By definition, for \(B=\bigoplus _{j=1}^g \mu _j A_j\) one has \(\mathcal G_{N,L}(B) =\mathcal G_{N,L}(\bigoplus _{j=1}^g \mu _j A_j)\).

For a family \(A_0,\ldots ,A_{r-1}\) and a chain length \(N\), let \(M_j=\operatorname{span}\{ V^{(N)}(A_j)\} \). Then

If \(1{\lt}L\) and \(N\ge L+1\), then for each normal tensor \(A_j\) in the basis, the vector \(V^{(N)}(A_j)\) lies in \(\mathcal G_{N,L}(\bigoplus _k \mu _k A_k)\).

For each block index \(j\) with \(\mu _j\neq 0\), and whenever \(0{\lt}N\) and \(L\le N\), the periodic-boundary ground space of the block \(A_j\) embeds into the corresponding ground space of the direct-sum tensor:

Assume \(\mu _j\neq 0\) for every block index \(j\). Whenever \(0{\lt}N\) and \(L\le N\), the linear sum of the blockwise ground spaces with periodic boundary conditions is contained in the direct-sum tensor’s ground space with periodic boundary conditions:

where \(j\) ranges over all block indices.

Let \(A^i=\bigoplus _j\mu _j A_j^i\) be in block-injective canonical form, where \(A_1,\ldots ,A_g\) is a basis of normal tensors. The linear sum of the blockwise ground spaces with periodic boundary conditions is contained in the ground space of the block-diagonal tensor on the \(N\)-site ring:

Let \(S\) be a family of virtual matrices. If each sector projection satisfies \(P_j\in \operatorname{span}S\) and a boundary matrix \(X\) commutes with every \(M\) in \(S\), so that \(XM=MX\), then \(XP_j=P_jX\) for every sector \(j\), and hence \(P_jXP_k=0\) for distinct sectors \(j\) and \(k\). Thus \(X\) is block diagonal with respect to the sector decomposition.

Let \(T_a=(T_a(i))_i\) be a family of tuples spanning the full product algebra \(\prod _i M_{n_i}(\mathbb {C})\), and let \(c_i\neq 0\) be nonzero scalars. Then, for every sector \(k\), the sector projection \(P_k\) lies in the span of the scaled dependent block-diagonal matrices \(\bigoplus _i c_i T_a(i)\).

If the simultaneous length-\(m\) block-word tuples \(\omega \mapsto (A_j^\omega )_j\) span the full product algebra \(\prod _j M_{D_j}(\mathbb {C})\) and every \(\mu _j\) is nonzero, then the pulled-back length-\(m\) word products of \(\bigoplus _j\mu _j A_j\) span every virtual sector projection \(P_j\). If a virtual matrix \(X\) commutes with the pulled-back word product for every word \(\omega \) of length \(m\),

then \(P_jXP_k=0\) for distinct sectors \(j\) and \(k\).

Let \(A\) and \(B\) be tensors with virtual bond dimensions \(D_A\) and \(D_B\). Assume that \(A\) is \(L\)-block injective. Let \(\Delta _A\in M_{D_A}(\mathbb {C})\) with \(\Delta _A\neq 0\), and let \(\Delta _B\in M_{D_B}(\mathbb {C})\). Suppose that, for all words \(u,v,t\) of length \(L\),

Then for every \(Z\in M_{D_A}(\mathbb {C})\) there is a matrix \(W\in M_{D_B}(\mathbb {C})\) such that, for all words \(t\) of length \(L\), \(\operatorname{tr}(ZA_t)+\operatorname{tr}(WB_t) =0\). This is the trace-dual algebraic step in the two-block case of the direct-sum lemma of [ PGVWC07 ] .

Under the hypotheses of Lemma 15.2.6, the length-\(L\) spaces \(\mathcal G_L^A\) and \(\mathcal G_L^B\) satisfy \(\mathcal G_L^A\subseteq \mathcal G_L^B\). If the corresponding nonzero middle matrix exists on both sides, then \(\mathcal G_L^A=\mathcal G_L^B\).

If \(A\) and \(B\) are length-\(L\) block-injective, the map \(Z\mapsto (t\mapsto \operatorname{tr}(A_tZ))\) has image dimension \(D_A^2\), and similarly for \(B\). Therefore an inclusion \(\mathcal G_L^A\subseteq \mathcal G_L^B\), together with \(D_B\le D_A\), forces \(D_A=D_B\) and \(\mathcal G_L^A=\mathcal G_L^B\). In particular, the three-block trace relation from Lemma 15.2.7 gives this equality whenever the ordering \(D_B\le D_A\) is available. If the inclusion comes from the strictly larger block, \(D_B{\lt}D_A\), then the same dimension step already contradicts the existence of the three-block trace relation. More generally, let \(H\) be a source hypothesis which excludes the simultaneous conclusion

Then the three-block intersection vanishes:

With injectivity at the three-block length, the strict-size branch gives

One such source hypothesis is non-proportionality of the periodic MPS vectors at a sufficiently long chain length: \(\mathbb {C}\, V^{(N)}(A)\ne \mathbb {C}\, V^{(N)}(B)\). Indeed,

by equality of the periodic-boundary constraint spaces, followed by injective uniqueness. For same-dimensional normalized BNT representatives that are not gauge-phase equivalent, the non-equivalence condition supplies this non-proportionality at arbitrarily large chain lengths; the direct-sum injectivity hypotheses remain explicit. Combining the equal-dimension and unequal-dimension branches over all ordered block pairs gives one common positive length \(S_0\) with

For an ordered pair of blocks, the trace-separation conclusion has the form

Equivalently, the homogeneous pair span is full:

Concatenating words propagates this equality to positive multiples of \(S\), and hence to a finite period window for every ordered pair. Combining that window with the conditional period-window homogenization theorem gives a single length \(S_0\) such that

The same conclusion applies to the corresponding normal canonical-form hypotheses with a basis of normal tensors once one-site injectivity is supplied explicitly, since those hypotheses carry the irreducibility and normalization assumptions used in the direct-sum comparison.

