \([0,B]\), the integral of the spectral-shift function is approximated by \[ \frac1\pi\int_0^B\arg F(x+iy)\,dx \quad\text{as }y\downarrow0, \tag{60}\] uniformly over these finite calculations. To see uniformity, interchange the integrals in Equation (58); the error is at most the \(L^1(\mathbb R)\) error in the Poisson approximation to \(1_{[0,B]}\), because \(0\leq n_P-n_Q\leq1\). More precisely, put \[p_y(u)=\frac{y}{\pi(u^2+y^2)},\qquad \delta_B(y)=\|p_y*1_{[0,B]}-1_{[0,B]}\|_{L^1(\mathbb R)}.\] For fixed \(B\), one has \(\delta_B(y)\to0\) as \(y\downarrow0\). Combining the smoothing error with Equation (59) gives, uniformly for the finite rooted calculations on the band, \[ \left|d_v-\frac1\pi\int_0^B\arg F_v(x+iy)\,dx\right| \leq\frac{C}{B}+\delta_B(y),\qquad B\geq1. \tag{61}\] The same estimate holds for each shifted virtual option. For fixed \(B,y\), convergence of the costs and locally uniform convergence of the quotients in Lemma 15 pass this estimate to every present band vertex of a bounded infinite cut, and to present unassigned band vertices for the virtual options. The quotient has imaginary part at least \(y\), so its argument is continuous throughout this passage. We use this estimate in terms of the cost and quotient on the infinite cut. For fixed \(B,y\), the arguments in its integral are controlled by the integrated quotient comparisons already proved. First choose \(B\) large, then \(y\) small for that \(B\), and finally the upper cutoff \(H_1\) large for the fixed segment \([0,B]+iy\). This proves the high-cutoff comparison for \(d\) and every \(d^i\).

It remains to remove the lower cutoff. For a fixed finite upper cutoff, Lemma 15 approximates the costs uniformly by bounded neighborhood calculations. A radius-\(R\) neighborhood of a root in the fixed band cannot contain a vertex of weight below \(K^{-R}\). Thus a sufficiently small lower cutoff is invisible to any one of these fixed neighborhood calculations. This proves uniform convergence as \(l\downarrow0\) with that upper cutoff fixed. For joint convergence, first compare both large upper cutoffs to one sufficiently large fixed upper cutoff by the preceding estimates, and then compare the lower cutoffs there. This gives uniform joint \(L^1\)-Cauchy estimates. Division by \(W\) does not change convergence on a fixed band.

The uniform bound asserted in the lemma follows directly from Equation (57) and Lemma 12: \(m_v/W_v\) is uniformly bounded, and \(d_v\leq2m_v+a_v\). The same bounds hold for all virtual options. In particular they are independent of the band, the cutoff, and the configuration.

We have constructed limits on every fixed band. To identify their height dependence, translate height by a real number \(b\). This scales every weight by \(e^{-b}\) and changes the cutoffs correspondingly. All outcome matrices and all costs scale by the same positive factor. Thus \[\frac{f^{l,H}(v,s+b)}{W(v,s+b)} =\frac{f^{e^b l,e^b H}(v,s)}{W(v,s)}.\] Joint convergence makes the limiting normalized field invariant almost everywhere under every rational height translation, simultaneously. A measurable function on \(\mathbb R\) invariant almost everywhere under all rational translations is constant almost everywhere; this follows, for example, by convolution. Applying this on almost every base fiber gives the functions \(\bar f(v)\) and their asserted independence of height. Their measurability also follows from the band limits, for instance by integrating a representative over a fixed height interval.

