The first four statements hold for the identities of the factors of a product of matrix algebras. For the last, let \(e=e_\lambda \), let \(e^\ast \) be its image under the involution, and let \(P\) be the image of \(e\) in the left regular representation, which is faithful and sends \(e^\ast \) to the adjoint \(P^\ast \). Then \(e^\ast \) is central, so \(e^\ast e\) is a central element of the factor \(\lambda \) and equals \(c\, e\) for a scalar \(c\). Hence \(P^\ast P=cP\). If \(c=0\) then \(P=0\), which is impossible. Taking adjoints gives \(P^\ast =uP\) with \(u=c/\bar c\), and idempotence of \(P^\ast \) forces \(u=1\). Thus \(P^\ast =P\), and faithfulness gives \(e^\ast =e\).

Apply the homomorphism to the relations of Lemma 22.1.2. The image of the involution of \(\mathbb {C}[G]\) is the adjoint, since \(U(g)^\ast =U(g^{-1})\), so each projector is self-adjoint. An operator commuting with every \(U(g)\) commutes with the image of every element of \(\mathbb {C}[G]\).

Compare the coefficients of \(h\, a\) and \(a\, h\) at \(hg\). A permutation and its inverse have the same cycle type, hence are conjugate.

For \(Q\subseteq Q'\) we have \(U_{Q'}(\pi )=U_Q(\pi )U_{Q'\setminus Q}(\pi )\), where the second factor commutes with every operator generated by \(Q\) and the first commutes with every central function of \(Q\). Disjoint commutation is immediate. On \(\mathcal S_k\), \(U_Q(\pi )U_{Q^c}(\pi )z=z\), so \(U_Q(\pi )z=U_{Q^c}(\pi ^{-1})z\). Writing a central element as \(\sum _\pi a(\pi )\pi \) with \(a(\pi )=a(\pi ^{-1})\) and reindexing by \(\pi \mapsto \pi ^{-1}\) shows that it acts in the same way through \(Q\) and through \(Q^c\) on \(z\).

For \(Q\subseteq Q'\), factor the restricted action on \(Q'\) into its actions on \(Q\) and \(Q'\setminus Q\). A central function of the first action commutes with both factors. For disjoint regions, every permutation from one restricted action commutes with every permutation from the other, and hence so do their group-algebra images.

Multiplicativity follows from the orthogonality relations. The homomorphism is continuous, so it commutes with the exponential series, which is computed pointwise. The remaining statements are direct.

Let \(E_{ij}\) be the images of the matrix units. For \(w\ne 0\) in the subspace, \(w=\sum _iE_{ii}w\), so some \(E_{ii}w\ne 0\). The vectors \(E_{ji}w\) lie in the subspace and are linearly independent: applying \(E_{ik}\) to a vanishing combination \(\sum _jc_jE_{ji}w\) gives \(c_kE_{ii}w=0\). For the trace, \(\operatorname{tr}E_{ii}=\operatorname{tr}E_{i1}E_{1i}=\operatorname{tr}E_{1i}E_{i1}=\operatorname{tr}E_{11}\), and the trace of an idempotent is its rank.

For each label choose a basis \((y_a)\) of the range of \(E_{11}\) and a nonzero row \(r\) with \(rE_{11}=r\). The operators \(\sum _jE_{j1}\, y_ar^{\mathsf T}E_{1j}\) lie in the commutant, and compressing a vanishing combination by \(E^\mu _{11}\) on both sides shows that they are linearly independent across all labels. A commuting operator has constant entries on orbits of pairs. An orbit of a pair of configurations is determined by the number of copies carrying each pair of basis labels, a function from \(q^2\) pairs to \(\{ 0,\dots ,k\} \).

The spectral projections \(E_u\) of \(\rho \) commute with the label projectors, so the products \(E_u\pi ^\lambda \) form an orthogonal resolution and all operators involved are functions of it. The range of a nonzero \(E_u\pi ^\lambda \) is an invariant subspace of the \(\lambda \)-isotypic component, so its rank \(n_{u,\lambda }\) is at least \(d_\lambda \); since \(\sum _{u,\lambda }u\, n_{u,\lambda }=\operatorname{tr}\rho =1\), this gives \(ud_\lambda \le 1\), so \(\log d_\lambda \le -\log u\) there. The trace equals \(\sum _{u,\lambda }(ud_\lambda )^{1-b}n_{u,\lambda }/d_\lambda \le \sum _\lambda \operatorname{tr}\pi ^\lambda /d_\lambda =\sum _\lambda m_\lambda \), and Lemma 22.4.3 bounds the last sum.

A permutation operator acts on \(E^\lambda _{jo}\) from the left by \(\sum _pc_{pj}E^\lambda _{po}\) and on \(E^\lambda _{oj}\) from the right by \(\sum _qc_{jq}E^\lambda _{oq}\). The lower bound repeats the proof of Lemma 22.4.2 with the products \(A_{ji}B_{ba}\).

Compatibility gives an irreducible block \(B\) of \(\lambda \) whose image \(Y\) under \(\pi ^\alpha \pi ^\beta \) is nonzero. \(Y\) is invariant under both groups and lies in the joint isotypic component, so \(d_\alpha d_\beta \le \dim Y\le \dim B\le d_\lambda \). For the upper bound choose a joint block \(Z\subseteq Y\) of dimension at most \(d_\alpha d_\beta \). The translates of \(Z\) by coset representatives of the Young subgroup span a nonzero invariant subspace of the \(\lambda \)-isotypic component, of dimension at least \(d_\lambda \) and at most \(\binom {k}{r}d_\alpha d_\beta \). Central elements of \(S_k\) commute with its subgroup algebras.

A joint block of \(\lambda ,\mu \) meeting the \(\nu \)-isotypic component is invariant under \(U_{QE}(\pi )=U_Q(\pi )U_E(\pi )\) and has dimension at most \(d_\lambda d_\mu \); its image under \(\pi ^\nu \) has dimension at least \(d_\nu \). For the second inequality, let \(S_k\) act on matrices by \(M\mapsto U_{QE}(\pi )MU_E(\pi )^{-1}\); with \(P=\pi ^\lambda \pi ^\mu \pi ^\nu \) this sends \(P\) to \(PU_Q(\pi )\), so the \(\lambda \)-projection of \(P\) is \(P\). Writing \(P=\sum _{j,b}E^\nu _{jo}NE^\mu _{ob}\) with \(N=E^\nu _{oj}PE^\mu _{bo}\), some span of the matrices \(E^\nu _{j'o}NE^\mu _{ob'}\) is invariant, of dimension at most \(d_\nu d_\mu \), and has nonzero \(\lambda \)-projection, of dimension at least \(d_\lambda \). This replaces the duality argument of the source.

With \(P=\pi ^\lambda \pi ^\mu \), the joint matrix units give \(\operatorname{Tr}(\rho Z)\operatorname{tr}P=\operatorname{Tr}(\rho P)\operatorname{tr}Z\) for \(Z\) in their span, in particular for \(Z=\pi ^\nu P\). Since \(\operatorname{tr}P\ge d_\lambda d_\mu \) when \(P\ne 0\) and \(\operatorname{tr}(\pi ^\nu P)\le \operatorname{tr}\pi ^\nu =d_\nu m_\nu \), the conditional probability is at most \(m_\nu d_\nu /(d_\lambda d_\mu )\). When it is nonzero the labels are compatible, so \(R=d_\lambda d_\mu /d_\nu \ge 1\) and \(R^b\le R\). Summing, the moment is at most \(\sum _\nu m_\nu \le (k+1)^{q^2}\).

The two separate actions change different coordinates of \((x,c)\). Applying both with the same permutation gives the simultaneous action.

