Dependencies

Let \(F\) be a finite set with finite coordinate sets \(I_f\), let \(A\) be another finite coordinate set, and let \(k\geq 0\). Write \(\mathcal C_k=(\prod _{f\in F}I_f)^k\). Separating the added coordinate in each copy gives the bijection

Let \(F\) be a finite set of sites with finite coordinate sets \(I_f\), let \(A\) be an additional coordinate set, and let \(k\geq 0\). Write \(\mathcal C_k\) for the configurations of \(k\) copies of the original sites, and \(\widetilde{\mathcal C}_k\) for the configurations after adjoining a site \(*\) with coordinate set \(A\). For any \(b:A\to \mathbb {C}\), define the rectangular matrix \(J_b\) by

Thus insertion retains every original coordinate and places the same vector at the additional site in every copy. This is the tensor insertion in [ Ope26b , Section 8, lines 283–308 ] .

Group the coordinates of \(X\) with \(C\), and leave \(R\) and all complementary physical coordinates separate. This gives

Applying \(e\) independently to the \(k\) copies defines \(e_k\). These are the coordinate changes in [ Ope26b , Section 8, lines 283–308 ] .

For natural numbers \(d,K,k\) and \(c\in \mathbb {R}\), put

The estimates below concern positive \(k\).

For a Hermitian matrix \(A\) and \(b\in \mathbb {R}\), let \(\Pi =\mathbf1_{[b,\infty )}(A)\) be the spectral projection onto the eigenvalues at least \(b\). An eigenvalue equal to \(b\) is included.

For a density matrix \(\sigma \) on the symmetric subspace of \(k\) copies, the coherent measure is \(d\mu _\sigma (\theta )=D_k\operatorname{Tr}(\sigma P_{\theta ,k})\, d\theta \), where \(\theta =Ue\) for Haar-distributed \(U\). It is a probability measure, and \(\bigl|\int \langle \theta ^{\otimes m},G\theta ^{\otimes m}\rangle \, d\mu _\sigma \bigr|\le \| G\| \) for every operator \(G\) on \(m\) copies.

Let \(V\) be a finite set of physical sites, with local coordinate sets \(I_v\), and let \(X\subseteq V\). For an auxiliary coordinate set \(R\), define

A physical region \(B\subseteq V\) becomes \(\widehat B=B\cap X^c\), included in \(\widehat V\) and excluding the auxiliary site. These are the exterior coordinates in [ Ope26b , 07-comparators.tex, lines 240–247 and 283–308 ] . They require no choice of a vector or spectral subspace, and no coordinate set is required to be nonempty.

For finite index sets \(N,M,Q\), let \(B:\mathbb {C}^N\to \mathbb {C}^M\) be rectangular and let \(J:\mathbb {C}^N\to \mathbb {C}^Q\) satisfy \(J^\dagger J=I_N\). Its adjoint domain extension is \(\overline B=BJ^\dagger \). It satisfies

Thus local contractions remain contractions on their common ambient domain. These equalities allow empty domains, empty codomains, and zero or subnormalized vectors.

The copy permutations of a subsystem, the symmetric subspace, and the averaging projector \(\Pi _k=(k!)^{-1}\sum _{\pi \in S_k}U(\pi )\) onto it.

Fix the original physical vector \(\Omega \), the two auxiliary labels, the copy number \(k\), and the cutoff \(\tau k\). Let \(P_0\) be the actual low-defect projection in the physical-copy and two auxiliary registers. Using the fixed regrouping \(G_k\) of Definition 23.20.1, define

This projection is defined without a regional cut, metric, or mean tree.

For the original physical Hamiltonian \(H\), define in the global coordinates

Here \(H_i\) acts on the \(i\)th physical copy. At \(k=0\), set \(\overline H_0=0\).

For an operator \(G\) on \(m\) copies and an ordered injection \(\iota \) of \(m\) copies into \(k\) copies, \(G_\iota \) is \(G\) on the copies \(\iota (1),\dots ,\iota (m)\) and the identity elsewhere. The injection average \(\mathcal T_{k,m}(G)\) is the average of \(G_\iota \) over all ordered injections; \(\| \mathcal T_{k,m}(G)\| \le \| G\| \).

Let \(G\) be a finite group. By Maschke’s theorem and the Artin–Wedderburn theorem over \(\mathbb {C}\) there is an algebra isomorphism

and we fix one. Its factors are the labels \(\lambda \) of \(G\), and \(d_\lambda \ge 1\) is the dimension of the irreducible representation with label \(\lambda \). The central idempotent \(e_\lambda \in \mathbb {C}[G]\) and the matrix units \(e^\lambda _{ij}\) are the preimages of the identity and of the matrix units of the factor \(M_{d_\lambda }(\mathbb {C})\).

Let \(G\) act on a finite set \(X\). The permutation operator \(U(g)\) on \(\mathbb {C}^X\) sends the basis vector \(e_x\) to \(e_{gx}\), and extends linearly to an algebra homomorphism \(\mathbb {C}[G]\to \mathrm{End}(\mathbb {C}^X)\). The label projector \(\pi ^\lambda \) is the image of \(e_\lambda \). For a real function \(f\) of the labels, the label observable is \(\sum _\lambda f(\lambda )\pi ^\lambda \); for \(f(\lambda )=\log d_\lambda \) it is the observable \(F=\sum _\lambda (\log d_\lambda )\pi ^\lambda \) of  [ Ope26b , Section 6.1 ] .

A weighted binary tree is either a leaf with a label \(j\) or an internal vertex with a parameter \(p\in [0,1]\) and two subtrees. Its root on inputs \((A_j)\) is \(A_j\) at a leaf and \(M_l\# _pM_r\) at an internal vertex with subtree roots \(M_l\), \(M_r\). The terminal weights are \(\delta _j\) at a leaf and \((1-p)w^{(l)}+p\, w^{(r)}\) at an internal vertex.

Let \(\mathcal H_{\mathrm{phys}}\) be a finite tensor product of local coordinate spaces, and fix auxiliary coordinate spaces \(C,R\). An added site is distinguished from the original sites; applying this construction twice retains two distinct auxiliary coordinates. For disjoint physical regions \(Q,Y\), write \(F=(Q\cup Y)^c\). Restriction of configurations gives the equivalence

The five coordinate factors, in order, are \(Q,Y,F,C,R\).

On \(k\) copies, let \(G_k\) extract the original physical copies and the two original auxiliary registers from the full configuration. For any physical coordinate equivalence \(e\), let \(E_k(e)\) apply \(e\) independently to those physical copies and act identically on both auxiliary registers. Finally let \(F_k\) be the five-factor regrouping into these three registers. The actual regional equivalence is

It maps the original full configurations to the grouped five-factor configurations. The auxiliary coordinate sets and their order are unchanged. Empty regions and zero copies are included. These are the coordinate identifications in [ Ope26b , Section 8, lines 20–37 and 80–110 ] .

Assign an original site to factor \(0,1,2,3,4\) according as it belongs to \(Q,Y,F,\{ c\} ,\{ r\} \). For a set \(S\subseteq \{ 0,1,2,3,4\} \), let \(\widetilde S\) consist of all original sites assigned to \(S\).

Let \(u_d=d^{-1/2}\sum _{i=0}^{d-1}|i,i\rangle \) be the uniform auxiliary pair for \(d{\gt}0\), and set \(u_0=0\). For \(k=0\), use the empty tensor product. Regroup its literal tensor power into the two auxiliary systems and write it as \(U_k\). For a central label projector \(\Pi _\lambda \) on \((\mathbb C^d)^{\otimes k}\), define

All products and projections here act on the actual finite copy spaces.

Fix \(k\geq 0\) and an irreducible label \(\ell \) of \(\mathfrak S_k\). Let \(v_{k,\ell }\) be the original prevector in the canonical coordinates \((A\times B)^k\times (R^k\times R^k)\). The regrouping

and the fixed bijection \(e_k\) of Definition 22.49.1 give

The original state and supplied label are unchanged, as in [ Ope26b , Section 8, lines 240–254 and 283–290 ] .

Group \(X\) with \(C\) in each copy, and then separate these \(k\) pairs from the exterior copies. The resulting bijection is

These are the fixed coordinates used in [ Ope26b , Section 8, lines 240–254 and 283–290 ] .

For \(a\in \mathbb R\), define

An alternant with two equal exponents vanishes, and \(a_\alpha e_r=\sum _{|S|=r}a_{\alpha +1_S}\). Consequently \(\sum _{|S|=r}V(\alpha +1_S)=\binom qrV(\alpha )\) for the Vandermonde product \(V(t)=\prod _{i{\lt}j}(t_i-t_j)\), and \(\sum _{|S|=r}W_{\lambda +1_S}=\binom qrW_\lambda \) for the right-hand side \(W\) of the Weyl dimension formula.

For \(a,b{\gt}0\), \(\int _0^\infty u^{a-1}(1+u)^{-(a+b)}\, du=\Gamma (a)\Gamma (b)/\Gamma (a+b)\).

For a permutation action of a finite group, an operator commuting with the permutation operators and with every operator of their commutant is \(\sum _\lambda c_\lambda \pi ^\lambda \).

Let \(X_k\) be Hermitian with eigenvalues in \([a,b]\) and coherent symbol \(f\), with \(f\) real with values in \([a,b]\) on unit vectors, and let \(\varphi \) be continuous on \([a,b]\). Then \(\varphi (X_k)\) has coherent symbol \(\varphi \circ f\). In particular, if \(X_k\) has coherent symbol \(f\), then \(\bigl(\Pi _kX_k^\ast X_k\bigr)^{1/2}\) has coherent symbol \(|f|\).

If \(|\langle u,Mz\rangle |\le c\| u\| \| z\| \) for all symmetric \(u,z\), then \(\| \Pi _kM\Pi _k\| \le c\). Let \(X_k\) commute with \(\Pi _k\), with \(\| X_k\Pi _k\| \) bounded. Suppose that for every \(\varepsilon {\gt}0\) there are \(X'_k\) with coherent symbol \(f'\) such that \(|f-f'|\le \varepsilon \) on unit vectors and eventually \(\| (X_k-X'_k)\Pi _k\| \le \varepsilon \). Then \(f\) is continuous on unit vectors and \(X_k\) has coherent symbol \(f\).

