Dependencies

For \(k\ge 0\), define the bijection

Its inverse places each physical configuration and auxiliary index in the same copy.

Let the local physical coordinate sets be finite. Given an original vector \(\Omega \), a spectral selection \(E\) on \(X\) with actual mass \(z\), and a copy number \(k\ge 0\), define for \(D\subseteq \widehat V\)

Here the formula is extended to \(z=0\) by setting \(a_k(D)=k\). Positive selected mass is required when this threshold is used in the exponential cutoff estimate.

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 every \(B\subseteq V\), evaluation of the auxiliary and physical coordinates gives a bijection

If \(X\cap B=\varnothing \), evaluation of the physical coordinates also gives a bijection

The inverses assign these values to the same auxiliary and physical sites. No local coordinate set is required to be nonempty or finite for these bijections.

If \(z{\gt}0\), define \(\chi =z^{-1/2}(J^\dagger \otimes I_B)\Omega \) on \(\mathcal H_R\otimes \mathcal H_B\).

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.

For a region \(D\subseteq \widehat V\), let

regarded as a subset of the original site set \(V\). The auxiliary site contributes no element to this set.

Let \(Q,Y,V,C,R\) be coordinate sets and let \(m\geq 0\). A configuration \(x\) assigns one coordinate in each of these five sets to each copy \(j\). The equivalence \(e_m\) groups these coordinates as

Thus its range is

The inverse places the five coordinates back in the same copy. This includes \(m=0\).

A fixed division \([k]\simeq [m]\sqcup [r]\) of the copies induces the coordinate decomposition of the full configuration space into the configurations on the first and second groups. All physical and auxiliary factors use this same division.

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.

For \(\sigma \in S_m\) and \((x,c)\in Q^m\times C^m\), put

Let \(U_Q,U_C,U_{QC}\) be their permutation operators. The separate actions commute, and \(u_{QC}(\sigma )=u_Q(\sigma )u_C(\sigma )\). Let \(\Pi ^Q_\lambda ,\Pi ^C_\mu ,\Pi ^{QC}_\nu \) be the corresponding central Schur projectors, with irreducible dimensions \(d_\lambda ,d_\mu ,d_\nu \).

For the \(QC\) pair set \(D_C=F_Q+F_C-F_{QC}\), and for the \(VR\) pair set \(D_R=F_V+F_R-F_{VR}\). Each operator is Hermitian. On the common space write \(\widehat D_C=D_C\otimes I_{VR}\) and \(\widehat D_R=I_{QC}\otimes D_R\).

The physical label operator and the two merge deficits are

The region \(Y\) is independent of \(Q\) and \(V\).

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 ] .

Regroup the same component \(w\) into retained and discarded factors:

The common density \(\Gamma _B(u)\) is \(\operatorname {Tr}_{\mathcal H_{\mathrm{tr}}}|w\rangle \langle w|\). It retains the four exterior good regions and is unnormalized. The discarded factors include the good middle copies and the middle coordinates within the bad physical copies, so every middle physical coordinate is traced.

Regroup \(w\) as a vector on \(\mathcal H_{\mathrm{ret}}\otimes \mathcal H_{\mathrm{tr}}\), where

The joint marginal \(\rho _{Q_GC_G}(w)\) is the partial trace of \(|w\rangle \langle w|\) over the second factor. Thus it retains the good physical \(Q\) coordinates and the good auxiliary \(C\) coordinates of the same component, without normalization.

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.

For \(k\ge 0\), define the bijection

Let \(U_k f=f\circ e_k\) be the induced unitary between coordinate 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 finite list \(\mathcal B=(B_1,\ldots ,B_m)\) and real thresholds \(a_B\), put \(C_j=C_{B_j}(a_{B_j})\) and set \(C=\prod _{j=1}^m C_j\), in the order of the list. The empty product is the identity. Repeated regions are permitted.

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

For every \(k\ge 0\), define the bijection

Its inverse joins the two restricted configurations in each copy.

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.

On \((\mathbb {C}^q)^{\otimes k}\), \(\sum _{a,b}E_{ab}E_{ba}=kq+T_k\). Consequently, for a label \(\lambda \) with at most \(q\) rows, \(\kappa _\lambda =\sum _a\lambda _a^2+\sum _{a{\lt}b}(\lambda _a-\lambda _b)-kq\).

For a permutation representation \(\rho \) of a finite group \(G\) with multiplicities \(m_\lambda \), \(\operatorname{tr}\rho (a)=\sum _\lambda m_\lambda \operatorname{tr}W_\lambda (a)\). In the regular representation \(m_\lambda =d_\lambda \), so \(|G|e_\lambda (g)=d_\lambda \operatorname{tr}W_\lambda (g^{-1})\), and \(\sum _g\operatorname{tr}\rho (g^{-1})\, g=\sum _\lambda (|G|m_\lambda /d_\lambda )e_\lambda \).

Let \(\mu \) be a partition of \(k\) padded to length \(q\). Place the copies of \((\mathbb {C}^q)^{\otimes k}\) in the boxes of the Young diagram of \(\mu \), let \(x\) assign to each copy the row of its box, and let \(C_\mu \) be the group of permutations of the copies preserving every column. Then \(\sum _{\sigma \in C_\mu }\operatorname {sgn}(\sigma )U(\sigma )e_x\) is a highest-weight vector of weight \(\mu \).

The number of labels of a finite group is its number of conjugacy classes. The conjugacy classes of \(S_k\) correspond to the partitions of \(k\), which correspond to the weakly decreasing vectors of \(k\) nonnegative integers summing to \(k\).

Let \(A_r=\frac1{r!}\sum _{\tau \in S_r}\operatorname {sgn}(\tau )\tau \). Then \(A_r^2=A_r\) and \(\operatorname{Tr}\bigl(A_r\operatorname {diag}(y)^{\otimes r}\bigr)=e_r(y)\). On \((\mathbb {C}^q)^{\otimes (m+r)}\) the permutations of the first \(m\) and of the last \(r\) copies act on the two tensor factors of \((\mathbb {C}^q)^{\otimes m}\otimes (\mathbb {C}^q)^{\otimes r}\).

Let \(\mu \) be a label of \(S_m\) and let \(c(\lambda )\) be the rank of the \(\lambda \) block of \(e^\mu _{00}\otimes A_r\) in \(\mathbb {C}[S_{m+r}]\). In every dimension \(q\) in which \(\lambda \) and \(\mu \) occur, \(c(\lambda )\le 1\), and \(c(\lambda )=0\) unless \(\lambda =\mu +1_S\) for a set \(S\) of \(r\) rows.

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 \(\lambda \) be a partition of \(k\ge 1\) padded to length \(q\) and \(D_\lambda \) the right-hand side of the formula for \(d_\lambda \). Then \(D_\lambda =\sum _iD_{\lambda -e_i}\), the sum over the rows from which a box can be removed, and \(D_{\lambda -e_i}/D_\lambda =(l_i/k)\prod _{j\ne i}(l_i-1-l_j)/(l_i-l_j)\). If the last row of \(\lambda \) is empty, \(\sum _iD_{\lambda +e_i}=(k+1)D_\lambda \), the sum over the rows to which a box can be added. Appending empty rows does not change \(D_\lambda \).

The central idempotents are nonzero, central, mutually orthogonal, and sum to the identity. Each is fixed by the conjugate-linear involution of \(\mathbb {C}[G]\) sending \(c\, g\) to \(\bar c\, g^{-1}\).

A nonzero subspace of the range of \(\pi ^\lambda \) invariant under all permutation operators has dimension at least \(d_\lambda \). Moreover \(\operatorname{tr}\pi ^\lambda =d_\lambda m_\lambda \), where \(m_\lambda \) is the rank of the image of the matrix unit \(e^\lambda _{11}\), the multiplicity of \(\lambda \).

For one action, the span of the vectors \(E^\lambda _{j1}v\) is invariant and has dimension at most \(d_\lambda \). For two commuting actions with labels \(\alpha ,\beta \) and matrix units \(A_{ij},B_{ab}\), the span of the vectors \(A_{p1}B_{q1}z\) is invariant under both and has dimension at most \(d_\alpha d_\beta \), and every nonzero subspace of the joint isotypic component invariant under both actions has dimension at least \(d_\alpha d_\beta \).

Let \(Z_k=\sum _{\tau \in S_k}f(\tau )\tau \), where \(f(\tau )\) is the number of configurations of \((\mathbb {C}^q)^{\otimes k}\) fixed by \(\tau \). Then \(Z_{k+1}=(q+X_{k+1})Z_k\), with \(X_{k+1}=\sum _{j\le k}(j\ k{+}1)\) and \(Z_k\) embedded through the permutations of the first \(k\) copies.

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 any permutation action, \(\sum _\lambda m_\lambda \) is at most the dimension of the commutant of the permutation operators, which is at most the number of orbits of pairs. For the copy permutations of \((\mathbb {C}^q)^{\otimes k}\) this number is at most \((k+1)^{q^2}\). Hence the total multiplicity, and the number of labels occurring, are at most \((k+1)^{q^2}\).

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 \(\sum _i\| P_iz\| ^2=\| z\| ^2\).

\(c(\lambda ,\nu )\le 1\), and \(c(\lambda ,\nu )=0\) unless \(\lambda =\nu +e_i\) for some row \(i\).

Let \(\Pi \) be the projection onto \(\mathcal S_k\) and let \(A\) be a band metric, which commutes with \(\Pi \). The operator \(\widetilde A=\Pi A+(I-\Pi )\) is positive definite, agrees with \(A\) on \(\mathcal S_k\) and commutes with the copy permutations. Two such operators \(\widetilde A,\widetilde B\) commute exactly when \(A\) and \(B\) commute on \(\mathcal S_k\). Moreover \(\widetilde A^{-1/2}\widetilde B\widetilde A^{-1/2}=\Pi A^{-1/2}BA^{-1/2}+(I-\Pi )\), so every lower bound \(Q\le A^{-1/2}BA^{-1/2}\) with \(\Pi Q\Pi =Q\) holds for the extended operators, and for every \(X\) commuting with \(\Pi \) the skew squares of \(\widetilde A\) and of \(A\) have the same expectation in every density matrix on \(\mathcal S_k\).

