For two block upper-triangular matrices \(X,Y\), \((XY)_{kk}=\sum _l X_{kl}Y_{lk}\); the terms with \(l{\lt}k\) vanish because \(X_{kl}=0\), and those with \(l{\gt}k\) vanish because \(Y_{lk}=0\), leaving \((XY)_{kk}=X_{kk}Y_{kk}\). Induction on the word length gives (1) and the persistence of block triangularity. The identity \(Q_kMP_k=M_{kk}\) and (4) are direct computations from the coordinate description of \(P_k\) and \(Q_k=P_k^{\mathsf T}\).

The action of \(w\) is conjugate to the matrix \(C^w\) under the identification of the module with \(\mathbb {C}^D\), and the trace of a linear map is the trace of its matrix.

Let \(K\subseteq \mathbb {C}^D\) be a nonzero invariant subspace and \(u\in K\) nonzero. Normality gives a length \(L\) for which the span of the length-\(L\) word evaluations is all of \(M_{D}(\mathbb {C})\); since \(K\) is invariant under every such word and hence under their span, \(Xu\in K\) for every \(X\in M_{D}(\mathbb {C})\). Every vector is \(Xu\) for a suitable \(X\) because \(u\neq 0\), so \(K=\mathbb {C}^D\).

Their difference agrees with \((\chi _1(1)-\chi _2(1))\cdot \mathrm{aug}\) on every basis word, by induction on the monoid-algebra presentation, hence on all of \(F\) by linearity; on \(F_+\) the augmentation vanishes, so the two functionals agree there.

If every generator acts as zero, \(F\) acts through the augmentation, so the \(F\)-submodules and the \(\mathbb {C}\)-subspaces coincide; simplicity as a \(\mathbb {C}\)-vector space forces dimension one.

By Schur’s lemma over the algebraically closed field \(\mathbb {C}\), the commutant \(\operatorname{End}_F(Q)\) consists of scalars, so \(Q\) is finite-dimensional over its commutant. The Jacobson density theorem then makes the action map \(F\to \operatorname{End}_{\mathbb {C}}(Q)\) surjective.

For the first claim, Burnside’s theorem realizes, for a basis \(\{ b_j\} \) of \(Q\) and a nonzero vector \(x_i\cdot q_0\), the maps sending \(x_i\cdot q_0\mapsto b_j\) and \(q\mapsto (\text{coordinate }j\text{ of }q)\cdot q_0\); assembling them into \(u=\sum _j\alpha _jx_i\beta _j\) gives an element of \(F_+\) acting as the identity on \(Q\). For the pairwise separation, Schur’s lemma shows every \(F\)-linear endomorphism of \(Q\times T\) preserves the two non-isomorphic simple factors, so \(Q\times T\) is finite-dimensional over its (scalar) commutant; Jacobson density on \(Q\times T\) produces \(a\in F\) acting as the identity on the first factor and zero on the second, and \(p=au\) acts as the identity on \(Q\) and as zero on \(T\). The finite-family statement follows by induction on \(S\), multiplying the separators for the successive members.

This is the additivity of the ordinary trace of an endomorphism over an invariant subspace, applied to the action of each word on \(M\).

The characteristic polynomial is determined by the power sums of its roots (Newton’s identities), so vanishing trace powers force it to equal \(X^{\dim }\); the Cayley–Hamilton theorem then gives nilpotency.

If \(M/K\) were isomorphic to no \(C_s\), Theorem 29.3.5 supplies \(p\in F_+\) acting as the identity on \(M/K\) and as zero on every \(C_s\). Since the word trace of \(M\) agrees with the sum of the word traces of the \(C_s\) on every nonempty word, and \(p\in F_+\), Lemma 29.3.2 shows that every positive power of the action of \(p\) on \(M\) has vanishing trace, so this action is nilpotent by Theorem 29.3.7. A nilpotent action of \(p\) on \(M\) descends to a nilpotent action on the quotient \(M/K\), contradicting that \(p\) acts there as the identity on a nonzero module.

Choose a basis of \(H_1\) and extend it successively to bases of \(H_2,\ldots ,H_r\). In the resulting basis, every \(H_k\) is a coordinate initial segment; since every \(H_k\) is invariant under every \(B^i\), all matrices \(B^i\) are block upper triangular in this basis, with \(k\)-th diagonal block the induced action on \(H_k/H_{k-1}\).

