Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras

Proof. Theorem 1.1 makes every self-adjoint element of M (\eps,r_\eps) so-pavable. The forward transfer in [PV15, Theorem 2.7(3)] gives the stated norm paving in M^\omega with the same error and number of projections. Taking the real part of the diagonal element preserves both estimates.□

Proof. Write \xi=h_\varphi^{1/2} and introduce the symmetric map

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

For z\in M, cyclicity and Hölder’s inequality give

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

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 (2.2) bound its variance by

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

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 (2.2), 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

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

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 (2.1) implies

Moreover

Inside q_0Mq_0, let q be the spectral projection of q_0yq_0 for [-\eps/2,\eps/2]. The spectral Markov inequality gives

Taking \kappa sufficiently small makes this less than \rho. With a=(1-1/r_{\rm p})b, we have \|a\|\leq1 and

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

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 2.1 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 \eps, and the diagonal norm is at most one. The choices can ensure \chi(1-q)<\delta^2, which gives

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 [Tak72] 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 (2.5). 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

Lemma 2.3 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 2.3 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

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 2.2.□

Proof. Use the standard direct integral decomposition of a type I algebra over its center into full operator algebras; see [KR86, Sections 6.6 and 14.1–14.2]. 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 [Haa79, Proposition 1.5 and Theorem 2.1]. 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

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

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 (3.3) 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 [Kec95, Theorem 18.10] supplies a measurable selection s\colon T\to X on a positive-measure Borel set T such that

For a fixed \mu-null Borel set S, this implies

Consequently

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

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

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

For partition step functions this follows from refinement in Equation (3.5); bounded strong approximation and normality of \theta extend it to all f\in A.

We show that Equation (3.6) forces b\in N. Write its polar decomposition as b=u|b|. The intertwining relations imply

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 [KR86, Sections 6.4–6.6] 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 3.1, 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

Set v=uw. Its range projection uu^*=\supp|b^*| and its initial projection e' belong to A. Since w commutes with \theta(A),

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

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

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 [RS17]. Real-rootedness is part of the mixed determinantal theory and also follows from Lemma 4.2 below. Equation (4.2) is [RS17, Theorem 25], whose hypotheses are zero-diagonal Hermitian contractions and k\geq2.

By [RS17, Proposition 16 and Theorem 15], for a tuple of zero-diagonal Hermitian matrices, replacing one member by zero cannot increase its largest root. Expansion in principal minors gives

a fixed principal minor on S occurs with probability 2^{-|S|}. Applying monotonicity while retaining either L or -L proves Equation (4.3). 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 (4.4), 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 (4.5).

For Hermitian data, the uniform-variable polynomials have real coefficients; half-plane nonvanishing and conjugation therefore show that all their roots are real. Equation (4.5) 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 (4.6).

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 (4.8), and principal-matrix interlacing proves the interlacing assertion. Finally, uniform specialization of Equation (4.4) gives

The right side is the Cauchy transform of the root-counting measure. Uniqueness of partial fractions proves the second identity in Equation (4.8).□

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

Both sides equal one for J=\varnothing. We also use the following consequence of multiaffineness and the deletion identity in Equation (4.4). For real coordinates b=(b_v)_{v\in I} and increments h_v\geq0,

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 4.1 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 (4.11) 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 4.1 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 (4.11), all the corresponding unweighted principal evaluations are positive at

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 (4.12) follows from the principal interlacing in Lemma 4.2; integration gives Equation (4.13). 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

Since L-DLD=EL+LE-ELE, it follows that

Here e_0 is the first unit vector of the arrow matrix. The left side is the tail integral in Equation (4.14). 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

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 (4.4),

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

Expanding the logarithmic derivative to order z^{-3} gives the first identity in Equation (4.15). 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 (4.15) is at most

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 4.4, proves Equation (4.16).