For a subgroup \(H\) of a finite group \(G\), \(d_\lambda =\sum _\nu c(\lambda ,\nu )d_\nu \) and \(\sum _\lambda d_\lambda ^2=|G|\). For a permutation representation \(\rho \) of \(G\), an element \(a\) of \(\mathbb {C}[G]\) with \(\lambda \) block \(W_\lambda (a)\), and an operator \(M\) commuting with \(\rho \), \(d_\lambda \operatorname{Tr}\bigl(M\rho (a)\pi ^\lambda \bigr)=\operatorname{tr}W_\lambda (a)\operatorname{Tr}(M\pi ^\lambda )\), and \(\rho (a)\pi ^\lambda =0\) exactly when \(\pi ^\lambda =0\) or \(W_\lambda (a)=0\).

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

The coefficients of a central element of \(\mathbb {C}[G]\) form a class function. For \(G=S_k\), a central element has the same coefficient at \(\pi \) and at \(\pi ^{-1}\).

If \(Z\) commutes with every input of a mean tree and fixes the vector \(v\), then \(Z\sigma _{j,u}=\sigma _{j,u}\) for every leaf \(j\) and every real \(u\).

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 \(a\ge 0\).

For every \(t{\gt}0\) there is a constant \(C\geq 0\), depending only on \(t\) and the five spaces, with the following property. Let \(k\geq 0\), let \(B\) be a subset of the copies, and put \(r=|B|\). Let \(u\) be fixed by simultaneous copy permutations and belong to the two whole-copy auxiliary label spaces \(\lambda _3,\lambda _4\). For any physical vector \(\Omega \), let \(w=(P_B(\Omega )\otimes I)u\) be its actual excitation component, and express \(w\) in the five-factor coordinates as \(f\).

Let \(L_g(S)\) be the label entropy of the subsystem \(S\) for permutations of the complementary copies. Set

Then

Neither \(\Omega \) nor the excitation component is required to be normalized.

Let \(t{\gt}0\) satisfy \(4t\leq 1\). There exists \(C\geq 0\), depending only on \(t\) and the five spaces, with the following property. Let \(k\geq 0\) and \(B\subseteq \{ 1,\ldots ,k\} \), and put \(r=|B|\) and \(m=k-r\). Let \(\Omega \) be a unit vector on the physical factors. Suppose \(u\) is invariant under simultaneous copy permutations and is fixed by the original whole-copy auxiliary label projections \(\lambda _3,\lambda _4\). Form the actual excitation component \(w=(P_B(\Omega )\otimes I)u\) and write it as \(f\) in five-factor coordinates. With \(W_{t,k}(S)\) the replica metric of \(S\), set

Then

The vector \(u\) and its component may be unnormalized or zero. This is the component estimate in [ Ope26b , Section 8 ] , before combining its scalar factors into a common exponential bound.

For every \(t{\gt}0\) with \(4t\leq 1\), there exists \(C\geq 0\), depending only on \(t\) and the five spaces, with the following property. Let \(k\geq 0\), let \(\Omega \) be a unit physical vector, and let \(u\) be invariant under simultaneous copy permutations and fixed by the two original whole-copy auxiliary labels \(\lambda _3,\lambda _4\). Suppose \(0\leq \tau \leq \tfrac 12\) and \(2kS-\varepsilon \leq \log \dim \lambda _3+\log \dim \lambda _4\). For any excitation subset \(B\) with \(|B|\leq \tau k\), let \(w=(P_B(\Omega )\otimes I)u\) and write it as \(f\) in five-factor coordinates. Define

where \(h\) is binary entropy. Then \(\operatorname {Re}\langle f,A_k^{-1}f\rangle \leq c_k\| w\| ^2\). No normalization of \(u\) or \(w\) is required.

Suppose the five spaces are nonzero, and fix \(t{\gt}0\). There exists \(C_0\geq 0\), depending only on \(t\) and these spaces, with the following property. Use the data of the preceding theorem, and suppose also that \(u\) belongs to the two original whole-copy auxiliary label spaces \(\lambda _C,\lambda _R\). Assume \(20t\leq r\) and \(20t\leq 1\). Express \(w\) in five-factor coordinates as \(f\) and put \(q=|B|\). With \(W_{t,k}(S)\) denoting the replica metric, set

Then

The constant is chosen before \(k,B,\Omega ,\lambda _C,\lambda _R,u,K,r\). The vector \(u\) and its component may be unnormalized or zero.

For every \(t{\gt}0\), there exists \(C_0\geq 0\), depending only on \(t\) and the five spaces, with the following property. Let \(k\geq 0\), let \(\Omega \) be a unit physical vector, and let \(u\) be invariant under simultaneous permutations of the copies and fixed by the two original auxiliary labels \(\lambda _C,\lambda _R\). Suppose \(K\geq 0\) and \(r\in \mathbb R\) satisfy \(20t\leq r\), \(20t\leq 1\), and, for every \(j\in \{ Q,Y,V\} \) and \(|v|\leq r\),

Suppose also that \(0\leq \tau \leq \frac12\) and \(2kS-\varepsilon \leq \log \dim \lambda _C+\log \dim \lambda _R\). For any \(B\subseteq \{ 0,\ldots ,k-1\} \) with \(|B|\leq \tau k\), let \(w=(P_B(\Omega )\otimes I)u\), and express \(w\) in five-factor coordinates as \(f\). Set

Then \(\operatorname {Re}\langle f,A_k^{-1}f\rangle \leq c_k\| w\| ^2\). The constant is chosen before all state, label, copy, moment and cutoff data. The vector \(u\) and its component may be zero.

Let \(\Omega \) be a unit physical vector, and let \(u\) be fixed by simultaneous permutations of the \(k\) copies of the physical and auxiliary factors. For \(B\subseteq \{ 1,\ldots ,k\} \), put \(m=|B^c|\), \(w=(P_B(\Omega )\otimes I)u\), and let \(\rho _g\) be the actual density on its good-copy coordinates. Thus \(\operatorname {Tr}\rho _g=\| w\| ^2\). Write \(\rho _Q,\rho _Y,\rho _V\) for the singleton densities of \(\Omega \) and \(S_Q,S_Y,S_V\) for their entropies.

For \(j\in \{ Q,Y,V\} \), let \(p_{j,\ell }\) be the eigenvalues of \(\rho _j\) and define

where a zero eigenvalue contributes zero. Suppose that \(K,r\in \mathbb R\) satisfy \(\log M_j(v)\leq vS_j+Kv^2\) for every \(j\) and every \(|v|\leq r\). Let \(a\geq 0\) satisfy \(10a\leq r\) and \(10a\leq 1\). Write \(F(S)\) for the label entropy of the subsystem \(S\) on the good copies, and put

Then

This includes \(w=0\) and \(m=0\).

For every \(t{\gt}0\) with \(4t\leq 1\), a constant \(C\geq 0\) depending only on \(t\) and the five spaces has the following property. Let \(k\geq 0\), let \(\Omega \) be a unit physical vector, and let \(\lambda _3,\lambda _4,\tau ,S,\varepsilon \) satisfy the label lower bound and \(0\leq \tau \leq \tfrac 12\) in Theorem 22.62.1. Let \(P\) be the orthogonal projection onto the intersection of simultaneous permutation symmetry, the two original auxiliary label spaces, and defect count at most \(\tau k\). Let \(U\) be the unitary change from five-factor coordinates to the physical-and-two-auxiliary copy coordinates, and set \(H=UA_k^{-1}U^*\). With \(c_k\) given by (156),

This is the rough inverse compression estimate in [ Ope26b , Section 8 ] .

For each \(t{\gt}0\), a constant \(C_0\geq 0\) depending only on \(t\) and the five spaces has the following property. Use the state, moment, label and cutoff assumptions of Theorem 22.61.1. Let \(P\) be the orthogonal projection onto the intersection of simultaneous copy symmetry, the two original auxiliary label spaces, and defect count at most \(\tau k\). Let \(U\) be the unitary change from five-factor coordinates to physical-and-two-auxiliary copy coordinates, and set \(H=UA_k^{-1}U^*\). With \(c_k\) as in (154),

This is the sharp inverse compression estimate in [ Ope26b , Section 8 ] , with the one-copy moment assumptions and the original label lower bound retained explicitly.

Fix five nonzero finite-dimensional spaces of dimensions \(d_0,\ldots ,d_4\). For every \(t{\gt}0\) with \(4t\leq 1\), there is \(C\geq 0\) depending only on \(t\) and these spaces with the following property. Let \(k\geq 0\), let \(\Omega \) be a unit vector on the three physical factors, and suppose that the two original auxiliary labels satisfy

Suppose \(0\leq \tau \leq \tfrac 12\). Set

Let \(P\) be the actual projection from the preceding theorem and let \(U\) be the five-factor grouping unitary. Then \(b_k^{-1}U^*PU\leq A_k\). For every finite rectangular \(Z\) with \(ZZ^*=S\), the matrix \(P_Z=Z^*U^*PUZ\) is an orthogonal projection and

This is the rough lower bound in [ Ope26b , Section 8 ] , with the polynomial and subset-counting factors displayed explicitly.

Fix five nonzero finite-dimensional spaces and \(t{\gt}0\). There exists \(C_0\geq 0\), depending only on these spaces and \(t\), with the following property. Use the original unit state, auxiliary labels, marginal moment assumptions, and parameters of Theorem 22.61.2. Let \(A_k\) be the original squared metric in five-factor coordinates and let \(P\) be the original symmetric, fixed-label low-defect projection. If \(U\) changes five-factor coordinates to the physical-and-two-auxiliary coordinates, put

where \(c_k\) is given by (154). Then

Moreover, let \(P_{\mathrm{sym}}\) be simultaneous copy symmetry in five-factor coordinates. For every matrix \(Z\) with \(ZZ^*=P_{\mathrm{sym}}\), the matrix \(Z^*\widehat PZ\) is an orthogonal projection and

The constant is chosen before all copy, state, label, moment and cutoff data. This is the sharp lower bound used in [ Ope26b , Section 8 ] .

Let \(C\) be a finite coordinate set with \(d=|C|\), and split the \(k\) copy coordinates into specified groups of sizes \(m\) and \(r\), where \(m+r=k\). Let \(S_r\) act on \(\mathbb C^{C^k}\) by permuting the second group and fixing the first group. If its central projection \(P_\beta \) for an irreducible label \(\beta \) is nonzero, then

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

For a label \(\mu \) of \(S_m\) with at most \(q\) rows, \(\chi _\mu e_r=\sum _\lambda \chi _\lambda \), the sum running over the partitions \(\lambda =\mu +1_S\) of \(m+r\) with at most \(q\) rows.

