Both sides are the span of the same family of vectors.

The definition by length-\(L\) windows on the ring reduces the inclusion to the local block-embedding statement that each ground vector of \(A_j\) on the window lifts to the direct sum by the identity

Here \(\iota _j\) inserts a matrix in the \(j\)-th diagonal block and has zero off that block. Thus \(\iota _j(\mu _j^{-L}X)\) has exactly the local vector of the \(j\)-th block on that window.

Take the linear sum of the inclusions \(\mathcal G_{N,L}(A_j) \subseteq \mathcal G_{N,L}(\bigoplus _k \mu _k A_k)\) over all block indices \(j\).

Since \(\Delta _A\neq 0\), Lemma C.1.5 writes \(Z\) as a linear combination of matrices \(A_u\Delta _A A_v\). For one such term, cyclicity of the trace rewrites the three-block relation as

Taking the same linear combination of the matrices \(B_u\Delta _BB_v\) gives the required \(W\), and linearity gives the general case.

Write a vector in \(\mathcal G_L^A\) as \(t\mapsto \operatorname{tr}(A_tZ)\). The preceding lemma supplies \(W\) with \(\operatorname{tr}(ZA_t)+\operatorname{tr}(WB_t)=0\) for every word \(t\). Cyclicity of the trace gives \(\operatorname{tr}(A_tZ) = \operatorname{tr}(ZA_t) = -\operatorname{tr}(WB_t) = \operatorname{tr}(B_t(-W))\), so the vector lies in \(\mathcal G_L^B\). The equality statement follows by applying the same argument with the two blocks exchanged.

Block injectivity makes the boundary-to-chain map injective, so \(\dim \mathcal G_L^A=D_A^2\) and \(\dim \mathcal G_L^B=D_B^2\). Inclusion gives \(D_A^2\le D_B^2\), while \(D_B\le D_A\) gives the reverse inequality. Hence \(D_A=D_B\), and equality of dimensions inside the inclusion gives \(\mathcal G_L^A=\mathcal G_L^B\). In the strict-size branch this contradicts \(D_B{\lt}D_A\). For the conditional directness statement, suppose \(\psi \in \mathcal G_{3L}^A\cap \mathcal G_{3L}^B\) has boundary representations \(\psi = \Gamma _{3L}^A(X) = \Gamma _{3L}^B(Y)\). If \(X=0\), injectivity of the \(A\) boundary-to-chain map gives \(\psi =0\). Otherwise equality of the coefficient formulas gives \(\forall |t|=3L,\qquad \operatorname{tr}(XA^t)-\operatorname{tr}(YB^t) =0\), equivalently, \(\forall |t|=3L,\qquad \operatorname{tr}(XA^t)+\operatorname{tr}((-Y)B^t) =0\). The dimension step applied to this homogeneous three-block trace relation gives \(D_A^2\le D_B^2\), contradicting \(D_B{\lt}D_A\) in the strict-size branch. In the equal-size branch, equal local spaces impose the same length-\(L\) constraints on the \(N\)-site ring:

The injective uniqueness theorem identifies these periodic ground spaces with the one-dimensional MPS spans,

and hence \(\mathbb {C}\, V^{(N)}(A)=\mathbb {C}\, V^{(N)}(B)\), contradicting non-proportionality of the two MPS vectors.

Write

The equality \(W_\infty (A,B)=M_{D_A}(\mathbb {C})\times M_{D_B}(\mathbb {C})\) holds if and only if

Since the product algebra is finite-dimensional and

there is an index \(S_0\) with \(W_{S_0}(A,B) = W_\infty (A,B) = M_{D_A}(\mathbb {C})\times M_{D_B}(\mathbb {C})\). Applying the finite trace-duality criterion at \(S_0\) gives

For a finite family of ordered block pairs, choose such an \(S_{kj}\) for each pair and replace them by \(S=\max _{k\ne j}S_{kj}\). Monotonicity of \(W_S\) gives the same finite cumulative trace separation for every pair. This is the cumulative span at all lengths at most \(S\), not a single homogeneous length, since the length-zero word already contributes the identity to the cumulative span.

Projecting the pair generators to one coordinate gives exactly the one-block generator family. Injectivity says that the linear span of this family is the full matrix algebra, and the linear span is contained in the unital algebra it generates. The pair-word membership follows by induction on the word, using closure of the generated algebra under multiplication. Conversely, the all-word linear span is closed under componentwise multiplication, so it contains the algebra generated by the pair generators. In the graph case, each generator satisfies \(B_i=\varphi (A_i)\); applying Skolem–Noether to \(\varphi \) gives an ordinary gauge, hence a gauge-phase equivalence with scalar \(1\).

The length-zero word gives the identity only at length zero, so it is not a positive-length padding theorem. The useful padding facts are homogeneous: if the identity pair is in the span at lengths \(m\) and \(n\), componentwise multiplication puts it in the span at length \(m+n\). A period witness and a complete residue window therefore give homogeneous padding at every sufficiently large length, which is the input needed by the all-length-to-fixed-length implication.

If the pair span were proper, a nonzero linear functional would vanish on it. Nondegeneracy of the matrix trace pairing represents that functional as \((\Delta _A,\Delta _B)\), contradicting pair trace separation. Once the pair span is the whole product algebra, the tuple \((I,0)\) is in the span. Using the same coefficients on the ambient block family gives the required pairwise block-separating equation.

For the base of induction over the target blocks, the empty word gives the identity tuple on the empty target set. This does not represent a physical chain of length zero. The span of word tuples is closed under pointwise multiplication because the product of a length-\(L\) word and a length-\(S\) word is the concatenated length-\((L+S)\) word. Inducting over the finite set of blocks different from \(k\) and multiplying the corresponding pairwise equations keeps the value \(I\) on \(k\) and introduces one additional zero block at each step. Extracting coefficients from the final tuple-span membership gives the coefficient form of the displayed block-separating equation.

Let \(M=(M_j)_j\) be a target tuple. Write

simultaneously for every block \(j\). Multiplying the two expansions gives

The words \(uv\) have length \(L+S\), so the target tuple lies in the length-\(L+S\) span.