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 (4.17). 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

for the old arrow matrix. By Equation (4.9), 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

Since the first diagonal of H_+ is m, comparison of the diagonal blocks in H=H_+-H_- gives

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

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

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 (4.20). 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

and at least three quarters satisfy

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

where Lemma 4.4 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

Conjugate Equation (4.21) 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,

Put L_1=e_-. If L_1>0, conjugating Equation (4.22) by (L_1+|\Lambda_-|)^{-1/2} in the same way gives

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 (4.22) 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 (4.24) remains valid.

Choose a color for which both residue sums in Equations (4.23) and (4.24) 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

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

Together with Equation (4.19), this gives

Both cases imply Equation (4.18). All arguments allow empty deletion blocks and empty sums, so they include a one-site matrix and every zero-residue case.□

Proof. By Lemma 4.5, 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,

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

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 4.5; 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

The identity that the sum of the rooted measures counts all polynomial roots gives

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

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

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 (5.5) 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 (5.2) 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 (5.3): 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

Choose the polynomial approximants to x_+ constructed in the proof of Lemma 5.1. Uniform function approximation controls the two rooted terms on the left; the L^1 derivative bound in Equation (5.5) controls both comparison increments on the right. Finite measure then permits passage to the limit and proves Equation (5.6). 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 4.2, 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

where the second transform at v=u is zero. Choosing the phases of \lambda_v to make all summands nonnegative real yields

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=\diag(\lambda_v) and D_W=\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

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

Choose a disk of positive radius depending only on y,b_2,K on which Equation (5.10) 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 4.2 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

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

The probability of that color at an unassigned vertex is 1/r. Dividing by P_I proves Equation (5.11), 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

The constants in Equations (5.9) and (5.12) 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 5.1, including the conditional transforms in Equation (5.7). 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

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

At a J_i target, the corresponding expansion is

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 4.5. 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 4.5 and multiplication of the series.

For each individual coefficient the ordered local telescoping argument from Lemma 5.2, followed by Equation (5.2), proves the coefficient version of Equation (5.13). 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 (5.13) 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 (5.13) 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 (5.13) 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 (5.8) 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 5.3 gives, on the appropriate sites of the band, integrated absolute differences of order 1/H_1 for

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 4.5 is

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

The residue bound in Lemmas 4.2 and 4.5 also bounds J_i uniformly on these compact sets. Division is thus controlled in the identities

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,

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 4.4 and Equation (6.9) imply that

uniformly on the band, including for the shifted virtual options. Indeed, the bound in that lemma is

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

uniformly over these finite calculations. To see uniformity, interchange the integrals in Equation (6.10); 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

For fixed B, one has \delta_B(y)\to0 as y\downarrow0. Combining the smoothing error with Equation (6.11) gives, uniformly for the finite rooted calculations on the band,

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 5.1 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 5.1 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 (6.9) and Lemma 4.4: 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

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 4.6 gives the single-site inequality on finite restrictions. Lemma 5.1 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 (6.8).□

Proof. In the strip integral there is the identity

Divide the height interval [-n,n] into unit intervals. Translating each to a fixed interval uses the homogeneity proved in Lemma 6.2. 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 (6.15).

For Equation (6.16), 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 6.2. On this bounded cut, Lemma 5.1 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 (6.16).

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 4.3 gives zero cost on every finite principal restriction. Finite restriction limits and Lemma 6.2 therefore give \mathcal S=0.

To prove Equation (6.17), 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 (6.6) 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 5.2, deletion cannot increase the total integrated cost, so

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 (6.18) equals the integral over the selected lifted centers of s(P_{\mathrm{ball}}).

For a bad ball, the second part of Lemma 4.3 gives s(P_{\mathrm{ball}})\geq(\alpha/2)\min_{\mathrm{ball}}W. The edge-ratio bound then gives

Divide Equation (6.18) by 2n and let n\to\infty. Equation (6.15) 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 (6.19) gives

Summing over the bounded number of classes proves Equation (6.17). 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 5.2 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

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 (6.21) 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

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 (6.21). 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 (6.21) by 2n and use Lemma 6.3. 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

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 (6.16) 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 6.2.

