The cyclic relation and orthogonality give \(F_u(\sigma )=Q_uA^{i_0}\cdots A^{i_{N-1}}\). Hence

For \(N{\gt}0\), multiplying the local matrices in (1) gives \(F_u(\sigma )\), proving the coefficient identity. For \(N=0\), the same trace calculation reduces to (4).

Evaluating the component product inserts \(Q_u\) before the original word. Summing over \(u\) and using \(\sum _uQ_u=\mathbb {1}\) gives the coefficient sum. If \(p\nmid N\), the component starts in sector \(u\) and ends in sector \(u+N\ne u\). Cyclicity of the trace places these two orthogonal projections next to each other, so each coefficient is zero.

Choose the cyclic projections supplied by periodicity. If \(p\mid N\), then \(Q_{u+N}=Q_u\), and Theorem 7.2 gives (6) at every length. For \(N{\gt}0\), the explicit component-chain decomposition follows by inserting \(\sum _uQ_u=\mathbb {1}\) and multiplying the local corner matrices; this proves (7). The separate coefficient identity in Theorem 7.2 identifies its summands with the boundary traces. If \(p\nmid N\), then \(Q_{u+N}\ne Q_u\); cyclicity of the trace and orthogonality give \(\operatorname{tr}F_u(\sigma )=0\) for every \(u\) and \(\sigma \).

Unitality fixes \(\mathbb {1}\) and gives spectral radius one. Together with the faithful adjoint fixed point \(\Lambda \) and uniqueness of positive fixed matrices up to scalars, these are the spectral characterization of irreducibility. The irreducible unital peripheral spectrum is cyclic. Its primitive-root powers are distinct, so its cardinality identifies the cyclic order with \(p\). Positivity and unitality exclude nontrivial peripheral Jordan blocks; the irreducible peripheral eigenspaces are one-dimensional. These are the peripheral spectral facts used in [ 228 , proof of Theorem 5 ] .

The cyclic decomposition theorem for irreducible finite Kraus maps [ 216 , “Cyclic decomposition for irreducible finite Kraus maps ] supplies orthogonal projectors with \(\mathcal{E}_A(Q_{u+1})=Q_u\). Their sum is the identity, and cyclicity makes each nonzero. The Kraus shift gives (9) for the original matrices \(A^i\). Inserting the resolution of the identity into the trace proves (10). At a non-divisible length, the starting and final projectors differ; their orthogonality and cyclicity of the trace kill each component.

Expand the transfer map using (4) and use \((\mathbb {1}-P)A^iP=0\) to reduce each Kraus term to the \(P\)-corner.

Diagonalize \(P\) by a unitary \(U\). Since \(P^2=P\), the diagonal form of \(P\) has only \(0\)- and \(1\)-eigenvalues, so the basis splits as \(n+m=D\). Conjugating \(A\) by \(U\) preserves the MPV family by Theorem D.1.1. In this basis the relation \((\mathbb {1}-P)A^iP=0\) makes each letter upper block triangular. Write the conjugated letters as

Since \(B^{i_1}\cdots B^{i_N}\) is again upper block triangular, its trace satisfies

Hence the block-diagonal tensor \(A_1\oplus A_2\) has the same MPV coefficients as the conjugated tensor.

From \(\mathcal{E}_A(\rho )=\rho \) and positivity of each summand \(A^i\rho (A^i)^\dagger \), one gets \((\mathbb {1}-P)A^i\rho (A^i)^\dagger (\mathbb {1}-P)=0\) for each \(i\). Since \(\rho \) is supported on the range of \(P\), this forces \((\mathbb {1}-P)A^iP=0\).

This is the finite-dimensional case of [ 244 , Proposition 6.10 ] : the support projection of any stationary state is sub-harmonic. The equivalence between irreducibility and absence of nontrivial invariant projections is [ 244 , Theorem 6.2 ] ; see also  [ 17 , Section IV.A.1 ] .

