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.

Group the summands into the MPV phase classes of Definition 8.6.1.7, and choose one representative from each class. By Theorem 8.1.3, 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 E.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 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 every coefficient is nonzero. 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.

If the bond dimension were zero, the bond-matrix space and its endomorphism algebra would each contain only zero. The transfer operator would therefore have spectral radius zero, contradicting the spectral-radius-one clause in the definition of a normal tensor.

Positive-definite right and left Perron eigenvectors have the same positive eigenvalue by the trace pairing. The spectral-radius identity makes this eigenvalue equal to one. Thus the left Perron eigenvector is an adjoint fixed point, so its square-root gauge is trace-preserving. Similarity preserves irreducibility.

Pure Perron normalization gives a trace-preserving gauge by a pure similarity. Irreducibility and primitivity are invariant under this similarity.

Apply Theorem 8.2.3. 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 8.2.3 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 8.2.6 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 8.2.8, \(|\zeta |=1\). Hence

Apply Theorem 8.2.9. 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

Since both self-overlaps tend to one, (4) gives \(|c_N|^2\to 1\) and therefore \(|c_N|\to 1\). Substituting into (3) now gives \(|O_{AB}(N)|\to 1\).

Normality gives \(O_{AA}(N)\to 1\) and \(O_{BB}(N)\to 1\), so Theorem 8.2.11 gives \(|O_{AB}(N)|\to 1\). The first alternative in Theorem 8.2.10 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 8.2.3. 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.

Given a target tuple \((M_j)_j\), conjugate it backwards to \((X_j^{-1}M_jX_j)_j\). If the word tuples of \((A_j)_j\) span the product algebra, one common coefficient family \((c_\omega )_\omega \) expresses this conjugated tuple. Gauge covariance of word evaluation then gives

for every \(j\). Thus the \((B_j)_j\) word tuples span the product algebra. Replacing every \(X_j\) by \(X_j^{-1}\) proves the reverse implication.

Lemma E.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 8.3.3 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.

Put every representative in its trace-preserving Perron gauge. Since \(D_j\leq \sum _kD_k\leq K\), quantum Wielandt and homogeneous-span propagation make every representative block-injective at \(K^4\), \(K^4+1\), and \(3(K^4+1)\). Conjugation preserves the simultaneous word span.

For \(g=0\), take \(L=1\); the product algebra has only its zero element. For \(g=1\), take \(L=K^4\). If \(g\geq 2\), the three-block separation theorem gives

Positivity of the \(D_j\) gives \(g=\sum _j1\leq \sum _jD_j\leq K\), and hence \(L+1\leq 3(K-1)(K^4+1)+1\leq 3K^5\), because \((K-1)(K^4+1)=(K-1)K^4+K-1{\lt}K^5\). In the cases \(g=0\) and \(g=1\) the bound \(L+1\leq 3K^5\) holds since \(K\geq 1\).

In every phase class, choose its first copy. The class-copy equivalence sends these choices to distinct displayed block indices, and the first copy is the chosen representative. Thus the sum of the representative dimensions is bounded by the sum over all displayed blocks, which is at most \(D\) by the canonical-form dimension bound.

Choose one representative from every matrix-product-vector phase class. These representatives form a basis of normal tensors, and their total bond dimension is at most \(D\) by Theorem 8.3.6. Apply Theorem 8.3.5 with \(K=D\). A word of length \(L\) in \(A_j\) is a one-letter evaluation of \(A_j^{[L]}\), so the resulting length-\(L\) span is exactly the asserted one-letter span.

For large \(N\), Lemma E.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 5.4.4 and 5.5.5 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 E.1.7, both families are linearly independent for all sufficiently large \(N\). Choose \(N\) large enough that both families are linearly independent and the assumed span equality holds. The common span then 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 8.4.1 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 (7) gives \(|c_N|\to 1\), and hence (8) 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.

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 8.5.1 gives \(m=0\), contradicting \(m\geq 1\).

These identities follow directly from the definition.

Partition the finite block 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 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 7.6.18 now gives the basis-of-normal-tensors conditions.

Enumerate the block 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 (14). The BNT characterization proves that the representatives form a basis of normal tensors. Reindexing the dependent block coordinates gives \(\Phi \) and (15). 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 (17); substitution into (11) gives (18) with the original coisometry \(U\). In the notation of [ 13 , Proposition 4.13 ] , these identities are

Apply Theorem 8.1.4 first to the direct-sum tensor of the first presentation, using the representatives of the second as a BNT, and then with the presentations interchanged. This gives a matching map in each direction. Eventual linear independence excludes gauge-phase-equivalent distinct representatives, so both maps are injective. Since the index sets are finite, they have equal cardinality, and either matching map is a bijection. Normal-tensor gauge rigidity gives the asserted gauge-phase relation for each matched pair.

The class-copy equivalence reindexes the listed blocks. (23) 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 8.3.4 give eventual simultaneous word-tuple separation for the representatives. Reindexing the finite sum by \((j,q)\) gives (24). The original coisometric reconstruction and simultaneous block gauge preserve every positive-length closed-chain coefficient, yielding (25).

Physical relabeling preserves irreducibility, left-canonical normalization, normalized self-overlaps, eventual linear independence, and gauge-phase separation of the representatives. Since the weights and all sector data are unchanged, the canonical-form normalization is also preserved.

The identity

shows that \(V^{(N)}(A)\) is nonzero for all sufficiently large \(N\). A family consisting of one nonzero vector is linearly independent. All remaining conditions are immediate from the hypotheses and the unit weight.

Choose the pure trace-preserving Perron gauge. Its tensor is irreducible and left-canonical, and primitivity gives self-overlap converging to one. The preceding theorem supplies the BNT canonical form. If \(P\) denotes the total tensor of its single-sector decomposition, then gauge invariance and the sector-decomposition formula give, for every \(N{\gt}0\) and word \(\sigma \),

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 5.9.3 applies to each block.