Identify a configuration with its two copy strings:

This bijection transports the density and all three copy actions. It therefore transports their central projectors and preserves every trace in 3. The separate invariances are unchanged, and the one-copy dimension is exactly \(q\). The proof of Theorem 22.5.4 uses \(d_\lambda d_\mu /d_\nu \ge 1\) on every nonzero joint block. Consequently, its bound applies for every \(b\le 1\), including negative \(b\).

Set \(r=\Re \operatorname{Tr}\rho \ge 0\). If the basis is empty then \(\rho =0\). Otherwise choose the total normalization

It is positive, has trace one and retains both separate invariances. Positivity implies \(\rho =0\) when \(r=0\), so \(r\eta =\rho \) in both cases. Apply 3 to \(\eta \) and multiply by \(r\). Trace linearity gives 4.

Each logarithmic dimension observable is a real linear combination of orthogonal central projectors. Sums and differences of Hermitian operators remain Hermitian, as do their extensions by an identity.

The two products \(\widehat D_C\widehat D_R\) and \(\widehat D_R\widehat D_C\) both equal \(D_C\otimes D_R\). For commuting operators the exponential of their sum is the product of their exponentials. Exponentiating either tensor-factor extension adjoins the same identity, giving (7).

The partial trace is adjoint to adjoining an identity under the trace pairing. Apply this identity to each local exponential, using the identity for its tensor-factor extension.

The three commuting families of central projections give a joint orthogonal resolution. On the sector with labels \(\lambda ,\mu ,\nu \), the deficit is

Every nonzero joint sector satisfies \(d_\nu \le d_\lambda d_\mu \), so its coefficient is nonnegative. Vanishing sectors contribute zero. Positivity therefore follows by summing the actual Hermitian projections with these coefficients. This also covers a zero-dimensional ambient space.

All eigenvalues of \(B\) are nonnegative, including its zero eigenvalues, so the closed nonnegative spectral projection is the identity. Functional calculus preserves the Hermitian intertwining relation, giving \(J_AT=TJ_B=T\).

The physical copy action is the commuting product of the \(Y\) and \(QV\) copy actions. On its fixed vectors, inversion-invariance of the central coefficients gives \(\Pi _Y^\lambda P=\Pi _{QV}^\lambda P\) for every label. Summing with coefficient \(\log d_\lambda \) gives \(F_YP=F_{QV}P\). The \(Q,V,QV\) central observables commute with \(P\) by nesting, which gives (13). The actual merge-dimension inequality gives \(D_{QV}\ge 0\). Functional calculus of (13) therefore gives (14).

The \(Q\) action is nested in \(QC\), while \(V\) and \(Y\) are disjoint from \(QC\). The analogous statements hold for \(V\) inside \(VR\), with \(Q\) and \(Y\) disjoint. Central-label commutation therefore gives commutation of \(G_{\mathrm{phys}}\) with both merged label observables. The auxiliary \(C,R\) actions are disjoint from all three physical actions. The same nesting and disjointness give commutation with the singleton terms in \(D_C,D_R\). Addition, subtraction and real functional calculus preserve these commutations.

The lower bound is Stirling’s inequality \(n!\ge \sqrt{2\pi n}(n/e)^n\). The upper bound follows because the Stirling sequence \(n!/(\sqrt{2n}(n/e)^n)\) decreases and equals \(e/\sqrt2\) at \(n=1\). For the second statement use \(\log (1+s/a)\le s/a\).

Write \(\log D_\lambda =\log k!+\sum _{i{\lt}j}\log (l_i-l_j)-\sum _i\log l_i!\) and \(kH(\lambda /k)=k\log k-\sum _i\lambda _i\log \lambda _i\). The difference is the sum of the Stirling errors, the Vandermonde term, the shifts \(\sum _i(l_i-\lambda _i)\le q^2\), and \(\sum _i(\lambda _i\log \lambda _i-l_i\log l_i)\); each is bounded by Lemma 22.10.1 using \(1\le l_i-l_j\le l_i\le k+q\).

Changing one shifted part changes the Vandermonde product by the factor \(\prod _{j\ne i}(c-l_j)/(l_i-l_j)\); terms for rows that cannot lose or gain a box vanish because two shifted parts coincide. The recursions reduce to the interpolation identities

for distinct nodes, which follow from Lagrange interpolation: the sum \(\sum _iP(t_i)/\prod _{j\ne i}(t_i-t_j)\) is the coefficient of \(X^{q-1}\) in any polynomial \(P\) of degree less than \(q\); for the first identity apply this to \(X\bigl(\prod _j(X-t_j-1)-\prod _j(X-t_j)\bigr)+q\prod _j(X-t_j)\), whose top coefficients are computed from \(e_1\) and \(e_2\) of the nodes.

The operators \(\Delta (A)\) commute with the copy permutations. Conversely, \(A^{\otimes k}=A^{(1)}\cdots A^{(k)}\) is the top elementary symmetric polynomial of the commuting family \(A^{(j)}\), hence by Newton’s identities a polynomial in the power sums \(\sum _j(A^{(j)})^m=\Delta (A^m)\). Polarization,

follows by expanding the right side and using \(\sum _{S\supseteq T}(-1)^{k-|S|}=[T=\{ 1,\dots ,k\} ]\). Finally an operator commuting with the copy permutations is the average of its conjugates, a combination of symmetrizations of tensor products of matrix units.

A weight vector of lexicographically maximal weight is annihilated by the raising operators. For irreducibility, the span \(L\) of the vectors \(E_{a_1b_1}\cdots E_{a_rb_r}v\) with all \(a_i{\gt}b_i\) is invariant under every \(E_{ab}\), by induction on \(r\) using \([E_{ab},E_{cd}]=\delta _{bc}E_{ad}-\delta _{ad}E_{cb}\); by Theorem 22.11.1 it is invariant under the commutant, hence equal to the subspace. Each nonempty lowering product strictly lowers the weight. Permuting the basis of \(\mathbb {C}^q\) on every copy commutes with the copy permutations and permutes the weights; if \(\mu _a{\lt}\mu _b\) for some \(a{\lt}b\), the transposition of \(a\) and \(b\) would produce a weight above \(\mu \).

The centre of \(\mathbb {C}[G]\) is the algebra of tuples of scalars of the Artin–Wedderburn decomposition, and also the space of class functions. Two permutations are conjugate exactly when they have the same cycle type, and every partition is a cycle type. A partition \(p\) gives the decreasing vector \(i\mapsto \# \{ \text{parts}{\gt}i\} \) of its conjugate, and a decreasing vector gives the partition formed by its nonzero entries; both maps are injective, since a decreasing vector takes values at least \(v\) on an initial segment.

Every term has weight \(\mu \), and the coefficient of \(e_x\) is one because a column-preserving permutation fixing \(x\) also preserves rows. For \(a{\lt}b\) the operator \(E_{ab}\) commutes with the antisymmetrization and changes the value of a copy in row \(b\) to \(a\); since \(\mu _a\ge \mu _b\) the same column has a copy in row \(a\), and the transposition of these two copies fixes the new configuration and reverses the sign, so the antisymmetrization vanishes.

The orbit of a configuration with distinct entries carries the regular representation, so every label occurs. A highest-weight vector of weight \(\mu \) with a nonzero component of label \(\lambda \) yields, through a matrix unit, a highest-weight vector of weight \(\mu \) in \(V_\lambda \), so \(\mu \) is the shape of \(\lambda \). Applied to Lemma 22.12.2, every partition is a shape. Lemma 22.12.1 gives equal cardinalities.

For \(q\le q'\) the inclusion \(\mathbb {C}^q\subseteq \mathbb {C}^{q'}\) induces an embedding of tensor powers commuting with the copy permutations and mapping highest-weight vectors of weight \(\mu \) to highest-weight vectors of weight \(\mu \) padded with zeros. Compare with \(q=k\), using Lemma 22.12.2 for the converse.