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 (6.22) by their values at c creates an error in absolute value at most

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 (6.22). By Equation (6.8), the resulting old-cost contribution is at most

For each prescribed accuracy in Equation (6.23), 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 (6.20) uniformly over the current configurations.□

Proof of Theorem 6.1. 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 6.4. Choose T_0>0 and h\in(0,1/2), and apply the construction of Lemma 6.5 with independent fresh coordinates for

steps. Write S_j for the cost after j steps. Iterating Equation (6.20) gives

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 6.4. First choose T_0 large, and then h small, so that

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 (6.4).

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

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 (6.3) 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 [Ued99, Section 2] and [Ued13, Section 2]. 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 [Tak72], the modular group of \Phi restricts to the modular group of each constituent. This also agrees with [Ued13, Equation (2.1)]. 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

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

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 (7.1).□

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

Choosing \alpha first proves the uniform assertion (7.2), 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 (7.3).

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

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

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 3.2.

Uniform approximation in fixed words. Lemma 7.3 controls bounded errors in the state norm when the right word is formed from fixed operators and bounded varying A-functions. Expanding each fixed finite normalizer sum when needed, telescoping and

bound scalar word errors uniformly over the evaluated diagonal functions. If a fixed N-letter is approximated by a bounded strongly convergent sequence from N_{\rm alg}, that sequence and its limit form a set precompact in the state norm, so the lemma applies to the whole sequence as right factors. Each normalizer sum is expanded only after that approximant is fixed; no uniform bound on all such expansions is needed. The order is to fix an approximant, pass to the partition limit, and then let its approximation error tend to zero. The same statements apply in the model, with \Phi and bounded factors from B, which lies in its centralizer.

Reduction to centered alternating words. We induct on the number of factors. A word of length one in M gives an identity. For one factor in \mathcal Y, average its cylinder coefficients. At almost every base point, distinct sites in a fixed finite translated list eventually belong to distinct atoms of \mathcal P_j. The averaged coefficient thus converges boundedly almost everywhere to its conditional label average. This is also state-norm convergence, and the fixed normalizers are allowed right multipliers. It proves Equation (7.4) in length one.

Adjacent factors from the same algebra can be folded into one factor. For an alternating word, subtract and add the N-expectation of each factor in the model. Make this same expansion on the evaluated side; it is not necessary that a centered model factor remain centered after evaluation. The expectations of \mathcal Y-factors lie in N_{\rm alg} and are fixed by evaluation. Every term containing an expectation can be reduced in length by folding it into a neighbor. If an arbitrary N-element from the expectation of an M-factor must be multiplied into a \mathcal Y-factor, first approximate it boundedly in the strong-* topology by N_{\rm alg}. Such approximation follows from generation and Kaplansky density, and is legitimate on both sides by the uniform word estimates above. The induction hypothesis therefore handles all terms except fully centered alternating words. Their model expectation is zero by freeness.

We may approximate each centered M-factor boundedly in the strong-* topology by an entire analytic element for \sigma^\varphi belonging to M^\circ. For example, convolve its modular orbit with Gaussian approximate identities; complex translations of the Gaussian give the entire continuation. The expectation E_N commutes with the modular group, so the convolutions remain centered. Their complex modular translates also remain in M^\circ, by analytic continuation after applying E_N.

Each centered \mathcal Y-factor is a finite sum of terms vb, where v is a base kernel groupoid normalizer and b is a cylinder function whose conditional label average is zero. To obtain this form, move each coefficient to the right of its normalizer by covariance on the relevant supports, incorporating bounded base coefficients into b, and subtract the expectation of each summand. The sum of the subtracted expectations is zero. Cut the initial support of each v as in Section 3 to make it entire analytic. These cuts preserve centering of the cylinder coefficients and have arbitrarily small state errors in the fixed words. The cylinder functions and their evaluations belong to the respective centralizers. Hence complex modular shifts of vb are compatible with evaluation: the shift only changes the cut normalizer by the same bounded base formula in both algebras. We fix these analytic approximants before passing to the partition limit; their complex-shift norms may enter all subsequent bounds.