Otherwise Theorem 29.3.5 applied to one member \(C_{s_0}\) of the family supplies \(u\in F_+\) acting as the identity on \(C_{s_0}\); vanishing total word trace forces every positive power of the action of \(u\) on the product of the modules in the family to have trace zero, so this action is nilpotent by Theorem 29.3.7. A nilpotent action of \(u\) contradicts its action as the identity on the nonzero summand \(C_{s_0}\).

Induction on \(\dim M\). If \(M=0\), Lemma 29.4.3 forces \(S=\emptyset \), and the empty chain is the required flag. If \(M\neq 0\), choose a maximal proper invariant submodule \(K\leq M\) (a coatom of the invariant-submodule lattice, which exists because \(M\) is finite-dimensional), so \(M/K\) is simple. If some generator acts nontrivially on \(M/K\), Theorem 29.3.8 identifies \(M/K\) with some block \(C_{s_0}\); Lemma 29.3.6 shows the word trace of \(K\) agrees with the sum over the remaining blocks \(S\setminus \{ s_0\} \), and the induction hypothesis (applied to \(K\), of strictly smaller dimension) supplies a flag of \(K\), extended by the matched final step \(K\subset M\) labelled \(s_0\). If instead every generator acts as zero on \(M/K\), Lemma 29.3.3 shows \(M/K\) is one-dimensional; Lemma 29.3.6 shows the word trace of \(K\) already agrees with the sum over all of \(S\), and the induction hypothesis supplies a flag of \(K\), extended by the unmatched final step \(K\subset M\).

By Theorem 29.4.4, applied with \(M\) the word-algebra module of \(B\), whose word traces agree with the sum of those of the \(C_s\) by hypothesis, \(\mathbb {C}^{D_B}\) carries a labelled invariant flag over \((S,C)\): an invariant chain \(0=H_0\leq \cdots \leq H_r=\mathbb {C}^{D_B}\) whose subquotients are, in some order, the \(C_s\) (each once) and \(z=r-|S|\) copies of the zero module. This chain is in particular a composition series, so Lemma 29.4.1 supplies \(G\in \mathrm{GL}_{D_B}(\mathbb {C})\) for which \(T^i=GB^iG^{-1}\) is block upper triangular with these subquotients as diagonal blocks, proving (i)–(iii).

Let \(P_{j(s)}\), \(Q_{j(s)}\) be the coordinate embedding and projection of the matched slot and set \(V_s=G^{-1}P_{j(s)}X_s\), \(W_s=X_s^{-1}Q_{j(s)}G\). Then \(W_sV_t=X_s^{-1}Q_{j(s)}P_{j(t)}X_t\), and (4) gives (iv). For a nonempty word \(w\), \(B^w=G^{-1}T^wG\), and Lemma 29.1.1 gives \(Q_{j(s)}T^wP_{j(s)}=X_sC_s^wX_s^{-1}\), so \(W_sB^wV_s=X_s^{-1}Q_{j(s)}T^wP_{j(s)}X_s=C_s^w\), proving (v).

For the remainder, \(G\widehat C^iG^{-1}=\sum _{s\in S}P_{j(s)}X_sC_s^iX_s^{-1}Q_{j(s)}\) is the block-diagonal matrix agreeing with \(T^i\) on every matched diagonal block; every unmatched diagonal block of \(T^i\) is already zero, so \(GR^iG^{-1}=T^i-G\widehat C^iG^{-1}\) is strictly block upper triangular for the \(r\)-block decomposition. A nonzero entry of a product of \(r\) strictly block upper-triangular \(r\)-block matrices would need a strictly increasing chain of \(r+1\) block indices, which does not exist among \(r\) positions; hence \(GR^{i_1}\cdots R^{i_r}G^{-1}=0\), proving (vi). Finally, dimensions add along the composition series, and each zero-module step contributes dimension one, giving (vii).

Apply Theorem 29.5.1 to the slots \(C_{k,q}=\mu _kA_{(k)}\): for a word of length \(N\), \(C_{k,q}^{i_1}\cdots C_{k,q}^{i_N}=\mu _k^NA_{(k)}^{i_1}\cdots A_{(k)}^{i_N}\), so clause (v) of Theorem 29.5.1 gives (5).