The separate label resolutions commute. Centrality of their projections also gives commutation with the combined action and its label projections. Hence the actual products \(P_Q^\lambda P_E^\mu P_{QE}^\nu \) form a joint orthogonal resolution. Its three marginals give the three logarithmic dimension observables. Thus \(D\) has value \(\log d_\lambda +\log d_\mu -\log d_\nu \) on each joint summand. Compute the exponential on each summand. Positivity of all irreducible dimensions identifies the coefficient with the stated ratio power.

Insert the preceding expansion and distribute multiplication and trace over the finite sums. The coefficients are real, so they factor out of the real trace pairing.

Set \(U=e^{aA}\) and \(V=e^{aB}\). These matrices are Hermitian and commute, so positivity of the weighted trace gives \(0\le \operatorname {Re}\operatorname{tr}(\rho (U-V)^2)\). Expanding the square yields \(2\operatorname {Re}\operatorname{tr}(\rho UV) \le \operatorname {Re}\operatorname{tr}(\rho U^2)+\operatorname {Re}\operatorname{tr}(\rho V^2)\). The identities \(UV=e^{a(A+B)}\), \(U^2=e^{2aA}\) and \(V^2=e^{2aB}\) then give (21).

The joint density identity gives \(\rho =\rho _Q^{\otimes m}\otimes \rho _{C_G}(w)\), where \(\rho _Q=\operatorname {Tr}_T|\Omega \rangle \langle \Omega |\). The tensor power is invariant under the \(Q\) permutations, and the actual auxiliary marginal is invariant under the \(C\) permutations. The latter invariance follows from the assumed simultaneous symmetry of \(u\), excitation projection, and partial trace. Thus the positive matrix \(\rho \) is invariant under both separate actions. The exponential trace expansion and the homogeneous paired merge-moment bound imply \(\operatorname {Re}\operatorname {Tr}(\rho e^{bD_C}) \leq (m+1)^{(|Q|\, |C|)^2}\operatorname {Tr}\rho \). Partial traces and coordinate bijections preserve the trace. Since \(\operatorname {Tr}\rho _Q=\| \Omega \| ^2=1\), the trace of the product density is \(\operatorname {Tr}\rho =\operatorname {Tr}|w\rangle \langle w|=\| w\| ^2\). This proves (22). No division by the component mass is needed.

The literal partial trace gives \(\rho \geq 0\), while \(D_C,D_R\) are commuting Hermitian matrices acting on separate tensor factors. The arithmetic-mean trace estimate bounds the left side by

Since \(2a\leq 1\), Theorem 13.71.1 bounds the two terms by \(p_{QC}\| w\| ^2\) and \(p_{VR}\| w\| ^2\), respectively. Their arithmetic mean gives (23).

Tracing the complementary physical coordinates gives exactly \(\rho _Q^{\otimes m}\otimes \rho _K(w)\), where \(\rho _K(w)=\operatorname {Tr}_{(Q\times T)^k}|w\rangle \langle w|\) is the literal exterior marginal of the same component. The exponential of the retained-factor action is the exponential on \(Q^m\) tensored with the identities on every other factor. The partial-trace pairing and the trace of a tensor product thus factor the left side into the iid regional moment and \(\operatorname {Tr}\rho _K(w)=\| w\| ^2\). This proves the identity without division by the component mass.

Sum the unused resolution to the identity. These identities require no positivity or Hermiticity of the resolutions.

The joint spectral projections \(P_iQ_j\) are Hermitian orthogonal projections, and their trace weights \(\mu _{ij}=\operatorname {ReTr}(\rho P_iQ_j)\) are nonnegative. The exponential traces are the corresponding weighted scalar sums. For nonnegative weights \(\mu _i\) and \(s_j\) with \(\sum _j s_j=1\), scalar weighted Hölder gives

Here is a direct proof of (28). If all \(\mu _i\) vanish, the left side is zero. Otherwise put \(M_j=\sum _i\mu _i e^{x_{ji}}{\gt}0\). Convexity of the exponential, applied to \(x_{ji}-\log M_j\), gives

Multiply by \(\mu _i\) and sum in \(i\); each normalized inner sum is one. This proves (28), and the two joint spectral coordinates give (27). A single spectral resolution proves nonnegativity of the exponential trace.

Induct on \(r\). The case \(r=1\) is equality. For \(r\ge 2\), set \(B=\sum _{j=2}^r A_j\) and apply (27) to \(raA_1\) and \(raB/(r-1)\) with weight \(t=1/r\). Apply the induction hypothesis to \(B\) with exponent \(ra/(r-1)\). Raising that bound to \(1-1/r\) gives exponent \((1/(r-1))(1-1/r)=1/r\) on every remaining factor. All bases are nonnegative, so this argument also includes zero traces.

Each \(F_A\) is a real linear combination of orthogonal central projections and is therefore Hermitian. The seven subsystems \(Q,Y,V,C,R,QC,VR\) are pairwise nested or disjoint, so their central observables commute. Addition, subtraction and negation give pairwise commutation of \(D_C,D_R,-F_Q,-F_V,F_Y\). Apply equal-weight trace Hölder to these five operators. Their sum is \(D_C+D_R-G\), giving (31).

Use the joint spectral resolution of \(\rho \) and the central projections. On a nonzero joint eigenspace with positive eigenvalue \(r\), invariance and normalization give \(r d_\lambda \le 1\), so \(\log d_\lambda \le -\log r\). The desired comparison follows by monotonicity of the scalar exponential. The terms with \(r=0\) have zero trace weight. Thus only the supported comparison is used; an inequality on the entire ambient space is unnecessary.

The Hermitian operators \(-2u(L-sI)\) and \(2u(L-F)\) commute, since both are functions of the joint spectral resolution. Their average is \(-u(F-sI)\). Apply the weighted trace inequality with weights \(1/2,1/2\). The individual exponential traces are nonnegative, so the product of their square roots is the square root of their product.

Apply the preceding comparisons and the Schur remainder estimate \(\operatorname {ReTr}(\rho e^{b(L-F)})\le (k+1)^{q^2}\) with \(b=2u\le 1\). The other factor is nonnegative, so multiplication and the square root preserve the bound.

Every Hermitian exponential is positive definite. Its trace pairing with \(\rho \ge 0\), \(\operatorname {Tr}\rho =1\), is therefore strictly positive, so logarithmic monotonicity applies to the preceding moment comparisons. Since \(\sqrt{\mathcal B}\ge 1\), the chosen range gives both \(2u\le 1\) and \(2u\le c_0/\sqrt{\mathcal B}\). The positive comparison immediately gives the first bound. For the negative comparison,

Finally, \(C_0k\mathcal B t^2\) and \(\log (k+1)\) are nonnegative, so the larger bound also covers the positive sign. All zero-copy and singular-state cases remain included.

The central idempotents of \(H\) sum to one, and the block of the central idempotent of \(\nu \) has trace \(d_\nu c(\lambda ,\nu )\). The Artin–Wedderburn isomorphism preserves dimension. Finally \(\rho (a)\pi ^\lambda =\sum _{ij}W_\lambda (a)_{ij}E^\lambda _{ij}\) and \(\operatorname{Tr}(ME^\lambda _{ij})=\delta _{ij}\operatorname{Tr}(ME^\lambda _{00})\).

Work in \((\mathbb {C}^k)^{\otimes k}=(\mathbb {C}^k)^{\otimes (k-1)}\otimes \mathbb {C}^k\). A highest-weight vector of weight \(\mu \) fixed by a minimal idempotent of \(\nu \) in the first \(k-1\) copies has components along the last copy in \(V_\nu \), and its last nonzero component, at row \(i\), is a highest-weight vector of weight \(\mu -e_i\); hence \(\mu =\nu +e_i\), and the component at row \(i\) determines the vector, so these vectors span at most a line. The images of the matrix units of \(\lambda \) applied to a highest-weight vector of \(V_\lambda \) embed the range of the block of the minimal idempotent into this space.