There is \(C\) such that \(\| W_{R,k}^{-1}(\rho _R^t)^{\otimes k}\| \le (k+2)^C\) for every region \(R\) and every density matrix \(\rho _R\) on \(R\). Label functions of \(Q\) commute with \(M^{\otimes k}\) for \(M\) supported on a region contained in or disjoint from \(Q\).

The von Neumann entropy is continuous along continuous families of Hermitian matrices with eigenvalues in \([0,1]\). Hence \(\theta \mapsto \eta _\theta =I(x:F|U)_\theta \) is continuous on unit vectors.

For \(t\ge 0\), \(\det [\Gamma (1+\delta _i+tl_j)]_{i,j}=\prod _j\Gamma (1+tl_j)\cdot a_\delta (tl)\), and \(a_\delta (tl)=t^Na_\delta (l)\).

Let \(\lambda \vdash k\) have length at most \(q\), and let \(D_\lambda \) be the right-hand side of the formula for \(d_\lambda \). Then

so \(\log D_\lambda =kH(\lambda /k)+O_q(\log (k+2))\). The right-hand side of the formula for \(\dim V^{(q)}_\lambda \) is at most \((k+q)^{q^2}\).

Let \(Y_k=\sum _i\mathcal T_{k,r_i}(G_i)\) for finitely many fixed operators \(G_i\), with symbol \(\psi (\theta )=\sum _i\langle \theta ^{\otimes r_i},G_i\theta ^{\otimes r_i}\rangle \). Then \(Y_k\) commutes with the copy permutations, \(Y_k^\ast \) has symbol \(\overline\psi \), the product of two such sums is, up to \(O(k^{-1})\) in operator norm, a third one whose symbol is the product of the symbols, and \(|\operatorname{Tr}\sigma Y_k-\int \psi \, d\mu _\sigma |=O(k^{-1})\) uniformly in symmetric density matrices \(\sigma \).

The label projectors are mutually orthogonal orthogonal projections with sum the identity, and they commute with every operator that commutes with all permutation operators. Every label observable is self-adjoint.

For \(u\ge 0\), \(|\log \Gamma (1+u)-(u\log u-u)|\le 3\log (u+2)+3\).

Let \(\theta \) be a unit vector, \(\rho _Q\) its marginal on \(Q\) lifted by the identity, \(0{\lt}t{\lt}1/2\) and \(0{\lt}\delta \). Then \(\| \rho _Q^t\| \le 1\), \(\| \rho _Q^{-t}\theta \| ^2\le d_Q\), and

with powers zero on kernels. Consequently, if \(|p-u_+^t|\le \eta \) and \(|q-\max (u,\delta )^{-t}|\le \eta \) on \([-1,1]\), then \(\| \rho _Q^ta\rho _Q^{-t}\theta -p(\rho _Q)aq(\rho _Q)\theta \| \le \| a\| \bigl(\eta \sqrt{d_Q}+(1+\eta )(\sqrt{d_Q\delta ^{1-2t}}+\eta )\bigr)\).

In the joint resolution by the label \(\lambda \) of all copies and the label \(\nu \) of the first \(k-1\) copies, \(R_{Q,k}\) and \(J_{Q,k}\) are diagonal. On a nonzero block, \(\nu =\lambda -e_i\), the ratio acts by \(r(k,l_i)\) and the star operator by \((l_i-d)/k\), where \(1\le l_i\le k+d\) is the shifted part of \(\lambda \) padded to length \(d\).

Let \(\pi _\iota \) exchange the copy \(\iota (j)\) with the copy \(k+j\) for every \(j\). Then \(\operatorname{Tr}[(\sigma \otimes G)U(\pi _\iota )]=\operatorname{Tr}(\sigma G_\iota )\).

\(C_{x,k}{\gt}0\), and for all \(z,w\), \(|\langle z,w\rangle |^2\le \| W_{Px,k}W_{xY_0,k}W_{P,k}^{-1}W_{Y_0,k}^{-1}z\| ^2 \langle w,C_{x,k}w\rangle \).

For a vector \(\theta \) on the sites and a partition \(P,x,Y_0,F\), the exponent of Lemma 23.11.3 is \(\eta _\theta =S_\theta (x|P)+S_\theta (x|Y_0)\), the marginals \(\rho _P,\rho _{Y_0}\) there are the states of \(\theta \) on \(P\) and \(Y_0\), and

Let \(u_i{\gt}-1\) and \(E=d+\sum _iu_i\). For \(b{\gt}0\), \(\int _{(0,\infty )^d}\prod _ix_i^{u_i}e^{-bx_i}\, dx=\prod _i\Gamma (1+u_i)b^{-1-u_i}\), and for \(0\le a{\lt}E\),

Let \((P_i)\) be mutually orthogonal idempotents with \(\sum _iP_i=1\). Then \(f\mapsto \sum _if(i)P_i\) is an algebra homomorphism from functions on the index set, and \(\exp \bigl(\sum _if(i)P_i\bigr)=\sum _ie^{f(i)}P_i\). Two commuting resolutions give the product resolution \(P_iQ_j\). If the \(P_i\) are self-adjoint and \(f(i)\ge 0\) whenever \(P_i\ne 0\), the image of \(f\) is positive semidefinite. The spectral projections of a self-adjoint matrix form a resolution, and its functional calculus is the image of the restriction of the function to the eigenvalues.

Let \(P_i\) be Hermitian idempotents with \(P_iP_j=\delta _{ij}P_i\) and \(\sum _iP_i=I\), and let \(g\) be real. Then \(f(\sum _ig_iP_i)=\sum _if(g_i)P_i\) for every real function \(f\). If \(|c_i|\le M\) whenever \(P_i\ne 0\), then \(\| \sum _ic_iP_i\| \le M\) in operator norm. For every vector \(z\), \(\| \sum _ic_iP_iz\| ^2=\sum _i|c_i|^2\| P_iz\| ^2\) and \(\sum _i\| P_iz\| ^2=\| z\| ^2\).

Let \(\mathcal V\) be partitioned as \(P=P_0P_1\), \(Y=xU\) and \(F\), and identify the configurations of \(\mathcal V\) with those of \((P_0P_1)((xU)F)\). Lifts from a region and marginals on a region agree with the corresponding tensor-factor lifts and partial traces. In particular, if \(h\) is supported on \(P_0x\), the symbol \(f_\theta \) at \(t\) is the function \(f(t)\) of the conditional skew estimate for the transported vector.

For a unit vector \(\theta \) on the sites and positive semidefinite \(\sigma \) on \(Px\), \(\tau \) on \(xY_0\) of trace at most one, the word \(\sigma ^{a/2}\hat\rho _P^{-a/2}\tau ^{a/2}\hat\rho _{Y_0}^{-a/2}\theta \), each factor acting on its region, satisfies the bound of Lemma 5.1. The kernel projections of \(\rho _P\) and \(\rho _{Y_0}\) annihilate \(\theta \).

Let \(0{\lt}t{\lt}1\). There is \(C\ge 1\) such that, for \(k\ge 1\) and \(1\le l\le k+d\), the ratio \(r(k,l)\) of the previous lemma satisfies \(C^{-1}(l/k)^t\le r(k,l)\le C\).

Let \(0{\lt}t{\lt}1\). There is a constant \(C\) such that, for \(k\ge 1\) and \(1\le l\le k+d\),

The operator \(h^{(j)}\) commutes with the copy permutations of every subsystem disjoint from the support of \(h\), and with every copy permutation of a subsystem that fixes the copy \(j\). A permutation \(\tau \) of entire copies conjugates \(h^{(j)}\) to \(h^{(\tau j)}\), and \(\| h^{(j)}\| \le \| h\| \). An operator supported on a disjoint union \(X\cup T\) is a finite sum of products \(a_ib_i\) with \(a_i\) supported on \(X\) and \(b_i\) supported on \(T\).

There is a constant \(C_{d,t}\) such that every \(\lambda \vdash k\) of length at most \(d\) satisfies \(|\log w_k(\lambda )+tkH(\lambda /k)|\le C_{d,t}\log (k+2)\).

The label function is positive, and \(w_0=1\). If \(\lambda \vdash k\) is obtained from \(\nu \vdash k-1\) by adding a box in row \(i\), then \(w_k(\lambda )=r_{\lambda ,i}\, w_{k-1}(\nu )\) with

where \(l_i\) is the shifted part of \(\lambda \).

Let \(t{\gt}0\). The density proportional to \(a_\delta (x)a_\delta (x^t)e^{-\sum _ix_i}\) on the positive orthant is a probability density. For \(x\) in the orthant and a unitary \(U\), the sample \(\sigma =U\operatorname {diag}(x/\sum _ix_i)U^\ast \) is a positive definite density matrix and \(\sigma ^t=U\operatorname {diag}((x/\sum _ix_i)^t)U^\ast \).

For a subsystem \(Q\), the star operator \(J_{Q,k}\) on \(k\) copies has marked symbol \(\theta \mapsto \rho _Q(\theta )\otimes 1\), where \(\rho _Q(\theta )\) is the marginal of \(|\theta \rangle \langle \theta |\) on \(Q\).

\(\| J_{Q,k}\| \le 1\), so every star eigenvalue lies in \([-1,1]\). If \(|\varphi -p|\le M\) on \([-1,1]\) for a function \(\varphi \) and a real polynomial \(p\), then \(\| \varphi (J_{Q,k})-p(J_{Q,k})\| \le M\).

For every \(n\ge 0\), \(0\le \log n!-(n\log n-n)\le \log (n+2)+1\). For natural numbers \(a,s\), \(0\le (a+s)\log (a+s)-a\log a\le s(\log (a+s)+1)\).

Under \(\mathcal V^{\otimes k}\cong Q^{\otimes k}\otimes (Q^c)^{\otimes k}\), the label projector \(\pi ^\lambda _{Q,k}\) is the label projector of \(Q^{\otimes k}\) tensored with the identity. Consequently a label with \(\pi ^\lambda _{Q,k}\ne 0\) has at most \(d_Q\) rows, and \(d_Q\le d\).