The trace is invariant under the change of basis and is the sum of the traces of the diagonal blocks; by Lemma 29.1.1 the diagonal blocks of \(B^w\) are the words of the diagonal blocks, which are the \(C_s^w\) on the matched slots and \(0\) on the one-dimensional zero slots. For \(C_s=\mu _sA_s\) a word of length \(|w|\) gives \(C_s^w=\mu _s^{|w|}A_s^w\).

Since \(R^i=0\), \(B^i=\sum _{t\in S}V_tC_t^iW_t\). Multiplying on the right by \(V_s\), or on the left by \(W_s\), and using the biorthogonality \(W_tV_s=\delta _{ts}\operatorname{id}_{D_s}\) of clause (iv) leaves only the term \(t=s\).

Conjugating by \(U\otimes \operatorname{id}\) turns each \(\Pi _{j(s)}\) into a matrix unit, so every letter becomes block diagonal with the blocks \(\mu _sC_s^i\) at the positions \(j(s)\) and zero blocks elsewhere; block diagonality gives the triangular form and the vanishing remainder, and Proposition 29.5.4 gives the intertwining relations.

A word in paired indices specifies separate ket and bra words, and its trace is the corresponding entry of \(O_N(M)\). Equality of these entries for all configurations gives equality of the operators. Rescaling a tensor by \(c\) multiplies each letter’s matrix by \(c\), hence a length-\(N\) word evaluation, and its trace, by \(c^N\).

Treat each pair \((i,j)\) as one letter. The operator identity is the equality of the periodic vectors of \(B\) and of \(\bigoplus _\gamma \lambda _\gamma M_\gamma \) on the pair alphabet, so Theorem 29.5.2 applies.

The stacked tensor of \(T_g\) and \(T_h\) generates the same periodic operators as \(T_{gh}\), which is normal; apply Corollary 29.9.1.

The periodic vectors of \(T\cdot A_x\) are the images of those of \(A_x\) under the periodic operators of \(T\), so the hypotheses of Corollary 29.9.1 and of Theorem 29.5.2 hold.

Apply Proposition 29.5.3 on the pair alphabet for the product and on the physical alphabet for \(T\cdot A_x\). The definitions of the periodic operator and of \(T\cdot A_x\) give the two identities.

The word traces of \(B\) are those of the family in which \(C_c\) occurs \(N_c\) times, so Theorem 29.5.1 applies to that family. The number of blocks plus the number of zero slots is at most \(D\), which bounds the nilpotency order.

The periodic vectors of the stacked product on the pair alphabet are the matrix entries of \(O_N(M_a)O_N(M_b)\), and those of \(T\cdot A_x\) are the entries of \(O_N(T)|V^{(N)}(A_x)\rangle \); apply Theorem 29.6.1.1.

Apply Theorem 29.11.28 on the pair alphabet.

List the distinct entries of the two multisets as \(z_0,\ldots ,z_{r-1}\), and let \(b_j\) be the difference of their multiplicities. For a threshold \(N_0\) and each \(0\leq i{\lt}r\),

The Vandermonde matrix is invertible, so \(b_jz_j^{N_0}=0\). Since \(z_j\neq 0\), all multiplicities agree. Comparison with the empty multiset proves the vanishing assertion.

Apply Theorem 7.1.1.2, normalize each nonzero irreducible factor to a periodic tensor, and group factors whose periodic vectors are proportional by a fixed nonzero scalar raised to the length. The irreducible-form framework is described in Ref.  [ 191 ] . This gives separated periodic representatives \(A_j\) of dimensions \(d_j\), positive multiplicities \(r_j\), and nonzero weights \(a_{jq}\) such that, for every \(N{\gt}0\),

Grouping preserves the dimension count. Indeed, if periodic tensors \(A,A'\) of periods \(m,m'\) have such proportional vectors, then \(|O_{AA'}(N)|^2=O_{AA}(N)O_{A'A'}(N)\). Different dimensions would force the left side to tend to zero by Theorem 22.2.3.42, whereas along common multiples of the periods the right side tends to \(mm'{\gt}0\).

Normalize the targets to period-one tensors \(C_\gamma \) and combine them with the representatives not related to any target by an invertible gauge and a nonzero scalar. The combined family has no repeated blocks. On sufficiently large multiples \(pn\) of a common period, it is linearly independent by Theorem 22.2.3.44. Subtracting (8) from (10), after absorbing matched terms into the targets, forces every unmatched representative to satisfy \(\sum _{q=0}^{r_j-1}(a_{jq}^{p})^n=0\) for all sufficiently large \(n\). Theorem 29.7.1 contradicts \(r_j{\gt}0\). Thus every representative matches a target.