By induction on \(k\). From Lemma 22.21.2, \(d_\lambda =\sum _\nu c(\lambda ,\nu )d_\nu \le \sum _iD_{\lambda -e_i}=D_\lambda \), and \(\sum _\lambda D_\lambda d_\lambda =\sum _\nu d_\nu \sum _\lambda c(\lambda ,\nu )D_\lambda \le \sum _\nu d_\nu \sum _iD_{\nu +e_i}=k\sum _\nu d_\nu ^2=k!=\sum _\lambda d_\lambda ^2\), using Lemma 22.10.3. Hence \(d_\lambda =D_\lambda \), and a missing removable row would make the first inequality strict.

A matrix unit summed over its index pairs acts on two distinct copies as their transposition and on one copy as \(q\) times the identity. On a highest-weight vector of weight \(\lambda \) in \(V_\lambda \) the diagonal terms give \(\lambda _a^2\), and for \(a{\lt}b\) the commutator \([E_{ab},E_{ba}]=E_{aa}-E_{bb}\) gives \(\lambda _a-\lambda _b\), while the terms with \(a{\gt}b\) vanish.

Compatibility and the probability follow from Lemma 22.21.1 and the branching rule of Theorem 22.21.3; the quotient is the recursion of Lemma 22.10.3, whose factors with \(j{\lt}i\) are at most two and with \(j{\gt}i\) at most one. Since \(2kJ=T_k-T_{k-1}\), the star operator acts on the branch by \((\kappa _\lambda -\kappa _\nu )/(2k)\), which Lemma 22.22.1 evaluates.

The first formula is Lemma 22.21.1 with \(M=1\). In the regular representation the image of a minimal idempotent is a left ideal of \(\mathbb {C}[G]\), which the Artin–Wedderburn isomorphism identifies with one row of the \(\lambda \) factor. The trace of the regular representation is \(|G|\) times the coefficient of the identity; apply it to \(e_\lambda g^{-1}\) and compare coefficients.

Every permutation of \(k+1\) copies is uniquely \(\sigma \) or \((j\ k{+}1)\sigma \) with \(\sigma \) permuting the first \(k\) copies; these fix \(qf(\sigma )\) and \(f(\sigma )\) configurations respectively, the last copy being free or determined by copy \(j\).

The labels and \(d_\lambda \) come from Theorems 22.12.4 and 22.21.3, and the asymptotics from Lemma 22.10.2. By Lemma 22.22.3 the fixed-point sum acts on \(\lambda \) by \(s_\lambda =k!m_\lambda /d_\lambda \). On a branch \(\nu =\lambda -e_i\), Lemma 22.22.4 and the star-operator eigenvalue of Theorem 22.22.2 give \(s_\lambda =(q+\lambda _i-i)s_\nu =l_is_\nu \). Combined with the quotient \(d_\nu /d_\lambda \) of Theorem 22.22.2 this is the recursion \(W_\lambda \prod _{j\ne i}(l_i-1-l_j)=W_\nu \prod _{j\ne i}(l_i-l_j)\) of the Weyl formula, and induction on \(k\) gives \(m_\lambda =W_\lambda \).

Expand the determinant over permutations \(\sigma \) and reindex \(e_r\) by \(\sigma \). At \(y=(1,z,\dots ,z^{q-1})\) the alternant is the Vandermonde determinant in the \(z^{\alpha _j}\), equal to \((z-1)^{\binom q2}\) times \(\prod _{i{\lt}j}(G_{\alpha _j}-G_{\alpha _i})\) with \(G_n=1+z+\cdots +z^{n-1}\). Cancel the power of \(z-1\) in the polynomial ring and set \(z=1\), where \(G_n=n\) and \(e_r=\binom qr\).

The diagonal entry of \(A_r\operatorname {diag}(y)^{\otimes r}\) at a configuration \(z\) is \(\frac1{r!}\prod _jy_{z_j}\) times the signed count of permutations fixing \(z\), which is one when the entries of \(z\) are distinct and zero otherwise, since a transposition of two equal entries reverses the sign. Each set of \(r\) distinct entries arises from \(r!\) configurations, which cancels the factor \(\frac1{r!}\).

A highest-weight vector \(w\) of weight \(\lambda \) fixed by \(e^\mu _{00}\otimes A_r\) has components \(w_z\in V_\mu \), indexed by the configurations \(z\) of the last \(r\) copies, which are antisymmetric in \(z\) and vanish unless \(z\) has distinct entries. Among the nonzero components choose one maximizing \(\sum _jz_j\). A raising operator \(E_{ab}\), \(a{\lt}b\), maps it to minus a sum of components whose configurations have a larger sum, so it is a highest-weight vector of \(V_\mu \) of weight \(\lambda -\mathrm{wt}(z)\); hence \(\lambda =\mu +1_S\) with \(S\) the set of entries of \(z\). Configurations with the same entries differ by a permutation, so the component at one such configuration determines \(w\), and it lies on the line of highest-weight vectors of \(V_\mu \). As in Lemma 22.21.2, the matrix units of \(\lambda \) applied to a highest-weight vector of \(V_\lambda \) embed the range of the block into this space.

By Lemma 22.21.1, \(\operatorname{Tr}\bigl(\rho (a)\operatorname {diag}(y)^{\otimes k}\bigr) =\sum _\lambda \operatorname{tr}W_\lambda (a)\chi _\lambda (y)\) for every \(a\in \mathbb {C}[S_k]\), and for \(a=e^\mu _{00}\otimes A_r\) the left side is \(\chi _\mu (y)e_r(y)\) by Lemma 22.23.2. Labels with more than \(q\) rows contribute nothing, since their characters vanish. For the other labels, Lemma 22.23.3 bounds the coefficients by one on these partitions and by zero elsewhere. At \(y=1\) the two sides are \(\binom qrm_\mu \) and \(\sum _\lambda c(\lambda )m_\lambda \), while the Weyl dimension formula and Lemma 22.23.1 give \(\sum _\lambda m_\lambda =\binom qrm_\mu \) over the same partitions; the summands with \(\mu +1_S\) not a partition vanish because two shifted parts coincide. Since \(m_\lambda {\gt}0\), every coefficient is one.

By induction on \(k\), and for fixed \(k\) downward on the number \(\ell \) of nonempty rows. For \(\ell =0\) both sides are \(a_\delta \). Otherwise let \(\mu \) be \(\lambda \) with its first column removed, a partition of \(k-\ell \). Multiply the rule of Theorem 22.23.4 for \(r=\ell \) by \(a_\delta \). By induction on \(k\) the left side is \(a_{\mu +\delta }e_\ell =\sum _{|S|=\ell }a_{\mu +\delta +1_S}\), in which the terms with \(\mu +1_S\) not a partition vanish. On the right, every partition \(\mu +1_S\) other than \(\lambda \) has a nonempty row below the first \(\ell \), hence more than \(\ell \) nonempty rows, and the inner induction applies to it. The remaining terms are \(a_\delta \chi _\lambda \) and \(a_{\lambda +\delta }\).

If \(\tau =0\), then \(r=0\) and both claims are equalities. Otherwise \(r\le k\) and each nonnegative term in the binomial expansion of \((\tau +(1-\tau ))^k=1\) is at most \(1\). Hence

The last inequality follows from \(\tau \le 1/2\) and \(r\le \tau k\). Exponentiating proves the second claim.

There are at most \(k+1\) admissible cardinalities, each contributing at most \(e^{kh(\tau )}\) by Theorem 22.24.1. Every admissible subset belongs to the family of subsets of cardinality \(r\) for one admissible \(r\), and that family has \(\binom {k}{r}\) members.

The entry at \((x,y)\) is \(d^{-k/2}P_{y,x}\). Thus \(\| (I\otimes P)w_{d,k}\| ^2=d^{-k}\operatorname {tr}(P^*P) =d^{-k}\operatorname {tr}P\).