Let \(z\in \mathcal S_k\). On a branch \(\nu =\lambda -e_i\), \(d_\lambda \| \pi ^\lambda \pi ^\nu z\| ^2=d_\nu \| \pi ^\lambda z\| ^2\); hence \(\| \pi ^\lambda \pi ^\nu z\| ^2\le 2^d(l_i/k)\| \pi ^\lambda z\| ^2\). Each label has at most \(d\) branches with nonzero projector.

For \(A\ge 0\) on \(\mathbb {C}^n\), an isometry \(E:\mathbb {C}^m\to \mathbb {C}^n\) given by an injection of basis vectors, and \(p\ge 1\), \(\operatorname{Tr}(E^\ast AE)^p\le \operatorname{Tr}A^p\).

The averaging element \(|G|^{-1}\sum _gg\) of \(\mathbb {C}[G]\) is one of the central idempotents \(e_\lambda \). Hence, for every permutation action, the projector onto the invariant vectors is a label projector. In particular it is an orthogonal projection.

An operator on \((\mathbb {C}^q)^{\otimes k}\) commuting with \(U^{\otimes k}\) for every unitary \(U\) commutes with \(\sum _jA^{(j)}\) for every matrix \(A\). If it also commutes with the copy permutations, it is a central label function.

Let \(A,B\) be positive definite. Then \(A\# _pB\) is positive definite, \(A\# _0B=A\), \(A\# _1B=B\), and:

1. \((S^*AS)\# _p(S^*BS)=S^*(A\# _pB)S\) for every invertible \(S\);

2. \((cA)\# _p(dB)=c^{1-p}d^p\, (A\# _pB)\) for \(c,d{\gt}0\);

3. \(A\# _pB=B\# _{1-p}A\);

4. for \(0\le p\le 1\) the mean is monotone in each argument, and for every vector \(z\), \(\langle z,(A\# _pB)z\rangle \le \langle z,Az\rangle ^{1-p}\langle z,Bz\rangle ^p\);

5. if \(A_1,B_1\) commute with \(A_2,B_2\), then \((A_1A_2)\# _p(B_1B_2)=(A_1\# _pB_1)(A_2\# _pB_2)\), and the two factors commute.

For \(x{\gt}0\) and \(0\le s\le 1\),

If moreover \(x\ge 1\), then \(x^s-s\le \Gamma (x+s)/\Gamma (x)\).

Suppose \(u\) is fixed by simultaneous permutation of the physical copies and both auxiliary copy registers. Let \(w=(E_B\otimes I_{C^k\otimes R^k})u\) and \(f=w\circ \Gamma _k\). Enumerate the complement of \(B\) as the good copies and \(B\) as the bad copies. Then \(f\) is simultaneously symmetric under permutation of the bad copies. No normalization of \(u\) or \(\Omega \) is required. This is the symmetry inheritance in [ Ope26b , Section 8, lines 441–456 ] .

With the same \(u,\Omega ,B,w,f\) as in Theorem 23.13.2, take \(P=Q\cup C\) and \(F=V\cup R\). Then (4) holds for this actual component \(f\), with these regions and every \(a\ge 0\).

Let \(U(\sigma )\) and \(\widetilde U(\sigma )\) denote simultaneous copy permutations before and after insertion. Let \(P_{\rm sym}\) and \(\widetilde P_{\rm sym}\) be their averaging projections. For every \(b\) and \(\sigma \in \mathfrak S_k\),

Consequently, \(J_bv\) is symmetric whenever \(v\) is symmetric. Neither vector is required to be normalized.

For every \(b:A\to \mathbb {C}\), the insertion matrix \(J_b\) satisfies

In particular, if \(\| b\| _2=1\), then \(J_b\) is an isometry. The identity includes \(k=0\) and empty coordinate sets.

For every \(v:\mathcal C_k\to \mathbb {C}\),

This identity does not require \(A\) to be finite.

For \(a=\sum _{\sigma \in S_k}a_\sigma \sigma \in \mathbb {C}[S_k]\), put \(U_Q(a)=\sum _\sigma a_\sigma U_Q(\sigma )\), and define \(\widetilde U_R(a)\) in the same way. Then

The same group-algebra element occurs on both sides.

Let \(f\) be any real-valued function on the irreducible labels of \(S_k\), and let \(L_Q(f)=\sum _\lambda f(\lambda )\pi _Q^\lambda \). With the analogous notation for the enlarged family,

In particular, these identities retain any chosen original-system label weights. The weights are not recomputed on the enlarged family.

Suppose also that \(A\) is finite. Let \(Q\subseteq F\) and \(\sigma \in S_k\). Denote the original subsystem action by \(U_Q(\sigma )\) and the enlarged subsystem action by \(\widetilde U_R(\sigma )\). Then

Both identities hold for every \(b\), including the zero vector, and every \(k\), including zero.

Let \(L\) be an invertible Hermitian matrix and let \(S\) be an idempotent matrix commuting with \(L\). Let \(B{\gt}0\). If \(S(L^2)^{-1}S\le BS\) in the positive-semidefinite order, then

Assume the original registers have nonempty finite coordinate sets, and let \(t\ge 0\). Define every replica metric \(W_Q\) using the original full-system dimension, including when \(Q\) lies entirely in the exterior. For arbitrary physical regions \(U,Y\) and any inserted Bell-coordinate vector,

When \(U,Y\) are disjoint from \(X\), these are precisely the two three-factor products in [ Ope26b , 07-comparators.tex, lines 299–308, comparator:pinned-metric ] . The statement includes zero copies.

Assume that the original site family is finite and each physical and auxiliary coordinate set is finite and nonempty. Retain the original full-system weights, \(t\ge 0\), and the actual Bell-inserted symmetric operator \(P_k=J_k\Pi _EJ_k^*\). For physical regions \(U_{jg},Y_{jg}\), assume

Let \(A_{jg},D_{jg}\) be the original and exterior metric factors of Theorem 23.24.7. Evaluate the same finite weighted mean tree \(\mathcal T\) in every band, and form the ordered products

Then \(N\) is positive definite, both \(M\) and \(N\) commute with \(P_k\), and

The inserted vector is arbitrary. The assertion includes \(K=0\) and \(k=0\) and does not assert that \(M\) is positive definite on the unrestricted original space.

Let the original finite site family consist of the physical registers and two auxiliary registers \(C,R\), all with finite nonempty coordinate sets. Fix a physical region \(X\), a replica parameter \(t\ge 0\), a copy number \(k\ge 0\), and an isometry \(Z:\mathbb {C}^n\to \mathcal H_k\) with \(ZZ^*=\Pi _0\), where \(\Pi _0\) is full simultaneous-copy symmetry. Let \(J_k\) insert the tensor power of an arbitrary vector on \(X\cup C\), let \(\Pi _E\) be exterior copy symmetry, and put \(P_k=J_k\Pi _EJ_k^*\).

Suppose the physical regions satisfy

All replica metrics \(W\) use the original full-family weights. Write \(E(B)=B\setminus X\), viewed as a subsystem of the original family, and set

Complements here are taken among physical sites. For the same finite weighted mean tree \(\mathcal T\) in every band, define

Then \(M\) is positive definite, and for every \(s\in \mathbb {R}\),

Empty band products are included. The inserted vector need not be unit.

Fix a finite family of nonempty physical coordinate sets, two finite nonempty auxiliary registers \(C,R\), a physical region \(X\), a finite status set, and \(t{\gt}0\). There is a constant \(c\ge 0\) with the following property, chosen before the copy number, the symmetric coordinates, the band regions, the mean tree, the cutoff and the vectors below.

Let \(Z:\mathbb {C}^n\to \mathcal H_k\) be any isometry onto the full simultaneous symmetric subspace. Let \(U_{jg},Y_{jg}\) be indexed by this status set and \(0\le g{\lt}K\), with

Use the original full-family metric weights, the same weighted mean tree in every band, and the compressed product \(M\) of Theorem 23.26.1. Insert any unit vector on \(X\cup C\), and let \(P=J_k\Pi _EJ_k^*\) be the resulting orthogonal projection, where \(\Pi _E\) denotes exterior simultaneous copy symmetry.

Put \(E(B)=B\setminus X\), viewed inside the original family. Let \(a\in \mathbb {R}\) and let \(\varphi \) be a unit vector in the original replica space such that the cutoff projectors satisfy

for every \(j,g\). Then, for every vector \(u\) and \(0\le 2s\le 1\),

Empty band products and zero copies are included. No normalization or symmetry condition is imposed on \(u\).

For every real function \(f\) of the irreducible copy-permutation label, let \(L_Q(f)=\sum _\lambda f(\lambda )\pi _Q^\lambda \). Then

The label function is unchanged, and the inserted vector is arbitrary. Finite coordinate sets may be empty. This is the far-region label identification in [ Ope26b , 07-comparators.tex, lines 299–308, comparator:pinned-metric ] .

The coordinate change \(e_k\) commutes with simultaneous copy permutations. Hence, for every \(b\) and \(\sigma \in \mathfrak S_k\),

If \(\| b\| _2=1\), then also \(J^*J=I\). These identities include zero copies and empty coordinate sets.

For every \(b\), \(U\subseteq V\), and real label function \(f\),

The same function \(f\) occurs on both sides of each equality. The registers \(C\) and \(R\) need not have the same dimension.

Let \(V\) and the physical coordinate sets \(D_v\) be finite, fix \(X\subseteq V\), and put \(A_X=\prod _{v\in X}D_v\) and \(B_X=\prod _{v\notin X}D_v\). Let \(C,D,R\) be finite coordinate sets and \(c:D\to C\) a bijection. For an arbitrary \(b:A_X\times C\to \mathbb {C}\), set \(b_D(a,d)=b(a,c(d))\). The map \(e_k\) from the family with register \(D\) to the family with register \(C\) applies \(c\) to that register in every copy and fixes all physical coordinates and \(R\). Let \(J_C(b)\) and \(J_D(b_D)\) be the corresponding Bell insertion matrices, with the same exterior columns \((B_X\times R)^k\). For every row \(x\) and exterior column \(y\),

This identity includes \(k=0\), empty coordinate sets, and \(b=0\). It expresses the change of the auxiliary enumeration in [ Ope26b , Section 8, lines 283–308 ] .