The support projection \(P\) of \(\rho \) is neither \(0\) nor \(\mathbb {1}\), since \(\rho \neq 0\) and \(\rho \) is not positive definite. Theorem 7.1.4 gives \((\mathbb {1}-P)A^iP=0\), so a unitary change of basis diagonalizing \(P\) puts \(A^i\) in upper-triangular block form with diagonal blocks \(B_{11}^i,B_{22}^i\). Theorem D.1.1 preserves the MPV under the conjugation, and the trace identity (11) shows that deleting the off-diagonal blocks preserves all MPV coefficients.

By strong induction on \(D\). If \(A\) is irreducible, take \(r=1\). Otherwise a nontrivial invariant projection \(P\) gives the upper-triangular splitting of Theorem 7.1.5; deleting the off-diagonal part preserves the MPV family and yields two blocks of strictly smaller bond dimension. Apply the induction hypothesis to each block.

Both  [ 228 , Theorem 4 ] and  [ 17 , Section IV.A.1 ] obtain the block decomposition by the same invariant-projection splitting.

The Perron–Frobenius TP-gauge construction is applied to each irreducible block using the gauge (3). Gauge equivalence preserves irreducibility and the MPV family, and the positive scaling factors are absorbed into the weights \(\mu _k\).

Theorem D.1.3 supplies the nonzero irreducible blocks \((C_k)_{k=1}^r\), with \(V^{(N)}(A)_\sigma =V^{(N)}(\bigoplus _{k=1}^rC_k)_\sigma \) for every \(N\ge 1\) and every \(\sigma \). Applying the Perron–Frobenius TP gauge (3) to each \(C_k\) gives a left-canonical block \(B_k\) with weight \(\mu _k\), and

The gauge preserves every block dimension, so the structural bound \(\sum _kD_k\le D\) is unchanged.

The peripheral cyclic-structure theorem identifies the period and its primitive root. The cyclic decomposition of the adjoint transfer map gives projections \((P_k)\) with \(\mathcal{E}_{A^\dagger }(P_{k+1})=P_k\). Iterating the cyclic relation \(m\) times gives \((\mathcal{E}_{A^\dagger })^m(P_k)=P_k\), and the blocked-transfer identity \(\mathcal{E}_{(A^{[m]})^\dagger }=(\mathcal{E}_{A^\dagger })^m\) identifies these with the adjoint transfer map of \(A^{[m]}\). Apply the block decomposition from adjoint-fixed projections.

Theorem 7.4.3 gives irreducibility of \((\mathcal{E}_A^\dagger )^m\) on each corner \(P_u\). The compression equivalences in (14) intertwine the adjoint transfer map of \(C_u\) with the corresponding corner restriction of \((\mathcal{E}_A^\dagger )^m\). The peripheral spectrum of the blocked adjoint map is

because \(\gamma \) is a primitive \(m\)th root of unity. Hence each nonzero irreducible corner restriction is primitive. Primitivity and irreducibility are preserved across \(\varphi _u\); then use Theorem D.4.4 to pass from the adjoint blocked map to the primal transfer map of \(C_u\).

Left-canonicality makes \(K_i:=(A^i)^\dagger \) unital, irreducibility passes to the adjoint map, and the quantum Perron–Frobenius theorem provides a positive-definite fixed point; the cyclic peripheral-spectrum theorem and Theorem 7.3.1.1 then produce the cyclic-sector decomposition. Keep the projections and compression equivalences, and apply Theorem 7.3.1.2 to the sector blocks.

Start from Theorem D.5.4. With \(p_A,p_B\) the least common multiples of the period-removal lengths on the two sides, use \(p=\operatorname{lcm}(p_A,p_B)\). The prescribed-common-multiple lemma constructs both one-sided sector families at this length, and the positive-length blocking lemmas carry the one-sided tensor/nonzero-part equalities over to the common alphabet, giving (16) and (17). The two-sided blocking lemma applied to (16) gives (18).