Let \(\Omega \) be a unit physical vector, let \(\lambda _3,\lambda _4\) be the original auxiliary labels, and let \(\tau \in \mathbb R\). Let \(P\) project onto simultaneous symmetry, those two label spaces, and defect count at most \(\tau k\) in grouped coordinates. With \(U,S\) as above, every rectangular matrix \(Z\) satisfying \(ZZ^*=S\) has \(Z^*U^*PUZ\) an orthogonal projection.

Let \(\sigma \in \mathfrak S_k\) act only on the copies of the auxiliary site. If \(e_k(x)=(b,a)\), then

Thus the physical-complement coordinates are fixed, and the auxiliary coordinates carry the ordinary copy action.

For any function \(\chi :A\times \Omega _{X^c}\to \mathbb C\), set \(\chi _{\mathrm{site}}=\chi \circ e\). Then for every \((b,a)\in \Omega _{X^c}^k\times A^k\),

Let \(\Omega \in \mathcal H_V\), let \(E\) select eigenvectors of its reduced matrix on \(X\), and let their total mass be \(z{\gt}0\). The compressed system has auxiliary coordinate set \(\{ 0,\ldots ,|E|-1\} \). Put \(\mathcal L_k=(\mathcal H_X\otimes \mathbb {C}^E)^{\otimes k}\), and let \(\varphi \in \mathcal L_k\otimes \widehat{\mathcal H}_{\widehat V}^{\otimes k}\). Let \(\mathcal R\) be a finite list of regions containing \(B\), and suppose that for every \(C\in \mathcal R\),

Suppose \(X\) is disjoint from \(B,Q\), these two regions are comparable, \(S(\rho _B^\Omega )\leq B_{\rm sh}\), and \(1+\log \dim \mathcal H_{B\triangle Q}\leq B_{\rm exc}\). Then

The vector \(\Omega \) need not be normalized. This is the matched-region implication in [ Ope26b , Section 8, lines 332–365 ] .

For every \(B\) disjoint from \(X\),

with the same convention at \(z=0\). No normalization of \(\Omega \) is needed for this identity.

Assume now that the local coordinate sets are finite. Let \(\Omega \) be any original physical vector, and let \(E\) select eigenvalue indices of its marginal on \(X\). Let \(\psi \) be its actual selected vector and let \(\chi \in \mathbb C^{|E|}\otimes \mathbb C^{\Omega _{X^c}}\) be its actual compressed vector, with the selected indices in their fixed order. Take \(A=\{ 0,\ldots ,|E|-1\} \) and define \(\chi _{\mathrm{site}}(x)=\chi (e(x))\).

If \(X\cap B=\varnothing \) and \(U_B\) is the unitary coordinate identification induced by \(e_B\), then

The identity does not require normalization of \(\Omega \) or positive selected mass.

For any function \(\chi :A\times \Omega _{X^c}\to \mathbb C\), define

For every irreducible label \(\lambda \) of \(\mathfrak S_k\),

In particular, take \(A=\{ 0,\ldots ,|E|-1\} \) and take \(\chi \) to be the actual compressed spectral vector of an original physical vector \(\Omega \), with the selected indices in their fixed order. The right side is then exactly the auxiliary-label mass of that same compressed vector in physical-complement-first coordinates. The identity applies to every member of an already chosen label sequence. It includes zero copies and zero selected mass and requires no normalization or symmetry assumption.

Use the augmented site family \(\widehat V=\{ r\} \sqcup X^c\), with finite auxiliary coordinate set \(A\) and finite local physical coordinate sets. Let \(T_k\) be the unitary induced by the bijection \(e_k\) of Definition 22.36.1. For every \(k\ge 0\) and every irreducible label \(\lambda \) of \(\mathfrak S_k\),

where the left projection is for the actual auxiliary subsystem action and the right projection is for the ordinary copy action on \(A^k\). The label \(\lambda \) is unchanged.

Assume that \(A\) and the local physical coordinate sets are finite. For every function \(\chi :A\times \Omega _{X^c}\to \mathbb C\), the vector

is fixed by every simultaneous permutation of its \(k\) copies. No normalization assumption is needed.

Suppose \(X\cap B=\varnothing \). Let \(b\) and \(c\) be configurations on \(\widehat B\) and \(\widehat B^c\), respectively, and write \(e_C(c)=(a,d)\). Joining \(b\) and \(c\) into one exterior-site configuration and then applying \(e\) gives

On the right, the union joins the physical configurations on \(B\) and \((X\cup B)^c\) to a configuration on \(X^c\).

Assume the same eventual lower bound \(q_k\ge 1/(2(k+1)^{d^2})\). Let \(\beta _E\) be the actual selected Bell vector on \(\mathcal H_X\otimes \mathbb C^E\), with projection \(\Pi _E=|\beta _E\rangle \langle \beta _E|\), and put \(\mathrm{post}_k=\beta _E^{\otimes k}\otimes \Psi _k\). For all sufficiently large \(k\), there is a unit vector \(\varphi _k\) such that

The vector \(\varphi _k\) is invariant under simultaneous copy permutations, using the canonical identification of copies of a product with the product of copy spaces. The left Bell coordinate is indexed by \(E\) and the right auxiliary coordinate by its fixed enumeration \(\{ 0,\ldots ,d-1\} \). The eventual threshold may depend on the original state, selection, region list, and prescribed label sequence.

Assume that \(q_k\ge 1/(2(k+1)^{d^2})\) for all sufficiently large \(k\). Then, for all sufficiently large \(k\),

The label sequence is the prescribed sequence in (77).

Retain the physical vector \(\Omega \), disjoint regions \(X,Q\), selected indices \(E\), their enumeration, and the compressed product vector \(\Psi _k\) from Theorem 22.45.6. Suppose the actual selected mass is \(z=\operatorname {Tr}(P_E\rho _X){\gt}0\), where \(\rho _X=\operatorname {Tr}_{X^c}|\Omega \rangle \langle \Omega |\). Write \(\rho _Q=\operatorname {Tr}_{Q^c}|\Omega \rangle \langle \Omega |\) for the original physical marginal. For every \(k\ge 0\) and \(u\in \mathbb R\),

The vector \(\Omega \) need not be normalized. The comparison includes zero copies and empty regions. It is the selected-marginal estimate used in the sharp upper comparison of [ Ope26b , Section 8 ] .

Let \(V\) be a finite set of sites with finite local coordinate sets, let \(\Omega \) be any physical vector, and let \(X,Q\subseteq V\) be disjoint. Choose spectral indices \(E\) of the marginal on \(X\), with their fixed enumeration. Let \(\psi \) and \(\chi \) be the selected physical vector and its compressed representative from Section 22.39, with both vectors set to zero when the selected mass is zero. Place \(\chi \) on the augmented sites \(\widehat V=\{ r\} \sqcup X^c\) and set \(\Psi _k=\chi _{\mathrm{site}}^{\otimes k}\). Write \(\widehat Q\) for the image of \(Q\) in this augmented family. Write \(\sigma _Q=\operatorname {Tr}_{Q^c}|\psi \rangle \langle \psi |\), and let \(F_{\widehat Q,k}\) and \(F^{\mathrm{reg}}_{Q,k}\) be the label entropies of the augmented subsystem and the ordinary copy action on the original regional space, respectively. For every \(k\ge 0\) and \(u\in \mathbb R\),

No normalization of \(\Omega \) or positivity of the selected mass is required. Empty regions and empty local coordinate sets are included. This is the compressed-state expectation identity used in the sharp upper comparison of [ Ope26b , Section 8 ] .

Let \(V\) be a finite set of sites with finite-dimensional coordinate spaces, let \(X\subseteq V\), and let \(\Omega \in \mathcal H_X\otimes \mathcal H_{X^c}\). Select eigenvalue indices \(E\) of its actual marginal \(\rho _X^\Omega \), with

In the fixed enumeration of \(E\), let \(J_E:\mathbb C^d\to \mathcal H_X\) have the selected orthonormal eigenvectors as columns. Put \(\mathcal H_R=\mathbb C^d\) and define

Thus \(\psi \in \mathcal H_{X^c}\otimes \mathcal H_R\) is unit. Its auxiliary density is

where the trace is over the physical first factor. Place the selected coordinate of \(\chi \) at the auxiliary site \(R\) and leave its physical coordinates at their sites in \(X^c\); write \(\chi _{\mathrm{site}}\) for this vector and \(\Psi _k=\chi _{\mathrm{site}}^{\otimes k}\) for its literal tensor power.

There is one sequence of irreducible labels \(\lambda _k\) of \(\mathfrak S_k\) such that, with \(Q_k\) the auxiliary isotypic projection of label \(\lambda _k\),

No normalization of the original vector \(\Omega \) is required. This is the selected-state conclusion of [ Ope26b , Section 7, lines 255–281 ] .

Use the preceding construction of \(\psi \), \(\rho _R\) and \(\Psi _k\), and assume that the original physical vector \(\Omega \) is unit. Fix a finite list \(\mathcal B=(B_1,\ldots ,B_m)\) of physical regions disjoint from \(X\), every two of which are comparable by inclusion. The list may be empty or contain repetitions. Let \(\mathcal H_C=\mathbb C^E\), and let the actual selected Bell vector and its projection be

Here \(u_i\) is the selected eigenvector of \(\rho _X^\Omega \); the \(C\) coordinate is indexed by \(E\), whereas the \(R\) coordinate uses its fixed enumeration by \(\{ 0,\ldots ,d-1\} \). Put \(\mathrm{post}_k=b_E^{\otimes k}\otimes \Psi _k\). For each \(B_j\), let \(\widehat B_j\) be its unchanged physical sites in the augmented family \(\{ R\} \sqcup X^c\), and define

The central label observable \(F_{\widehat B_j,k}\) has value \(\log \dim \mu \) on the isotypic subspace of label \(\mu \). The product follows the prescribed list order; for the empty list it is the identity.

There is one sequence \(\lambda _k\), with its auxiliary projection \(Q_k\), such that

Define the masses using this same sequence by

For all sufficiently large \(k\), there is a unit vector \(\varphi _k\in (\mathcal H_X\otimes \mathcal H_C)^{\otimes k} \otimes \mathcal H_{\{ R\} \sqcup X^c}^{\otimes k}\) satisfying

The vector \(\varphi _k\) is invariant under simultaneous permutations of all copies, using the canonical identification between copies of the product space and the product of the copy spaces. The eventual threshold may depend on the fixed state, selection and region list. This is the common-cutoff overlap construction in [ Ope26b , Section 7, lines 283–298 and 332–354 ] .