Since \(d{\gt}0\), the coefficient \(d^{-k/2}\) is nonzero. The coordinate identity \(((I\otimes P)w_{d,k})(x,y)=d^{-k/2}P_{y,x}\) proves both directions.

Every permutation in \(S_k\) is conjugate to its inverse, so a central element of \(\mathbb C[S_k]\) has inversion-invariant coefficients. Transposing the permutation representation sends each permutation to its inverse; hence \((\pi ^\lambda )^T=\pi ^\lambda \). The two coordinate expressions are therefore \(d^{-k/2}\pi ^\lambda _{x,y}=d^{-k/2}\pi ^\lambda _{y,x}\).

The subgroup permutes precisely its specified copies and fixes every other coordinate. Extend the identity from permutations to their group algebra, and then to the central label observables.

Sum the coordinate identities with their specified coefficients. Exponentiation commutes with a change of basis and with adjoining an identity factor.

Disjointness gives \(U_{A\cup B}(\sigma )=U_A(\sigma )U_B(\sigma )\). If \(v\) belongs to the range of \(\Pi _{A\cup B,k}\), then \(U_B(\sigma )v=U_A(\sigma ^{-1})v\). The coefficients of \(e_\lambda \) are invariant under inversion in \(S_k\). Summing the last identity with these coefficients gives \(\pi ^\lambda _{B,k}v=\pi ^\lambda _{A,k}v\). Apply this equality to every column of \(\Pi _{A\cup B,k}\).

Multiply (46) by \(\log d_\lambda \) and sum over the irreducible labels.

For each \(\sigma \in S_k\), nesting gives \(U_B(\sigma )=U_A(\sigma )U_{B\setminus A}(\sigma )\). The image of the central idempotent \(e_\lambda \) commutes with \(U_A(\sigma )\), while disjointness of \(A\) and \(B\setminus A\) makes it commute with \(U_{B\setminus A}(\sigma )\). Thus \(\pi ^\lambda _{A,k}\) commutes with every \(U_B(\sigma )\). Sum first over \(\sigma \) to obtain commutation with \(\Pi _{B,k}\), and then over \(\lambda \) with coefficients \(f(\lambda )\).

If \(a\in S\), then \(\{ a\} \subseteq S\); otherwise \(\{ a\} \) and \(S\) are disjoint. In either case, the one-site label projector commutes with each \(\pi ^\mu _{S,k}\). Multiply by \(f(\mu )\) and sum over \(\mu \).

The label projector of \(A\) commutes with each copy permutation of \(B\). Averaging these permutations gives the asserted commutation with \(\Pi _{B,k}\).

Put \(X=\pi ^\lambda _{\{ a\} ,k}\). Each band factor satisfies \(X\widetilde A_{j,g}=\widetilde A_{j,g}X\). Repeated multiplication gives \(X\prod _g\widetilde A_{j,g}=(\prod _g\widetilde A_{j,g})X\) in the same order. For an old terminal label this is the old metric; for a new terminal label it is the new metric.

The terminal inputs are positive definite. Put \(X=\pi ^\lambda _{\{ a\} ,k}\). Terminal commutation gives \(XA_j=A_jX\), and commutation passes through each weighted geometric mean of the same tree. Hence \(XM(p)=M(p)X\). Functional calculus gives

Multiplication by the scalar normalization preserves this equality, including when \(x=0\).

The partition gives \(Y^c=P\cup F\). Complementary logarithmic labels on their symmetric union satisfy \(F_{Y,k}\Pi _k=F_{P\cup F,k}\Pi _k\). Define \(D_k=F_{P,k}+F_{F,k}-F_{P\cup F,k}\). The actions on \(P\) and \(F\) commute, and their pointwise product is the action on \(P\cup F\), so the merge-deficit inequality gives \(D_k\geq 0\). Nested label observables commute with \(\Pi _k\). Consequently,

The label observables are self-adjoint. Apply the nonnegative spectral intertwining theorem to (52) to obtain (51). The closed interval includes the zero eigenspace.

For fixed \(c{\gt}0\) and \(d\ge 0\), the quantity \(x^d e^{-cx}\) tends to zero as \(x\to \infty \). Consequently

It is therefore eventually at most one. Division by the positive quantity \(4(k+1)^d\) gives the conclusion.

If \(I\) is empty, the left side is zero. Otherwise write \(m=|I|{\gt}0\). For each \(i\), exponential decay gives \(m e^{-c_i k}\le 1/(4(k+1)^d)\) eventually. Finiteness gives a common threshold after which all these inequalities hold. Sum them and cancel the positive factor \(m\).

Intersect the finitely many eventual tail bounds and apply Theorem 22.30.1. Thus, with \(t_k=1/(4(k+1)^d)\), the total loss is eventually at most \(t_k\), while \(q_k\ge 2t_k\). The retained-mass inequality gives

The quantity \(q_k\) is the same original mass; no new selection is involved.

Every function on the finite spectrum of \(A\) is continuous in its relative topology, and the same holds on its image under \(-\ell \). The composition rule for spectral functional calculus therefore applies. At a zero eigenvalue, both sides have the value \(f(0)\).

Define \(X(i)=-\log p_i\) when \(p_i{\gt}0\), and \(X(i)=0\) otherwise. Since \(p_i\le 1\), the variable \(X\) lies in \([0,M]\), where \(M=1+\sum _i|X(i)|{\gt}0\). Its expectation under the law \((p_i)_i\) is \(H\). For independent samples \(X_0,\ldots ,X_{k-1}\), Hoeffding’s inequality gives

On every tuple of positive probability, the sum of the surprisals equals the surprisal of the product weight. The strict event in the statement is contained in the displayed event. Thus \(c=2/M^2\) is admissible, including when the probability law is concentrated at a single point.

The eigenvalues of \(\rho \) form a finite probability distribution whose entropy is \(S\). The product spectral distribution identifies the trace with the preceding finite-product tail. Zero eigenvalues have zero trace weight, so the value of the logarithm on the kernel causes no additional term.

Nestedness makes the \(C_j\) commute, and disjointness makes each \(C_j\) commute with \(Q\). For an orthogonal projection \(P\), \(\| Pv\| ^2+\| (I-P)v\| ^2=\| v\| ^2\). Apply this identity successively. At each step the removed component is a commuting orthogonal projection of \((I-C_j)\psi \), so its norm is at most \(\| (I-C_j)\psi \| \). Summing the losses gives the inequality.

Set \(v=CQ\psi \). Theorem 22.33.2 gives \(v\ne 0\), so take \(\varphi =v/\| v\| \). The commuting projection identities imply the range conditions. Each subsystem central projection commutes with simultaneous copy permutations, which preserves the symmetry. Since \(C\) is an orthogonal projection, \(\langle v,Q\psi \rangle =\langle v,v\rangle \); normalization gives the overlap.

Take \(\Phi =\beta _E^{\otimes k}\otimes \varphi \), where \(\varphi \) is given by Theorem 22.33.3. Positive spectral mass implies \(E\ne \varnothing \), hence \(\| \beta _E\| =1\). Tensor products give the range conditions, the symmetry, and the exact overlap identity.

A physical site in \(\widehat B\) belongs to \(B\). Conversely, disjointness places every site of \(B\) in \(X^c\), so that it occurs in \(\widehat B\).

Intersecting with \(X^c\) preserves inclusion, and the lift contains only physical sites. Apply preservation of inclusion to each comparison in the original list.

Substitute \(\operatorname {phys}_X(\widehat B)=B\) in the definition.

Every physical site lies outside \(\{ r\} \), while the auxiliary site lies inside it. Substitution in the definition of the subsystem action gives the asserted action formula.

In each copy, the inverse coordinate identification has auxiliary value \(a_j\) and physical-complement configuration \(b_j\). The factors on the two sides are therefore equal term by term.

The permutation changes only the order of the factors in the finite product.