Choose its matching label \(\gamma (j)\) and a nonzero \(\theta _j\) so that \(|V^{(N)}(M_{\gamma (j)})\rangle =\theta _j^N|V^{(N)}(A_j)\rangle \). For each target, collect the weights \(a_{jq}/\theta _j\) over its matching representatives. This proves (9). Since \(d_j=D_{\gamma (j)}\), the dimension bound follows. Normalize the normal targets and use their eventual linear independence to compare arbitrary expansion coefficients with these power sums. The nonzero normalization scalars can be cancelled, giving a positive threshold for coefficient uniqueness. Theorem 29.7.1, applied to each target, then proves uniqueness against any tail expansion.

Form the product tensor \(P^{ij}=\sum _s M_\alpha ^{is}\otimes M_\beta ^{sj}\), of bond dimension \(D_\alpha D_\beta \). Theorem 21.3.9 gives \(O_L(P)=O_L(M_\alpha )O_L(M_\beta )\). Equality of operators is equivalent, entry by entry, to equality of MPS coefficients on the combined physical alphabet. Apply Theorem 29.7.2 to \(P\) and take \(\chi _{\alpha ,\beta ,\gamma }\) to have diagonal entries \(\{ \mu _{\gamma k}\} _{k=0}^{n_\gamma -1}\). The same entrywise equivalence transfers the expansion, eventual coefficient uniqueness, dimension bound, and uniqueness of nonzero entries to operators.

Since the algebra is closed under conjugate transposition, the orthogonal complement of an invariant subspace is invariant. Thus every invariant subspace has an invariant complement and the module is semisimple. Every step of the flag of Theorem 29.4.4 then has an invariant complement, the module is the direct sum of these complements, and a basis adapted to the direct sum makes every \(B^i\) block diagonal with the blocks prescribed by the flag.

With vanishing remainder, \(B^i=\sum _sV_sC_s^iW_s\), and the sitewise relations follow from the biorthogonality of the pairs.

An invariant subspace has an orthogonal projection whose lower off-diagonal Kraus blocks vanish. Invariant-projector closure forces the upper blocks to vanish as well, so its orthogonal complement is invariant. Every invariant subspace therefore has an invariant complement. Semisimplicity splits the composition flag, and the zero-remainder compression gives the asserted intertwiners.

A vanishing remainder would make the right map \(V_s\) a sitewise intertwiner, and \(W_sV_s=\operatorname{id}\) forces \(V_s\neq 0\). By Theorem 29.8.1, closure under conjugate transposition would give a compression pair whose right map is a nonzero sitewise intertwiner.

Induction on the length of a word \(w\) gives \(A^wX=XA'^w\). If \(Xv=0\), then \(X(A'^wv)=A^w(Xv)=0\) for every word \(w\). At a length where the words of \(A'\) span the full matrix algebra, this gives \(X(Mv)=0\) for every matrix \(M\), and a nonzero \(v\) makes \(Mv\) an arbitrary vector, so \(X=0\). Symmetrically, if \(X^{\mathsf T}u=0\), then \(X^{\mathsf T}(A^w)^{\mathsf T}u=(XA'^w)^{\mathsf T}u =(A'^w)^{\mathsf T}X^{\mathsf T}u=0\) for every word \(w\), and the words of \(A\) at a spanning length give \(X=0\) unless \(u=0\). If \(X\neq 0\), both \(X\) and \(X^{\mathsf T}\) are injective, so \(D'\leq D\) and \(D\leq D'\). Then \(X\) is invertible, and the pair \((X,X^{-1})\) witnesses that \(A\) and \(A'\) are gauge related.