For an odd alternating word of length at least three, the first and last factors lie in the same constituent algebra. Apply the KMS identity

to the first evaluated analytic factor, and the corresponding identity for \Phi in the model. Fold the last factor with the shifted first factor and apply the induction hypothesis to the shorter word. Its model value equals that of the original centered word and is therefore zero.

For an even alternating word, rotate in the same way if necessary to start in M^\circ. Expand the other-side factors as vb and absorb each v into the preceding centered M-factor. Modular shifts of normalizers, if present, only add fixed base coefficients. Writing the length of this word as 2m, it remains to prove that the averaged evaluations of

tend to zero. Here the y_h are fixed bounded entire analytic elements of M^\circ, and each d_h\in A is the substitution in a centered finite cylinder function. All constants below may depend on this fixed word and on the fixed analytic approximants.

Fourier expansion and index equalities. Split each cylinder domain according to the domains of its translated sites and the equality pattern among those sites. On each piece, its distinct label symbols are independent. Fourier expansion on the finite product of cyclic groups expresses the centered coefficient as a finite sum of monomials with fixed bounded A-coefficients. Centering removes the constant Fourier term. Delete occurrences having exponent zero modulo s. Each remaining monomial has at least one occurrence, and all its exponents are nonzero modulo s. Within each diagonal block d_h, the sites of its occurrences are pointwise distinct on that block’s indicated support.

We may, after further finite splitting, regard every translation used at an occurrence as an automorphism of the full abelian algebra A. Here the extension away from the specified support need not belong to \mathcal R. Proposition 3.2 shows that the y_h annihilate the atomic part of A on both sides, so only diffuse supports contribute to Equation (7.5). The partial isomorphisms map these supports into the diffuse part modulo null sets. Split a nonnull common domain piece of a block into two positive-measure pieces. On either smaller piece, both its domain and the range of each restricted map have nonnull diffuse complements in the diffuse part: the other piece and its image provide such complements. Normalize the measures on each pair of complements separately and use the isomorphism theorem for atomless standard probability spaces [Kec95, Theorem 17.41]. It extends each restricted map to a measure-class automorphism of the diffuse part. Use the identity on the atomic part. This gives finitely many terms, and for an occurrence o we can write its translation as

where \gamma_o is a nonsingular measurable bijection modulo null sets.

Expand Equation (7.5) in partition indices a_o, one for each occurrence. Averaging the independent symbols z_a keeps precisely the index equality patterns for which the sum of the Fourier exponents in each equality class is zero modulo s. Since individual exponents are nonzero, every such class has at least two occurrences. Inclusion–exclusion over the finitely many possible coarsenings removes inequalities between distinct classes. Thus it suffices to prove vanishing for a sum with prescribed equalities within the classes of a partition of the occurrences, every class having size at least two, and with no inequalities imposed between classes. An occurrence contributes the projection \lambda_o(e_{a_o}).

We will keep selected pair equalities as explicit summation indices. Whenever the selected pairs form a forest in each equality class, extend them to a spanning tree of that class. Encode the other tree edges with independent auxiliary phases. For an edge between indices a,b, its equality indicator is

where the \zeta_a are independent uniform circle phases; use a fresh phase vector for each edge. At an occurrence o, let w_o(a) be the product of its incident auxiliary edge phases, with conjugates according to their orientations and with empty product one. Thus |w_o(a)|=1. Set

The projections \{\lambda_o(e_a)\}_a form a partition, so U_o is a diagonal unitary. Summing an unretained index gives U_o, while at a retained index j one has

Thus each U_o, as an operator, is independent of the retained numerical indices. Different U_o may be correlated through the auxiliary phases; the estimates below hold uniformly over their joint realizations as diagonal unitaries before averaging. Products of these factors commute within each diagonal block.