Choose a positive-definite fixed point \(\rho \) of \(\mathcal{E}_A\). The weighted trace \(X\mapsto \operatorname{tr}(\rho X)\) is invariant under \(\mathcal{E}_{A^\dagger }\) and its powers. Corner preservation gives

The invariant weighted trace of this positive semidefinite difference is zero. Since \(\rho {\gt}0\), the difference vanishes.

Irreducibility passes to the adjoint transfer map. The cyclic relation gives equality in both Kadison–Schwarz inequalities:

Hence every \(P_k\) lies in the multiplicative domain, so for every \(Z\),

If \(X\in P_kM_{D}(\mathbb {C})P_k\) is fixed by \((\mathcal{E}_{A^\dagger })^m\), then its orbit sum

is fixed by \(\mathcal{E}_{A^\dagger }\). By Theorem 7.8.1, \(R_X=c\mathbb {1}\). Orthogonality of the cyclic sectors and the multiplicative-domain identities give \(X=P_kR_XP_k=cP_k\). Similarly, \(Y=c'P_k\). Since \(\mathcal{E}_{A^\dagger }(P_k)=P_{k-1}\), both sides of the required identity equal \(cc'P_{k-1}\).

Follow [ 228 , Theorem 4 ] . The nonzero irreducible decomposition is obtained by strict dimension induction. On a nonzero current block, Perron–Frobenius gives a positive semidefinite eigenvector at a positive spectral radius. If it is singular, positivity makes its support invariant: \((\mathbb {1}-P)A^iP=0\). After diagonalizing \(P\), every letter is upper block triangular and the finite-ring trace splits as

Discard all-zero blocks and repeat on the two strictly smaller diagonal blocks. This gives the finite nonzero irreducible family of Theorem D.1.3.

For each retained irreducible block, divide by the square root of its positive spectral radius and conjugate by the square root of its positive-definite Perron eigenvector. Theorem 4.5.4 gives the unital identity

Nonzero scalar rescaling preserves irreducibility. The proof passes from tensor irreducibility to irreducibility of the matrix action, carries invariant subspaces through the gauge similarity, and returns to tensor irreducibility using Theorem 6.7.3.4 and Lemma C.2.2. Hence Theorem 7.8.1 makes every fixed point scalar. Theorem 7.10.2 then diagonalizes a positive-definite dual fixed point by a unitary without changing unitality.

Apply this single-block construction to every member of the finite family. The square roots of the block spectral radii become positive weights, gauge invariance preserves every positive-length MPV coefficient, and the block dimensions are unchanged. Combining this blockwise result with the initial decomposition yields the arbitrary-input unnormalized form and its bound \(\sum _k\dim (A_k)\le D\).

If the decomposition had no blocks, its weighted block tensor would have zero MPV coefficients at every positive length, contrary to the assumed nonzero coefficient. Thus \(r{\gt}0\). Since the finite family \((a_k)\) is nonempty and every \(a_k{\gt}0\), its maximum \(s\) is positive. The definition \(\nu _k=a_k/s\) gives \(0{\lt}\nu _k\le 1\), and any maximizing index has \(\nu _k=1\). Finally, \(a_k=s\nu _k\) for every \(k\), so homogeneity of a length-\(N\) matrix product vector gives (19).

Choose the blocks and weights from Theorem 7.5.2. The assumed nonzero coefficient makes its block family nonempty, so Theorem 7.5.3 supplies \(s{\gt}0\) and the normalized positive weights. Equation (19) and homogeneity give

The decomposition satisfies all required block conditions and the bond-dimension bound.

Split according to whether there exist \(N{\gt}0\) and \(\sigma \) with \(V^{(N)}(A)_\sigma \neq 0\). In that case apply Theorem 7.5.4. If no such pair exists, every positive-length coefficient vanishes.