Finally Lemma 14 gives the single-site inequality on finite restrictions. Lemma 15 passes it to each bounded cut. Joint band convergence then permits a pointwise convergent subsequence for the finite list of normalized fields. Dividing by the positive root weight and taking this limit yields Equation (56). ◻

Proof. In the strip integral there is the identity \[|f_{\Sigma_n}-W\bar f|\,d\nu =|f_{\Sigma_n}/W-\bar f|\,d\mu\,ds.\] Divide the height interval \([-n,n]\) into unit intervals. Translating each to a fixed interval uses the homogeneity proved in Lemma 19. For intervals a sufficiently large fixed distance from both strip ends, the translated lower and upper cutoffs are uniformly close to zero and infinity, respectively. Uniform joint convergence on the fixed band bounds their average errors by any prescribed positive number. The remaining bounded number of end intervals has uniformly bounded error, and its contribution divided by \(2n\) tends to zero. This proves Equation (63).

For Equation (64), first approximate both limiting normalized fields on a fixed unit height band in \(L^1\) by common, fixed lower and upper cutoffs. The errors are uniform by Lemma 19. On this bounded cut, Lemma 15 approximates each field uniformly by a finite-radius calculation. Agreement of the base colorings on a sufficiently large ball makes these lifted local data identical. Since the normalized limit fields are independent of height, integration on the unit band bounds the required base integral. Choosing the cutoff and then the radius for each prescribed accuracy gives Equation (64).

Finally, all bounded-cut calculations are measurable, being limits of rooted finite calculations. They remain so with additional measurable parameters. An auxiliary parameter held fixed along the relation does not alter the degree, edge ratios, or any spectral estimate. The preceding uniform estimates therefore hold on its product probability space as well, and the \(L^1\) limit construction gives jointly measurable limit fields there. ◻

Proof. For the all-unassigned coloring, Lemma 11 gives zero cost on every finite principal restriction. Finite restriction limits and Lemma 19 therefore give \(\mathcal S=0\).

To prove Equation (65), partition the bad centers into a bounded finite number of measurable classes so that two distinct centers in one class have graph distance greater than \(2m+1\). Here is a measurable construction. A bounded-degree Borel graph has a countable measurable partition into independent sets: assign to each vertex a finite prefix of an injective standard-space code long enough to separate it from all of its neighbors, and include the prefix length in the assigned code. Equal assigned codes cannot occur at adjacent vertices. Visit these independent sets in sequence, and give each vertex the first color not already used by one of its previously colored neighbors, from a palette of one more than the degree bound. This is a measurable proper coloring. Apply this construction to the graph whose edges join vertices at distance at most \(2m+1\), and intersect its color classes with \(\operatorname{Bad}_m(c)\). The number of resulting classes is bounded in terms of \(D_0,m\).

Fix one such class \(C\) of bad centers. In the invariant lift retain the strip \(\Sigma_n\), and select the lifted centers above \(C\) whose height is at distance at least \(m\log K\) from both ends of \([-n,n]\). Every radius-\(m\) ball around a selected center lies entirely in the strip. Moreover these balls are disjoint and nonadjacent along each lifted orbit. Indeed Equation (54) and the cocycle law give exactly one lifted site above each base site in a fixed lifted orbit. Distinct centers there therefore project to distinct centers of \(C\), whose distance is greater than \(2m+1\).

Delete every strip site outside these balls. By Lemma 16, deletion cannot increase the total integrated cost, so \[ \int_{\Sigma_n}m_{\Sigma_n}\,d\nu \geq\int m_{\mathrm{retained}}\,d\nu. \tag{66}\] Each retained graph component is one of the selected finite balls. Invariant mass transport sends the rooted costs in each such ball to its center, and hence the right side of Equation (66) equals the integral over the selected lifted centers of \(s(P_{\mathrm{ball}})\).

