Write \(x+s=(1-s)x+s(x+1)\) and \(x+1=s(x+s)+(1-s)(x+s+1)\), apply log-convexity of \(\Gamma \) to each, and use \(\Gamma (y+1)=y\Gamma (y)\). For the last claim, \((x+s)^{s-1}\ge x^s/(x+s)\), so the lower bound is at least \(x^s-sx^s/(x+s)\ge x^s-s\) when \(x\ge 1\).

Let \(n=\lfloor u\rfloor \) and \(s=u-n\). Wendel’s inequalities at \(x=n+1\) give \(|\log \Gamma (1+u)-\log n!|\le \log (u+2)\). The Stirling bounds control \(\log n!-(n\log n-n)\), and \(u\log u-u\) differs from \(n\log n-n\) by at most \(\log (u+2)+2\).

Every argument of \(\Gamma \) is positive. Adding the box raises the number of copies by one and the \(i\)-th shifted part by one, leaving the other shifted parts unchanged.

Apply the Stirling estimate to each of the \(d+2\) Gamma factors. The terms containing \(\log t\) cancel up to a constant because \(\sum _il_i=k+\binom d2\). The shifts \(0\le l_i-\lambda _i\le d-1\) change \(\sum _il_i\log l_i\) by \(O_d(\log (k+2))\), and \(kH(\lambda /k)=k\log k-\sum _i\lambda _i\log \lambda _i\).

The ratio is a quotient of two Wendel ratios, with \(x=1+t(l-1)\) and \(y=c+t(k-1)\), both at least one. Wendel’s inequalities give \((1-t)(x/y)^t\le r(k,l)\le (x/y)^t/(1-t)\). On the stated range, \(tl\le x\le (1+d)y\) and \(y\le (c+t)k\).

Wendel’s inequalities give \(|r(k,l)-(x/y)^t|\le Ck^{-t}\), since \(y\ge tk\) and \((x/y)^t\le 1+d\). Moreover \(|x/y-l/k|=O(1/k)\) and \(|l/k-\max ((l-d)/k,0)|\le d/k\), and \(|a^t-b^t|\le |a-b|^t\) for \(a,b\ge 0\).

Polynomials act pointwise, and the spectrum lies in the range of \(g\), so interpolating \(f\) on that finite set by a polynomial proves the first claim. For the second, \(A^\ast A=\sum _i|c_i|^2P_i\le M^2I\) and \(\| A\| ^2=\| A^\ast A\| \).

Intertwined permutation actions have the same group-algebra image up to relabelling the basis, and an action on the first factor of a product gives the image tensored with the identity. The row bound is the corresponding statement for \((\mathbb {C}^{d_Q})^{\otimes k}\).

The weights are positive and the label projectors form an orthogonal resolution of the identity. The other claims hold for every central label function, by the commutation and complement properties of the label projectors.

Both sides are functions of the label projectors, so it suffices to compare the coefficients of the labels that occur. Such a label has at most \(d\) rows, \(d_\lambda \) equals the dimension formula, and the entropy asymptotics of the dimension formula and of the label function give \(|\log w_k(\lambda )+t\log d_\lambda |\le C\log (k+2)\).

The label projections of disjoint regions commute. Their products form a common orthogonal resolution. On a nonzero product with labels \(\alpha ,\beta ,\gamma \) for \(P,F,Y\), respectively, the inverse metric has coefficient \((w_k(\alpha )w_k(\beta )/w_k(\gamma ))^2\). If \(C_0\) is the constant in Theorem 23.5.3, the logarithm of this coefficient is at most

Exponentiating and summing over the common orthogonal resolution proves the assertion with \(C=6C_0\).

The extended metric is the central label function of the copy permutations fixing the last copy. On a nonzero block the labels are compatible, so \(\nu =\lambda -e_i\); the star eigenvalue is the one of the Schur-label estimates, and the ratio of label functions is the exact marked ratio.

Both operators are diagonal in the branch decomposition; apply the limit and the bounds of the marked ratio blockwise and the operator-norm bound on an orthogonal resolution.

The average of \(|z\rangle \langle z|\) over the copy permutations of \(Q^c\) commutes with the copy permutations of \(Q\), because \(z\) is symmetric, and pairs with every operator generated by them as \(k!\, |z\rangle \langle z|\); this is the matrix form of the permutation invariance of the \(Q\) marginal. The branch-probability identity of the Schur-label estimates applies to it. Distinct branches of one label differ in their removed row.

Both operators are diagonal in the branch decomposition. On a branch, \(r^{-2}\| \pi ^\lambda \pi ^\nu z\| ^2\le C(l_i/k)^{1-2t}\| \pi ^\lambda z\| ^2\), and summing over at most \(d\) branches and over labels gives the first bound. For the second, on \(J\ge \delta \) the clipped inverse differs from \(r^{-1}\) by \(O(k^{-t}\delta ^{-2t})\). On \(J{\lt}\delta \), \(l_i/k{\lt}\delta +d/k\), and both inverses contribute at most \(C\bigl((\delta +d/k)^{1-2t}+\delta ^{-2t}(\delta +d/k)\bigr)\| \pi ^\lambda z\| ^2\). Let \(k\to \infty \) and take square roots.

The first two statements compare matrix entries: \(h^{(j)}\) is nonzero only between configurations that agree outside the support factors of copy \(j\), and the permutations in question preserve this relation and these factors. The operator \(h^{(j)}\) is \(h\otimes I\) after splitting off copy \(j\). For the decomposition, expand a local operator on \(X\cup T\) in matrix units.

The square root of \(A_k\) is \(W_{P,k}^{-1}W_{F,k}^{-1}W_{Y,k}\), the \(F\) factors commute with \(\bar h\) and cancel, and \(O_k\) preserves \(\mathcal S_k\). Between symmetric vectors, \(h^{(k)}\) may replace \(\bar h\). Write \(h=\sum _ib_ia_i\) with \(b_i\) supported on \(Y\) and \(a_i\) on \(P\). Since the metric of the first \(k-1\) copies commutes with operators on the last copy,

Hence \(\langle w,O_kz\rangle =\sum _i\langle R_Yb_i^{(k)\ast }R_Y^{-1}w, R_Pa_i^{(k)}R_P^{-1}z\rangle \) for symmetric \(w,z\), in which both inverses act on symmetric vectors. The forward and inverse bounds on the marked ratio bound each term by \(C_i\| w\| \| z\| \); take \(w=O_kz\).

