At every sufficiently large length the matrix product vectors of the basis members are linearly independent. A gauge-phase equivalence between two distinct members would give \(|V^{(N)}(A_k)\rangle =\zeta ^N|V^{(N)}(A_j)\rangle \) for a nonzero scalar \(\zeta \). Thus one vector would be a scalar multiple of the other, contradicting linear independence.

First discard the zero-weight summands; they contribute zero at every positive length. Reindex the remaining summands, group them into the MPV phase classes of Definition 10.6.1.7, and choose one representative from each class. By Theorem 10.1.3, eventual linear independence of a BNT excludes gauge-phase equivalence between two distinct \(A_j\).

Suppose that some phase-class representative is not gauge-phase equivalent to any \(A_j\). Partition all phase classes into the unmatched classes \(T\) and the remaining classes. Every remaining representative is a nonzero scalar multiple of some \(|V^{(N)}(A_j)\rangle \) at each positive length. Distinct unmatched representatives have vanishing mutual overlaps, and their overlaps with every \(A_j\) also vanish. Explicitly, for \(t\ne t'\) in \(T\) and every \(j\),

By Lemma F.1.7, the combined family

is linearly independent for all sufficiently large \(N\). Comparing the canonical-form expansion with the BNT expansion gives a relation

Linear independence forces every \(c_{N,t}\) to vanish. Hence the coefficient of every unmatched phase class vanishes for all sufficiently large \(N\). For an unmatched class \(t\), let \(q\) range over its active canonical-form summands, write \(\mu _q\) for the corresponding canonical-form weight, and write \(\zeta _q\) for the phase relating that summand to the chosen representative. Then \(c_{N,t}=\sum _{q\in t}(\zeta _q\mu _q)^N\). Every base \(\zeta _q\mu _q\) is nonzero because only active summands occur. Eventual vanishing of this finite geometric sum would imply vanishing at \(N=0\), where its value is the positive number of summands in the class. This contradiction proves coverage. The normal-tensor phase theorem turns the resulting MPV phase relation into the stated gauge-phase relation.

Conversely, coverage gives, for each canonical-form summand, \(|V^{(N)}(\widetilde A_k)\rangle =\zeta _k^N|V^{(N)}(A_{j(k)})\rangle \). Hence

This is the required BNT span at every positive length. Pairwise gauge-phase distinction and normality give eventual linear independence, completing the converse.

Take physical dimension two and bond dimension one. Let \(A_0\) be supported on physical symbol zero and let \(A_1\) be supported on physical symbol one:

Both transfer maps are the identity on the one-dimensional matrix algebra: \(\mathcal{E}_{A_j}(X)=X\) for \(j=0,1\). Hence both tensors are normal. At every positive length,

Thus \((A_0)\) and \((A_0,A_1)\) are both bases of normal tensors for \(A_0\). Indeed, for every \(N\geq 1\), the second expansion is \(|V^{(N)}(A_0)\rangle =1\cdot |V^{(N)}(A_0)\rangle +0\cdot |V^{(N)}(A_1)\rangle \). Their index sets have cardinalities one and two, so they cannot be related by a bijection.

For each \(A_j\), choose an active canonical-form block \(\widetilde A_{\ell (j)}\) representing it, and then choose a member \(B_{f(j)}\) representing that block. If \(f(j)=f(j')\), then

Minimality of the first basis gives \(j=j'\), so \(f\) is injective. Reversing the roles of the two bases gives an injection in the other direction. Since both index sets are finite, \(f\) is bijective; set \(e=f\). Write the two relations through the chosen active block as

Solving the second identity for \(B_{e(j)}^i\) and using the first gives

Denote the scalar and gauge in this identity by \(\zeta _j\) and \(X_j\). The gauge-phase relation gives

Normality gives \(O_{A_jA_j}(N)\to 1\) and \(O_{B_{e(j)}B_{e(j)}}(N)\to 1\); hence the last identity forces \(|\zeta _j|=1\).