The Euclidean division of \(n-s\) by \(p\) gives \(n=s+r+qp\) with \(0\le r{\lt}p\). The assumed residue-window span gives the full product algebra at length \(s+r\). Applying homogeneous padding with the length-\(p\) span once gives the full product algebra at length \(s+r+p\); iterating gives the full product algebra at length \(s+r+qp\).

For a target tuple \((M_j)_{j\in J}\), block injectivity gives coefficients \(a_u^{(k)}\) such that \(M_k=\sum _{|u|=L}a_u^{(k)}A_k^u\). Multiply this expansion by the selector equation for \(k\) and sum over \(k\). For every block \(j\) this gives

Each product \(A_j^uA_j^v\) is the evaluation of the concatenated word \(uv\), which has length \(L+S\). Hence the length-\((L+S)\) word tuples span the full product algebra.

Full pair span gives coefficients satisfying \((X,0) = \sum _{|w|=S}c_w(A^k_w,A^j_w)\). Define \(M_\ell =\sum _{|w|=S}c_wA^\ell _w\) for every block \(\ell \). Then \(M\) lies in the simultaneous word-tuple span, and its \(k\)- and \(j\)-components are \(X\) and \(0\), respectively.

For every word \(w\) of length \(m\),

Nondegeneracy of the product trace pairing, together with the spanning hypothesis, gives \(X_j-Y_j=0\) for every block \(j\).

Write \(\psi =\sum _j\phi _j\) with \(\phi _j\in G_{m+2}(A^j)\). For each \(j\), choose \(E_j\) whose image under the length-\(m+2\) ground-space map for \(A^j\) is \(\phi _j\). Comparing the coefficient of the word \(awb\) in the two expressions for \(\psi \) gives

Lemma 13.2.1.2 gives the displayed blockwise identities.

For fixed \(a,b\), set \(\Delta _j=A^j_bC^j_a-D^j_bA^j_a\). The assumed trace equality says that \(\sum _j \operatorname{tr}\! \left(\Delta _jA^j_w\right) = 0\) for every word \(w\) of length \(m\). Since the tuples \((A^1_w,\ldots ,A^g_w)\) span the product algebra, the product trace pairing is nondegenerate and hence every \(\Delta _j\) is zero.

Fix \(\beta \) and \(\rho \), and set \(\Delta _j=A^j_\beta C^j_\rho -D^j_\beta A^j_\rho \). The trace equality says that \(\sum _j\operatorname{tr}(\Delta _jA^j_w) = 0\) for every word \(w\) of length \(m\). The same nondegenerate product trace-pairing argument as in Lemma 13.2.1.4 gives \(\Delta _j=0\) for every \(j\). The specialization \(D^j_\beta =X_jA^j_\beta \) is immediate.

Fix \(\beta \) and set \(\Delta _j=A^j_\beta C^j_\rho -(X_jA^j_\beta )A^j_\rho \). The trace equality gives \(\sum _j\operatorname{tr}(\Delta _jA^j_w) = 0\) for every word \(w\) of length \(m\). The common word-span property and the product trace pairing imply \(\Delta _j=0\) for every \(j\).

Multiply \(D_b\) by the normalization:

Substituting this identity into \(A_bC_a=D_bA_a\) gives \(A_bC_a=A_bEA_a\). For two alphabets, the corresponding step is

where \(E=\sum _a C_aB_a^\dagger \). Substituting this identity into \(A_bC_a=D_bB_a\) gives \(A_bC_a=A_bEB_a\).

Lemma 13.2.1.5 gives \(A^j_\beta C^j_\rho = D^j_\beta A^j_\rho \). The single-letter normalization gives the word normalization \(\sum _\rho A^j_\rho \left(A^j_\rho \right)^\dagger = \mathbb {1}\). Lemma 13.2.1.7, applied to the two word alphabets, gives

Apply Lemma 13.2.1.7 with \(D_\beta =X A_\beta \). It gives \(A_\beta C_\rho = A_\beta E A_\rho \) for \(E=\sum _\gamma C_\gamma A_\gamma ^\dagger \). With \(E_\rho =EA_\rho \), the compatibility hypothesis gives

Lemma 13.2.1.5, applied with \(D^j_\beta =X_jA^j_\beta \), gives \(A^j_\beta C^j_\rho = (X_jA^j_\beta )A^j_\rho \). The hypothesis \(\sum _a A^j_a A^{j\, \dagger }_a=\mathbb {1}\) extends to words of length \(M\): \(\sum _\rho A^j_\rho \left(A^j_\rho \right)^\dagger = \mathbb {1}\). Lemma 13.2.1.9 gives the matrices \(E_{j,\rho }\).

The boundary identity gives

Hence, by cyclicity of the trace,

which is the ground-space parametrization of \(E\).

Fix \(j\). By the assumed boundary identity and cyclicity of the trace,

Thus \(\alpha _j\) is the image of \(E^j\) under the parametrization of \(G_{n+2}(A^j)\). Hence each \(\alpha _j\) lies in \(G_{n+2}(A^j)\), and the finite sum lies in the supremum of these subspaces.

The equality of the two boundary trace expressions and the common word-span hypothesis give \(A^j_bC^j_a = D^j_bA^j_a\) for every \(j,a,b\). By the normalization,

Hence \(A^j_bC^j_a=A^j_bE^jA^j_a\). Lemma 13.2.1.12 then gives \(\psi = \sum _j\alpha _j \in \bigvee _j G_{n+2}(A^j)\).

Choose finite decompositions of the fixed-boundary vectors:

For every middle word \(w\),

and the second decomposition gives

Hence

For an arbitrary word \((i_1,\ldots ,i_{n+2})\), the first decomposition gives

by cyclicity of the trace.

Choose finite decompositions of the fixed-boundary vectors:

Evaluating these two decompositions on \((a,w,b)\), where \(w\) has length \(n\), gives

The first decomposition also gives

Lemma 13.2.1.13 therefore gives \(\psi \in \bigvee _jG_{n+2}(A^j)\).

The reverse implication is Lemma 13.2.1.15. For the forward implication, write

Each \(\gamma _j\) has fixed-boundary restrictions in \(G_{n+1}(A^j)\):

Taking the finite sum over \(j\) gives the two restrictions of \(\psi \) in \(\bigvee _jG_{n+1}(A^j)\).