Assume \(\| \Omega \| =1\) and \(X\cap B=\varnothing \). Let \(P_{\widehat B}^{(k)}(a)\) be the closed central-label cutoff \(\log \dim \lambda \le a\) for the actual subsystem copy action on \(\widehat B\). There is a constant \(c{\gt}0\) such that for every integer \(k\ge 1\),

The rate may depend on the fixed original state, selection, and region. The region may be empty and its marginal may be singular.

Use the canonical physical coordinate split \(\mathcal H_{X^c}=\mathcal H_B\otimes \mathcal H_{(X\cup B)^c}\) to write \(\chi (r,b,c)\). Place the discarded coordinates first by setting \(\tau ((r,c),b)=\chi (r,b,c)\). Then

There is no normalization assumption on \(\Omega \) and no positive-mass assumption on \(E\).

Assume in addition that \(\| \Omega \| =1\). For every physical region \(B\) disjoint from \(X\), regard \(B\) as the exterior-site region \(\widehat B\). Then

The actual compressed vector satisfies

Only \(z{\gt}0\) is required; the original vector \(\Omega \) need not be unit.

Suppose \(z{\gt}0\) and put \(\Omega _E=z^{-1/2}(P_E\otimes I_B)\Omega \). Then \(\| \chi \| =1\) and

In particular, every reduced state on a subsystem of \(B\) is preserved by this relabeling, as required in the marginal comparison of Section 7.

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

For a partition \(\lambda \vdash k\) padded to any length \(q\) at least its number of rows, \(d_\lambda =k!\prod _{i{\lt}j}(l_i-l_j)/\prod _il_i!\) with \(l_i=\lambda _i+q-i\). Moreover \(c(\lambda ,\nu )=1\) if \(\lambda =\nu +e_i\) for some row \(i\) and \(c(\lambda ,\nu )=0\) otherwise.

Let \(k,r\) be nonnegative integers and \(0\le \tau \le 1/2\). If \(r\le \tau k\), then

These are the binomial estimates used before the inverse-compression inequality in  [ Ope26b , Section 7 ] .

Let \(\Omega \) be a physical one-copy vector and let \(B\) be a subset of the \(k\) copies. Suppose a vector \(u\) is fixed separately by the two original auxiliary label projections. Form its actual excitation component \(w=(E_B(\Omega )\otimes I)u\) and express it as \(f=w\circ \eta \) in five-factor coordinates. With \(r=|B|\) and the good group \(B^c\), inequality (144) holds for \(f\). Neither \(u\) nor \(\Omega \) needs normalization, and no symmetry of \(u\) is assumed.

Let \(S\) be a finite set with \(k\) elements and \(0\le \tau \le 1/2\). Then

This is the subset-count estimate preceding the inverse-compression inequality in  [ Ope26b , Section 7 ] .

For every \(b\in \mathbb R\),

This identity includes empty ambient coordinate sets and absent labels.

Let \(c{\gt}0\), let \(m\in \mathbb R\), and let \(d\) be a nonnegative integer. For all sufficiently large integers \(k\),

In particular, the statement applies when \(m\) counts a finite family of cutoffs, including the empty family.

Give the physical sites finite coordinate sets \(I_v\) and the auxiliary site any finite coordinate set \(R\). If \(X\) is disjoint from both \(B\) and \(Q\), then \(\dim \widehat{\mathcal H}_{\widehat B\triangle \widehat Q} =\dim \mathcal H_{B\triangle Q}\).

For a finite site set \(V\) and \(X,B,Q\subseteq V\), write \(\widehat B\) for the exterior lift of \(B\) to \(\widehat V=\{ *\} \sqcup X^c\). Then \(\widehat{B\triangle Q}=\widehat B\triangle \widehat Q\).

If \(B\subseteq V\) is disjoint from \(X\), then

Let \(I\) be a finite set, let \(c_i{\gt}0\) for every \(i\in I\), and let \(d\ge 0\) be an integer. For all sufficiently large integers \(k\),

The set \(I\) may be empty.

Let \(N\ge 0\). For each \(0\le i{\lt}N\), let \(I_i\) be a finite set and let \(A_i,B_i\) be complex matrices indexed by \(I_i\). If \(0\le A_i\le B_i\) for every \(i\), then

The matrices act on \(\bigotimes _{i=0}^{N-1}\mathbb C^{I_i}\). When \(N=0\), both products are the identity on the one-dimensional empty tensor product.

Let \(\theta \colon H\to S_k\) be a group homomorphism and let \(a=\sum _h a_hh\in \mathbb C[H]\). The auxiliary action of \(a\) on \(C\) becomes \(I\otimes ((\sum _h a_hU_C(\theta (h)))\otimes I)\) under \(\eta \). On \(R\) it becomes \(I\otimes (I\otimes \sum _h a_hU_R(\theta (h)))\). The sums are finite; injectivity of \(\theta \) is not required.

Under \(\eta \), permuting the copies of \(C\) by \(\sigma \in S_k\) becomes \(I\otimes (U_C(\sigma )\otimes I)\), and permuting those of \(R\) becomes \(I\otimes (I\otimes U_R(\sigma ))\).

The matrix \(P_0\) is an orthogonal projection and \(bP_0\leq P_0L_0P_0\).

If \(a\geq 0\) and \(P_0v=v\), then

Let \(U\) identify the \(k\) copies of five finite coordinate sets with the grouped physical copies of factors \(0,1,2\) and the two auxiliary copy spaces of factors \(3,4\). Let \(S\) be the average of simultaneous copy permutations before grouping, and let \(S'\) be the corresponding average in the three grouped spaces. Then \(U^*S'U=S\).

Let \(c\geq 0\), let \(M_i\geq 0\) and \(A_i\in \mathbb R\) for \(0\leq i{\lt}5\), and suppose that \(M_i\leq c e^{A_i}\). Then \(\prod _{i=0}^4M_i^{1/5}\leq c\exp (\frac15\sum _{i=0}^4A_i)\). The common mass \(c\) may vanish.

Let \(A\) and \(P\) be Hermitian, let \(B\) be a complex matrix, and assume \(AP=PA\) and \(AB=BA\). If \(Pe^BP\leq cP\) for \(c\in \mathbb R\), then every vector \(v\) with \(Pv=v\) satisfies \(\operatorname {Re}(v^*e^{A+B}v)\leq c\operatorname {Re}(v^*e^Av)\).

Fix a finite physical site set \(V\), nonzero finite-dimensional spaces \(H_v\), and two nonzero finite-dimensional auxiliary spaces \(H_c,H_r\). For every \(t{\gt}0\) with \(4t\leq 1\), there is \(c\geq 0\), depending only on these spaces and \(t\), with the following property.

Let \(k\geq 0\), let \(\Omega \in \bigotimes _{v\in V}H_v\) be a unit vector, and let \(\lambda _c,\lambda _r\) be the two auxiliary labels. Suppose \(0\leq \tau \leq \tfrac 12\), and let \(S_0,\varepsilon \in \mathbb R\) satisfy

Let \(P_{\mathrm{glob}}\) be the common projection of Definition 23.38.1, formed from this same vector, these labels, and the defect cutoff \(\tau k\). Set

where \(h\) is binary entropy. Let \(Q_k\) denote the projection onto simultaneous copy symmetry of the full physical and auxiliary space. For every finite rectangular matrix \(Z\) with \(ZZ^*=Q_k\), the matrix \(P_Z=Z^*P_{\mathrm{glob}}Z\) is an orthogonal projection. For every pair of disjoint physical regions \(Q,Y\subseteq V\), put \(F=(Q\cup Y)^c\) and

Here every metric retains the weights of the original full site family. Then

Thus the same constant, projection, and matrix \(Z\) serve every physical cut. This is the uniform regional form of the rough lower bound in [ Ope26b , Section 8 ] .

Fix a finite physical site set \(V\), nonzero finite-dimensional spaces \(H_v\), and two nonzero finite-dimensional auxiliary spaces \(H_c,H_r\). For every \(t{\gt}0\), there is \(c\geq 0\), depending only on these spaces and \(t\), with the following property.

Let \(k\geq 0\), let \(\Omega \in \bigotimes _{v\in V}H_v\) be a unit vector, and let \(\lambda _c,\lambda _r\) be the auxiliary labels. Suppose \(0\leq \tau \leq \tfrac 12\) and

Let \(P_{\mathrm{glob}}\) be the common projection of Definition 23.38.1, formed from this vector, these labels, and the defect cutoff \(\tau k\). For every rectangular matrix \(Z\) satisfying \(ZZ^*=Q_k\), where \(Q_k\) is the projection onto simultaneous copy symmetry, the matrix \(P_Z=Z^*P_{\mathrm{glob}}Z\) is an orthogonal projection.

Let \(Q,Y\subseteq V\) be disjoint and put \(F=(Q\cup Y)^c\). Write \(\rho _0,\rho _1,\rho _2\) for the reductions of the original pure state on \(Q,Y,F\), respectively. Suppose \(K\geq 0\), \(20t\leq r\), \(20t\leq 1\), and, for \(j=0,1,2\) and \(|v|\leq r\),

where eigenvalues are counted with multiplicity. Set

Every metric retains the weights of the original full site family. Then

The same constant, projection, and symmetric coordinates serve all cuts. This is the regional sharp lower bound under the original marginal moment hypotheses in [ Ope26b , Section 8 ] .

Let \(C\) be a finite coordinate set, put \(d=|C|\), and specify groups of \(m\) good and \(r\) bad copies with \(m+r=k\). For the actual copy-permutation action on \(\mathbb C^{C^k}\), write \(F_g\) for the good-copy logarithmic label observable and \(P_\lambda \) for the whole-copy central projection of an irreducible label of dimension \(d_\lambda \). Then \(F_g\) commutes with \(P_\lambda \), and

No occurrence assumption is imposed on \(\lambda \). Empty coordinate sets and zero copy groups are included, with the convention \(\log 0=0\).

Let \(C\) be a finite coordinate set, put \(d=|C|\), and specify a split of the \(k\) copies into groups of sizes \(m\) and \(r\), where \(m+r=k\). For the actual copy-permutation action on \(\mathbb C^{C^k}\), let \(F_g,F_b,F_w\) denote the central logarithmic label observables for the good, bad and whole groups. Using the real logarithm extended by \(\log 0=0\), one has, on the whole space,

Empty coordinate sets and zero copy groups are included.

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.

Let \(S_j\) be a finite family of subsystems and let \(a_j\in \mathbb R\). Writing \(F_{S_j,G}\) for the first-subgroup label entropy on the full space, the same coordinates give

No disjointness of the subsystems is required. This is a coordinate identity, as used in [ Ope26b , Section 8, lines 454–560 ] .