If no nonzero trace functional vanishes on all finite pair words, then there is a finite \(S\) such that vanishing on every word of length at most \(S\) already forces the test pair to be zero. For a finite family of ordered block pairs, these bounds can be replaced by one common bound.

If one of the two blocks is injective, then the corresponding coordinate projection of the pair algebra generated by the simultaneous one-letter pairs is the full matrix algebra. Every finite pair word lies in this pair algebra, and the all-word linear span is the submodule generated by the pair algebra. If the pair algebra is the graph of an algebra automorphism of the matrix algebra, Skolem–Noether turns this graph case into gauge-phase equivalence, so under non-gauge-equivalence the graph branch is impossible. For two blocks of unequal bond dimensions, the graph branch is also impossible because it would identify matrix algebras with different finite dimensions over \(\mathbb {C}\).

Homogeneous identity-padding witnesses multiply by concatenation of words. Therefore, if one has a positive period and a full residue window of homogeneous pair spans, then the simultaneous identity pair lies in all sufficiently large homogeneous pair spans. Combined with all-length pair trace separation, this period-window input gives one homogeneous pair trace-separation length. This does not prove the block-separation span condition of [ PGVWC07 ] by itself: fullness of spans up to a bounded length is weaker than the exact homogeneous input used here.

At a fixed length \(S\), the pair trace-separation criterion is dual to full pair product-span. Therefore a pair trace-separation witness supplies a linear combination of word evaluations that is identity on the first block and zero on the second. A common witness for all ordered distinct pairs gives pairwise block-separating words at the same length.

If every ordered pair of distinct blocks admits a common-length pairwise block-separating equation, then for each \(k\) there are coefficients \(c_\omega ^{(k)}\) with

Hence common block injectivity plus pairwise separation implies the finite-length blockwise physical–virtual correspondence of the block-injectivity proposition in [ CPGSV16 , Section 2.3 ] and the block-injective canonical-form trace-separation condition.

Suppose the length-\(L\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\), and suppose the simultaneous identity tuple \((I,\ldots ,I)\) lies in the span of the length-\(S\) simultaneous block-word tuples. Then the length-\(L+S\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\). In particular, two full homogeneous spans at lengths \(L\) and \(S\) give a full homogeneous span at length \(L+S\).

Let \(p{\gt}0\). Suppose the length-\(p\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\), and suppose that for every \(0\le r{\lt}p\), the length-\(s+r\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\). Then there is a length \(N\) such that, for every \(n\ge N\), the length-\(n\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\). More concretely, one may take \(N=s\): writing \(n=s+r+qp\), with \(0\le r{\lt}p\), the span at length \(s+r\) propagates to the span at length \(n\) by multiplying \(q\) times by the period-\(p\) span.

Let \(J\) be a finite set of \(r\) blocks, and let \((A_j)_{j\in J}\) be a finite family of tensors with common block injectivity at length \(L\). Suppose there are length-\(S\) block-separating equations: for each block \(k\),

Then the simultaneous length-\((L+S)\) block-word evaluations span \(\prod _j M_{D_j}(\mathbb {C})\). If the block-separating equations are first obtained from pairwise block-separating equations at length \(S\), the same conclusion holds at length \(L+(r-1)S\).

Let \((A^\ell )_{\ell =1}^r\) be a finite family of tensors. Suppose that the length-\(S\) pair evaluations of blocks \(k\) and \(j\) span their product matrix algebra. For every \(X\in M_{D_k}(\mathbb {C})\), there is a tuple \(M=(M_\ell )_\ell \) in the span of the simultaneous length-\(S\) word evaluations such that \(M_k=X\) and \(M_j=0\).

Let \(A^1,\ldots ,A^g\) be block tensors whose length-\(m\) simultaneous word tuples span the product algebra. If, for every word \(w\) of length \(m\), two blockwise matrix families \(X_j\) and \(Y_j\) satisfy \(\sum _j\operatorname{tr}(X_jA^j_w) = \sum _j\operatorname{tr}(Y_jA^j_w)\), then \(X_j=Y_j\) for every block \(j\).

Let \(A^1,\ldots ,A^g\) be block tensors whose length-\(m\) simultaneous word tuples span the product algebra. Let \(C^j_a\in M_{D_j}(\mathbb {C})\) be matrices indexed by blocks \(j\) and physical letters \(a\). Suppose that, for every pair of boundary letters \(a,b\) and every word \(w\) of length \(m\), a vector \(\psi \) has the left-boundary trace form \(\psi (a,w,b) = \sum _j\operatorname{tr}\! \left(A^j_b C^j_a A^j_w\right)\). If \(\psi \in \bigvee _j G_{m+2}(A^j)\), then there are matrices \(E_j\) such that, for every block \(j\) and all \(a,b\), \(A^j_bC^j_a = A^j_bE_jA^j_a\).