Lemma 13.2.1.16 gives the displayed equivalence for each vector \(\psi \). Interpreting the two sides of that equivalence as subspaces gives the stated equality.

Lemma 13.2.12 gives a length \(N\) such that the simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\) at every length \(n\ge N\). For each such \(n\), Lemma 13.2.1.17 applies and gives the displayed subspace equality.

Lemma 8.3.3 gives \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w): |w|=n\} = \prod _jM_{D_j}(\mathbb {C})\). Substituting this span into Lemma 13.2.1.17, together with the normalization, gives the stated subspace equality.

The direct-sum dimension step gives, for each ordered pair of distinct blocks \(k\ne j\), coefficients \(c^{k,j}_\omega \) of length \(S\) such that

Multiplying these equations over the \(r-1\) blocks \(j\ne k\) gives coefficients \(c^{(k)}_\omega \) of total length \(q\) satisfying

Combining this block-separating equation with block injectivity at length \(p\) gives \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=p+q\} = \prod _j M_{D_j}(\mathbb {C})\). Combining the same separating equations with the assumed block-injectivity window gives the full product span at lengths \(s_0+q+s\), for \(0\le s{\lt}p+q\). The period-window form of Lemma 13.2.2.1 then gives the displayed equality for all sufficiently large \(n\).

Given matrices \(M_j\), first represent the tuple \(((A^j_a)^\dagger M_j)_j\) by words of length \(q\). Prefixing the letter \(a\) and summing over \(a\) gives \(\left(\sum _aA^j_aA^{j\, \dagger }_aM_j\right)_j =(M_j)_j\). This proves the step from \(q\) to \(q+1\); induction gives every \(n\ge q\).

By Theorem 6.8.1, the normalization propagates the length-\(L\) span equality to every larger homogeneous length:

For \(n\ge L+(r-1)S\), take \(m=n-(r-1)S\). Combining the length-\(m\) block span with the \(r-1\) block-separating equations gives the displayed product span at length \(n\). In the BNT case, the injectivity of \(\Gamma _{L_0}^{A^j}\) gives the full span at length \(L_0\), and the normalization propagates it to the lengths \(L_0+1\) and \(3(L_0+1)\). Lemma 8.3.3 then supplies the block-separating equations with \(S=3(L_0+1)\).

Tensor irreducibility is the absence of a nontrivial invariant matrix projection, which is precisely the corresponding irreducibility condition for the transfer map.

The square-root gauge conjugates the transfer map by an invertible similarity. Invariant faces therefore correspond bijectively before and after gauging.

If \(X,Y,Z\) are the left gauge, gauge-phase intertwiner, and right gauge, respectively, then \(Z^{-1}YX\) is the required intertwiner from \(A\) to \(B\); it carries the same unit-modulus phase as \(Y\).

Compose the left gauge, the gauge-phase intertwiner, and the inverse right gauge. After transporting across the bond-dimension identification, the composite is the required gauge-phase intertwiner.

Define the prepared blocks by

Positive definiteness makes this an invertible gauge, while the dual fixed-point equation gives

Gauge transport preserves irreducibility, the three fixed-length injectivity statements, and pairwise inequivalence up to gauge and phase. Thus every \(B^j\) is algebraically normal and left-canonical, so

The sharp separation theorem now gives

Theorem 8.3.2 transports this equality back to the source representatives.

Theorem 6.8.1 first gives

Theorem 13.2.2.10 gives the full simultaneous span at \(q=3(r-1)(L_0+1)\) on the source representatives. The source unital identity and Lemma 13.2.2.4 propagate this equality to every \(n\ge q\).

Write \(\phi _j=\Gamma _n^{A^j}(X_j)\) and suppose that \(\sum _j\phi _j=0\). The simultaneous spanning equality gives

Hence every \(\phi _j\) vanishes.

Let a vector belong to both restricted spaces. The two endpoint decompositions give boundary matrices \(C_a^j\) and \(D_b^j\). For every pair of physical indices \(a,b\) and every middle word \(w\) of length \(m-1\), their coefficients obey

The simultaneous product span at length \(m-1\) and nondegeneracy of the trace pairing therefore give, block by block,

Set \(E^j=\sum _a C_a^jA_a^{j\, \dagger }\). Identity (9) and the right-unital normalization imply \(A_b^jC_a^j=A_b^jE^jA_a^j\). Thus the component in block \(j\) lies in \(G_{m+1}(A^j)\). The reverse inclusion follows by splitting a word of length \(m+1\) at either endpoint.

Propagate injectivity to the two longer lengths used by the sharp block separation theorem, obtain the simultaneous span at \(3(r-1)(L_0+1)\), and propagate it using the right-unital identity.

Apply nondegeneracy of the trace pairing to the simultaneous product span.

Use the simultaneous product span at length \(m-1\) in the endpoint restriction calculation.

Write \(\phi _j(\sigma )=\operatorname{tr}(A^j_\sigma X_j)\). Evaluating \(\sum _j\phi _j=0\) on each word \(w\) of length \(n\) gives \(\sum _j\operatorname{tr}(A^j_wX_j)=0\). Since \(\operatorname{tr}(A^j_wX_j)=\operatorname{tr}(X_jA^j_w)\), the product-span equation gives, for every \((M_1,\ldots ,M_r)\in \prod _jM_{D_j}(\mathbb {C})\), \(\sum _j\operatorname{tr}(X_jM_j)=0\). Taking only the \(j\)-th component nonzero and using nondegeneracy of the trace pairing yields \(X_j=0\), hence \(\phi _j=0\). The BNT statement follows by substituting Lemma 13.2.2.5.

Lemma 13.2.2.5 gives \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=n\} =\prod _jM_{D_j}(\mathbb {C})\) for every \(n\ge N\). Substituting this product-span equation into Lemma 13.2.1.17, together with the normalization equation, gives the displayed intersection identity. Lemma 13.2.2.17 at lengths \(n+1\) and \(n+2\) gives the two directness statements.

Apply Lemma 13.3.1 and take \(N=(L_0+1)+(r-1)3(L_0+1)\).