For a bad ball, the second part of Lemma 11 gives \(s(P_{\mathrm{ball}})\geq(\alpha/2)\min_{\mathrm{ball}}W\). The edge-ratio bound then gives \[ s(P_{\mathrm{ball}}) \geq\frac\alpha2 K^{-m}W_{\mathrm{center}}. \tag{67}\]

Divide Equation (66) by \(2n\) and let \(n\to\infty\). Equation (63) makes the left side tend to \(\mathcal S(c)\). On the right, \(W\,d\nu=d\mu\,ds\), and the allowed height interval has length \(2n-2m\log K\) for all sufficiently large \(n\). Thus Equation (67) gives \[\frac\alpha2 K^{-m}\mu(C)\leq\mathcal S(c).\] Summing over the bounded number of classes proves Equation (65). Every part of the argument uses only the stated probability-space, cocycle, degree, and matrix bounds, so it applies unchanged to the indicated extensions. ◻

Proof. Apply Lemma 16 to the change from \(c\) to \(c^h\) on the invariant lift restricted to \(\Sigma_n\). Order the changed sites by their times \(U_v\). These times are almost surely distinct along every countable orbit, after discarding a saturated null set. The order need not be a well-order, as allowed in that lemma. At a comparison site \(v\) the pre-change configuration is the restriction of \(c^{U_v}\), with the number \(U_v\) held fixed as a threshold along that orbit.

We claim that the resulting integral of the single-site increments is \[ \begin{split} &\int_{\Sigma_n} \bigl(m_{\Sigma_n}(c^h)-m_{\Sigma_n}(c)\bigr)\,d\nu\\ &\quad=\int_0^h\frac{d\theta}{1-\theta} \int_{\Sigma_n}1_{\{v\in E_0,\ U_v\geq\theta\}} \left[ d_{\Sigma_n}^{i(v)}(c^\theta) -d_{\Sigma_n}(c^\theta) \right]d\nu. \end{split} \tag{69}\] Here \(\nu\) includes all label laws as well as the height measure, and \(v\) denotes the current base site of a lifted root.

To justify the claim, condition at a root in \(E_0\) on the old coordinates, all tie coordinates, and every fresh time except the time at that root. Hold the height fixed as well. For a fixed threshold \(\theta\), leave the root site unassigned, and denote the bracketed difference in Equation (69) by \(D(\theta)\) for these fixed other data. Whenever the root time \(u_0\) is at least \(\theta\), this difference is independent of its actual value. The chosen colors were determined before the fresh times, other sites use only their own fresh times in the threshold test, and neither the graph nor the restricted weight data use this new root coordinate. This independence holds for the deterministic rooted finite-subset calculations and hence for the bounded-strip costs obtained from them. The same description applies at \(u_0=\theta\) because the update condition is strict.

When \(u_0<h\), the ordered comparison increment at the root is therefore \(D(u_0)\), outside the null event of a tie with another site. Integrating over this last uniform time uses precisely the identity \[\int_0^h D(u_0)\,du_0 =\int_0^h\frac{d\theta}{1-\theta} \int_\theta^1 D(\theta)\,du_0.\] The bounded-strip computations are measurable jointly in the threshold and in the remaining data, again by their finite rooted definitions. Fubini’s Theorem, first for almost every root and height and then over the remaining coordinates, proves Equation (69). This argument takes place entirely before choosing any homogeneous limit version depending on \(\theta\).

We can now pass to that limit in an integrated expression. Divide Equation (69) by \(2n\) and use Lemma 20. For the right side, adjoin \(\theta\) with normalized Lebesgue probability on \([0,h]\), holding it fixed along the equivalence relation, and regard \(c^\theta\) as one configuration on this augmented base. The lemma applies uniformly there. Its absolute integrated errors also control the errors after multiplication by the displayed mask, after selection among the finite list of options \(i(v)\), and after multiplication by \((1-\theta)^{-1}\leq2\). For a height-independent limit field, cancellation of \(W\) against the height density leaves a height interval of length exactly \(2n\). Consequently \[ \begin{split} \mathcal S(c^h)-\mathcal S(c) =\int_0^h\frac{d\theta}{1-\theta} \int_X1_{\{v\in E_0,\ U_v\geq\theta\}} \left[ \bar d^{i(v)}(c^\theta)-\bar d(c^\theta) \right]d\mu. \end{split} \tag{70}\] All fields in this formula have jointly measurable versions on the augmented space. No evaluation of an unspecified limit at the random diagonal \(\theta=U_v\) has been used.