Fix any \(U\subseteq V\) and write \(\widehat U\) for its exterior part \(U\setminus X\). Denote the new bundled site \(A_X\times C\) by \(*\). For every \(\sigma \in \mathfrak S_k\) and original configuration \(x\),

For a real function \(f\) on the irreducible labels of \(\mathfrak S_k\), let \(L_Q(f)=\sum _\lambda f(\lambda )\Pi _{Q,\lambda }\). If \(W_kv=v\circ e_k\), then

No disjointness condition on \(X\) and \(U\) is required.

Let \(A_j,D_j\) be the preceding factors for a family of pairs \(U_j,Y_j\) with \(U_j\cap Y_j=\varnothing \), and let \(\mathcal T\) be any finite weighted mean tree whose leaves have these labels. Then \(\mathcal T(A)\) and \(\mathcal T(D)\) commute with \(P_k\), and

The statement includes negative powers and \(k=0\).

Use the original full-family replica metrics and regions

from the actual Bell coordinate identity. Suppose the original coordinate sets are finite and nonempty, \(t\ge 0\), and \(U\cap Y=\varnothing \). Then both matrices

are positive definite, for arbitrary \(X\) and every copy number \(k\ge 0\).

Assume that the original registers have nonempty finite coordinate sets, \(t\ge 0\), and \(U\cap Y=\varnothing \). All replica metrics below use the original full-system dimension. Set

Then

The physical region \(X\) and the inserted vector are arbitrary.

Let \(d,k\) be positive integers, \(K\) a nonnegative integer, \(z{\gt}0\), \(c\ge 0\), and \(0\le 2s\le 1\). Let \(t,B,r\in \mathbb {R}\). Suppose that

Then \(r{\gt}0\) and

The remainder is independent of \(s,t,B,z\).

Retain the physical family, unit state \(\Omega \), region \(X\), positive-mass spectral selection \(E\), auxiliary registers \(C=E\) and \(R=[d]\), and supplied label sequence of Theorem 22.51.1. In particular, \(d=|E|\), the selected mass is \(z{\gt}0\), and the actual compressed tensor powers satisfy

for all sufficiently large \(k\). Fix a finite nested list \(\mathcal R\) of physical regions disjoint from \(X\), a finite status set \(S\), and \(t{\gt}0\).

There is a constant \(c\ge 0\) such that, for all sufficiently large \(k\), the following holds. Let \(Z:\mathbb {C}^n\to \mathcal H_k\) be any isometry onto the full simultaneous symmetric subspace. Choose \(K\ge 0\) and regions \(U_{jg},Y_{jg}\), indexed by \(j\in S\) and \(0\le g{\lt}K\), such that

Let \(T\) be any weighted matrix mean tree on \(S\). Suppose that real bounds \(B_{\mathrm{sh}},B_{\mathrm{exc}}\) satisfy \(S_\Omega (D)\le B_{\mathrm{sh}}\) for \(D\in \mathcal R\), and that every \(Q\in \{ U_{jg},U_{jg}\cup Y_{jg}\} \) has a region \(D\in \mathcal R\) comparable to \(Q\) with

Use the original full-family weights and put

Let \(v_k\) be the same supplied-label prevector as in Theorem 22.51.1. For every \(0\le 2s\le 1\), the squared norm \(N_k^2=\| M^{-s}Z^*v_k\| ^2\) is positive, and

Both \(c\) and the eventual threshold precede all the choices of \(n,Z,K,U,Y,T,B_{\mathrm{sh}},B_{\mathrm{exc}},s\) above.

Let \(b\) be a unit vector in the coordinates of the physical region \(X\) together with its auxiliary coordinate \(C\). Let \(J_k\) be its literal \(k\)-copy insertion into the original coordinates, leaving the distinct auxiliary coordinate \(R\) and the physical complement of \(X\) as the exterior system. If \(\Pi _E\) is the simultaneous-copy symmetric projection on this exterior system, then

is an orthogonal projection. The assertion includes \(k=0\) and arbitrary finite coordinate sets.

For a finite family of finite coordinate spaces, let \(L_Q(f)\) be the central label observable of a subsystem \(Q\), with arbitrary real label function \(f\). If \(\Pi \) is the full simultaneous-copy symmetric projection, then \(L_Q(f)\Pi =\Pi L_Q(f)\).

Let \(A\) be Hermitian and \(b\in \mathbb {R}\). The projection \(\Pi \) is the functional calculus of \(\mathbf1_{[b,\infty )}\) at \(A\), and satisfies \(\Pi ^*=\Pi \), \(\Pi ^2=\Pi \), and \(A\Pi =\Pi A\).

For a unit basis vector \(e\), \(\int P_{Ue,k}\, dU=D_k^{-1}\Pi _k\), where \(P_{\theta ,k}=|\theta ^{\otimes k}\rangle \langle \theta ^{\otimes k}|\), \(\Pi _k\) is the projector onto the symmetric subspace and \(D_k=\operatorname{Tr}\Pi _k\).

Say that operators \(X_k\) have coherent symbol \(f\) if they commute with \(\Pi _k\), are uniformly bounded on \(\mathcal S_k\), \(f\) is continuous on unit vectors, and for every \(\varepsilon {\gt}0\) some sum of injection averages \(Y_k\) with symbol \(\psi \) satisfies \(|f-\psi |\le \varepsilon \) on unit vectors and \(\| (X_k-Y_k)\Pi _k\| \le \varepsilon \) for large \(k\). This property is preserved by sums, scalar multiples, adjoints with the conjugate symbol, and products with the product symbol. Consequently, for every noncommutative polynomial \(q\) in two variables, \(q(X_k,X_k^\ast )\) has coherent symbol \(q(f,\overline f)\), and

over symmetric density matrices \(\sigma \).

Let \(A,B\) be commuting Hermitian matrices. If \(A\leq B\), then \(e^A\leq e^B\).

Let \(\phi ,\chi ,\psi \) be actions of \(S_r\) on the same finite coordinate set. If \(\psi (\sigma )=\phi (\sigma )\chi (\sigma )\) for every \(\sigma \), then \(F_\phi \Pi _\psi =F_\chi \Pi _\psi \). This is the complementary-label argument of [ Ope26b , Lemma 6.1(2) ] .

Let \(\theta \) be a unit vector on four finite-dimensional systems \(P,x,U,F\), and let \(\rho _D\) denote its marginal on \(D\). Set \(d=\dim x\), \(\ell =\log (ed)\) and \(\eta =S(x|P)_\theta +S(x|U)_\theta \). There are universal constants \(c,C{\gt}0\) such that, for \(0{\lt}a\ell \le c\) and arbitrary positive semidefinite \(\sigma \) on \(xP\) and \(\tau \) on \(xU\) of trace at most one,

One may take \(C=200000\) and \(c=1/8\).

There is a universal constant \(C\) such that for \(0{\lt}a\ell _h\le \frac18\) and \(t=a/2\),

In particular the left side is zero if \(\eta =0\). One may take \(C=18\pi ^2+6144\, e^{\pi ^2+1}\).

Let \(P\) be an orthogonal projection and let \(Z\) be a rectangular matrix such that \(PZZ^*=P\). Then \(Z^*PZ\) is an orthogonal projection.

Suppose that the local spaces are nonempty. For \(t\geq 0\), \(k\geq 0\) and pairwise disjoint regions \(P,F,Y\), the matrix \(R=W_{P,k}^{-1}W_{F,k}^{-1}W_{Y,k}\) and its square \(R^2\) are positive definite on the full configuration space.

For every physical region \(Q\) and every real label function \(f\), the original-system central observable on \(E(Q)=Q\setminus X\) satisfies

The original and exterior label functions are identical.

For \(\sigma \) on \(k\) copies and \(G\) on \(m\) copies, \(\int \operatorname{Tr}(\sigma P_{Ue,k})\operatorname{Tr}(GP_{Ue,m})\, dU=D_{k+m}^{-1}\operatorname{Tr}[(\sigma \otimes G)\Pi _{k+m}]\).

Under the matrix hypotheses of Theorem 7.16.5, \(Pe^BP\leq P\) implies \(\operatorname {Re}(v^*e^{A+B}v)\leq \operatorname {Re}(v^*e^Av)\) for every \(Pv=v\).

For \(t\ge 0\) and \(0\le a{\lt}D\),

the integrand is integrable, and \(a_\delta (x)a_\delta (x^t)\ge 0\) on the orthant.

Suppose \(\| \Omega \| =1\), \(\gamma {\gt}0\), and

Let \(k{\gt}0\) and \(\tau {\gt}0\). Let \(v\) be a unit vector with simultaneous copy symmetry and the two original auxiliary labels \(\ell _C,\ell _R\). For the common projection \(\Pi \) of Definition 23.38.1,

This is the actual energy expectation in the cutoff estimate of [ Ope26b , Section 8, lines 421–443 ] .

Let \(\mathcal H\) be the original finite physical tensor product, and let \(C,R\) be finite auxiliary coordinate spaces. For a copy number \(k\), let \(G_k\) be the fixed regrouping of Definition 23.20.1, with codomain \(\mathcal H^{\otimes k}\otimes C^{\otimes k}\otimes R^{\otimes k}\). For any original labels \(\ell _C,\ell _R\), let \(P_C,P_R\) denote their whole-copy auxiliary projections. Then

On the right, \(C\) and \(R\) denote the two distinguished original auxiliary sites. The identities hold for zero copies and empty coordinate spaces; no state or regional cut is needed. They give the auxiliary-coordinate identification used in [ Ope26b , Section 8, lines 421–443 ] .

Let a group \(G\) act by permutations on finite sets \(Z\) and \(B\), with actions \(\alpha \) and \(\beta \). Let \(A\) be any set and \(e:Z\to A\times B\) a bijection satisfying

for every \(g\in G\) and \(z\in Z\). Let \(\pi _\alpha \) and \(\pi _\beta \) be the linear extensions of the two permutation representations to \(\mathbb C[G]\). For every \(h\in \mathbb C[G]\), \(a,a'\in A\), and \(b,b'\in B\),

Equivalently, the matrix in product coordinates is \(I_A\otimes \pi _\beta (h)\). Neither \(G\) nor \(A\) is required to be finite.

Commuting permutations of the copies induce commuting actions on any two physical subsystems, including overlapping subsystems. Consequently, after a division of the copies into groups of sizes \(m\) and \(r\), every real label observable of the first group commutes with every such observable of the second group. It also commutes with the simultaneous symmetric projection of the second group. These are the subgroup commutations used in [ Ope26b , Section 8, lines 454–480 ] .