For each fixed list position, Theorem 22.40.3 bounds its loss by \(e^{-c_jk}\) for some \(c_j{\gt}0\) and every \(k\ge 1\). Theorem 22.30.2 gives the two bounds on \(q_k-\ell _k\), and its lower bound is strictly positive. The lifted regions remain nested and avoid the auxiliary site by Theorem 22.34.3. Thus the finite projection union bound gives \(q_k-\ell _k\le r_k\), proving the remaining inequalities.

Theorem 22.37.1 makes \(q_k-\ell _k\) eventually positive and gives the bounds on \(r_k\). The vector \(\Psi _k\) is symmetric by Theorem 22.36.4. Apply Theorem 22.33.4 to this actual exterior vector, with the original selected Bell vector and the same label \(\lambda _k\). Its exact overlap identity gives \(r_k\) from (77).

The auxiliary copy-action identity supplies the product equivariance. Apply Theorem 22.42.1 to the central idempotent of the given irreducible label.

The product-coordinate identity gives \(T_k\Psi _k=w_k\). Combine this with Theorem 22.38.1 and use preservation of norm by \(T_k\). Substitution of the actual compressed spectral vector gives the last assertion without making another selection of a vector or a label.

Embedding the selected coordinate space by \(J_E\) takes \(\chi \) to \(\psi \). Since \(J_E\) is an isometry, this embedding preserves the entire marginal on \(X^c\). Trace that marginal further over \((X\cup B)^c\). In coordinates, this amounts to replacing the sum over the selected index \(r\) by the sum over the original \(X\) coordinates and then interchanging the two finite sums. The resulting matrix is precisely the regional marginal of the same selected vector \(\psi \).

The coordinate identification first separates the auxiliary site and then identifies its ordered indices with \(E\). It is an isometry from the selected coordinate space to the site description. The selected compressed vector has squared norm \(z^{-1}z=1\).

The actual compressed regional marginal, after the canonical coordinate identification, equals the marginal of the actual selected physical vector. Entropy is unchanged by this identification. The selected-state entropy inequality bounds the latter entropy by \(S(\rho _B^\Omega )/z\). The mass \(z\) is the same original spectral-selection mass throughout.

The actual regional marginal of \(\chi _{\mathrm{site}}\) is positive and has trace one by Theorem 22.40.1. Its entropy is bounded by Theorem 22.40.2. Apply Theorem 22.43.1 with \(s=S(\rho _B^\Omega )/z\), then use Theorem 22.45.4 to identify the trace tail with the squared cutoff failure norm of \(\chi _{\mathrm{site}}^{\otimes k}\).

At the auxiliary site, both sides have value \(a\). At a physical site in \(B\), both have the value prescribed by \(b\); at every other site of \(X^c\), both have the value prescribed by \(d\).

The split identity identifies the discarded coordinates with the selected index and the physical complement of \(X\cup B\). Reindex the finite partial-trace sum by \(e_C\), and apply preservation of the regional marginal under the actual spectral compression.

For a single group element, the permutation-matrix entry at \((x,y)\) is one precisely when \(\alpha (g)y=x\). Equivariance expresses this as equality of the first coordinates and the corresponding permutation condition on the second coordinates. This gives the identity for group elements. Extend it by linearity over the finite support of \(h\).

The actual tensor power \(\rho _k=\rho ^{\otimes k}\) is positive, has trace one, and commutes with all copy permutations. The supported Schur comparison therefore bounds its removed label mass at threshold \(a\) by

where the logarithm is zero on the kernel. The functional-calculus composition identity gives precisely this cutoff expression. Since \(k(S(\rho )+1)\le k(s+1)\), the strict upper tail at the latter threshold is contained in that at the former. The eigenvalue weights are nonnegative, so this inclusion also orders their trace masses. Apply the exponential surprisal bound at \(k(S(\rho )+1)\).

Compute the functional calculus in the orthogonal label decomposition. The coefficient on the \(\lambda \) subspace is one precisely when \(\log d_\lambda \le a\).

Use the joint orthogonal decomposition for \(\rho \) and the label projectors. On a nonzero joint subspace with eigenvalue \(u{\gt}0\) and label \(\lambda \), the trace-one condition gives \(u d_\lambda \le 1\). Consequently \(\log d_\lambda \le -\log u\), so the event \(\log d_\lambda {\gt}a\) is contained in \(-\log u{\gt}a\). Multiply this scalar comparison by \(u\) and sum the traces of the joint projections. Subspaces with \(u=0\) contribute zero to both sides.

Restriction to \(B^c\) sees only fixed coordinates. Restriction to \(B\) sees exactly the copy permutation, with the inverse permutation in the coordinate formula for the left action.

Apply the group-algebra product-coordinate identity to the actual subsystem action and the central idempotent of \(\lambda \). Theorem 22.45.2 supplies its equivariance hypothesis. Summing the resulting projection identities with \(\log \dim \lambda \le a\) gives the cutoff identity.

In the split coordinates, the cutoff complement is the identity on the complementary copies tensored with the regional cutoff complement. Both are orthogonal projections. Hence its squared norm on the vector is its expectation in the corresponding pure-state matrix. Tracing the complementary copies gives \(\rho _B^{\otimes k}\), which proves the identity.

Expand the exponential in the orthogonal label projections and apply Theorem 22.45.3. This gives

Put \(\xi =T_B\psi ^{\otimes k}\). By Theorem 13.38.1, tracing the complementary copies of \(|\xi \rangle \langle \xi |\) gives \(\rho _B^{\otimes k}\). The defining trace identity for the partial trace therefore gives the claimed equality. The coordinate identification preserves the inner product.

Theorem 22.45.5 expresses the left side as the trace against the tensor power of the actual compressed regional marginal \(\widehat\sigma _Q\). Let \(T_k\) be the unitary induced, in each copy, by the regional coordinate bijection. Theorem 22.41.3 and the covariance of the copy action give, with \(\widehat F^{\mathrm{reg}}_{Q,k}\) the label entropy on the compressed regional copy space,

For the second identity, the same central group-algebra element defines both label entropies. Conjugation therefore also identifies their exponentials. Multiplicativity of conjugation and invariance of the trace prove the assertion.

Let \(\sigma _Q\) be the marginal of the same selected physical vector. Since \(Q\) is disjoint from the region of spectral selection, the original marginal is the sum of \(z\sigma _Q\) and a positive semidefinite remainder. Hence \(0\le \sigma _Q\le z^{-1}\rho _Q\). Theorem 1.7.2 gives

The label entropy is Hermitian, so its exponential \(e^{uF^{\mathrm{reg}}_{Q,k}}\) is positive for every real \(u\). Taking its trace pairing with this operator inequality preserves the order. Theorem 22.45.6 identifies the resulting left side with the actual compressed expectation. At \(k=0\), both tensor powers are the scalar identity, so the same argument applies.

The range \(W\) of \(\pi ^\lambda \) is a nonzero invariant subspace on which \(\pi ^\lambda \) acts as the identity. Hence \(\dim \lambda \leq \dim W\leq \dim \mathbb {C}^Y=|Y|\).

In regional and complementary coordinates the projector is \(I\otimes \pi ^\lambda \) on the \(k\) copies of \(\mathcal H_B\). A nonzero subsystem projector therefore gives a nonzero regional projector. Apply the dimension estimate to its coordinate space.

Resolve the identity into labels on the difference region. On a nonzero joint block, the compatible-label dimension inequalities and the difference-region dimension estimate give

Thus every nonzero joint block retained by \(P_B(a)\) is retained by \(P_Q(t)\). Summing the joint blocks proves the identity.

The hypotheses give \(k(s/z+1)+k\log \dim \mathcal H_{B\triangle Q} \leq k(B_{\rm sh}/z+B_{\rm exc})\). Tensor the absorption identity with \(I_{\mathcal D}\) and apply it to \(\varphi \).

The auxiliary site belongs to neither side. At each exterior physical site, both memberships mean membership in exactly one of \(B,Q\).