The irreducible Perron–Frobenius theorem gives positive definite right and left eigenvectors with positive eigenvalues \(r\) and \(t\). The spectral normalization gives \(r=1\). Adjointness with respect to the trace pairing gives \(r\, \operatorname{Tr}(\sigma \rho )=t\, \operatorname{Tr}(\sigma \rho )\). Since \(\sigma \) and \(\rho \) are positive definite, their pairing is nonzero, hence \(t=1\). Thus the Perron gauge is a pure similarity. Irreducibility and primitivity are invariant under this similarity.

Apply Theorem 10.2.1. The trace-preserving, primitive, irreducible representative is eventually block-injective, and gauge invariance transports this conclusion back to the original tensor.

Every block has positive bond dimension, so spectral normality implies eventual block injectivity for each block. The spanning identities and eventual linear independence are unchanged.

If the bond dimensions differed, the rectangular overlap-decay theorem would give \(O_{AB}(N)\to 0\), and hence \(|O_{AB}(N)|\to 0\), contradicting the assumed limit.

If every letter vanished, then the transfer map would vanish and its spectral radius would be zero, contradicting the spectral-radius-one normalization of a CPSV normal tensor.

Apply Theorem 10.2.1 to both tensors. Gauge invariance of matrix product vectors preserves the phase relation, whose self-overlap identities imply unit modulus for the phase. The resulting rectangular cross-overlap has norm tending to one. If the bond dimensions differed, Theorem 10.2.4 would give a contradiction. Thus the bond dimensions agree, and the unit-overlap rigidity theorem gives a gauge-phase equivalence between the trace-preserving representatives. Conjugating by the two Perron gauges gives the asserted relation for \(A\) and \(B\).

Put both tensors in their pure trace-preserving gauges. If their bond dimensions differ, rectangular overlap decay gives \(O_{AB}(N)\to 0\). If their dimensions agree but they are not gauge-phase equivalent, the equal-dimension overlap-decay theorem gives the same conclusion. In the remaining case, \(B^i=\zeta X A^iX^{-1}\). By Theorem 10.2.6, \(|\zeta |=1\). Hence

Apply Theorem 10.2.7. In its second alternative, gauge-phase covariance gives \(V^{(N)}(B)_\sigma =\zeta ^N V^{(N)}(A)_\sigma \) for every configuration \(\sigma \), and hence the asserted equality of vectors.

The proportionality relation gives

Hence \(|c_N|^2\to 1\) and therefore \(|c_N|\to 1\). The first identity now gives \(|O_{AB}(N)|\to 1\).

Normality gives \(O_{AA}(N)\to 1\) and \(O_{BB}(N)\to 1\), so Theorem 10.2.9 gives \(|O_{AB}(N)|\to 1\). The first alternative in Theorem 10.2.8 gives \(O_{AB}(N)\to 0\), hence \(|O_{AB}(N)|\to 0\), a contradiction. In the remaining alternative, \(|V^{(N)}(B)\rangle =\zeta ^N|V^{(N)}(A)\rangle \) with \(|\zeta |=1\); take \(a=\zeta ^{-1}\).

Put every tensor in the pure trace-preserving gauge of Theorem 10.2.1. Distinct gauge-phase classes have overlaps tending to zero, while each self-overlap tends to one. The finite Gram matrix therefore tends to the identity and is invertible for all sufficiently large lengths. Gauge invariance of matrix product vectors gives the conclusion for the original tensors.

Lemma F.3.1 gives homogeneous pair-trace separation at \(S=3(L_0+1)\). For a chosen anchor block, pair separation prescribes an arbitrary matrix there and zero on one other block. Multiplying \(g-1\) such homogeneous combinations removes the other blocks one at a time. Induction on the number of removed blocks therefore realizes every tuple supported on the anchor at length \((g-1)S\). Summing over the anchors gives the full product algebra at \(3(g-1)(L_0+1)\).

Put each BNT representative in its trace-preserving Perron gauge. The gauged family is irreducible and left-canonical, has normalized self-overlaps, and remains gauge-phase separated. Choose a common positive length \(p\) at which every gauged block is block-injective. Propagation of homogeneous word spans gives block injectivity at \(p+1\) and \(3(p+1)\).