Let \(A^1,\ldots ,A^g\) be block tensors with the following common word-span property: for a fixed \(m\), the simultaneous tuples \(w \longmapsto \left(A^1_w,\ldots ,A^g_w\right), \qquad |w|=m\), span the full product algebra \(\prod _j M_{D_j}(\mathbb {C})\). Suppose that matrices \(C^j_a,D^j_b\in M_{D_j}(\mathbb {C})\) satisfy, for all physical indices \(a,b\) and all words \(w\) of length \(m\),

Then \(A^j_bC^j_a = D^j_bA^j_a\) for every block \(j\) and all \(a,b\).

Let \(A^1,\ldots ,A^g\) be block tensors with the common word-span property at length \(m\). Suppose that \(C^j_\rho \in M_{D_j}(\mathbb {C})\) is indexed by a block \(j\) and a word \(\rho \) of some fixed length \(M\), and that \(D^j_\beta \in M_{D_j}(\mathbb {C})\) is indexed by a block \(j\) and a word \(\beta \) of some fixed length \(K\). If, for every word \(\rho \) of length \(M\), every word \(\beta \) of length \(K\), and every middle word \(w\) of length \(m\),

then \(A^j_\beta C^j_\rho = D^j_\beta A^j_\rho \) for every block \(j\) and all words \(\beta ,\rho \). In particular, taking \(D^j_\beta =X_jA^j_\beta \) gives \(A^j_\beta C^j_\rho = (X_jA^j_\beta )A^j_\rho \).

Let \(A^1,\ldots ,A^g\) be block tensors with the common word-span property at length \(m\). Fix a complementary word \(\rho \) of length \(M\) and matrices \(C^j_\rho ,X_j\in M_{D_j}(\mathbb {C})\). If, for every wrapped word \(\beta \) of length \(K\) and every middle word \(w\) of length \(m\),

then \(A^j_\beta C^j_\rho = (X_jA^j_\beta )A^j_\rho \) for every block \(j\) and every wrapped word \(\beta \).

Let \((A_a)_{a\in I}\), \((C_a)_{a\in I}\), and \((D_b)_{b\in I}\) be matrices in \(M_D(\mathbb {C})\), indexed by a finite alphabet. If

for all \(a,b\in I\), and if \(E=\sum _a C_a A_a^\dagger \), then

for all \(a,b\in I\).

In particular, the same calculation applies when the alphabet is the finite set of words of a fixed length and \(A_\beta \) denotes \(A_{\beta _1}\cdots A_{\beta _K}\).

The same argument also applies when the normalized family is indexed by a finite set \(J\), while \((A_b)_{b\in I}\) and \((D_b)_{b\in I}\) are indexed by a set \(I\). If \((B_a)_{a\in J}\) and \((C_a)_{a\in J}\) are indexed by \(J\), and if, for all \(a\in J\) and \(b\in I\),

then, with \(E=\sum _a C_aB_a^\dagger \), one has

This includes the case where \(I\) and \(J\) are word sets of different lengths.

Let \(A^1,\ldots ,A^g\) be block tensors with the common word-span property at length \(m\). Fix matrices \(X_j\), and put \(D^j_\beta =X_jA^j_\beta \). Suppose that \(C^j_\rho \) is indexed by a block \(j\) and a complementary word \(\rho \) of length \(M\), and assume that, for every wrapped word \(\beta \) of length \(K\), every complementary word \(\rho \), and every middle word \(w\) of length \(m\),

If each block satisfies \(\sum _a A^j_aA^{j\, \dagger }_a = \mathbb {1}\), then, for every block \(j\), there is a matrix \(E^j = \sum _\rho C^j_\rho \left(A^j_\rho \right)^\dagger \) such that, for every wrapped word \(\beta \) and complementary word \(\rho \),

This is the calculation in [ PGVWC07 , Theorem 12, proof lines 1446–1451 ] , with the adjoint on \(A^j_\rho \) required by the displayed normalization.

Let \(A^1,\ldots ,A^g\) be block tensors with the common word-span property at length \(m\). Fix \(N\in \mathbb {N}\), complex numbers \(\mu _j\), and block-diagonal boundary matrices \(X_j\), and put \(D^j_\beta =(\mu _j^NX_j)A^j_\beta \). Suppose that, for every wrapped word \(\beta \) of length \(K\), every complementary word \(\rho \) of length \(M\), and every middle word \(w\) of length \(m\),

If each block satisfies \(\sum _a A^j_aA^{j\, \dagger }_a = \mathbb {1}\), then, for every block \(j\), there is a matrix \(E^j = \sum _\rho C^j_\rho \left(A^j_\rho \right)^\dagger \) such that, for every wrapped word \(\beta \) and complementary word \(\rho \),

This is the boundary-matrix calculation in [ PGVWC07 , Theorem 12, proof lines 1446–1451 ] , after substituting the block-diagonal boundary matrix \(D^j_\beta =(\mu _j^NX_j)A^j_\beta \).

Let \(A_\beta \) be the products over wrapped words of length \(K\), let \(A_\rho \) be the products over complementary words of length \(M\), and let \(X\in M_D(\mathbb {C})\) be a matrix. Suppose that the complementary-word products satisfy \(\sum _\rho A_\rho A_\rho ^\dagger = \mathbb {1}\) and that there are matrices \(C_\rho \), indexed by complementary words, such that, for every wrapped word \(\beta \) and complementary word \(\rho \), \(A_\beta C_\rho = (X A_\beta )A_\rho \). Then, for every complementary word \(\rho \), with

one has, for every wrapped word \(\beta \), \((X A_\beta )A_\rho = A_\beta E_\rho \).