The set \(B\triangle Q\) is disjoint from \(X\). Its exterior configurations are in bijection with its original physical configurations, and the auxiliary coordinate does not occur.

Use the supplied range equation for \(B\) and substitute \(a_k(\widehat B)=k(S(\rho _B^\Omega )/z+1)\). The exterior lifts remain comparable, and their difference has exactly the original physical dimension. Apply the spectator-factor cutoff transfer with \(\mathcal D=\mathcal L_k\) to the same \(\varphi \).

Positive selected mass normalizes the actual compressed vector. Regroup its factors so that the physical complement comes first, and apply the pure-tensor-power label-sequence theorem to \(\psi \). The auxiliary projection-mass identity preserves the selected label and its squared mass when returning to the site description. The auxiliary dimension is \(d\), giving the displayed polynomial. The same selection supplies the logarithmic-dimension asymptotic for the actual density \(\rho _R\).

Apply the preceding theorem to the actual compressed selected state, obtaining its one label sequence, eventual mass bound and entropy asymptotic. Keep that sequence in the common-cutoff Bell-vector theorem for the prescribed region list. The exponentially small regional losses leave at least half of its auxiliary mass, hence the displayed quarter-polynomial bound. Normalizing the common projection and adjoining the actual selected Bell power gives the four range and invariance properties. Its squared overlap with the unnormalized projected \(\mathrm{post}_k\) is exactly the retained mass \(r_k\).

The physical coordinate identification preserves the norm of \(\Omega \). Apply Theorem 13.80.2 to \(\Omega _X\), its selected spectral indices, and its original eigenvector equation. Retain the normalized selected vector, diagonal auxiliary marginal, and the single sequence supplied there. This gives every positive-copy initial-vector conclusion, the eventual projected-mass bound, and the entropy asymptotic before any regional list is chosen.

Theorem 22.38.2 identifies the selected mass with \(q_k\) for each copy number and the same label. Thus \(q_k\ge 1/(2(k+1)^{d^2})\) eventually. For an arbitrary fixed nested list \(\mathcal R\) disjoint from \(X\), apply Theorem 22.37.2 with this supplied sequence. It yields the unit vector, simultaneous-copy invariance, all Bell and cutoff projection equations, the exact overlap \(r_k\), and both lower bounds. No further label selection is made.

In each Bell factor, enumeration followed by its inverse returns the original selected index. The identity on the exterior factors remains the identity under the bijection. The second equality is the auxiliary copy-action identity, with the auxiliary and physical factors interchanged.

Substitution of \(e_k\) into the two finite products cancels \(\iota ^{-1}\iota \) in every left auxiliary coordinate.

Apply \(U_k\) to the selected Bell identity and substitute the two projection identities and the product-vector identity. The coordinate change preserves both the label and the scalar \((\sqrt z/d)^k\).

The Bell projection is self-adjoint. Thus \(\langle \varphi ,v^{\mathrm{site}}_{k,\ell }\rangle =\langle P\varphi ,v^{\mathrm{site}}_{k,\ell }\rangle =\langle \varphi ,Pv^{\mathrm{site}}_{k,\ell }\rangle \). Apply the exact Bell identity and use linearity in the second argument. The scalar is consequently not conjugated.

The regional exponential losses are eventually smaller than half the supplied inverse-polynomial mass. The common projection therefore has positive mass at least \(1/[4(k+1)^{d^2}]\) before normalization. Apply the common Bell-vector construction with the given label and pull its vector back through \(\tau _k\). This preserves its norm, symmetry, auxiliary label and Bell range. The exact Bell-pinning coefficient is \((\sqrt z/d)^k\), whose squared modulus is \((z/d^2)^k\). Self-adjointness of \(P_k\) gives \(\langle \Phi _k,P_kv_k\rangle =\langle \Phi _k,v_k\rangle \) because \(P_k\Phi _k=\Phi _k\). Each matched-cutoff estimate applies to the same common vector, and its coordinate pullback gives the two stated physical cutoff equations. Finally, a nonzero coordinate of the unit vector \(\Omega \) and the positive selected mass give the asserted nonemptiness; the selected Bell vector has unit norm by its defining spectral embedding.

The three families of central label projections commute. Their products \(P^g_\alpha P^b_\beta P^w_\lambda \) form an orthogonal resolution of the identity. On every nonzero product, the two subgroup dimension bounds give

All irreducible dimensions are positive, and \(r\le k\) implies \(\binom {k}{r}{\gt}0\). Taking logarithms proves both inequalities on this joint component. Vanishing products make no contribution. Summing over the joint resolution proves 139.

Write \(x=(x_g,x_b)\) and \(y=(y_g,y_b)\) in the specified split of the coordinates. The central projection has entries

where \(P^{(r)}_\beta \) is the same label projection for the actual permutation action on \(\mathbb C^{C^r}\). Thus nonvanishing of \(P_\beta \) implies nonvanishing of \(P^{(r)}_\beta \). Its range is invariant under \(S_r\), is nonzero, and is contained in the \(\beta \)-isotypic component. Consequently its dimension is at least \(d_\beta \) and at most the ambient dimension \(d^r\).

Every nonzero bad-label projection has irreducible dimension \(d_\beta \le d^r\). Positivity of \(d_\beta \) permits taking logarithms, which gives \(\log d_\beta \le r\log d\). Summing over the actual orthogonal resolution gives the first inequality; zero projections contribute nothing. The subgroup operator bound gives \(F_w\le F_g+F_b+\log \binom {k}{r}\, I\). Adding the first inequality proves the second one. The empty cases use the stated total logarithm convention.

Every permutation of the good copies is also a whole-copy permutation, so it commutes with the whole-copy central projection. Summing the central projections of the good-copy representation proves the stated commutation. Compress the preceding whole-space upper bound for \(F_w\) by \(P_\lambda \). Since \(P_\lambda \) is an orthogonal projection and \(F_wP_\lambda =(\log d_\lambda )P_\lambda \), the compressed inequality is exactly the stated lower bound. This argument also includes \(P_\lambda =0\).

The identity physical permutation and the auxiliary operator \(P_\lambda \otimes I_D\) give \((I\otimes P_\lambda \otimes I_D)w=w\). The coordinate split has exactly \(k-r\) good and \(r\) bad copies. Lift the positive semidefinite difference \(P_\lambda F_gP_\lambda -\tau P_\lambda \) by the identity on the physical and additional auxiliary registers. Its quadratic form on \(w\) is nonnegative. The Hermitian label projection fixes \(w\), and hence cancels on both sides of that quadratic form, giving the required inequality.

The coordinate bijection retains the physical configuration and rearranges the auxiliary coordinates. Applying either permutation before or after this rearrangement gives the same coordinate tuple. The corresponding permutation matrix entries therefore agree.

The claim holds for a single group element by the permutation identity, and both sides preserve scalar multiplication and addition.

Apply the good auxiliary bound separately to \(C\) and \(R\). Tensor each positive difference with the other positive label projection and with \(I_A\), then add. Their two scalar floors sum to (142). The label projections are orthogonal and each commutes with its own good-copy observable, which also proves the stated projection and commutation identities.

The matrix \(L\) is Hermitian. Apply the negative exponential inequality on the invariant range of \(P\) to \(bP\leq PLP\).

The coordinate bijection sends \(P_0\) and \(L_0\) to \(P\) and \(L\) with spectator register \((Q\times Y\times V)^k\). Transport the joint auxiliary entropy bound through this bijection.

Nesting and disjointness of the indicated factors imply that \(G-L_0\) commutes with both \(P_0\) and \(L_0\). The auxiliary floor gives \(P_0e^{-aL_0}P_0\leq e^{-ab}P_0\). Apply the fixed-vector exponential comparison with generators \(-a(G-L_0)\) and \(-aL_0\).