It remains to compare the intermediate costs with the old costs, for which \(i(v)\) was chosen. Let \(D_R\) be a finite bound, depending only on \(D_0,R\), for the cardinality of a radius-\(R\) graph ball. On the event that this ball has no update from \(c\) to \(c^\theta\), the two partial colorings agree there. Equation (64) controls the integrated difference of their cost fields on that event; summing over the finite option list also controls the selected field. Conditional on the old and tie coordinates, the probability of an update anywhere in the ball is at most \(D_R h\), by the union bound for the fresh uniform times. On the event that an update does occur, the cost differences are uniformly bounded by Lemma 19.

These observations apply either for each fixed \(\theta\), or jointly with normalized measure on \([0,h]\) as above. They show that replacing the two costs in the brackets of Equation (70) by their values at \(c\) creates an error in absolute value at most \[ C h\bigl(\gamma(R)+D_Rh\bigr), \tag{71}\] where \(C\) depends only on the fixed parameters. The factor \((1-\theta)^{-1}\) is bounded by \(2\).

The old costs and the minimizing choice \(i(v)\) are independent of the fresh root time. The survival probability \(1-\theta\) therefore cancels the denominator in Equation (70). By Equation (56), the resulting old-cost contribution is at most \[h\int_{E_0}\bigl(\bar d^{i(v)}(c)-\bar d(c)\bigr)\,d\mu(v) \leq24h\int_{E_0}\bar m(c)\,d\mu \leq24h\mathcal S(c).\] For each prescribed accuracy in Equation (71), first take \(R\) large enough to control \(\gamma(R)\) and then take \(h\) small enough to control \(D_Rh\). Equivalently, taking the infimum of these uniform error bounds over integer \(R\) gives a nonnegative function \(g(h)\to0\). This proves Equation (68) uniformly over the current configurations. ◻

Proof of Theorem 18. Work on the relative Bernoulli extension of the original base, reserving independent coordinates at every site for each step. Start with the all-unassigned coloring, whose cost is zero by Lemma 21. Choose \(T_0>0\) and \(h\in(0,1/2)\), and apply the construction of Lemma 22 with independent fresh coordinates for \[N=\lceil T_0/h\rceil\] steps. Write \(S_j\) for the cost after \(j\) steps. Iterating Equation (68) gives \[ S_N\leq\frac{g(h)}{24}\bigl((1+24h)^N-1\bigr). \tag{72}\] For fixed \(T_0\) this tends to zero as \(h\downarrow0\), since \((1+24h)^N\leq\exp(24(T_0+h))\).

An unassigned site receives a color at its first successful hazard. Conditional on the original base point, its probability of remaining unassigned after all \(N\) steps is \((1-h)^N\leq e^{-T_0}\). This conclusion does not depend on the color selected at a successful step. If \(D_m\) bounds the size of a radius-\(m\) ball, the conditional probability that any site in that ball remains unassigned is at most \(D_m e^{-T_0}\). The unconditional probability that the ball is fully assigned but violates the required norm bound is at most \(C_4 S_N\), by Lemma 21. First choose \(T_0\) large, and then \(h\) small, so that \[D_m e^{-T_0}+C_4S_N<\eta.\] Complete the partial coloring by fresh independent uniform colors at the remaining sites. Every previously complete ball is unchanged. The resulting coloring therefore satisfies Equation (52).

It remains to verify the conditional marginals. Fix a permutation \(\sigma\) of \(\{1,\ldots,r\}\), and let \(\Theta_\sigma\) apply it to the color names in every tie-breaking permutation and every final uniform color input, leaving all times unchanged. This transformation fixes the original base point, commutes with rerooting, and preserves each conditional product measure \(\kappa_v\). On the height lift it also fixes the height, so it acts isometrically on every band \(L^1\) space.