On the corner of the matrix unit \(E^\lambda _{00}\) the operator commutes with every compressed matrix, because the averages \(\sum _jE^\lambda _{j0}BE^\lambda _{0j}\) lie in the commutant; hence it is a multiple of the corner projection there. The matrix units \(E^\lambda _{i0}\) carry this multiple to the whole block.

Differentiate along the one-parameter unitary groups \(1+(\cos \theta -1)P+(\sin \theta )J\) with \((P,J)=(e_{aa}+e_{bb},e_{ab}-e_{ba})\), \((e_{aa}+e_{bb},i(e_{ab}+e_{ba}))\) and \((e_{aa},ie_{aa})\), whose derivatives at \(\theta =0\) span all matrices. By Schur–Weyl duality the diagonal actions generate the commutant of the copy permutations.

Left invariance of Haar measure makes \(T(A)\) commute with every unitary tensor power, and \(T(A)\) commutes with the copy permutations because tensor powers and \(A\) do. The previous lemma gives the central form. The label projectors commute with unitary tensor powers, so \(\operatorname{Tr}(\pi ^\lambda T(A))=\operatorname{Tr}(\pi ^\lambda A)\).

The averaging element \(\varepsilon \) satisfies \(\varepsilon a=\operatorname {aug}(a)\varepsilon \). The augmentations of the \(e_\lambda \) are idempotent, sum to one and have vanishing pairwise products, so exactly one \(e_{\lambda _0}\) has \(\varepsilon e_{\lambda _0}=\varepsilon \). In the simple block of \(\lambda _0\) the central idempotent \(\varepsilon e_{\lambda _0}\) is nonzero, hence equal to \(e_{\lambda _0}\). Label projectors are self-adjoint idempotents.

The integral is the twirl of \(P_{e,k}\), a central label function with coefficients fixed by \(\langle e^{\otimes k},\pi ^\lambda e^{\otimes k}\rangle \). The symmetric vector \(e^{\otimes k}\) lies in the range of the trivial label projector \(\Pi _k\) and is orthogonal to the other label projectors.

\(P_{\theta ,k+m}=P_{\theta ,k}\otimes P_{\theta ,m}\); pair the coherent-state resolution for \(k+m\) copies with \(\sigma \otimes G\).

The coherent-state resolution gives total mass \(\operatorname{Tr}(\sigma \Pi _k)=\operatorname{Tr}\sigma =1\). The density \(D_k\operatorname{Tr}(\sigma P_{\theta ,k})\) is nonnegative, and \(|\langle \theta ^{\otimes m},G\theta ^{\otimes m}\rangle |\le \| G\| \) for a unit vector \(\theta \).

Write configurations of \(k+m\) copies as a configuration \(u\) of the first \(k\) copies and \(w\) of the last \(m\). The permutation \(\pi _\iota \) sends \((u,w)\) to the configuration whose first part is \(u\) with \(w\) inserted at \(\iota \) and whose last part is \(u\circ \iota \); the trace is the resulting partial trace.

Expand \(\Pi _{k+m}\) into permutations in the finite-copy identity. A permutation sending each of the last \(m\) copies into the first \(k\) is uniquely \(\pi _\iota \gamma \) with \(\gamma \) fixing the last \(m\) copies; symmetry of \(\sigma \) absorbs \(\gamma \), and the previous lemma evaluates \(\pi _\iota \). These permutations therefore contribute their proportion times \(\operatorname{Tr}\sigma \mathcal T_{k,m}(G)\). The other permutations send a prescribed one of the last copies to a prescribed one of them for some pair, so they form a fraction at most \(m^2/(k+m)\), and each term is at most \(d^m\| G\| \) in absolute value because \(|\operatorname{Tr}(\rho M)|\le \operatorname{Tr}\rho \, \| M\| \) for \(\rho \ge 0\). The normalization \(D_k/D_{k+m}\) is eliminated with the case \(G=1\), where the coherent measure has total mass one. For small \(k\) both terms are bounded by \(\| G\| \).

If \(\iota ,\kappa \) are the restrictions of an injection \(\lambda \) of \(m+n\) copies, then \(G_\iota H_\kappa =(G\otimes H)_\lambda \), and pairs of injections with disjoint images correspond to injections of \(m+n\) copies. A pair has \(\iota (a)=\kappa (b)\) for prescribed \(a,b\) with probability \(1/k\), so pairs with intersecting images form a fraction at most \(mn/k\). These pairs and the change of normalization each cost at most that fraction times \(\| G\| \| H\| \).

Relocating a placement by a copy permutation conjugates it by that permutation, and relocation permutes the injections. Product vectors factor: \(\langle \theta ^{\otimes (m+n)},(G\otimes H)\theta ^{\otimes (m+n)}\rangle =\langle \theta ^{\otimes m},G\theta ^{\otimes m}\rangle \langle \theta ^{\otimes n},H\theta ^{\otimes n}\rangle \). The product estimate and the trace estimate follow termwise from the two previous theorems.

For products, write \(XX'\Pi =X\Pi (X'-Y')\Pi +(X-Y)\Pi Y'\Pi +YY'\Pi \), using that all factors commute with \(\Pi \), and approximate \(YY'\) by the previous lemma. Symbols are bounded on the compact unit sphere, so the symbol errors combine in the same way. For the trace limit, \(|\operatorname{Tr}\sigma (X_k-Y_k)|\le \| (X_k-Y_k)\Pi _k\| \), the coherent measure is a probability measure, and the previous lemma treats \(Y_k\). Polynomials follow by induction.

An injection average commutes with \(a^{(j)}\) except for the fraction \(r/k\) of placements meeting copy \(j\), so \(\| [\mathcal T_{k,r}(G),a^{(j)}]\| \le 2r\| G\| \| a\| /k\). Products are therefore reordered as \((AA')^{(k+1)}\mathcal T(G)\mathcal T(G')\) up to \(O(k^{-1})\), and the product of injection averages is handled by the product theorem. For compression, all placements of one copy are conjugate under copy permutations, so \(\Pi a^{(j)}\Pi =\mathcal T_{k,1}(a)\Pi \), and \(\mathcal T_{k,1}(a)\mathcal T_{k,r}(G)\) is an injection average of \(a\otimes G\) up to \(O(k^{-1})\).