Lemma 10.3.2 gives pairwise separation at length \(S=3(p+1)\). Write \(g\mathbin {\dot-}1=\max \{ g-1,0\} \). Multiplying one separator for each of the other blocks gives a selector of length \((g\mathbin {\dot-}1)S\) for every block. Transport these selectors back from the Perron gauges. Gauge invariance also gives block injectivity of every original block at length \(p\). Concatenating the injective prefix with the selectors yields the full product-algebra span at

The truncated factor makes the same construction valid for empty and singleton families. The same span provides selectors and block injectivity at length \(L\). For every sufficiently large \(n\), concatenate an \((n-L)\)-letter injective prefix with these selectors to obtain the full product-algebra span at length \(n\). Finally, length-\(L\) words of \(A_j\) are the one-letter evaluations of \(A_j^{[L]}\), which proves the blocked statement.

For large \(N\), Lemma F.2.2 gives an invertible matrix \(U_N\) relating the two MPV families. Fix \(j\) and write \(|V^{(N)}(B_j)\rangle =\sum _i u_{ij}^{(N)}|V^{(N)}(A_i)\rangle \). With \(G_A(N)=(O_{A_iA_\ell }(N))_{i,\ell }\) and \(b_j(N)=(O_{A_iB_j}(N))_i\), this gives

Moreover,

If all mixed overlaps \(O_{A_iB_j}(N)\) tended to \(0\), then \(b_j(N)\to 0\); since \(G_A(N)\to \mathbb {1}\), one obtains \(\overline{u_j(N)}\to 0\) and hence \(u_j(N)\to 0\). Thus \(O_{B_jB_j}(N)\to 0\), contradicting \(O_{B_jB_j}(N)\to 1\). Thus each \(B_j\) has a matched block \(A_{f(j)}\) with non-decaying mixed overlap. Theorems 7.5.3 and 7.6.3 then force equal bond dimension and gauge-phase equivalence for each match. Off-diagonal decay within the \(B\)-family makes \(f\) injective, hence bijective, and its inverse is the required permutation.

By Lemma F.1.10, both families are linearly independent for all sufficiently large \(N\). At such an \(N\), the common span has the same dimension when computed from the \(A\)-family or the \(B\)-family, so \(g_A = g_B\). After identifying the two index sets, Theorem 10.4.1 yields the permutation conclusion.

Suppose that for all sufficiently large \(N\) one had \(|V^{(N)}(A)\rangle =c_N|V^{(N)}(B)\rangle \). Then

Since \(O_{AA}(N)\to 1\) and \(O_{BB}(N)\to 1\), the identity (??) gives \(|c_N|\to 1\), and hence (??) gives \(|O_{AB}(N)|\to 1\). The irreducible trace-preserving overlap decay theorem gives \(O_{AB}(N)\to 0\) when \(A\not\sim _{\mathrm{gp}}B\), a contradiction.

Let \(e_m\) denote the \(m\)th elementary symmetric polynomial in the eigenvalues, with \(e_0 = 1\). The Newton–Girard recursion

expresses the coefficients of the characteristic polynomial in terms of the power-sum traces \(\operatorname{tr}(A^i)\). Equal power sums for \(A\) and \(B\) therefore give equal coefficients, hence equal characteristic polynomials.

Apply the finite-range Newton–Girard theorem to \(\operatorname{diag}(\alpha )\) and \(\operatorname{diag}(\beta )\) and extract the roots of \(\prod _i (X - \alpha _i)\) and \(\prod _i (X - \beta _i)\). For unequal cardinalities, pad both lists with zeros to length \(\max (m,n)\); the nonzero hypothesis makes the padded-zero counts \(\max (m,n)-m\) and \(\max (m,n)-n\), forcing \(m=n\). Without that hypothesis, delete the zero entries first (unchanged for positive \(N\)); the equality at \(N=0\) gives \(x_a=x_b\).