For finite local configuration spaces and every subsystem \(S\), the copy action of the first subgroup on \(S\) becomes its usual copy action on the first \(m\) copies, tensored with the identity on the remaining \(r\) copies. The same coordinate identity holds for every real label observable.

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 \(r=0\).

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

Let \(q_k,r_k\) and \(\ell _{i,k}\) be real sequences indexed by integers \(k\ge 0\), with \(i\) in a finite set \(I\), and let \(d\ge 0\) be an integer. Suppose \(q_k\ge 1/(2(k+1)^d)\) for all sufficiently large \(k\) and \(r_k\ge q_k-\sum _{i\in I}\ell _{i,k}\) for every \(k\). Suppose also that, for each \(i\in I\), there is a constant \(c_i{\gt}0\) for which \(\ell _{i,k}\le e^{-c_i k}\) for all sufficiently large \(k\). Then, for all sufficiently large \(k\),

No additional sign assumptions on the sequences are required.

Let \(\psi \in \mathcal H_S\otimes \mathcal H_R\) be a unit vector, let \(\rho =\operatorname {tr}_S|\psi \rangle \langle \psi |\), and let \(v_k\) be its literal tensor power regrouped into \(\mathcal H_S^{\otimes k}\otimes \mathcal H_R^{\otimes k}\). There is one sequence \((\lambda _k)\) for which, for all sufficiently large \(k\),

and \(\log d_{\lambda _k}=kS(\rho )+o(k)\).

Every nonzero subspace of \((\mathbb {C}^q)^{\otimes k}\) invariant under the commutant of the copy permutations contains a highest-weight vector. If the subspace is irreducible and contains a highest-weight vector \(v\) of weight \(\mu \), then its vectors of weight \(\mu \) are the multiples of \(v\), all its weights are lexicographically at most \(\mu \), every highest-weight vector in it has weight \(\mu \), and \(\mu \) is weakly decreasing.

Let \(\rho \ge 0\) be a finite matrix with \(\operatorname {Tr}\rho =1\), and let \(s\in \mathbb R\) satisfy \(S(\rho )\le s\). There is a constant \(c{\gt}0\) such that, for every integer \(k\ge 1\),

Singular density matrices are included. The permutation invariance is that of the actual tensor power.

Let \(\rho \ge 0\) be a matrix on a finite-dimensional Hilbert space with \(\operatorname {Tr}\rho =1\). Write \(S=S(\rho )\), \(\rho _k=\rho ^{\otimes k}\), and \(L_k=-\log \rho _k\), taking the logarithm to be zero on the kernel. There is a constant \(c{\gt}0\), depending only on \(\rho \), such that for every integer \(k\ge 1\),

The matrix \(\rho \) may be singular.

Let \((p_i)_{i\in I}\) be a finite family with \(p_i\ge 0\) and \(\sum _i p_i=1\), and set \(H=-\sum _i p_i\log p_i\). There is a constant \(c{\gt}0\), depending only on this family, such that for every integer \(k\ge 1\),

Tuples containing a zero weight contribute zero to the sum.

For every real function \(f\) and \(k\ge 0\),

This identity also holds for any positive semidefinite \(\rho \) whose trace need not be one. A real function on a finite spectrum defines its spectral functional calculus without a global continuity assumption.

Under the hypotheses on \(A,P,b\) in Theorem 7.16.2, every \(a\geq 0\) satisfies \(Pe^{-aA}P\leq e^{-ab}P\).

The matrix \(P\) is an orthogonal projection, \(LP=PL\), and \(bP\leq PLP\).

For every \(a\geq 0\), \(Pe^{-aL}P\leq e^{-ab}P\).

The shape of a label of \(S_k\) in \((\mathbb {C}^k)^{\otimes k}\) is a partition of \(k\), and this is a bijection between the labels of \(S_k\) and the partitions of \(k\).

For arbitrary physical regions \(B,C\subseteq V\),

Consequently, a finite list whose regions are pairwise comparable by inclusion remains pairwise comparable after lifting. The list may contain repeated regions.

In the same finite tensor product, put \(P_B(a)=\sum _{\log \dim \lambda \leq a}\pi _B^\lambda \). Suppose \(B\subseteq Q\) or \(Q\subseteq B\). For real \(a,t\) satisfying \(a+k\log \dim \mathcal H_{B\triangle Q}\leq t\), one has \(P_Q(t)P_B(a)=P_B(a)\). This is the cutoff comparison in [ Ope26b , Section 8, lines 356–365 ] .

Let \(\mathcal D\) be any finite-dimensional coordinate space, and let \(\varphi \in \mathcal D\otimes \mathcal H_F^{\otimes k}\). Suppose \(B,Q\) are comparable, \(z{\gt}0\), and the real parameters satisfy \(s\leq B_{\rm sh}\) and \(1+\log \dim \mathcal H_{B\triangle Q}\leq B_{\rm exc}\). Then

No normalization of \(\varphi \) is required.

Let \(J\) be finite and let \(g\) range over \(0,\ldots ,G-1\). Let \(T_g\) be weighted binary trees with terminal weights \(w_{j,g}\) and positive definite inputs \(A_{j,g}\) on one finite-dimensional complex Hilbert space. Suppose their roots \(M_g=T_g(A_{\cdot ,g})\) commute pairwise, set \(M=M_0\cdots M_{G-1}\), and let \(P_g\) be orthogonal projections. Let \(b_g{\gt}0\), \(\kappa _g\ge 0\), and let \(L_{j,g},U_g,r_g\in \mathbb R\) satisfy

For \(s\ge 0\) and \(0{\lt}\| p\| \le 1\), put \(u=M^{-s}p\) and \(v=u/\| u\| \). Suppose \(\delta _g\ge 0\) and \(1-\delta _g\le \langle v,P_gv\rangle \). Define \(R_g=2|r_g|+|\log b_g|+\log 2+\log (3/2)\). Then

The remainders are independent of the deficits and terminal weights; no upper bound on \(\delta _g\) is imposed. This gives explicit errors in the analytic lower comparison of [ Ope26b , Section 8 ] .

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.

The actual merge deficit is positive semidefinite:

In particular, this holds for the separate and simultaneous actions on the literal paired copy space, at every number of copies.

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

Let \(A,B\subseteq F\) satisfy \(A\subseteq B\). For every irreducible label \(\lambda \) of \(S_k\),

This includes empty coordinate sets and \(k=0\), and follows from [ Ope26b , Lemma 6.1(2) ] .

For a Hermitian matrix \(A\), let \(J_A=\mathbf1_{[0,\infty )}(A)\). If \(B\ge 0\), then \(J_B=I\). If \(T\) is any rectangular matrix satisfying \(AT=TB\), then

The matrix \(T\) may have zero or noninjective range, and the coordinate sets may be empty.

Let a finite group \(G\) act by permutations on a finite set \(Y\). If the projector \(\pi ^\lambda \) for an irreducible label \(\lambda \) is nonzero, then \(\dim \lambda \leq |Y|\).

Let \(F\) be a finite set of sites with finite coordinate sets \(I_f\), and write \(\mathcal H_B=\bigotimes _{f\in B}\mathbb {C}^{I_f}\) for \(B\subseteq F\). For \(k\geq 0\), let \(\pi _B^\lambda \) denote the projector for the \(B\)-subsystem copy action of \(S_k\). Then \(\pi _B^\lambda \ne 0\) implies \(\dim \lambda \leq (\dim \mathcal H_B)^k\).

The lifted deficits commute. For all real \(a,b\),

Let \(\rho \) be any matrix on the common space, and let \(\rho _{QC}=\operatorname {Tr}_{VR}\rho \) and \(\rho _{VR}=\operatorname {Tr}_{QC}\rho \). For all real \(a,b\),

The matrix need not be normalized, invariant under copy permutations, or independent between the two pairs. The identities are complex trace equalities, and hence also hold for their real parts.

Suppose \(\rho \ge 0\), \(\operatorname{Tr}\rho =1\), and \(\rho \) commutes with \(U_Q(\sigma )\) and \(U_C(\sigma )\) for every \(\sigma \in S_m\). For every real \(b\le 1\),

Suppose \(\rho \ge 0\) and \(\rho \) commutes with both separate copy actions. No normalization or positive-mass condition is imposed. For every real \(b\le 1\),

The statement includes \(m=0\), empty bases and zero-trace components.

For an arbitrary real function \(f\) and independently chosen real label functions \(g_A\), with \(A\in \{ QC,VR,C,R\} \), write \(L_A(g_A)=\sum _\lambda g_A(\lambda )\Pi _A^\lambda \). Then

In particular, \(J\) commutes with both merge deficits.

Set \(D_{QV}=F_Q+F_V-F_{QV}\) and \(J=\mathbf1_{[0,\infty )}(G_{\mathrm{phys}})\). Then

Zero copies and zero-dimensional factors are included.

For every real \(a\),

The weight \(\rho \) need not be normalized or commute with any of the observables. Zero copies, zero weights and zero-dimensional factors are included. This is the five-factor Hölder step in [ Ope26b , Section 7, lines 556–560 ] .

If \(\| \Omega \| =1\), then the three original marginals satisfy

The marginals may be singular. This is the nonnegativity of the physical entropy difference used in [ Ope26b , Section 8 ] .

For every \(\psi \) and every \(k\ge 0\),

For every complex matrix \(\rho \) on \(X\) and every \(b\in \mathbb R\),

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 ] .

Under the hypotheses of Theorem 13.77.2, let \(j\in \{ 0,1,2\} \). For a positive semidefinite matrix \(\rho \) with eigenvalues \(\lambda _a\), define

A zero eigenvalue contributes zero to both expressions. For every real \(v\),

Let \(V\) be finite, let each \(\beta _v\) be a finite coordinate set, and let \(Q,Y\subseteq V\) be disjoint. Put \(F=(Q\cup Y)^c\). The restriction map

is the actual regional coordinate bijection. Every complex-valued function \(\Omega \) on the original configurations satisfies

Empty regions and empty coordinate sets are included.

Fix nonzero finite physical site spaces, nonzero auxiliary spaces \(C,R\), and \(0{\lt}t\leq 1/4\). There are \(c,C_f\geq 0\), depending only on these spaces and \(t\), with the following property.