Let \(A^1,\ldots ,A^g\) be block tensors with the common word-span property at length \(m\). Fix matrices \(X_j\). Suppose that \(C^j_\rho \) is indexed by a block \(j\) and a complementary word \(\rho \) of length \(M\). Assume that, for every wrapped word \(\beta \) of length \(K\), every complementary word \(\rho \), and every middle word \(w\) of length \(m\),

If each block satisfies \(\sum _a A^j_aA^{j\, \dagger }_a = \mathbb {1}\), then, for every block \(j\) and every complementary word \(\rho \), there is a matrix \(E_{j,\rho }\) such that, for every wrapped word \(\beta \), \((X_jA^j_\beta )A^j_\rho = A^j_\beta E_{j,\rho }\).

Let \(A\) be a block tensor, and let \(C_a,E\in M_D(\mathbb {C})\) satisfy \(A_b C_a = A_bEA_a\) for all physical indices \(a,b\). For a word \(\sigma =(\sigma _1,\ldots ,\sigma _{n+2})\), define

Then \(\alpha \in G_{n+2}(A)\).

Let \(A^1,\ldots ,A^g\) be block tensors, and let \(C^j_a,E^j\in M_{D_j}(\mathbb {C})\) satisfy \(A^j_b C^j_a = A^j_bE^jA^j_a\) for every block \(j\) and all physical indices \(a,b\). For a word \(\sigma =(\sigma _1,\ldots ,\sigma _{n+2})\), define

Then \(\sum _j \alpha _j \in \bigvee _j G_{n+2}(A^j)\).

Let \(A^1,\ldots ,A^g\) be block tensors with common word-span separation at length \(n\), and suppose that \(\sum _a A^j_a A^{j\, \dagger }_a = \mathbb {1}\) for every block \(j\). Suppose a vector \(\psi \in (\mathbb {C}^d)^{\otimes (n+2)}\) has the left-boundary trace decomposition

and suppose that the two trace decompositions agree for every middle word:

Then \(\psi \in \bigvee _j G_{n+2}(A^j)\).

Let \(A^1,\ldots ,A^g\) be block tensors. If \(\psi \in (\mathbb {C}^d)^{\otimes (n+2)}\) satisfies

for every physical index \(a,b\), then there are matrices \(C^j_a\) and \(D^j_b\) such that

and, for every middle word \(w\) of length \(n\),

Let \(A^1,\ldots ,A^g\) be block tensors with common word-span separation at length \(n\), and suppose that \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) for every block \(j\). If \(\psi \in (\mathbb {C}^d)^{\otimes (n+2)}\) satisfies

for every physical index \(a,b\), then \(\psi \in \bigvee _jG_{n+2}(A^j)\).

Under the hypotheses of Lemma 15.2.1.16, an \((n+2)\)-site vector satisfies \(\psi \in \bigvee _jG_{n+2}(A^j)\) if and only if

for every physical index \(a,b\).

Under the hypotheses of Lemma 15.2.1.16, set \(S_n:=\bigvee _jG_{n+1}(A^j)\). Then

Let \(p{\gt}0\). Suppose the length-\(p\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\), and suppose that, for every \(0\le r{\lt}p\), the length-\((s+r)\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\). Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) for every block \(j\). Then there is a length \(N\) such that, for every \(n\ge N\), if \(S_n:=\bigvee _jG_{n+1}(A^j)\), then

Under the hypotheses of Lemma 10.3.2, assume in addition the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) for every block \(j\). Let

Then

Let \(A^1,\ldots ,A^r\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent. Assume \(L{\gt}1\), and assume that, for every block \(j\),

Set \(S=3L\) and \(q=(r-1)S\). Suppose that \(p{\gt}0\) and let \(s_0\) be a starting length. Assume that, for every block \(j\), \(\operatorname{span}\{ A^j_u:|u|=p\} =M_{D_j}(\mathbb {C})\), and that, for every \(0\le s{\lt}p+q\) and every block \(j\), \(\operatorname{span}\{ A^j_v:|v|=s_0+s\} =M_{D_j}(\mathbb {C})\). Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) for every block \(j\). Then there is \(N\) such that, for every \(n\ge N\), if \(S_n:=\bigvee _jG_{n+1}(A^j)\), then

Suppose that the length-\(q\) simultaneous products span the product algebra, \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=q\} = \prod _jM_{D_j}(\mathbb {C})\), and that every block satisfies \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\). Then the simultaneous products span the product algebra at every length \(n\ge q\). This is the simultaneous-tuple form of the unital propagation in [ PGVWC07 , lines 893–898 ] .

Let \(A^1,\ldots ,A^r\) be blocks satisfying the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) and assume that every block is injective at length \(L\): \(\operatorname{span}\{ A^j_u:|u|=L\} =M_{D_j}(\mathbb {C})\). If there are block-separating equations of length \(S\), then, for every \(n\ge L+(r-1)S\), \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=n\} = \prod _j M_{D_j}(\mathbb {C})\). In particular, for normalized BNT representatives satisfying the displayed unital normalization, if \(L_0{\gt}0\) and \(\Gamma _{L_0}^{A^j}\) is injective for every block \(j\), the same conclusion holds with \(S=3(L_0+1)\) and \(n\ge (L_0+1)+(r-1)3(L_0+1)\).

