Eventual linear independence makes the matrix product vector of each block nonzero at some positive length. A tensor with zero bond dimension has zero matrix product vector at every length, which is a contradiction.

The only orthogonal projections on a one-dimensional complex vector space are zero and the identity.

Irreducibility follows from Lemma E.1.2. The identity transfer map has spectral radius one, and its only eigenvalue is one.

The two transfer maps are similar through the invertible congruence \(X\mapsto \sigma ^{-1/2}X\sigma ^{-1/2}\). Similar linear maps have the same spectrum, so the condition that \(1\) is the only eigenvalue on the unit circle is unchanged.

Substitute the positive-length MPV equality into the spanning identity for \(A\). Normality and eventual linear independence depend only on the family \((A_j)_{j=1}^{g}\).

Positive-length nonzero proportionality implies eventual nonzero proportionality. Apply Theorem 8.2.12.

The Gram matrix of the family \(\{ |V^{(N)}(A_i)\rangle \} _{i \in I}\) converges entrywise to the identity matrix. Hence for all sufficiently large \(N\) it is invertible, so the MPV states are linearly independent.

This is Lemma E.1.7, restated with an explicit threshold \(N_0\).

Apply the finite-index Gram-matrix criterion to the disjoint union of the two index sets. The within-family hypotheses give the diagonal and off-diagonal limits inside each block, and the mixed overlap hypothesis gives the off-diagonal limits between the two blocks.

Move the two linear combinations to one side. Linear independence forces each coefficient difference to vanish at every sufficiently large length.

Because the two families span the same subspace, each \(w_j\) has a unique expansion in the basis \(\{ v_i\} _{i=1}^g\); these coefficients form the columns of \(U\). If \(U a = 0\), then the corresponding linear relation among the \(w_j\) is trivial because the \(w_j\) are linearly independent. Thus \(U\) is injective, hence invertible.

By Lemma E.1.7, both families are linearly independent for all sufficiently large \(N\). Hence, for all sufficiently large \(N\), both families are linearly independent and the assumed span equality holds. Lemma E.2.1 applies at each such \(N\).

Induct on the number of terms. Subtracting \(w_1\) times the equality at exponent \(N\) from the equality at exponent \(N+1\) gives \(\sum _{i=2}^{n} c_i(w_i-w_1)w_i^N=0\). The induction hypothesis yields this equality at every exponent, so the original sum \(S_N=\sum _i c_iw_i^N\) satisfies \(S_{N+1}=w_1S_N\) for every \(N\). Since \(S_N\) eventually vanishes and \(w_1\ne 0\), it follows that \(S_0=0\) and hence \(S_N=0\) for every \(N\).

Set \(w_l=a_l\) for \(l\leq m\) and \(w_{m+l}=b_l\) for \(l\leq n\), with coefficients \(c_l=1\) for \(l\leq m\) and \(c_l=-1\) for \(l{\gt}m\). Then \(\sum _{l=1}^{m+n}c_lw_l^k=\sum _i a_i^k-\sum _j b_j^k\). The hypothesis makes this expression zero for every sufficiently large \(k\). Lemma E.2.3 makes it zero for every \(k\), which is the required power-sum equality.

Construct diagonal matrices \(\operatorname{diag}(\alpha )\) and \(\operatorname{diag}(\beta )\). The power-sum hypothesis gives \(\operatorname{tr}(\operatorname{diag}(\alpha )^k) = \operatorname{tr}(\operatorname{diag}(\beta )^k)\). By the corresponding theorem in the companion quantum-channel volume  [ 216 , “Newton–Girard trace recursion” ] , the characteristic polynomials agree: \(\prod _i (X - \alpha _i) = \prod _i (X - \beta _i)\). Extracting roots gives multiset equality.

Since \(r_j{\gt}0\), choose one copy of \(A_j\). Its contribution \(D_j\) is one non-negative summand in the total dimension of the flattened copy family, so \(D_j\leq D\).