Let \(\Omega \) be a unit physical vector, let \(\gamma {\gt}0\), and suppose

Fix \(k{\gt}0\), an isometry \(Z\) onto simultaneous copy symmetry, and disjoint physical regions \(Q_{j,g},Y_{j,g}\) such that \(Q_{j,g}\cup Y_{j,g}\subseteq Q_{j',h}\) whenever \(g{\lt}h\). Let \(T\) be one finite weighted mean tree, used for all \(G\) bands. Suppose the nonzero vector \(\xi \) is simultaneously symmetric, has norm at most one, and has the two original auxiliary labels \(\lambda _C,\lambda _R\), where

In the full original site coordinates, put

For \(s\geq 0\), define

where \(\overline H\) is the averaged original physical Hamiltonian with the two auxiliary identities. Then \(w\ne 0\), \(\| v\| =1\), and \(E_{\rm def}\geq 0\). For every \(0{\lt}\tau \leq 1/2\), set

The same vector satisfies

The constants are chosen before the copy number, vector, labels, tree, regions and symmetric coordinates. This is an intermediate form of the rough lower comparison in [ Ope26b , Section 8 ] ; the selected common initial vector supplies its hypotheses in the physical application.

In the setting of Theorem 22.70.1, suppose that both auxiliary spaces have dimension \(d\geq 1\), that the two auxiliary labels are the same label \(\lambda \), and that the number of bands is \(K\geq 8\). Choose

There are constants \(c,C_f\geq 0\), chosen before the physical state, copy number, mean tree, regions, symmetric coordinates and label. Retain the same band product \(M\), initial vector \(\xi \), and vectors

In particular, \(\xi \ne 0\), \(\| \xi \| \leq 1\), simultaneous symmetry, the two auxiliary label equations, the global gap and the nested region assumptions are those of that theorem. Assume \(2kS-\varepsilon \leq 2\log \dim \lambda \). Put

Then \(w\ne 0\), \(\| v\| =1\), and \(\delta _k\geq 0\). For every \(0{\lt}\tau \leq 1/2\),

No sign condition is imposed on \(S\) or \(\varepsilon \), and no upper bound on \(\delta _k\) is required. This is a consequence for a supplied initial vector, used in the physical comparison of [ Ope26b , Section 8 ] .

Fix nonzero finite physical site spaces, nonzero auxiliary spaces \(C,R\), and \(t{\gt}0\). There are \(c,C_f\geq 0\), depending only on these spaces and \(t\), with the following property.

Let \(\Omega \) be a unit physical vector, let \(\gamma {\gt}0\), and suppose

Fix \(k{\gt}0\), an isometry \(Z\) onto simultaneous copy symmetry, and disjoint physical regions \(Q_{j,g},Y_{j,g}\) such that \(Q_{j,g}\cup Y_{j,g}\subseteq Q_{j',h}\) whenever \(g{\lt}h\). Suppose that \(K\geq 0\), \(20t\leq r_0\) and \(20t\leq 1\). For each of the three regions \(Q_{j,g}\), \(Y_{j,g}\) and \((Q_{j,g}\cup Y_{j,g})^c\), suppose its reduced density \(\rho \) has eigenvalues \((p_i)\) satisfying

Define

Let \(T\) be one finite weighted mean tree, used for all \(G\) bands. Suppose the nonzero vector \(\xi \) is simultaneously symmetric, has norm at most one, and has the two original auxiliary labels \(\lambda _C,\lambda _R\), where

In the full original site coordinates, put

For \(s\geq 0\), define

where \(\overline H\) is the averaged original physical Hamiltonian with the two auxiliary identities. Then \(w\ne 0\), \(\| v\| =1\), and \(E_{\rm def}\geq 0\). For every \(0{\lt}\tau \leq 1/2\), set

Here \(w_j\) denotes the terminal weight of \(T\). The same vector satisfies

The constants are chosen before the copy number, vector, labels, tree, regions and symmetric coordinates. This is an intermediate form of the sharp lower comparison in [ Ope26b , Section 8 ] ; the selected common initial vector supplies its hypotheses in the physical application.

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.

Let \(A\) be positive definite, let \(P\) be an orthogonal projection, let \(U\) be a permutation unitary identifying their coordinate spaces, and let \(c{\gt}0\). If \(PUA^{-1}U^*P\leq cP\), then every rectangular matrix \(Z\) with rows indexed by the coordinates of \(A\) satisfies

For every real \(b\leq 1\),

This is the unnormalized form of the first exponential moment in [ Ope26b , Section 7, equation (comparator:merge-moments) ] . It includes \(w=0\), \(m=0\) and \(k=0\).

Let \(k,r\) be nonnegative integers with \(r\leq k\). Suppose that \(0\leq \tau \leq \frac12\), \(r\leq \tau k\), and \(a,d,g,K,\Lambda \geq 0\). Let \(C,S,\varepsilon ,L\in \mathbb R\) satisfy \(2kS-\varepsilon \leq L\). Write \(h\) for binary entropy and \(m=k-r\). Then

Suppose \(\sigma B=B\) and \((U_\sigma \otimes M)v=v\). Then the component \(v_B=(R_B\otimes I_{\mathcal K})v\) satisfies \((U_\sigma \otimes M)v_B=v_B\).

Let \(A,C,D\) be finite coordinate sets, let \(\Omega \in \mathbb C^A\), and let \(B\subseteq \{ 1,\ldots ,k\} \) have size \(r\). Write \(R_B\) for the literal excitation operator, with factor \(I-|\Omega \rangle \langle \Omega |\) on \(B\) and \(|\Omega \rangle \langle \Omega |\) on its complement. Enumerate the complement first and \(B\) second to define the good-copy auxiliary observable \(F_g\). Let \(P_\lambda \) be the actual whole-copy central projection on \(\mathbb C^{C^k}\), with irreducible dimension \(d_\lambda \), and put

For \(v\in \mathbb C^{A^k}\otimes \mathbb C^{C^k}\otimes \mathbb C^D\) satisfying \((I\otimes P_\lambda \otimes I_D)v=v\), define \(w=(R_B\otimes I)v\). Then

No normalization is required. Empty coordinate sets, zero copies and zero components are included, with the total logarithm convention \(\log 0=0\).

In the preceding definitions, suppose that \(v\) is fixed by every simultaneous copy permutation of \(A^k,C^k,D^k\). Then \(\rho _C^{\mathrm{good}}\) commutes with every copy-permutation operator on \(C^m\). No normalization or independence assumption is made. The vector \(\Omega \) is arbitrary, and zero components and zero copy groups are included.

Retain the same \(\Omega ,u,B,w,f\) and density \(\rho _G\), and let \(m=|B^c|\). For every finite family of subsystems \(S_j\) and real coefficients \(a_j\),

Here the left-hand observables are the actual good-subgroup entropies on the full space; those on the right act on the \(m\) good copies. For \(H_g=F_{QC,g}+F_{VR,g}-F_{Y,g}-F_{C,g}-F_{R,g}\), \(\Delta _{QC}=F_{Q,m}+F_{C,m}-F_{QC,m}\), \(\Delta _{VR}=F_{V,m}+F_{R,m}-F_{VR,m}\) and \(G_m=F_{Q,m}+F_{V,m}-F_{Y,m}\), this gives

for every real \(a\). These are exact complex equalities, without normalization, symmetry or commutation assumptions on the region family, as used in [ Ope26b , Section 8, lines 454–560 ] .

Let the original one-copy physical vector have norm one, and let the original vector be fixed by simultaneous permutations of the physical and two auxiliary copy registers. Let \(w\) be its excitation component, let \(m\) be the number of good copies, and let \(\rho _G\) be its actual five-factor good density. For every real \(a\leq 1\),

Each deficit is defined from the actual subsystem copy actions, as in Theorem 13.68.2. The component may vanish and \(m\) may be zero. These are the two separate estimates in [ Ope26b , Section 8 ] .

Let \(I_0,\ldots ,I_4\) be finite coordinate sets, and let \(\Omega \in \mathbb C^{I_0}\otimes (\mathbb C^{I_1}\otimes \mathbb C^{I_2})\) be a unit vector. Fix \(k\geq 0\), a set \(B\subseteq \{ 0,\ldots ,k-1\} \), and an arbitrary vector \(u\) on the \(k\) physical copies and the two auxiliary copy registers. Put \(m=|B^c|\), let \(w\) be the component obtained by applying the excitation projection for \(B\) to the physical copies of \(u\), and let \(\rho _G\) be its good-copy density. Fix \(j\in \{ 0,1,2\} \). Write \(\rho _j\) for the original singleton density, \(d_j=|I_j|\), and \(S_j=S(\rho _j)\).

Let \(p_i\) be the eigenvalues of \(\rho _j\), and define \(M_j(v)=\sum _i p_i e^{-v\log p_i}\), with zero weights contributing zero. Suppose \(K,r\in \mathbb R\) satisfy \(\log M_j(v)\leq vS_j+Kv^2\) whenever \(|v|\leq r\). Let \(F_j\) be the entropy observable of the permutation action on the \(m\) good copies of factor \(j\). If \(a\geq 0\), \(2a\leq r\), and \(2a\leq 1\), then

The vector \(u\) need not be a unit vector or permutation invariant, and \(w\) may vanish. These are the signed physical moments used in [ Ope26b , Section 8 ] .

Under the preceding hypotheses, suppose that \(u\) is fixed by simultaneous permutations of the physical and both auxiliary copies. Let \(w=(R_B\otimes I)u\), and write \(d_X=|X|\) for each coordinate set. If \(a\geq 0\) and \(2a\leq 1\), then

The component \(w\) may vanish. No normalization of \(u\) or \(w\) is required, and the conclusion includes \(m=0\). This is the rough good-copy moment estimate in [ Ope26b , Section 8, lines 520–555 ] .

Suppose \(\| \Omega \| =1\). Set \(\rho _Q=\operatorname {Tr}_T|\Omega \rangle \langle \Omega |\), and let \(\rho _{C_G}(w)\) be the actual good auxiliary marginal of \(w\). Then

The original vector and its component may be unnormalized or zero. The identity includes zero copies and empty good sets.

Let the one-copy physical vector \(\Omega \) have norm one, and let \(u\) be fixed by simultaneous permutations of the physical and two auxiliary copy registers. For a set \(B\) of bad copies, put \(m=|B^c|\) and \(w=(R_B\otimes I)u\). Let \(\rho _{\mathrm{pair}}\) be the marginal of \(|w\rangle \langle w|\) on the good \(QC\) and \(VR\) coordinates. Write \(d_Q,d_V,d_C,d_R\) for the one-copy dimensions and \(D_{QC},D_{VR}\) for the two lifted merge deficits. For every real \(a\leq 1\),

The component may vanish. These are the separate bounds before their combination in [ Ope26b , Section 8 ] .