The partial swap of \(Q\) between distinct copies \(i,j\) is \(\sum _{a,b}(E_{ab})^{(i)}(E_{ba})^{(j)}\) with matrix units \(E_{ab}\) on \(Q\). Averaging over the donor copy gives \(J_{Q,k}=\sum _{a,b}\mathcal T_{k,1}(E_{ab})E_{ba}^{(k)}+O(k^{-1})\), whose marked symbol is \(\sum _{a,b}\langle \theta ,E_{ab}\theta \rangle E_{ba}=\rho _Q(\theta )\otimes 1\).

For the first statement put \(w=\Pi _kM\Pi _kv\); then \(\| w\| ^2=\langle w,M\Pi _kv\rangle \le c\| w\| \| v\| \). For the second, \(f\) is a uniform limit of continuous functions on the unit sphere, and the approximations combine by the triangle inequality.

\(J_{Q,k}\) is \(k^{-1}\) times a sum of \(k-1\) permutation operators. On a nonzero branch projection its eigenvalue is bounded by its norm, and functions of \(J_{Q,k}\) act on the branch decomposition by their values at the eigenvalues.

For a real function \(g\), \(\| g(\rho _Q)\theta \| ^2=\operatorname{Tr}\rho _Qg(\rho _Q)^2=\sum _i\lambda _ig(\lambda _i)^2\) over the eigenvalues \(\lambda _i\in [0,1]\) of the marginal. For the clipping, \(\lambda (\max (\lambda ,\delta )^{-t}-\lambda ^{-t})^2\) vanishes for \(\lambda \ge \delta \) and is at most \(\lambda ^{1-2t}\le \delta ^{1-2t}\) for \(0{\lt}\lambda {\lt}\delta \).

Write \(h=\sum _ib_ia_i\). Given \(\varepsilon \), choose \(\delta \), then polynomials \(p,q\) approximating \(u_+^t\) and \(\max (u,\delta )^{-t}\) on \([-1,1]\), and compare \(O_k\) with the compression of \(W_k=\sum _i(q(J_Y)b_i^{(k)}p(J_Y))(p(J_P)a_i^{(k)}q(J_P))\). Between symmetric vectors, the forward-ratio estimate, the clipped inverse estimate and the polynomial approximation bound the difference of the pairings by a multiple of the chosen accuracy for large \(k\). The marked symbol of \(W_k\) replaces each star by the corresponding marginal, and the clipping is removed on the one-copy side by the preceding lemma. The trace limit then follows from the coherent symbol calculus.

Approximate \(\varphi \) uniformly on \([a,b]\) by a polynomial \(p\). Then \(\| \varphi (X_k)-p(X_k)\| \le \sup _{[a,b]}|\varphi -p|\), and \(p(X_k)\) has coherent symbol \(p\circ f\) by the product calculus. For the absolute value, \(\Pi _kX_k^\ast X_k =(X_k\Pi _k)^\ast (X_k\Pi _k)\) is positive with norm at most \(\sup _k\| X_k\Pi _k\| ^2\), and its symbol is \(|f|^2\).

Both \(|O_k|\) and \(|O_k^\ast |\) have coherent symbol \(|f_\theta |\) by the previous lemma, whether or not \(O_k\) is normal, and \(O_k+O_k^\ast \) has symbol \(2\operatorname {Re}f_\theta \). Apply the trace limit of the coherent symbol calculus.

Compare matrix entries: a lift from \(D\) is nonzero only between configurations that agree off \(D\), and the marginal on \(D\) sums over the configurations off \(D\). The real powers through the complex functional calculus agree with the real powers on positive marginals.

\(S(\rho )=\operatorname{Tr}(-\rho \log \rho )\) is the trace of a continuous function of \(\rho \) on \([0,1]\), and the continuous functional calculus is continuous in the operator.

The pointwise bound is the conditional skew estimate read on regions. Integrate it against the coherent measure, which is a positive measure, using the skew symbol and the continuity of both integrands. In the coordinates of the conditional skew estimate the marginals of the transported vector on \(xU\), \(U\) and \(UF\) are the states of \(\theta \) on \(xU\), \(U\) and, by purity, the complement \(Px\), and its full marginal on \(xUF\) has the entropy of the state on \(P\); hence \(I(x:F|U)=S_\theta (x|P)+S_\theta (x|U)\). The factors \(W_P^{-1}\) and \(W_F^{-1}\) commute, so the partition metric, and with it the similarity transform and \(\mathcal D_k\), is unchanged when the outer parts are exchanged.

Substitute \(x=u/(1+u)\) in the Beta integral \(\int _0^1x^{a-1}(1-x)^{b-1}\, dx\).

The first integral factorizes into one-dimensional gamma integrals. For the second, insert \(\Gamma (a)r^{-a}=\int _0^\infty v^{a-1}e^{-rv}\, dv\) with \(r=\sum _ix_i\), exchange the integrals, evaluate the inner one by the first formula with \(b=1+v\), and apply the Beta integral.

\(\Gamma (1+tl+n)=\Gamma (1+tl)(1+tl)(2+tl)\cdots (n+tl)\), a monic polynomial of degree \(n\) in \(tl\). A determinant of monic polynomials of degrees \(\delta _i\) evaluated at the points \(tl_j\) equals the Vandermonde determinant with the same ordering of degrees, which is \(a_\delta (tl)\).

Expand both alternants over permutations; each product is a monomial with exponents \(\delta _{\sigma ^{-1}i}+tl_{\tau ^{-1}i}\), whose sum is \(N+t\sum _jl_j\). The previous lemma integrates each term, and the double sum over \(\sigma ,\tau \) collapses to \(d!\) copies of \(\det [\Gamma (1+\delta _i+tl_j)]\). Finally \(a_\delta (x)a_\delta (x^t)=\prod _{i{\lt}j}(x_j-x_i)(x_j^t-x_i^t)\).

Here \(D=c+tk\) and \(D-tk=c\) for \(l\), while \(D=c\) for \(\delta \). The factors \(d!\, t^N\) cancel, and \(a_\delta (l)/a_\delta (\delta )=W_\lambda \) because \(a_\delta \) is a fixed sign times the Vandermonde product and \(a_\delta (\delta )\) is the product of the differences \(j-i\).

Write \(E^\ast AE=\sum _i\beta _i\psi _i\psi _i^\ast \). Then \(\beta _i=\langle E\psi _i,AE\psi _i\rangle \), and Jensen’s inequality for the convex function \(x^p\) gives \(\beta _i^p\le \langle E\psi _i,A^pE\psi _i\rangle \). Summing over \(i\) gives \(\operatorname{Tr}(E^\ast A^pE)\), a partial sum of the nonnegative diagonal entries of \(A^p\).