For the second assertion, every representative has multiplicity \(r_j\geq 1\), and each copy has positive dimension \(D_j{\gt}0\). Therefore

The coefficient identity in (1) is \((\mu ^p)^N=\mu ^{pN}\). The tensor identity is Theorem D.4.12 applied to the flattened copy family.

Choose one original block index. Its occurrence in the phase-class enumeration supplies a phase class containing it.

Evaluate both sides at a pair of matrix indices and apply the bijective enumeration of the finite phase-class quotient.

Expanding the sector decomposition gives the left-hand side of (2), which is the finite block-sum expression for \(\bigoplus _k\mu _kB_k\).

If the coefficient vanished for every \(N\geq M\), the finite geometric-sum extrapolation would make it vanish at every non-negative exponent. However, at exponent zero, \(c_{P,j}^{(0)}=\sum _q\mu _{j,q}^0=r_j{\gt}0\), which is impossible. For the subsequence, apply the same argument to the nonzero bases \(\mu _{j,q}^p\).

Suppose the chosen sector decays against every sector of \(Q\), and separate all such \(P\)-sectors from the remaining ones. For each remaining sector, choose a \(Q\)-sector with non-decaying overlap. Both blocks are normal by the canonical-form hypotheses. Theorem 8.2.9 rules out its decay alternative and supplies equal bond dimensions and a gauge-phase relation with a unit phase. Gauge-phase covariance therefore expresses that \(P\)-sector MPV as a geometric scalar multiple of the chosen \(Q\)-sector MPV.

Substitute these relations into the eventual proportionality identity. The combined-family Gram criterion makes the decaying residue together with the \(Q\) family eventually linearly independent. Exact coefficient extraction then makes the chosen sector coefficient eventually zero, contradicting its power-sum nonvanishing. Thus a non-decaying partner exists. A second application of Theorem 8.2.9 rules out decay and gives the required dimension equality and gauge-phase equivalence; no modulus-one overlap limit is assumed.

Reverse the eventual proportionality and apply Theorem E.2.1.1 to each sector of \(Q\); symmetry of the overlap converts the resulting witness to the stated orientation.

Basis separation makes each partner map injective: two sectors with the same partner would be gauge-phase equivalent to one another. The injections in both directions force equal finite cardinalities, so the forward map is bijective.

Put \(L=L_0+1\) and \(S=3L\). For a block \(A\), write

Block injectivity at \(L\) and \(S\) makes both boundary maps injective; in particular,

Fix \(k\ne j\) and first order the pair so that \(D_j\leq D_k\). Suppose that a nonzero vector belongs to \(\mathcal G_S(A^k)\cap \mathcal G_S(A^j)\). There are boundary matrices \(X\) and \(Y\) such that

Injectivity at \(S\) gives \(X\ne 0\) and \(Y\ne 0\). Split every length-\(S\) word as \(uvw\) with \(|u|=|v|=|w|=L\). Relation (4) gives

Because the length-\(L\) words of \(A^k\) span \(M_{D_k}(\mathbb {C})\) and \(X\ne 0\), the two-sided ideal generated by \(X\) is all of \(M_{D_k}(\mathbb {C})\):

Indeed, expand arbitrary left and right matrix units in length-\(L\) words; a nonzero entry of \(X\) then produces every matrix unit. Thus, for each \(Z\in M_{D_k}(\mathbb {C})\), choose coefficients \(a_{uv}\) with \(Z=\sum _{u,v}a_{uv}A^k_uXA^k_v\) and set \(R(Z)=\sum _{u,v}a_{uv}A^j_uYA^j_v\). Cyclicity of the trace and (5), applied to the word \(vwu\), yield

Hence \(\Gamma _L^{A^k}(Z)=\Gamma _L^{A^j}(R(Z))\) for every \(Z\), and therefore

Equations (3) and (7) give

Together with \(D_j\leq D_k\), this forces \(D_j=D_k\) and equality in (7). This proves the unequal-dimension branch immediately: if \(D_j{\lt}D_k\), a nonzero intersection is impossible. Reversing the initial ordering treats \(D_k{\lt}D_j\) in the same way.