This is the diagonal-block trace formula specialized to a block-diagonal boundary matrix. The \(j\)-th diagonal block of \(\bigoplus _kX_k\) is \(X_j\), giving the displayed sum.

Choose \(\phi _j\in G_L(A_j)\) such that

Choose \(Y_j\) with \(\phi _j=\Gamma _L^{A_j}(Y_j)\), and set \(X_j=\mu _j^{-L}Y_j\). Since \(\mu _j\ne 0\), one has, for every \(j\),

Lemma 13.3.3 now gives

A vector in \(G_L(A_j)\) has the form \(\Gamma _L^{A_j}(X)\). Put \(\mu _j^{-L}X\) in the \(j\)-th diagonal block of a boundary matrix for \(B\) and put zero in every other block. If \(\iota _j(Z)\) denotes this block insertion, Lemma 13.3.3 gives

Hence \(\Gamma _L^{A_j}(X)\in G_L(B)\).

A boundary matrix \(X\) for \(B\) has diagonal blocks \(X_j\), and block-diagonality gives \(\Gamma _L^B(X) =\sum _{j=1}^g\Gamma _L^{A_j}(\mu _j^LX_j)\). This proves \(G_L(B)\subseteq \bigvee _jG_L(A_j)\). The reverse inclusion is Lemma 13.3.5 for each block \(j\). Applying \(\iota _L\), which preserves finite sums of subspaces, gives \(G_L^{\mathrm{ES}}(B)=\bigvee _jG_L^{\mathrm{ES}}(A_j)\).

Theorem 13.3.6 gives \(G_L(B)=S_L\). Apply Lemma 11.5.5.18 to the sequence \(S_M\).

Put \(T=(L_0+1)+(g-1)3(L_0+1)\). For \(M\ge L\), one has \(M-1\ge T\). Applying Lemma 13.3.1 at \(n=M-1\) gives \(\mathbb {C}^d\otimes S_M\cap S_M\otimes \mathbb {C}^d =S_{M+1}\) for every \(M\ge L\). Lemma 13.3.7 then propagates the local constraint from \(L\) to \(N\).

The inclusion is Lemma 13.3.8. Since \(N\ge L\) and \(L\ge (L_0+1)+(g-1)3(L_0+1)+1\), the product-span directness range of Lemma 13.2.2.17 applies at length \(N\):

Lemma 13.3.9 gives

and the spaces \(G_N(A_j)\) are independent: for every \(j\) and every \(x_j\in G_N(A_j)\), \(\sum _jx_j=0 \quad \Longrightarrow \quad x_j=0\). Thus every \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\) has a unique decomposition \(\psi =\sum _j\psi _j\) with \(\psi _j\in G_N(A_j)\).

Lemma 13.3.8 places \(\psi \) in \(\bigvee _{j=1}^gG_N(A_j)\). Applying Lemma 13.3.4 gives boundary matrices \(X_j\) for which, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j)\in G_N(A_j)\), and

Put \(S_M=\sum _jG_M(A_j)\). For every \(M\) with \(L\le M{\lt}N\), Theorem 13.2.2.13 gives

Induction through the cyclic local constraints therefore gives

Since \(N\ge L\ge 3(g-1)(L_0+1)+1\), Theorem 13.2.2.12 shows that the spaces \(G_N(A_j)\) form an internal sum. Lemma 13.3.4 then supplies matrices \(X_j\) such that \(\Gamma _N^{A_j}(\mu _j^NX_j)\in G_N(A_j)\) for every \(j\) and

Put \(S_M=\sum _jG_M(A_j)\). For every \(M\) with \(L\le M{\lt}N\), Theorem 13.2.2.16 gives

Induction through the cyclic local constraints therefore yields

At the stated length, the block-space independence theorem makes this sum internal. Lemma 13.3.4 then supplies the matrices \(X_j\).

Multiplication on the left by \((A^b)^\dagger \) and summation over \(b\) gives \(C_a=\sum _b(A^b)^\dagger D_b A^a=EA^a\).

The two restrictions provide boundary matrices \(C_a^j,D_b^j\) (Lemma 13.2.1.14). The simultaneous word span gives \(A^{j,b}C_a^j=D_b^j A^{j,a}\) (Lemma 13.2.1.4). Set \(E_j=\sum _b(A^{j,b})^\dagger D_b^j\). Then \(C_a^j=E_jA^{j,a}\) by Theorem 13.3.14, so each boundary summand lies in \(G_{n+2}(A^j)\) (Lemma 13.2.1.12). The reverse inclusion follows by restricting a boundary-condition vector at its first or last physical letter (Lemmas 11.5.2.1 and 11.5.2.2):

By Definition 12.3.7.1, the Hamiltonian is a sum of orthogonal projectors,

Hence, by Theorem 12.3.7.12,

so, by Lemma 12.3.11, every contiguous length-\(L\) restriction of a zero-energy vector lies in \(G_L(B)\) under the identification of Lemma 11.3.2. The block intersection identity of Theorem 13.3.15 extends these constraints one site at a time (Lemma 11.5.5.17), giving membership in \(G_N(B)\). Conversely, \(G_N(B)\subseteq \ker H^{\mathrm{open}}_{N,L}(B)\) by Theorem 12.3.7.14. Nonzero block weights identify \(G_N(B)\) with the sum of the block spaces (Theorem 13.3.6).

Fix \(k\ne j\). If \(D_k\ne D_j\), the strict-size branch of Lemma 13.2.5 at the comparison length \(L_0+1\) gives

If \(D_k=D_j\), the blocks are not related by a virtual gauge and a nonzero scalar, so their periodic matrix product vectors are not proportional at arbitrarily long chain lengths, and the same lemma gives \(\mathcal G_S^{A_k}\cap \mathcal G_S^{A_j}=\{ 0\} \) and the same pair span. In both cases the pair is trace separated at length \(S\), which is Lemma E.3.1, and Lemma 13.2.9 converts the pair span into the displayed equations.

Take \(L_0=s+p\) and let \(L\ge L_0\). Since \(L-s\ge p\), every \(A^j\) is block injective at length \(L-s\) by Theorem 6.1.1.24. Lemma 13.2.13 with the selectors of length \(s\) gives the full product span at length \((L-s)+s=L\).