The normalization is the gamma integral at \(a=0\), which is positive. The power of the sample follows from the functional calculus of a diagonal matrix.

The average over \(U\) is the twirl of \(\operatorname {diag}(s^t)^{\otimes k}\), which by Theorem 23.8.3 acts on \(\lambda \) by \(\chi _\lambda (s^t)/\dim V_\lambda \). Since \(\chi _\lambda \) is homogeneous of degree \(k\), \(\chi _\lambda (s^t)=(\sum _ix_i)^{-tk}\chi _\lambda (x^t)\), and by the Weyl character formula \(a_\delta (x^t)\chi _\lambda (x^t)=a_l(x^t)\). The average over \(x\) is therefore \(J_{tk}(l)/(J_0(\delta )W_\lambda )=w_k(\lambda )\) by Theorem 23.9.5.

Choose an injection \(j\) of the basis configurations of \(Q\) into those of \(\mathbb {C}^d\), \(d=\dim \mathcal V\), with isometry \(E\). A permutation operator of \((\mathbb {C}^d)^{\otimes k}\) compressed by \(E^{\otimes k}\) is the same permutation operator of \(Q^{\otimes k}\), so the compression of \(\sum _\lambda w_k(\lambda )\pi ^\lambda \) is \(W_{Q,k}\) restricted to \(Q\); the label function is the same on every subsystem. On each sample the compression of \((\sigma ^t)^{\otimes k}\) is \(B^{\otimes k}\) with \(B=E^\ast \sigma ^tE{\gt}0\). Put \(\tau =B^{1/t}\); then \(\tau ^t=B\), and Lemma 23.10.1 with \(p=1/t\ge 1\) gives \(\operatorname{Tr}\tau \le \operatorname{Tr}\sigma =1\). The measure \(\zeta _Q\) is the law of \(\tau \).

For \(C{\gt}0\), Cauchy–Schwarz gives \(|\langle z,w\rangle |^2\le \langle z,C^{-1}z\rangle \langle w,Cw\rangle \). With \(A_k^{1/2}=W_P^{-1}W_F^{-1}W_{xY_0}\) and \(B_k^{1/2}=W_{Px}^{-1}W_F^{-1}W_{Y_0}\), \(\langle z,C_{x,k}^{-1}z\rangle =\| B_k^{-1/2}A_k^{1/2}z\| ^2\); the \(F\) factors cancel and the remaining nested or disjoint central factors commute.

\((\rho _R^{\otimes k})^t=(\rho _R^t)^{\otimes k}\). In the joint resolution of \(\rho _R^{\otimes k}\) and the label projectors, an eigenvalue \(u\) occurring in the \(\lambda \) block satisfies \(ud_\lambda \le 1\), and \(w_k(\lambda )^{-1}\le (k+2)^Cd_\lambda ^t\) by the entropy asymptotics; hence each weight \(w_k(\lambda )^{-1}u^t\) is at most \((k+2)^C\). Copy permutations of \(Q\) act on the copies of \(Q\), on which \(M^{\otimes k}\) is symmetric; for \(M\) off \(Q\) write them as full copy permutations times inverse copy permutations of the complement.

The configurations of the sites split as those of \(x\), \(Y_0\), \(P\) and \(F\); in these coordinates the operators are those of Lemma 5.1.

The coordinates of Lemma 23.11.3 split the configurations along \(xY_0\) and along \(Px\), so the marginals on these regions are the regional states, and their partial traces are the states on \(Y_0\) and \(P\). Complementary regions of a vector have equal entropy, which turns \(S_\theta (x|P)+S_\theta (x|Y_0)\) into either conditional mutual information; strong subadditivity gives \(I_\theta (x:F|P)\ge 0\).

By Lemma 23.11.1 it suffices to bound \(\| W_{Px}W_{xY_0}W_P^{-1}W_{Y_0}^{-1}\theta ^{\otimes k}\| \). Since \(\rho _R^t\hat\rho _R^{-t}\) is the support projection of \(\rho _R\), \(W_P^{-1}W_{Y_0}^{-1}\theta ^{\otimes k}=T_PT_{Y_0}y\) with \(T_R=W_R^{-1}(\rho _R^t)^{\otimes k}\) and \(y=(\hat\rho _P^{-t}\hat\rho _{Y_0}^{-t}\theta )^{\otimes k}\); the compensators commute with \(W_{Px}\) and \(W_{xY_0}\) and have polynomial norm. Expanding both numerators by the integral representation, \(\| W_{Px}W_{xY_0}y\| \) is at most the supremum of \(\| \sigma ^t\tau ^t\hat\rho _P^{-t}\hat\rho _{Y_0}^{-t}\theta \| ^k\), which Lemma 23.11.3 bounds by \(\exp (-ka\eta _\theta /2+kCa^{5/4}\ell ^2)\), with \(\ell =\log (e\dim x)\) and \(C\) the constant of Lemma 5.1. Squaring gives the claim with twice that constant, and Lemma 23.11.4 identifies the exponent. For a move to \(F\), apply the statement to the partition \(F,x,Y_0,P\): the partition metric is symmetric in its outer parts, since \(W_P^{-1}\) and \(W_F^{-1}\) commute.

On a vector fixed by \(\psi \), the action of \(\chi (\sigma )\) equals that of \(\phi (\sigma ^{-1})\). The coefficients of every central idempotent of \(\mathbb {C}[S_r]\) are invariant under inversion. Apply this identity to each central idempotent, then sum with coefficients \(\log \dim \lambda \).

Centrality on a smaller region and commutation with its disjoint complement give the first assertion. Complementary labels replace \(F_Y^\theta \Pi \) by \(F_{P\cup F}^\theta \Pi \). Thus \(G\Pi =D\Pi \), where \(D=F_P^\theta +F_F^\theta -F_{P\cup F}^\theta \ge 0\) is the actual merge deficit. Hence \(\Pi G\Pi =\Pi D\Pi \ge 0\).

On the intersection of the two subsystems, the induced actions commute because the underlying copy permutations commute. Outside the intersection, at most one of the actions moves a coordinate. Pass from these permutation identities to their group-algebra images and to the average defining the symmetric projection.