Multiplying (13) on the left by \(V_{c,\mu }\), or on the right by \(W_{c,\mu }\), gives the sitewise relations \(V_{c,\mu }B^i=C_c^iV_{c,\mu }\) and \(B^iW_{c,\mu }=W_{c,\mu }C_c^i\), and likewise for the second compression. Hence \(C_c^i\, \widetilde V_{c,\mu }W_{e,\nu }=\widetilde V_{c,\mu }B^iW_{e,\nu } =\widetilde V_{c,\mu }W_{e,\nu }\, C_e^i\), and Lemma 29.8.1.2 makes this intertwiner zero for \(e\neq c\). For \(e=c\) it commutes with every \(C_c^i\), and since the words of \(C_c\) of a suitable length span the full matrix algebra, it is a scalar \((Y_c)_{\mu \nu }\). Compressing \(B^i\) with the second compression on both sides and expanding \(B^i\) by the first gives \(\delta _{\mu \kappa }C_c^i=\sum _\nu (Y_c)_{\mu \nu }(Y'_c)_{\nu \kappa }C_c^i\), where \(V_{c,\nu }\widetilde W_{c,\kappa }=(Y'_c)_{\nu \kappa }\operatorname{id}\); a nonzero \(C_c^i\) gives \(Y_cY'_c=\operatorname{id}\). Expanding \(\widetilde V_{c,\mu }B^i\) by the first compression gives (14), and multiplying it on the right by \(W_{c,\kappa }\) recovers \((Y_c)_{\mu \kappa }\), which gives uniqueness.

Apply Theorem 29.8.1.3 to the stacked product on the pair alphabet.

Apply Theorem 29.8.1.3 to \(T_a\cdot A_x\).

Apply Theorem 29.5.1 to the single-block target \(C=A\). Clauses (iv) and (v) give the compression pair and (15), clause (vii) gives \(D_B=D_A+z\geq D_A\), and clause (vi) gives (16) with \(r=|S|+z=1+(D_B-D_A)\).

Put \(E=\sum _sV_sW_s\). Zero remainder and biorthogonality give \(B^iE=B^i\), hence \(B^i(\operatorname{id}-E)=0\). Normalization yields

Thus \(E=\operatorname{id}\). Biorthogonality gives \(LR=\operatorname{id}\), and \(RL=E=\operatorname{id}\). The sitewise intertwiners give \(W_sB^iV_t=\delta _{st}C_s^i\), which is the stated simultaneous similarity. The inverse maps give dimension equality; comparison with \(D_B=\sum _sD_s+z\) then gives \(z=0\).

Faithful stationary splitting supplies a compression with zero remainder. The normalized split-compression theorem then supplies the inverse maps, dimension equality, and simultaneous similarity.

The coefficient identity follows by applying the chosen right inverse to a matrix unit. Expanding the power of the finite sum defining \(K\), multiplicativity of the Kronecker product collects the lower factors in site order. Transposing the upper product reverses its factors, which gives the displayed entry formula. Specializing the lower tensor to \(\widetilde A\) and applying the coefficient identity gives the stated rank-one matrix.

Expand \(K^n\) by Lemma 29.11.2 and use the trace formula for a Kronecker product. Positive-length MPV equality replaces the trace of each nonempty \(\widetilde B\) word by the corresponding \(\widetilde A\) trace. The resulting scalar sum is therefore the trace of the target cross matrix power. The coefficient-dual identity gives that target cross matrix its rank-one form, whose \(n\)th trace is \(D_A^n\).

Expanding the lower trace into matrix entries gives the cross-sum formula for arbitrary \(X\). For the target tensor, the rank-one form of \(K(\widetilde A,C)\) evaluates this sum as \(D_A^{n-1}X_{a'a}\). Specializing \(X\) to a word evaluation gives the self-evaluation formula.

For blocked tensors, Lemma 2.7.2 identifies each blocked word with the evaluation of its flattened word. Since \(n,p{\gt}0\), this flattened word is nonempty. Append the unblocked tail \(w\), apply positive-length MPV equality to the resulting original-site word, and use the target self-evaluation with \(X=A^w\).

Choose an isomorphism from the scalar division ring to the one-dimensional range of \(f\). Composing its inverse with \(f\) gives \(\varphi \), while the image of \(1\) gives \(y\). Choose a preimage \(x\) of \(y\); the construction gives \(\varphi (x)=1\), \(f(x)=y\), and the stated factorization of \(f\).

In standard bases this factorization is an outer product of two nonzero coordinate vectors. For the product-index specialization, reshape the left vector as a \(D_B\times D_A\) matrix and the right vector as a \(D_A\times D_B\) matrix, preserving the displayed index order.