Apply Theorem 7.5.5. If every positive-length coefficient vanishes, take \(s=1\) and the empty retained family; the block conditions and bond-dimension bound are vacuous, and the MPV identity is the zero identity. In the nonzero branch, use the nonempty normalized family supplied by the second alternative.

Algebraic normality together with \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\) makes the transfer map strongly irreducible. It consequently has a positive-definite fixed point, no nontrivial invariant projection, and peripheral spectrum \(\{ 1\} \). The fixed-point eigenvalue and left-canonical normalization give \(r(\mathcal{E}_A)=1\). Thus no scalar rescaling remains, and the two conditions in Definition 7.6.1 hold for \(A\) itself.

Entrywise conjugating \(B^i=X A^i X^{-1}\) gives \(\overline{B^i}=\overline X\, \overline{A^i}\, (\overline X)^{-1}\). Conjugating the left-canonical identity gives

Entrywise conjugation maps the span of the length-\(N\) products of \(A\) onto the span of those of \(\overline A\), so it preserves positive-length block injectivity and hence algebraic normality. For the normal-tensor assertion, the transfer map of \(\overline A\) is the anti-linear conjugate \(\sigma \circ \mathcal{E}_A\circ \sigma \) by (25). By the corresponding theorem in the companion quantum-channel volume  [ 216 , “Entrywise conjugation transports peripheral eigenvalues” ] this conjugation preserves the spectral radius and primitivity, and by Theorem 7.6.6 entrywise conjugation preserves irreducibility. Hence both clauses of Definition 7.6.1 hold for \(\overline A\).

Entrywise conjugation commutes with finite sums, matrix products, and conjugate transposition, which gives (25) on every matrix, and (26) follows since \(\sigma \) is an involution. The diagonal entries of a positive-definite complex matrix are real, while all off-diagonal entries of a diagonal matrix vanish. This proves (27).

Suppose \(P\) is a nontrivial invariant orthogonal projection for \(\overline A\), so \((\mathbb {1}-P)\, \overline{A^i}\, P=0\) for every \(i\). Conjugating entrywise gives \((\mathbb {1}-\overline P)A^i\overline P=0\), and \(\overline P\) is again a Hermitian idempotent distinct from \(0\) and \(\mathbb {1}\). Hence \(\overline P\) is a nontrivial invariant orthogonal projection for \(A\), contradicting the irreducibility of \(A\).

A bijection of the physical alphabet only permutes the summands in \(\mathcal{E}_A(X)=\sum _iA^iX(A^i)^\dagger \). Hence the transfer map, its spectral radius, its peripheral spectrum, and its invariant orthogonal projections are unchanged. Formula (32) is obtained from Formula (28) by replacing \(i\) with \(e(j)\).

Expand both transfer maps and take \(C=X^{-1}\).

Theorem 7.6.9 identifies the two transfer maps by an algebra automorphism. Such an automorphism preserves composition and scalar multiplication, and therefore preserves the displayed equation.

The two maps are conjugate by the invertible linear transformation \(Y\mapsto CYC^\dagger \), and therefore have the same spectrum.

Congruence similarity is an algebra isomorphism on the endomorphism algebra and hence preserves the peripheral spectrum.

With \(S_X(Y)=XYX^\dagger \), the transfer maps satisfy

Similarity preserves the spectral radius and the peripheral spectrum. For irreducibility, start with an invariant orthogonal projection \(Q\) for one transfer map and form the support projection of the corresponding positive congruence \(XQX^\dagger \) (using \(X^{-1}\) in the reverse direction). The intertwining in Formula (33) makes this support projection invariant for the other transfer map. Irreducibility forces its support to be zero or full, and invertibility of \(X\) then forces \(Q=0\) or \(Q=\mathbb {1}\). Thus both conditions in Definition 7.6.1 are transported across the gauge.