The preceding corollary gives normality of every basis block. The sector coefficient formula (10) 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.

Positive blocking preserves normality of each representative. Fix a blocked length \(N\). If the \(m\)-th blocked letter is \(J_m=(i_{m,1},\ldots ,i_{m,p})\), define the configuration reindexing \(\tau _p\) by concatenation:

For every blocked configuration \(\sigma \) and every representative \(A_j\), the coefficient identities are

The spanning coefficients at length \(Np\) therefore give those at blocked length \(N\), and the same configuration reindexing preserves eventual linear independence.

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 E.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 8.3.3 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 8.6.1.26), and the inter-family decay hypothesis into the two-family overlap-orthonormality criterion (Lemma E.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 E.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 chosen representatives inherit irreducibility and trace preservation, while primitivity gives normalized self-overlap. Distinct phase classes give gauge-phase-distinct representatives. The collapsed phase-class construction supplies the eventual linear independence and retains every original block as one copy. Its phase multipliers have unit modulus, so the adjusted weights still have modulus at most one, and at least one has modulus one. Thus Definition 8.6.1.13 holds.

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 E.3.4 identifies the bond dimensions. If the resulting same-dimensional blocks were not gauge-phase equivalent, Theorem 8.4.3 would give a positive length at which their matrix product vectors are not proportional. This contradicts the assumed scalar-power relation. The normalized self-overlaps required there follow from Theorem C.5.2.

Theorem 8.6.3.4 gives an invertible \(G\) and a nonzero scalar \(\eta \) such that \(Y^i=\eta G X^iG^{-1}\). By Theorem 2.4.20, \(V^{(N)}(Y)_\sigma =\eta ^N V^{(N)}(X)_\sigma \). Theorem C.5.2 gives consecutive positive lengths \(M\) and \(M+1\) with nonzero coefficients of \(V^{(M)}(X)\) and \(V^{(M+1)}(X)\). Comparing the two scalar-power identities gives \(\zeta ^M=\eta ^M\) and \(\zeta ^{M+1}=\eta ^{M+1}\). Hence \(\zeta =\eta \), which proves (33).

Use the phase-class sector decomposition from Theorem 8.6.3.2. Its flattened copies are in bijection with the original blocks, and phase-equivalent blocks have equal bond dimension by Theorem 8.6.3.4. Summing the copy dimensions over this bijection gives (34).

Evaluate both sides on two flattened block coordinates. When their block labels agree, (35) gives the matrix entry. When the labels differ, both block-diagonal entries vanish. This proves (36).

Partition the prepared blocks into matrix-product-vector phase classes. For the copy \(q\) in class \(j\), retain the precise class scalar \(\zeta _{j,q}\) chosen in the weight \(\zeta _{j,q}\mu _{j,q}\). The normal-tensor phase theorem supplies an invertible \(G_{j,q}\) with

Thus the original weighted block equals the corresponding sector copy after conjugation, since its sector weight is \(\zeta _{j,q}\mu _{j,q}\). Assemble the \(G_{j,q}\) into a block-diagonal matrix \(X\) and apply Theorem 8.6.3.7. This gives (38); summing the matched copy dimensions gives (37).

Choose per-block gauge matrices via the hypothesis; then apply Lemma F.4.3.

Multiply the reconstruction identities around a positive-length cycle. Each adjacent pair \(UU^\dagger \) cancels, and cyclicity of the trace removes the remaining outer factors.

Apply Theorem 8.6.3.12. Its reconstruction writes \(A^i=U^\dagger B^iU\), where \(UU^\dagger =\mathbb {1}\) and \(B^i=XP^iX^{-1}\). Theorem 8.6.3.10 shows that \(A\) and \(B\) generate the same matrix product vectors at every positive length. Theorem 2.4.19 gives the same conclusion for \(B\) and \(P\). Transitivity yields \(\mathcal{V}^+(A)=\mathcal{V}^+(P)\), while the dimension identity is already part of the exact reconstruction theorem.

The canonical-form data give the exact reconstruction

Put every normal block into a trace-preserving primitive irreducible gauge \(Y\), and apply Theorem 8.6.3.8. Let \(Q\) be the permutation matrix of its flattened coordinate equivalence, so \(QQ^\dagger =\mathbb {1}\). Reindexing \(Y\) by the same equivalence gives an invertible matrix \(W\) on the bond space of \(P\) and the identities

If \(X_0\) is the prepared-block gauge, set \(X=W^{-1}X_0\) and \(U=QV\). Substitution in (44) gives (43), while

This proves (42). The block enumeration gives (41).

Apply Theorem 8.6.3.12 to obtain BNT canonical forms \(P,Q\), coisometries \(U_A,U_B\), and retained-space gauges \(X_A,X_B\) with

Theorem 8.6.3.10, gauge invariance, and the hypothesis give

Theorem 9.4.2.1 therefore identifies the retained dimensions and gives \(Y\) such that \(Q^i=YP^iY^{-1}\). After identifying the retained spaces, the two middle tensors in (49) are conjugate by \(X_BYX_A^{-1}\). Theorem 7.8.5 aligns the two coisometries inside the common ambient space and yields (48).

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 8.6.3.1 is nonempty.

Normalize its weights by their maximal modulus \(m\) (Theorem E.3.7). By the weight-scaling identity of Lemma E.3.8,

where the last identity is the prepared supplier of Theorem 8.6.3.3 applied to the normalized weights, which gives the BNT canonical form \(P\).

Expand both sides of the coefficient identity (52) 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 8.5.1), 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 8.7.1.1 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 E.2.1.1 to the chosen sector.

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

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