It remains to exclude the intersection when \(D_j=D_k\). Gauge-phase separation and Theorem 8.4.3 give, beyond any prescribed threshold, a chain length \(N{\gt}L\) for which the two periodic MPVs are not proportional. On the other hand, equality \(\mathcal G_L(A^k)=\mathcal G_L(A^j)\) gives identical translated length-\(L\) constraints on the \(N\)-site ring. The parent-Hamiltonian uniqueness mechanism identifies the corresponding chain ground spaces with the one-dimensional MPV spans:

Equality of the local spaces therefore forces these two MPV lines to be equal, contradicting the choice of \(N\). Consequently, for every distinct pair, including the unequal-dimensional pairs,

Finally, let \(\Phi _{kj}\) be the displayed length-\(S\) word-evaluation map from coefficient families to \(M_{D_k}(\mathbb {C})\oplus M_{D_j}(\mathbb {C})\). Pair the target by

A pair \((X,Y)\) annihilates \(\operatorname {im}(\Phi _{kj})\) precisely when

or equivalently \(\Gamma _S^{A^k}(X)=\Gamma _S^{A^j}(-Y)\). There is therefore a linear map

It is injective because both boundary maps are injective. Conversely, if \(\psi \) lies in the intersection, there are unique \(X\) and \(Z\) with \(\Gamma _S^{A^k}(X)=\psi =\Gamma _S^{A^j}(Z)\); the pair \((X,-Z)\) annihilates \(\operatorname {im}(\Phi _{kj})\) and maps to \(\psi \). This construction is the inverse of \(\Theta \), so the two spaces are isomorphic:

The right-hand side vanishes by (8), and hence so does the annihilator. The trace pairing is nondegenerate and all spaces are finite-dimensional, so \(\operatorname {im}(\Phi _{kj})^\perp =\{ 0\} \) implies that \(\Phi _{kj}\) is surjective. Surjectivity first gives an arbitrary value on one block and zero on an anchor block; the selector construction then chooses \((\mathbb {1},0)\) for the remaining pairs, and products of these homogeneous selectors give the simultaneous word-tuple span in Lemma 8.3.3.

For each block, Theorem 6.9.1 gives a positive length at which it becomes one-site injective; the product of these lengths is a common length. Injectivity persists under positive further blocking, and Theorem D.4.11 preserves the other hypotheses.

The collapsed decomposition counts one representative dimension for each assigned copy. Let \(e(j,q)\) denote the original CF summand assigned to the \(q\)th copy of representative \(P_j\). Since \((j,q)\mapsto e(j,q)\) reindexes the original summands and each assigned summand has the representative’s bond dimension,

This is the equality between the collapsed total dimension and the original direct-sum dimension.

Choose \(\zeta \neq 0\) with \(|V^{(N)}(Y)\rangle =\zeta ^N|V^{(N)}(X)\rangle \) for every positive length \(N\). The normalized self-overlaps give \(|O_{XX}(N)|\to 1\), and the phase relation gives

Since the same normalization forces \(|\zeta |=1\), this gives \(|O_{YX}(N)|\to 1\). If the two bond dimensions differed, the rectangular overlap-decay theorem would force \(O_{YX}(N)\to 0\), contradicting the preceding limit.

Apply Lemma E.3.4 to each representative and each assigned summand.

Every assigned summand shares the bond dimension of its representative, by Lemma E.3.5. The displayed regrouping identity in Lemma E.3.3 then identifies the collapsed sum over pairs \((j,q)\) with \(\sum _{k=1}^{r}D_k\), the original direct-sum dimension.

Take \(m=\max _k|\mu _k|\); the maximum over a nonempty finite family is attained, positive because every weight is nonzero, an upper bound for every modulus, and equal to the modulus of the attaining weight.

Expand the block-diagonal MPV coefficient as a finite sum over blocks and factor \(c^N\) out of the sum.