Primitivity gives normality (Theorem 6.7.4.4), hence a common positive length \(p\) at which every block is injective (Theorem 6.5.8). Injectivity persists at every length at least \(p\) (Theorem 6.1.1.24), in particular at \(p+1\) and \(3(p+1)\). Each block is irreducible and has primitive transfer map, so its self-overlap satisfies \(O_{A^jA^j}(N)\to 1\) (Theorem C.5.2). Theorem 13.3.17 with \(L_0=p\) gives, for every ordered pair \(k\ne j\), a word polynomial of length \(3(p+1)\) equal to \(I\) on block \(k\) and to \(0\) on block \(j\). Lemma 13.2.10 multiplies these polynomials into selectors of the common length \(s=(r-1)\cdot 3(p+1)\):

Theorem 13.3.18 then gives the full product span at every length \(L\ge s+p\).

Choose \(L_0\) one larger than the eventual simultaneous word-span threshold of Theorem 13.3.19 and apply Theorem 13.3.16.

If \(eB^i=\zeta X(cA^i)X^{-1}\) for every \(i\), with \(\zeta \ne 0\), then

for every \(i\), and \(\zeta c/e\ne 0\).

Normalize each block independently by Theorem 12.8.8. The resulting nonzero rescaling and virtual gauge leave its local ground spaces unchanged. Thus \(B^{i,a}=\zeta _iX_iA^{i,a}X_i^{-1}\) for every physical index \(a\), with \(\zeta _i\ne 0\) and \(X_i\) invertible. If \(D_i=D_j\) and \(B^{i,a}=\zeta YB^{j,a}Y^{-1}\) for every \(a\), with \(\zeta \ne 0\) and \(Y\) invertible, then composing the gauges gives

for every \(a\) (Lemma 13.2.2.9 transports the gauges, and Theorem 13.3.21 removes the nonzero rescalings), contrary to the hypothesis.

The assembled local ground spaces are the sums of the respective block spaces. Equality of these spaces gives equality of their orthogonal complement projections and hence of every translated parent interaction.

Normalize each block while retaining its local ground spaces and the inequivalence of distinct blocks. The assembled parent Hamiltonians therefore coincide at each interaction range and volume.

By Theorem 12.8.8, normalize each tensor by a nonzero scalar and an invertible virtual conjugation. These operations preserve block injectivity at each fixed word length, preserve all local ground spaces, and preserve inequivalence. By Theorem 13.3.23, equal block ground spaces give equal ground spaces of the weighted block sums, hence every local parent projection, and therefore every periodic parent Hamiltonian, is unchanged.

For an arbitrary tuple \((M_j)_j\), represent \((M_j(A^{j,a})^\dagger )_j\) as a linear combination of length-\(L\) word tuples. Append the letter \(a\) and sum over \(a\). The resulting tuple is \((M_j)_j\) by trace-preserving normalization. This proves the one-site extension, and induction gives every larger length.

The simultaneous-span estimate of Lemma 8.3.3, the direct-sum lemma of [ 228 , lines 1346–1408 ] , gives the full product matrix algebra at length \(3(r-1)(L_0+1)\). Trace-preserving propagation keeps the span full at every larger length. The block intersection theorem therefore gives the kernel identity at the stated interaction range.

Replace each block by a normalized primitive representative. This preserves inequivalence and leaves every parent Hamiltonian unchanged. The uniform gap theorem for primitive block sums therefore applies with the same interaction ranges and the same constants.

Normalize the blocks to primitive representatives without changing the specified injectivity length \(L_0\). Their periodic parent Hamiltonians agree with the original ones at every range and chain length. Apply the primitive-block gap theorem at the prescribed range.

Propagating the intertwining identity through the word gives \(Y_iA^w=A^wY_{Ns+i}\). Since \(Ns=0\) in the cyclic index group, the two boundary matrices coincide.

Choose the boundary matrices \(X_i^j\) of the hypothesis at each cyclic cut \(i\). Comparing adjacent cuts against all complementary words of length \(N-1\) gives \(X_i^j A^{j,a}=A^{j,a}X_{i-1}^j\). Iterating this identity once around the ring yields

Each block is injective at length \(N-1\), hence also at length \(N\). Its boundary matrix therefore commutes with the full matrix algebra and is scalar. The representation at cut zero is consequently a linear combination of the periodic block vectors, that is, an element of the sum of their lines (Lemma 13.1.2).

Trace-preserving normalization propagates simultaneous injectivity to every length at least \(S\), in particular to \(N-1\). The open-chain intersection property expresses each periodic ground vector as a sum of block boundary vectors at every cut. Cyclic closure places this vector in the span of the periodic block vectors. Conversely, each periodic block vector satisfies every local parent constraint.

Rescaling the \(j\)th block multiplies its length-\(S\) words by \(\zeta _j^S\). For any target tuple \((M_j)_j\), first express \((\zeta _j^{-S}M_j)_j\) in the original word span. The same coefficients express \((M_j)_j\) in the rescaled word span.

Projection of the simultaneous span onto each matrix summand proves injectivity of every block at length \(S\). Each block is therefore normal and admits a primitive representative obtained by a rescaling by some \(\zeta _j\neq 0\) and an invertible virtual conjugation. Both operations preserve the simultaneous word span at the specified length. They preserve every component ground space, and the periodic component vectors satisfy

Since \(\zeta _j^{\, N}\neq 0\), each block spans the same line for \(A^j\) and \(B^j\), and the two spans, being the sums of these lines (Lemma 13.1.2), agree. Taking sums of the component ground spaces proves equality of the assembled parent Hamiltonians.

Normalize the blocks while retaining simultaneous injectivity at length \(S\). The parent Hamiltonians and the span of the periodic component vectors are unchanged. Apply the trace-preserving block closure theorem. For positive operators \(H_i\),

Replacing the local projectors by positive terms with the same local kernels therefore preserves the ground space.

The representative bond dimensions sum to at most \(D\). The simultaneous span estimate supplies \(S{\gt}0\) with \(S+1\leq 3D^5\). Take \(R=S+1\) and apply periodic closure for canonical tensors at every \(R\leq L\leq N\).