Let \(\Pi _b\) be simultaneous symmetry on the bad copies. Its compression of \(G_b\) is nonnegative and it commutes with \(G_b\). Functional calculus on its range gives \(\Pi _b e^{-aG_b}\Pi _b\le \Pi _b\). The good operator commutes with \(\Pi _b\) and \(G_b\). Apply Theorem 7.16.6 with \(A=-aG_g\) and \(B=-aG_b\).

The equivalence \(\Gamma _k\) carries simultaneous copy permutation to its physical and two auxiliary tensor factors. A permutation within \(B\) preserves \(B\), so covariance of the excitation operators gives \(U_\sigma E_B=E_BU_\sigma \). Thus the physical excitation operation preserves the original simultaneous symmetry for every bad-copy permutation. Pulling back by \(\Gamma _k\) proves the claim.

The three regions are disjoint and cover all five factors. Apply Theorem 23.13.1 to the bad-copy symmetry just derived for \(f\).

Apply restricted-copy commutation to the identity homomorphism, reversing the order of the two observables in the second inclusion case.

All regions occurring in the product are nested or disjoint. Their central metrics and inverse metrics therefore commute. Each metric is positive definite, as is its inverse; products of commuting positive definite matrices are positive definite.

If \(g{\lt}h\), every region from the earlier factor lies in \(Q_{j'h}\). It is therefore contained in both later inner regions and disjoint from the later band. All constituent central observables commute. The case \(h{\lt}g\) follows by symmetry; positivity is the single-band case above.

Every subsystem central observable commutes with simultaneous symmetry. Complementary label projections agree after multiplication by \(\Pi \); summing them with the reciprocal weights gives the corresponding inverse metric identity. Multiply the three intertwining identities and square.

Since \(U\subseteq U\cup Y\), \(Y\subseteq U\cup Y\), and \(U\cap Y=\varnothing \), the three families of label projections commute. Their products form a common orthogonal resolution. On a nonzero product with labels \(\alpha ,\beta ,\gamma \) for \(U,U\cup Y,Y\), respectively, the squared metric factor has coefficient \(q=(w_k(\gamma )/(w_k(\alpha )w_k(\beta )))^2\). Let \(C_0\ge 0\) be the constant in the logarithmic weight comparison of Theorem 23.5.3. Applying its lower bound to the first two weights and its upper bound to the third gives

All weights are positive. Exponentiating and summing over the common orthogonal resolution proves the assertion with \(C=6C_0\). Zero products of label projections contribute nothing to either side.

For \(g{\lt}h\), the set \(R\) lies in \(U_h\), so it is contained in \(U_h\) and \(U_h\cup Y_h\) and disjoint from \(Y_h\). The case \(h{\lt}g\) follows by symmetry. For \(g=h\), use the disjointness of \(U_g,Y_g\) and their inclusions in their union.

On each nonzero label subspace, the replica-weight comparison gives

Since \(d_\lambda \ge 1\), this implies \(w_k(\lambda )\le (k+2)^C\). If \(\log d_\lambda \le T\), it also gives

The metric, its inverse, and the cutoff are diagonal with respect to the same orthogonal label decomposition. Their operator norms are therefore bounded by these scalar coefficients. Labels absent from the representation contribute zero; labels outside the cutoff also contribute zero to the second operator.

Write \(C_0\) for the exponent in the metric cutoff estimate and put \(p=(k+2)^{C_0}\) and \(e=\exp (tT)\). The region relations make every pair among the regions \(U_g,Y_g,U_g\cup Y_g\) nested or disjoint. Their central label functions commute. Hence each metric and inverse metric preserves the common cutoff space.

On this space, the two inverse factors have norm at most \(pe\), and the third factor has norm at most \(p\). Each squared factor therefore has norm at most \(p^6e^4\), and their ordered product has norm at most \(p^{6K}e^{4K}\). Cauchy–Schwarz gives the stated expectation bound with \(C=6C_0\). The same argument applies to the empty product and to zero copies. It also applies to negative \(T\), for which the cutoff may have zero range.

Vector Jensen and monotonicity of nonnegative real powers give

Apply concavity of \(x\mapsto x^p\) to the spectral probability measure of \(M\) at \(\varphi \), and then apply the uniform root bound:

Cauchy–Schwarz with opposite powers gives \(|\langle \varphi ,u\rangle |^2 \le \langle \varphi ,M^{2s}\varphi \rangle \| M^{-s}u\| ^2\). Apply the uniform power bound with \(p=2s\).

Each \(D_{j,g}\) is positive definite by the one-band case of the preceding positivity result. If \(g{\lt}h\), every region in band \(g\) lies inside \(U_{j',h}\). It is consequently nested with \(U_{j',h}\) and \(U_{j',h}\cup Y_{j',h}\) and disjoint from \(Y_{j',h}\). The central label functions therefore commute across distinct bands, even for different leaf labels. Apply the factorization of a weighted mean tree across commuting bands.

The homogeneous leaf estimate gives the same positive bound \(L=(k+2)^{CK}\exp (4tKa)\) for the quadratic form at every ordered leaf product. Each such product is positive definite. Vector Jensen, spectral Jensen, and Cauchy–Schwarz with opposite powers yield \(|\langle \varphi ,u\rangle |^2\le L^{2s}\| \mathcal T(A)^{-s}u\| ^2\). The preceding exact factorization replaces this root by the actual ordered product \(M\) of the band roots.

Complementary label agreement gives \(A_{jg}\Pi =\Pi D_{jg}\), and \(D_{jg}\) preserves the symmetric space. Each \(D_{jg}\) is positive definite. Thus the actual compression is positive definite and both intertwining identities hold. Cross-status nesting makes every pair of regions occurring in different laminar bands either nested or disjoint, so the corresponding central label observables commute. Consequently \(D_{jg}\) and \(D_{j'h}\) commute when \(g\ne h\), and commutation descends through the common isometry \(Z\).

Positivity is preserved by each weighted geometric mean. A matrix commuting with every leaf commutes with the mean root; applying this observation to both trees gives the assertion.

The actual identities \(A_{j,g}Z=ZB_{j,g}\) multiply in the original order, and multiplication by \(Z^*\) gives the first equality. The compressed band factors are positive definite and different bands commute for every pair of statuses. The product identity for finite mean trees gives the second equality.

The compressed leaves are positive definite, and the roots of distinct bands commute. Their ordered product is therefore positive definite. Since \(\xi \) is nonzero and belongs to the range of \(Z\), both \(Z^*\xi \) and \(M^{-s}Z^*\xi \) are nonzero. The isometry then gives normalization and simultaneous symmetry of \(v\).