Suppose inductively that the current coloring satisfies \(c\circ\Theta_\sigma=\sigma\circ c\). Exact pullback of the finite-cut calculations and their color covariance give \[\bigl((d^{\sigma(i)})^{l,H}(c)\bigr)\circ\Theta_\sigma =(d^i)^{l,H}(c),\qquad 1\leq i\leq r.\] The band \(L^1\) isometry and uniqueness of the homogeneous limits give \((\bar d^{\sigma(i)}(c))\circ\Theta_\sigma=\bar d^i(c)\) almost everywhere. Thus the set of minimizing colors at the transformed input is the \(\sigma\)-image of the original minimizing set. The transformed permutation tie-breaker chooses the corresponding color, and the update test is unchanged because the times are unchanged.

There are only finitely many color permutations and, with \(T_0,h\) fixed, finitely many steps. Choose a common conull set for the required identities, intersect its images under the finite color action, and discard the saturation of its complement under the relation. The induction just described then holds throughout the construction on that set, starting from the all-unassigned coloring. The final uniform color inputs transform in the same way, so the completed coloring also obeys \(c\circ\Theta_\sigma=\sigma\circ c\).

Since \(\Theta_\sigma\) preserves the conditional product measure over each original base point, disintegration shows that all output colors at the root have equal conditional probabilities for almost every base point. They sum to one, which proves Equation (51) and completes the proof. ◻

Proof. Use the right-counting direct-integral representations recalled in Section 3. Under the bijection between extended and base classes, a pulled-back base kernel acts by the same orbit operator on each fiber. Pullback is a normal faithful representation: it is an amplification over the conditional probability spaces of labels. Equivalently, matrix coefficients against vector fields on the extension disintegrate to normal functionals on the original decomposable algebra. The pulled-back copy of \(N\) lies in \(\widetilde N\), since this holds for the generating kernel normalizers.

The isometry from the original representation space to the extension space that makes a vector field constant in the labels intertwines these copies of \(N\) and sends the original diagonal state vector to the new one. Compression by this isometry maps \(\widetilde N\) normally into \(N\). To see the range assertion, first consider finite sums of \(B\)-functions times lifted base groupoid normalizers. Compression integrates the coefficient functions over the labels at the relevant sites and hence has its range in \(N\). These finite sums form a weakly dense \(*\)-algebra: the lifted partial isomorphisms enumerate the extended relation, and their cuts by extension-measurable sets, together with scalar phase functions, give its kernel normalizers. Normality of compression and weak closure of \(N\) give the range assertion for all of \(\widetilde N\).

The compression is unital, \(N\)-bimodular, and equal to the identity on \(N\), so it gives the asserted normal conditional expectation. It preserves the diagonal state. That state is faithful by the faithful normal diagonal expectation for a countable nonsingular relation algebra. Consequently \(\widetilde E_N\) is faithful as well. The diagonal expectation is \(B\)-bimodular and \(B\) is abelian, so \(B\subset\widetilde N_{\widetilde\varphi}\).

Both expectations onto \(N\) are faithful and normal, and the constituent algebras are \(\sigma\)-finite. The von Neumann algebra reduced amalgamated free-product construction therefore gives \(\mathcal K\), \(E_N^{\mathcal K}\), generation by the two copies, and their freeness over \(N\); see [18] and [19]. These hypotheses are the faithful-normal conditional-expectation hypotheses of that construction, and do not require tracial states. Faithfulness and normality of \(\Phi\) follow from those of the expectation and of \(\varphi|_N\).