Choose a trace-preserving gauge \(B^i=XA^iX^{-1}\). Positive blocking gives

The trace-preserving primitive and irreducible structure of \(B\) is preserved by positive blocking, so Theorem 7.6.3 makes \(B^{[L]}\) CPSV normal. Gauge transport (Theorem 7.6.13) then gives CPSV normality of \(A^{[L]}\).

Write \(I=(i_1,\ldots ,i_L)\). Since \(L\geq 1\), the word \(I\) is nonempty. Repeatedly inserting \(UU^\dagger =\mathbb {1}\) into \(A^{i_1}\cdots A^{i_L}\) gives

The preceding theorem gives CPSV normality of every blocked block \(A_k^{[L]}\), and the displayed identity is exactly the retained direct-sum reconstruction.

Apply Theorem 7.11.1.3 to \(A\) and itself. It gives \(p\ge 1\), weights \(\mu _k\), and left-canonical, primitive, tensor-irreducible blocks \(C_k\) such that, for every \(N\ge 1\),

First, Theorem 7.11.1.1 makes each \(C_k\) algebraically normal. Second, Theorem 7.6.3 combines this conclusion with the left-canonical identity to make each \(C_k\) a CPSV normal tensor. Hence

satisfies the block-diagonal identity (28), so \(B\in \mathrm{CF}\). Equation (37) gives the second part of (36).

Induction on a nonempty physical word \(w\) of length \(N\), using (28) at each letter and \(UU^\dagger =\mathbb {1}_S\) between adjacent letters, gives

Applying cyclicity of the trace and \(UU^\dagger =\mathbb {1}_S\) to (40) gives

Applying Theorem 2.6.9 to the right-hand side of (41) yields \(\operatorname{tr}(A^w)=\sum _{k=1}^{r}\mu _k^N\operatorname{tr}(A_k^w)\), which is (39).

The canonical-form data supply normality of every \(A_k\) and the positive-length identity \(|V^{(N)}(A)\rangle =\sum _k\mu _k^N|V^{(N)}(A_k)\rangle \). Substitution of these two facts into Theorem 8.1.4 gives (42).

Write \(E\) for the transfer map and \(E^*\) for its trace adjoint. Invariance and normalization give \(PE^*(P)=P=E^*(P)P\), whence

Stationarity gives \(\operatorname{tr}(\rho (E^*(P)-P))=\operatorname{tr}((E(\rho )-\rho )P)=0\). Faithfulness of the positive weight implies \(E^*(P)=P\). Since \(P^\dagger P=P\), the fixed-point commutant criterion gives \(PA^i=A^iP\). Applying this conclusion to \(\mathbb {1}-P\) gives the stated projector closure in the opposite orientation.

Proceed by strong induction on the bond dimension. If \(A\) is irreducible, the single corner \(V=\mathbb {1}\) suffices. Otherwise, \(A\) has a nontrivial invariant projection \(P\), which commutes with every \(A^i\) by Theorem D.2.1. Factoring \(P=V_1V_1^\dagger \) and \(\mathbb {1}-P=V_2V_2^\dagger \) through support isometries by Lemma D.2.3 splits \(A\) into two corner tensors of strictly smaller bond dimension. For \(j=1,2\), commutation gives \(A^iV_j=V_j(V_j^\dagger A^iV_j)\), and each corner inherits invariant-projector closure by Lemma D.2.4. The two recursively obtained families are combined. Ranges from different corners are orthogonal because

Equality of matrix product vectors follows from Lemma D.2.6.