Let \(m+r=k\), fix a partition of the \(k\) copies into specified groups of sizes \(m\) and \(r\), and let \(\phi :S_k\to \operatorname {Perm}(X)\) be any permutation representation on a finite coordinate set \(X\). For the restrictions to the two groups and for the whole group, let

be the central label observables on \(\mathbb C^X\). Then, on the whole space,

The statement includes \(m=0\) and \(r=0\).

If \(P,F,Y\) are pairwise disjoint and \(a\geq 0\), then \(e^{-aG}\leq \binom {k}{r}^{a}e^{-aG'}\). This statement permits arbitrary finite local coordinate spaces, including empty ones.

Fix \(t{\gt}0\) and finite nonempty local spaces. There is \(C\geq 0\), chosen before the copy number, its decomposition and the three regions, such that every such decomposition and pairwise disjoint \(P,F,Y\) satisfy

Divide \(k\) copies into groups of sizes \(m\) and \(r\). For pairwise disjoint regions \(P,Y,F\) covering the one-copy space, define \(G_g=F_{P,g}+F_{F,g}-F_{Y,g}\) and \(G_b=F_{P,b}+F_{F,b}-F_{Y,b}\). If \(v\) is simultaneously symmetric on the bad copies, then every \(a\ge 0\) satisfies

Both expectations are real. No normalization of \(v\) is assumed.

For operators \(G\) on \(m\) copies and \(H\) on \(n\) copies and every \(k\ge 1\),

Let \(Z\) be a rectangular complex matrix with \(Z^*Z=I\). Suppose \(MZ=ZB\), \(NZ=ZC\), and \(MN=NM\). Then \(BC=CB\). No Hermitian hypothesis is required. This is the elementary restriction argument used in [ Ope26b , Section 7, lines 268–276 ] .

For an arbitrary status set \(J\), suppose

for every \(j,j'\in J\). Set

Every \(D_{jg}\) is positive definite, and \(D_{jg}\) commutes with \(D_{j'h}\) whenever \(g\ne h\). This is the cross-status commutation in [ Ope26b , Section 7, lines 268–276; Section 8, lines 20–37 ] .

For \(R,S\subseteq F\), suppose that \(R\subseteq S\), \(S\subseteq R\), or \(R\cap S=\varnothing \). Then every real central label observable of the copy action on \(R\) commutes with every such observable on \(S\). This is the laminar form of [ Ope26b , Lemma 6.1(2) ] .

Let \(0{\lt}t{\lt}1\). There are constants such that, on the entire tensor space and for every subsystem, \(\| R_{Q,k}-(J_{Q,k})_+^t\| \le Ck^{-t}\) and \(\| R_{Q,k}\| \le C\). In particular \(\| R_{Q,k}-(J_{Q,k})_+^t\| \to 0\).

Let \(0{\lt}t{\lt}1/2\). There is \(C\) such that \(\| R_{Q,k}^{-1}z\| \le C\| z\| \) for every \(z\in \mathcal S_k\), and, for every \(0{\lt}\delta {\lt}1\),

Let \(0{\lt}t{\lt}1/2\). For fixed \(P,Y,F\) and \(h\) there is \(C\), depending on \(\mathcal V\), \(t\), the partition and \(h\), such that \(\| O_kz\| \le C\| z\| \) for every \(k\) and every \(z\in \mathcal S_k\).

Let \(0{\lt}t{\lt}1/2\), let \(P,Y,F\) be disjoint and let \(h\) be supported on \(P\cup Y\). Then \(O_k\) has coherent symbol \(f_\theta =\langle \theta ,\rho _P^t\rho _Y^{-t}h\rho _P^{-t}\rho _Y^t\theta \rangle \). In particular, for every noncommutative polynomial \(q\) in two variables,

over symmetric density matrices \(\sigma \).

On \(k+1\) copies with the last copy marked, say that operators \(X_k\) have marked symbol \(F\), a function from vectors to operators on one copy, if for some finitely many one-copy operators \(A_i\) and fixed operators \(G_i\),

Marked symbols are preserved by sums, scalar multiples, products (with the ordered product of the symbols), adjoints and real polynomials. If \(X_k\) has marked symbol \(F\), then \(\Pi _{k+1}X_k\Pi _{k+1}\) has coherent symbol \(\theta \mapsto \langle \theta ,F(\theta )\theta \rangle \).

Let \(T,S\) be complex matrices on a finite-dimensional space and let \(c\in \mathbb R\). If \(S\le I\), \(c\le 1\) and \(cS\le T\), then

In particular, this applies to extension by the identity off the range of an orthogonal projection. The inequality itself does not require \(S\) to be a projection. This auxiliary order inequality extends a lower bound to the whole space.

Let \(R\) be a semiring, let \(I,J\) be finite index sets, and fix \(Z\in R^{I\times J}\). For \(0\leq g{\lt}K\), let \(A_g\in R^{I\times I}\) and \(B_g\in R^{J\times J}\) satisfy \(A_gZ=ZB_g\). Then

where both products are taken in increasing order. The assertion includes \(K=0\) and empty index sets.

Let \(A\ge 0\) and \(B\ge 0\) satisfy \(AJ=JB\). For every real exponent \(s\), \(A^sJ=JB^s\). For negative exponents the spectral power is zero on the kernel.

Let \(A_1,B_1{\gt}0\) on \(\mathcal H_1\) and \(A_2,B_2{\gt}0\) on \(\mathcal H_2\) satisfy \(A_1J=JA_2\) and \(B_1J=JB_2\). For every real \(p\),

Let \(T\) be a fixed weighted binary tree whose vertex weights lie in \([0,1]\). If its positive definite inputs \(A_j,B_j\) satisfy \(A_jJ=JB_j\) for every label \(j\), and its roots are \(M_A=T(A)\) and \(M_B=T(B)\), then \(M_AJ=JM_B\). Repeated labels and zero vertex weights are permitted.

Let a weighted binary tree have terminal weights \(w_j\) and positive definite inputs \(A_j\), and let \(M\) be its root. Then \(M\) is positive definite, commutes with invertible congruence, is monotone in the inputs, and satisfies \(\mathcal T(c_jA_j:j)=\bigl(\prod _jc_j^{w_j}\bigr)M\) for \(c_j{\gt}0\). A tree with constant positive definite input \(X\) has root \(X\). For every vector \(z\),

If \(P\) is positive semidefinite (in particular, an orthogonal projection) and \(A_j\ge c_jP\) with \(c_j{\gt}0\), then \(M\ge \bigl(\prod _jc_j^{w_j}\bigr)P\). Finally, if \(A_j=\prod _{g=1}^KA_{j,g}\) with positive definite factors and every factor from band \(g\) commutes with every factor from band \(g'\ne g\), then \(M=\prod _{g=1}^KM_g\), where \(M_g\) is the root of the same tree with inputs \(A_{j,g}\), and the factors \(M_g\) commute.

Let \(L{\gt}0\) and let \(\varphi \) be any vector. If \(\langle \varphi ,A_j\varphi \rangle \le L\) for every \(j\), then \(\langle \varphi ,M\varphi \rangle \le L\).

Under the same hypotheses with \(\| \varphi \| =1\), for every \(s\in [0,1/2]\) and every vector \(u\),

Under the same hypotheses, suppose additionally that \(\| \varphi \| =1\). For every \(p\in [0,1]\), \(\langle \varphi ,M^p\varphi \rangle \le L^p\).

Let \(A,B\subseteq F\) satisfy \(A\subseteq B\), and let \(f\) be any real-valued function on the irreducible labels of \(S_k\). Then

For \(g\in \{ 0,\ldots ,K-1\} \), let \(Q_g,Y_g\subseteq F\) satisfy

Then the ordered product

is positive definite. In particular, each single squared factor is positive definite. The empty product is included. This is the positive laminar product used in [ Ope26b , Section 8, lines 367–378 ] .

Let \(\mathcal H,\mathcal K\) be finite-dimensional complex Hilbert spaces, let \(Z:\mathcal K\to \mathcal H\), and let \(M,N,P\) be operators on \(\mathcal H\). Suppose that \(N\) is positive definite and

Then the actual compression \(B=Z^*MZ\) satisfies

No Hermitian assumption on \(M\) is required, and the spaces may have dimension zero. This is the finite-dimensional compression used for the restricted replica metrics in [ Ope26b , Section 8, lines 80–110, 299–308 and 370–381, equation (comparator:pinned-metric) ] .

Let \(\mathcal H\) and \(\mathcal K\) be finite-dimensional complex Hilbert spaces, let \(Z:\mathcal K\to \mathcal H\) satisfy \(Z^*Z=I\), and let \(P\) be an orthogonal projection on \(\mathcal H\). Suppose that \(A\) on \(\mathcal K\) and \(N\) on \(\mathcal H\) are self-adjoint and

Then every \(u\in \mathcal H\) satisfies

Neither matrix is required to be positive, and either Hilbert space may have dimension zero.

Every element of the permutation group algebra has grouped matrix \(UXU^*\), where \(X\) is its matrix for the corresponding original subsystem. In particular, for any real function \(f\) of the same irreducible labels,

Let the physical and two auxiliary coordinate spaces be finite and nonzero. Suppose that \(\| \Omega \| =1\), \(\gamma {\gt}0\), and

Fix \(k{\gt}0\), an isometry \(Z\) onto simultaneous copy symmetry, and a nonzero symmetric vector \(\xi \) in the original auxiliary sectors \(\ell _C,\ell _R\). For terminal labels \(j\) and bands \(0\leq g{\lt}G\), let \(Q_{jg},Y_{jg}\) be disjoint physical regions such that, whenever \(g{\lt}h\),

for every pair of terminal labels \(j,j'\). Let \(t\geq 0\), retain the original full-system replica weights, and set

where every band uses the same weighted mean tree \(T\). The symbols \(C\) and \(R\) in these regions denote the two auxiliary sites; the complement in the second factor is taken among physical sites. Let \(\overline H_k\) be the physical replica average with both auxiliary identities, in the global coordinates of Definition 23.39.1. For any real \(s\), define

Then \(M\) is positive definite, \(w\ne 0\), \(\| \phi \| =\| v\| =1\), and \(\delta \geq 0\). For every \(\tau {\gt}0\), the common projection \(\Pi _\tau \) from Definition 23.38.1 has an orthogonal projection \(P_\tau =Z^*\Pi _\tau Z\) as its compression, and

The cutoff parameter is chosen after the normalized vector. Repeated terminal labels, zero tree weights, and \(G=0\) are allowed. This is the mass estimate used in [ Ope26b , Section 8, lines 421–443 ] .