The coisometric reconstruction preserves the local space on every positive-length interval. The local space of the resulting direct sum is the sum of its block spaces. Each repeated copy differs from its representative by a virtual similarity and a nonzero scalar, both of which preserve the local space. Taking the sum once over each distinct representative therefore gives the same space.

Replace the original canonical tensor by the direct sum containing one copy of each normal representative. Their local spaces, hence their periodic local constraints, coincide. Apply the simultaneous block-injectivity closure theorem at the same supplied length \(S\). For positive operators \(H_i\),

Replacing each orthogonal projection by any positive interaction with the same kernel therefore leaves the periodic ground space unchanged.

Apply Lemma 13.3.3 at length \(N\), then use linearity of the cyclic restriction \(R_{i,\tau }\).

The hypothesis \(\psi \in \mathcal G_{N,L}(B)\) says that the left-hand side before decomposition lies in \(G_L(B)\). Lemma 13.3.1.1 rewrites this local vector as the displayed sum, and Theorem 13.3.6 identifies \(G_L(B)\) with \(\bigvee _jG_L(A_j)\).

When the window stays inside the chosen linear interval, the boundary matrix remains outside the window. Write \(A_{\mathrm{left}}=A^j_{\tau _0}\cdots A^j_{\tau _{i-1}}\) and \(A_{\mathrm{right}}=A^j_{\tau _{i+L}}\cdots A^j_{\tau _{N-1}}\). Trace cyclicity gives

Hence the restricted vector lies in \(G_L(A_j)\).

Let \(\omega \) be the \(N\)-site word obtained by inserting \(\sigma \) in the cyclic interval beginning at \(i\) and by using \(\tau \) outside that interval. Since the interval crosses the cut, \(\omega =\beta \rho \alpha \). Hence

This is exactly \(R_{i,\tau }\! \left(\Gamma _N^{A_j}(\mu _j^NX_j)\right)(\sigma )\).

Decompose the local vector in \(\bigvee _jG_L(A_j)\) as a sum of vectors \(\Gamma _L^{A_j}(C^j_{i,\tau })\). Evaluating at \(\sigma \) gives \(\sum _j\operatorname{tr}\! \left(A^j_{\alpha \beta }C^j_{i,\tau }\right)\). Trace cyclicity rewrites this as \(\sum _j\operatorname{tr}\! \left(A^j_\beta C^j_{i,\tau }A^j_\alpha \right)\). On the other hand, Lemma 13.3.1.4 gives the second trace expression for the same local coefficient sum.

Since \(d{\gt}0\), extend the outside word \(\rho \) to a full outside configuration \(\tau \) whose entries between the end of the interval and the cut are the word \(\rho \). By the nonzero-weight hypothesis, Lemma 13.3.1.2 applies and puts the sum of the corresponding block restrictions in \(\bigvee _jG_L(A_j)\). Applying Lemma 13.3.1.5 gives the displayed trace equality after rewriting the opened cyclic word as \(\beta \rho \alpha \).

Apply Lemma 13.3.1.6 to each boundary-crossing start \(i\) and each outside word \(\rho \), and choose the resulting matrices \(C^j_{i,\rho }\).

Lemma 13.3.1.5 gives matrices \(C^j_{i,\tau }\) with

for every word \(\alpha \) of length \(N-i\) and every word \(\beta \) of length \(a\). The fixed complementary-word trace comparison, applied with \(X_j\) replaced by \(\mu _j^NX_j\), gives (18).

Lemma 13.3.1.7 gives matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing start \(i\), outside word \(\rho \) of length \(N-L\), word \(\beta \) of length \(i+L-N\) before the cut, and wrapped tail word \(\alpha \) of length \(N-i\),

Apply Lemma 13.2.1.6 with the stated product-algebra span at length \(N-i\). Equality of the traces for all \(\alpha \) then gives \(A^j_\beta C^j_{i,\rho }=((\mu _j^NX_j)A^j_\beta )A^j_\rho \) for every block \(j\).

Theorem 13.3.1.8 gives matrices \(C^j_{i,\tau }\) satisfying (18). Thus \(E=C^j_{i,\tau }\) satisfies the boundary matrix identity required in Lemma 13.3.1.11, and the displayed membership follows for every block \(j\).

Write a length-\(L\) word in the boundary-crossing interval as \(\alpha \beta \), where \(\alpha \) is the segment inside \(\{ i,\ldots ,N-1\} \) and \(\beta \) is the segment after the interval wraps past the cut, inside \(\{ 0,\ldots ,a-1\} \). The full \(N\)-site word appearing in the restriction is then \(\beta \, \tau _a\cdots \tau _{i-1} \alpha \). Trace cyclicity gives

applying (19) to the inside product. The last expression is the coefficient of \(\Gamma _L^{A_j}(E)\) at the word \(\alpha \beta \). Hence the restricted vector lies in \(G_L(A_j)\).

The cyclic interval beginning at \(M\) reads the last site first and then wraps to the sites \(0,\ldots ,m\). Thus, for the local word \(awb\), the full word in the trace is \(w\, b\, \rho \, a\). Hence the \(j\)-th coefficient is

where the equality follows from trace cyclicity. This is the displayed left-boundary trace form.

Sum Lemma 13.3.1.12 over the block index.

Lemma 13.3.1.13 writes the local block sum in the form

Applying Lemma 13.2.1.3 gives (20).

Lemma 13.3.1.14 gives matrices \(E_j\) satisfying (20). By Lemma 13.3.1.12, the \(j\)-th cyclic restriction is the left-boundary component determined by \(C^j_a\). The displayed identity makes that component equal to \(\Gamma _{m+2}^{A_j}(E_j)\), hence it lies in \(G_{m+2}(A^j)\).

The local constraint of \(\psi \) at the cyclic interval beginning at \(M\) gives

by Lemma 13.3.1.2. Applying Lemma 13.3.1.14 gives the displayed equations.

Lemma 13.3.1.2, applied to the window beginning at the last site, gives

Lemma 13.3.1.15 then applies to each block component.