For completeness, the product has \(\Phi\)-preserving conditional expectations onto each constituent algebra. Fix one constituent and compress the state representation of \(\mathcal K\) to the closure of its state vectors. The linear span of \(N\) and alternating centered words is weakly dense. Every reduced word containing a centered letter from the other constituent compresses to zero: test between vectors from the retained algebra, expand at the two ends, and absorb the resulting \(N\)-coefficients into adjacent letters; freeness then applies. The retained algebra acts normally and faithfully on its state-vector space. Compression is thus a normal bimodular state-preserving map onto that represented algebra. By Takesaki’s Theorem [17], the modular group of \(\Phi\) restricts to the modular group of each constituent. This also agrees with [19]. Since \(B\) is in the centralizer of \(\widetilde\varphi\), it is in \(\mathcal K_\Phi\). ◻

Proof. In \(L^2(\mathcal K,\Phi)\), let \(P\) be the orthogonal projection onto the closed span of reduced-word vectors whose first letter is a centered \(M\)-letter. The orthogonal complement is spanned densely by the \(N\)-vectors and the reduced-word vectors whose first letter is a centered \(\widetilde N\)-letter. Both the spanning assertion and the orthogonality follow from generation and freeness over \(N\).

Left multiplication by \(w\) sends each of these generators of the orthogonal complement into the range of \(P\). Thus \[(1-P)w(1-P)=0.\] On a word beginning with a centered \(M\)-letter, left multiplication by \(p-\lambda1\) inserts a centered \(\widetilde N\)-letter at the beginning. Its image is orthogonal to the range of \(P\). Hence \[PpP=\lambda P, \qquad \|Pp\|^2=\|PpP\|\le\lambda.\] Insert \(P+(1-P)\) on both sides of \(w\) in \(pwp\). The term with \((1-P)w(1-P)\) vanishes. The remaining three terms have norms at most \(\lambda\|w\|\), \(\sqrt\lambda\|w\|\), and \(\sqrt\lambda\|w\|\), respectively. This proves Equation (73). ◻

Proof. Let \(\Omega\) be the GNS state vector. Faithfulness makes it separating and hence cyclic for \(M'\). For any \(\alpha>0\), choose \(c'\in M'\) with \(\|z\Omega-c'\Omega\|<\alpha\). Then every \(\|D\|\leq C\) satisfies \[\|Dz\|_\varphi \leq C\alpha+\|Dc'\Omega\| = C\alpha+\|c'D\Omega\| \leq C\alpha+\|c'\|\|D\|_\varphi.\] Choosing \(\alpha\) first proves the uniform assertion (74), including families indexed by additional parameters.

The left multiplier bound is immediate. For \(b\in M_\varphi\) and \(D_0\geq0\), centrality gives \(\varphi(b^*D_0b)=\varphi(D_0bb^*)\). The functional \(c\mapsto\varphi(D_0c)\) is positive on \(M_\varphi\), since for \(c\geq0\) there it equals \(\varphi(c^{1/2}D_0c^{1/2})\). It follows that \(\varphi(b^*D_0b)\leq\|b\|^2\varphi(D_0)\), proving Equation (75).

For a norm-bounded state-norm precompact set \(\mathcal C\), take a finite state-norm approximation \(z_1,\ldots,z_m\) at a prescribed accuracy. If \(z\in\mathcal C\) is close to \(z_i\), then \[\|Dz\|_\varphi \leq\|Dz_i\|_\varphi+C\|z-z_i\|_\varphi.\] The finitely many fixed-factor moduli give one modulus for \(\mathcal C\). The set consisting of a convergent sequence and its limit is precompact, so this applies to every bounded sequence \(z_j\) with \(\|z_j-z\|_\varphi\to0\). Apply the right multiplier bounds successively in the order in which the factors occur. At each step the errors remain uniformly bounded in operator norm, so the next modulus continues to apply. For a scalar word with a middle error \(D\), the estimate \[|\varphi(UDV)|\leq\|U\|\,\|DV\|_\varphi\] and telescoping reduce the approximation to the preceding bounds. Strong-* approximation ensures state-norm convergence of a replaced letter and its adjoint whenever both occur. For each fixed approximant the finite right words may be expanded as needed; its approximation index can then be sent to infinity after the uniform error estimates. ◻

Proof. We first record how operator approximation will be used. We then reduce to centered cyclic words. Their label averages are sums of terms with prescribed equality relations among partition indices. A shortest pair of linked occurrences will give a state Cauchy–Schwarz estimate whose small factor is controlled by Proposition 8.