Assume \(\| \Omega \| =1\), and let \(Q_k\) denote simultaneous symmetry on the original full configuration space. Then \(P_{\mathrm{glob}}\) is an orthogonal projection and

For every \(Z\) with \(ZZ^*=Q_k\), the matrix \(Z^*P_{\mathrm{glob}}Z\) is an orthogonal projection.

For every disjoint physical cut \(Q,Y\), transport the same vector by \(e_{Q,Y}\) and form its actual five-factor low-defect projection \(P_{Q,Y}\). Keep both original auxiliary labels. Then

Thus all cuts have exactly the same projection in the same fixed symmetric coordinates. See [ Ope26b , Section 8, lines 20–37, 421–456 and 603–615 ] .

The whole one-copy dimension agrees before and after grouping. Consequently, for every real \(t\), copy number \(k\) and irreducible label \(\lambda \), the replica label weight \(w_{t,k}(\lambda )\) is the same for the original and grouped site families.

For every real \(t\), every \(k\geq 0\) and every set \(S\) of grouped factors,

Empty coordinate sets are included, and no positivity condition on \(t\) is required.

For arbitrary subsets \(Q,Y\subseteq V\), assign a physical site to factor \(0\) if it belongs to \(Q\), otherwise to factor \(1\) if it belongs to \(Y\), and otherwise to factor \(2\). Assign \(c\) to factor \(3\) and \(r\) to factor \(4\). The inverse images of \(\{ 0,3\} \), \(\{ 1\} \) and \(\{ 2,4\} \) are pairwise disjoint and cover every original physical and auxiliary site. This assertion includes empty physical factors.

Let \(V\) be a finite set, let \(Q,Y\subseteq V\) be disjoint, and adjoin the original auxiliary sites \(c,r\). Group the sites into the five factors \(Q,Y,(Q\cup Y)^c,\{ c\} ,\{ r\} \), in that order. The inverse images of the grouped regions \(\{ 0,3\} \), \(\{ 2,4\} \), and \(\{ 1\} \) are respectively

These are the original metric regions of [ Ope26b , Section 8 ] .

For any three sets \(S_1,S_2,S_3\) of grouped factors, the actual product satisfies

The sets \(S_i\) need not be disjoint. The identity is valid with arbitrary real \(t\) and empty coordinate spaces, with inversion extended by zero on singular matrices.

For every copy number \(k\geq 0\), copy permutation \(\sigma \) and set \(S\) of grouped factors, the coordinate actions satisfy \(\phi _S(\sigma )U=U\phi _{\widetilde S}(\sigma )\).

There are universal constants \(C,c{\gt}0\) such that, for \(0{\lt}t\le 1\), \(a=2t\) and \(a\log (e\dim x)\le c\), there is a polynomial \(p(k)\), uniform in \(\theta \), with

where \(\eta _\theta =S_\theta (x|P)+S_\theta (x|Y_0)=I_\theta (x:F|P)=I_\theta (x:F|Y_0)\). The same statement holds for a move to \(F\) after exchanging the outer parts: \(A_k^{-1/2}A_k(P,Y_0,Fx)A_k^{-1/2}\) satisfies the bound with \(\eta _\theta =S_\theta (x|F)+S_\theta (x|Y_0)\).

Let \(F\) be a set and let \(U_g,Y_g\) be finite subsets indexed by \(0\le g{\lt}K\). Suppose

If \(R\) is one of \(U_g,Y_g,U_g\cup Y_g\) and \(S\) is one of \(U_h,Y_h,U_h\cup Y_h\), then \(R\subseteq S\), \(S\subseteq R\), or \(R\cap S=\varnothing \).

Let \(t{\gt}0\) and let \(U\) be Haar distributed, independent of \(x\). Then \(\int (\sigma ^t)^{\otimes k}=\sum _\lambda w_k(\lambda )\pi ^\lambda \) on \((\mathbb {C}^d)^{\otimes k}\) for every \(k\).

Retain the same auxiliary spaces \(C,R\) and labels \(\ell _C,\ell _R\), and write \(\widehat U=U^{\otimes k}\otimes I_{C^{\otimes k}}\otimes I_{R^{\otimes k}}\). The simultaneous symmetry projections satisfy \(S'_k=\widehat U S_k\widehat U^*\). If \(\| \Omega \| =1\), the actual low-defect projections, at any real cutoff \(\tau k\), satisfy

The same original vector and labels are used on both sides. This is the coordinate invariance required by [ Ope26b , Section 8, lines 421–456 and 603–615 ] .

If \(\| \Omega \| =1\), then \(P\) is an orthogonal projection. Every vector \(u\) satisfying \(Pu=u\) lies in the range of \(Q\), has both original auxiliary labels, and is fixed by every simultaneous copy permutation. No commutation with a replica metric is assumed.

Suppose \(\| \Omega \| =1\), \(g{\gt}0\), \(k{\gt}0\), \(\tau {\gt}0\), and

For every unit vector \(v\) with simultaneous copy symmetry and both original auxiliary labels,

This is the retained-mass estimate in [ Ope26b , Section 8, (comparator:defect-mass) ] .

Let \(t\ge 0\) and assume every factor is nonzero. The metric \(W_{Q,k}=\sum _\lambda w_k(\lambda )\pi ^\lambda _{Q,k}\) is positive definite, commutes with the copy permutations of \(Q\) and with the metrics of nested and disjoint subsystems, and \(W_{Q,0}=I\). On the symmetric subspace, \(W_{Q,k}=W_{Q^c,k}\).

Let \(t{\gt}0\). There is \(C\), depending only on \(d\) and \(t\), such that for every subsystem \(Q\) and every \(k\),

Let the original system be a finite tensor product of nonzero finite-dimensional spaces, and fix \(t{\gt}0\). Write \(W_{Q,k}\) for its replica metric, using the same label weights on every subsystem. For \(T\in \mathbb {R}\), let

There is \(C\ge 0\), depending on the original system and \(t\), such that for every subsystem \(Q\), every integer \(k\ge 0\), and every real threshold \(T\),

These estimates supply the individual metric factors in the rough upper comparison of [ Ope26b , Section 8, upper comparisons ] . They retain the label weights of the original full system.

Let \(0{\lt}t\le 1\). For each subsystem \(Q\) there is a probability measure \(\zeta _Q\) on positive definite subdensities \(\tau \) on \(Q\), with \(\operatorname{Tr}\tau \le 1\), such that \(W_{Q,k}=\int (\tau ^t)^{\otimes k}\, d\zeta _Q(\tau )\) for every \(k\ge 0\), the identity on the complement of \(Q\) being implicit. The measure does not depend on \(k\).

Let \(Q_{jg},Y_{jg}\) be physical regions, indexed by arbitrary terminal labels \(j\) and bands \(0\leq g{\lt}G\). Assume that every pair is disjoint and that \(Q_{jg}\cup Y_{jg}\subseteq Q_{j'h}\) for all \(j,j'\) whenever \(g{\lt}h\). Adjoin distinct auxiliary sites \(c,r\), and write

where the complement on the right is taken among physical sites. Then \(L_{jg}\) and \(E_{jg}\) are disjoint, \(F_{jg}=(L_{jg}\cup E_{jg})^c\) in the enlarged site set, and \(L_{jg}\cup E_{jg}\subseteq L_{j'h}\) whenever \(g{\lt}h\). These are the lifted regions used in [ Ope26b , Section 8, lines 80–110 and 421–443 ] .

Fix a finite family of nonzero finite-dimensional local Hilbert spaces and \(t{\gt}0\). There is \(C\ge 0\) such that, for every number of copies \(k\ge 0\) and every three pairwise disjoint regions \(P,Y,F\),

The inequality holds on the whole copy space. The constant is independent of \(k\) and the three regions; it may depend on the fixed local spaces and \(t\). Empty regions and zero copies are included.

Let \(P,Y,F\) be pairwise disjoint subsets whose union is \(V\), and put \(H_k=F_{P,k}+F_{F,k}-F_{Y,k}\). Then

This includes \(k=0\), empty regions, and an empty family of factors.

Let \(U\) be the coordinate unitary of an equivalence of finite physical coordinate sets. For any physical matrix \(H\) and vector \(\Omega \), put \(H'=UHU^*\) and \(\Omega '=U\Omega \). Then

Normalization is unnecessary for these identities.

Fix a finite tensor product of nonzero finite-dimensional Hilbert spaces and \(t{\gt}0\). On every subsystem \(Q\), use the replica metric \(W_{Q,k}\) defined by the common weights of the original full system, and put \(F_{Q,k}=\sum _\lambda (\log d_\lambda )\pi _{Q,k}^\lambda \). There is \(C\ge 0\), depending on the fixed local spaces and \(t\), such that for every integer \(k\ge 0\) and every pair of disjoint regions \(U,Y\),

The inequality holds on the whole copy space. Empty regions and zero copies are included. With \(a=2t\), it gives the exponential comparison in the proof of the sharp upper bound in [ Ope26b , Section 8 ] .

Fix a finite tensor product of nonzero finite-dimensional spaces and \(t{\gt}0\). Use the replica weights of this original full system to define \(W_{Q,k}\) on every subsystem. There is \(C\ge 0\) such that the following holds for every \(k,K\ge 0\) and every real number \(T\).

Let \(U_g,Y_g\), \(0\le g{\lt}K\), be regions satisfying

Suppose that a vector \(\varphi \) lies in the closed label cutoffs \(F_{U_g}\le T\) and \(F_{U_g\cup Y_g}\le T\) for every \(g\). Set

with the product taken in increasing order. Then

The vector \(\varphi \) is arbitrary in the stated cutoffs. This is the physical-region product in [ Ope26b , 07-comparators.tex, comparator:pinned-metric, lines 356–373 ] ; its identification with the Bell restriction is separate.