A singleton is contained in or disjoint from every region. Its original label projection \(P\) consequently commutes with every regional factor and with \(\Pi \). The relation \(PZ=Z(Z^*PZ)\) passes this commutation to each compressed leaf, hence to each band root and to \(M^{-s}\). The equation \(P\xi =\xi \) is thus preserved by the inverse power, by normalization, and by the lift through \(Z\).

Expand the matrix-vector product. The indicator restricts the finite sum to the single original configuration \(x|_F\).

Apply both sides to an arbitrary vector. On the original coordinates the two actions agree. In the first identity, the action on the new coordinates permutes the factors of \(\prod _j b(x_j(*))\), leaving the product unchanged. In the second identity these coordinates are fixed.

Sum the corresponding permutation identities with coefficients \(a_\sigma \) and use bilinearity of matrix multiplication.

Apply the group-algebra identities to \(\sum _\lambda f(\lambda )e_\lambda \), where \(e_\lambda \) is the central idempotent for the label \(\lambda \).

Reindex the sum defining the Gram matrix by \(s_k\). The original-coordinate indicators give \(\mathbf1_{\{ x=y\} }\). The remaining sum is \(\sum _{a\in A^k}\prod _j|b(a_j)|^2 =(\sum _{a\in A}|b(a)|^2)^k\).

Specialize the subsystem identity to all original sites. Averaging over \(\mathfrak S_k\) gives the projection identity. Applying the permutation identity to a fixed vector proves the last assertion.

On the bundled site, the first action permutes all coordinates of \(X\) and \(C\) together; the second action fixes them. Both actions fix \(R\) and agree on each complementary physical site. Transporting the same central group-algebra element \(\sum _\lambda f(\lambda )e_\lambda \) gives the observable identities.

Use the subsystem coordinate identities and the two insertion identities for label observables. For the last equality, the original register \(R\) becomes the retained exterior register \(R\).

The map \(e_k\) acts separately in each copy, so it commutes with their simultaneous permutation. Transport the corresponding rectangular identity and average over \(\mathfrak S_k\). Finally, \(W_k\) is unitary, and the Gram identity gives \(J^*J=J_b^*J_b=I\) when \(\| b\| _2=1\).

At a site in \(Q\), both sides have coordinate \(c_f(x_{\sigma ^{-1}(i),f})\); outside \(Q\), both have coordinate \(c_f(x_{i,f})\).

Each matrix entry is the product of the \(k\) inserted coefficients and the indicator that the exterior row coordinates equal \(y\). The coefficients agree by the definition of \(b_D\), and \(e_k\) fixes the exterior coordinates.

Apply the containing-subsystem commutation theorem with the full system as the containing subsystem. This is the symmetry used in [ Ope26b , 05-replicas.tex, lines 101–102; 07-comparators.tex, lines 299–308 ] .

The actual insertion intertwines the original and exterior symmetry projections. On the original symmetric subspace, the complementary regions \(F(U)\) and \(P(U)\) have the same label projectors. The coordinate identity for the insertion changes the \(P(U)\) label to the exterior \(E(U)\) label. Finally, this exterior central label observable commutes with the full exterior symmetry projection.

The original replica weights are strictly positive, so the inverse metric is the central label observable with reciprocal original weights. Apply the insertion identities to the first and third factors, and the preceding complementary-label identity to the second factor. The corresponding exterior observables commute with the exterior symmetry projection. Thus each factor intertwines through \(K_k\). Multiplying the three rectangular identities and then squaring proves the result.

No replica weight is recomputed from the compressed family. The auxiliary-label insertion identity preserves any already chosen \(R\) label through the same map; the displayed equality holds on the entire inserted symmetric range. Restriction of the weighted mean tree to this range remains a separate step.

The regions \(P(U)\), \(F(U\cup Y)\) and \(E(Y)\) are pairwise disjoint. Their replica metrics are positive definite and commute; their inverses retain these properties. Their product and its square are therefore positive definite. For the second matrix, the regions \(E(U)\) and \(E(Y)\) are disjoint and \(E(U\cup Y)=E(U)\cup E(Y)\). Apply positivity of a single pinned band. Every metric throughout uses the original full-family weights, as in [ Ope26b , 07-comparators.tex, lines 299–308, comparator:pinned-metric ] .

The unit norm of \(b\) gives \(J_k^*J_k=I\) for the actual insertion. The finite-group average \(\Pi _E\) is self-adjoint and idempotent. Conjugating it by this isometry therefore gives a self-adjoint, idempotent operator on the original coordinates. This is the projection onto the inserted symmetric subspace used in [ Ope26b , 07-comparators.tex, lines 283–308, comparator:pinned-metric ] .

The actual insertion identity gives \(L_{E(Q)}(f)J_k=J_kL_E(f)\). Self-adjointness gives its adjoint identity. The exterior central observable \(L_E(f)\) commutes with \(\Pi _E\), so multiplication of these identities proves the assertion. This is the invariant pin in [ Ope26b , 07-comparators.tex, lines 291–308, comparator:pinned-metric ] .

The preceding lemma applies to the original replica weights and their reciprocals, proving that \(D\) commutes with \(P_k\). The actual three-factor insertion identity, multiplied on the right by \(J_k^*\), gives \(AP_k=DP_k\). Both factors are positive definite and \(P_k\) is self-adjoint. Taking adjoints yields the remaining equality. This supplies the concrete factor restriction in [ Ope26b , 07-comparators.tex, lines 299–308 ] .

Each actual factor is positive definite. The weighted mean preserves a common rectangular intertwining, so the preceding identity passes through every node of the same tree. Functional calculus then gives every real power. Commutation passes through the tree by the same established mean identities. This is the single-band restriction used before the product of band roots in [ Ope26b , 07-comparators.tex, lines 370–381 ] .

The actual single-band intertwining multiplies in band order, without requiring commutation between the original roots. Each root commutes with \(P_k\), so its ordered product does as well.

Removing \(X\) and including the remaining physical sites in the original family preserves unions, inclusions, and disjointness. Thus the exterior factors satisfy the actual nested-band hypotheses. Their product of tree roots equals the tree of their positive ordered product leaves and is therefore positive definite. This identifies the ordered product used in [ Ope26b , 07-comparators.tex, lines 370–381 ] . Real powers of the original product still require its actual symmetric restriction.