Let \(\| \Omega \| =1\), \(0\le \tau \le \tfrac 12\), \(c\ge 0\), and \(H\ge 0\). For \(B\subseteq \{ 0,\ldots ,k-1\} \), let \(R_B\) be the physical projection onto precisely the copies in \(B\) being orthogonal to \(\Omega \). Suppose that, for every vector \(u\) with simultaneous copy symmetry and both original auxiliary labels and every \(B\) with \(|B|\le \tau k\), the components \(w_B=(R_B\otimes I_{CR})u\) satisfy

Then

where \(h(\tau )=-\tau \log \tau -(1-\tau )\log (1-\tau )\), with \(0\log 0=0\). The component estimate remains an explicit hypothesis; this is the subsequent implication used in [ Ope26b , Section 8, (comparator:inverse-compression) ] .

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 ] .

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.

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

No normalization or nonvanishing of \(u\) or \(w\) is required; zero copies are included.

Write \(\rho _Q=\operatorname {Tr}_T|\Omega \rangle \langle \Omega |\) and \(\rho _{\mathcal K}(w)=\operatorname {Tr}_{(QT)^{\otimes k}} |w\rangle \langle w|\). After regrouping the retained coordinates as \(Q^{\otimes m}\otimes \mathcal K\), one has

Neither \(u\) nor \(w\) need be normalized or nonzero. Zero copies and empty good sets are included.

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 ] .

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.

For every real \(a\) with \(2a\leq 1\),

No normalization or nonvanishing of \(w\) is required. Zero copies and an empty good set are included. This is an auxiliary polynomial bound for the rough estimate in [ Ope26b , Section 8 ] .

Let \((P_i)_{i\in I}\) and \((Q_j)_{j\in J}\) be finite orthogonal resolutions of the identity, and suppose their products \(P_iQ_j\) also form an orthogonal resolution. For arbitrary complex functions \(f\) on \(I\) and \(g\) on \(J\),

Let \(S\in \mathbb R\), let \(D,h,\delta \geq 0\), and suppose that \(\tau {\gt}0\) and \(0{\lt}a\leq 1/2\). Then

This numerical estimate is used in the rough lower comparison of [ Ope26b , Section 8 ] .

Let \(V\) be a finite set with finite local coordinate sets, let \(\Omega \) be a unit vector in the corresponding physical Hilbert space, and fix \(X\subseteq V\). Write \(A\) and \(B\) for the configuration sets on \(X\) and \(X^c\), respectively, and let \(\Omega _X\in \mathbb C^A\otimes \mathbb C^B\) be \(\Omega \) in these product coordinates. In the fixed spectral decomposition of the actual marginal \(\rho _X^\Omega \), write \(u_i\) for the eigenvectors and \(p_i\) for the eigenvalues. Choose spectral indices \(E\) with \(d=|E|\) and \(z=\sum _{i\in E}p_i{\gt}0\). Put \(D=\{ 0,\ldots ,d-1\} \), let \(\iota :E\to D\) be the fixed enumeration, and define

Let \(H\) be a matrix on \(\mathbb C^A\otimes \mathbb C^B\), let \(E_0\in \mathbb R\), and suppose \(H\Omega _X=E_0\Omega _X\). Here \(\psi \in \mathbb C^B\otimes \mathbb C^D\), with the physical complement as the first factor. The actual selected vector and its auxiliary marginal satisfy

There is a sequence \((\lambda _k)_{k\ge 0}\) of irreducible labels of \(\mathfrak S_k\) with all the following properties. Denote the central label projection on \((\mathbb C^d)^{\otimes k}\) by \(\Pi _k=\Pi ^{d,k}_{\lambda _k}\). In the original product coordinates \((A\times B)^k\times (D^k\times D^k)\), let \(v_k\) be the initial vector of (499), using \(\Omega _X\) and \(\lambda _k\). Explicitly, with the uniform pair regrouped into \(D^k\times D^k\),

For every \(k\ge 1\),

For a finite coordinate set \(K\), write \(U^K_\sigma \) for the operator which permutes its \(k\) copies by \(\sigma \). For every \(\sigma \in \mathfrak S_k\),

The same label sequence satisfies, for all sufficiently large \(k\),

and \(\log \dim \lambda _k-kS(\rho _R)=o(k)\) as \(k\to \infty \). Here the copies of \(\psi \) are regrouped into \(B^k\times D^k\); the partial trace defining \(\rho _R\) is over the physical first factor.

Place the coordinate \(r\) of \(\chi (r,b)\) at the auxiliary site of \(\widehat V=\{ r\} \sqcup X^c\), retaining all physical coordinates in \(X^c\). Write \(\chi _{\mathrm{site}}\) for this vector, \(\Psi _k=\chi _{\mathrm{site}}^{\otimes k}\), and \(Q_k=\Pi _{\{ r\} ,\lambda _k}\) for the actual auxiliary-site label projection. In particular,

With this sequence fixed, let \(\mathcal R=(B_1,\ldots ,B_m)\) be any finite list of physical regions disjoint from \(X\), pairwise comparable by inclusion. The list may be empty or contain repetitions. For its actual lifted regions \(\widehat B_j\subseteq \widehat V\), define the closed label cutoffs and their ordered product by

The empty product is the identity. The thresholds use the original physical marginal entropies and the same selected mass \(z\). For the left auxiliary space indexed by \(E\), set

The right auxiliary space remains indexed by \(\{ 0,\ldots ,d-1\} \). For all sufficiently large \(k\), there is a unit vector \(\varphi _k\) in \((\mathbb C^A\otimes \mathbb C^E)^{\otimes k}\otimes \widehat{\mathcal H}^{\otimes k}\), where \(\widehat{\mathcal H}\) is the Hilbert space of the augmented site family, such that

For every position \(j\) of the list, \((I\otimes C_{j,k})\varphi _k=\varphi _k\). The vector \(\varphi _k\) is fixed by every simultaneous copy permutation of its Bell and augmented site factors, after the canonical regrouping into \(k\) copies of \((\mathbb C^A\otimes \mathbb C^E)\otimes \widehat{\mathcal H}\). Its overlap with the actual projected post vector is

The sequence \((\lambda _k)\) is chosen before \(\mathcal R\). The eventual threshold and the vectors \(\varphi _k\) may depend on \(\mathcal R\).

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 \(k\ge 0\) and any actual central label \(\lambda \) on \(R^k\), let

Regroup the factors into \((AC)^k\otimes (R^k\otimes B^k)\) and put \(Q_k=(|\beta \rangle \langle \beta |)^{\otimes k}\otimes I_{R^kB^k}\). Then

The right label acts on a factor disjoint from the Bell projection.

Suppose in addition that \(\| \Omega \| =1\) and \(H\Omega =E_0\Omega \), where \(E_0\in \mathbb R\). Write \(\psi (b,r)=\chi (r,b)\), so that

There is one sequence \((\lambda _k)\) such that, for every \(k\ge 1\), the actual initial vector in 499 is nonzero, has norm at most one, and satisfies

It has label \(\lambda _k\) on both \(C\) and \(R\), and is fixed by simultaneous copy permutations of the physical and both auxiliary systems. For all sufficiently large \(k\),

The Bell identity 500 applies to this same sequence.

Suppose the selected mass \(z=\sum _{i\in E}p_i\) is positive. Let \(v_{k,\lambda }\) be the original initial vector of (499), expressed in the coordinates \((AC)^k\), \(R^k\), \(B^k\). For every \(k\ge 0\) and every \(\lambda \),

This is the exterior-site form of the Bell identity in [ Ope26b , Section 8 ] , equation (comparator:bell-pin).

Define

Then \(U_k(\beta _d^{\otimes k}\otimes \chi _d^{\otimes k}) =\operatorname {post}_{E,k}\), where the coordinates of the vector on the left are ordered as \((AC)^k\), \(R^k\), \(B^k\).

Let \(P_{E,k}=(|\beta _E\rangle \langle \beta _E|)^{\otimes k}\) and \(P_{d,k}=(|\beta _d\rangle \langle \beta _d|)^{\otimes k}\). For every irreducible label \(\lambda \) of \(\mathfrak S_k\),

Here \(\Pi ^{d,k}_\lambda \) is the central projection for the ordinary copy action on \(\{ 0,\ldots ,d-1\} ^k\), and \(\Pi _{\{ r\} ,\lambda }\) acts on the auxiliary exterior site.

Suppose the actual selected mass \(z=\sum _{i\in E}p_i\) is positive. Let \(P=(|\beta _E\rangle \langle \beta _E|)^{\otimes k}\otimes I\) be the selected Bell projection, let \(Q_\ell =I\otimes \Pi _{R,\ell }\), and let \(\operatorname {post}_{E,k}\) be the actual selected product vector in the same exterior-site coordinates. For every supplied \(\varphi \) satisfying \(P\varphi =\varphi \),

Neither normalization nor copy symmetry of \(\varphi \) is required. The identity holds also for \(k=0\) and retains the supplied label. It is the overlap consequence of [ Ope26b , Section 8, lines 283–290 and 346–354 ] .

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 ] .

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.

Let \(\psi _0\in \mathbb C^A\otimes \mathbb C^B\), where \(A\) and \(B\) are finite sets. Select spectral indices \(E\) of its actual \(A\)-marginal, and suppose their spectral mass \(z_E\) is positive. Let \(J_E:\mathbb C^E\to \mathbb C^A\) be the corresponding eigenvector isometry, and set

Retain all the hypotheses on \(F,k,\mathcal B,R,\lambda ,\psi \) and \(q\) of Theorem 22.33.3. Then there is a unit vector \(\Phi \) in \((\mathbb C^A\otimes \mathbb C^E)^{\otimes k}\) tensored with the exterior copy space, such that

After the canonical identification of copies of a product with the product of copy spaces, \(\Phi \) is invariant under simultaneous copy permutations. The original vector \(\psi _0\) need not be normalized.

Suppose that the regions in \(\mathcal B\) are pairwise comparable by inclusion, and let \(R\subseteq F\) be disjoint from every \(B_j\). For any irreducible label \(\lambda \) of \(\mathfrak S_k\), put \(Q=P_R^\lambda \). Then every vector \(\psi \) satisfies

Under the hypotheses of Theorem 22.33.2, suppose that \(\psi \) is invariant under simultaneous copy permutations and

There is a unit vector \(\varphi \), invariant under simultaneous copy permutations, with \(Q\varphi =\varphi \) and \(C_j\varphi =\varphi \) for every \(j\), and

No normalization of \(\psi \) is required.