Decompose \(A\) into irreducible corners \(B_k\) along isometries \(V_k\) by Theorem 7.7.3; every matrix product vector coefficient of \(A\) is the sum of those of the corners. A corner whose matrices all vanish contributes nothing at positive length and is dropped. Every remaining corner has a Perron eigenpair \(\mathcal{E}_{B_k}(\rho _k)=r_k\rho _k\) by Lemma D.3.1. The absence of \(p\)-periodic vectors, applied to the corner \(B_k\), gives that every eigenvalue of \(\mathcal{E}_{B_k}\) of modulus \(r_k\) equals \(r_k\), so \(A_k:=r_k^{-1/2}B_k\) is normal by Lemma D.3.2. Setting \(\mu _k:=r_k^{1/2}\) gives \(\mu _kA_k=B_k\) at the level of matrices, so for every word \(w\) of positive length \(N\) the weight cancels:

Summing over \(k\) gives (43), while the corner reconstruction gives (44). The retained dimensions satisfy \(\sum _kD_k\le \sum _k\operatorname{tr}(V_kV_k^\dagger )=\operatorname{tr}(\mathbb {1})=D\).

Let \(\lambda _{\max }\) be the largest eigenvalue of \(X\). Then \(\rho :=\lambda _{\max }\mathbb {1}-X\) is positive semidefinite, singular, and nonzero because \(X\) is not scalar. Unitality and the fixed-point equation for \(X\) give \(\mathcal{E}_A(\rho )=\rho \). Apply Theorem 7.1.5. The non-scalar hypothesis is needed because scalar multiples of \(\mathbb {1}\) are fixed by every unital map and yield no nontrivial support projection.

Tensor irreducibility gives irreducibility of the completely positive transfer map. By Wolf’s Theorem 6.6 for irreducible unital Kraus maps, every fixed point is scalar.

The positive square root of \(\rho \) is invertible. Substituting \(B^i=r^{-1/2}\rho ^{-1/2}A^i\rho ^{1/2}\) and using \(\mathcal{E}_A(\rho )=r\rho \) gives

which is (45). For \(r=1\), the change from \(A\) to \(B\) is a similarity, so cyclicity of the trace gives equality of all MPV coefficients.

Write the retained part of the canonical-form decomposition as \(C^i=\bigoplus _k\mu _kA_k^i\), and let \(U\in \mathbb {C}^{S\times D}\) satisfy \(UU^\dagger =\mathbb {1}_S\) and \(A^i=U^\dagger C^iU\). For each normal block, first choose its pure trace-preserving Perron gauge and then diagonalize its positive definite fixed point by a unitary. This gives \(X_k\in \mathrm{GL}(D_k,\mathbb {C})\) and blocks \(B_k\) satisfying the two canonical-form-II identities. Put \(X=\bigoplus _kX_k\) and extend it to the omitted zero coordinates by

Its inverse is

The identity \(UU^\dagger =\mathbb {1}_S\) gives \(GG^{-1}=G^{-1}G=\mathbb {1}_D\). It also gives

and

Therefore

which has the same ambient reconstruction and satisfies canonical form II.

Choose \(X\in \mathrm{GL}(n,\mathbb {C})\) such that, for every physical index \(i\), \(E^i=XC^iX^{-1}\). Equality of the Gram matrices \(U_AU_A^\dagger \) and \(U_BU_B^\dagger \) gives a unitary \(W\in \mathbb {C}^{D\times D}\) with

Extend the retained gauge through \(U_A\) by

Its inverse is

The coisometry identity gives \(GU_A^\dagger =U_A^\dagger X\) and \(U_AG^{-1}=X^{-1}U_A\). Hence, with \(Y=WG\),

The TP hypothesis makes \(\mathcal{E}_A\) a channel with a nonzero PSD fixed point. Irreducibility makes \(\mathcal{E}_A\) an irreducible CP map, and Theorem 4.1.1 upgrades the fixed point to positive definite. The spectral theorem diagonalizes it by a unitary conjugation, which preserves the TP normalization.

If the bond dimensions differ, Theorem 5.4.4 gives the decay. If the bond dimensions agree, the non-equivalence hypothesis excludes gauge-phase equivalence, so Theorem 5.5.5 gives the decay.

Injective blocks are normal. The block-diagonal decomposition theorem gives the MPV expansion of \(\bigoplus _k \mu _k A_k\) in terms of the block MPVs. The self-overlap limits are part of the hypotheses, and Theorem 7.9.1.3 supplies the off-diagonal decay. Therefore Lemma E.1.7 gives the eventual linear independence required in Definition 8.1.1.