Let \(r\ge 2\). Suppose that the normalized BNT blocks are injective at the common length \(L_0{\gt}0\) and satisfy \(\sum _aA^j_aA^{j\, \dagger }_a=\mathbb {1}\). For every \(n\ge 3(r-1)(L_0+1)\), one has \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=n\} =\prod _jM_{D_j}(\mathbb {C})\). Consequently the spaces \(G_n(A^j)\) form an internal sum. This is the simultaneous spanning estimate used in [ PGVWC07 , Theorem 12, proof lines 1424–1456 ] .

Suppose \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=n\} = \prod _jM_{D_j}(\mathbb {C})\). Then the sum of the local spaces \(G_n(A^j)\) is direct: if

then \(\phi _j=0\) for every \(j\). In particular, for the normalized BNT hypotheses above, this directness holds for every \(n\ge (L_0+1)+(r-1)3(L_0+1)\).

Let \(A^1,\ldots ,A^r\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent. Suppose \(L_0{\gt}0\), and, for every block \(j\), the map

is injective. Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\). Set \(N=(L_0+1)+(r-1)3(L_0+1)\). For every \(n\ge N\), set \(S_n:=\bigvee _jG_{n+1}(A^j)\). The two consecutive sums are internal direct sums:

With this notation,

Let \(A^1,\ldots ,A^r\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent. Suppose \(L_0{\gt}0\), and, for every block \(j\), the map

is injective. Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\). Then there is a length \(N\) such that, for every \(n\ge N\), with \(S_n:=\bigvee _jG_{n+1}(A^j)\), one has

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\). For block-diagonal boundary conditions \(X_j\),

Let \(B=\bigoplus _{k=1}^g\mu _kA_k\). If \(\mu _j\ne 0\), then, for every length \(L\), one has \(G_L(A_j)\subseteq G_L(B)\).

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\) for \(1\le j\le g\). Then, for every length \(L\), \(G_L(B)=\bigvee _{j=1}^gG_L(A_j)\).

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\) for \(1\le j\le g\), and set \(S_M:=\bigvee _{j=1}^gG_M(A_j)\). If, for every \(M\ge L\), \(\mathbb {C}^d\otimes S_M\cap S_M\otimes \mathbb {C}^d =S_{M+1}\), then \(\mathcal G_{N,L}(B)\subseteq S_N\).

Let \(A_1,\ldots ,A_g\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent. Suppose \(L_0{\gt}0\), and, for every block \(j\), the map

is injective. Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\). For nonzero scalars \(\mu _j\), set

If \(L\ge (L_0+1)+(g-1)3(L_0+1)+1\), then, for every \(N\ge L\), \(\mathcal G_{N,L}(B)\subseteq S_N\).

With the hypotheses and notation of Lemma 15.3.7, set \(S_N:=\bigvee _{j=1}^gG_N(A_j)\). Then, for every \(N\ge L\), \(\mathcal G_{N,L}(B)\subseteq S_N\), and the sum defining \(S_N\) is internal: if

then \(\phi _j=0\) for every \(j\).

With the hypotheses and notation of Lemma 15.3.7, every vector \(\psi \in \mathcal G_{N,L}(\bigoplus _{j=1}^g\mu _jA_j)\) has a unique family \((\psi _j)_{j=1}^g\) such that

With the hypotheses and notation of Lemma 15.3.7, every vector \(\psi \in \mathcal G_{N,L}(\bigoplus _{j=1}^g\mu _jA_j)\) admits block-diagonal boundary conditions \(X_j\) such that

for every \(j\).

Let \(g\ge 2\). Let \(A_j\) be irreducible left-canonical blocks with normalized self-overlap, pairwise inequivalent up to gauge and phase, injective at the common length \(L_0{\gt}0\), and satisfying \(\sum _aA^j_aA^{j\, \dagger }_a=\mathbb {1}\). Let every weight \(\mu _j\) be nonzero. If \(0{\lt}L\le N\) and \(L\ge 3(g-1)(L_0+1)+1\), then

In particular, every vector \(\psi \) in the left-hand side admits matrices \(X_j\) such that

This is the open-boundary decomposition used in [ PGVWC07 , Theorem 12, proof lines 1424–1456 ] .

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), and fix block-diagonal boundary conditions \(X_j\). For every cyclic restriction \(R_{i,\tau }\) from \(N\) sites to \(L\) sites,

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), with \(\mu _j\ne 0\), and suppose

Then every local constraint of \(\psi \) gives

Fix block-diagonal boundary conditions \(X_j\). If the cyclic window beginning at \(i\) stays inside the chosen linear interval, \(i+L\le N\), then for every block \(j\),

Assume \(0{\lt}N\), \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Write \(a=i+L-N\). For a local word \(\sigma \) of length \(L\), set

Then the corresponding coefficient of the \(j\)-th block component is

Assume \(0{\lt}N\), \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Put \(a=i+L-N\). For a local word \(\sigma \) of length \(L\), write

These are the word coordinates obtained by opening the periodic boundary at the chosen cut, in the comparison with the source expressions involving \(C^j_{i_1}\) and \(D^j_{i_{m+1}}\). If

then there are matrices \(C^j_{i,\tau }\) such that, for every \(\sigma \),

Assume that the physical dimension \(d\) is positive, \(0{\lt}N\), \(L\le N\), and \(\mu _j\ne 0\) for every block \(j\). Let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Let

For every outside word \(\rho \) of length \(N-L\), there are matrices \(C^j_{i,\rho }\) such that, for every word \(\beta \) of length \(i+L-N\) before the cut and every wrapped tail word \(\alpha \) of length \(N-i\),

Assume that the physical dimension \(d\) is positive, \(0{\lt}N\), \(L\le N\), and \(\mu _j\ne 0\) for every block \(j\). Let

Then there are matrices \(C^j_{i,\rho }\), indexed by the boundary-crossing start \(i\) and by an outside word \(\rho \) of length \(N-L\), such that whenever \(N{\lt}i+L\),

for every word \(\beta \) of length \(i+L-N\) before the cut and every wrapped tail word \(\alpha \) of length \(N-i\).