Suppose that every power sum with \(1\leq N\leq m\) vanishes. Comparing \(\lambda _0,\ldots ,\lambda _{m-1}\) with the empty family in Theorem 10.5.2 gives \(m=0\), contradicting \(m\geq 1\).

These identities follow directly from the definition.

Discard the zero-weight blocks and partition the remaining finite family by matrix-product-vector phase equivalence. Choose one representative from each class. The representatives are normal and have positive bond dimensions. Distinct representatives are not gauge-phase equivalent. For every active block, the normal-tensor phase theorem identifies its bond dimension with that of its representative and realizes their phase relation by a similarity and a unit-modulus scalar. Theorem 9.6.7 now gives the basis-of-normal-tensors conditions.

Enumerate the active indices by their phase classes and choose one normal representative in each class. The normal-tensor phase theorem gives the dimension equality, unit phase, and invertible gauge in (??). The BNT characterization proves that the representatives form a basis of normal tensors. Adjoin the inactive complement to the class-copy order. Reindexing the dependent block coordinates gives \(\Phi \) and (??). The exact heterogeneous block-diagonal reindexing identity identifies the grouped tensor with the representative copies in the original retained-coordinate space. Block-diagonal conjugation then gives (??); substitution into (??) gives (??) with the original coisometry \(U\).

The active class-copy equivalence reindexes the nonzero listed blocks. (??) combines the original coefficient with the unit phase relating its block to \(C_j\); both factors are nonzero. The basis-of-normal-tensors condition and Theorem 10.3.3 give eventual simultaneous word-tuple separation for the representatives. At positive length, every inactive contribution contains a positive power of its zero coefficient and vanishes. Reindexing the remaining finite sum by \((j,q)\) gives (??).

This is the unit-modulus existential in the canonical-form normalization.

Each basis block has positive bond dimension, is irreducible and left-canonical, and has normalized self-overlap converging to \(1\). Theorem 7.10.3 applies to each block.

The preceding corollary gives normality of every basis block. The sector coefficient formula (??) expresses the total MPV as a linear combination of the basis MPVs at each positive length, and the BNT sector data give their eventual linear independence.

For each representative \(A_j\), normalized self-overlap gives peripheral period one, hence a primitive transfer map. Positive blocking therefore preserves irreducibility and left-canonical normalization. Moreover, \(O_{A_j^{[p]}A_j^{[p]}}(N)=O_{A_jA_j}(Np)\longrightarrow 1\). Let \(F_N\) be the linear isomorphism induced by concatenating blocked words, from the blocked configuration space at length \(N\) to the original one at length \(Np\). For every representative, \(F_N|V^{(N)}(A_j^{[p]})\rangle =|V^{(Np)}(A_j)\rangle \). Hence eventual linear independence is preserved. If two blocked representatives were gauge-phase equivalent, then for some nonzero \(\zeta \) and every \(N\), \(|V^{(N)}(A_k^{[p]})\rangle =\zeta ^N|V^{(N)}(A_j^{[p]})\rangle \). Writing these two vectors as \(f_k\) and \(f_j\) gives \(1\cdot f_k=\zeta ^N\cdot f_j\) in an eventually linearly independent family, and therefore \(k=j\). Finally, \(|\mu _{j,q}^p|=|\mu _{j,q}|^p\leq 1\), and a unit-modulus weight remains of unit modulus after taking its \(p\)-th power.

The BNT hypotheses make \(A_j\) irreducible and left-canonical, with \(O_{A_jA_j}(N)\to 1\). The preceding theorem therefore makes \(A_j\) normal, and quantum Wielandt gives \(S_{D_j^4}(A_j)=M_{D_j}(\mathbb {C})\). By Theorem F.2.6, \(D_j^4\leq D^4\). Write \(D^4=D_j^4+k\) and apply positive-length propagation \(k\) times.

Every representative is block-injective at \(D^4\), and homogeneous-span propagation gives the same conclusion at \(D^4+1\) and \(3(D^4+1)\). If \(g=1\), take \(N=D^4\). If \(g\geq 2\), apply Lemma 10.3.2 with \(L_0=D^4\) and take

The positive bond dimension of every representative gives \(g\leq D\). Consequently \(N\leq 3D^5\); the same estimate also covers the one-block case. The unit-modulus weight required by the BNT normalization ensures \(D{\gt}0\).