The normal-canonical hypotheses already give the block-diagonal MPV expansion, the self-overlap normalization, and the eventual linear independence coming from irreducible trace-preserving overlap decay. Supplying algebraic normality for each block adds the remaining clause in Definition 8.1.1.

Pass to the conjugate-transposed Kraus family \(K_i:=(A^i)^\dagger \), which is unital with irreducible Kraus map. The TP fixed-point theorem for \(\mathcal{E}_A\) provides a positive definite fixed point for the adjoint of \(K\), so the corresponding theorem in the companion quantum-channel volume  [ 216 , “Roots of unity via adjoint fixed point” ] shows that every peripheral eigenvalue is a root of unity. Since \(\sigma _\partial (\mathcal{E}_A)\) is finite (the corresponding theorem in the companion quantum-channel volume  [ 216 , “Finiteness of peripheral eigenvalues” ] ), the corresponding lemma in the companion quantum-channel volume  [ 216 , “Common exponent for a finite set of roots of unity” ] gives a common exponent \(p\) with \(\mu ^p=1\) for every \(\mu \in \sigma _\partial (\mathcal{E}_A)\), and the corresponding lemma in the companion quantum-channel volume  [ 216 , “Powering collapses peripheral eigenvalues” ] gives \(\sigma _\partial (\mathcal{E}_A^p)=\{ 1\} \). Blocking by \(p\) takes the \(p\)th power of the transfer map, so \(\mathcal{E}_{A^{[p]}}\) is primitive in the peripheral-spectrum sense of the corresponding definition in the companion quantum-channel volume  [ 216 , “Primitive map” ] . The blocked map need not remain irreducible: eigenvalue \(1\) may have several cyclic corners. The adjoint-fixed projections are therefore compressed before irreducibility and primitivity are asserted sector by sector in Theorem 7.3.1.3. This is the periodicity-removal step of [ 13 , Section 2.3 ] for an irreducible block already in TP gauge.

Theorem 7.8.6 provides the unitary and the diagonal positive-definite fixed point, and unitary conjugation preserves trace preservation and the MPV family (Theorem D.1.1).

Apply Theorem 7.10.1 to the conjugate-transposed family \(i\mapsto (A^i)^\dagger \): unitality of \(A\) is trace preservation for this family, whose transfer map is the dual of \(\mathcal{E}_A\). Translating back gives (61).

The primitive peripheral spectrum, tensor irreducibility, and left-canonical normalization give primitive fixed-point data. The fixed point is positive definite by Lemma D.4.9. Theorem 6.7.4.4 then gives normality.

Theorem 7.2.1.1 supplies the left-canonical nontrivial blocks, the positive-length equality, and the bound \(\sum _kD_k\le D\). Apply Theorem D.4.2 directly to this finite irreducible family. Its proof first applies Theorem 7.9.2.1 to each block and then takes the least common multiple of the resulting periods through Theorem D.4.1. This gives one positive length \(p\) at which every blocked transfer map is primitive. Lemma D.4.18 keeps the blocked weights nonzero, and Lemma D.4.16 carries the positive-length equality through blocking, giving (64).

Apply Theorem 7.3.1.4 to choose the common blocking length and the two common cyclic-sector families. For either family, write \(G_k\) for its cyclic-sector blocks and \((\nu _x,C_x)\) for the weights and blocks of the flattened common-sector family from Lemma D.5.10. The common-alphabet nonzero-part identity gives, for every length \(N\) and blocked word \(\sigma \),

Applying this identity on the two sides of the comparison yields a positive length \(p\) and weighted primitive block families such that, for every \(N\ge 1\) and blocked word \(\sigma \),

The two weighted nonzero parts are equal at every positive length. The structural properties and the nonvanishing of the weights come from the common-sector families.