Assume \(0{\lt}N\), \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Put \(a=i+L-N\). Suppose that there is a matrix \(E\) such that for every word \(\beta \) of length \(a\),

Then

Write \(N=M+1\) and let the local length be \(m+2\le M+1\). Consider the cyclic interval beginning at the last site \(M\). For an outside configuration \(\tau \), put

Then the \(j\)-th component of the last crossing-window restriction has the left-boundary trace form

Write \(N=M+1\) and let the local length be \(m+2\le M+1\). Consider the cyclic interval beginning at the last site \(M\). For an outside configuration \(\tau \), put

Then the block sum of the last crossing-window restrictions has the left-boundary trace form

In the setting of Lemma 15.3.1.13, assume that the length-\(m\) simultaneous word tuples span the product algebra and that

Then there are matrices \(E_j\) such that, for every block \(j\) and all boundary letters \(a,b\),

In the setting of Lemma 15.3.1.13, assume that the length-\(m\) simultaneous word tuples span the product algebra and that

Then, for every block \(j\),

Let \(B=\bigoplus _j\mu _jA^j\), and assume \(\mu _j\ne 0\) for every block \(j\). Suppose

Assume also that the length-\(m\) simultaneous word tuples span the product algebra. Then, for every outside configuration \(\tau \), with \(\rho =\tau _{m+1}\cdots \tau _{M-1}\), there are matrices \(E_j\) such that \(A^j_b(A^j_\rho A^j_a(\mu _j^{M+1}X_j)) = A^j_bE_jA^j_a\) for every block \(j\) and all boundary letters \(a,b\).

Let \(B=\bigoplus _j\mu _jA^j\), and assume \(\mu _j\ne 0\) for every block \(j\). Suppose

If the length-\(m\) simultaneous word tuples span the product algebra, then for every outside configuration \(\tau \) and every block \(j\),

Assume \(0{\lt}N\) and \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Put \(a=i+L-N\). Suppose that there is a matrix \(Y\) such that, for every word \(\beta \) of length \(a\), \(\mu _j^N X_j A^j_\beta = A^j_\beta Y\). Then

Assume \(0{\lt}N\) and \(L\le N\). Fix block-diagonal boundary conditions \(X_j\). Suppose that, for every block \(j\), every cyclic interval beginning at \(i\) with \(N{\lt}i+L\), and every configuration \(\tau \),

Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\) and \(L\le N\). Fix block-diagonal boundary conditions \(X_j\). Suppose that, for every block \(j\), every boundary-crossing interval beginning at \(i\), and every configuration \(\tau \), there is a matrix \(Y_{j,i,\tau }\) such that, for every word \(\beta \) of length \(a=i+L-N\), \(\mu _j^N X_j A^j_\beta = A^j_\beta Y_{j,i,\tau }\). Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\) and \(L\le N\), and assume that, for every block \(j\), the length-\(N-L\) word products of \(A_j\) span the full matrix algebra. Suppose that, for every block \(j\), every boundary-crossing interval beginning at \(i\), and every outside word \(\rho \) of length \(N-L\), there is a matrix \(E_{j,i,\rho }\) such that, for every word \(\beta \) before the cut of length \(i+L-N\), \(\mu _j^N X_j A^j_\beta A^j_\rho = A^j_\beta E_{j,i,\rho }\). Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies the normalization equation \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Suppose also that the boundary-crossing matrix identities of Lemma 15.3.1.21 hold for every outside word \(\rho \) of length \(N-L\). Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

If \(\sum _a A_aA_a^\dagger =\mathbb {1}\), then, for every length \(M\), \(\sum _\rho A_\rho A_\rho ^\dagger = \mathbb {1}\), where \(\rho \) ranges over all words of length \(M\).

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Fix block-diagonal boundary conditions \(X_j\). For every boundary-crossing interval beginning at \(i\), suppose there are matrices \(C^j_{i,\rho }\), indexed by outside words \(\rho \) of length \(N-L\), such that, for every word \(\beta \) before the cut of length \(i+L-N\), \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \). After opening the periodic boundary at the chosen cut, this is the source comparison

Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Let

If every boundary-crossing interval has the corresponding tail-word spanning property, then there are block-diagonal boundary conditions \(X_j\) with the displayed representation of \(\psi \) and, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Fix block-diagonal boundary matrices \(X_j\). Suppose the opened-boundary matrices \(C^j_{i,\rho }\) satisfy, for every boundary-crossing interval \(i\), every outside word \(\rho \), and every word \(\beta \) before the cut, \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \). Then, for every middle word \(w\),