Fix a finite tensor product of nonzero finite-dimensional spaces and \(t\ge 0\). Use its original full-system weights to define the replica metrics \(W_{Q,k}\). Let \(\mathcal T\) be one weighted geometric-mean tree with leaf labels in \(J\). Suppose the regions \(U_{j,g},Y_{j,g}\) satisfy

for every \(j,j'\). Set

Then, with all products in increasing band order,

This is the actual band factorization in [ Ope26b , 06-transport.tex, Lemma 7.1; 07-comparators.tex, lines 367–375 ] .

Fix the original finite tensor product, \(t{\gt}0\), and a finite leaf-label set \(J\). There is \(C\ge 0\) such that the following holds for every copy number \(k\), every number \(K\) of bands, and every weighted mean tree \(\mathcal T\) on \(J\).

Let the regions satisfy the preceding disjointness and cross-label nesting conditions, and put \(M=\prod _{g{\lt}K}M_g\). Let \(a\in \mathbb {R}\) and let \(\varphi \) be a unit vector in all the closed cutoffs \(F_{U_{j,g}}\le a\) and \(F_{U_{j,g}\cup Y_{j,g}}\le a\). For every vector \(u\) and every \(0\le s\le 1/2\),

The constant is chosen before \(k,K\), the regions, the tree, the cutoff, and the vectors. The statements include \(k=0\), \(K=0\), and negative cutoff thresholds. This is the inverse-power step of [ Ope26b , 07-comparators.tex, lines 367–381 ] .

Retain the physical family, unit vector \(\Omega \), region \(X\), spectral selection \(E\), positive mass \(z\), supplied label sequence \(\ell _k\), nested list \(\mathcal R\) and \(t{\gt}0\) of Theorem 23.28.1, including its eventual auxiliary-mass hypothesis. Put \(d=|E|\) and now take both auxiliary registers to be \([d]\).

There exists \(c\ge 0\) such that, for every sufficiently large \(k\), the following statement holds. Choose a symmetric isometry \(Z\) and band data \(K,U,Y,\mathcal T,B_{\mathrm{sh}},B_{\mathrm{exc}},s\) satisfying all the disjointness, nesting, entropy, matching and exponent hypotheses of Theorem 23.28.1. Form the same ordered compressed product \(M\) with the full-family weights on the enumerated registers. In the notation of Theorem 23.30.1, put \(v_k=\mathrm{pre}_{k,\ell _k}^{\Omega }\circ G_k\) and \(N_k^2=\| M^{-s}Z^*v_k\| ^2\). Then \(N_k^2{\gt}0\) and

The constant and eventual threshold precede all the choices of symmetric coordinates and band data. This is the physical-coordinate form of the upper estimate in [ Ope26b , Section 8, lines 130–163 and 332–381 ] .

For every \(j,g\), the actual compression \(B_{jg}\) is positive definite and

If \(g\ne h\), then \(B_{jg}B_{j'h}=B_{j'h}B_{jg}\) for all \(j,j'\). This gives the symmetric-space commutation required in [ Ope26b , Section 7, lines 268–276; Section 8, lines 20–37 ] .

Let \(F\) be a finite set of sites with finite nonempty local configuration sets, let \(k\ge 0\) and \(t\ge 0\), and let \(W(S)\) be the regional replica metric with the original full-system weights. Write \(\Pi \) for simultaneous copy symmetry and fix \(Z\) with \(Z^*Z=I\) and \(ZZ^*=\Pi \). For terminal labels \(j\in J\) and bands \(g\in \{ 0,\ldots ,G-1\} \), assume

for all \(j,j'\). For one weighted mean tree \(T\), put

Then \(M\) is positive definite. For every real \(s\) and every nonzero \(\xi \) satisfying \(\Pi \xi =\xi \), the vectors

satisfy \(w\ne 0\), \(\lVert v\rVert =1\) and \(\Pi v=v\). Moreover, for every site \(a\) and representation label \(\ell \),

The same \(Z,T,\xi \) and original labels occur throughout. Repeated terminal labels, zero tree weights and \(G=0\) are allowed. This is the label-preservation step used in [ Ope26b , Section 7, lines 388–401; Section 8, lines 421–443 ] .

Every weighted mean tree \(T\) with terminal labels in \(J\) satisfies \(T(B_{\cdot g}){\gt}0\). For \(g\ne h\) and any two such trees \(T,T'\), one has

The isometry \(Z\) is the same for all bands and trees. Repeated terminal labels and zero edge weights are allowed.

Fix a finite family of nonzero local spaces. Let \(Q_{j,g},Y_{j,g}\) be disjoint regional pairs satisfying \(Q_{j,g}\cup Y_{j,g}\subseteq Q_{j',h}\) for \(g{\lt}h\) and all statuses \(j,j'\). Fix \(t\ge 0\) and an isometry \(Z\) onto simultaneous copy symmetry. Use the original full-family Schur weights and set

Then

for every weighted mean tree \(T\). Every band uses the same tree and therefore the same full terminal-path probabilities, exactly as in [ Ope26b , Section 8, lines 114–121 ] . The empty band family and zero edge weights are included.

Let \(V\) be a finite set of sites, and let \(\iota _v\) be a finite nonempty coordinate set for every \(v\in V\). Put \(\mathcal V=\bigotimes _{v\in V}\mathbb {C}^{\iota _v}\) and \(d=\dim \mathcal V=\prod _{v\in V}|\iota _v|\). Fix \(t{\gt}0\). For every subsystem \(Q\subseteq V\), use the common replica metric \(W_{Q,k}=\sum _\lambda w_k(\lambda )\pi ^\lambda _{Q,k}\) from Theorem 23.5.2, with every label padded to the same full dimension \(d\).

There exists \(C\ge 0\), depending on the fixed system and \(t\), such that the following holds for every \(k\ge 0\) and every pairwise disjoint cover \(P\cup Y\cup F=V\). Let \(\Pi _k\) be the orthogonal projection onto \(\operatorname {Sym}^k\mathcal V\), and define the full-space operators and the scalar

Then \(0{\lt}c_k\le 1\), and, in the positive-semidefinite order,

The exponent \(C\) is independent of \(k\) and of \(P,Y,F\). Empty parts, \(k=0\), and the empty set of sites are included, with the empty tensor product equal to \(\mathbb {C}\).

Let \(J_a(l)\) denote the integral of the previous theorem and \(W_\lambda \) the Weyl dimension formula. For \(t{\gt}0\) and a partition \(\lambda \vdash k\) with at most \(d\) rows, \(w_k(\lambda )=J_{tk}(l)/\bigl(J_0(\delta )W_\lambda \bigr)\).

Retain the finite coordinate sets, selected set \(E\), enumeration \(\iota :E\to D\) and bijection \(e_k:\mathcal H_D^k\to \mathcal H_E^k\) of Theorem 23.30.1. Fix \(k,n\geq 0\) and an arbitrary matrix \(Z_D:\mathbb {C}^n\to \ell ^2(\mathcal H_D^k)\). Define \(Z_E\) by \(Z_E(x,a)=Z_D(e_k^{-1}x,a)\), retaining the same columns.

Let \(t\in \mathbb {R}\). Write \(W^D_{Q,k}\) and \(W^E_{Q,k}\) for the replica metrics on the two full families, using their full one-copy dimensions in the common label weights. Fix an integer \(K\geq 0\), a set \(J\), regions \(U_{j,g},Y_{j,g}\subseteq V\) for \(j\in J\) and \(0\leq g{\lt}K\), and a weighted binary mean tree \(\mathcal T\) with labels in \(J\). In the augmented site set \(V\sqcup \{ c,r\} \) put

Here complements are taken in \(V\), and physical subsets are identified with their images in the augmented site set. For \(\sigma \in \{ D,E\} \) define the ordered factors and their compressions by

with the last product in increasing order of \(g\). Then \(B^D_{j,g}=B^E_{j,g}\) for every \(j,g\), and \(M_D=M_E\). A singular matrix is assigned inverse zero. No isometry, positivity, disjointness or nesting assumption is needed. The statement includes zero columns, an empty band family and empty coordinate sets. It identifies the compressed matrices used in [ Ope26b , Section 8, lines 80–110, 283–308 and 370–381 ] .

Retain the physical vector \(\Omega \), selected set \(E\), copy number \(k\) and irreducible label \(\ell \) of Theorem 23.30.1. Set \(v_E=v^{\mathrm{site}}_{k,\ell }(\Omega _X,E)\circ \tau _k\) and \(v_D=\mathrm{pre}_{k,\ell }^{\Omega }\circ G_k\). For any \(Z_D:\mathbb {C}^n\to \ell ^2(\mathcal H_D^k)\) define \(Z_E(x,a)=Z_D(e_k^{-1}x,a)\). Then \(Z_D^*v_D=Z_E^*v_E\). No normalization or positive selected-mass hypothesis is imposed. This is the compressed form of the coordinate identification in [ Ope26b , Section 8, lines 130–147, 240–254 and 283–290 ] .

Let \(V\) be finite with finite coordinate sets \(D_v\), and fix \(X\subseteq V\). Put \(D_V=\prod _{v\in V}D_v\), \(A=\prod _{v\in X}D_v\) and \(B=\prod _{v\notin X}D_v\). Let \(\Omega \in \ell ^2(D_V)\) be arbitrary, and let \(\Omega _X\) be the same vector in \(\ell ^2(A\times B)\) under restriction of configurations. Fix \(E\subseteq A\), put \(d=|E|\) and \(D=\{ 0,\ldots ,d-1\} \), and retain the fixed enumeration \(\iota :E\to D\).

The two descriptions of the full configuration space are \(\mathcal H_E=D_V\times E\times D\) and \(\mathcal H_D=D_V\times D\times D\). For every integer \(k\geq 0\), the change of the left auxiliary coordinate and the grouping of copies are

Let \(\tau _k:\mathcal H_E^k\to (A\times E)^k\times (B\times D)^k\) be the fixed witness coordinates of Definition 23.29.1. For any irreducible label \(\ell \) of \(\mathfrak S_k\), let \(v^{\mathrm{site}}_{k,\ell }(\Omega _X,E)\) be the prevector of Definition 22.50.1, and let \(\mathrm{pre}_{k,\ell }^{\Omega }\) be the original prevector in \(D_V^k\times (D^k\times D^k)\) with that same label. Then

No normalization or positive selected-mass hypothesis is required. Empty \(E\) and zero copies are included. This is the change of coordinates between the initial vector and the selected Bell coordinates in [ Ope26b , Section 8, lines 130–147, 240–254 and 283–290 ] .