Let \(\rho \) be a permutation-invariant density matrix on \(Q^{\otimes k}\), where \(\dim Q=q\). For every \(s\in \mathbb R\) and \(0\le u\le 1/2\),

Let \(\rho \) be a permutation-invariant density matrix on \(Q^{\otimes k}\), where \(\dim Q=q\). Write \(F\) for its Schur-label observable and \(L=-\log \rho \) on the support of \(\rho \), with zero value on the kernel. For a real center \(s\), put

Suppose \(C_0\ge 0\), \(c_0{\gt}0\), and \(\mathcal B\ge 1\), and assume the actual centered surprisal estimate

Then \(c=\min (c_0,1)/2\) is positive. For \(0\le u\le c/\sqrt{\mathcal B}\),

In particular, throughout the symmetric interval \(|t|\le c/\sqrt{\mathcal B}\),

The quadratic coefficient and the interval constant are independent of \(q\). The center is arbitrary; the specialization \(s=kS(Q)\) requires the actual independent-copy surprisal estimate at that center.

Without requiring \(\operatorname {Tr}\rho =1\), for all \(s,u\in \mathbb R\),

If \(\operatorname {Tr}\rho =1\), then for every \(s\in \mathbb R\) and \(u\ge 0\),

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.

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

The cutoff includes labels exactly at the threshold.

Split the \(k\) copies into specified groups of sizes \(k-r\) and \(r\). If their labels \(\alpha ,\beta \) on a fixed subsystem are compatible with the full label \(\lambda \), then \(d_\alpha d_\beta \le d_\lambda \le \binom {k}{r}d_\alpha d_\beta \). The group label projectors commute with the full label projector.

If \(Q,E\) are disjoint and their labels \(\lambda ,\mu \) are compatible with the label \(\nu \) on \(QE\), then \(d_\nu \le d_\lambda d_\mu \) and \(d_\lambda \le d_\nu d_\mu \).

Let \(\rho \) be a density matrix on \((Q\otimes E)^{\otimes k}\) invariant under the copy permutations of \(Q\) and of \(E\), for instance a tensor product of permutation-invariant density matrices, and \(q=d_Qd_E\). The conditional probability of \(\nu \) given \(\lambda ,\mu \) is at most \((k+1)^{q^2}d_\nu /(d_\lambda d_\mu )\), and for \(0\le b\le 1\),

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 \(0\le b\le 1\),

Let \(\rho \) be a positive semidefinite matrix on \(\mathbb C^X\) with \(\operatorname {Tr}\rho =1\), commuting with every permutation operator of the action. For every \(a\in \mathbb R\),

The logarithm is assigned value zero on the kernel of \(\rho \).

The labels occurring in \((\mathbb {C}^q)^{\otimes k}\) are the partitions of \(k\) with at most \(q\) rows. For them, with \(l_i=\lambda _i+q-i\), \(\dim V^{(q)}_\lambda =\prod _{i{\lt}j}(l_i-l_j)/(j-i)\), \(d_\lambda =k!\prod _{i{\lt}j}(l_i-l_j)/\prod _il_i!\), and \(|\log d_\lambda -kH(\lambda /k)|\le (q^2+q+1)(\log (k+q+2)+1)\). The number of labels and the sum of the multiplicities are at most \((k+1)^{q^2}\) (Lemma 22.4.3).

On \((\mathbb {C}^q)^{\otimes k}\), the full label \(\lambda \) and the label \(\nu \) of the first \(k-1\) copies are compatible precisely when \(\lambda \) occurs and \(\nu =\lambda -e_i\) for a removable row \(i\). For every operator \(\rho \) commuting with the copy permutations, \(d_\lambda \operatorname{Tr}(\rho \pi ^\lambda \pi ^\nu )=d_\nu \operatorname{Tr}(\rho \pi ^\lambda )\) on that branch, so in a permutation-invariant state the branch has conditional probability

The normalized star operator \(J=\frac1k\sum _{j{\lt}k}U((j\ k))\) acts on the branch by \((\lambda _i-i)/k\), with rows indexed from one.

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

The label projector \(\pi ^\lambda \) on \((\mathbb {C}^q)^{\otimes k}\) is nonzero exactly when the partition \(\lambda \) has at most \(q\) rows, and then the highest weight of \(V^{(q)}_\lambda \) is \(\lambda \).

Let \(I\) be a finite set of sites with local dimensions \(n_i\ge 0\), let \(k\ge 0\), and let \(t\in \mathbb {R}\). Write \(W_{S,k}(t)\) for the regional replica metric, using the label weights of the full one-copy space, and let \(\Pi \) project onto simultaneous copy symmetry. For arbitrary subsets \(P,Y,F\subseteq I\), set

Here the inverse of a singular matrix is interpreted as zero. For every site \(a\in I\) and irreducible label \(\lambda \) of \(S_k\),

Empty local coordinate sets and zero replicas are included. The regional commutation is a consequence of [ Ope26b , Lemma 6.1(2) ] , applied to the band metric of [ Ope26b , Proposition 7.4 ] .

Let \(a\in F\), let \(S\subseteq F\), and let \(f\) be a real-valued function on the irreducible labels of \(S_k\). For every label \(\lambda \),

Empty local coordinate sets and \(k=0\) are allowed. This is the one-site consequence of  [ Ope26b , Lemma 6.1(2) ] .

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 ] .

For any vector \(\psi \in \mathbb C^\Omega \), let \(\rho _B=\operatorname {Tr}_{B^c}|\psi \rangle \langle \psi |\) be its actual regional marginal. For every \(k\ge 0\) and \(a\in \mathbb R\),

No normalization of \(\psi \) is required. The identity includes empty regions, zero copies, and singular marginals.

For any vector \(\psi \in \mathbb C^\Omega \), let \(\rho _B=\operatorname {Tr}_{B^c}|\psi \rangle \langle \psi |\). For every \(k\ge 0\), define the label observables

Then, for every \(u\in \mathbb R\),

No normalization of \(\psi \) is required. The identity includes zero copies, empty regions, and empty local coordinate sets. It is the regional trace identity used in the sharp upper comparison of [ Ope26b , Section 8 ] .

For every irreducible label \(\lambda \) of \(\mathfrak S_k\) and every \(a\in \mathbb R\),

Let \(\sigma \in \mathfrak S_k\) act on the copies at sites in \(B\), fixing the coordinates at sites in \(B^c\). If \(e_B(x)=(c,r)\), then

For every finite Hermitian matrix \(A\) and every function \(f:\mathbb R\to \mathbb R\),

No positivity or invertibility of \(A\), and no continuity of \(f\) on the whole real line, is required. In particular, for any real \(a\), the strict upper spectral cutoff of \(-\ell (A)\) equals the functional calculus on \(A\) of the function \(u\mapsto \mathbf1_{\{ a{\lt}-\ell (u)\} }\).

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 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

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 \(Z\).

Let \(A,B\subseteq F\) be disjoint. For every irreducible label \(\lambda \) of \(S_k\),

Let \(A,B\subseteq F\) be disjoint. Then

Let the site dimensions be positive, let \(k\ge 0\), and let \(t\ge 0\). Fix admissible transport data: the history and conditional weights are positive, the old triples are partitions, and every move transfers a subset of the middle region to an outer region. For any terminal labels \(j,j'\) and distinct bands \(g,g'\), assume that the raw band metrics \(A_{j,g}\) and \(A_{j',g'}\) commute on the copy-symmetric subspace \(\mathcal S_k\). Put \(\widetilde A_{j,g}=\Pi A_{j,g}+(I-\Pi )\) and \(A_j=\prod _{g=0}^{K-1}\widetilde A_{j,g}\) in increasing band order.

For \(p\in \mathbb {R}\), set \(\bar p=\max (0,\min (1,p))\) and let \(T_{\bar p}\) be the interpolated tree formed from the supplied history and conditional trees. For a vector \(x\), define

where \(v(p)=0\) when \(x=0\). If \(\pi ^\lambda _{\{ a\} ,k}x=x\) for a site \(a\) and a label \(\lambda \), then \(\pi ^\lambda _{\{ a\} ,k}v(p)=v(p)\) for every real \(p\). The label and initial vector are the same throughout. The filtering is that of  [ Ope26b , Proposition 7.4 ] .

Let \(I\) be a finite set of sites with local dimensions \(n_i\ge 0\), and fix \(k,K\ge 0\) and \(t\in \mathbb {R}\). Fix transport data consisting of a finite history tree with labels in a set \(H\), a finite conditional choice tree at each \(h\in H\), old triples in \(K\) bands, and their prescribed moves. The history and choice sets may be arbitrary. For a terminal label \(j=(h,\mathrm{old})\) or \(j=(h,c,\mathrm{new})\), let \(\widetilde A_{j,g}\) be the extended band metric of its prescribed triple and put

with the factors in increasing band order. For every site \(a\in I\) and irreducible label \(\lambda \) of \(S_k\),

Empty local coordinate sets, zero replicas, and the empty band product are included. This is a consequence of the one-site commutation in  [ Ope26b , Lemma 6.1(2) ] for the terminal products of  [ Ope26b , Proposition 7.4 ] .

For every irreducible label \(\lambda \) of the copy-permutation representation,

For every orthogonal projector \(P\) on \((\mathbb C^d)^{\otimes k}\),

For every operator \(P\) on \((\mathbb C^d)^{\otimes k}\), \((I\otimes P)w_{d,k}\ne 0\) if and only if \(P\ne 0\).

Let \(\rho \ge 0\), and let \(A,B\) be commuting Hermitian matrices on a finite-dimensional space. For \(0\le t\le 1\),

Each exponential trace is nonnegative. The matrix \(\rho \) need not have trace one or commute with \(A\) or \(B\); zero weights and a zero-dimensional space are included.

Let \(r\ge 1\) be an integer, let \(A_1,\ldots ,A_r\) be pairwise commuting Hermitian matrices, and let \(\rho \ge 0\). For every real \(a\),

In particular, \(r=5\) gives the fifth-root product in the sharp component estimate. No normalization or commutation condition is imposed on \(\rho \).

Let \(\rho \) be a positive semidefinite complex matrix on a finite coordinate set, and let \(A,B\) be Hermitian matrices on the same set with \(AB=BA\). For every \(a\in \mathbb {R}\),

The weight need not have trace one or commute with \(A\) or \(B\).

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.