The two original label equations imply the equation for their product. Acting only on the physical factors, \(E_B(\Omega )\otimes I\) preserves this equation. The coordinate identities then give \(P_0f=f\). Apply Theorem 22.57.6 with the specified good/bad split.

Choose the constant in the whole-space inverse comparison before choosing \(k\), \(B\), the labels, or \(u\). Evaluate that operator inequality on \(f\). The symmetry inherited by the excitation component removes the bad-copy contribution. Its two original auxiliary labels allow the joint auxiliary comparison, with the summed entropy floor \(b\). Multiply these inequalities by their nonnegative scalar factors. The resulting expression is the stated bound.

Apply the inverse-to-good-expectation comparison with its constant chosen before the copy number and the vector. Express its quadratic form as a trace against the actual reduced good-copy density. Remove the physical deficit on its invariant spectral support and apply the paired merge moment estimate with parameter \(2t\). The hypothesis \(4t\leq 1\) is exactly the range of that estimate. Multiplication by the nonnegative prefactor gives the stated bound.

Apply the five-factor trace inequality at parameter \(a\). Its factors are the two merge moments, the negative physical moments for \(Q\) and \(V\), and the positive physical moment for \(Y\), all at parameter \(5a\). Apply the separate merge bounds and the signed singleton bounds. Each factor has the same mass \(\| w\| ^2\); the product of their fifth roots retains this mass once, including when it vanishes. Summing the five scalar exponents and dividing by five gives the displayed exponent. The two negative moments each contribute \(2mK(5a)^2\), and the positive moment contributes \(mK(5a)^2\); their sum divided by five is \(25mKa^2\).

Apply the inverse-to-good-expectation comparison with its constant chosen before the component data. Express the resulting quadratic form as a trace in the actual good-copy density and apply the preceding theorem with \(a=2t\). The two restrictions on \(20t\) give its parameter range. Multiplication by the nonnegative comparison factor yields the assertion.

Purity and subadditivity give \(g\geq 0\). The auxiliary dimensions give \(d\geq 0\), and \(\Lambda \geq 0\). Apply the uniform scalar estimate to the sharp component bound. The number of good copies is at least \((1-\tau )k\), while \(K\geq 0\) permits replacing it by \(k\) in the quadratic loss. All three binomial factors are bounded using \(\log \binom {k}{|B|}\leq kh(\tau )\). Multiplication by the nonnegative squared component norm proves the assertion, including when that norm vanishes.

Positivity of the inverse squared metric follows from the disjointness of its three subsystems. Every vector in the range of \(P\) has the required copy symmetry and the two original labels. Apply the uniform bound to all its actual excitation components. The compression inequality then contributes the number of retained subsets, at most \((k+1)e^{kh(\tau )}\).

The average of the two merge polynomials is bounded by \((k-|B|+1)^D\). Apply the uniform scalar estimate with zero physical entropy and quadratic moment terms, retaining polynomial degree \(D\). The original label lower bound supplies \(2kS-\varepsilon \), and \(\log \binom {k}{|B|}\leq kh(\tau )\) accounts for all three binomial contributions. Multiply this scalar bound by the actual squared component norm.

The inverse squared metric is positive, and this remains true after changing coordinates. Every vector in the range of \(P\) satisfies the original symmetry and two label equations. Apply the uniform bound to each of its actual excitation components. The excitation compression inequality bounds the sum by the number of retained subsets, at most \((k+1)e^{kh(\tau )}\). No commutation between the metric and the excitation projections is used.

The matrix is Hermitian, and \((Z^*PZ)^2=Z^*(PZZ^*)PZ=Z^*P^2Z=Z^*PZ\).

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

Apply the compressed-inverse inequality to \(UAU^*\) to obtain \(c^{-1}P\leq UAU^*\). Conjugation first by \(U\) and then by \(Z\) gives the conclusion. No commutation of \(P\) and \(UAU^*\) is required.

The grouping map is equivariant for every copy permutation. Apply the identity for the finite-group average.

The literal projection satisfies \(PS'=P\). Thus \((U^*PU)ZZ^*=U^*PU\). Apply the contained-projection compression identity.

The actual inverse-compression estimate gives \(PUA_k^{-1}U^*P\leq b_kP\). The squared disjoint-region metric is positive definite and \(b_k{\gt}0\). The fixed-coordinate lower bound therefore applies, first with the identity matrix and then with \(Z\). The compressed projection property follows from the actual symmetry and label projection identities.

The sharp inverse compression gives \(P(UA_k^{-1}U^*)P\leq b_kP\) with \(b_k{\gt}0\). Positivity of \(A_k\) follows from its three disjoint subsystems. The inverse-compression inequality therefore gives \(b_k^{-1}\widehat P\leq A_k\), and compression by \(Z\) preserves this inequality. The actual low-defect projection is supported on simultaneous symmetry; together with \(ZZ^*=P_{\mathrm{sym}}\), this proves that \(Z^*\widehat PZ\) is an orthogonal projection.

Reindex the finite sum of squared absolute values along the bijection \(e_{Q,Y}\). Taking square roots gives the identity.

Apply the five-factor rough lower bound to each physical cut pair. Its polynomial exponent has the form \(c_{Q,Y}+D_{Q,Y}\), where

The finite maximum of these exponents and zero gives \(c\) before the copy number, vector, labels, cutoffs, and matrix \(Z\) are chosen. Since \(k+2\geq 1\), replacing a cut-dependent exponent by \(c\) increases \(b_k\) and weakens the lower bound.

The actual regional equivalence pulls the regional projection back to \(P_{\mathrm{glob}}\) and the regional metric product back to \(A_{Q,Y}\). Pull back the five-factor inequality and conjugate by \(Z\). The common projection theorem supplies that \(P_Z\) is an orthogonal projection. The estimate retains the three binary-entropy losses and the subset-counting factor \((k+1)e^{kh(\tau )}\).

Apply the five-factor sharp lower bound after grouping the original physical sites into \(Q,Y,F\). The grouping preserves the norm, and each grouped marginal has exactly the entropy and surprisal moments of the corresponding original reduction.

The polynomial exponent has the form \(c_{Q,Y}+\Lambda _{Q,Y}\). Take the maximum of these finitely many exponents and zero before choosing the copy number, state, labels, moments, cutoff, or coordinates. Replacing each exponent by this maximum increases the positive coefficient \(b_k\) and weakens its reciprocal lower bound.

Pull back the resulting matrix inequality along the actual regional coordinate equivalence and conjugate by \(Z\). The projection and metric identities give \(P_{\mathrm{glob}}\) and \(A_{Q,Y}\), respectively. The common projection theorem gives the projection property of \(P_Z\).

Choose the uniform regional and symmetric-floor constants before choosing any copy or state data. The original regions form a disjoint partition, and nesting makes the band roots commute. Each compressed leaf is bounded below by \(2b_kI\) and by \(e^{2tk\ell _{j,g,\tau }-r_k}P\), where \(\ell _{j,g,\tau }=2S+(1-\tau )g_{j,g}-\tau d- (3+(2t)^{-1})h(\tau )-50Kt\), and \(P\) is the same compressed global low-defect projection. The original labels and simultaneous symmetry are preserved by the actual inverse power and normalization. Applying the original gap gives projection mass at least \(1-E_{\rm def}/\tau \); its upper bound by one also gives \(E_{\rm def}\geq 0\). The explicit mean-tree lower norm estimate now applies with \(\kappa =2tk\) and upper rate \(2S+g_{\max }\). Since \(0\leq 1-\tau \leq 1\) and \(g_{j,g}\leq g_{\max }\) with \(g_{\max }\geq 0\), one has \((1-\tau )g_{j,g}\leq g_{\max }\). The remaining losses are nonnegative, so \(\ell _{j,g,\tau }\leq 2S+g_{\max }\). Since terminal weights sum to one, their average of \(\ell _{j,g,\tau }\) is \(L_{g,\tau }\). Summing over the bands gives the inequality, also when \(G=0\).