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Assume also that the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\) at the middle-word length \(m\) span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Fix complex numbers \(\mu _j\) and block-diagonal boundary conditions \(X_j\). For every boundary-crossing interval beginning at \(i\), suppose there are matrices \(C^j_{i,\rho }\), indexed by outside words \(\rho \) of length \(N-L\), such that, for every word \(\beta \) before the cut, outside word \(\rho \), and middle word \(w\),

Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Assume also that the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\) at the middle-word length \(m\) span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Let

for some block-diagonal boundary conditions \(X_j\). Suppose that whenever \(\psi \) has such a representation, there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) of length \(i+L-N\) before the cut, and every word \(w\) of length \(m\),

Then there are block-diagonal boundary conditions \(X_j\) with

and, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Assume also that the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\) at the middle-word length \(m\) span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Let \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\) for some block-diagonal boundary conditions \(X_j\). Suppose that whenever \(\psi \) has such a representation, there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), and every word \(\beta \) of length \(i+L-N\) before the cut, \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \). Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

With the hypotheses and notation of Lemma 15.3.10, assume in addition that \(L+L_0\le N\). Assume also that, for some middle-word length \(m\), the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\), where \(w\) ranges over words of length \(m\) in \(\{ 0,\ldots ,d{-}1\} \), span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Suppose that whenever \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\), there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) before the cut, and every word \(w\) of length \(m\),

Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

With the hypotheses and notation of Lemma 15.3.10, assume in addition that \(L+L_0\le N\) and \(L \ge (L_0+1) +(g-1)((L_0+1)+((L_0+1)+(L_0+1)))+1\). Put \(m = (L_0+1) +(g-1)((L_0+1)+((L_0+1)+(L_0+1)))\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Suppose that whenever \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\), there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) before the cut, and every word \(w\) of length \(m\),

Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

With the hypotheses and notation of Lemma 15.3.10, assume in addition that \(L+L_0\le N\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Suppose that whenever \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\), there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), and every word \(\beta \) before the cut, \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \). Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

If \(\psi \in \mathcal G_{N,L}(A)\), then \(T_s\psi \in \mathcal G_{N,L}(A)\) for every cyclic translation \(s\).

If \(\beta \) has length \(K\) and \(\alpha \) has positive length \(S\), then \(T_K(\beta \alpha )=\alpha \beta \) as configurations on the periodic chain of length \(K+S\).

Let \(X_j,Y_j\in M_{D_j}(\mathbb {C})\), and suppose

If the simultaneous block words of length \(K\) span \(\prod _jM_{D_j}(\mathbb {C})\), then, for every block \(j\) and every word \(\alpha \) of length \(S\), \(X_jA^j_\alpha = A^j_\alpha Y_j\).

With the hypotheses and notation of Lemma 15.3.10, assume in addition that \(L+L_0\le N\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Suppose that whenever \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\), the corresponding boundary conditions satisfy the block-diagonal boundary-condition equations: for every boundary-crossing interval \(i\), every word \(\beta \) before the cut, and every outside word \(\rho \) of length \(N-L\), \(\mu _j^N X_jA^j_\beta A^j_\rho = A^j_\beta E_{j,i,\rho }\). Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

With the hypotheses and notation of Lemma 15.3.7, one has

and the sum defining the right-hand side is internal.

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\) for \(1\le j\le g\). If the periodic-boundary comparison with block-diagonal boundary conditions gives

then \(\mathcal G_{N,L}(B) = \bigvee _{j=1}^g\mathcal G_{N,L}(A_j)\).

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\). Suppose that every vector \(\psi \in \mathcal G_{N,L}(B)\) can be written as

Then

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\). Fix block-diagonal boundary conditions \(X_j\). Suppose that, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^N X_j)\in \mathcal G_{N,L}(A_j)\). Then

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\). Suppose every \(\psi \in \mathcal G_{N,L}(B)\) admits block-diagonal boundary conditions \(X_j\) such that \(\psi = \Gamma _N^B\! \left(\bigoplus _{j=1}^gX_j\right)\), and, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^N X_j)\in \mathcal G_{N,L}(A_j)\). Then

With the hypotheses and notation of Lemma 15.3.7, suppose every vector \(\psi \in \mathcal G_{N,L}(B)\) has block-diagonal boundary conditions \(X_j\) such that \(\psi = \Gamma _N^B\! \left(\bigoplus _{j=1}^gX_j\right)\), and, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^N X_j)\in \mathcal G_{N,L}(A_j)\). Then \(\mathcal G_{N,L}(B) = \bigvee _{j=1}^g\mathcal G_{N,L}(A_j)\), and the sum defining \(\bigvee _jG_N(A_j)\) is internal.

With the hypotheses and notation of Lemma 15.3.2.6, assume in addition that \(L+L_0\le N\). Assume further that, for every \(\psi \in \mathcal G_{N,L}\! \left(\bigoplus _j\mu _jA_j\right)\) and every block-diagonal boundary representation

the complementary-word matrix identities hold: for every block \(j\), boundary-crossing interval \(i\), word \(\beta \) before the cut, and outside word \(\rho \) of length \(N-L\), there is a matrix \(E_{j,i,\rho }\) such that \(\mu _j^N X_jA^j_\beta A^j_\rho = A^j_\beta E_{j,i,\rho }\). Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum.