A pair in one diagonal block. Suppose two occurrences in one equality class belong to the same block. Retain their pair equality and encode the remaining tree edges as just described. Summing the pair index produces the projection

The two sites are distinct on the fixed support, while the partitions separate points. Hence \mu(p_{\mathcal P})\to0. Use KMS rotation to place this projection first in the word. Only fixed analytic letters are shifted; diagonal factors are in the centralizer and remain unchanged. The rotated word has the form \varphi(p_{\mathcal P}Z) with a uniform norm bound on Z. Centrality gives

so these sums tend to zero uniformly in the auxiliary phases.

A shortest pair between different blocks. It remains to treat equality patterns with at most one occurrence of any class in each diagonal block. This case has m\geq2. Number the blocks by \mathbb Z/m\mathbb Z, and for distinct positions h,k define

where both residues are taken in \{1,\ldots,m-1\}. Among pairs of occurrences in the same class, choose one minimizing this cyclic distance, and call the minimum \ell. Choose the endpoint order so that one shortest arc runs forward in the cyclic order of the word, and label its blocks 0,1,\ldots,\ell; either endpoint order is allowed in an antipodal tie.

For each interior block h=1,\ldots,\ell-1, choose one occurrence and one other occurrence of its class. Its mate lies outside the entire chosen arc, including the endpoints. Indeed, a mate in another block k on that arc would satisfy d_m(h,k)\leq|h-k|<\ell, contradicting minimality. The same inequality shows that the interior classes are distinct from each other and from the endpoint class. When \ell=1 there are no interior blocks.

Retain the endpoint pair and these interior-to-mate pairs. These pairs lie in distinct equality classes, so they form a forest and can be included in the spanning trees used above. The retained indices are J=(j_0,j_1,\ldots,j_{\ell-1}), with j_0 linking the endpoints. Write

Each j_h for h\ge1 occurs in one further retained projection in an exterior block. Several such mate projections may lie in the same exterior block.

Rotate the summands by KMS to begin at E_0, and insert E_0 at the far end inside the state, using centrality. Absorb fixed diagonal coefficients into neighboring fixed letters. More explicitly, within a diagonal block place the retained projections first, then the auxiliary unitaries, then its fixed coefficient; absorb that coefficient into the following intervening letter. The fixed intervening letters, including any modular translates created by rotation, remain bounded analytic elements of M^\circ. Relabel them as y_h, and write V_h\in A for the auxiliary unitaries, which are independent of the explicit indices. Split the word at E_\ell' using (E_\ell')^2=E_\ell'. Before averaging the auxiliary unitaries, the resulting summands have the form \varphi(X_JY_J), where

The product in X_J is in increasing order and is omitted when \ell=1. In B_{j_1,\ldots,j_{\ell-1}}, each exterior mate projection for these indices occurs exactly once; every other factor is independent of them. Diagonal factors in the same block commute.

Figure 1 illustrates the selected links in one allowed cyclic pattern.

State Cauchy–Schwarz gives

The second square sum is uniformly bounded. First drop E_\ell' from the positive operator bound for Y_J^*Y_J. Order the exterior mate occurrences from left to right, using commutativity for those in one diagonal block. Write \mathbf j=(j_1,\ldots,j_{\ell-1}). At the first remaining mate occurrence, factor

where j is that occurrence’s index, \mathbf j' lists the others, \{F_j\}_j is its projection partition, and C_0,B'_{\mathbf j'} are independent of j. Then

Iterating over the remaining mate occurrences, with the fixed norm bounds on the intervening multipliers, gives

for a fixed constant C. When \ell=1, this is simply the fixed norm bound for the exterior word. Finally sum the E_0 corners in the state. As \{\lambda_0(e_{j_0})\}_{j_0} is a partition in the centralizer, this leaves a bound independent of \mathcal P and of all auxiliary unitaries.