Let \(E:=Y A^j_{\tau _a}\cdots A^j_{\tau _{i-1}}\). Then, for every word \(\beta \) on the segment \(0,\ldots ,a-1\),

Lemma 13.3.1.11 gives the stated local constraint.

By definition, membership in \(\mathcal G_{N,L}(A_j)\) is the collection of all cyclic length-\(L\) local constraints. For an interval beginning at \(i\), either \(i+L\le N\) or \(N{\lt}i+L\). In the first case, use Lemma 13.3.1.3. In the second case, use the displayed hypothesis.

Lemma 13.3.1.18 turns the displayed boundary identity into the local constraint for each boundary-crossing interval. Lemma 13.3.1.19 then yields \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\), handling both boundary-crossing intervals \(N{\lt}i+L\) and non-crossing intervals \(i+L\le N\).

For fixed \(j,i\), apply Theorem 11.10.2 to the family

Since the length-\(N-L\) outside-word products span the full matrix algebra, there is a single matrix \(Y_{j,i}\) such that, for every \(\beta \), \(\mu _j^NX_jA^j_\beta = A^j_\beta Y_{j,i}\). Lemma 13.3.1.20 then gives the periodic-boundary constraints for each block.

Since \(L+L_0\le N\), the complementary length satisfies \(L_0\le N-L\). Theorem 6.8.1 therefore gives, for every block \(j\), \(S_{N-L}(A_j) = M_{D_j}(\mathbb {C})\). Apply Lemma 13.3.1.21.

The case \(M=0\) is the identity matrix. For the induction step, write a word of length \(M+1\) as \(a\rho \). Since \(A_{a\rho }=A_aA_\rho \),

The normalization equation gives, by induction on word length, \(\sum _\rho A^j_\rho A^{j\dagger }_\rho = \mathbb {1}\) for words \(\rho \) of length \(N-L\). Applying Lemma 13.2.1.9 with \(X=\mu _j^NX_j\) gives, for every outside word \(\rho \), a matrix \(E_{j,i,\rho }\) such that, for every word \(\beta \) before the cut, \(((\mu _j^NX_j)A^j_\beta )A^j_\rho = A^j_\beta E_{j,i,\rho }\). Lemma 13.3.1.22 then gives the periodic-boundary constraint for each block.

Theorem 13.3.1.9 supplies matrices \(C^j_{i,\rho }\) satisfying \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \). Lemma 13.3.1.24 then gives the single-block periodic constraints.

Substituting the displayed comparison identity in each summand gives

Summing over \(j\) gives the conclusion.

Applying Lemma 13.2.1.8 with the boundary matrix \(D^j_\beta =(\mu _j^NX_j)A^j_\beta \), for every block \(j\) and every boundary-crossing interval \(i\), gives a matrix \(E^j_i\) such that, for every word \(\beta \) before the cut, \((\mu _j^NX_j)A^j_\beta = A^j_\beta E^j_i\). Hence, for every outside word \(\rho \), taking \(E_{j,i,\rho }=E^j_iA^j_\rho \) gives \(((\mu _j^NX_j)A^j_\beta )A^j_\rho = A^j_\beta E_{j,i,\rho }\). These are precisely the block-diagonal boundary-condition equations of Lemma 13.3.1.22, which gives the periodic-boundary constraint for each block.

Choose a block-diagonal boundary representation of \(\psi \). For that representation, the boundary trace comparison supplies the matrices \(C^j_{i,\rho }\). Applying Lemma 13.3.1.27 gives the desired containment \(\Gamma _N^{A_j}(\mu _j^NX_j)\in \mathcal G_{N,L}(A_j)\) for every block \(j\).

Choose a block-diagonal boundary representation of \(\psi \). For that representation, Lemma 13.3.1.26 gives, for every boundary-crossing interval \(i\), outside word \(\rho \), word \(\beta \) before the cut, and middle word \(w\),

Applying Lemma 13.3.1.28 to these trace comparisons gives \(\Gamma _N^{A_j}(\mu _j^NX_j)\in \mathcal G_{N,L}(A_j)\) for every \(j\).

By Lemma 13.3.11, there are matrices \(X_j\) with \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\). Lemma 13.3.1.28 applies to this representation and the assumed equality of boundary trace expressions. Thus, \(\Gamma _N^{A_j}(\mu _j^NX_j)\in \mathcal G_{N,L}(A_j)\) for every \(j\).

Lemma 13.2.2.5 gives \(\operatorname{span}\{ (A^1_w,\ldots ,A^g_w):|w|=m\} = \prod _jM_{D_j}(\mathbb {C})\). Applying Lemma 13.3.1.30 with this middle-word span gives the stated block-diagonal boundary conditions and the periodic-boundary constraint in each block.

By Lemma 13.3.11, there are matrices \(X_j\) with \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\). For these boundary conditions, the assumed \(C^j\) boundary comparison supplies the matrices \(C^j_{i,\rho }\). Lemma 13.3.1.24 gives \(\Gamma _N^{A_j}(\mu _j^NX_j)\in \mathcal G_{N,L}(A_j)\) for every \(j\).

Lemma 13.3.11 supplies block-diagonal boundary conditions \(X_j\), and Lemma 13.3.1.25, applied with the assumed tail-word spans, places each \(\Gamma _N^{A_j}(\mu _j^NX_j)\) in \(\mathcal G_{N,L}(A_j)\).

For a boundary-crossing interval beginning at \(i\), either \(N-i {\lt} (L_0+1) +(g-1)((L_0+1)+((L_0+1)+(L_0+1)))\) or \(N-i \ge (L_0+1) +(g-1)((L_0+1)+((L_0+1)+(L_0+1)))\). In the first case, the assumed short-tail span applies; in the second case, Lemma 13.2.2.5 gives the simultaneous product span at length \(N-i\). Hence every boundary-crossing interval has the tail-word spanning property, and Theorem 13.3.1.33 gives the block-diagonal boundary conditions and \(\Gamma _N^{A_j}(\mu _j^NX_j)\in \mathcal G_{N,L}(A_j)\) for every \(j\).

For every configuration \(\sigma \), the two sides evaluate \(\psi \) on the same configuration because \(T_t(T_s\sigma )=T_{s+t}\sigma \).