First suppose that \(A\) is injective; equivalently, use blocking length one. More generally, choose any positive blocking length \(p\) for which the blocked target \(\widetilde A=A^{[p]}\) is injective, and put \(\widetilde B=B^{[p]}\). Positive blocking preserves equality of the MPV families. Let \(C\) be the coefficient dual of \(\widetilde A\) and set

Regard \(K\) as an endomorphism and take its Jordan–Chevalley decomposition \(K=N+S\), where \(N\) is nilpotent, \(S\) is semisimple, and both belong to the algebra generated by \(K\). Their membership in this commutative generated algebra implies \(NS=SN\). The commuting nilpotent part does not change traces of powers. Since \(\operatorname{tr}K^n=D_A^n\) for every \(n{\gt}0\), the semisimple endomorphism \(D_A^{-1}S\) has trace one at every power above one. The semisimple trace-power theorem gives \(\operatorname{rank}S=1\); commutation and nilpotence then imply \(NS=SN=0\) and

for some \(J{\gt}0\).

Factor the matrix of \(S\) with the mixed-bond orientation

Its first trace is \(D_A\), so multiplication of outer products gives

Apply the padded open-boundary identity with an arbitrary original-scale word \(w\). Rewriting its cross sum with the preceding factorization yields

Since \(D_A{\gt}0\), cancel the nonzero scalar and conclude \(VB^wW=A^w\) for every word \(w\). The empty word gives \(VW=1_{D_A}\). For normal \(A\), choose the positive injective blocking supplied by normality and repeat the same padded argument; the lower tail remains the unblocked word \(w\), so the conclusion is already at the original physical scale.

Scaling both tensors multiplies each length-\(|w|\) word by the same factor \(c^{|w|}\), so the reduction identities are preserved. For a reduction to \(cA\), expand \((cA)^w=c^{|w|}A^w\) in the defining word identity. The retraction \(VW=1_{D_A}\) is unchanged.

Set \(\widetilde B=c^{-1}B\). Expand its word evaluations with Lemma 2.4.12, take the trace, and use (19). The positive-length MPV families of \(\widetilde B\) and \(A\) are equal. Apply Theorem 29.11.5 to obtain a reduction from \(\widetilde B\) to \(A\). Scale both tensors by \(c\) to obtain a reduction from \(B\) to \(cA\), and then apply Lemma 29.11.6 to get (20).

A generator has the required one-factor expression. The product of two such expressions is represented by their concatenation, which is again nonempty and has every factor in \(S\).

Each residual letter is one of the generators of the residual algebra.

Apply \(VB^{\mathbf w}W=A^{\mathbf w}\) to \(\mathbf p\), \(\mathbf q\), and \(\mathbf p\, i\, \mathbf q\). The two expressions for \(A^{\mathbf p}A^iA^{\mathbf q}\) give

Induction on the nonempty residual word \(\mathbf u\) proves (22).

Take both exterior words empty in (22).

Induct on the word and split the first letter as \(B^i=N^i+WA^iV\). The induction hypothesis supplies the recursive windows in the remaining word. By (23), \(VN^{\mathbf u}W=0\) for every nonempty \(\mathbf u\), so prefixing by \(V\) collapses \(\mathcal W(\mathbf w)\) to \(\mathcal I(\mathbf w)\); adjacent selected blocks combine by \(VW=1\).

Prove both identities by induction on the word. For the left contraction, decompose the first letter and apply (28) to the tail; the sandwiched terms \(VN^{\mathbf u}W\) vanish by (23), and \(VW=1\) merges adjacent selected blocks. For the right contraction, decompose the first letter and apply the reduction identity to the tail.

Induct on the length of the word. The summands with \(r=0\) constitute \(\widehat{\mathcal I}\). By the left all-cut identity,

where the last equality is the initial-window case of the recursive expansion. Deleting the first letter identifies the summands with \(r{\gt}0\) with \(N^{i_1}\widehat{\mathcal W}(i_2,\ldots ,i_n)\).

Substitute \(\mathcal W(\mathbf i)=\widehat{\mathcal W}(\mathbf i)\) into (28), and then reindex the positive central length by \(s=r+m\).

Take the trace of the corrected residual expansion. For a mixed term in the explicit strict-window sum, cyclicity gives

by (23). In the recursive sum, the corresponding step is

by (22), for every nonempty residual word \(\mathbf u\). The unique term with no residual letters is \(WA^{\mathbf i}V\), whose trace is \(\operatorname{tr}(A^{\mathbf i})\) because \(VW=1\). Thus

Equality of the two closed chains now gives \(\operatorname{tr}(N^{\mathbf i})=0\). Linearity extends this identity to the generated nonunital algebra. Vanishing traces of positive powers make each algebra element nilpotent.