We have reduced the moment estimate to the first square sum in Equation (7.7). Its smallness will come from a collision between the two endpoint indices. The interior pinchings first converge at fixed coarse endpoint data; only afterward will those coarse endpoint data be refined.

Remove V_0 from the state by centrality and sum the interior indices in X_JX_J^*. The result is

where the positive map C_{\mathcal P}:A\to M is given by the nested operations

For \ell=1 this means simply C_{\mathcal P}(f)=y_1fy_1^*.

Fix a coarse partition \mathcal Q from the generating sequence and let \mathcal P refine it. Positivity permits each input projection in Equation (7.8) to be enlarged to its coarse piece. Regrouping the output corners by centrality then bounds that expression by

For each fixed f\in A, the nested expression defining C_{\mathcal P}(f) converges in state norm to the expression in which each pinching is replaced by E_A and the V_h disappear. This convergence is uniform over the auxiliary unitaries. Indeed, work from the innermost pinching outward. Each translated sequence \lambda_h(\mathcal P_j) is generating, so Lemma 2.3 gives its state-norm convergence to E_A on a fixed input. Conjugation by V_h commutes with the pinching, preserves the state norm, and fixes the limiting A-valued datum. Pinching is contractive in the state norm. Its error can thus be propagated through the fixed left and right multipliers y_h,y_h^* using Lemma 7.3, uniformly over all the remaining phases. This proves the asserted inside-out convergence.

Define positive kernel maps on A by

For each fixed input f=\lambda_\ell(e), e\in\mathcal Q, the preceding convergence applies. Since \mathcal Q is finite, the limit of Equation (7.9), with \mathcal Q fixed, is

Proposition 3.2, applied to the adjoints of the y_h, says that these kernels and their composition are atomless almost everywhere. In taking compositions, a preceding kernel avoids the fixed null exceptional set where a following kernel may fail to be atomless.

Finally refine \mathcal Q along the generating sequence. The integrand sum in Equation (7.10) tests whether \gamma_0 of the output site and \gamma_\ell of the input site lie in the same partition piece. These conditions decrease to equality of those two points. Since \gamma_\ell is one-to-one, this is a graph having zero conditional mass for each atomless row of the composed kernel. Nonsingularity makes all fixed null exceptions irrelevant. The kernel masses are bounded, so bounded convergence shows that Equation (7.10) tends to zero.

For each fixed coarse \mathcal Q, the limsup of the first square sum, even after taking the supremum over the auxiliary unitaries at each fine \mathcal P, was bounded by Equation (7.10). First take that fine-partition limsup and then refine \mathcal Q. The first square sum therefore vanishes uniformly in the auxiliary phases. The second square sum was uniformly bounded, so Equation (7.7) proves vanishing of the equality-pattern sum. Averaging the auxiliary phases, undoing the finite inclusion–exclusion and Fourier expansions, and then removing the fixed analytic approximations proves the required vanishing in Equation (7.5). The induction now proves Equation (7.4).□

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 7.3. 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

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

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 2.4, 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

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.

Finite-support Cartan approximations. Choose self-adjoint contractions in N_{\mathrm{alg}} converging strongly to E_Nx. To obtain them, first apply Kaplansky density to the norm closure of the finite normalizer algebra, then approximate its self-adjoint contractions in norm by self-adjoint elements of N_{\mathrm{alg}} and rescale. Bounded strong approximation can be taken sequentially in the faithful separable state representation.

For each approximant, only finitely many partial isomorphisms occur in its support. We may also arrange a bound on their edge Radon–Nikodym ratios. Indeed, for such a finite list of maps \gamma, remove from X each domain set where \Delta(\gamma v,v) lies outside [b^{-1},b], and let e_b\in A be the indicator of the remaining set. As b\to\infty, these projections increase strongly to 1. Compressing the approximant by e_b preserves self-adjointness and its contraction bound, and restricts every surviving edge to the required ratio range. Enlarge the finite list by its inverses if necessary. Choosing b sufficiently large for each approximant preserves strong convergence.