Let \(\beta \) be a finite family of finite coordinate sets, let \(\Omega \) be a unit vector on its product, and let \(X\) be a set of physical sites. Fix a spectral selection \(E\) of the reduced density on \(X\), of mass \(z{\gt}0\), and put \(d=|E|\). The two auxiliary coordinate sets are \(C=E\) and \(R=[d]=\{ 0,\ldots ,d-1\} \). Let \(\psi \) be the corresponding normalized compressed selected vector on \(X^c\times R\), and let \(\Psi _k\) be its tensor power in exterior-site coordinates. Let \(\ell _k\) be a supplied sequence of auxiliary labels. Suppose that, for all sufficiently large \(k\),

Fix a finite nested list \(\mathcal R\) of physical regions disjoint from \(X\), and an arbitrary index set \(S\). Let \(b\) be the selected Bell vector on \(X\times E\). Then every original and auxiliary coordinate set is nonempty and \(\| b\| =1\).

Write \(\tau _k\) for the selected-Bell witness coordinates, \(J_k\) for the actual insertion of \(b^{\otimes k}\) in physical coordinates, and \(\Pi _{\mathrm{ext},k}\) for simultaneous copy symmetry on the exterior sites. Put \(P_k=J_k\Pi _{\mathrm{ext},k}J_k^*\), and let \(v_k\) be the pullback through \(\tau _k\) of the supplied-label prevector in Definition 22.50.1. For all sufficiently large \(k\), there is a vector \(\Phi _k\) such that

The vector \(\Phi _k\) is invariant under simultaneous copy permutations.

This same vector has the following property for every \(K\ge 0\), every pair of families \(U,Y:S\times \{ 0,\ldots ,K-1\} \to 2^V\), and all real \(B_{\mathrm{sh}},B_{\mathrm{exc}}\). Suppose that \(S_\Omega (D)\le B_{\mathrm{sh}}\) for every \(D\in \mathcal R\), that \(X\) is disjoint from every \(U_{jg}\cup Y_{jg}\), and that each \(Q\in \{ U_{jg},U_{jg}\cup Y_{jg}\} \) has a region \(D\in \mathcal R\) comparable to \(Q\) with

Set \(a_k=k(B_{\mathrm{sh}}/z+B_{\mathrm{exc}})\). If \(\Pi _{Q,\le a_k}\) denotes the original physical exterior cutoff, omitting \(X\) and both auxiliary sites, then

for all \(j,g\). The vector and the eventual threshold are chosen before these matched families and bounds; no common threshold over different physical states or ambient systems is asserted.

For every irreducible label \(\ell \) of \(\mathfrak S_k\), let \(\Pi ^e_{R,\ell }\) and \(\Pi _{R,\ell }\) be the label projections on the retained register \(R\) in the exterior and original coordinates. Then

For any \(U\subseteq V\) and \(a\in \mathbb R\), let \(Q_{U\setminus X}(a)\) be the sum of the label projections with \(\log d_\ell \leq a\) on the indicated original physical sites, and let \(Q^e_{U\setminus X}(a)\) denote the same cutoff in the exterior coordinates. Then

The label and threshold are unchanged. This is the coordinate identity used for the supplied common vector in [ Ope26b , Section 8, lines 346–365 ] .

The insertion satisfies

No normalization or positive selected-mass hypothesis is required.

Let \(\varphi \in \ell ^2((A\times C)^k\times \mathcal H_e^k)\) be invariant under simultaneous permutation of the \(k\) copies in both factors, and suppose \((P_E^{\otimes k}\otimes I)\varphi =\varphi \). Let \(P_{e,\mathrm{sym}}\) average the simultaneous copy permutations on \(\mathcal H_e^k\). Then the same vector in physical coordinates, \(\Phi =W_k\varphi \), is symmetric and satisfies

The vector \(\varphi \) need not be normalized.

Central label functions commute for nested or disjoint subsystems: if \(Q\subseteq Q'\), a central function of the copy permutations of \(Q\) commutes with every operator generated by the copy permutations of \(Q'\), and operators generated by the copy permutations of disjoint subsystems commute. On \(\mathcal S_k\), complementary label projectors agree: \(\pi ^\lambda _{Q,k}\Pi _k=\pi ^\lambda _{Q^c,k}\Pi _k\), even when \(Q\) and \(Q^c\) have different dimensions.

Let \(\rho \) be any permutation-invariant density matrix on \(Q^{\otimes k}\), \(q=d_Q\), and \(L_\rho =-\log \rho \) on its support. Then \(F_Q\) commutes with \(\rho \), \(0\le F_Q\le L_\rho \) on the support of \(\rho \), and for \(0\le b\le 1\),

The operators on \((\mathbb {C}^q)^{\otimes k}\) commuting with the copy permutations form the algebra generated by the operators \(\Delta (A)\).

Every real central label observable on \(S\) commutes with \(\Pi _T^\theta \) when \(S\subseteq T\). If \(P,Y,F\) are pairwise disjoint and cover \(V\), then, with \(G=F_P^\theta +F_F^\theta -F_Y^\theta \) and \(\Pi =\Pi _V^\theta \),

The signed operator need not be nonnegative on the entire space.

Let \(P=P_0P_1\), let \(h\) be supported on \(P_0x\) with \(0\le h\le I\), \(d_h=(\dim P_0)(\dim x)\), \(\eta _\theta =S_\theta (x|P)+S_\theta (x|U)=I_\theta (x:F|U)\), and let \(0{\lt}a\) with \(a\log (ed_h)\le 1/8\) and \(t=a/2\). Then \(|f_\theta |-\operatorname {Re}f_\theta \le Ca^2\log ^4(ed_h)\eta _\theta ^{1/8}\) on unit vectors, and uniformly in symmetric density matrices \(\sigma \),

The same bound holds with the two outer parts of the partition metric exchanged.

Let \(\mathcal D_k=|O_k|+|O_k^\ast |-O_k-O_k^\ast \), with absolute values taken on \(\mathcal S_k\). Uniformly in symmetric density matrices \(\sigma \),

Let \(G,H\) be commuting Hermitian matrices on a finite-dimensional complex coordinate space, let \(a\ge 0\), and put \(J=\mathbf1_{[0,\infty )}(G)\). Then

The coordinate space may have dimension zero. Positivity of \(G\) on the whole space is not required. This gives the spectral-compression step used in the rough estimate of  [ Ope26b , Section 7 ] .

Let \(\theta :S_m\to S_k\) be a group homomorphism, and restrict every subsystem copy action along \(\theta \). For a real function \(f\) on the irreducible labels of \(S_m\), write \(F_{Q,\theta }(f)=\sum _\lambda f(\lambda )\pi ^\lambda _{Q,\theta }\). For all real label functions \(f,g\), if \(Q\subseteq Q'\) or \(Q\cap Q'=\varnothing \), then

The local spaces are arbitrary finite-dimensional spaces, and \(\theta \) need not be injective. This is the restricted-copy form of [ Ope26b , Lemma 6.1(2), Section 8 ] .

Let \(F\) be a set of sites, let \(A_f,B_f\) be coordinate sets, and let \(c_f:B_f\to A_f\) be bijections. For \(k\geq 0\), define \(e_k:(\prod _{f\in F}B_f)^k\to (\prod _{f\in F}A_f)^k\) by \((e_kx)_{i,f}=c_f(x_{i,f})\). If \(Q\subseteq F\) is finite and \(\sigma \in \mathfrak S_k\), then

The coordinate sets need not be finite or nonempty. This is the coordinate invariance of the replica action in [ Ope26b , Section 6, lines 24–27 ] .

Under the hypotheses of Theorem 23.43.1, let \(\rho \ge 0\) satisfy \(J\rho =\rho \). Then

No normalization of \(\rho \) is assumed.

Let a finite group act on two finite coordinate sets, and let \(U\) be the permutation unitary induced by an equivariant equivalence of these sets. If \(S_1,S_2\) are the respective group-average projections, then

Let \(\Pi \) project onto simultaneous copy symmetry. For arbitrary regions \(Q,Y\subseteq F\), with no disjointness assumption, put

Both matrices commute with \(\Pi \), and \(A\Pi =\Pi D\). Thus the original far-region factor and the laminar factor agree on the actual symmetric space, as used in [ Ope26b , Section 8, lines 299–308 ] .

For every \(t{\gt}0\) there is \(C\geq 0\), depending only on \(t\) and the original local spaces, such that for every \(k\geq 0\) and every disjoint partition \((P,Y,F)\) of \(V\),

Empty regions and zero copies are included. This is the common symmetric-space floor used before the tree comparison in [ Ope26b , Section 8 ] .

For every \(t{\gt}0\) there is \(C\geq 0\), depending only on \(t\) and the original local spaces, with the following property. For every \(k\geq 0\), disjoint partition \((P,Y,F)\) of \(V\), and rectangular matrix \(Z\) satisfying \(Z^*Z=I\) and \(ZZ^*=Q_k\),

The constant is independent of the regions and of the fixed coordinates \(Z\).

For a density matrix \(\sigma \) on the symmetric subspace of \(k\ge 1\) copies and an operator \(G\) on \(m\) copies,

where \(d\) is the one-copy dimension.

Let \(A\) commute with the copy permutations, and let \(T(A)=\int U^{\otimes k}A(U^{\otimes k})^\ast \, dU\) over the Haar probability measure of the compact unitary group. Then \(T(A)=\sum _\lambda c_\lambda \pi ^\lambda \) with \(c_\lambda \operatorname{Tr}\pi ^\lambda =\operatorname{Tr}(\pi ^\lambda A)\).

For a label \(\lambda \) of \(S_k\) with at most \(q\) rows, \(a_\delta (y)\chi _\lambda (y)=a_{\lambda +\delta }(y)\) with \(\delta _j=q-j\); that is, \(\Delta (y)\chi _\lambda (y)=\det [y_i^{l_j}]\) with \(l_j=\lambda _j+q-j\) and \(\Delta \) the Vandermonde determinant.