The actual band factors intertwine with their compressions through \(Z\). The same is true of their matrix means and their ordered products. The compressed band roots are positive and commute across different bands, so \(M\) is positive definite. The exterior product \(N\) is also positive definite.

Full copy symmetry contains the inserted symmetric range, hence \(ZZ^*P_k=P_k\). Combining the band intertwiners with the Bell restriction gives

Functional calculus gives the same equality for every real power. Multiplying by \(Z\) and using \(ZZ^*P_k=P_k\) proves the claim. These are the fixed symmetric coordinates used in [ Ope26b , 07-comparators.tex, lines 80–110, 299–308 and 370–381 ] .

Apply the exterior cutoff estimate to the test vector \(\varphi \) and the vector \(Pu\). The map \(B\mapsto E(B)\) preserves unions, inclusions and disjointness, so the stated hypotheses give precisely the geometry and cutoffs required there. Its constant depends on the fixed replica family and \(t\) and therefore has the asserted order of quantification.

Write \(N\) for the ordered exterior product. The real-power identity in Theorem 23.26.1, evaluated at \(-s\), gives

Both powers are Hermitian and \(P\) is an orthogonal projection. The projected norm comparison consequently gives

Substitution into the exterior cutoff estimate proves the result. This is the inverse-power step in [ Ope26b , 07-comparators.tex, lines 299–308 and 370–381 ] . In the physical application, the unit inserted vector and the common cutoff conditions must be obtained for the selected state.

The physical-witness theorem gives nonempty coordinate sets and a unit selected Bell vector. Apply the compressed inverse-norm estimate to obtain \(c\). Intersect the eventual witness statement with \(k\ge 1\). The witness \(\Phi _k\) is chosen before the matched regions and bounds, and hence serves every subsequent choice in the statement.

Put \(B=B_{\mathrm{sh}}/z+B_{\mathrm{exc}}\). The witness satisfies the actual exterior cutoffs at \(a_k=kB\). Its overlap lower bound and the inverse-norm estimate, applied with the same insertion projection \(P_k\), give

Positive selected mass implies \(d{\gt}0\). The logarithmic overlap bound therefore proves positivity of \(N_k^2\) and the displayed inequality. The matrix \(M\), its coordinates \(Z\), the prevector and its label are unchanged throughout. This proves the upper estimate in [ Ope26b , Section 8, lines 153–163 and 332–381 ] for the supplied common-label sequence.

The matrix entry of \(J\) is the product of the \(k\) Bell coefficients, multiplied by the indicator that the exterior configurations agree. In \(JJ^*\) the sum over the exterior configuration collapses to that indicator. The remaining product is the corresponding entry of \(P_E^{\otimes k}\).

The map \(\tau _k\) commutes with simultaneous copy permutations, so \(\Phi \) is symmetric. The Bell-range hypothesis gives \(JJ^*\Phi =\Phi \). The insertion identity \(JP_{e,\mathrm{sym}}=P_{\mathrm{sym}}J\) then gives \(JP_{e,\mathrm{sym}}J^*\Phi =P_{\mathrm{sym}}\Phi =\Phi \).

A permutation on \(R\) fixes the selected Bell coordinates and acts on the exterior register \(R\). Transport the central idempotent for the same label \(\ell \) along this product action.

An exterior subsystem permutation fixes every Bell coordinate. Transport the central group-algebra element \(\sum _\ell \mathbf1_{\{ \log d_\ell \leq a\} }e_\ell \) along this product action. The coefficients and subsystem are the same on both sides.

Write \(a_{k,\ell }(c,r)\) for the coefficient of the labelled auxiliary Bell vector \((I_C\otimes \Pi _{R,\ell })U_k\) at \((c,r)\in D^k\times D^k\). In the composition with \(e_k\), the enumeration of the left auxiliary coordinate cancels \(\iota ^{-1}\), so this coefficient remains \(a_{k,\ell }(c,r)\). The physical coordinates are the restrictions to \(X\) and \(X^c\). Therefore both sides at \(((x_j,c_j,r_j))_j\) are identified by

The equality uses only the definition of the split vector \(\Omega _X\).

The one-copy configuration bijection preserves the full dimension, hence the label weights. It intertwines every subsystem copy action, so \(W^D_{Q,k}(x,y)=W^E_{Q,k}(e_kx,e_ky)\). Inverse, ordered multiplication and squaring preserve this identity. Changing both summation variables by \(e_k\) in each compressed matrix entry therefore gives

Equal leaf inputs give equal evaluations of the same tree, and then equal ordered products.

The prevector identity gives \(v_D=v_E\circ e_k\). Changing the summation variable by \(e_k\) gives, for each column \(a\),

Let \(e_k\) carry enumerated configurations to selected configurations, and define \(Z_E(x,a)=Z(e_k^{-1}x,a)\). The bijection intertwines the simultaneous copy permutations. It therefore transports the symmetric projection, so \(Z_E\) is an isometry onto the same symmetric space in selected coordinates. The two compressed identities give

Thus \(\| M_E^{-s}Z_E^*v_E\| ^2=N_k^2\). Apply Theorem 23.28.1 with \(Z_E\).

Each coordinate lies in exactly one of \(Q,Y,F,\{ c\} ,\{ r\} \). The two actions permute its copy index precisely when its factor belongs to \(S\).

Extend the intertwining identity linearly to the group algebra, then apply it to the linear combination of central idempotents with coefficients \(f(\lambda )\).

The one-copy grouping equivalence gives equality of cardinalities. The weight formula depends on the site family only through that full dimension.

The central projections transform by \(U\) and their original replica weights are unchanged. Apply the central-observable identity.

Conjugation by the coordinate permutation preserves inversion, multiplication and squaring.

Evaluate the grouping map at each physical site and at the two auxiliary sites. Disjointness identifies the second physical block with all of \(Y\).

The two auxiliary sites are distinct and lie outside the physical sites. The induced partition gives disjointness. The complement identity follows by separating the two auxiliary sites from a physical site. Finally, \(L_{jg}\cup E_{jg}=\{ c\} \cup (Q_{jg}\cup Y_{jg})\); adjoining the same site to the given physical inclusion proves the nesting assertion.

The three displayed subsets partition the five factor indices. Their inverse images therefore form a partition of the original sites.

The regrouping intertwines the permutation of the \(C\) copies with the same permutation of the original \(C\) site, and similarly for \(R\). Taking the central idempotent of each label gives the two identities. The other coordinate factors carry the identity.