Apply the finite-dimensional nil-matrix theorem to the residual algebra. Its dimension-sharp matrix form makes every product of \(D_B\) algebra elements zero. Residual letters lie in this algebra, so \(D_B\) is a bound. Well-ordering gives the least bound, and prefix factorization gives vanishing for all longer residual words.

Split the residual word after its first \(L\) letters. The prefix is zero by the bound, so the full word is zero.

Start from (29) and (30). In the left sum, every omitted term has residual suffix length at least \(L\). In the right sum, every omitted term has residual prefix length at least \(L\), so (36) makes all these terms zero.

First use induction along the right exterior word to show that a positive residual block preceded by \(V\) vanishes once its length together with the right buffer reaches \(L\). Expand only the left exterior word \(B^{\mathbf p}\) by Theorem 29.11.15. The left buffer bound removes the pure residual term, while the preceding induction and (22) remove the strict-window terms. Hence \(B^{\mathbf p}N^iB^{\mathbf q}=0\) under the two buffer bounds.

Separately, induction on a word \(\mathbf a\) gives \(VB^{\mathbf a\mathbf q}=A^{\mathbf a}VB^{\mathbf q}\) from the right buffer bound. Write the nonempty central word as \(\mathbf c=i\mathbf a\) and decompose the corresponding letter as \(B^i=N^i+WA^iV\). The residual term is zero by the first step, and the second induction identifies the remaining term with (40).

The bounds \(N\leq |\mathbf p|+1\) and \(N\leq |\mathbf q|+1\) in Theorem 29.11.23 read \(N-1\leq |\mathbf p|\) and \(N-1\leq |\mathbf q|\), and Theorem 29.11.20 makes the least length a bound under equality of the positive-length closed chains. Word evaluation of \(B^{[L]}\) on a blocked word is evaluation of \(B\) on the flattened word, and evaluation of \(B\circ f\) on a word is evaluation of \(B\) on the mapped word, so the reduction identities transfer. For the exterior identity, the flattened exterior words have length at least \(kL\geq m\) and the flattened central word has positive length \(|\mathbf c|L\).

By Theorem 29.11.25, \(N_i-1\) is an exterior buffer length, and \(N_i-1\leq L=1\cdot L\) with \(L{\gt}0\) by (42). The blocking statement of the same theorem with \(k=1\) gives the conclusion.

If the target bond dimension is zero, both dressed identities hold entrywise for \(\lambda =1\) because the relevant target row or column index is empty. Hence assume that the target bond dimension is positive.

Choose \(L{\gt}0\) such that \(A\) is \(L\)-block injective. For exterior words \(\mathbf p,\mathbf q\) satisfying the \(N_0\) buffer inequalities, compare the two exterior-buffer expansions of the mixed contraction \(VB^{\mathbf p}\widetilde W\). Inserting a nonempty central \(A\)-word gives

Since the length-\(L\) words span the full target matrix algebra, linearity replaces \(A^{\mathbf c}\) by an arbitrary matrix. Insert matrix units and choose a nonzero target word at a positive multiple of \(L\). Entrywise cancellation then yields one scalar \(\lambda \) with \(VB^{\mathbf p}\widetilde W=\lambda A^{\mathbf p}\) for every sufficiently buffered \(\mathbf p\). Reversing the two reductions gives a scalar \(\mu \) for the opposite mixed boundary.

Choose a longer nonzero target word, split it into two buffered exterior pieces and a nonempty injective central block, and apply the exterior-buffer identity once more. The two mixed-boundary formulas give \((\lambda \mu )A^{\mathbf a}=A^{\mathbf a}\); nonvanishing of the chosen word implies \(\lambda \mu =1\), so in particular \(\lambda \ne 0\).

Finally, if \(|\mathbf a|{\gt}2N_0\), split \(\mathbf a\) into exterior words of length \(N_0\) and a nonempty center. Substitution of the two mixed-boundary formulas into the corresponding exterior-buffer identities gives (44) and first gives \(B^{\mathbf a}\widetilde W=\lambda B^{\mathbf a}W\) on the right. Multiplication by \(\lambda ^{-1}\) yields (45).

For the least-length corollary, positive-length closed-chain equality makes each least residual nilpotency length an actual bound. Prefix factorization propagates vanishing from that least length to \(N_0\), so the direct-bound theorem applies.