With the hypotheses and notation of Lemma 15.3.2.6, assume in addition that \(L+L_0\le N\). Assume further that, for every \(\psi \in \mathcal G_{N,L}\! \left(\bigoplus _j\mu _jA_j\right)\) and every block-diagonal boundary representation

there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), and every word \(\beta \) before the cut, \(A^j_\beta C^j_{i,\rho } = ((\mu _j^N X_j)A^j_\beta )A^j_\rho \). This is the \(C^j,D^j\) compatibility input; the subsequent normalized \(E^j\) identities follow from Lemma 15.2.1.10. Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum.

With the hypotheses and notation of Lemma 15.3.2.8, one has

This records only the ground-space equality conclusion obtained after the \(C^j,D^j\) boundary-condition comparison input in [ PGVWC07 , Theorem 12, proof lines 1446–1456 ] ; the local normalized \(E^j\) calculation is Lemma 15.2.1.10. The independence of the \(N\)-site spaces is a separate conclusion of Lemma 15.3.2.8. The statement is still in the length-\(L_0\) injectivity range displayed in that lemma, not the shorter range in [ PGVWC07 , Theorem 12 ] .

With the hypotheses and notation of Lemma 15.3.2.6, assume in addition that \(L+L_0\le N\). Assume also that, for every boundary-crossing interval beginning at \(i\), the simultaneous products

span the product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum. The present statement is the span-dependent consequence, after opening the boundary, of the \(C^j,D^j\) compatibility input in [ PGVWC07 , Theorem 12, proof lines 1436–1451 ] in the length-\(L_0\) injectivity range \(L \ge (L_0+1)+3(g-1)(L_0+1)+1\). The source theorem has the shorter bound \(L \ge 3(g-1)(L_0+1)+1\).

With the hypotheses and notation of Lemma 15.3.2.10, one has

This records the ground-space equality conclusion of the conditional crossing-span version above. It is the corresponding part of [ PGVWC07 , Theorem 12 ] , but only under the visible tail-word span hypothesis and in the longer stated range. The independence of the \(N\)-site spaces is a separate conclusion of Lemma 15.3.2.10. Thus the statement remains conditional on the short tail-word spans; it is not the unconditional boundary-condition comparison in the source theorem.

With the hypotheses and notation of Lemma 15.3.2.6, assume in addition that \(L+L_0\le N\). Assume also that, for some middle-word length \(m\), the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\), where \(w\) ranges over words of length \(m\) in \(\{ 0,\ldots ,d{-}1\} \), span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Assume further that, for every \(\psi \in \mathcal G_{N,L}\! \left(\bigoplus _j\mu _jA_j\right)\) and every block-diagonal boundary representation

there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) before the cut, and every word \(w\) of length \(m\),

Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum.

With the hypotheses and notation of Lemma 15.3.2.6, assume in addition that \(L+L_0\le N\) and \(L \ge (L_0+1)+(g-1)((L_0+1)+((L_0+1)+(L_0+1)))+1\). Put \(m = (L_0+1)+(g-1)((L_0+1)+((L_0+1)+(L_0+1)))\). Assume that, for every \(\psi \in \mathcal G_{N,L}\! \left(\bigoplus _j\mu _jA_j\right)\) and every block-diagonal boundary representation

there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) before the cut, and every word \(w\) of length \(m\),

Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum.

Let \(A_1,\ldots ,A_g\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent, and let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\). Assume each block is injective at length \(L_0\), assume \(\sum _aA^j_aA^{j\dagger }_a=\mathbb {1}\) for every \(j\), and suppose

If every vector \(\psi \in \mathcal G_{N,L}(B)\) can be written as

then \(\mathcal G_{N,L}(B) = \bigvee _{j=1}^g\mathcal G_{N,L}(A_j)\), and the sum \(\bigvee _jG_N(A_j)\) is internal.

Let \(B^i=\bigoplus _j\mu _j A_j^i\), and assume \(1{\lt}L\), \(N\ge L+1\), and \(0{\lt}m\). Assume that every virtual sector projection \(P_j\) lies in the pulled-back span of the length-\(m\) words \(B^\omega \), and assume that every vector \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _j A_j)\) has a boundary matrix \(X\) with

for the boundary-crossing words \(\omega \). Then

with equality after the forward inclusion. This is a conditional form of the block-diagonal boundary-condition step in [ CPGSV21 , lines 2126–2128 ] .

Let \(A^i=\bigoplus _j\mu _j A_j^i\) be in block-injective canonical form, and assume \(1{\lt}L\), \(N\ge L+1\), and \(0{\lt}m\). If

and every vector in \(\mathcal G_{N,L}(\bigoplus _j \mu _j A_j)\) comes with a boundary matrix representation commuting with all length-\(m\) direct-sum words, then

Combined with the forward inclusion, this gives the corresponding conditional equality. In the parent-ground-space formulation, the same hypotheses imply containment in \(\operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \).

Let \(A^i=\bigoplus _j\mu _j A_j^i\) be in block-injective canonical form, with \(A_1,\ldots ,A_g\) a basis of normal tensors. Assume \(1{\lt}L\) and \(N\ge L+1\), and suppose that for each block \(k\) there are coefficients \(c_\omega ^{(k)}\) such that

If every vector in \(\mathcal G_{N,L}(\bigoplus _j \mu _j A_j)\) has a boundary matrix representation commuting with all length-\((1+S)\) direct-sum words, then

Together with the forward inclusion, this gives the corresponding conditional equality. In the parent-ground-space formulation, the same hypotheses imply containment in \(\operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \). This is the block-injective ground-space argument of [ CPGSV21 , lines 2120–2128 ] , where the re-growing step is restricted to block-diagonal boundary conditions.