Subtracting the diagonal now gives a sequence

Each y_n/2 is represented by measurable zero-diagonal Hermitian contractions on orbit \ell^2 spaces, supported on an undirected graph of finite degree with bounded edge cocycle ratios. These bounds may depend on n. Theorem 6.1 permits this dependence: it does not affect r.

One partition for the two operator terms. Use the fixed relative Bernoulli extension \widetilde N and amalgamated product

from Lemma 7.1. Apply Theorem 6.1 to y_n/2 with

Denote the color projections by p_1^n,\ldots,p_r^n\in B. They form a partition, belong to the centralizer of \Phi, and satisfy

With \beta=80, the norm bound supplied by the coloring theorem, after restoring the factor two, obeys

Put

At a root where the radius-n ball has the asserted pinching bound, \widehat Y_n^n applied to the root vector agrees with the nth power of that principal matrix applied to the same vector. Every path of length n starting at the root stays in its radius-n ball. At all other roots, \|\widehat Y_n\|\le2. Integrating the squared norms of these vectors gives

This uses diagonal integration in the relation representation, not a trace on the nonsingular relation algebra.

For the original operators set

Pinching by a centralizer partition is contractive in the state norm, so Equation (8.1) implies

By Lemma 7.2 and Equation (8.3),

Here orthogonality of the p_i^n makes the norm of the block sum the maximum of its block norms. Thus the same r projections control the complementary term; there is no second partition refinement.

Clipping before fixing a transferred moment. Let f:\mathbb R\to[-\eps/2,\eps/2] be the continuous clipping function

Spectral calculus and Equation (8.5) show that

Since \|\widehat Y_n\|\le2, it follows that

We apply Lemma 8.1 in \mathcal K. Each sequence (p_i^n) lies in \mathscr R_\Phi, and r is fixed. Hence (Y_n) and (W_n) belong to that algebra, being finite sums of products of centralizer sequences and fixed operators. Equation (8.6) also places (\widehat Y_n) in it. Equations (8.6)–(8.8) and the continuous-calculus conclusion of the lemma yield

The self-adjoint operators

therefore differ by a bounded state-norm-null sequence from

Both sequences belong to \mathscr R_\Phi. For every fixed integer k\ge1, telescoping their powers gives

Only a fixed power is used at this stage. In particular, Equation (8.1) required no prescribed rate of convergence.

Transfer and the final spectral cut. Fix \rho>0. First choose k so large that (3/4)^{2k}<\rho/4. Then choose n sufficiently large in Equation (8.10) that

With n and k fixed, approximate the partition (p_i^n) in B by a finite-symbol cylinder partition, using Lemma 7.5. Bounded state-norm approximation of these self-adjoint projections is strong-* approximation, so the fixed polynomial moment changes arbitrarily little. We may retain the strict upper bound \rho\eps^{2k} with positive slack.

Proposition 7.4 applies to the expansion of this moment. For a sufficiently fine base partition, its averaged evaluated moment remains strictly less than \rho\eps^{2k}. Every evaluation is a genuine projection partition in A, and its even moment is nonnegative. Some deterministic evaluation consequently gives a partition P=(p_1,\ldots,p_r) with

Let z=\sum_i p_i(x-a)p_i=z^* and take q=1_{[-\eps,\eps]}(z)\in M. Then

Because a\in A commutes with the partition, z=S_P(x)-a. This proves the proposition with the number of colors fixed before \rho.□

Proof of Theorem 1.1. If x=0, take p_1=1, p_2=\cdots=p_{r_\eps}=0, a=0, and q=1. Otherwise replace x by x/\|x\|. Proposition 8.2, together with Proposition 2.2, supplies the required paving in the original representation using

projections. This number depends only on \eps. The supporting corner and all partitions, diagonal elements, and compression projections may depend on F and \delta, as permitted in Definition 1.1. The construction has diagonal norm at most one, and

Multiplying the diagonal by the original \|x\| restores scale and gives every assertion of Theorem 1.1.□