Proof. The extension \(\sigma\)-algebra is generated by base measurable sets and countably many label-coordinate readings along the partial graphings. Dyadic approximations to the label coordinates therefore show that the finite cylinder set algebra, allowing arbitrary base measurable sets, is dense in probability. Approximate each of the measurable sets corresponding to the \(p_i\) by sets in this algebra. Disjointize the approximations in order and put the remaining complement into one piece, retaining exactly \(r\) pieces and covering the identity. The resulting symmetric-difference measures can be made arbitrarily small, which gives the displayed state-norm bounds for their projections. There are only finitely many pieces and coordinates, so all of them use one common finite dyadic symbol size after refinement.

The resulting identities \(b_i=b_i^*=b_i^2\), \(b_ib_h=0\) for \(i\ne h\), and \(\sum_i b_i=1\) are identities of finite cylinder functions. The full-support argument in the construction of \(\mathrm{ev}_{\mathcal P}\) shows that substitution preserves all these identities. Thus their evaluations form a genuine projection partition in \(A\). ◻

Proof. The assertions for constant sequences and centralizer sequences follow from Lemma 25. If \((R_n),(S_n)\in\mathscr R_\omega\) and \((D_n)\) is a bounded state-norm-null sequence, then \((D_nR_n)\) is another such sequence, so \(\|D_nR_nS_n\|_\omega\to0\). Addition is immediate. Closure in the supremum norm follows by approximating the right multiplier uniformly before taking the limit in \(n\).

For the perturbation assertion, if \((E_n)\) is bounded and state-norm-null, then \[\|D_nE_n\|_\omega\le\|D_n\|\,\|E_n\|_\omega\longrightarrow0.\] Thus \((R_n+E_n)\in\mathscr R_\omega\) whenever \((R_n)\in\mathscr R_\omega\).

For a polynomial \(f\), the assertion about membership follows from the algebra property. Polynomial differences are controlled by the telescoping identity \[U_n^j-V_n^j=\sum_{h=0}^{j-1}U_n^h(U_n-V_n)V_n^{j-1-h}.\] Every right multiplier on the right belongs to \(\mathscr R_\omega\), while the left multipliers are uniformly bounded. Each summand therefore tends to zero in state norm. Uniform polynomial approximation on the common spectral interval proves both assertions for continuous \(f\). ◻

Proof. By Lemma 6, it suffices to work with separable predual. The conditional expectation in that reduction is the restriction of \(E_A\), so proving the stated diagonal choice in the smaller inclusion proves it in the original one. We henceforth use the notation of Sections 3 and 7.

Fix \(r=r_{\mathrm e}\) and write \[a=E_Ax,\qquad y=E_Nx-E_Ax\in N, \qquad w=x-E_Nx\in\ker E_N.\] The three elements are self-adjoint, \(\|a\|\le1\), and \(\|y\|,\|w\|\le2\). Moreover \(E_Ay=0\): uniqueness of the \(\varphi\)-preserving expectation onto \(A\) gives \(E_AE_N=E_A\).

Proof of Theorem 1. If \(x=0\), take \(p_1=1\), \(p_2=\cdots=p_{r_\varepsilon}=0\), \(a=0\), and \(q=1\). Otherwise replace \(x\) by \(x/\|x\|\). Proposition 29, together with Proposition 4, supplies the required paving in the original representation using \[r=\left\lceil400000000\varepsilon^{-2}\right\rceil +\left\lceil4/\varepsilon\right\rceil+1\] projections. This number depends only on \(\varepsilon\). The supporting corner and all partitions, diagonal elements, and compression projections may depend on \(F\) and \(\delta\), as permitted in Definition 1. The construction has diagonal norm at most one, and \[r\le400000000\varepsilon^{-2}+4\varepsilon^{-1}+3 \le400000007\varepsilon^{-2}<500000000\varepsilon^{-2}.\] Multiplying the diagonal by the original \(\|x\|\) restores scale and gives every assertion of Theorem 1. ◻