The claim follows from the definition of the sector coefficient.

Case on the bond dimensions. If they differ, the dimension-mismatch overlap-decay theorem applies. If they agree, the basis-distinctness clause rules out gauge-phase equivalence, so the equal-dimension overlap-decay theorem applies.

Feed the per-family normalized self-overlaps, the within-family cross-overlap decay (Lemma 10.6.1.19), and the inter-family decay hypothesis into the two-family overlap-orthonormality criterion (Lemma F.1.9).

Apply the common primitive irreducible block decomposition to \((A,A)\), giving a positive blocking length and a nonzero weighted family of trace-preserving primitive irreducible blocks with the positive-length MPV identity. Theorem F.3.2 supplies one common further blocking making every block one-site injective while preserving the other hypotheses; finally identify the iterated blocked alphabet with the direct one.

The phase-class quotient groups gauge-phase-equivalent blocks into sectors with unit-modulus per-copy scalars. Under the TP, primitive, and irreducible hypotheses the phase-class construction applies, and for distinct representatives \(B_j\), \(B_{j'}\) (\(j \neq j'\)) the cross-overlap decays,

which gives the eventual linear-independence condition with no injectivity required. The resulting sector decomposition inherits the span and multiplicity conditions, and the bounded-weight and global-unit hypotheses give the weight-norm conditions.

Lemma F.3.4 identifies the bond dimensions. If no gauge-phase relation existed after this identification, Theorem 10.4.3 would give a positive length \(N\) at which the two MPV states are not proportional. This contradicts the assumed scalar-power identity at every positive length.

Choose a word \(w\) with \(\operatorname{tr}(A^w)\neq 0\). For every \(p\geq 1\), there is a \(q\geq 1\) such that \(\operatorname{tr}((A^w)^{pq})\neq 0\). Otherwise Newton–Girard applied to \((A^w)^p\) and the zero matrix would show that \((A^w)^p\), and hence \(A^w\), is nilpotent, contradicting \(\operatorname{tr}(A^w)\neq 0\). Repeating \(w\) exactly \(pq\) times gives a nonzero coefficient of the \(p\)-blocked tensor. Thus the prepared family at the blocking length supplied by Theorem 10.6.3.1 is nonempty.

Normalize its weights by their maximal modulus \(m\) (Theorem F.3.8). By the weight-scaling identity of Lemma F.3.9,

where the last identity is the prepared supplier of Theorem 10.6.3.2 applied to the normalized weights, which gives the BNT canonical form \(P\).

Expand both sides of the coefficient identity (??) into per-copy power sums \(\sum _q \mu _{j_0, q}^N\) and \(\sum _{q'} \nu _{\tilde k_0, q'}^N\), and distribute \(\zeta ^N\) through the right-hand sum to obtain, for every \(N {\gt} N_0\),

Extend to all positive exponents and apply the unequal-cardinality multiset-recovery lemma (Theorem 10.5.2), using \(\zeta \neq 0\) together with the weight non-vanishing assumption to ensure the input weights are nonzero. This yields the multiplicity match and the multiset equality.

Invoke Lemma 10.7.1.1 to obtain the cardinality match and the multiset equality. From this multiset-map equality extract a permutation matching the elements pointwise (proved by induction on the cardinality). Compose with the cardinality identification and cancel \(\zeta \) via \(\zeta \neq 0\) to convert the gauge factor on the right of the pointwise identity to \(\zeta ^{-1}\) on the left.

Equality at every positive length gives eventual nonzero proportionality with scalar \(1\). Apply Theorem F.2.1.1 to the chosen sector.

Convert equality to eventual nonzero proportionality with scalar \(1\) and apply Theorem F.2.1.2.

Apply the full-basis matcher in both directions, using symmetry of the same-MPV relation. Lemma F.2.1.3 converts these two families of matches into the required bijection.