Each permutation matrix transforms by conjugation with \(U\). Average these identities over the same finite group.

The one-copy rank-one matrix transforms as \(|\Omega '\rangle \langle \Omega '|=U|\Omega \rangle \langle \Omega |U^*\). Each tensor summand transforms by the same coordinate equivalence; sum over the copy index.

Functional calculus transports the spectral cutoff of the Hermitian defect count. The two auxiliary-label projectors remain unchanged. The physical coordinate equivalence intertwines simultaneous copy permutations, so their average transports as well. Multiply these three factor identities.

The fixed regrouping intertwines the actual simultaneous permutations. The native cutoff ends with their symmetry projection, giving the containment after pullback. Projection compression then gives the fixed-coordinate assertion. Finally physical coordinate covariance identifies the regional cutoff with the same original cutoff, and the three equivalences in \(\Xi _{Q,Y,k}\) cancel in their stated order.

Transport \(v\) to the fixed physical-copy coordinates. The exact symmetry and auxiliary identities put it in the original joint sector, and the regrouping preserves its norm. Apply the replica gap estimate, then transport the projection and energy quadratic forms back by \(G_k\).

The actual lifted regions satisfy the hypotheses of the label-preservation theorem. Hence the same normalized vector \(v\) has simultaneous symmetry and both original auxiliary labels. Apply the original gap estimate to \(v\) for each \(\tau {\gt}0\). The equality \(v=Z\phi \) identifies its projection mass with the displayed compressed quadratic form. Since the common projection is contained in the symmetric range, its compression is an orthogonal projection. Its mass on the unit vector \(\phi \) is at most one. Taking \(\tau =1\) in the gap estimate therefore gives \(1-\delta \leq 1\), and hence \(\delta \geq 0\).

Each metric is positive definite. Its inverse is a label observable with reciprocal positive weights. Disjoint subsystems give commuting permutation actions and hence commuting label observables. A product of commuting positive definite matrices is positive definite, as is its square.

The subgroup dimension bounds give \(L^g_S+L^b_S\leq L_S\leq L^g_S+L^b_S+\log \binom {k}{r}I\). Apply the lower bound to \(P,F\) and the upper bound to \(Y\) to obtain \(G'-\log \binom {k}{r}I\leq G\). The whole and subgroup observables in this inequality commute: within one region this follows from centrality, and between the three regions it follows from disjointness. Exponentiating after multiplication by \(-a\) gives the result, since \(e^{a\log \binom {k}{r}}=\binom {k}{r}^a\).

The whole-space inverse comparison gives the upper bound \((k+2)^Ce^{-2tG}\) with \(C\) depending only on \(t\) and the fixed local family. Apply the grouped label exponential theorem with \(a=2t\) and multiply by the nonnegative factor \((k+2)^C\).

On simultaneous symmetry, the signed label entropy \(G=F_P+F_F-F_Y\) is nonnegative and commutes with \(Q_k\). Thus \(Q_ke^{-2tG}Q_k\leq Q_k\). The whole-space inverse comparison gives \(A_{t,k}^{-1}\leq (k+2)^Ce^{-2tG}\), which proves the first inequality after compression. The matrix \(A_{t,k}\) is positive definite, so the noncommuting compressed-inverse inequality gives the reciprocal lower bound.

Since \(Q_kZ=Z\) and \(Z^*Q_kZ=I\), conjugating the two preceding inequalities by \(Z\) gives the result.

The spectral projectors of \(G\) and \(H\) commute and give a common orthogonal resolution. On a joint block with eigenvalues \(g,h\), the compressed exponential is \(e^{a(h-g)}\) if \(g\ge 0\) and zero otherwise. When \(g\ge 0\), the inequalities \(a\ge 0\) and \(g\ge 0\) imply \(e^{a(h-g)}\le e^{ah}\). On every other block the difference is \(e^{ah}{\gt}0\). Summing these nonnegative differences against the joint projectors proves the matrix inequality.

Taking adjoints gives \(\rho J=\rho \). For \(E=\exp (a(H-G))\), cyclicity of the trace gives \(\operatorname{tr}(\rho JEJ)=\operatorname{tr}(\rho E)\). Theorem 23.43.1 says that \(\exp (aH)-JEJ\ge 0\). Its trace pairing with the positive matrix \(\rho \) is nonnegative; substituting the trace identity gives the claimed inequality.

Congruence by \(L\) preserves the matrix inequality. Commutation, idempotence, and invertibility give

Thus \(S\le BSL^2S\). Multiplication by the positive scalar \(B^{-1}\) proves the assertion.

Choose \(C\ge 0\) from Theorem 23.5.4 before choosing \(k\) or the partition. Set \(H_k=F_{P,k}+F_{F,k}-F_{Y,k}\), where \(F_{Q,k}=\sum _\lambda (\log d_\lambda )\pi ^\lambda _{Q,k}\), and let \(J_k=\mathbf1_{[0,\infty )}(H_k)\). The whole-space inverse comparison and Theorem 22.28.1 give

The second equality in (42) is the adjoint of the first. Both \(J_k\) and \(\Pi _k\) are orthogonal projections; for \(\Pi _k\) this also follows from Lemma 23.8.4. Theorem 23.43.1, applied to \(H_k\) and zero with parameter \(2t\ge 0\), gives \(J_ke^{-2tH_k}J_k\le I\). Congruence by \(\Pi _k\) and (42) therefore yield

Apply congruence by \(\Pi _k\) to (41) and then use (43). This gives \(\Pi _kA_k^{-1}\Pi _k\le (k+2)^C\Pi _k\), which is (38).

The three metric factors are positive definite by Theorem 23.5.2. Their inverses are the label observables with reciprocal weights. Label observables of disjoint subsystems commute, so \(L_k\) is invertible and Hermitian. Theorem 22.27.3, with the containing subsystem equal to \(V\), shows that all three factors commute with \(\Pi _k\). Thus \(L_k\Pi _k=\Pi _kL_k\). Theorem 23.44.1 applies to (38): explicitly, congruence by \(L_k\) gives

Since \((k+2)^C{\gt}0\) and \(c_k=((k+2)^C)^{-1}\), this proves (39). Finally, \(C\ge 0\) and \(k+2\ge 1\) imply \(0{\lt}c_k\le 1\). Theorem 7.20.2 applies; equivalently,

giving (40).