A length-\(L\) interval of \(T_s\psi \) beginning at \(i\) is the corresponding interval of \(\psi \) beginning at \(i-s\), with the outside configuration translated by \(s\).

This is addition modulo \(K+S\), separated according to whether the site lies before or after the cut.

Evaluate the translated equality on the word \(\beta \alpha \), where \(|\beta |=K\). Cyclicity of the trace gives

The simultaneous spanning hypothesis separates the blocks and gives the asserted matrix identities.

Lemma 13.3.12 gives boundary matrices \(X_j\). Translate the periodic state so that the cut moves past the final \(L_0\) sites. By Lemma 13.3.1.38, the translated state remains in the periodic chain space, so applying Lemma 13.3.12 again yields boundary matrices \(Y_j\) for the translated state. For a word \(\alpha \) of length \(L_0\), comparison of the two decompositions against every complementary word \(w\) of length \(N-L_0\) gives

The length bound and Theorem 13.2.2.11 imply that the simultaneous products of length \(N-L_0\) span the product algebra. Therefore, Lemma 13.3.1.40 gives, block by block, \((\mu _j^NX_j)A^j_\alpha = A^j_\alpha (\mu _j^NY_j)\). Since the words of length \(L_0\) span each block matrix algebra, evaluation at the identity gives

Thus (21) says that \(\mu _j^NX_j\) commutes with every word of length \(L_0\), and consequently with every one-site matrix. The boundary condition may then be moved across every interval crossing the cut, which places each block component in its periodic chain space. This is the change-of-cut argument of [ 228 , Theorem 12, proof lines 1276–1289 and 1454–1456 ] .

Obtain block-diagonal boundary matrices at the chosen cut. Move the cut past the final \(L_0\) sites and compare the two boundary representations. The doubly normalized simultaneous product-span theorem separates the blocks and gives

for every word \(\alpha \) of length \(L_0\). Injectivity identifies the two boundary matrices and permits the cut to move across every local interval, proving the periodic single-block constraints.

This is the equality component of Theorem 13.3.1.44.

Theorem 13.3.1.41 gives the forward inclusion in the displayed equality. Every periodic single-block vector is a periodic vector for the block-diagonal tensor, which gives the reverse inclusion. For independence, write \(\phi _j=\Gamma _N^{A^j}(X_j)\) and suppose that \(\sum _j\phi _j=0\). Theorem 13.2.2.12 and nondegeneracy of the trace pairing give

The last implication follows by choosing a tuple supported in one block. Thus every \(\phi _j\) vanishes, and the open-boundary spaces form an internal sum.

Apply Theorem 13.3.1.44 with \(\Lambda _j=\mathbb {1}\). Left canonicity gives the required dual fixed-point identity, while the remaining hypotheses are unchanged.

This is the equality component of Theorem 13.3.1.45.

The source-length open-boundary theorem places \(\psi \) in \(\sum _jG_N(A_j)\) and proves that the spaces \(G_N(A_j)\) form an internal sum. Existence follows from membership in the sum, and internality makes the components unique.

The parent-Hamiltonian kernel is the periodic chain space. Apply the preceding block decomposition and then the assumed single-block containments.

Theorem 13.3.1.46 identifies the periodic chain space of the block sum with the sum of the single-block chain spaces. Apply the assumed containment to each summand, and use the identification of the parent-Hamiltonian kernel with the periodic chain space.

By Lemma 13.3.11, there are matrices \(X_j\) with \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\). These \(X_j\) satisfy the assumed identities, so Lemma 13.3.1.22 gives \(\Gamma _N^{A_j}(\mu _j^NX_j)\in \mathcal G_{N,L}(A_j)\) for every \(j\).

The first inclusion is Lemma 13.1.4. The second inclusion and internal directness are Lemma 13.3.9.

The reverse inclusion is Lemma 13.1.4; combine the two inclusions.

For \(\psi \in \mathcal G_{N,L}(B)\), choose the displayed decomposition. Each summand lies in the corresponding subspace \(\mathcal G_{N,L}(A_j)\), hence their finite sum lies in \(\bigvee _j\mathcal G_{N,L}(A_j)\).

The block-diagonal boundary formula gives

Each summand belongs to the corresponding block periodic-boundary space by assumption, so the finite sum lies in the linear span of these spaces.

Apply Lemma 13.3.2.4 to the block-diagonal boundary representation of each \(\psi \in \mathcal G_{N,L}(B)\).

The block-diagonal boundary representation gives

by Lemma 13.3.2.5. The equality follows from Lemma 13.3.2.2; the internality statement is Lemma 13.3.9.

For each \(\psi \), Lemma 13.3.1.50 gives matrices \(X_j\) with

and, for every \(j\), the membership \(\Gamma _N^{A_j}(\mu _j^N X_j)\in \mathcal G_{N,L}(A_j)\) holds. Lemma 13.3.2.6 then gives

and the internality of the sum of the \(G_N(A_j)\).

For each \(\psi \), Lemma 13.3.1.32 gives matrices \(X_j\) with

and, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^N X_j)\in \mathcal G_{N,L}(A_j)\). Lemma 13.3.2.6 then gives

and the internality of the sum of the \(G_N(A_j)\).

The claim is the equality part of Lemma 13.3.2.8.

Lemma 13.3.2.6 reduces the claim to producing, for each \(\psi \), block-diagonal boundary conditions \(X_j\) whose single-block components lie in \(\mathcal G_{N,L}(A_j)\). The crossing comparison theorem supplies this: it first uses the boundary-comparison-free BNT representation to obtain \(X_j\), and then uses the stated tail-word spans and Theorem 13.3.1.9 to produce matrices \(C^j_{i,\rho }\) satisfying \(A^j_\beta C^j_{i,\rho } = ((\mu _j^N X_j)A^j_\beta )A^j_\rho \). Lemma 13.3.1.24 then gives \(\Gamma _N^{A_j}(\mu _j^N X_j)\in \mathcal G_{N,L}(A_j)\) for every \(j\). Applying Lemma 13.3.2.6 to these boundary conditions gives the displayed equality and internality.

The claim is the equality part of Lemma 13.3.2.10.